Keywords:Estimation, Kalman filtering, Autonomous robots Abstract: In this tutorial, a new class of attitude estimation solutions is presented to determine the orientation of a robotic platform resorting only to a set of triaxial gyroscopes and an additional measurement of a constant inertial vector. This approach poses a significant advantage in terms of setup design and reduced mission costs since only one inertial measurement is required, in contrast with existing solutions that require at least two of these. Pertinent examples of viable applications include navigation of submarines or large ships, assisting autonomous robots in precise maneuvers, inertial platform stabilization, etc. The novelty of the solutions proposed herein exploit the well-known inertial information about the Earth’s angular velocity by using high-grade gyroscopes that are sensitive to the planet’s spin. Experimental results are presented to demonstrate the good performance achieved by the algorithms.

Keywords:Observers for nonlinear systems, Sensor fusion, Autonomous vehicles Abstract: The purpose of this tutorial is to expose results developed over the last three years for state estimation problems arising with unmanned mobile robots equipped with a monocular camera and a 3-axis gyrometer, complemented with either a velocity sensor, or a 3-axis accelerometer, or an optical flow sensor. Definition and characterization of uniform observability for linear time-varying systems are first recalled. An observer design framework exploiting first-order approximations of a class of nonlinear systems is then reported. The resulting Riccati observers are locally uniformly exponentially stable when associated uniform observability conditions are satisfied. This framework is subsequently applied, with detailed explanations, to a set of practical problems, namely i) classical PnP camera pose estimation using known source points and bearing measurements, ii) the adaptation of this problem to unknown source points by using epipolar constraints, iii) camera pose and velocity estimation using bearing and IMU measurements, and iv) camera velocity and depth estimation using optical flow and IMU measurements from the observation of a planar target. The observer solutions proposed for these last two problems are validated with experimental data.

Keywords:Autonomous robots, Observers for nonlinear systems, Filtering Abstract: Visual Odometry is a core problem in robotics. Recent advances in equivariant systems theory, and the discovery of symmetries for the visual odometry problem, has provided a new perspective that is leading to new algorithms. This paper provides an overview of the geometry of visual odometry problem and an algorithm that results from this perspective.

Keywords:Observers for nonlinear systems Abstract: The design of navigation observers able to simultaneously estimate the position, linear velocity and orientation of a vehicle in a three-dimensional space is crucial in many robotics and aerospace applications. This problem was mainly dealt with using the extended Kalman filter and its variants which proved to be instrumental in many practical applications. Although practically efficient, the lack of strong stability guarantees of these algorithms motivated the emergence of a new class of geometric navigation observers relying on Riemannian geometry tools, leading to provable strong stability properties. The objective of this brief tutorial is to provide an overview of the existing estimation schemes, as well assome recently developed geometric nonlinear observers, for autonomous navigation systems relying on inertial measurement unit (IMU) and landmark measurements.

Keywords:Smart grid, Agents-based systems, Optimization algorithms Abstract: This paper proposes online algorithms for dynamic matching markets in power distribution systems. These algorithms address the problem of matching flexible loads with renewable generation, with the objective of maximizing social welfare of the exchange in the system. More specifically, two online matching algorithms are proposed for two generation-load scenarios: (i) when the mean of renewable generation is greater than the mean of the flexible load, and (ii) when the condition (i) is reversed. With the intuition that the performance of such algorithms degrades with increasing randomness of the supply and demand, two properties are proposed for assessing the performance of the algorithms. First property is convergence to optimality (CO) as the underlying randomness of renewable generation and customer loads goes to zero. The second property is deviation from optimality, which is measured as a function of the standard deviation of the underlying randomness of renewable generation and customer loads. The algorithm proposed for the first scenario is shown to satisfy CO and a deviation from optimality that varies linearly with the variation in the standard deviation. %The same algorithm, however, is shown to not satisfy CO for the second scenario. We then show that the algorithm proposed for the second scenario satisfies CO and a deviation from optimality that varies linearly with the variation in standard deviation plus an offset under certain condition.

Keywords:Smart grid, Machine learning, Estimation Abstract: As electric grids experience high penetration levels of renewable generation, fundamental changes are required to address real-time situational awareness. This paper uses unique traits of tensors to devise a model-free situational awareness framework for distribution networks. This work formulates the state of the network at multiple time instants as a three-way tensor; hence, recovering full state information of the network is tantamount to estimating all the values of the tensor. Given measurements received from muphasor measurement units and/or smart meters, the recovery of unobserved quantities is carried out using the low-rank canonical polyadic decomposition of the state tensor. Two structured sampling schemes are considered: slab sampling and fiber sampling. For both schemes, we present sufficient conditions on the number of sampled slabs and fibers that guarantee identifiability of the factors of the state tensor. Numerical results demonstrate the ability of the proposed framework to achieve high estimation accuracy in multiple sampling scenarios.

Keywords:Smart grid, Power systems, Network analysis and control Abstract: We consider the problem of stability analysis of an inverter-based microgrid where higher order models are used for the inverter and line dynamics. Decentralized conditions are derived through which stability of the network can be deduced, with these formulated as input/output conditions on locally defined subsystems. The conditions derived allow to exploit the natural passivity properties of lines when these are represented in a common reference frame, but reduce the conservatism by additionally taking into account the coupling with neighbouring buses. Examples are given to demonstrate the results presented.

Keywords:Smart grid, Power systems, Power generation Abstract: The rising penetration of renewable energy sources (RESs) has led to increased research interest in the optimal scheduling and management of multi-microgrid (MMG) systems. This is due to the potential economic benefits which can be derived through the sharing of resources between the MGs which constitute the MMG. Advanced optimization procedures such as robust optimization (RO) frameworks have been proposed in the literature to handle any uncertainties caused by the stochastic nature of the RES generation, the load demand, and the electricity prices. However, the existing works in the literature do not consider the nonlinear network constraints of individual MGs which might significantly impact the power balance and the power sharing between MGs in MMG systems. This paper proposes a cooperative trading scheme including a two-stage network-constrained robust energy management system to optimize the resource utilization within each MG and the sharing of resources between the constituent MGs of the MMG system. The results clearly highlight the performance of the proposed approach and the impact of including the AC network constraints on the optimal dispatch of the MMG system under varying uncertainties.

Keywords:Smart grid, Optimization, Power systems Abstract: The locational marginal price scheme, though widely adopted, fails to reflect the startup costs for many generators in the real-time market. To solve this issue, the extended locational marginal price (eLMP in short, a.k.a. convex hull price) has been proposed. The key idea is to convexify the non-convex generation cost functions due to the startup costs, and then design the uplift payment whenever necessary. While eLMP partially solves the incentive issue, there are still chances for generators to manipulate the market prices (as well as the uplift payments) by strategic bidding. In this paper, we first propose a profit decomposition method to evaluate different bidding strategies' impacts on individual payoffs. This decomposition allows us to better identify the potential strategic generators and investigate their best strategies. We further propose to use the maximal markup as an index for eLMP scheme to quantify market power in the whole system. Numerical studies further highlight the existence of market power in practice.

Keywords:Smart grid, Optimization algorithms, Power systems Abstract: The charging processes of a large number of electric vehicles (EVs) require coordination and control for the alleviation of their impacts on the distribution network and for the provision of various grid services. However, the scalability of existing EV charging control paradigms are limited by either the number of EVs or the distribution network dimension, largely impairing EVs' aggregate service capability and applicability. To overcome the scalability barrier, this paper, motivated by the optimal scheduling problem for the valley-filling service, (1) proposes a novel dimension reduction methodology by grouping EVs (primal decision variables) and establishing voltage (global coupled constraints) updating subsets for each EV group in the distribution network and (2) develops a novel decentralized shrunken primal multi-dual subgradient (SPMDS) optimization algorithm to solve this reduced-dimension problem. The proposed SPMDS-based control framework requires no communication between EVs, reduces over 43% of the computational cost in the primal subgradient update, and reduces up to 68% of the computational cost in the dual subgradient update. The efficiency and efficacy of the proposed algorithm are demonstrated through simulations over a modified IEEE 13-bus test feeder and a modified IEEE 123-bus test feeder.

Keywords:Smart grid, Power systems, Statistical learning Abstract: Residential loads have great potential to enhance the efficiency and reliability of electricity systems via demand response (DR) programs. One major challenge in residential DR is how to learn and handle unknown and uncertain customer behaviors. In this paper, we consider the residential DR problem where the load service entity (LSE) aims to select an optimal subset of customers to optimize some DR performance, such as maximizing the expected load reduction with a financial budget or minimizing the expected squared deviation from a target reduction level. To learn the uncertain customer behaviors influenced by various time-varying environmental factors, we formulate the residential DR as a contextual multi-armed bandit (MAB) problem, and develop an online learning and selection (OLS) algorithm based on Thompson sampling to solve it. This algorithm takes the contextual information into consideration and is applicable to complicated DR settings. Numerical simulations are performed to demonstrate the efficiency and learning effectiveness of the proposed algorithm.

Keywords:Autonomous systems, Markov processes, Uncertain systems Abstract: Multi-agent partially observable Markov decision processes (MPOMDPs) provide a framework to represent heterogeneous autonomous agents subject to uncertainty and partial observation. In this paper, given a nominal policy provided by a human operator or a conventional planning method, we propose a technique based on barrier functions to design a minimally interfering safety-shield ensuring satisfaction of high-level specifications in terms of linear distribution temporal logic (LDTL). To this end, we use sufficient and necessary conditions for the invariance of a given set based on discrete-time barrier functions (DTBFs) and formulate sufficient conditions for finite time DTBF to study finite time convergence to a set. We then show that different LDTL mission/safety specifications can be cast as a set of invariance or finite time reachability problems. We demonstrate that the proposed method for safety-shield synthesis can be implemented online by a sequence of one-step greedy algorithms. We demonstrate the efficacy of the proposed method using experiments involving a team of robots.

Keywords:Stochastic systems, Optimization, Constrained control Abstract: We present a general framework for risk semantics on Signal Temporal Logic (STL) specifications for stochastic dynamical systems using axiomatic risk theory. We show that under our recursive risk semantics, risk constraints on STL formulas can be expressed in terms of risk constraints on atomic predicates. We then show how this allows a (stochastic) STL risk constraint to be transformed into a risk-tightened deterministic STL constraint on a related deterministic nominal system, enabling the application of existing STL methods. For affine predicate functions and a (coherent) Distributionally Robust Value at Risk measure, we show how risk constraints on atomic predicates can be reformulated as tightened deterministic affine constraints. We demonstrate the framework using a Model Predictive Control (MPC) design with an STL risk constraint.

Keywords:Markov processes, Formal Verification/Synthesis, Machine learning Abstract: Probabilistic Computation Tree Logic (PCTL) is frequently used to formally specify control objectives such as probabilistic reachability and safety. In this work, we focus on model checking PCTL specifications statistically on Markov Decision Processes (MDPs) by sampling, e.g., checking whether there exists a feasible policy such that the probability of reaching certain goal states is greater than a threshold. We use reinforcement learning to search for such a feasible policy for PCTL specifications, and then develop a statistical model checking (SMC) method with provable guarantees on its error. Specifically, we first use upper-confidence-bound (UCB) based Q-learning to design an SMC algorithm for bounded-time PCTL specifications, and then extend this algorithm to unbounded-time specifications by identifying a proper truncation time by checking the PCTL specification and its negation at the same time. Finally, we evaluate the proposed method by case studies.

Keywords:Switched systems, Networked control systems, Lyapunov methods Abstract: This paper focuses on the single-agent indirect herding problem where a herder agent is tasked with regulating a group of target agents to a desired goal state. To achieve this goal, the herder must switch between controlling different targets, resulting in a switched dynamical system. Lyapunovbased switched system analysis methods are used to develop sufficient dwell-time conditions. These dwell-time conditions are then encoded into Metric Temporal Logic (MTL) specifications that constitute the constraints in a switched nonlinear Model Predictive Control (MPC) problem that is formulated to synthesize a desired switching protocol. The synthesized switching protocol ensures the herding objective is achieved while simultaneously satisfying the dwell-time MTL specifications.

Keywords:Formal Verification/Synthesis, Constrained control, LMIs Abstract: For a nonlinear system (e.g. a robot) with its continuous state space trajectories constrained by a linear temporal logic specification, the synthesis of a low-level controller for mission execution often results in a non-convex optimization problem. We devise a new algorithm to solve this type of non-convex problems by formulating a rapidly-exploring random tree of barrier pairs, with each barrier pair composed of a quadratic barrier function and a full state feedback controller. The proposed method employs a rapidly-exploring random tree to deal with the non-convex constraints and uses barrier pairs to fulfill the local convex constraints. As such, the method solves control problems fulfilling the required transitions of an automaton in order to satisfy given linear temporal logic constraints. At the same time it synthesizes locally optimal controllers in order to transition between the regions corresponding to the alphabet of the automaton. We demonstrate this new algorithm on a simulation of a two linkage manipulator robot.

Keywords:Autonomous systems, Robotics, Intelligent systems Abstract: Recent years have seen the increasing use of Signal Temporal Logic (STL) as a formal specification language for symbolic control, due to its expressiveness and closeness to natural language. Furthermore, STL specifications can be encoded as cost functions using STL's robust semantics, transforming the synthesis problem into an optimization problem. Unfortunately, these cost functions are non-smooth and non-convex, and exact solutions using mixed-integer programming do not scale well. Recent work has focused on using smooth approximations of robustness, which enable faster gradient-based methods to find local maxima, at the expense of soundness and/or completeness. We therefore propose a novel robustness approximation that is smooth everywhere, sound, and asymptotically complete. Our approach combines the benefits of existing approximations, while enabling an explicit tradeoff between conservativeness and completeness.

Keywords:Autonomous vehicles, Formal Verification/Synthesis, Automotive control Abstract: In this paper, we propose an approach for ensuring the safety of a vehicle throughout a parking task, even when the vehicle is being operated at varying levels of automation. We start by specifying a vehicle parking task using linear temporal logic formulae that can be model checked for feasibility. The model-checking is facilitated by the construction of a temporal logic tree via Hamilton-Jacobi reachability analysis. Once we know the parking task is feasible for our vehicle model, we utilize the constructed temporal logic tree to directly synthesize control sets. Our approach synthesizes control sets that are least-restrictive in the context of the specification, since they permit any control inputs that are guaranteed not to violate the specification. This least-restrictive characteristic allows for the application of our approach to vehicles under different modes of operation (e.g., human-in-the-loop shared autonomy or fully-automated schemes). Implementing in both simulation and on hardware, we demonstrate the approach's potential for ensuring the safety of vehicles throughout parking tasks, whether they are operated by humans or automated driving systems.

Keywords:Formal Verification/Synthesis, Autonomous robots, Constrained control Abstract: Temporal logics provide a formalism for expressing complex system specifications. A large body of literature has addressed the verification and the control synthesis problem for deterministic systems under such specifications. For stochastic systems or systems operating in unknown environments, however, only the probability of satisfying a specification has been considered so far, neglecting the risk of not satisfying the specification. Towards addressing this shortcoming, we consider, for the first time, risk metrics, such as (but not limited to) the Conditional Value-at-Risk, and propose risk signal temporal logic. Specifically, we compose risk metrics with stochastic predicates to consider the risk of violating certain spatial specifications. As a particular instance of such stochasticity, we consider control systems in unknown environments and present a determinization of the risk signal temporal logic specification to transform the stochastic control problem into a deterministic one. For unicycle-like dynamics, we then extend our previous work on deterministic time-varying control barrier functions.

Keywords:Robust adaptive control, Uncertain systems, Linear systems Abstract: We consider the problem of stabilization of a linear system, under state and control constraints, and subject to bounded disturbances and unknown parameters in the state matrix. First, using a simple least square solution and available noisy measurements, the set of admissible values for parameters is evaluated. Second, for the estimated set of parameter values and the corresponding linear interval model of the system, two interval predictors are recalled and an unconstrained stabilizing control is designed that uses the predicted intervals. Third, to guarantee the robust constraint satisfaction, a model predictive control algorithm is developed, which is based on solution of an optimization problem posed for the interval predictor. The conditions for recursive feasibility and asymptotic performance are established. Efficiency of the proposed control framework is illustrated by numeric simulations.

Keywords:Adaptive control, Uncertain systems, Optimization algorithms Abstract: This paper proposes a Newton seeking control design that achieves finite-time stability of the optimum of unknown multivariable static maps. The Newton seeking system is shown to yield an averaged finite-time stable Newton descent algorithm with a finite-time stable equilibrium of the optimum. A classical averaging theorem due to Krasnosel'skii and Krein is used to demonstrate that the trajectories of the Newton seeking system approximate the trajectories of the averaged finite-time stable system. The analysis shows that the finite-time Newton-seeking technique achieves finite-time practical stability of the optimum of the cost function. A simulation study is used to demonstrate the effectiveness of the design method.

Keywords:Adaptive control, Robust control, LMIs Abstract: The two-degree-of-freedom (2DOF) structure achieves stabilization and good tracking independently by feedback (FB) and feedforward (FF) control, respectively. The latter often suffers from model uncertainty, which is remedied by tuning the FF control on-line. A tuning law ensures that the tracking error converges to zero for any target signals under a strictly positive real (SPR) condition, but this is rather restrictive. This paper proposes a robust design method for the tuning law in FF control, which is effective for any minimum phase plant with relative degree one, by using a Lyapunov solution derived in the FB control design. The effectiveness of the proposed method is illustrated via numerical examples.

Keywords:Adaptive systems, Robust adaptive control, Optimization Abstract: This study proposes a design of extremum seeking controllers that guarantees precise convergence of the control system to the unknown optimizer of a measured unknown cost function. The approach introduces an uncertainty estimation technique that provides an estimate of the worst case uncertainty associated with the identification of the optimal conditions. An uncertainty set update algorithm is proposed to reduce the radius of uncertain set as new data is received. The radius of the uncertainty set is used to gradually reduce the required dither amplitude. The technique yields the simultaneous convergence to the unknown optimum and the removal of the dither signal. Asymptotic convergence of the extremum seeking control system to the unknown minimizer is achieved. A simulation example is provided to demonstrate the effectiveness of the approach.

Keywords:Direct adaptive control, Robust adaptive control, Adaptive control Abstract: This paper presents a novel solution to the problem of designing an implementable (i.e., differentiator-free) model-reference output-feedback direct-adaptive controller for single-input-single-output linear time-invariant systems with relative degree possibly larger than one. The new paradigm is based on a version of the Dynamic Certainty Equivalence (DyCE) principle. The approach proposed in this work consists in realizing the DyCE control through surrogate parameter derivatives, made available by a Nonlinear Parameter Filter (NPF), instead of feeding the DyCE controller with the derivatives of the estimates produced High-Order Tuner (HOT). The proposed adaptive controller does not require error augmentation or normalization, allowing the use of large adaptation gains for fast convergence speed. Moreover, the proposed architecture can be easily equipped with well-known robust modifications of tuning laws. The performance of the proposed algorithm is demonstrated via comparative simulations with an error augmentation-based method and a simplified HOT algorithm.

Keywords:Predictive control for linear systems, Indirect adaptive control, Uncertain systems Abstract: This paper addresses the regulation problem of discrete-time linear time-invariant (LTI) systems with parametric uncertainties in the presence of hard state and input constraints. An indirect adaptive identifier is strategically fused with MPC to ensure constraint satisfaction in presence of parametric uncertainties. The estimated plant model, based on indirect adaptive identifier, is used for future state prediction in the MPC algorithm. An estimated model of the actual uncertain plant is used for predictions of the future states. The estimated plant parameters belong to a convex model set, whose vertices are updated using a gradient descent based adaptive update law, the convex combination of which yields the plant parameter estimates. The errors arising due to model mismatch between the estimated plant model and the actual uncertain plant are accounted for in the MPC algorithm, using a constraint tightening method, which is dependent upon the diameter of the model set. Therefore, the magnitude of constraint tightening is reduced whenever better knowledge of the updated model set is obtained. The proposed adaptive MPC strategy is proved to be recursively feasible and the closed-loop system states are proved to be bounded and asymptotically converging to the origin.

Keywords:Adaptive control, Decentralized control, Delay systems Abstract: In this paper, the problem of decentralized robust adaptive control design is addressed for a class of large-scale system with interconnected time-varying delay. We decompose subsystem into cascade system by using state coordinate transformation and introducing state feedback. By choosing new Lyapunov Krasovskii functional, we prove that the states of the closed-loop system converge asymptotically to zero. The nonlinear gain function, which is proposed in this paper, is used in the controller and adaptive laws of the system. %This function does not have to be derived by solving LMIs, but is expressed explicitly. At last, The effectiveness and feasibility of this proposed design technique is showed by the simulation example.

Keywords:Robust adaptive control, Uncertain systems, Nonholonomic systems Abstract: A function approximation technique based immersion and invariance (FATII) control method is proposed in this letter. Firstly, an unknown control system is restructured as the combination of an auxiliary system and a variation term from the original system. The variation term is treated as a time-varying uncertainty and parameterized by a group of weighted chosen basis functions. These weights are estimated at every time instant and the change of the estimates is governed by an update law. The update law is defined based on an immersion and invariance approach such that both the system state and the estimation error converge to zero. The FATII method is model-free and thus applicable to a wide range of systems. The asymptotic stability of the proposed method is established and its feasibility is verified under simulations.

Keywords:Aerospace, Control applications, Optimal control Abstract: In this paper, a near fuel-optimal, receding horizon based station-keeping strategy is designed and implemented to estimate a two-year average station-keeping cost on a L_1 Halo orbit in Sun-Earth Circular Restricted Three-Body problem. Impulsive Model Predictive Static Programming, a finite time, non-linear optimal control technique is used to obtain optimal station-keeping manoeuvres. The technique involves calculation of sensitivity matrices which are done in a computationally efficient manner owing to their recursive nature. The algorithm is iterative where the guess station-keeping manoeuvres are optimally updated by a closed-form equation until an output terminal constraint as well as an optimal cost function are satisfied. The effect of station-keeping horizon time on the total manoeuvre cost was analyzed in the presence of Solar Radiation Pressure and uncertainties associated with orbit insertion, tracking and manoeuvre execution errors.

Keywords:Aerospace, Predictive control for linear systems, Control applications Abstract: The removal of orbital debris by means of dedicated space missions has been recently identified as a priority for the sustainability of the space environment. Electrically propelled spacecraft, in particular, are seen as a cost-effective solution for such type of missions. This paper develops an MPC strategy for space debris rendezvous, which is able to account for mission-specific performance and safety requirements, while satisfying on-off constraints inherent to the electric propulsion technology. The proposed design requires to solve a mixed integer linear program at each time step. In order to limit the computational burden, a linear programming relaxation tailored to a realistic thrusting configuration is devised. A rendezvous case study demonstrates the effectiveness of the proposed solution.

Keywords:Aerospace, Predictive control for linear systems, Constrained control Abstract: In this paper a spacecraft rendezvous policy is developed that yields safe rendezvous trajectories under various thruster failure scenarios. The policy makes use of polytopic robust backwards reachable sets to characterize the state-space that under a given thruster failure scenario would lead to collision between a deputy and a chief spacecraft no matter the remaining available thrust. That is, this region of state-space is such that no feasible evasive abort maneuver exists for the given failure scenario. Abort-safety constraints are formulated as local hyperplanes separating the deputy spacecraft and the unsafe state-space. These constraints are incorporated in a model predictive control-based online trajectory generation scheme in order to guide the deputy to rendezvous with its chief through an inherently safe approach. Simulations demonstrate the effectiveness of the safety constraints in altering a nominally unsafe rendezvous to one that is abort-safe.

Keywords:Constrained control, Autonomous systems, Aerospace Abstract: This paper presents a new methodology for ensuring forward invariance of sublevel sets of high relative degree functions, and convergence of the state to these sets. We introduce the notion of the boundary layer of a set defined by multiple constraints, and develop polynomially-derived trajectory constraints as means to enforce set invariance by redirecting trajectories that enter this boundary layer. This strategy is then extended to achieve convergence to and invariance of goal sets that are specified using Signal Temporal Logic. A quadratic program computes control inputs online that yield trajectories that achieve high level objectives specified by these sets, such as obstacle avoidance and target observation. We present a case study utilizing this controller for a spacecraft position and attitude control problem requiring observation of targets on the surface of a small body.

Keywords:Observers for nonlinear systems, Robotics, Nonlinear output feedback Abstract: In this paper, we propose a novel controller to stabilize the shape (joints) of an orbital robot about a set-point, i.e. a regulation task, in the specific setting that its spacecraft velocity is unmeasured. For this output feedback stabilization problem, the controller is presented as a synthesis of an observer for the spacecraft's motion states and a shape control law. To this end, we exploit the inertia-decoupled reduced Euler-Lagrange equations. The main advantage of this approach is that the block-diagonal inertia avoids the need for joint acceleration measurements, and the well-partitioned Coriolis/Centrifugal matrix highlights useful properties, which aid in the stability analysis. Additionally, the proposed controller uses only a minimal set of measurements in form of shape state-space, i.e. positions and velocities, and the exteroceptive pose (attitude and position) of the spacecraft. Thus, the need for inertial sensors (velocity measurements) on the spacecraft is also avoided. For the error dynamics of the resulting system, we prove uniform almost global asymptotic stability. Furthermore, we validate the analysis and prove the effectiveness of the method though simulations.

Keywords:Algebraic/geometric methods, Lyapunov methods, Observers for nonlinear systems Abstract: A geometric estimator is proposed for the rigid body attitude under multi-rate measurements using discrete-time Lyapunov stability analysis in this work. The angular velocity measurements are assumed to be sampled at a higher rate compared to the attitude. The attitude determination problem from two or more vector measurements in the body-fixed frame is formulated as Wahba's problem. In the case when measurements are absent, a discrete-time model for attitude kinematics is assumed in order to propagate the measurements. A discrete-time Lyapunov function is constructed as the sum of a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential energy-like term obtained from Wahba's cost function. A filtering scheme is obtained by discrete-time stability analysis using a suitable Lyapunov function. The analysis shows that the filtering scheme is exponentially stable in the absence of measurement noise and the domain of convergence is almost global. For a realistic evaluation of the scheme, numerical experiments are conducted with inputs corrupted by bounded measurement noise. Simulation results exhibit convergence of the estimated states to a bounded neighborhood of the actual states.

Keywords:Cooperative control, Distributed control, Robust control Abstract: Distributed fixed-time attitude tracking consensus is investigated in the paper for rigid spacecraft systems, with a directed communication graph containing a single leader which is the root of a directed spanning tree. The evolution of each follower is subject to unknown but bounded disturbances. First, a distributed fixed-time observer with heterogeneous coefficients is presented for each follower allowing cooperative estimation of the state of the leader at each follower. Then an observer-based fixed-time tracking protocol is designed to make the state of each spacecraft exactly track the state of the leader by the aid of backstepping technique. Finally, a simulation example is conducted to verify the analytical results.

Keywords:Hybrid systems, Lyapunov methods, Maritime control Abstract: This paper presents a hybrid feedback controller suitable for orientation control of ships. A hybrid kinematic controller on the unit circle is constructed from the gradient of a synergistic potential function, which globally asymptotically stabilizes a desired orientation, with yaw rate viewed as control input. While this idea is not new, the potential function is novel and possesses some desired properties. The kinematic controller generates smooth reference signals for the desired velocity and acceleration, except at instances when the controller switches. Continuity of velocity and acceleration is achieved by controlling the yaw rate through a double integrator. Moreover, the velocity and acceleration converge to their desired values exponentially. The resulting closed-loop system is stable, provided the controller gains satisfy mild constraints. This is shown using a hybrid Lyapunov function.

Keywords:Autonomous robots, Robotics, Game theory Abstract: We study a variant of pursuit-evasion game in the context of perimeter defense. In this problem, the intruder aims to reach the base plane of a hemisphere without being captured by the defender, while the defender tries to capture the intruder. The perimeter-defense game was previously studied under the assumption that the defender moves on a circle. We extend the problem to the case where the defender moves on a hemisphere. To solve this problem, we analyze the strategies based on the breaching point at which the intruder tries to reach the target and predict the goal position, defined as optimal breaching point, that is achieved by the optimal strategies on both players. We provide the barrier that divides the state space into defender-winning and intruder-winning regions and prove that the optimal strategies for both players are to move towards the optimal breaching point. Simulation results are presented to demonstrate that the optimality of the game is given as a Nash equilibrium.

Keywords:Autonomous robots, Robotics, Algebraic/geometric methods Abstract: This work presents a single-step diffeomorphic transformation from a known star world to a trivial domain called the point world, where the navigation task is reduced to connecting the images of the initial and destination configurations by a straight line. Obstacle potential functions – derived using Zenkin's formulas – are used to define a transformation activation region around each obstacle. Configurations in this region are radially mapped with respect to the center of the corresponding obstacle. The proposed transformation guarantees almost global navigation. The provided theoretical results are backed by analytical proofs, while the effectiveness of the method is demonstrated by a series of simulation studies.

Keywords:Autonomous robots, Robotics, Autonomous vehicles Abstract: Vector field guided path following (VF-PF) algorithms are fundamental in robot navigation tasks, but may not deliver the desirable performance when robots encounter singular points where the vector field becomes zero. The existence of singular points prevents the global convergence of the vector field's integral curves to the desired path. Moreover, VF-PF algorithms, as well as most of the existing path following algorithms, fail to enable following a self-intersected desired path. In this paper, we show that such failures are fundamentally related to the mathematical topology of the path, and that by "stretching" the desired path along a virtual dimension, one can remove the topological obstruction. Consequently, this paper proposes a new guiding vector field defined in a higher-dimensional space, in which self-intersected desired paths become free of self-intersections; more importantly, the new guiding vector field does not have any singular points, enabling the integral curves to converge globally to the "stretched" path. We further introduce the extended dynamics to retain this appealing global convergence property for the desired path in the original lower-dimensional space. Both simulations and experiments are conducted to verify the theory.

Keywords:Autonomous systems, Agents-based systems, Autonomous robots Abstract: This paper considers the distributed localization problem for multi-robot systems in the plane with bearing measurements. A necessary and sufficient condition for individual robot to check its localizability is provided in terms of a geometric condition. Then a distributed orthogonal algorithm is presented for verifying whether a robot is localizable via local computation. In addition to the distributed localizability test, this paper also develops a distributed conjugate residual algorithm, which can solve the localization problem in a more efficient manner. Numerical simulations demonstrate the effectiveness of the approach developed in this paper.

Keywords:Autonomous robots, Numerical algorithms, Neural networks Abstract: We introduce LRT-NG, a set of techniques and an associated toolset that computes a reachtube (an over-approximation of the set of reachable states over a given time horizon) of a nonlinear dynamical system. LRT-NG significantly advances the state-of-the-art Langrangian Reachability and its associated tool LRT. From a theoretical perspective, LRT-NG is superior to LRT in three ways. First, it uses for the first time an analytically computed metric for the propagated ball which is proven to minimize the ball's volume. We emphasize that the metric computation is the centerpiece of all bloating-based techniques. Secondly, it computes the next reachset as the intersection of two balls: one based on the Cartesian metric and the other on the new metric. While the two metrics were previously considered opposing approaches, their joint use considerably tightens the reachtubes. Thirdly, it avoids the ``wrapping effect'' associated with the validated integration of the center of the reachset, by optimally absorbing the interval approximation in the radius of the next ball. From a tool-development perspective, LRT-NG is superior to LRT in two ways. First, it is a standalone tool that no longer relies on CAPD. This required the implementation of the Lohner method and a Runge-Kutta time-propagation method. Secondly, it has an improved interface, allowing the input model and initial conditions to be provided as external input files. Our experiments on a comprehensive set of benchmarks, including two Neural ODEs, demonstrates LRT-NG's superior performance compared to LRT, CAPD, and Flow*.

Keywords:Autonomous robots, Control applications, Optimal control Abstract: In-flight docking between unmanned aerial systems (UASs) is an essential capability for extending collaborative long-range missions. This work presents a planning strategy for a smaller multirotor UAS to autonomously dock with a non-stationary carrier/leader UAS in forward flight. Our method assumes the leader aircraft to be another multirotor, and first projects the hypotheses for its pose forward in time. Using a multi-objective cost function, we then solve an optimal control problem to obtain an interception trajectory to all these possible locations. We employ a cost formulation that allows us to generate piecewise smooth curves that favor different objectives during the course of the mission. Through a greedy strategy, the paths are iteratively refined online as the prediction is improved with new observations. We demonstrate and evaluate our method through a series of physics-based simulations with different operating conditions for both vehicles.

Keywords:Autonomous vehicles, Optimal control, Autonomous robots Abstract: Consider a set of autonomous vehicles, each one with a preassigned task to start at a given region. Due to energy constraints, and in order to minimize the overall task completion time, these vehicles are deployed from a faster carrier vehicle.

This paper develops a dynamic programming based solution for the problem of finding the optimal deployment location and time for each vehicle, and for a given sequence of deployments, so that the global mission duration is minimal. The problem is specialized for ocean-going vehicles operating under time-varying currents. The solution approach involves solving a sequence of optimal stopping problems that are transformed into a set variational inequalities through the application of the dynamic programming principle. The optimal trajectory for the carrier and the optimal deployment location and time for each vehicle to be deployed are obtained in feedback-form from the numerical solution of the variational inequalities. The solution is computed with our open source parallel implementation of the fast sweeping method. The approach is illustrated with two numerical examples.

Keywords:Robust adaptive control, Stability of nonlinear systems, Robust control Abstract: Autonomous robots that are capable of operating safely in the presence of imperfect model knowledge or external disturbances are vital in safety-critical applications. In this paper, we present a planner-agnostic framework to design and certify safe tubes around desired trajectories that the robot is always guaranteed to remain inside. By leveraging recent results in contraction analysis and L1-adaptive control we synthesize an architecture that induces safe tubes for nonlinear systems with state and time-varying uncertainties. We demonstrate with a few illustrative examples how contraction theory-based L1-adaptive control can be used in conjunction with traditional motion planning algorithms to obtain provably safe trajectories.

Keywords:Identification, Nonlinear systems identification, Optimization Abstract: In this paper, we develop a nonparametric system identification method for nonlinear gradient-flow dynamics. In these systems, the vector field is the gradient field of a potential energy function. This fundamental fact about the dynamics of system plays the role of a structural prior knowledge as well as a constraint in the proposed identification method. While the nature of the identification problem is an estimation in the space of functions, we derive an equivalent finite dimensional formulation, which is a convex optimization in the form of a quadratic program. This gives scalability to the problem and provides the opportunity for utilizing recently developed large-scale optimization solvers. The central idea in the proposed method is representing the energy function as a difference of two convex functions and learning these convex functions jointly. Based on necessary and sufficient conditions for function convexity, the identification problem is formulated, and then, the existence, uniqueness and smoothness of the solution is addressed. We also illustrate and evaluate the method numerically with two demonstrative examples.

Keywords:Identification, Nonlinear systems identification, Optimization Abstract: We consider the problem of nonlinear system identification when prior knowledge is available on the region of attraction (ROA) of an equilibrium point. We propose an identification method in the form of an optimization problem, minimizing the fitting error and guaranteeing the desired stability property. The problem is approached by joint identification of the dynamics and a Lyapunov function verifying the stability property. In this setting, the hypothesis set is a reproducing kernel Hilbert space, and with respect to each point of the given subset of the ROA, the Lie derivative inequality of the Lyapunov function imposes a constraint. The problem is a non-convex infinite-dimensional optimization with an infinite number of constraints. To obtain a tractable formulation, only a suitably designed finite subset of the constraints are considered. The resulting problem admits a solution in form of a linear combination of the sections of the kernel and its derivatives. An equivalent optimization problem with a quadratic cost function subject to linear and bilinear constraints is derived. A suitable change of variable gives a convex reformulation of the problem. The method is demonstrated by several examples.

Keywords:Nonlinear systems identification, Adaptive systems, Distributed parameter systems Abstract: The reproducing kernel Hilbert space (RKHS) embedding method is a recently introduced estimation approach that seeks to identify the unknown or uncertain function in the governing equations of a nonlinear set of ordinary differential equations (ODEs). While the original state estimate evolves in Euclidean space, the function estimate is constructed in an infinite dimensional RKHS and must be approximated in practice. When a finite dimensional approximation is constructed using a basis defined in terms of shifted kernel functions centered at the observations along a trajectory, the RKHS embedding method can be understood as a data-driven approach. This paper derives sufficient conditions that ensure that approximations of the unknown function converge in a Sobolev norm over a submanifold that supports the dynamics. Moreover, the rate of convergence for the finite dimensional approximations is derived in terms of the fill distance of the samples in the embedded manifold. A numerical simulation of an example problem is carried out to illustrate the qualitative nature of convergence results derived in the paper.

University of Applied Sciences and Arts of Southern Switzerland

Keywords:Nonlinear systems identification, Identification, Neural networks Abstract: This paper introduces a novel neural network architecture, called Integrated Neural Network (INN), for direct identification of nonlinear continuous-time dynamical models in state-space representation. The proposed INN is used to approximate the continuous-time state map, and it consists of a feed-forward network followed by an integral block. The unknown parameters are estimated by minimizing a properly constructed dual-objective criterion. The effectiveness of the proposed methodology is assessed against the Cascaded Tanks System benchmark

Keywords:Nonlinear systems identification, Modeling, Reduced order modeling Abstract: This paper derives rates of convergence of certain approximations of the Koopman operators that are associated with discrete, deterministic, continuous semiflows on a complete metric space. Approximations are constructed in terms of reproducing kernel bases that are centered at samples taken along the system trajectory. It is proven that when the samples are dense in a certain type of smooth manifold, the derived rates of convergence depend on the fill distance of samples along the trajectory in that manifold. Error bounds for projection-based and data-dependent approximations of the Koopman operator are derived in the paper. A discussion of how these bounds are realized in intrinsic and extrinsic approximation methods is given. Numerical examples are given that illustrate qualitatively the convergence guarantees derived in the paper.

Keywords:Nonlinear systems identification, Optimal control, Numerical algorithms Abstract: Optimal experiment design (OED) aims to optimize the information content of experimental observations for various types of applications by designing the experimental conditions. In Bayesian OED for parameter estimation, the design selection is based on an expected utility metric that accounts for the joint probability distribution of the uncertain parameters and the observations. This work presents an approximation of the Bayesian OED problem based on Kullback–Leibler divergence that is amenable to global optimization. The experiment design adopts a parsimonious input parametrization that reduces the number of design variables. This leads to a tractable polynomial optimization problem that can be solved to global optimality via the concept of sum-of-squares polynomials.

University of Applied Sciences and Arts of Southern Switzerland

Keywords:Identification, Hybrid systems Abstract: Learning PieceWise Affine Output-Error (PWA-OE) models from data requires to estimate a finite set of affine output-error sub-models as well as a partition of the regressors space over which the sub-models are defined. For an output-error type noise structure, the algorithms based on standard least squares (LS) fail to compute a consistent estimate of the sub-model parameters. On the other hand, the prediction error methods (PEMs) provide a consistent parameter estimate, however, they require to solve a non-convex optimization problem for which the numerical algorithms may get trapped in a local minimum, leading to inaccurate estimates. In this paper, we propose a recursive bias-correction scheme for identifying PWA-OE models, retaining the computational efficiency of the standard LS algorithms while providing a consistent estimate of the sub-model parameters, under suitable assumptions. The proposed approach allows one to recursively update the estimates of the sub-models parameters and to cluster the regressors. Linear multi-category techniques are then employed to estimate a partition of the regressor space based on the estimated clusters. The performance of the proposed algorithm is demonstrated via an academic example.

IDSIA Dalle Molle Institute for Artificial Intelligence

Keywords:Nonlinear systems identification, Switched systems, Learning Abstract: We propose two optimization-based heuristics for structure selection and identification of PieceWise Affine (PWA) models with exogenous inputs. The first method determines the number of affine sub-models assuming known model order of the sub-models, while the second approach estimates the model order for a given number of affine sub-models. Both approaches rely on the use of regularization-based shrinking strategies, that are exploited within a coordinate-descent identification algorithm. This allows us to estimate the structure of the PWA models along with its model parameters. Starting from an overparameterized model, the key idea is to alternate between an identification step and structure refinement. The performance of the presented strategies is assessed over two benchmark examples.

Keywords:Game theory, Resilient Control Systems, Markov processes Abstract: Motivated by recent works addressing adversarial attacks on deep reinforcement learning, a deception attack on linear quadratic Gaussian control is studied in this paper. In the considered attack model, the adversary can manipulate the observation of the agent subject to a mutual information constraint. The adversarial problem is formulated as a novel dynamic cheap talk game to capture the strategic interaction between the adversary and the agent, the asymmetry of information availability, and the system dynamics. Necessary and sufficient conditions are provided for subgame perfect equilibria to exist in pure strategies and in behavioral strategies; and characteristics of the equilibria and the resulting control rewards are given. The results show that pure strategy equilibria are informative, while only babbling equilibria exist in behavioral strategies. Numerical results are shown to illustrate the impact of strategic adversarial interaction.

Keywords:Game theory, Learning, Identification Abstract: In this work, a meta-learning framework for games between adapting players is proposed. An agent with increased cognitive abilities is augmented with a structure that allows them to identify the way that their opponents learn during the game. This is achieved via approximators that are tuned online leveraging only observed actions from the environment. We show that knowledge of the utilities of the opponents enable asymptotic convergence of the approximation weights. We, then, extend the framework via backpropagation through time such that knowledge of the utilities is not necessary and we show convergence of the errors to a residual set. Finally, simulations of players learning in a penny matching game demonstrate the efficacy of our approach.

Keywords:Game theory, Variational methods, Machine learning Abstract: Generative adversarial networks (GANs) are a class of generative models, known for producing accurate samples. The key feature of GANs is that there are two antagonistic neural networks: the generator and the discriminator. The main bottleneck for their implementation is that the neural networks are very hard to train. One way to improve their performance is to design reliable algorithms for the adversarial process. Since the training can be cast as a stochastic Nash equilibrium problem, we rewrite it as a variational inequality and introduce an algorithm to compute an approximate solution. Specifically, we propose a stochastic relaxed forward–backward algorithm for GANs. We prove that when the pseudogradient mapping of the game is monotone, we have convergence to an exact solution or in a neighbourhood of it.

Keywords:Game theory, Machine learning, Decentralized control Abstract: Motivated by applications of multi-agent learning in noisy environments, this paper studies the robustness of gradient-based learning dynamics with respect to disturbances. While disturbances injected along a coordinate corresponding to any individual player's actions can always affect the overall learning dynamics, a subset of players can be disturbance decoupled---i.e., such players' actions are completely unaffected by the injected disturbance. We provide necessary and sufficient conditions to guarantee this property for games with quadratic cost functions, which encompass quadratic one-shot continuous games, finite-horizon linear quadratic (LQ) dynamic games, and bilinear games. Specifically, disturbance decoupling is characterized by both algebraic and graph-theoretic conditions on the learning dynamics, the latter is obtained by constructing a game graph based on gradients of players' costs. For LQ games, we show that disturbance decoupling imposes constraints on the controllable and unobservable subspaces of players. For two player bilinear games, we show that disturbance decoupling within a player's action coordinates imposes constraints on the payoff matrices. Illustrative numerical examples are provided.

Keywords:Game theory, Optimal control Abstract: In this paper, the problem of guarding a circular target wherein the Defender(s) is constrained to move along its perimeter is posed and solved using a differential game theoretic approach. Both the one-Defender and two-Defender scenarios are analyzed and solved. The mobile Attacker seeks to reach the perimeter of the circular target, whereas the Defender(s) seeks to align itself with the Attacker, thereby ending the game. In the former case, the Attacker wins, and the Attacker and Defender play a zero sum differential game where the payoff/cost is the terminal angular separation. In the latter case, the Defender(s) wins, and the Attacker and Defender play a zero sum differential game where the cost/payoff is the Attacker’s terminal distance to the target. This formulation is representative of a scenario in which the Attacker inflicts damage on the target as a function of its terminal distance. The state-feedback equilibrium strategies and Value functions for the Attacker-win and Defender(s)-win scenarios are derived for both the one- and two-Defender cases, thus providing a solution to the Game of Degree. Analytic expressions for the separating surfaces between the various terminal scenarios are derived, thus providing a solution to the Game of Kind.

Keywords:Game theory, Distributed control, Agents-based systems Abstract: A popular formalism for multiagent control applies tools from game theory, casting a multiagent decision problem as a cooperation-style game in which individual agents make local choices to optimize their own local utility functions in response to the observable choices made by other agents. When the system-level objective is submodular maximization, it is known that if every agent can observe the action choice of all other agents, then all Nash equilibria of a large class of resulting games are within a factor of 2 of optimal; that is, the price of anarchy is 1/2. However, little is known if agents cannot observe the action choices of other relevant agents. To study this, we extend the standard game-theoretic model to one in which a subset of agents either become blind (unable to observe others' choices) or isolated (blind, and also invisible to other agents), and we prove exact expressions for the price of anarchy as a function of the number of compromised agents. When k agents are compromised (in any combination of blind or isolated), we show that the price of anarchy for a large class of utility functions is exactly 1/(2+k). We then show that if agents use marginal-cost utility functions and at least 1 of the compromised agents is blind (rather than isolated), the price of anarchy improves to 1/(1+k). We also provide simulation results demonstrating the effects of these observation denials in a dynamic setting.

Keywords:Game theory, Stability of nonlinear systems, Compartmental and Positive systems Abstract: This paper studies the problem of a large group of individuals that has to get to a safety exit in the context of high-stress emergency evacuations. We model this problem as a discrete-state continuous-time game, where the players update their strategies to reach the exit within a defined time horizon, whilst avoiding undesirable situations such as congestion and being trampled. The proposed model builds on crowd dynamics in a two-strategy game theoretic context, which we extend to include aspects of crowd behavior originating in sociology and psychology, and in the analogous studies performed in immersive virtual environments. The main contribution of this paper is threefold: i) we propose a novel game formulation of the model in terms of the population distribution across three strategies, and provide a link with prospect theory; ii) we study the equilibria of the system and their stability via Lyapunov stability theory of nonlinear systems; iii) we extend the model to a multi-population setting, where each population represents the group of players at a certain distance from the exit.

Keywords:Game theory, Lyapunov methods, Agents-based systems Abstract: While the rate of convergence has been a long-standing topic in optimization, it has received a less systematic treatment in games. This paper provides convergence rates to the Nash equilibrium for large classes of game dynamics in games with (relatively) strongly and weakly concave potential functions. Simulations are performed for representative games.

Keywords:Optimization, Power systems, Smart grid Abstract: Consider an optimization problem with a convex cost function but a non-convex compact feasible set X, and its relaxation with a compact and convex feasible set X̂⊃X. We prove that if from any point x∈X̂\X there is a path connecting x to X along which both the cost function and a Lyapunov-like function are improvable, then any local optimum in X for the original non-convex problem is a global optimum. We use this result to show that, for AC optimal power flow problems, a well-known sufficient condition for exact relaxation also guarantees that all its local optima are globally optimal. This helps explain the widespread empirical experience that local algorithms for optimal power flow problems often work extremely well.

Keywords:Optimization, Smart grid, Power systems Abstract: The optimal power flow (OPF) problem is a well-known non-convex optimization problem that aims to minimize the cost of electric power generation subject to consumer demand, the physics of power flow, and technological constraints. To find an optimal solution to this problem, local search techniques such as interior point methods are typically used. However, due to the non-convex nature of the problem, these methods are likely to result in a sub-optimal solution. The goal of this paper is to characterize the worst-case performance of local search on the OPF problem. To accomplish this, we formulate the OPF problem as a canonical quadratically-constrained quadratic program (QCQP). Then, we study the problem of finding the worst-case local minimum of this QCQP, which is non-convex and hard to solve in general. We find a relaxation of this problem into a semidefinite program (SDP) and show that it is exact for certain cases. Using some test cases which are known to have multiple local minima, we demonstrate the effectiveness of the proposed relaxation to bound the worst-case local minimum. We compare the obtained upper bound on local minima to the lower bound provided by the standard SDP relaxation of the OPF problem to understand how much SDP outperforms local search for a given problem.

Keywords:Optimal control, Robotics, Machine learning Abstract: Mixed-integer convex programming (MICP) has seen significant algorithmic and hardware improvements with several orders of magnitude solve time speedups compared to 25 years ago. Despite these advances, MICP has been rarely applied to real-world robotic control because the solution times are still too slow for online applications. In this work, we present the CoCo (Combinatorial Offline, Convex Online) framework to solve MICPs arising in robotics at very high speed. CoCo encodes the combinatorial part of the optimal solution into a strategy. Using data collected from offline problem solutions, we train a multiclass classifier to predict the optimal strategy given problem-specific parameters such as states or obstacles. Compared to previous approaches, we use task-specific strategies and prune redundant ones to significantly reduce the number of classes the predictor has to select from, thereby greatly improving scalability. Given the predicted strategy, the control task becomes a small convex optimization problem that we can solve in milliseconds. Numerical experiments on a cart-pole system with walls, a free-flying space robot, and task-oriented grasps show that our method provides not only 1 to 2 orders of magnitude speedups compared to state-of-the-art solvers but also performance close to the globally optimal MICP solution.

Keywords:Optimization, Sensor networks, Large-scale systems Abstract: This paper studies selecting a subset of the system's output to minimize the state estimation mean square error (MSE). This results in the maximization problem of a set function defined on possible sensor selections subject to a cardinality constraint. We consider to solve it approximately by a greedy search. Since the MSE function is not submodular nor supermodular, the well-known performance guarantees for the greedy solutions do not hold in the present case. Thus, we use the quantities---the submodularity ratio and the curvature---to evaluate the degrees of submodularity and supermodularity of the objective unction. By using the properties of the MSE function, we approximately compute these quantities and derive a performance guarantee for the greedy solutions. It is shown that the guarantee is less conservative than those in the existing results.

Keywords:Optimization, Stability of nonlinear systems, Lyapunov methods Abstract: The problem of finding a point in R^n, from which the sum-of-distances to a finite number of non-empty, closed and convex sets is minimum is called generalized Fermat-Torricelli Problem (FTP). In applications, along with the point that minimizes sum-of-distances, it is important to know the points in the convex sets at which the minimum sum-of-distances is achieved. Various formulations existing in literature do not involve finding the optimal points on the convex sets. In this work, we formulate a non-smooth convex optimization problem, with both the point/set of points which yields the minimum sum-of-distances as well as the corresponding points in the convex sets as primal variables. We term this problem as extended FTP (eFTP). We adopt non-smooth projected primal-dual dynamical approach to solve this problem. The proposed dynamical system can exhibit a continuum of equilibria. Hence we show semistability of the set of optimal points, which is the pertinent notion of stability for such systems. A distributed implementation of the primal-dual dynamical system is also presented in this work. Four illustrative examples are considered for the simulation based validation of the solution proposed for eFTP.

Keywords:Optimal control, Distributed control, Linear systems Abstract: System Level Synthesis (SLS) parametrization facilitates controller synthesis for large, complex, and distributed systems by incorporating system level constraints (SLCs) into a convex SLS problem and mapping its solution to stable controller design. Solving the SLS problem at scale efficiently is challenging, and current attempts take advantage of special system or controller structures to speed up the computation in parallel. However, those methods do not generalize as they rely on the specific system/controller properties.

We argue that it is possible to solve general SLS problems more efficiently by exploiting the structure of SLS constraints. In particular, we derive dynamic programming (DP) algorithms to solve SLS problems. In addition to the plain SLS without any SLCs, we extend DP to tackle infinite horizon SLS approximation and entrywise linear constraints, which form a superclass of the locality constraints. Comparing to convex program solver and naive analytical derivation, DP solves SLS 4 to 12 times faster and scales with little computation overhead. We also quantize the cost of synthesizing a controller that stabilizes the system in a finite horizon through simulations.

Keywords:Estimation Abstract: In recent years the Hawkes model has been applied in diverse application areas ranging from neural coding, genomics, econometrics to social networks. In the original Hawkes model, the parameters are time-invariant. But a number of studies indicate the need for time-variant parameters. In this paper, we extend previous work on time-variant scalar Hawkes models to the vector case. In addition, we develop a computational procedure that is exact and so is very fast. We demonstrate the algorithm using some neural data.

Keywords:Optimal control, Linear systems, Optimization algorithms Abstract: In this paper we propose a new objective function for the discrete minimum time control problem, where constraints in control are allowed. With this objective function, the control problem is transformed into a tractable optimization problem where traditional numerical algorithms can be applied. We proved that under some conditions, the optimal solution of the newly transformed problem yields the minimum time of the original problem. We propose a Sliding Window Algorithm which dynamically adjusts a time window aiming to catch the optimal solution, and for problems with constraints we propose a constrained interior search algorithm. Numerical examples are provided to show the effectiveness of our algorithms.

Keywords:Stochastic optimal control, Computational methods, Markov processes Abstract: In this paper, we study the numerical solution of a class of elliptic type of Hamilton-Jacobi-Bellman (HJB) equations. These equations arise naturally from one or multiplayer stochastic control problems with random terminal time. Two closely related algorithms are proposed. One of them is value iteration on approximating Markov decision processes, and the other is deep learning approach on its Bellman equations. The convergence is shown by using a viscosity solution approach.

Keywords:Stochastic optimal control, Markov processes, Computational methods Abstract: Policy Iteration (PI) is a classical family of algorithms to compute an optimal policy for a Markov Decision Problem (MDP). The basic idea in PI is to begin with some initial policy and to repeatedly update the policy to one from an improving set until an optimal policy is reached. Different variants of PI result from the (switching) rule used for improvement. An important theoretical question is how many iterations a specified PI variant will take to terminate as a function of the number of states n and the number of actions k in the input MDP. While there has been considerable progress towards upper-bounding this number, there are fewer results on lower bounds. In particular, existing lower bounds primarily focus on the special case of k = 2. In this paper, we devise lower bounds for k geq 3. Our main result is that a particular variant of PI can take Omega(k^{n/2}) iterations to terminate. We also generalise existing constructions on 2-action MDPs to scale lower bounds by a factor of k for some common deterministic variants of PI, and by log(k) for corresponding randomised variants.

Keywords:Stochastic optimal control, Information theory and control, Distributed control Abstract: We consider a discrete-time linear-quadratic-Gaussian control problem in which we minimize a weighted sum of the directed information from the state of the system to the control input and the control cost. The optimal control and sensing policies can be synthesized jointly by solving a semidefinite programming problem. However, the existing solutions typically scale cubic with the horizon length. We leverage the structure in the problem to develop a distributed algorithm that decomposes the synthesis problem into a set of smaller problems, one for each time step. We prove that the algorithm runs in time linear in the horizon length. As an application of the algorithm, we consider a path-planning problem in a state space with obstacles under the presence of stochastic disturbances. The algorithm computes a locally optimal solution that jointly minimizes the perception and control cost while ensuring the safety of the path. The numerical examples show that the algorithm can scale to thousands of horizon length and compute locally optimal solutions.

Keywords:Stochastic optimal control, Kalman filtering, Constrained control Abstract: We consider the problem of steering, via output feedback, the state distribution of a discrete-time, linear stochastic system from an initial Gaussian distribution to a terminal Gaussian distribution with prescribed mean and maximum covariance, subject to probabilistic path constraints on the state. The filtered state is obtained via a Kalman filter, and the problem is formulated as a deterministic convex program in terms of the distribution of the filtered state. We observe that, in the presence of constraints on the state covariance, and in contrast to classical Linear Quadratic Gaussian (LQG) control, the optimal feedback control depends on both the process noise and the observation model. The effectiveness of the proposed approach is verified using a numerical example.

Keywords:Stochastic optimal control, Game theory, Linear systems Abstract: This paper addresses the problem of steering a discrete-time linear dynamical system from an initial Gaussian distribution to a final distribution in a game-theoretic setting. One of the two players strives to minimize a quadratic payoff, while at the same time tries to meet a given mean and covariance constraints at the final time-step. The other player maximizes the same payoff, but it is assumed to be indifferent to the terminal constraint. At first, the unconstrained version of the game is examined, and the necessary conditions for the existence of a saddle point are obtained. We show that obtaining a solution for the one-sided constrained dynamic game is not guaranteed, and subsequently the players' best responses are analyzed. Finally, we propose to numerically solve the problem of steering the distribution under adversarial scenarios using the Jacobi iteration method.

Keywords:Stochastic optimal control, Optimal control, Stochastic systems Abstract: In this paper, we propose a minimax linear-quadratic control method to address the issue of inaccurate distribution information in practical stochastic systems. To construct a control policy that is robust against errors in an empirical distribution of uncertainty, our method adopts an adversary, which selects the worst-case distribution. The opponent receives a penalty proportional to the amount (measured in the Wasserstein metric) of deviation from the empirical distribution. In the finite-horizon case, using a Riccati equation, we derive a closed-form expression of the unique optimal policy and the opponent’s policy that generates the worst-case distribution. This result is then extended to the infinite-horizon setting by identifying conditions under which the Riccati recursion converges to the unique positive semi-definite solution to an associated algebraic Riccati equation (ARE). The resulting optimal policy is shown to stabilize the expected value of the system state under the worst-case distribution. We also discuss that our method can be interpreted as a distributional generalization of the H_infty-method.

Keywords:Stochastic optimal control, Optimization, Control Systems Privacy Abstract: We consider a counter-adversarial sequential decision-making problem where an agent computes its private belief (posterior distribution) of the current state of the world, by filtering private information. According to its private belief, the agent performs an action, which is observed by an adversarial agent. We have recently shown how the adversarial agent can reconstruct the private belief of the decision-making agent via inverse optimization. The main contribution of this paper is a method to obfuscate the private belief of the agent from the adversary, by performing a suboptimal action. The proposed method optimizes the trade-off between obfuscating the private belief and limiting the increase in cost accrued due to taking a suboptimal action. We propose a probabilistic relaxation to obtain a linear optimization problem for solving the trade-off. In numerical examples, we show that the proposed methods enable the agent to obfuscate its private belief without compromising its cost budget.

Keywords:Statistical learning, Learning, Communication networks Abstract: In this paper, we present a non-Bayesian model of learning over a social network where a group of agents with insufficient and heterogeneous sources of information share their experiences to learn an underlying state of the world. Inspired by a recent body of research in cognitive science on human decision making, we presume two behavioral assumptions. Motivated by the coarseness of communication, our first assumption posits that agents only share samples taken from their belief distribution over the set of states, to which we refer as their actions.

This situation is to be contrasted with that of sharing the full belief, i.e. probability distribution over the entire set of states. The second assumption is limited cognitive power, based on which individuals incorporate their neighbors' actions into their beliefs following a simple DeGroot-like social learning rule which suffers from redundancy neglect and imperfect recall of the past history. We show that so long as all the individuals trust their neighbors' actions more than their private signals, they may end up mislearning the state with positive probability. Learning, on the other hand, requires that the population includes a group of self-confident experts in different states. This means that for each state, there is an agent whose signaling function for her state of expertise is distinguishable from the convex hull of the remaining signaling functions, and that her private signals sufficiently weigh in her social learning rule.

Keywords:Stability of nonlinear systems, Lyapunov methods, Network analysis and control Abstract: The desire to better understand the behaviour of an epidemic has never been more important. Not only do epidemic models provide vital tools for predicting the spread of a disease, but they have also seen other uses in modelling the propagation of rumour spreading on online networks and the distribution of computer viruses over the internet. In this paper we use the Susceptible-Exposed-Infected-Recovered (SEIR) model, along with an extended multi-population model that can display richer dynamics. Previously, a diagonal Lyapunov function of a log-linear structure has been shown to exist under some assumptions, for both the single and multi-population cases. However, when the reproduction number is less than unity or the network between subsystems is not strongly connected, no multi-population Lyapunov function is known. We propose an alternative Lyapunov function structure for a two-population model, that has been shown to be valid for many instances with Sum of Squares programming. These instances include conditions where the original log-linear Lyapunov function fails, meaning that we have constructed a Lyapunov function for an SEIR multi-population model for the first time. This approach can then be scaled up to the general multi-population case using advanced techniques.

Keywords:Distributed control, Network analysis and control, Identification for control Abstract: In this paper, the problem of synthesizing a distributed controller from data is considered, with the objective to optimize a model-reference control criterion. We establish an explicit ideal distributed controller that solves the model-reference control problem for a structured reference model. On the basis of input-output data collected from the interconnected system, a virtual experiment setup is constructed which leads to a network identification problem. We formulate a prediction-error identification criterion that has the same global optimum as the model-reference criterion, when the controller class contains the ideal distributed controller. The developed distributed controller synthesis method is illustrated on an academic example network of nine subsystems and the influence of the controller interconnection structure on the achieved closed-loop performance is analyzed.

Keywords:Decentralized control, Optimal control Abstract: The optimal decentralized control (ODC) is an NP-hard problem with many applications in real-world systems. There is a recent trend of using local search algorithms for solving optimal control problems. However, the effectiveness of these methods depends on the connectivity property of the feasible region of the underlying optimization problem. In this paper, we develop the notion of stable expandability and use it to obtain a novel criterion for certifying the connectivity of the feasible region for ODC problems with state feedback and an identity input matrix. This criterion can be checked via an efficient algorithm. Based on the developed mathematical technique, we prove that the feasible region is guaranteed to be connected in presence of only a small number of communication constraints. We also show that among the exponential number of possible communication networks (named patterns), a square root of them lead to connected feasible regions. A by-product of this result is that a high-complexity ODC problem may be approximated with a simpler one by replacing its pattern with a favorable pattern that makes the feasible region connected.

Keywords:Game theory, Large-scale systems, Randomized algorithms Abstract: We propose a hierarchical approach for the stochastic stability analysis of evolutionary dynamics. Each layer in the hierarchy represents a compromise between computational effort and the resolution of information about the long-run behavior of evolutionary dynamics. Previously, we proposed a graphical reformulation of Evolutionarily Sable Strategy (ESS) analysis through which we identified a set of strategies that cannot be ESS. Moreover, we also computed a set of strategies that was guaranteed to contain the set of stochastically stable strategies. The previous analysis was developed by considering transitions resulting from single mutations only. We extend the graphical approach to higher order analysis by incorporating mutations of higher order and show that we can refine our solution estimate by identifying smaller subsets of strategies that contain the set of stochastically stable strategies. However, this refinement comes at a cost of increase in computational budget.

Keywords:Sampled-data control, Networked control systems Abstract: This paper presents a novel approach towards synchronization analysis of nonlinear oscillatory systems, bidirectionally coupled via a networked communication channel. The system under consideration is a two-agent nonlinear system, under the constraint that information is transmitted between the two systems using a sampled-data communication strategy that could be periodic or aperiodic. The networked system dynamics is remodelled as a feedback-interconnection of a continuous-time system, and an operator that accounts for the communication constraints. By studying the properties of this feedback-interconnection in the framework of dissipativity theory, we provide a novel criterion that guarantees exponential synchronization. The provided criterion also aids in deciding the trade-off between a bound on the sampling intervals, the coupling gain, and the desired transient rate of synchronization. Finally, the theoretical results are illustrated using a two-agent Fitzhugh-Nagumo system.

Keywords:Network analysis and control, Uncertain systems, Stability of linear systems Abstract: We give a sufficient and a necessary condition for the topology-independent robust stability of networked systems formed by uncertain MIMO systems. Both conditions involve constants associated with the nominal node dynamics and arc interconnection matrices, the uncertainty bounds, and the maximum connectivity degree of the network; they are scalable (they can be checked locally), independent of the network topology and even of the number of nodes and arcs, and hold for networks of heterogeneous MIMO systems and interconnection matrices, with heterogeneous uncertainties. The dual cases of 1-norm and infinity-norm bounds are considered. In both cases, if the systems at the nodes are diagonal, we get a necessary and sufficient condition. We apply our results to the topology-independent robust stability analysis of a case-study from cancer biology.

Keywords:Agents-based systems, Distributed control, Autonomous systems Abstract: The Fiedler vector of a graph is the eigenvector corresponding to the smallest non-trivial eigenvalue of the corresponding Laplacian matrix, i.e, the algebraic connectivity. We propose and prove the convergence properties of a novel continuous-time distributed control protocol to drive the value of the state variables of a network toward the Fiedler vector, up to a scale factor, assuming known algebraic connectivity. The proposed strategy is unbiased and robust with respect to the initial network state. The proposed strategy does not require initialization of state variables to particular values. By exploiting the proposed control protocol we design a local state feedback that achieves desynchronization on arbitrary undirected connected networks of diffusively coupled harmonic oscillators. We provide numerical simulations to corroborate the theoretical results.

Keywords:Information theory and control, Networked control systems Abstract: We study a linear quadratic Gaussian (LQG) control problem, in which the optimal LQG cost is unsatisfactory. To achieve the desired LQG cost, we introduce a communication link from the system to the controller, in addition to a noisy observation of the system state that is already available to the controller. We investigate the trade-off between the improved LQG cost and the consumed communication (information) resources that are measured with the conditional directed information. The objective is to minimize the directed information over all encoding-decoding policies subject to a constraint on the LQG cost. The main result is a semidefinite programming formulation for the optimization problem in the finite-horizion scenario where the dynamical system may have time-varying parameters. This result extends the seminal work by Tanaka et al., where the direct noisy measurement of the system state at the controller is assumed to be absent. As part of our derivation to show the optimality of an encoder that transmits a Gaussian measurement of the state, we show that the presence of the noisy measurements at the encoder can not reduce the minimal directed information, extending a prior result of Kostina and Hassibi to the vector case. Finally, we show that the results in the finite-horizon case can be extended to the infinite-horizon scenario when assuming a time-invariant system, but possibly a time-varying policy. We show that the solution for this optimization problem can be realized by a time-invariant policy whose parameters can be computed explicitly from a finite-dimensional semidefinite programming.

Keywords:Constrained control, Networked control systems, Control over communications Abstract: In the context of uncertain control systems, the notion of invariance feedback entropy (IFE) quantifies the state information required by any controller to render a subset Q of the state space invariant. IFE equivalently also quantifies the smallest bit rate, from the coder to the controller in the feedback loop, above which Q can be made invariant over a digital noiseless channel. In this work, we consider discrete-time uncertain control systems described by difference inclusions and establish three results for IFE. First, we show that the IFE of a discrete-time uncertain control system Sigma and a nonempty set Q is upper bounded by the largest possible IFE of Sigma and any member of any finite partition of Q. Second, we consider two uncertain control systems, Sigma_1 and Sigma_2, which are identical except for the transition function, such that the behavior of Sigma_1 is included within that of Sigma_2. For a given nonempty subset of the state space, we show that the IFE of Sigma_2 is larger or equal to the IFE of Sigma_1.

Third, we establish an upper bound for the IFE of a network of uncertain control subsystems in terms of the IFEs of smaller subsystems. Further, via an example, we show that the upper bound is tight for some systems. Finally, to illustrate the effectiveness of the results, we compute an upper bound and a lower bound of the IFE of a network of uncertain, linear, discrete-time subsystems describing the evolution of temperature of 100 rooms in a circular building.

Keywords:Networked control systems, Stability of nonlinear systems, Stochastic systems Abstract: We present a controller and transmission policy design procedure for nonlinear wireless networked control systems. Our objective is to ensure the stability of the closed-loop system, in a stochastic sense, together with given control performance, while minimizing the average power used to generate the communication instants. The controller is designed by emulation, i.e. ignoring the network, and the transmission instants are generated by a so-called time-based threshold policy. The latter consists in waiting a given amount of time since the last successful transmission instant before using a constant power to transmit. We explain how to select the waiting time and the power to minimize the induced average communication power while ensuring the desired control objectives.

Keywords:Networked control systems, Communication networks, Optimization algorithms Abstract: In this paper, we consider a wireless networked control system (W-NCS) and seek to optimize the performance of the system by adapting the transmission throughput of the communication link, which is assumed to be a quasi-static fading channel. Towards this end, an optimization problem is formulated and solved, herein called Maximum Throughput with Energy Constraints (MaxTEC), in which the optimal achievable throughput is selected subject to the limited available energy per transmission. It is demonstrated that the larger the available energy, the higher the throughput, and, subsequently, the better the control performance. The performance of our proposed scheme is illustrated via simulations of an inverted pendulum on a cart.

Keywords:Networked control systems, Control over communications, Stability of nonlinear systems Abstract: We study emulation-based stabilisation of nonlinear networked control systems communicating over multiple wireless channels subject to packet loss. Specifically, we establish sufficient conditions on the rate of transmission that guarantee Lp stability-in-expectation of the overall closed-loop system. These conditions depend on the cumulative dropout probability of the network nodes for static protocols. We use the obtained stability results to study power control, where we show there are interesting trade-offs between the transmission rate, transmit power, and stability. Lastly, numerical examples are presented to illustrate our results.

Keywords:Networked control systems, Stochastic optimal control Abstract: An optimal control law for networked control systems with a discrete-time linear time-invariant plant and networks between sensors and controller as well as between controller and actuators is proposed. The networks are characterized by random delays and dropouts of transmitted data packets. The optimal control law is linear in an extended state vector; the gains can be calculated offline in advance.

Keywords:Networked control systems Abstract: A stability criterion for networked feedback loops with time-varying measurement delays is proposed. It allows to check stability under the presence of different packetized network transmission policies where, e.g., newer packets may arrive at the same time or before older packets. The criterion is based on the small gain theorem and explicitly takes into account a packet selection and hold mechanism at the receiver side. It can easily be checked in frequency domain and does not involve any computationally expensive optimization algorithms.

Keywords:Networked control systems, Stochastic systems, Distributed control Abstract: By using dissipativity approach, we establish the stability condition for the feedback connection of a deterministic dynamical system and a stochastic memoryless map. After that, we extend the result to the class of large-scale systems in which: the deterministic system consists of many sub-systems; and the stochastic map consists of many stochastic actuators. We will demonstrate the proposed approach by showing the design procedures to globally stabilize the manufacturing systems while locally balance the stock levels in any production process.

Keywords:Machine learning, Lyapunov methods, Constrained control Abstract: Optimal control problems with constraints ensuring safety can be mapped onto a sequence of real time optimization problems through the use of Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). One of the main challenges in these approaches is ensuring the feasibility of the resulting quadratic programs (QPs) if the system is affine in controls. In this paper, we improve the feasibility robustness (i.e., feasibility maintenance in the presence of time-varying and unknown unsafe sets) through the definition of a High Order CBF (HOCBF); this is achieved by a proposed feasibility-guided learning approach using machine learning techniques. The effectiveness of the proposed feasibility-guided learning approach is demonstrated on a robot control problem.

Keywords:Machine learning, Optimal control, Learning Abstract: In recent years, a wide number of theoretical papers have focused on reinforcement learning approaches to the linear quadratic regulator (LQR) problem. However, nearly all of these papers assume that an initial stabilizing controller is given. This paper gives a model-free, off-policy reinforcement learning algorithm for computing a stabilizing controller for deterministic LQR problems with unknown dynamics and cost matrices. When the system is stabilizable, a controller which is guaranteed to stabilize the system is computed after finitely many steps. Furthermore, the solution converges to the optimal LQR gain.

Keywords:Machine learning, Lyapunov methods, Uncertain systems Abstract: In this paper we seek to quantify the ability of learning to improve safety guarantees endowed by Control Barrier Functions (CBFs). In particular, we investigate how model uncertainty in the time derivative of a CBF can be reduced via learning, and how this leads to stronger statements on the safe behavior of a system. To this end, we build upon the idea of Input-to-State Safety (ISSf) to define Projection-to-State Safety (PSSf), which characterizes degradation in safety in terms of a projected disturbance. This enables the direct quantification of both how learning can improve safety guarantees, and how bounds on learning error translate to bounds on degradation in safety. We demonstrate that a practical episodic learning approach can use PSSf to reduce uncertainty and improve safety guarantees in simulation and experimentally.

Keywords:Machine learning, Information theory and control, Statistical learning Abstract: Learning-based techniques are increasingly effective at controlling complex systems. However, most work done so far has focused on learning control laws for individual tasks. Simultaneously learning multiple tasks on the same system is still a largely unaddressed research question. In particular, no efficient state space exploration schemes have been designed for multi-task control settings. Using this research gap as our main motivation, we present an algorithm that approximates the smallest data set that needs to be collected in order to achieve high performance across multiple control tasks. By describing system uncertainty using a probabilistic Gaussian process model, we are able to quantify the impact of potentially collected data on each learning-based control law. We then determine the optimal measurement locations by solving a stochastic optimization problem approximately. We show that, under reasonable assumptions, the approximate solution converges towards the exact one. Additionally, we provide a numerical illustration of the proposed algorithm.

Keywords:Boolean control networks and logic networks, Iterative learning control, Biological systems Abstract: In this paper, we study the control of probabilistic Boolean control networks (PBCNs) by leveraging a model-free reinforcement learning (RL) technique. In particular, we propose a Q-learning (QL) based approach to address the feedback stabilization problem of PBCNs, and we design optimal state feedback controllers such that the PBCN is stabilized at a given equilibrium point. The optimal controllers are designed for both finite-time stability and asymptotic stability of PBCNs. In order to verify the convergence of the proposed QL algorithm, the obtained optimal policy is compared with the optimal solutions of model-based techniques, namely value iteration (VI) and semi-tensor product (STP) methods. Finally, some PBCN models of gene regulatory networks (GRNs) are considered to verify the obtained results.

Keywords:Machine learning, Formal Verification/Synthesis, Autonomous systems Abstract: We consider the problem of reward learning for temporally extended tasks. For reward learning, inverse reinforcement learning (IRL) is a widely used paradigm. Given a Markov decision process (MDP) and a set of demonstrations for a task, IRL learns a reward function that assigns a real-valued reward to each state of the MDP. However, for temporally extended tasks, the underlying reward function may not be expressible as a function of individual states of the MDP. Instead, the history of visited states may need to be considered to determine the reward at the current state. To address this issue, we propose an iterative algorithm to learn a reward function for temporally extended tasks. At each iteration, the algorithm alternates between two modules, a task inference module that infers the underlying task structure and a reward learning module that uses the inferred task structure to learn a reward function. The task inference module produces a series of queries, where each query is a sequence of subgoals. The demonstrator provides a binary response to each query by attempting to execute it in the environment and observing the environment's feedback. After the queries are answered, the task inference module returns an automaton encoding its current hypothesis of the task structure. The reward learning module augments the state space of the MDP with the states of the automaton. The module then proceeds to learn a reward function over the augmented state space using a novel deep maximum entropy IRL algorithm. This iterative process continues until it learns a reward function with satisfactory performance. The experiments show that the proposed algorithm significantly outperforms several IRL baselines on temporally extended tasks.

Keywords:Delay systems, Networked control systems, Observers for Linear systems Abstract: This paper studies the consensus tracking control for multi-agent systems (MASs) of general linear dynamics considering heterogeneous constant known input and communication delays under a directed communication graph containing a spanning tree. First, for open-loop stable MASs, a distributed predictive observer is proposed to estimate the consensus tracking error and to construct the control input that does not involve any integral term (which is time-efficient in calculation). Then, using the generalized Nyquist criterion, we derive the conditions for asymptotic convergence of the closed-loop system and show that is delay-independent. Subsequently, another observer is designed that allows the MASs to be open-loop unstable. Next, we use the generalized Nyquist criterion to compute the observer’s gain matrix. Towards this end, we choose a specific structure with which the problem boils down to computing a single parameter, herein called the predictive observer parameter. Two algorithms are proposed for choosing this parameter: one for general linear systems and one for monotone systems. To the best of the authors' knowledge, this is the first work for which asymptotic convergence of consensus is proven for general linear MASs with arbitrary heterogeneous delays. Finally, the validity of our results is demonstrated via a vehicle platooning example.

Keywords:Delay systems, Algebraic/geometric methods Abstract: In this paper, it is shown that the two notions of weak observability and strong observability may not be sufficient to describe the link between the input/output equation associated to the behaviour of a system and its state space realization. A new notion, called regular observability, is introduced, which is shown to capture essential features of nonlinear time delay systems and the existence of some realization.

Keywords:LMIs, Delay systems, Linear systems Abstract: This paper deals with the problem of input-output admissibility analysis of descriptor systems with time-varying delay, where besides stability also, regularity and causality are considered. By using the matrix decomposition technique and auxiliary system, a new transformation model is obtained. Using the input-output approach, generalized Lyapunov function and introducing slack variables, a new input-output admissibility condition is proposed in terms of linear matrix inequality (LMI). Finally, a numerical example is presented to show the effectiveness and the merit of the proposed approach.

Keywords:Delay systems, Time-varying systems, Lyapunov methods Abstract: For linear systems with a time-varying input delay, the predictor feedback controller and exponential stability have been established. However, the now-classical approach of representing the delay by a transport partial differential equation (PDE) on a strictly positive and constant spatial domain precludes the possibility of the delay assuming the zero value at any time instant. To eliminate this limitation, we provide a new representation of the delay by a transport equation with a time-varying spatial domain. The resulting backstepping approach leads to the same predictor feedback that was previously designed by the last author. However, the controller derivation and the stability analysis are quite different, even though both the controller and the assumptions are the same. A representative example is provided to illustrate the methodology and results.

Keywords:Agents-based systems, Decentralized control, Delay systems Abstract: Consensus of autonomous agents is a benchmark problem in cooperative control. In this paper, we consider standard continuous-time averaging consensus policies (or Laplacian flows) over time-varying graphs and focus on robustness of consensus against communication delays. Such a robustness has been proved under the assumption of uniform quasi-strong connectivity of the graph. It is known, however, that the uniform connectivity is not necessary for consensus. For instance, in the case of undirected graph and undelayed communication consensus requires a much weaker condition of integral connectivity. In this paper, we show that the latter results remain valid in presence of unknown but bounded communication delays, furthermore, the condition of undirected graph can be substantially relaxed and replaced by the conditions of non-instantaneous type-symmetry. Furthermore, consensus can be proved for any feasible solution of the delay differential inequalities associated to the consensus algorithm. Such inequalities naturally arise in problems of containment control, distributed optimization and models of social dynamics

Keywords:Delay systems, Stability of nonlinear systems, Network analysis and control Abstract: In the present paper, our previous proposed method for analyzing the stability of a class of non-weakly reversible chemical reaction networks ( CRNs ) is generalized to be applied to CRNs with time delays. This method is based on the decomposition of the whole network into weakly reversible subnetworks, and application of the Deficiency Zero Theorem to them. On the basis of this method, it is proven that any positive solution to the delay differential equations that describe the dynamics of non-weakly reversible single linkage class CRNs, each pair of complexes of which has disjoint supports, modeled by mass action kinetics with time delays converges to an equilibrium point.

Institute of Problems of Mechanical Engineering Russian Academy

Keywords:Lyapunov methods, Stability of nonlinear systems, LMIs Abstract: Theorems on Implicit Lyapunov-Razumikhin functions (ILRF) for asymptotic, exponential, finite-time and nearly fixed-time stability analysis of nonlinear time-delay systems are presented. Based on these results, finite-time stabilization of a special class of such systems is addressed. These systems are represented by a chain of integrators with a time-delay term multiplied by a function of instantaneous state vector. Possible explicit restriction on nonlinear time-delay terms is discussed. Simple procedure of control parameters calculation is given in terms of linear matrix inequalities (LMIs). Some aspects of digital implementations of the presented nonlinear control law are touched upon. Theoretical results are illustrated by numerical simulations.

Keywords:Stability of nonlinear systems, Delay systems, Lyapunov methods Abstract: This paper aims to study the problem of stability for discrete-time fully nonlinear time-delay systems with constrained time-varying delays. A delays digraph is used to model the topology of the delay signals. By exploiting the Halanay technique and suitable Lyapunov functions, some sufficient conditions for the global asymptotic stability, uniform global asymptotic stability and global exponential stability are established. A matrix inequality is derived, and it is employed to prove the global exponential stability of linear discrete-time delay systems with delay signals obeying to a delays digraph. Finally, examples are given to illustrate the results.

Keywords:Constrained control, Uncertain systems, Adaptive control Abstract: A new framework is developed for control of constrained nonlinear systems with structured parametric uncertainty. Forward invariance of a safe set is achieved through online parameter adaptation and data-driven model estimation. The new adaptive and data-driven safety paradigm is merged with a recent adaptive controller for systems nominally contracting in closed-loop. This unification is more general than other safety controllers as contraction does not require the system be invertible or in a particular form. The method is tested on the pitch dynamics of an aircraft with uncertain nonlinear aerodynamics.

Keywords:Constrained control, Stability of nonlinear systems, Robust control Abstract: In this paper we address the problem of tracking control of nonlinear systems via contraction analysis. The necessary conditions of the system which can achieve universal asymptotic tracking are studied under several different cases. We show the links to the well developed control contraction metric, as well as its invariance under dynamic extension. In terms of these conditions, we identify a differentially detectable output, based on which a simple differential controller for trajectory tracking is designed via damping injection. To demonstrate the effectiveness we apply to electrostatic microactuators.

Keywords:Algebraic/geometric methods, Stability of nonlinear systems, Optimization Abstract: In this paper, we present a simple geometric attitude controller that is globally, exponentially stable. To overcome the topological restriction, the controller is designed to follow a reference trajectory that in turn converges to the desired equilibrium (making it discontinuous in the initial conditions, but continuous in time). The system and reference dynamics are studied as a single augmented system that can be analyzed and tuned simultaneously. The controller’s stability is proved using contraction analysis (on the manifold), and the bounds on the convergence rate can be found via a semi-definite program with linear matrix inequalities. Additionally, our approach allows the use of the Nelder-Mead algorithm to automatically select controller gains and reference trajectory parameters by optimizing the aforementioned bounds. The resulting controller is verified through simulations.

Keywords:Stochastic systems, Stability of nonlinear systems, Optimization algorithms Abstract: A method is presented to obtain inner estimates of the region of transverse contraction (ROTC) which are invariant regions in which trajectories of a stochastic system converge to a stochastic limit cycle. Using the framework of Polynomial Chaos Expansions (PCE) the stochastic system is represented by a higher dimensional deterministic system. First, the connection between the stability of the periodic orbits of the stochastic system and the stability of the limit cycle of its PCE system is established. Then transverse contraction criteria, as well as invariance conditions, are formulated for the PCE system to certify an ROTC estimate for the PCE system. From this, and by leveraging the established stability connection, an ROTC estimate of the stochastic system is retrieved. Finally, an optimization program, based on matrix sum-of-squares verification techniques, to implement the contraction and invariance criteria is proposed.

Keywords:Machine learning, Observers for nonlinear systems, Optimal control Abstract: This paper presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between perturbed and unperturbed system trajectories. We demonstrate how to exploit NCMs to design an online optimal estimator and controller for nonlinear systems with bounded disturbances utilizing their duality. The performance of our framework is illustrated through Lorenz oscillator state estimation and spacecraft optimal motion planning problems.

Keywords:Direct adaptive control, Uncertain systems, Stability of nonlinear systems Abstract: In this paper, a new direct model reference adaptive control (DMRAC) for nonlinear control affine systems is proposed to track a contracting nonlinear reference model. The structure of the uncertainty is assumed, including a multiplicative unknown parameter on the input. The contracting reference model enables the use of the certainty equivalence principle (CEP) for designing the simple controller, a tool which often cannot be used for nonlinear systems due to its weakness on these systems. The proposed controller is constructed to minimize the distance from the estimated plant vector field to the desired contracting vector field, which leverages the benefits of the contracting reference model. Even though the proposed controller does not perfectly match the true contracting vector field, the combined efforts of the adaptive controller and the adaptation law make the Riemannian distance asymptotically approach zero with bounded parameter error. In addition, the proposed algorithm is extended to the case of using a control contraction metric (CCM) based controller to construct the contracting reference model. A fully actuated pendulum example is studied to illustrate the effectiveness of the algorithm.

Keywords:Control applications, Stability of nonlinear systems, Mechatronics Abstract: In this work, we propose a contraction-based variable gain nonlinear control scheme for the laser-beam stabilizing (LBS) servo-system, which guarantees that the closed-loop system is convergent. With the variable gain acting on the velocity error, the well-known waterbed effect of the low-frequency/bandwidth trade-off can be overcome. Moreover, the contraction-based framework allows us to extend the linear control performance metrics for analyzing the closed-loop nonlinear system behavior. The closed-loop system's performance is evaluated in numerical simulations under input disturbances and/or white noise measurements and its efficacy is compared to that using PID and LQG controllers.

Keywords:Predictive control for nonlinear systems, Uncertain systems, Constrained control Abstract: An assured controller is one that enforces safety online by filtering a desired control input at runtime, and control barrier functions (CBFs) provide an assured controller that renders a safe subset of the statespace forward invariant. In this work, we present a problem formulation for CBF-based runtime assurance for systems with disturbances, and controllers that solve this problem must, in some way, incorporate the online computation of reachable sets. In general, computing reachable sets in the presence of disturbances is computationally costly and cannot be directly incorporated in a CBF framework. To that end, we present a particular solution to the problem, whereby reachable sets are approximated via the mixed-monotonicity property. Efficient algorithms exist for over-approximating reachable sets for mixed-monotone systems with hyperrectangles, and we show that such approximations are suitable for incorporating into a CBF-based runtime assurance framework.

Keywords:Autonomous systems, Constrained control, Optimization Abstract: Many control applications require that a system be constrained to a particular set of states, often termed as safe set. A practical and flexible method for rendering safe sets forward-invariant involves computing control input using Control Barrier Functions and Quadratic Programming methods. Many prior results however require the resulting control input to be continuous, which requires strong assumptions or can be difficult to demonstrate theoretically. In this paper we use differential inclusion methods to show that simultaneously rendering multiple sets invariant can be accomplished using a discontinuous control input. We present an optimization formulation which computes such control inputs and which can be posed in multiple forms, including a feasibility problem, a linear program, or a quadratic program. In addition, we discuss conditions under which the optimization problem is feasible and show that any feasible solution of the considered optimization problem which is measurable renders the multiple safe sets forward invariant.

Keywords:Lyapunov methods Abstract: Control Lyapunov functions (CLFs) and control barrier functions (CBFs) have been used to develop provably safe controllers by means of quadratic programs (QPs), guaranteeing safety in the form of trajectory invariance with respect to a given set. In this letter, we show that this framework can introduce equilibrium points (particularly at the boundary of the safe set) other than the minimum of the Lyapunov function into the closed-loop system. We derive explicit conditions under which these undesired equilibria (which can even appear in the simple case of linear systems with just one convex unsafe set) are asymptotically stable. To address this issue, we propose an extension to the QP-based controller unifying CLFs and CBFs such that the resulting system trajectories avoid the undesirable equilibria problem on the boundary of the safe set. The solution is illustrated in the design of a collision-free controller.

Keywords:Lyapunov methods, Autonomous systems, Robotics Abstract: We propose a hybrid feedback control law that guarantees both safety and asymptotic stability for a class of Lagrangian systems in environments with obstacles. Rather than performing trajectory planning and implementing a trajectory-tracking feedback control law, our approach requires a sequence of locations in the environment (a path plan) and an abstraction of the obstacle-free space. The problem of following a path plan is then interpreted as a sequence of reach-avoid problems: the system is required to consecutively reach each location of the path plan while staying within safe regions. Obstacle-free ellipsoids are used as a way of defining such safe regions, each of which encloses two consecutive locations. Feasible Control Barrier Functions (CBFs) are created directly from geometric constraints, the ellipsoids, ensuring forward-invariance, and therefore safety. Reachability to each location is guaranteed by asymptotically stabilizing Control Lyapunov Functions (CLFs). Both CBFs and CLFs are then encoded into quadratic programs (QPs) without the need of relaxation variables. Furthermore, we also propose a switching mechanism that guarantees the control law is correct and well-defined even when transitioning between QPs. Simulations show the effectiveness of the proposed approach in two complex scenarios.

Keywords:Lyapunov methods, Optimal control, Constrained control Abstract: Control Barrier Functions (CBFs) have become a popular tool for enforcing set invariance in safety-critical control systems. While guaranteeing safety, most CBF approaches are myopic in the sense that they solve an optimization problem at each time step rather than over a long time horizon. This approach may allow a system to get too close to the unsafe set where the optimization problem can become infeasible. Some of these issues can be mitigated by introducing relaxation variables into the optimization problem; however, this compromises convergence to the desired equilibrium point. To address these challenges, we develop an approximate optimal approach to the safety-critical control problem in which the cost of violating safety constraints is directly embedded within the value function. We show that our method is capable of guaranteeing both safety and convergence to a desired equilibrium. Finally, we compare the performance of our method with that of the traditional quadratic programming approach through numerical examples.

Keywords:Cooperative control, Constrained control, Distributed control Abstract: This paper studies a distributed collision avoidance control problem for a group of rigid bodies on a sphere. A rigid body network, consisting of multiple rigid bodies constrained to a spherical surface and an interconnection topology, is first formulated. In this formulation, it is shown that motion coordination on a sphere is equivalent to attitude coordination on the 3-dimensional Special Orthogonal group. Then, an angle-based control barrier function that can handle a geodesic distance constraint on a spherical surface is presented. The proposed control barrier function is then extended to a relative motion case and applied to a collision avoidance problem for a rigid body network operating on a sphere. Each rigid body chooses its control input by solving a distributed optimization problem to achieve a nominal distributed motion coordination strategy while satisfying constraints for collision avoidance. The proposed collision-free motion coordination law is validated via simulation.

Keywords:Constrained control, Optimization Abstract: Verifying set invariance has classical solutions stemming from the seminal work by Nagumo, and defining sets via a smooth barrier function constraint inequality results in computable flow conditions for guaranteeing set invariance. While a majority of these historic results on set invariance consider flow conditions on the boundary, this paper fully characterizes set invariance through minimal barrier functions by directly appealing to a comparison result to define a flow condition over the entire domain of the system. A considerable benefit of this approach is the removal of regularity assumptions of the barrier function. This paper also outlines necessary and sufficient conditions for a valid differential inequality condition, giving the minimum conditions for this type of approach. We also show when minimal barrier functions are necessary and sufficient for set invariance.

Keywords:Power systems, Smart grid, Energy systems Abstract: Distributed algorithms enable private Optimal Power Flow (OPF) computations by avoiding the need in sharing sensitive information localized in algorithms sub-problems. However, adversaries can still infer this information from the coordination signals exchanged across iterations. This paper seeks formal privacy guarantees for distributed OPF computations and provides differentially private algorithms for OPF computations based on the consensus Alternating Direction Method of Multipliers (ADMM). The proposed algorithms attain differential privacy by introducing static and dynamic random perturbations of OPF sub-problem solutions at each iteration. These perturbations are Laplacian and designed to prevent the inference of sensitive information, as well as to provide theoretical privacy guarantees for ADMM sub-problems. Using a standard IEEE 118-node test case, the paper explores the fundamental trade-offs among privacy, algorithmic convergence, and optimality losses.

Keywords:Power systems, Smart grid, Energy systems Abstract: We consider the problem of stability analysis for distribution grids with linearized droop-controlled inverter and line dynamics. The inverters are modeled as voltage sources with controllable frequency and amplitude. This problem is very challenging for large networks as numerical simulations and detailed eigenvalue analysis are impactical. Motivated by the above limitations, we present in this paper a systematic and computationally efficient framework for stability analysis of inverter-based distribution grids. To design our framework,we use tools from singular perturbation and Lyapunov theories.Interestingly, we show that stability of the fast dynamics of the power grid depends only on the voltage droop gains of the inverters while, stability of the slow dynamics, depends on both voltage and frequency droop gains. Finally, by leveraging these timescale separation properties, we derive sufficient conditions on the frequency and voltage droop gains of the inverters that warrant stability of the full system. We illustrate our theoretical results through a numerical example on the IEEE 13-bus distribution grid.

Keywords:Smart grid, Power systems, Optimization Abstract: Load side participation can provide support to the power network by appropriately adapting the demand when required. In addition, it may allow for an economically improved power allocation. In this study, we consider the problem of providing an optimal power allocation among generation and on-off loads within the secondary frequency control timeframe. In particular, we consider a mixed integer optimization problem which ensures that secondary frequency control objectives, i.e. generation-demand balance and frequency attaining its nominal value at steady state, are satisfied. We present analytic conditions on generation and on-off load profiles such that an epsilon-optimality interpretation of the steady state power allocation is obtained, providing a non-conservative bound for epsilon. Moreover, we develop a hierarchical control scheme that provides on-off load values that satisfy the proposed conditions. Furthermore, we study the interaction of the proposed control scheme with the physical dynamics of the power network and provide analytic stability guarantees. Our results are verified with numerical simulations on the Northeast Power Coordinating Council (NPCC) 140-bus system, where it is demonstrated that the proposed algorithm enables an optimality interpretation of the steady state power allocation.

Keywords:Networked control systems, Power systems, Stochastic systems Abstract: Based on stochastic differential equations (SDEs), we analyse the overall performance of heterogeneous power systems network, subject to spatially distributed and correlated noises with random initial conditions. We determine bounds on the H2 norm of the heterogeneous system based on a closedform of the norm of the homogeneous power system. Then, we formulate possible scenarios for performance optimization and link these to applications for network design and control problems in power systems. Our results are corroborated by numerical simulations from Kundur’s four-machine two-area network after adaption to our setup.

Keywords:Formal Verification/Synthesis, Discrete event systems, Robust control Abstract: We consider the computation of resilient controllers for perturbed non-linear dynamical systems w.r.t. linear-time temporal logic specifications. We address this problem through the paradigm of Abstraction-Based Controller Design (ABCD) where a finite state abstraction of the perturbed system dynamics is constructed and utilized for controller synthesis. In this context, our contribution is twofold: (I) We construct abstractions which model the impact of occasional high disturbance spikes on the system via so called disturbance edges. (II) We show that the application of resilient reactive synthesis techniques to these abstract models results in closed loop systems which are optimally resilient to these occasional high disturbance spikes. We have implemented this resilient ABCD workflow on top of SCOTS and showcase our method through multiple robot planning examples.

Keywords:Formal Verification/Synthesis, Computational methods Abstract: We present a novel approach to compute non-convex inner-approximations of reachable sets for nonlinear continuous systems. The concept of our approach is to extract inner-approximations of reachable sets from pre-computed outer-approximations, which makes our method computationally very efficient as we demonstrate with several numerical examples. Since our approach has polynomial complexity with respect to the system dimension, it is well-suited for high-dimensional systems.

Keywords:Formal Verification/Synthesis, Hybrid systems, Computational methods Abstract: We introduce a new way of specifying rich behaviors for discrete-time dynamical systems called control programs. Essentially, a control program consists of a set of elementary control tasks with a scheduler. A control task is described by a discrete-time hybrid automaton and a termination semantics, specifying if the task must terminate in finite time or if it is allowed to run forever. The scheduler provides a set of rules that is used to sequence the control tasks. Control programs also have external inputs, which makes it possible to specify how a system must react to instructions provided by a human user or by another system. We define the set of executions that are accepted by the control program. Then, we consider the problem of synthesizing a controller for a dynamical system such that the closed-loop behavior is an execution of the control program. Building on our recent work on formal synthesis from specifications given by hybrid automata, we propose two algorithms for computing controllers based on contracting and expanding fixed-point computations. The first algorithm computes the maximal controllable set but needs to reach the fixed-point to provide a valid controller. The second algorithm may not converge to the maximal controllable set but provides a valid controller at each iteration. We illustrate our methodology with an autonomous vehicle control example.

Keywords:Switched systems, Discrete event systems, Large-scale systems Abstract: In this work, we propose a compositional framework for the verification of approximate initial-state opacity for networks of discrete-time switched systems. The proposed approach is based on a notion of approximate initial-state opacity-preserving simulation functions (InitSOPSFs), which characterize how close concrete networks and their finite abstractions are in terms of the satisfaction of approximate initial-state opacity. We show that such InitSOPSFs can be obtained compositionally by assuming some small-gain type conditions and composing so-called local InitSOPSFs constructed for each subsystem separately. Additionally, for switched systems satisfying certain stability property, we provide an approach to construct their finite abstractions together with the corresponding local InitSOPSFs. Finally, the effectiveness of our results is illustrated through an example.

Keywords:Robust control, Uncertain systems, Predictive control for linear systems Abstract: In this paper, we consider the robust closed-loop model predictive control (MPC) of a linear time-variant (LTV) system with norm bounded disturbances and LTV model uncertainty, wherein a series of constrained optimal control problems (OCPs) are solved. Guaranteeing robust feasibility of these OCPs is challenging due to disturbances perturbing the predicted states, and model uncertainty, both of which can render the closed-loop system unstable. As such, a trade-off between the numerical tractability and conservativeness of the solutions is often required. We use the System Level Synthesis (SLS) framework to reformulate these constrained OCPs over closed-loop system responses, and show that this allows us to transparently account for norm bounded additive disturbances and LTV model uncertainty by computing robust state feedback policies. We further show that by exploiting the underlying linear fractional structure of the resulting robust OCPs, we can significantly reduce the conservativeness of existing SLS-based and tube-MPC-based robust control methods while also improving computational efficiency. We conclude with numerical examples demonstrating the effectiveness of our methods.

Keywords:Robust control, Process Control, Predictive control for linear systems Abstract: Convex-lifting-based robust control evaluates manipulated variable by (i) solving linear programming, when the input/state constraints are active, or (ii) by linear state feedback control law when no constraints are active. This paper addresses the problem of switching linear feedback when multiple linear control laws are considered. As the sudden switching might be undesirable, the approximated control law is introduced into the real-time convex-lifting-based robust control approach, to avoid switching. This approximation of control law replaces the switching control law and ensures smoother, gain-scheduling-like, evaluation of the state feedback gain matrix. Linear interpolation is considered to approximate the switching control law. The properties of the proposed control strategy are experimentally investigated and compared to the original control approach without approximation of the control law. The laboratory device Flexy serves to experimentally demonstrate that the proposed approach outperforms the original approach.

Keywords:Robust control, Autonomous vehicles, Lyapunov methods Abstract: This paper addresses the design of a global saturated trajectory tracking controller for a quadcopter in the presence of external disturbances. The proposed control law guarantees that (i) the thrust force input generated by the propellers is bounded with respect to the position and linear velocity errors; (ii) the quadcopter globally converges to a neighborhood of a desired smooth trajectory, achieving global practical stability. Disturbance observers are employed to estimate and compensate for external constant and slowly time-varying disturbances, improving the robustness performance of the controller. A projection operator ensures that the estimates do not wind-up and that they remain within a prescribed bound. To validate the efficiency and robustness of the proposed controller, we present and analyze both simulation and experimental results. The proposed global saturated controller is also compared with a standard backstepping-based trajectory tracking controller to demonstrate the merits and improvements of the proposed controller.

Keywords:Robust control, Uncertain systems, LMIs Abstract: This paper presents a new algorithm for designing robust output-feedback controllers in the presence of structured uncertainty. The proposed approach is an alternative to the popular D-K Iterations, consisting of a pair of LMIs (Linear Matrix Inequalities) and an associated iterative algorithm. The new algorithm is illustrated by examples and its performance is compared with the D-K iterations by solving instances of the COMPleib benchmark library.

Keywords:Robust control, Distributed control, Distributed parameter systems Abstract: In this paper, the Banach duality structure of the optimal H^infty-problem for distributed spatially invariant systems is provided. Under specific assumptions, it is shown that an optimal feedback spatially invariant H^infty controller exists, and can be computed through the Youla parametrization. It also shown that the optimal H^infty cost is equal to the induced norm of a novel operator defined on a Banach projective tensor space. An operator identity is deduced to compute the optimal Youla parameter, and thus the optimal controller provided a maximizing vector exists.

Keywords:Neural networks, Optimal control, Machine learning Abstract: Adjoint methods are used in both control theory and machine learning (ML) to efficiently compute gradients of functionals. In ML, the adjoint method is a popular approach for training multilayer neural networks and is commonly referred to as backpropagation. Despite its importance in ML, the adjoint method suffers from two well documented shortcomings: (i) gradient decay/explosion and (ii) excessive training time. Until now, the gradient decay problem has primarily been addressed through modification to the network architecture with gating units that add additional parameters. This results in additional computational costs during evaluation and training which further exacerbates the excessive training time. In this work, we introduce a powerful framework for addressing the gradient decay problem based on second-order sensitivity concepts from control theory. As a result, we are able to robustly train arbitrary network architectures without suffering from gradient decay. Furthermore, we demonstrate that this method is able to speed up training with respect to both wall-clock time and data efficiency. We demonstrate our method on a synthetic long time gap experiment and a language processing task with a simple recurrent neural network architecture.

Keywords:Neural networks, Optimization, Uncertain systems Abstract: In this paper, we consider the problem of certifying the robustness of neural networks to perturbed and adversarial input data. Such certification is imperative for the application of neural networks in safety-critical decision-making and control systems. Certification techniques using convex optimization have been proposed, but they often suffer from relaxation errors that void the certificate. Our work exploits the structure of ReLU networks to improve relaxation errors through a novel partition-based certification procedure. The proposed method is proven to tighten existing linear programming relaxations, and asymptotically achieves zero relaxation error as the partition is made finer. We develop a finite partition that attains zero relaxation error and use the result to derive a tractable partitioning scheme that minimizes the worst-case relaxation error. Experiments using real data show that the partitioning procedure is able to issue robustness certificates in cases where prior methods fail. Consequently, partition-based certification procedures are found to provide an intuitive, effective, and theoretically justified method for tightening existing convex relaxation techniques.

Keywords:Neural networks, Machine learning, Computer-aided control design Abstract: In this paper, we consider the problem of automatically designing a Rectified Linear Unit (ReLU) Neural Network (NN) architecture (number of layers and number of neurons per layer) with the guarantee that it is sufficiently parametrized to control a nonlinear system. Whereas current state-of-the-art techniques are based on hand-picked architectures or heuristic-based search to find such NN architectures, our approach exploits a given model of the system to design an architecture; as a result, we provide a guarantee that the resulting NN architecture is sufficient to implement a controller that satisfies an achievable specification. Our approach exploits two basic ideas. First, we assume that the system can be controlled by a Lipschitz-continuous state-feedback controller that is unknown but whose Lipschitz constant is upper-bounded by a known constant; then using this assumption, we bound the number of affine functions needed to construct a Continuous Piecewise Affine (CPWA) function that can approximate the unknown Lipschitz-continuous controller. Second, we utilize the authors' recent results on the Two-Level Lattice (TLL) NN architecture, a novel NN architecture that was shown to be parameterized directly by the number of affine functions that comprise the CPWA function it realizes. We also evaluate our method by designing a NN architecture to control an inverted pendulum.

Keywords:Neural networks, Optimal control, Adaptive control Abstract: In this paper, online optimal adaptive tracking control of nonlinear discrete-time systems in affine form with uncertain internal dynamics is presented. The augmented system and the cost function over infinite horizon for the augmented state are defined. Two-layer neural network (NN) - based actor-critic framework is introduced to estimate the optimal control input and value function. The temporal difference (TD) error is derived as a function of the difference between actual and estimated value function. The NN weights of critic and actor are tuned at every sampling instant as a function of the instantaneous temporal difference errors and control policy errors, respectively. The proposed scheme ensures the closed-loop stability in the form of boundedness. Simulation results are provided to illustrate the effectiveness of the proposed approach.

Keywords:Neural networks, Optimal control, LMIs Abstract: An approach is proposed for the design of feedback controllers for processes described by dynamic artificial neural networks (DANNs) using linear matrix inequalities. This approach uses the fact that DANNs are subsets of the standard nonlinear operator form. We then reformulate the DANN written in standard nonlinear operator form as a diagonal norm-bounded linear differential inclusion. The latter formulation is applied to linear matrix inequality-based optimizations derived using the Lyapunov method for analysis and controller synthesis. The approach is demonstrated for the design of a feedback control system for a highly nonlinear pH control problem.

Keywords:Optimization algorithms, Randomized algorithms Abstract: In this paper we study variational inequalities (VI) defined by the conditional value-at-risk (CVaR) of uncertain functions. We introduce stochastic approximation schemes that employ an empirical estimate of the CVaR at each iteration to solve these VIs. We investigate convergence of these algorithms under various assumptions on the monotonicity of the VI and accuracy of the CVaR estimate. Our first algorithm is shown to converge to the exact solution of the VI when the estimation error of the CVaR becomes progressively smaller along any execution of the algorithm. When the estimation error is nonvanishing, we provide two algorithms that provably converge to a neighborhood of the solution of the VI. For these schemes, under strong monotonicity, we provide an explicit relationship between sample size, estimation error, and the size of the neighborhood to which convergence is achieved. A simulation example illustrates our theoretical findings.

Keywords:Stability of nonlinear systems, Randomized algorithms Abstract: We consider the problem of set invariance verification in black-box nonlinear systems without analytic dynamical models. A data-driven set invariance verification approach relying on the observation of trajectories is proposed to determine almost-invariant sets, which are invariant almost everywhere except possibly in a small subset. With these observations, scenario optimization problems are formulated. We show that probabilistic invariance guarantees on the almost-invariant sets can be established. To get explicit expressions of such sets, a set identification procedure is designed by the use of a polynomial classifier. The practical performance of the proposed data-driven framework is illustrated by numerical examples.

Keywords:Randomized algorithms, Statistical learning, Optimization Abstract: We consider the scenario approach theory to deal with convex optimization programs affected by uncertainty, which is in turn represented by means of scenarios. An approach to deal with such programs while trading feasibility to performance is known as sampling and discarding in the scenario approach literature. Existing bounds on the probability of constraint satisfaction for such programs are not tight. In this paper we use learning theoretic concepts based on the notion of compression to show that for a particular class of convex scenario programs, namely, the so called fully-supported ones, and under a particular scenario discarding scheme, a tight bound can be obtained. We illustrate our developments by means of an example that admits an analytic solution.

Keywords:Optimization, Uncertain systems, Randomized algorithms Abstract: Uncertain optimisation problems often require satisfaction of possibly infinite constraints, corresponding to each realisation of the uncertain phenomena influencing the problem setup. To find an approximate solution to such problems, randomised approaches such as the scenario approach can be employed where only a finite sample of these constraints are looked at. However, to have a strong probabilistic guarantee on the feasibility of the scenario solution for the original problem, we still need a large number of constraints. This leads to intractability of the scenario problems as well. In this paper we propose a method to remove redundant constraints in the scenario problem, prior to solving the problem itself. We consider a specific class of scenario problems with linear inequality constraints subject to one additive and one multiplicative uncertain parameter. The proposed method exploits the system structure to identify the supporting constraints and it is based on rigorous theoretical footings. The working of the method is also illustrated with the help of a numerical problem.

Keywords:Identification for control, Identification, Optimization algorithms Abstract: MIMO (multi-input multi-output) system identification is a particular instance of a parsimonious model selection problem. If the observed data is assumed to arise from a stable and low order plant, then the representing model should also be stable and have few poles in its realization. These constraints are challenging to impose in an L1 or nuclear norm framework, especially when observations are non-uniformly sampled. This paper implements MIMO identification by randomized active-set methods, as realized by Fully Corrective Frank-Wolfe (FCFW). Reweighting pole-group penalties allow for further system sparsification while monotonically decreasing the regularized fitting error. Efficacy of the approach is shown on two examples.

Keywords:Identification for control, Learning, Modeling Abstract: In the context of regularization methods for linear system identification, we introduce a new kernel design procedure that accounts for control objectives. We consider a model-reference control setup and assume data from one experiment is available. Exploiting the frequency response of the reference model, we design a new kernel that is able to extract the least amount of information from the data to the purpose of matching the desired closed-loop, with particular attention to user-defined frequency bands. Unlike the recently proposed CoRe algorithm, the proposed method is non-iterative and does not require any preliminary controller estimation. Simulation results on a benchmark example show that, when the model is used for control design, the proposed regularization procedure outperforms traditional kernel-based techniques as well as bias-shaping through data prefiltering.

Keywords:Identification for control, Predictive control for linear systems, Nonlinear systems identification Abstract: We present a Model Predictive Control (MPC) strategy for unknown input-affine nonlinear dynamical systems. A non-parametric method is used to estimate the nonlinear dynamics from observed data. The estimated nonlinear dynamics are then linearized over time-varying regions of the state space to construct an Affine Time-Varying (ATV) model. Error bounds arising from the estimation and linearization procedure are computed by using sampling techniques. The ATV model and the uncertainty sets are used to design a robust Model Predictive Controller (MPC) which guarantees safety for the unknown system with high probability. A simple nonlinear example demonstrates the effectiveness of the approach where commonly used estimation and linearization methods fail.

Keywords:Identification for control, Optimal control, Robust control Abstract: The paper proposes a dual control finite horizon LQR synthesis procedure for unknown systems characterized by mean and covariance estimates. The optimized policy comprises time-varying state-feedback and dithering components, and the control problem is framed as a multiobjective synthesis which seeks a balance between exploitation and exploration costs. It is shown that classic experiment design problems can be recast in this framework by replacing the exploitation cost with an information reward. Numerical examples demonstrate the different dual control trade-offs on plants with different properties.

Keywords:Identification for control, Robust control, Closed-loop identification Abstract: We present a novel strategy for robust dual control of linear time-invariant systems based on gain scheduling with performance guarantees. This work relies on prior results of determining uncertainty bounds of system parameters estimated through exploration. Existing approaches are unable to account for changes of the mean of system parameters in the exploration phase and thus to accurately capture the dual effect. We address this limitation by selecting the future (uncertain) mean as a scheduling variable in the control design. The result is a semi-definite program-based design that computes a suitable exploration strategy and a robust gain-scheduled controller with probabilistic quadratic performance bounds after the exploration phase.

Keywords:Mean field games, Learning, Linear systems Abstract: In this paper, we study large population multi-agent reinforcement learning (RL) in the context of discrete-time linear-quadratic mean-field games (LQ-MFGs). Our setting differs from most existing work on RL for MFGs, in that we consider a non-stationary MFG over an infinite horizon. We propose an actor-critic algorithm to iteratively compute the mean-field equilibrium (MFE) of the LQ-MFG. There are two primary challenges: i) the non-stationarity of the MFG induces a linear-quadratic tracking problem, which requires solving a backwards-in-time (non-causal) equation that cannot be solved by standard (causal) RL algorithms; ii) Many RL {algorithms} assume that the states are sampled from the stationary distribution of a Markov chain (MC), that is, the chain is already mixed, an assumption that is not satisfied for real data sources. We first identify that the mean-field trajectory follows linear dynamics, allowing the problem to be reformulated as a linear quadratic Gaussian problem. Under this reformulation, we propose an actor-critic algorithm that allows samples to be drawn from an unmixed MC. Finite-sample convergence guarantees for the algorithm are then provided. To characterize the performance of our algorithm in multi-agent RL, we have developed an error bound with respect to the Nash equilibrium of the finite-population game.

Keywords:Game theory, Agents-based systems, Optimization Abstract: This work considers an aggregative game over time-varying graphs, where each player’s cost function depends on its own strategy and the aggregate of its competitors’ strategies. Though the aggregate is unknown to any given player, each player may interact with its neighbors to construct an estimate of the aggregate. We design a distributed iterative Tikhonov regularization method in which each player may independently choose its steplengths and regularization parameters while meeting some overall coordination requirements. Under a monotonicity assumption on the concatenated player-specific gradient map, we prove that the generated sequence converges to the least-norm Nash equilibrium (i.e., a Nash equilibrium with the smallest two-norm) and validate the proposed method on a networked Nash-Cournot equilibrium problem.

Keywords:Game theory Abstract: In this paper, we study a Nash equilibrium (NE) seeking problem in a multi-player non-cooperative game over a directed communication graph. Specifically, the players' costs are functions of all players' actions, but only part of which are directly accessible. Moreover, we assume the explicit form/expression/model information of the cost function is unknown, but its value can be measured by the local player. To solve this problem, a non-model based distributed NE seeking algorithm is proposed, which requires no gradient information but the measurements of player's local cost function. A leader-following consensus technique is adopted with a row-stochastic adjacency matrix, which simplifies the implementation and increases the application range of the algorithm as compared to the doubly-stochastic matrix. Moreover, the algorithm is able to work with uncoordinated step-sizes, allowing the players to choose their own preferred step-sizes, which makes the algorithm more distributive. The convergence of the proposed algorithm is rigorously studied for both scenarios of diminishing and constant step-sizes, respectively. It is shown that players' actions converge to the exact NE almost surely for the case of diminishing step-size, and to an approximated NE with a gap depending on the step-size selection for the case of constant step-size. Numerical examples are provided to verify the algorithm's effectiveness.

Keywords:Game theory, Agents-based systems, Networked control systems Abstract: In this paper, we consider distributed Nash equilibrium seeking in (non-strictly/strongly) monotone games. We assume first that each player has full access to the opponents’ decisions and propose a new higher-order gradient-play dynamics, constructed by a passivity-based modification of a standard scheme. We show that this technique allows relaxation of strict monotonicity of the pseudo-gradient and, unlike other methods, can ensure exact asymptotic convergence in merely monotone regimes. We consider next that players have only partial-decision information, and can communicate with their neighbours over an arbitrary undirected graph. To distribute the problem we augment the variables, so that each player has local decision and auxiliary state estimates. We modify the higher-order gradient dynamics via a distributed Laplacian feedback and show how we can exploit equilibrium-independent passivity properties to achieve convergence to a Nash equilibrium in monotone regimes, under different assumptions on the game map.

Keywords:Optimization algorithms, Game theory, Variational methods Abstract: We address the Nash equilibrium problem in a partial-decision information scenario, where each agent can only observe the actions of some neighbors, while its cost possibly depends on the strategies of other agents. Our main contribution is the design of a fully-distributed, single-layer, fixed-step algorithm, based on a proximal best-response augmented with consensus terms. To derive our algorithm, we follow an operator-theoretic approach. First, we recast the Nash equilibrium problem as that of finding a zero of a monotone operator. Then, we demonstrate that the resulting inclusion can be solved in a fully-distributed way via a proximal-point method, thanks to the use of a novel preconditioning matrix. Under strong monotonicity and Lipschitz continuity of the game mapping, we prove linear convergence of our algorithm to a Nash equilibrium. Furthermore, we show that our method outperforms the fastest known gradient-based schemes, both in terms of guaranteed convergence rate, via theoretical analysis, and in practice, via numerical simulations.

Keywords:Optimization algorithms, Distributed control, Sensor fusion Abstract: We study distributed composite optimization over networks: agents minimize a sum of smooth (strongly) convex functions, the agents’ sum-utility, plus a nonsmooth (extended-valued) convex one. We propose a general unified algorithmic framework for such a class of problems, whose unified convergence analysis leverages the theory of operator splitting. Distinguishing features of our scheme are: (i) When the agents' functions are strongly convex, the algorithm converges at a linear rate, whose dependence on the agents’ functions and network topology is decoupled, matching the typical rates of centralized optimization; the rate expression improves on existing results; (ii) When the objective function is convex (but not strongly convex), similar separation as in (i) is established for the coefficient of the proved sublinear rate; and (iii) A by-product of our analysis is a tuning recommendation for several existing (non-accelerated) distributed algorithms yielding the fastest provably (worst-case) convergence rate. This is the first time that a general distributed algorithmic framework applicable to composite optimization enjoys all such properties.

Keywords:Optimization algorithms Abstract: This paper considers the decentralized consensus optimization problem defined over a network where each node holds a twice continuously differentiable local objective function. Our goal is to minimize the summation of local objective functions and find the exact optimal solution using only local computation and neighboring communications. We propose a novel Newton tracking algorithm, in which each node updates its local variable along a local Newton direction modified with neighboring and historical information. We investigate the connections between the proposed Newton tracking algorithm and several existing methods, including gradient tracking and second-order algorithms. Under the strong convexity assumption, we prove that our proposed algorithm converges to the exact optimal solution at a linear rate. We also present numerical results to demonstrate the efficacy of Newton tracking and validate the theoretical findings.

Keywords:Optimization algorithms, Agents-based systems, Communication networks Abstract: In this paper, we study the distributed optimization problem over a peer-to-peer network, where nodes optimize the sum of local objective functions via local computation and communicating with neighbors. Most existing algorithms cannot achieve globally superlinear convergence since they rely on either asymptotic consensus methods with linear convergence rates that bottleneck the global rate, or the pure Newton method that converges only locally. To this end, we introduce a finite-time set-consensus method, and then incorporate it into Polyak's adaptive Newton method, leading to our distributed adaptive Newton algorithm (DAN). Then, we propose a communication-efficient version of DAN called DAN-LA, which adopts a low-rank approximation idea to compress the Hessian and reduce the size of transmitted messages from O(p^2) to O(p), where p is the dimension of decision vectors. We show that DAN and DAN-LA can emph{globally} achieve emph{quadratic} and emph{superlinear} convergence rates, respectively. Numerical experiments are conducted to show the advantages over existing methods.

Keywords:Optimization algorithms, Network analysis and control, Agents-based systems Abstract: In this paper, we consider the problem of distributed consensus optimization over multi-agent networks with directed network topology. Assuming each agent has a local cost function that is smooth and strongly convex, the global objective is to minimize the average of all the local cost functions. To solve the problem, we introduce a robust gradient tracking method (R-Push-Pull) adapted from the recently proposed Push-Pull/AB algorithm. R-Push-Pull inherits the advantages of Push-Pull and enjoys linear convergence to the optimal solution with exact communication. Under noisy information exchange, R-Push-Pull is more robust than the existing gradient tracking based algorithms; the solutions obtained by each agent reach a neighborhood of the optimum in expectation exponentially fast under a constant stepsize policy. We provide a numerical example that demonstrate the effectiveness of R-Push-Pull.

Keywords:Optimal control, Optimization, Process Control Abstract: Systems with fast and slow dynamics give rise to objectives in different time scales which may not be aligned. The existing dynamic optimal control methods might become computationally infeasible due to the fine discretization required to capture the fast dynamics. On the other hand, a real time optimization (RTO) method based on steady-state models, which is computationally efficient, can greedily drive the plant towards optimal operation. The drawback of the RTO approach is that it may yield actions that only focus on near future goals and the objectives involving the slower dynamics are neglected. In this paper, we propose to extend RTO with a lookahead strategy by introducing a predictor to capture the effect of changing the current controls on the long-term objective. In this way, we introduce the long-term objectives in RTO while maintaining its computational efficiency and not losing focus of short-term objectives. The proposed approach is demonstrated in a simulation study from offshore petroleum production, that compares the proposed method with both an "industry-standard" RTO method, and a full fledged dynamic optimization method that takes both slow and fast dynamics into account. The proposed methodology performs almost as well as the dynamic optimization method while maintaining a low computational effort.

Keywords:Optimal control, Optimization algorithms, Computational methods Abstract: This paper presents analyses for the maximum hands-off control using the geometric methods developed for the theory of turnpike in optimal control. First, a sufficient condition is proved for the existence of the maximum hands-off control for linear time-invariant systems with arbitrarily fixed initial and terminal points using the relation with L^1 optimal control. Next, a sufficient condition is derived for the maximum hands-off control to have the turnpike property, which may be useful for approximate design of the control.

Keywords:Optimal control Abstract: Solving the Hamilton-Jacobi-Bellman (HJB) equation for nonlinear optimal control problems usually suffers from the so-called curse of dimensionality. In this letter, a nested sparse successive Galerkin method is presented for HJB equations, and the computational cost only grows polynomially with the dimension. Based on successive approximation techniques, the nonlinear HJB partial differential equation (PDE) is transformed into a sequence of linear PDEs. Then the nested sparse grid methods are employed to solve the resultant linear PDEs. The designed method is sparse in two aspects. Firstly, the solution of the linear PDE is constructed based on sparse combinations of nested basis functions. Secondly, the multi-dimensional integrals in the Galerkin method are efficiently calculated using nested sparse grid quadrature rules. Once the successive approximation process is finished, the optimal controller can be analytically given based on the sparse basis functions and coefficients. Numerical results also demonstrate the accuracy and efficiency of the designed nested sparse successive Galerkin method.

Keywords:Optimal control, Biological systems, Numerical algorithms Abstract: This paper is devoted to the numerical analysis of the recently obtained theoretic results for a class of optimal control processes governed by Volterra integro-differential equations (see [5]). Using the specific structure of the delayed Volterra integro-differential equations under consideration, we reduce the initially given Optimal Control Problem (OCP) to a numerically tractable separate convex-concave program. This reduction involves optimization techniques in real Hilbert spaces and makes it possible to apply the first-order solution algorithms to the sophisticated Volterra type OCPs. We next consider the celebrated Armijo gradient method for the purpose of a concrete computation and establish the numerical consistency of the resulting algorithm. Finally, we consider an illustrative example.

Keywords:Optimal control, Optimization Abstract: In this work, we study the question of selecting in minimal-time a microalgae species of interest using the Droop model to describe the dynamics of two distinct populations competing for a limiting nutrient. This amounts to consider an optimal control problem governed by a five-dimensional affine control system in which the control is the dilution rate. Throughout this paper, the optimal control strategies allowing the strain of interest to dominate the population in minimal-time is discussed. These results are illustrated using a numerical direct optimization method (implemented in Bocop).

Keywords:Cooperative control, Distributed control, Networked control systems Abstract: In this paper, the output feedback consensus control problem is investigated for a class of multi-agent systems with Lur’e non-linear dynamics over directed topology. Based on the distributed observer, a novel distributed event-triggered consensus protocol is proposed. The proposed event-triggered protocol removes the limitations of continuous communication and saves the on-board resources, that is, the continuous communication is avoided for both controller updating and event-triggered condition monitoring. Furthermore, the consensus is proved by Lyapunov method and Zeno behavior is excluded. Finally, simulation results attests to the properties of the proposed protocols.

Keywords:Flexible structures, Adaptive control, Neural networks Abstract: Vibration and displacement control are of critical importance for both high-rise and ultra high-rise building systems. A single-floor building-like structure equipped with an active mass damper (AMD) is investigated in this paper. Optimal vibration control, while dealing with system uncertainties, is realized by the reinforcement learning (RL) technique. When the unexpected natural disasters (such as strong wind excitation) occur, the proposed controller applying to the active mass damper can compensate the increase of the system vibration caused by external disturbances. In addition, a Lyapunov candidate is used to derive a semi-global uniformly ultimately bounded (SGUUB) property. The experimental platform is mainly composed of one flexible floor and a linear cart system. Both the acceleration and the displacement responses of the floor are provided and compared separately for passive mode, proportional-velocity (PV) control and RL control. The experimental results in the form of graphics and tables have shown the effectiveness of the proposed control algorithm.

Keywords:Control of networks, Distributed control, Sensor networks Abstract: This study concerns distributed spatial filtering over networked systems, i.e., transforming signal values given for nodes to those with a desired spatial frequency characteristic via a distributed computation. An existing filtering algorithm can achieve only low-pass filter characteristics, which limits its range of applications. To address this limitation, we extend the aforementioned filtering algorithm using an additional design parameter. We then present a characterization of all the realizable filter characteristics as a necessary and sufficient condition for achieving distributed spatial filtering. As a result, it is shown that the extended algorithm increases the range of the realizable filter characteristics. The proposed method is verified not only by simulation but also by denoising experiments for a real sensor network. The results show that the proposed method effectively reduces spatial noise and achieves higher performance than an average consensus algorithm and an average filter.

Keywords:Networked control systems, Predictive control for nonlinear systems, Constrained control Abstract: This paper proposes a new event-triggered adaptive horizon model predictive control for discrete-time nonlinear systems with additive disturbance. With the event-triggered control scheme, the optimization problem is solved only at triggering instant and the event is triggered if the difference between the actual state and the predicted state exceeds the triggering threshold. The triggering threshold depends on the prediction horizon and becomes larger as the state approaches the terminal constraint set. Therefore, larger triggering intervals can then be obtained. Finally, a numerical example shows the effectiveness of the proposed scheme.

Keywords:Stability of linear systems, Sampled-data control, Linear systems Abstract: In this paper, we study the problem of stability and stabilization of a sampled-data system under stochastic sampling. A dynamic output feedback control method is proposed in which it is assumed that the sampling interval is disturbed by noisy perturbations obeying a certain probability density. A stochastic stability condition is derived and presented in terms of a linear matrix inequality (LMI) using the Kronecker product operation and the Vandermonde matrix. On the basis of the obtained stochastic stability condition, a new approach to obtain the controller gain matrices is developed. The effectiveness of the proposed method is demonstrated by two examples.

Keywords:Sensor networks, Estimation Abstract: The cooperative localization problem consists of a group of networked agents aiming to find the true probability density function(pdf) of their states. Unlike existing algorithms such as Distributed Kalman filters or non-Bayesian social learning, our algorithm restricts each agent's estimates to a local pdf on its own and its neighbors' state variables. The agents update these pdfs via local observations of their neighbors and their shared messages. This partial state estimation problem is formulated as a distributed constrained optimization in the space of probability density functions. Consistent estimates across the agents are enforced with a constraint requiring equal estimated densities over common states in every communicating agent pair. Stochastic mirror descent steps are then computed to develop a novel cooperative estimation algorithm with geometric averaging over the common marginals to enforce the constraint. We specialize this algorithm to update rules with Gaussian observation models and density estimates. The Gaussian relative position observations are simulated and accuracy is compared to Belief propagation and full state consensus algorithms in varying graph topology.

Keywords:Sensor networks, Estimation, Distributed parameter systems Abstract: We derive the Kullback-Leibler average of von Mises distributions to fuse circular informations in multi-agent systems. Similar to the covariance intersection for Gaussian distributions, the derived fusion protocol does not require the independence among incoming distributions but maintains the estimation consistency, if those von Mises distributions represent estimates. Therefore, this fusion protocol is especially useful in the distributed estimation problem to avoid the over-confidence problem. For example, together with von Mises filters, the derived fusion protocol can estimate the dynamic circular term in a distributed manner. In addition, we apply this fusion protocol to determine the consensus of von Mises distributions over a network. Since the fusion protocol can be easily achieved by calculating the weighted average of the associated complex numbers, the corresponding convergent conditions of the consensus algorithm are then elegantly determined.

Keywords:Sensor networks, Kalman filtering, Estimation Abstract: This paper studies the problem of distributed state estimation (DSE) over sensor networks. Unlike the existing filtering algorithms that only consider targets moving in two-dimension (2-D) environments, we address this problem in three-dimension (3-D) scenarios where each agent equipped with the communication and sensing capabilities cooperatively track the state of a 3-D moving object. First, it is shown that the existing distributed Kalman filter (DKF) algorithms cannot solve the quaternion-based six degree-of-freedom (6-DoF) motion tracking. Then, a novel DKF applicable for the 3-D tracking is introduced for a general nonlinear system. The proposed algorithm is fully distributed and robust to time-varying communication topologies and changing blind agents (the agents that lose sight of the whole target object). Finally, we apply the proposed algorithm to a camera network to track the 6-DoF pose (position and orientation) of a moving target object. The effectiveness of our approach is demonstrated through Monte-Carlo simulations.

Keywords:Sensor networks, Randomized algorithms, Optimization Abstract: Tracking of multiple targets is a classical problem in signal processing that arises in many applications, e.g. air, maritime and road traffic control. Networks of autonomous sensors serve as desirable platforms for multi-target tracking in view of their redundancy and reconfigurability. The networked implementation, however, makes it impossible to use classical centralized approaches to filtering, since each sensor has limited computational capabilities and restricted access to the measurements of other sensors. Besides topological constraints (each sensor can interact only to a few adjacent nodes of a network), communication between sensors can be restricted, due to e.g. limited capacity of communication channels, delays and data distortions.

In this paper, we propose a new algorithm for distributed multi-target tracking in a sensor network. The algorithm is based on the seminal idea of simultaneous perturbation stochastic approximation (SPSA), being a special case of stochastic gradient descent algorithm. The important feature of the SPSA method is the ability to solve optimization (in particular, optimal tracking) problems in the presence of emph{arbitrary} unknown (but bounded) disturbances and time-varying parameters of the system. These uncertainties need not be random, and even if they are random, one need not know their statistical characteristics. We provide the mathematical results on stabilization of the mean-square estimation error and analyze its dependence on the choice of step-size parameters. Theoretical results are illustrated by numerical simulations.

University of Electronic Science and Technology of China

Keywords:Sensor networks, Linear systems, Distributed control Abstract: In this paper, we study the problem of localizing the sensors' positions in presence of denial-of-service (DoS) attacks. We consider a general attack model, in which the attacker action is only constrained through the frequency and duration of DoS attacks. We propose a distributed iterative localization algorithm with an abandonment strategy based on the barycentric coordinate of a sensor with respect to its neighbors, which is computed through relative distance measurements. In particular, if a sensor's communication links for receiving its neighbors' information lose packets due to DoS attacks, then the sensor abandons the location estimation. When the attacker launches DoS attacks, the AS-DILOC algorithm is proved theoretically to be able to accurately locate the sensors regardless of the attack strategy at each time. The effectiveness of the proposed algorithm is demonstrated through simulation examples.

Keywords:Learning Abstract: This paper considers the problem of learning control laws for nonlinear polynomial systems directly from the data, which are input-output measurements collected in an experiment over a finite time period. Without explicitly identifying the system dynamics, stabilizing laws are directly designed for nonlinear polynomial systems using experimental data alone. By using data-based sum of square programming, the stabilizing state-dependent control gains can be constructed.

Keywords:Predictive control for nonlinear systems, Learning, Robust control Abstract: The complex and uncertain dynamics of emerging systems pose several unique challenges that need to be overcome in order to design high-performance controllers. A key challenge is that safety is often achieved at the expense of closed-loop performance. This is particularly important when the uncertainty description is provided in the form of a bounded set that is estimated offline from limited data. Replacing this bounded set with a learned state- and input-dependent uncertainty enables representing the the variation of uncertainty in the model throughout the state space, thus improving closed-loop performance. Gaussian process (GP) models are a good candidate for learning such a representation; however, they produce a nonlinear and nonconvex description of the uncertainty set that is difficult to incorporate into currently available robust model predictive control (MPC) frameworks. In this work, we present a learning- and scenario-based MPC (L-sMPC) strategy that systematically accounts for feedback in the prediction using a state- and input-dependent scenario tree computed from a GP uncertainty model. To ensure that the closed-loop system evolution remains safe, we also propose a projection-based safety certification scheme that ensures the control inputs keep the system within an appropriately defined invariant set. The advantages of the proposed L-sMPC method in terms of improved performance and an enlarged feasible region are illustrated on a benchmark double integrator problem.

Keywords:Optimal control, Learning Abstract: We study the problem of optimal state-feedback tracking control for unknown discrete-time deterministic systems with input constraints. To handle input constraints, state-of-art methods utilize a certain nonquadratic stage cost function, which is sometimes limiting real systems. Furthermore, it is well known that Policy Iteration (PI) and Value Iteration (VI), two widely used algorithms in data-driven control, offer complementary strengths and weaknesses. In this work, a two-step transformation is employed, which converts the constrained-input optimal tracking problem to an unconstrained augmented optimal regulation problem, and allows the consideration of general stage cost functions. Then, a novel multi-step VI algorithm based on Q-learning and linear programming is derived. The proposed algorithm improves the convergence speed of VI, avoids the requirement for an initial stabilizing control policy of PI, and computes a constrained optimal feedback controller without the knowledge of a system model and stage cost function. Simulation studies demonstrate the reliability and performance of the proposed approach.

Keywords:Uncertain systems, Randomized algorithms, Statistical learning Abstract: Scenario optimization is by now a well established technique to perform designs in the presence of uncertainty. It relies on domain knowledge integrated with first-hand information that comes from data and generates solutions that are also accompanied by precise statements of reliability. In this paper, following recent developments in [A], we venture beyond the traditional set-up of scenario optimization by analyzing the concept of constraints relaxation. By a solid theoretical underpinning, this new paradigm furnishes fundamental tools to perform designs that meet a proper compromise between robustness and performance. After suitably expanding the scope of constraints relaxation as proposed in [A], we focus on various classical Support Vector methods in machine learning – including SVM (Support Vector Machine), SVR (Support Vector Regression) and SVDD (Support Vector Data Description) – and derive new results that attest the ability of these methods to generalize.

[A] S. Garatti and M.C. Campi. Risk and complexity in scenario optimization. Mathematical Programming, 2019. Published on-line. DOI: https://doi.org/10.1007/s10107-019-01446-4.

Keywords:Delay systems, Adaptive control, Uncertain systems Abstract: This paper proposes a discrete-time adaptive regulator for a MIMO linear time-invariant system with unknown, constant input time delays that may differ across the input channels. It is assumed the delay has a known upper-bound. In addition, the plant is subject to an unmeasurable exogeneous disturbance. To mitigate the effect of the disturbance, a second-order delay disturbance observer is used. A stability analysis shows that the proposed regulator drives the plant state to zero asymptotically with an O(T^2) bound on the regulation error and simulation results are shown to verify the approach.

Keywords:Delay systems, Distributed parameter systems, Time-varying systems Abstract: This paper deals with the problem of prescribed-time stabilization of controllable linear systems with input delay.The problem is reformulated under a cascade PDE-ODE setting from which a prescribed-time predictor feedback is designed based on the backstepping approach, and whose transformation makes use of time-varying kernels. The bounded invertibility of the transformation is guaranteed. It is proved that the solution converges to the equilibrium in a prescribed-time. A simulation example is presented to illustrate the results.

Keywords:Delay systems, Lyapunov methods, Estimation Abstract: This paper provides a practical control solution to remote control systems with unknown time-varying delays. An external signal which can be considered as a specific communication loop is added to estimate the unknown round-trip delay in finite time. Then the system is stabilized with the practical delay estimation technique and a predictor-based controller. The main results of this paper also provide an alternative way to design predictor-based controller for systems with known input and output time-varying delays. The theoretical results of this paper are illustrated by simulation results.

Keywords:Delay systems, Stability of linear systems, Lyapunov methods Abstract: This paper presents an improved necessary and sufficient condition of exponential stability for LTI delay systems. By this criterion, the stability analysis of any system of the class can be reduced to verification of the positive definiteness of a special matrix (which is based on the delay Lyapunov matrix). The dimension of the matrix depends on the parameters of the system, and the goal of this paper is to make it smaller. We applied a new technique, and reduced the dimension by about half.

Keywords:Linear systems, Delay systems Abstract: This paper proposes a fractional order internal model controller (FO-IMC) for a fractional order plus time delay (FrOPTD) process model to satisfy desired gain margin (Am) and phase margin (Pm) specifications. A method is proposed to solve four equations arising from the Am and Pm specifications, which also generates solution of the system gain-cross-over frequency (wg) and phase-cross-over frequency (wp) as intermediate variables. The designed FO-IMC is implemented on laboratory DC servo-system and compared with other integer order controller and fractional order controller.

Slovak University of Technology in Bratislava, Faculty of Mechan

Keywords:Predictive control for linear systems, Optimal control, Mechatronics Abstract: In this paper we revisit the problem of finding an equivalent inverse optimality formulation to a given linear model predictive control (MPC) problem using the technique of inverse parametric convex programming based on convex lifting. In particular, we show that the parametric solution to a typical QP-based MPC problem can be used to formulate an equivalent LP problem with a significantly reduced space of decision variables and a new set of constraints, the solution of which in both explicit and implicit fashion implies a lower online implementation effort. This is also demonstrated in a practical case study focused on active vibration control, using various optimization methods to solve both the nominal and the reformulated MPC problem. The obtained results suggest a potential of the convex lifting based inverse optimality technique for example in embedded MPC applications where, based on hardware specifications, either explicit or implicit solution is more suitable.

Keywords:Predictive control for linear systems, Optimization, Optimal control Abstract: This paper proposes a framework for the high accuracy, low-precision, and memory-efficient embedded model predictive control (MPC) using the positTM numbers and its implementation on the ARM-based embedded platform. A quadratic programming (QP) problem in posit-based linear MPC is solved by the active set method (ASM) with a Cholesky factorization-based linear solver. The main idea of this paper is to encode all data associated with the QP problem as posit numbers and employ posit number arithmetic to synthesis the ASM algorithm. We provide a detailed analysis of a posit number that acts as a memory-efficient replacement of the IEEE 754 floating-point standard numbers. We show the posit based ASM algorithm employed in MPC and its implementation on STM32 Nucleo-144 development board with STM32F746ZG MCU. The results of hardware-in-loop (HIL) simulations with the detailed analysis of memory utilization and performance of the posit-based ASM algorithm is shown with two case studies. HIL results show that the proposed approach can reduce memory footprints by 50% to 75% without losing control accuracy and performance.

Keywords:Stochastic optimal control, Uncertain systems, Predictive control for linear systems Abstract: Risk-sensitive model predictive control (MPC) is a promising technique since it can balance the main drawbacks of robust MPC and stochastic MPC: conservative decision making and vulnerability to extreme events. The main challenge is that the risk measures accompany high nonlinearity making the technique difficult for various applications. This paper presents a fast risk-sensitive MPC for linear time-invariant systems that use time-series forecasting. The risk measure involving high nonlinearity is converted into a weighted square-sum of decision variables with the assumption of Gaussian system disturbances. The error statistics update rule is designed to consider the characteristics of time-series forecasting. The proposed formulation is applied to energy-efficient car-following control and compared with the robust and the stochastic MPC.

Keywords:Predictive control for linear systems, Reduced order modeling, Robust control Abstract: Model predictive control is a powerful framework for enabling optimal control of constrained systems. However, for systems that are described by high-dimensional state spaces this framework can be too computationally demanding for real-time control. Reduced order model predictive control (ROMPC) frameworks address this issue by leveraging model reduction techniques to compress the state space model used in the online optimal control problem. While this can enable real-time control by decreasing the online computational requirements, these model reductions introduce approximation errors that must be accounted for to guarantee constraint satisfaction and closed-loop stability for the controlled high-dimensional system. In this work we propose an offline methodology for efficiently computing error bounds arising from model reduction, and show how they can be used to guarantee constraint satisfaction in a previously proposed ROMPC framework. This work considers linear, discrete, time-invariant systems that are compressed by Petrov-Galerkin projections, and considers output-feedback settings where the system is also subject to bounded disturbances.

Keywords:Predictive control for linear systems, Robust control, Estimation Abstract: This work addresses the problem of output feedback Model Predictive Control (MPC) of constrained, linear, discrete-time systems corrupted by additive perturbations on both state and output. The use of estimated variables in MPC is challenging due to the need of guaranteeing robust constraint satisfaction. Many of the existing solutions for this problem are either computationally expensive or conservative. To overcome this issue and cope with uncertainty, the proposed approach incorporates interval observers on the MPC scheme, leading to a novel, simple and very intuitive methodology providing robust constraint satisfaction with reduced computational complexity.

Keywords:LMIs, Quantized systems, Lyapunov methods Abstract: This paper studies feedback system with a state-feedback law with a ternary nonlinearity in the feedback loop. We propose a novel Lyapunov-based approach that takes advantage of a framework recently developed to assess positivity of generalized piece-wise quadratic forms. Such an approach is used to compute implicitly defined piece-wise quadraticfunction as stability certificates of the non-linear system

Keywords:Stability of nonlinear systems, Hybrid systems, LMIs Abstract: This paper addresses L2-stability analysis of discrete-time continuous piecewise affine systems described in input-output form by linear combinations of basis piecewise affine functions. The proposed approach exploits an equivalent representation of these systems as the feedback interconnection of a linear system and a diagonal static block with a repeated scalar nonlinearity. The transformation makes it possible to apply analysis results for systems with repeated nonlinearities based on integral quadratic constraints. This leads to a sufficient condition for L2-stability that can be checked via the solution of a single linear matrix inequality, whose dimension grows linearly with the number of basis piecewise affine functions defining the system. Numerical examples corroborate the proposed approach by providing a comparison with an alternative approach based on the computation of piecewise polynomial storage functions.

Keywords:Lyapunov methods, LMIs, Algebraic/geometric methods Abstract: Computing control invariant sets is paramount in many applications. The families of sets commonly used for computations are ellipsoids and polyhedra. However, searching for a control invariant set over the family of ellipsoids is conservative for systems more complex than unconstrained linear time invariant systems. Moreover, even if the control invariant set may be approximated arbitrarily closely by polyhedra, the complexity of the polyhedra may grow rapidly in certain directions. An attractive generalization of these two families are piecewise semi-ellipsoids. We provide in this paper a convex programming approach for computing control invariant sets of this family.

Keywords:Nonlinear output feedback, Lyapunov methods, LMIs Abstract: This paper is concerned with the problem of static output feedback stabilization of discrete-time Lur'e systems. By using a quadratic Lyapunov function, new design conditions are provided in terms of sufficient linear matrix inequalities where the control gains appear affinely. Using some relaxations, the search for the stabilizing control gains is performed through an iterative algorithm. The approach can be considered as more general than the existing ones thanks to the fact that the gains are treated as decision variables in the optimization problem. Therefore, the technique can handle state or output feedback indistinctly and can deal with magnitude or structural constraints (such as decentralization) on the gains. Numerical examples illustrate that the proposed method can provide less conservative results when compared with other techniques from the literature.

Keywords:Power systems, Smart grid, Game theory Abstract: A market consisting of a generator with thermal and renewable generation capability, a set of non-preemptive loads (i.e., loads which cannot be interrupted once started), and an independent system operator (ISO) is considered. Loads are characterized by durations, power demand rates and utility for receiving service, as well as disutility functions giving preferences for time slots in which service is preferred. Given this information, along with the generator’s thermal generation cost function and forecast renewable generation, the social planner solves a mixed integer program to determine a load activation schedule which maximizes social welfare. Assuming price taking behavior, we develop a competitive equilibrium concept based on a relaxed version of the social planner’s problem which includes prices for consumption and incentives for flexibility, and allows for probabilistic allocation of power to loads. Considering each load as representative of a population of identical loads with scaled characteristics, we demonstrate that the relaxed social planner’s problem gives an exact solution to the original mixed integer problem in the large population limit.

Keywords:Transportation networks, Optimization, Autonomous systems Abstract: This paper studies optimal pricing and rebalancing policies for Autonomous Mobility-on-Demand (AMoD) systems. We adopt a macroscopic planning perspective to tackle a profit maximization problem while ensuring that the system is load-balanced. We describe the system using a dynamic fluid model to show the existence and stability of an equilibrium (i.e., load balance) through pricing policies. We then develop an optimization framework that allows us to find optimal policies in terms of both pricing and rebalancing. We first maximize profit by only using pricing policies, then incorporate rebalancing, and finally we consider whether the solution is found sequentially or jointly. We apply each approach to a data-driven case study using real taxi data from New York City. Depending on which benchmarking solution we use, the joint problem (i.e., pricing and rebalancing) increases profits by 7% to 40%.

Keywords:Agents-based systems, Game theory Abstract: We consider the problem of steering the aggregative behavior of a set of noncooperative price-taking agents to a desired point. Different from prevalent pricing schemes where the price is available for design, we resort to suitable "nudge" mechanisms to influence the behavior of the agents. In particular, a regulator sends a price prediction signal to the agents, based on which the agents decide on their actions. This prediction is potentially different from the actual price, which brings the issue of reliability. We take this into account by associating trust variables to the agents, implying that the agents do not blindly follow the prediction signal. These trust variables are updated depending on the history of the discrepancy between the actual and the predicted price. We carefully examine the resulting multi-components model and analyse its convergence properties. We show that under the proposed nudge mechanisms, the regulator gains agents’ trust fully, and the aggregative behavior provably converges to a desired set point. The effectiveness of the approach is demonstrated by numerical examples.

Keywords:Emerging control applications, Smart cities/houses, Optimization Abstract: This paper considers off-street parking for the cruising vehicles of transportation network companies (TNCs) to reduce the traffic congestion. We propose a novel business that integrates the shared parking service into the TNC platform. In the proposed model, the platform (a) provides interfaces that connect passengers, drivers and garage operators (commercial or private garages); (b) determines the ride fare, driver payment, and parking rates; (c) matches passengers to TNC vehicles for ride-hailing services; and (d) matches vacant TNC vehicles to unoccupied parking garages to reduce the cruising cost. A queuing-theoretic model is proposed to capture the matching process of passengers, drivers, and parking garages. A market-equilibrium model is developed to capture the incentives of the passengers, drivers, and garage operators. An optimization-based model is formulated to capture the optimal pricing of the TNC platform. Through a realistic case study, we show that the proposed business model will offer a Pareto improvement that benefits all stakeholders, which leads to higher passenger surplus, higher drivers surplus, higher garage operator surplus, higher platform profit, and reduced traffic congestion.

Keywords:Finance, Stochastic systems, Sampled-data control Abstract: In this paper, we consider a discrete-time portfolio with m grater than or equal to 2 assets optimization problem which includes the rebalancing frequency as an additional parameter in the maximization. The so-called Kelly Criterion is used as the performance metric; i.e., maximizing the expected logarithmic growth of a trader’s account, and the portfolio obtained is called the frequency-based Kelly optimal portfolio. The focal point of this paper is to extend upon the results of our previous work to obtain various optimality characterizations on the portfolio. To be more speciﬁc, using Kelly’s criterion in our frequency-based formulation, we ﬁrst prove necessary and sufﬁcient conditions for the frequency-based Kelly optimal portfolio. With the aid of these conditions, we then show several new optimality characterizations such as expected ratio optimality and asymptotic relative optimality, and a result which we call the Extended Dominant Asset Theorem. That is, we prove that the ith asset is dominant in the portfolio if and only if the Kelly optimal portfolio consists of that asset only. The word “extended” on the theorem comes from the fact that it was only a sufﬁciency result that was proved in our previous work. Hence, in this paper, we improve it to involve a proof of the necessity part. In addition, the trader’s survivability issue (no bankruptcy consideration) is also studied in detail in our frequency-based trading framework. Finally, to bridge the theory and practice, we propose a simple trading algorithm using the notion called dominant asset condition to decide when should one triggers a trade. The corresponding trading performance using historical price data is reported as supporting evidence.