An LMI approach to stabilization of linear discrete-time periodic systems

The problem of stabilization of linear discrete-time periodic systems is considered. LMI based conditions for stabilization via static periodic state feedback as well as via static periodic output feedback are presented. In the case of state feedback, the conditions are necessary and sufficient whereas for output feedback the result is only sufficient as it depends on the particular state-space representation used to describe the system. The problem of quadratic stabilization in the presence of either norm-bounded or polytopic parameter uncertainty is also treated. As an application of the output feedback stabilization technique, we consider the problem of designing a stabilizing (respectively, quadratically stabilizing) static periodic output feedback controller for linear time-invariant discrete-time systems which are not stabilizable (respectively, quadratically stabilizable) by static constant output feedback.     

An LMI approach to stabilization of linear port-controlled Hamiltonian systems

In this paper, it is shown that controllers for stabilizing linear port-controlled Hamiltonian (PCH) systems via interconnection and damping assignment can be obtained by solving a set of linear matrix inequalities (LMIs). Two sets of (almost) equivalent LMIs are proposed. In the first set, the interconnection and damping matrices do not appear explicitly, which makes it more difficult to directly manipulate those matrices. By requiring the system to have no uncontrollable pole at s=0, the second set of LMIs, explicitly containing the interconnection and damping matrices, can be obtained. Taking into account the physical properties of the system, some prespecified structures can be imposed directly on those matrices.     

Non-smooth simultaneous stabilization of nonlinear systems: An interpolation method

In this paper, we introduce a method which enables us to construct a continuous simultaneous stabilizer for pairs of systems in which cannot be simultaneously stabilized by means of C¹ feedback. We extend this method to higher-dimensional systems and show that any pair of asymptotically stabilizable nonlinear systems can be simultaneously stabilized (not asymptotically) by means of continuous feedback.     

Transitivity in simultaneous stabilization for a family of time-varying systems

This paper develops necessary and sufficient conditions for the so-called transitivity and strong transitivity (precise definitions later) for a family of discrete time-varying linear systems, which when satisfied lead to characterizations for simultaneous stabilizability and simultaneously stabilizing feedback controllers. Some criteria for the strong transitivity are established in terms of single coprime factorizations of only one plant and one controller, the resulting characterization involves coprime factorizations of fewer systems compared to previous work.     

Solution to the Generalized Champagne Problem on simultaneous stabilization of linear systems

The well-known Generalized Champagne Problem on simultaneous stabilization of linear systems is solved by using complex analysis and Blondel’s technique. We give a complete answer to the open problem proposed by Patel et al., which auto-matically includes the solution to the original Champagne Problem. Based on the recent development in automated inequality-type theorem proving, a new stabiliz-ing controller design method is established. Our numerical examples significantly improve the relevant results in the literature.     

线性系统的同时镇定问题

线性系统的同时镇定问题,是系统与控制理论中的基本问题,有着广泛的理论意义和应用价值.本文介绍了线性系统同时镇定问题的研究现状和最新进展.首先回顾了同时镇定问题的研究内容、基本方法及相关结果.其次,从理论求解和控制器设计的角度对线性系统同时镇定研究中著名的"香槟问题","比利时巧克力问题"和"威士忌问题"进行了分析探讨,并基于不等式型定理机器证明理论,给出了线性系统同时镇定问题的相关工作.最后指出了对于同时镇定多个线性系统方面的若干研究方向.     

Reliable decentralized stabilization via extended linear matrix inequalities and constrained dissipativity

This paper considers the problem of reliable decentralized stabilization with multicontroller configurations when some of the controllers are faulty in the sense that they fail to act optimally or do not function in the way that they were originally intended to function. Specifically, we introduce a solution concept that requires controllers to respond optimally (i.e. in the sense of best‐response correspondences) to the nonfaulty controllers regardless of the identity or actions of the faulty controllers. At any time, we assume that the nonfaulty controllers know only that there can be at most one faulty controller in the system, but they know neither the identity of the faulty controller nor how this faulty controller behaves. We present a design framework using an extended linear matrix inequality technique for deriving reliable stabilizing state‐feedback gains; whereas a set of filters whose estimation‐error dynamics satisfy certain quadratic integral constraints is used as decentralized observers within the subsystems for extending the result to the output‐feedback case. Moreover, a sufficient condition for solvability of the problem is provided in terms of the minimum‐phase condition of the subsystems. We also present an application of the results to a power system problem. Copyright © 2013 John Wiley & Sons, Ltd.     

基于LMI的分布时滞系统输出动态反馈镇定

研究分布时滞系统的输出动态反馈镇定问题. 基于闭环系统的中立型变换及相应Lyapunov-Krasovskii泛函的构造与解析技巧, 建立了与时滞相关的控制器存在性判据. 在此基础上通过控制器参数化设计方法, 将控制器参数的求解归结为线性矩阵不等式解的形式. 仿真算例验证了方法的有效性     

LMI approach to state-feedback stabilization of multidimensional systems

An LMI approach to design an appropriate state feedback controller for linear multidimensional discrete systems in Roesser's state-space model structure is described. A numerical example is employed to illustrate the results developed in the paper and the numerical advantages of using LMI techniques.     

Robust satisfactory fault-tolerant control of uncertain linear discrete-time systems: an LMI approach

Fault-tolerant control is an important issue in practical systems. Based on satisfactory control and estimation theory, a passive fault-tolerant control strategy is proposed for a class of uncertain linear discrete-time systems in this article. Manipulating linear matrix inequality (LMI) technique, robust fault-tolerant state-feedback controllers are designed which take the possible actuator faults and sensor faults into consideration, respectively. The closed-loop systems are guaranteed by the designed controllers to meet the required constraints on regional pole index φ(q, r), steady-state variance matrix X index and control-cost function V ²(u) index simultaneously. Then, whether possible faults occur or not, the closed-loop systems would maintain the three desirable performance indices accordingly. Meanwhile, the consistency of the performance indices mentioned earlier is also discussed for fault-tolerant control.     

Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances

We consider the problem of finding an ellipsoidal reachable set that bounds the states of time-delayed linear systems with bounded peak disturbances. The considered time-delay is time-varying but it has an upper bound in magnitude and rate. Based on the modified Lyapunov-Krasovskii(L-K) type functional, we derive a delay-dependent result expressed in the form of matrix inequalities containing only one non-convex scalar. The result is extended to uncertain time-delayed linear systems. Finally, we show the usefulness of our result by a comparative numerical example.     

Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design

We derive conditions for quadratic stabilizability of linear networked control systems by dynamic output feedback and communication protocols. These conditions are used to develop a simultaneous design of controllers and protocols in terms of matrix inequalities. The obtained protocols do not require knowledge of controller and plant states but only of the discrepancies between current and the most recently transmitted values of nodes’ signals, and are implementable on controller area networks. We demonstrate on a batch reactor example that our design guarantees quadratic stability with a significantly smaller network throughput than previously available designs.     

积分二次约束线性系统的H∞控制器设计:一种LMI方法

基于动态耗散理论, 给出了具有积分二次约束线性系统稳定和具有L₂有限增益的充分条件. 导出了积分二次约束线性系统H_∞ 状态反馈和动态输出反馈降阶控制器的设计方法, 并转化为多个线性矩阵不等式, 利用LMI工具箱求解.     

Simultaneous pole-placement/stabilization of pairs of linear time-invariant plants using a continuous-time periodic controller

Given a pair of single input single output (SISO), linear time-invariant (LTI), and strictly proper plants of relative order r, this paper employs a continuous-time periodic controller to achieve 1) simultaneous pole-placement for r = 1 and 2) guaranteed simultaneous stabilization for r = 2; 3, and 4, which jobs LTI controllers cannot, in general, do. The controller also ensures insignificant output ripples. The analysis is based on averaging principle. The computational steps for controller synthesis are linear algebraic in nature. An example illustrates the design procedure.     

Robust stabilization of switched discrete-time systems with actuator saturation

This paper studies the robust stabilization problem of switched discrete-time linear systems subject to actuator saturation. New switched saturation-dependent Lyapunov functions are exploited to design a robust stabilizing state feedback controller that maximizes an estimation of the domain of attraction. The design problem of controller （coefficient matrices） is then reduced to an optimization problem with linear matrix inequality （LMI） constraints. A numerical example is given to show the effectiveness of the proposed method.     

Semi-global stabilization of linear systems subject to output saturation

It is established that a SISO linear stabilizable and detectable system subject to output saturation can be semi-globally stabilized by linear output feedback if all its invariant zeros are in the closed left-half plane, no matter where the open loop poles are. This result complements a recent result that such systems can always be globally stabilized by discontinuous nonlinear feedback laws, and can be viewed as dual to a well-known result: a linear stabilizable and detectable system subject to input saturation can be semi-globally stabilized by linear output feedback if all its poles are in the open left-half plane, no matter where the invariant zeros are.     

Semiglobal stabilization of linear systems using constrained controls: a parametric optimization approach

A bounded feedback control for asymptotic stabilization of linear systems is derived. The designed control law increases the feedback gain as the controlled trajectory converges towards the origin. A sequence of invariant sets of decreasing size, associated with a (quadratic) Lyapunov function, are defined and related to each of them, the corresponding possible highest gain is chosen, while maintaining the input bounded. Gains as functions of the position are designed by explicitly solving a c-parameterized programming problem. The proposed method allows global asymptotic stabilization of open-loop stable systems, with inputs subject to magnitude bounds and globally bounded rates. In the general case of linear systems that are asymptotic null controllable with bounded input, the semiglobal stabilization is also addressed taking into account the problem of semiglobal rate-limited actuators. The method is illustrated with the global stabilization of an inertial navigator, and the stabilization of a nonlinear model of a crane with hanging load. Copyright © 1999 John Wiley & Sons, Ltd.     

Simultaneous stabilizability of three linear systems is rationally undecidable

We show that the simultaneous stabilizability of three linear systems, that is the question of knowing whether three linear systems are simultaneously stabilizable, is rationally undecidable. By this we mean that it is not possible to find necessary and sufficient conditions for simultaneous stabilization of the three systems in terms of expressions involving the coefficients of the three systems and combinations of arithmetical operations (additions, subtractions, multiplications, and divisions), logical operations (and and or), and sign test operations (equal to, greater than, greater than or equal to,...).