Accelerating iterative algorithms for symmetric linear complementarity problems

We propose a method for accelerating iterative algorithms for solving symmetric linear complementarity problems. The method consists in performing a one-dimensional optimization in the direction generated by a splitting method even for non-descent directions. We give strong convergence proofs and present numerical experiments that justify using this acceleration.     

Fixed‐order robust H∞ control and control‐oriented uncertainty set shaping for systems with ellipsoidal parametric uncertainty

In this paper, sufficient conditions for robust output feedback controller design for systems with ellipsoidal parametric uncertainty are given in terms of solutions to a set of linear matrix inequalities. A polynomial method is employed to design a fixed‐order controller that assigns closed‐loop poles within a given region of the complex plane and that satisfies an H_∞ performance specification. The main feature of the proposed method is that it can be extended easily for control‐oriented uncertainty set shaping using a standard input design approach. Consequently, the results can be extended to joint robust control/input design procedure whose controller structure and performance specifications are translated into the requirements on the input signal spectrum used in system identification. This way, model uncertainty set can be tuned for the robust control design procedure. The simulation results show the effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Confidence regions of two-stage stochastic linear complementarity problems

Two-stage stochastic linear complementarity problems (TSLCP) model a large class of equilibrium problems subject to data uncertainty, and are closely related to two-stage stochastic optimization problems. The sample average approximation (SAA) method is one of the basic approaches for solving TSLCP and the consistency of the SAA solutions has been well studied. This paper focuses on building confidence regions of the solution to TSLCP when SAA is implemented. We first establish the error-bound condition of TSLCP and then build the asymptotic and nonasymptotic confidence regions of the solutions to TSLCP by error-bound approach, which is to combine the error-bound condition with central limit theory, empirical likelihood theory, and large deviation theory.     

New results on robust stability for differential-difference systems with affine linear parametric uncertainty

This article presents sufficient conditions to verify the robust stability property of convex combinations for quasipolynomials that represent the characteristic equation of differential-difference dynamics systems. It considers affine linear parametric uncertainty structure in the coefficients of quasipolynomials and also, interval uncertainty in the time delay. First of all, a transformation of the delay's operator is performed in order to get a two variable polynomial; after this, to obtain the robust stability property, a result based on the Hurwitz matrix is applied.     

Robust stability analysis of linear systems with parametric uncertainty

This article is concerned with the problem of robust stability analysis of linear systems with uncertain parameters. By constructing an equivalent system with positive uncertain parameters and using the properties of these parameters, a new stability analysis condition is derived. Due to making use of the properties of uncertain parameters, the new proposed method has potential to give less conservative results than the existing approaches. A numerical example is given to illustrate the effectiveness of the proposed method.     

Popov–Belevitch–Hautus type controllability tests for linear complementarity systems

It is well-known that checking certain controllability properties of very simple piecewise linear systems are undecidable problems. This paper deals with the controllability problem of a class of piecewise linear systems, known as linear complementarity systems. By exploiting the underlying structure and employing the results on the controllability of the so-called conewise linear systems, we present a set of inequality-type conditions as necessary and sufficient conditions for controllability of linear complementarity systems. The presented conditions are of Popov–Belevitch–Hautus type in nature.     

The distributionally robust complementarity problem

We investigate the linear complementarity problem with uncertain parameters (ULCP) which affect the linear mapping affinely or quadratically. Assuming that the distribution of the uncertain parameters belongs to some ambiguity set with prescribed partial information, we formulate the ULCP as a distributionally robust optimization reformulation named as the distributionally robust complementarity problem (DRCP), which minimizes the worst case of an expected complementarity measure with a joint chance constraint that the probability of the linear mapping being nonnegative is not less than a given level. Applying the cone dual theory and S-procedure, we conservatively approximate the DRCP as a nonlinear semidefinite programming (NSDP) with bilinear matrix inequalities, which can be solved by the NSDP solver PENLAB. The preliminary numerical test on a constrained stochastic linear quadratic control problem shows that the DRCP as well as the corresponding solution method is promising.     

Full‐complexity polytopic robust control invariant sets for uncertain linear discrete‐time systems

This paper presents an algorithm for the computation of full‐complexity polytopic robust control invariant (RCI) sets, and the corresponding linear state‐feedback control law. The proposed scheme can be applied for linear discrete‐time systems subject to additive disturbances and structured norm‐bounded or polytopic uncertainties. Output, initial condition, and performance constraints are considered. Arbitrary complexity of the invariant polytope is allowed to enable less conservative inner/outer approximations to the RCI sets whereas the RCI set is assumed to be symmetric around the origin. The nonlinearities associated with the computation of such an RCI set structure are overcome through the application of Farkas' theorem and a corollary of the elimination lemma to obtain an initial polytopic RCI set, which is guaranteed to exist under certain conditions. A Newton‐like update, which is recursively feasible, is then proposed to yield desirable large/small volume RCI sets.     

Robust stability and stabilization of discrete singular systems with interval time-varying delay and linear fractional uncertainty

The robust stability and robust stabilization problems for discrete singular systems with interval time-varying delay and linear fractional uncertainty are discussed. A new delay-dependent criterion is established for the nominal discrete singular delay systems to be regular, causal and stable by employing the linear matrix inequality (LMI) approach. It is shown that the newly proposed criterion can provide less conservative results than some existing ones. Then, with this criterion, the problems of robust stability and robust stabilization for uncertain discrete singular delay systems are solved, and the delay-dependent LMI conditions are obtained. Finally, numerical examples are given to illustrate the e?ectiveness of the proposed approach.     

How scaling of the disturbance set affects robust positively invariant sets for linear systems

This paper presents new results on robust positively invariant (RPI) sets for linear discrete‐time systems with additive disturbances. In particular, we study how RPI sets change with scaling of the disturbance set. More precisely, we show that many properties of RPI sets crucially depend on a unique scaling factor, which determines the transition from nonempty to empty RPI sets. We characterize this critical scaling factor, present an efficient algorithm for its computation, and analyze it for a number of examples from the literature. Copyright © 2017 John Wiley & Sons, Ltd.     

多胞约束LPV系统多面体不变集RMPC算法

针对一类输入和输出受约束且具有多胞结构的离散LPV 系统, 提出一种基于多面体不变集的鲁棒模型预测控制(RMPC) 算法. 选取一系列收敛于原点的离散状态点, 计算每个状态的反馈控制率, 构建相应的多面体不变集. 在每一个采样时刻, 确定包含当前状态的最小多面体不变集, 通过计算与相邻两个多面体不变集的位置关系, 执行连续的状态反馈控制率. 仿真结果表明, 相比椭圆不变集离线RMPC算法, 所提出算法扩大了系统的稳定区域, 取得了保守性较小的结果.

     

Robust performance analysis for systems with structured uncertainty

This paper analyses robust performance measures for linear time-invariant systems with norm-bounded time-varying structured uncertainty. We consider two robust performance measures. One is the worst-case peak value of the error signal in response to the disturbance with a known energy. The other is the worst-case energy of the error signal in response to impulsive disturbance. In both cases, the ‘worst case’ is taken over all admissible uncertainties and disturbances. The notion of robust stability is Q-stability, or the scaled H_∞ norm bound. Our main results provide an upper bound for each of the robust performance measures in terms of a positive definite matrix which satisfies a linear matrix inequality (LMI) together with a scaling matrix. Hence, the best bound in this LMI formulation can be computed by convex programming.     

Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1‐gain and L∞‐gain characterization

Copositive linear Lyapunov functions are used along with dissipativity theory for stability analysis and control of uncertain linear positive systems. Unlike usual results on linear systems, linear supply rates are employed here for robustness and performance analysis using L₁‐gain and L_∞‐gain. Robust stability analysis is performed using integral linear constraints for which several classes of uncertainties are discussed. The approach is then extended to robust stabilization and performance optimization. The obtained results are expressed in terms of robust linear programming problems that are equivalently turned into finite dimensional ones using Handelman's theorem. Several examples are provided for illustration. Copyright © 2012 John Wiley & Sons, Ltd.     

Identification and robust control for systems with ellipsoidal parametric uncertainty by convex optimization

In this paper, sufficient conditions for robust output feedback controller design for systems with ellipsoidal parametric uncertainty are given in terms of solutions to a set of linear matrix inequalities (LMIs) Performance specifications are in terms of combined pole placement with sensitivity function shaping in the H₂ or H_∞ norm. Furthermore, an optimal input design technique for parameter estimation that is integrated into the robust control design is employed in this paper. This means that performance specifications on the closed‐loop transfer functions are translated into the requirements on the input signal spectrum. The simulation results show the effectiveness of the proposed method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Filtering of discrete‐time Markov jump linear systems with uncertain transition probabilities

This article addresses the filtering design problem for discrete‐time Markov jump linear systems (MJLS) under the assumption that the transition probabilities are not completely known. We present the methods to determine ??₂‐ and ??_∞‐norm bounded filters for MJLS whose transition probability matrices have uncertainties in a convex polytope and establish an equivalence with the ones with partly unknown elements. The proposed design, based on linear matrix inequalities, allows different assumptions on Markov mode availability to the filter and on system parameter uncertainties to be taken into account. Under mode‐dependent assumption and internal model knowledge, observer‐based filters can be obtained and it is shown theoretically that our method outperforms some available ones in the literature to date. Numerical examples illustrate this claim. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust MPC for the constrained system with polytopic uncertainty

An improved method for synthesising the constrained robust model predictive controller is proposed in this study. It constructs a continuum of terminal constraint sets off-line, and achieves robust stability with a variable control horizon on-line from the very beginning and a time-varying terminal constraint set, by solving the min–max optimisation problem, which can be formulated as a linear matrix inequality problem. This algorithm not only dramatically reduces the on-line computation burden, but also guarantees the control performance by reserving at least one free control move in the whole process. Simulation results for the three-tank system with uncertain dynamic behaviour on flux coefficients are given.     

A test for stability robustness of linear time‐varying systems utilizing the linear time‐invariant ν‐gap metric

A stability robustness test is developed for internally stable, nominal, linear time‐invariant (LTI) feedback systems subject to structured, linear time‐varying uncertainty. There exists (in the literature) a necessary and sufficient structured small gain condition that determines robust stability in such cases. In this paper, the structured small gain theorem is utilized to formulate a (sufficient) stability robustness condition in a scaled LTI ν‐gap metric framework. The scaled LTI ν‐gap metric stability condition is shown to be computable via linear matrix inequality techniques, similar to the structured small gain condition. Apart from a comparison with a generalized robust stability margin as the final part of the stability test, however, the solution algorithm implemented to test the scaled LTI ν‐gap metric stability robustness condition is shown to be independent of knowledge about the controller transfer function (as opposed to the LMI feasibility problem associated with the scaled small gain condition which is dependent on knowledge about the controller). Thus, given a nominal plant and a structured uncertainty set, the stability robustness condition presented in this paper provides a single constraint on a controller (in terms of a large enough generalized robust stability margin) that (sufficiently) guarantees to stabilize all plants in the uncertainty set. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust controller design for input‐delayed systems using predictive feedback and an uncertainty estimator

This paper deals with the problem of stabilizing a class of input‐delayed systems with (possibly) nonlinear uncertainties by using explicit delay compensation. It is well known that plain predictive schemes lack robustness with respect to uncertain model parameters. In this work, an uncertainty estimator is derived for input‐delay systems and combined with a modified state predictor, which uses current available information of the estimated uncertainties. Furthermore, based on Lyapunov–Krasovskii functionals, a computable criterion to check robust stability of the closed‐loop is developed and cast into a minimization problem constrained to an LMI. Additionally, for a given input delay, an iterative‐LMI algorithm is proposed to design stabilizing tuning parameters. The main results are illustrated and validated using a numerical example with a second‐order dynamic system. Copyright © 2016 John Wiley & Sons, Ltd.     

A new approach to constrained state estimation for discrete‐time linear systems with unknown inputs

This paper addresses the problem of estimating the state for a class of uncertain discrete‐time linear systems with constraints by using an optimization‐based approach. The proposed scheme uses the moving horizon estimation philosophy together with the game theoretical approach to the filtering to obtain a robust filter with constraint handling. The used approach is constructive since the proposed moving horizon estimator (MHE) results from an approximation of a type of full information estimator for uncertain discrete‐time linear systems, named in short ‐MHE and –full information estimator, respectively. Sufficient conditions for the stability of the ‐MHE are discussed for a class of uncertain discrete‐time linear systems with constraints. Finally, since the ‐MHE needs the solution of a complex minimax optimization problem at each sampling time, we propose an approximation to relax the optimization problem and hence to obtain a feasible numerical solution of the proposed filter. Simulation results show the effectiveness of the robust filter proposed.     

Robust non‐fragile H ∞ ∕ L2 − L ∞ control of uncertain linear system with time‐delay and application to vehicle active suspension

This paper presents an approach to design robust non‐fragile H _∞ ∕ L₂ ? L _∞ static output feedback controller, considering actuator time‐delay and the controller gain variations, and it is applied to design vehicle active suspension. According to suspension design requirements, the H _∞ and L₂ ? L _∞ norms are used, respectively, to reflect ride comfort and time‐domain hard constraints. By employing a delay‐dependent Lyapunov function, existence conditions of delay‐dependent robust non‐fragile static output feedback H _∞ controller and L₂ ? L _∞ controller are derived, respectively, in terms of the feasibility of bilinear matrix inequalities. Then, a new procedure based on LMI optimization and a hybrid algorithm of the particle swarm optimization and differential evolution is used to solve an optimization problem with bilinear matrix inequality constraints. Simulation results show that the designed active suspension system still can guarantee their own performance in spite of the existence of the model uncertainties, the actuator time‐delay and the controller gain variations. Copyright © 2014 John Wiley & Sons, Ltd.