
WeIP02 Interactive Session, Elizabeth A,B,C,D,E,F 
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Linear Systems II 



15:3017:30, Paper WeIP02.1  Add to My Program 
An Improved Algorithm for the NonIterative Solution of the DiscreteTime FiniteHorizon LQ Control Problem with Fixed Final State 
Zattoni, Elena  Univ. of Bologna 
Keywords: Linear systems, Optimal control, Optimization algorithms
Abstract: The noniterative solution through MoorePenrose inverse which applies to discretetime finitehorizon LQ optimal control problems with fixed final state is subject to a constraint on the maximal length of the control time interval. This is a consequence of the limitation on the computational power available for processing the generalized inverse of properly constructed matrices. In this work, a computational framework where the dimensionality restriction is completely removed is presented. The core of the proposed algorithm consists in a procedure where the time interval taken into account doubles at each step. The routine guarantees a fast convergence to the solution, as can be easily checked by comparing the number of floating point operations with those required by other available routines. Moreover, the solution of the corresponding infinitehorizon problem is retrievable with arbitrary accuracy by setting the final state to zero and welding a sufficient number of arcs. The procedure returns an arbitrarily accurate solution of the infinitehorizon problem, with no dditional complications, also when the tobecontrolled system is nonleftinvertible.


15:3017:30, Paper WeIP02.2  Add to My Program 
Indefinite LQControl of Discrete Time Descriptor Systems by Nested PseudoInversion 
Nyman, PerOle  Narvik Univ. Coll. 
Keywords: Linear systems, Optimal control, Optimization
Abstract: A generalized pseudoinverse approach to discretetime finite horizon linear quadratic control of descriptor systems with final state constraint is considered. Necessary and sufficient conditions for existence of a unique optimal solution are provided. Both the test for existence of a solution, as well its computation, decompose hierarchically to problems of small size defined on subdivision of the time axis, thus rendering computational feasibility for problems too large to be handled directly.


15:3017:30, Paper WeIP02.3  Add to My Program 
Asymptotic Behavior of Imaginary Zeros of Linear Systems with Commensurate Delays 
Chen, Jie  Univ. of California at Riverside 
Fu, Peilin  Univ. of California, Riverside 
Niculescu, SilviuIulian  Univ. de Tech. de Compiegne 
Keywords: Stability of linear systems, Delay systems
Abstract: This paper addresses the problem of asymptotic stability of linear timedelay systems with commensurate delays. We study the asymptotic behavior of the critical characteristic zeros of such systems on the imaginary axis. This behavior determines whether the imaginary zeros cross from one half plane into another, and hence plays a critical role in determining the stability of a timedelay system. We consider timedelay systems given in both statespace form and as a quasipolynomial. Our results reveal that in the former case the zero asymptotic behavior can be characterized by solving a simple eigenvalue problem, and in the latter case, by computing the derivatives of the quasipolynomial. To perform such an analysis, we make use of an operator perturbation approach.


15:3017:30, Paper WeIP02.4  Add to My Program 
Homogeneous Polynomial Forms for Simultaneous Stabilizability of Families of Linear Control Systems: A Tensor Product Approach 
Altafini, Claudio  SISSA 
Keywords: Stability of linear systems, LMIs, Robust control
Abstract: The paper uses the formalism of tensor products in order to deal with the problem of simultaneous stabilizability of a family of linear control systems by means of Lyapunov functions which are homogeneous polynomial forms. While the feedback synthesis seems to be nonconvex, the simultaneous stability by means of homogeneous polynomial forms of the uncontrollable modes yields (convex) necessary but not sufficient conditions for simultaneous stabilizability.


15:3017:30, Paper WeIP02.5  Add to My Program 
Pole Placement Via Recursive Delayed Feedback Control for Discrete Systems with Unknown Equilibrium Points 
Zhu, Jiandong  Nanjing Normal Univ. 
Tian, YuPing  Southeast Univ. 
Keywords: Linear systems, Stability of linear systems, Uncertain systems
Abstract: In this paper, the pole placement problem for a class of singleinput discrete systems with unknown equilibrium points is investigated. The problem is completely resolved by developing a recursive delayed feedback control method. Two examples show the effectiveness of the proposed method in stabilizing uncertain equilibrium points or equilibrium sets.


15:3017:30, Paper WeIP02.6  Add to My Program 
Gain Scheduling of ObserverBased Controllers with Integral Action 
Trangbaek, Klaus  Aalborg Univ. 
Stoustrup, Jakob  Aalborg Univ. 
Bendtsen, Jan Dimon  Aalborg Univ. 
Keywords: Linear system observers, Switched systems, Stability of hybrid systems
Abstract: This paper presents a method for continuous gain scheduling of observerbased controllers with integral action. Given two stabilising controllers for a given system, explicit state space formulae are presented, allowing to change gradually from one controller to the other while preserving stability. As opposed to previous results, the method allows for nonsquare systems.


15:3017:30, Paper WeIP02.7  Add to My Program 
Parametric Approaches for Eigenstructure Assignment in HighOrder Descriptor Linear Systems 
Duan, GuangRen  Harbin Inst. of Tech. 
Yu, HaiHua  Harbin Inst. of Tech. 
Keywords: Linear systems
Abstract: This paper considers eigenstructure assignment in highorder descriptor linear systems via proportional plus derivative state feedback. It is shown that the problem is closely related with a type of socalled highorder Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closedloop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closedloop eigenvalues to be set undetermined and sought via certain optimization procedures.


15:3017:30, Paper WeIP02.8  Add to My Program 
Exact, Approximate, Optimized and Constrained Output Feedback Pole Assignment 
Tarokh, Mahmoud  San Diego State Univ. 
Keywords: Linear systems, Stability of linear systems
Abstract: The paper proposes a new formulation of the output feedback pole assignment problem. In this formulation, a unified approach is presented for solving the pole assignment problem with various additional objectives. These objectives include optimizing a variety of performance indices, and imposing constraints on the output feedback matrix structure, e.g. decentralized structure. Conditions for the existence of the output feedback are discussed. However, the thrust of the paper is on the development of a convergent pole assignment algorithm. It is shown that when exact pole assignment is not possible, the method can be used to place the poles close to the desired locations. Examples are provided to illustrate the method


15:3017:30, Paper WeIP02.9  Add to My Program 
On FloquetFourier Realizations of Linear TimePeriodic Impulse Responses 
Sandberg, Henrik  California Inst. of Tech. 
Keywords: Linear systems, Time varying systems, Reduced order modeling
Abstract: We show that a linear timeperiodic system with a smooth impulse response can be arbitrarily well approximated by a linear timeperiodic statespace representation in socalled FloquetFourier form. The FloquetFourier form has a constant state matrix and the input and output matrices have only finitely many nonzero Fourier coefficients. Such representations are easier to use for analysis and control design than the impulse response or fully general statespace forms. The construction of the FloquetFourier form is done using methods for model approximation of infinitedimensional linear timeinvariant systems. We also propose a method for constructing minimal realizations from impulse responses of a special structure.


15:3017:30, Paper WeIP02.10  Add to My Program 
Imposing FIR Structure on H2 Preview Tracking and Smoothing Solutions 
Mirkin, Leonid  Tech.  IIT 
Tadmor, Gilead  Northeastern Univ. 
Keywords: Linear systems, Optimal control, Linear system observers
Abstract: In this paper we study continuoustime H² feedforward tracking and estimation problems with FIR (finite impulse response) solutions. Both causal and noncausal solutions are considered. It is shown that these problems can be treated as special cases of static problems in the lifted domain, which, in turn, can be solved in terms of a twopoint boundary value (HamiltonJacobi) system. This results in fixed dimension, closedform solutions.


15:3017:30, Paper WeIP02.11  Add to My Program 
Parametric Approach for the Normal Luenberger Function Observer Design in SecondOrder Linear Systems 
Zhou, Bin  Harbin Inst. of Tech. 
Duan, GuangRen  Harbin Inst. of Tech. 
Keywords: Linear systems, Linear system observers
Abstract: In this paper, the normal Luenberger function observer design for secondorder linear systems is considered. It is shown that the main procedure of the design is to solve a socalled secondorder generalized Sylvesterobserver matrix equation. Based on an explicit parametric solution to this equation, a parametric solution to the normal Luenberger function observer design problem is given. The design degrees of freedom presented by explicit parameters can be further utilized to achieve some additional design requirements.


15:3017:30, Paper WeIP02.12  Add to My Program 
Rejection of Bounded Disturbances Via Invariant Ellipsoids Technique 
Polyak, Boris T.  Inst. of Control Sciences, RAS 
Nazin, Alexander V.  Inst. of Control Sciences, RAS 
Topunov, Michael  Inst. for Control Science, RAS 
Nazin, Sergey A.  Inst. of Control Sciences, RAS / Univ. catholique de Louvain 
Keywords: Linear systems, Optimal control
Abstract: In this paper an approach based on invariant ellipsoids is applied to the problem of persistent disturbance rejection by means of static statefeedback control. Dynamic system is supposed to be linear timeinvariant and affected by unknownbutbounded exogenous disturbances. Synthesis of an optimal controller that returns a minimum of the size of the corresponding invariant ellipsoid is reduced to onedimensional convex minimization with LMI constraints. The problem is considered in continuous and discrete time cases.


15:3017:30, Paper WeIP02.13  Add to My Program 
Stabilization of a Class of Uncertain "wave" Discrete Linear Repetitive Processes 
Galkowski, Krzysztof  Univ. of Zielona Gora 
Cichy, Blazej  Univ. of Zielona Gora 
Rogers, Eric  Univ. of Southampton UK 
Lam, James  Univ. of Hong Kong 
Keywords: Linear systems, Stability of linear systems, Robust control
Abstract: Repetitive processes are a~distinct class of twodimensional (2D) systems (i.e. information propagation in two independent directions occurs) of both systems theoretic and applications interest. They cannot be controlled by direct extension of existing techniques from either standard (termed 1D here) or (most often) 2D systems theory. In this paper we continue the development of a systems theory for a recently proposed model of these processes necessary to represent terms which arise in some applications areas but are not included in the currently used models. The new results are on the design of control laws for stabilization in the case when there is uncertainty associated with the defining statespace model.


15:3017:30, Paper WeIP02.14  Add to My Program 
Some Further Results on Optimal Scaling 
Roy, Sandip  Washington State Univ. 
Saberi, Ali  Washington State Univ. 
Keywords: Network Analysis and Control, Cooperative control, Optimization
Abstract: We present some new results on optimal scalingi.e., the design of a diagonal matrix K so as optimize the product KG, G in R^{n times n}, with respect to a performance measure. In particular, we extend our previous work on optimizing the dominant eigenvalue ratio to three other performance measures: 1) the condition number, 2) the ratio of the minimum eigenvalue to the infinitynorm of KG, and 3) a Lyapunov exponentbased measure. We also give some preliminary explorations into eigenvalue placement through scaling.


15:3017:30, Paper WeIP02.15  Add to My Program 
An H_Infinity Approach for Robust Calibration of Cascaded Sigma Delta Modulators 
Yang, Fuwen  King's Coll. London, Univ. of London 
Gani, Mahbub  King's Coll. London, Univ. of London 
Keywords: H infinity control, Robust control, Quantized systems
Abstract: In this paper, we propose a novel robust H_{infty} optimisation approach to the design of digital filters for calibrating 21 cascaded Sigma Delta modulators. The main contribution of this paper consists of two parts. First, we develop a new filter design technique based on a linear matrix inequality (LMI) approach to modelmatching problem with polytopic uncertainties in parameters. The advantage of the proposed method is that it leads to an optimal, less conservative, solution to the robust H_{infty} filtering problem. The second contribution involves the application of the proposed robust filter design scheme to correct for inevitable and unwelcome analog imperfections typically associated with cascaded Sigma Delta modulators. For numerical illustration we concentrate on the 21 architecture. Simulation results for a range of parameter excursions suggest that our robust H_{infty} filter guarantees an improved signaltonoise ratio (SNR) performance over the nominal filter.


15:3017:30, Paper WeIP02.16  Add to My Program 
OutputFeedback Model Predictive Control for LPV Systems with Input Saturation Based on QuasiMinMax Algorithm 
Kim, TaeHyoung  Kyoto Univ. 
Park, JeeHun  Samsung Electronics Co. 
Sugie, Toshiharu  Kyoto Univ. 
Keywords: Predictive control for linear systems
Abstract: Model predictive control (MPC) is one of the most promising and significant control approaches because of its ability to handle control problems for constrained systems. In this paper, we propose a robust outputfeedback MPC scheme for polytopic linear parameter varying (LPV) systems based on a quasiminmax algorithm. We first show the offline design procedure of a robust state observer for LPV systems using linear matrix inequality (LMI). Then, the online robust outputfeedback MPC algorithm using the estimated state is developed based on LMI technique. At each time instant, the control input is computed by solving the proposed infinite horizon optimization problem involving LMI constraints. Finally, a numerical example is given to demonstrate its effectiveness.


15:3017:30, Paper WeIP02.17  Add to My Program 
Model Predictive Control of the Hybrid Ventilation for Livestock 
Wu, Zhuang  Aalborg Univ. 
Stoustrup, Jakob  Aalborg Univ. 
Trangbaek, Klaus  Aalborg Univ. 
Heiselberg, Per  Aalborg Univ. 
RiisgaardJensen, Martin  SKOV 
Keywords: Predictive control for linear systems, Constrained control, Nonlinear systems
Abstract: In this paper, design and simulation results of model predictive control (MPC) strategy for livestock hybrid ventilation systems and associated indoor climate through variable valve openings and exhaust fans, are presented. The design is based on thermal comfort parameters for poultry in barns and a dynamic model describing the nonlinear behavior of ventilation and associated climate, by applying a socalled conceptual multizone method and the theory of conservation of energy and mass. A series of coupled linearized state space models corresponding to different interzonal airflow direction are developed to be adaptive to control. The simulation results illustrate the high potential of MPC in dealing with nonlinearities, handling constraints and performing offset free tracking. The purpose of this paper is to apply MPC taking into account of the random disturbances from animals and weather condition to calculate the optimal ventilation rate and air flow distribution and the prediction of indoor horizontal variation of temperature through a optimum energy approach.


15:3017:30, Paper WeIP02.18  Add to My Program 
Some NonStandard LinearQuadratic Problems for Descriptor Systems 
Kurina, Galina  Voronezh State Forestry Acad. 
Keywords: Linear systems, Optimal control
Abstract: The work deals with linearquadratic optimal control problems with constant coefficients when the state equation is unresolved with respect to the derivative. In the first two problems, the performance index is the sum of the integral of a quadratic form with respect to the control and quadratic forms with respect to the differences between the output variable values in the fixed points and given values. Two cases are researched, namely, when the initial value for the part of the state variable is given and when additional constraints for boundary points of the state variable are absent. For the third problem, output variable values in the fixed points are given. The sufficient and necessary control optimality conditions have been established for considered problems. The solvability of these optimal control problems has been proved. The examples are given which show that the necessary control optimality conditions are not valid under general assumptions. It has been pointed out that the problems on smoothing and interpolating splines are the particular cases of the studied optimal control problems.


15:3017:30, Paper WeIP02.19  Add to My Program 
The Performance Improvements of Train Suspension Systems with Inerters 
Wang, FuCheng  National Taiwan Univ. 
Yu, ChungHuang  National YangMing Univ. 
Chang, MongLon  National Taiwan Univ. 
Hsu, Mowson  National Taiwan Univ. 
Keywords: Linear systems, Modeling, Optimization
Abstract: This paper investigates the performance benefits of train suspension systems employing a new mechanical network element, called Inerter. Combined with traditional passive suspension elements  dampers and springs, Inerter is shown to be capable of improving the performance, in terms of the passenger comfort, system dynamics and stability (safety), of the train suspension systems. Furthermore, a motordriven platform is constructed to test the properties of suspension struts with inerters.


15:3017:30, Paper WeIP02.20  Add to My Program 
A Dual FrequencySelective Bounded Real Lemma and Its Applications to IIR Filter Design 
Tuan, Hoang Duong  The Univ. of New South Wales 
Hoang, Nguyen Thien  Nagoya Univ. 
Ngo, Q. Hung  SUNY at Buffalo 
Hoang, Tuy  NSC 
Vo, BaNgu  The Univ. of Melbourne 
Keywords: LMIs, Linear systems, Filtering
Abstract: Given a transfer function H(s) of order n, the celebrated bounded real lemma characterises the untractable semiinfinite programming (SIP) condition H(jmath omega)^2leqgamma^2 forallomegain R of function bounded realness (BR) by a tractable semidefinite programming (SDP). Some recent results generalise this result for the SIP condition H(jmath omega)^2leqgamma^2 forallomegaleqbar{omega} of frequencyselective bounded realness (FSBR). The SDP characterisations are given at the expense of an introduced Lyapunov matrix variable of dimension ntimes n. As a result, the dimension of the resultant SDPs grows so quickly in respect to the function order, making them much less computationally tractable and practicable. Moreover, they do not allow to formulate synthesis problems as SDPs. In this paper, a completely new SDP characterizations for general FSBR for allpole transfer functions is proposed. Our motivation is the design of infiniteimpulseresponse(IIR) filters involving a few of simutaneous FSBRs. Our SDP characterizations are of moderate size and free from Lyapunov variables and thus allow to address problems involving transfer functions of arbitrary order. Examples are also provided to validate the effectiveness of the resulting SDP design formulation. Finallly we also raise some issues arising with practicability of SDP for multidimensional filter design problems. In particular, any bilinear matrix inequality (BMI) optimization is shown to be solved by a SDP with any prescribed tolerance but the issue is dimensionality of this SDP.


15:3017:30, Paper WeIP02.21  Add to My Program 
Existence of Homogeneous Polynomial Solutions for ParameterDependent Linear Matrix Inequalities with Parameters in the Simplex 
Bliman, PierreAlexandre  INRIARocquencourt 
Oliveira, Ricardo C. L. F.  Univ. of Campinas 
Montagner, Vinícius F.  Univ. of Campinas 
Peres, Pedro L. D.  Univ. of Campinas 
Keywords: LMIs, Uncertain systems, Linear parametervarying systems
Abstract: This paper presents some general results concerning the existence of homogeneous polynomial solutions to parameterdependent linear matrix inequalities whose coefficients are continuous functions of parameters lying in the unit simplex. These results are useful in the context of robust analysis and synthesis of parameterdependent feedback gains (gainscheduling) for uncertain linear systems in polytopic domains. A result showing the generality of the class of static gains with homogeneous polynomial dependence and a result dealing with the solutions of parameterdependent linear matrix inequalities with slowly timevarying parameters are also given.


15:3017:30, Paper WeIP02.22  Add to My Program 
Rank Minimization Using Sums of Squares of Nonnegative Matrices 
Sadati, Nasser  Sharif Univ. of Tech. 
Isvand Yousefi, Mansoor  Sharif Univ. of Tech. 
Keywords: Computational methods, Optimization, LMIs
Abstract: Recently, moment dual approach of sums of squares relaxations developed for polynomial optimization problems was successfully extended to optimization problems with polynomial matrix inequality constraints. In this paper , we first derive an efficient polynomial formulization for matrix rank minimization problem which does not add any slack variable or additional equality or inequality constraint. Using the aforementioned theory, then we propose a hierarchy of convex LMI relaxations to provide a sequence of increasingly tight lower bounds on the global minimum rank of an arbitrary matrix under linear and polynomial matrix inequality constraints. Surprisingly enough, these lower bounds are usually exact and the optimal rank is often attained at early stages of the algorithm, make it possible to extract all optimal minimzers using CurtoFialkow flat extension results. Special issues on complexity, implementation and numerical results are also properly addressed.


15:3017:30, Paper WeIP02.23  Add to My Program 
Ellipsoidal Toolbox (ET) 
Kurzhanskiy, Alex A.  Univ. of California, Berkeley 
Varaiya, Pravin P.  Univ. of California at Berkeley 
Keywords: Computational methods, Uncertain systems, Linear systems
Abstract: Ellipsoidal Toolbox is the first free MATLAB package that implements the operations of ellipsoidal calculus: geometric (Minkowski) sums and differences of ellipsoids, intersections of ellipsoids, and ellipsoids with hyperlanes and polyhedra. The toolbox uses ellipsoidal methods to compute forward and backward reach sets of continuous and discretetime piecewise affine systems. Forward and backward reach sets can be also computed for continuoustime piecewise linear systems with disturbances.


15:3017:30, Paper WeIP02.24  Add to My Program 
Reverse Search for Parametric Linear Programming 
Jones, Colin Neil  Swiss Federal Inst. of Tech. 
Maciejowski, Jan M.  Univ. of Cambridge 
Keywords: Computational methods, Predictive control for linear systems, Optimization algorithms
Abstract: This paper introduces a new enumeration technique for (multi)parametric linear programs (pLPs) based on the reversesearch paradigm. We prove that the proposed algorithm has a computational complexity that is linear in the size of the output (number of socalled critical regions) and a constant space complexity. This is an improvement over the quadratic and linear computational and space complexities of current approaches. Current implementations of the proposed approach become faster than existing methods for large problems. Extensions of this method are proposed that make the computational requirements lower than those of existing approaches in all cases, while allowing for efficient parallelisation and bounded memory usage.


15:3017:30, Paper WeIP02.25  Add to My Program 
Generalized Linear Quadratic Control Theory 
Gattami, Ather  Lund Univ. 
Keywords: Stochastic optimal control, Optimal control, Agents and autonomous systems
Abstract: We consider the problem of stochastic control under power constraints. Problems such as linear quadratic optimal control with information constraints are instances of the problem considered. The calculations of the optimal control law can be done offline as in the classical linear quadratic Gaussian control theory using dynamic programming, which turns out to be a special case of the new theory developed in this paper.
