Stability robustness bounds for discrete-time linear regulators

Stability robustness analysis and design for linear multivariable discrete-time systems with bounded uncertainties are discussed. Robust stability of the full-state feedback linear quadratic (LQ) regulator in the presence of perturbations (modelling errors) of the system matrices is investigated. These results are based on a recently developed bound on elemental (structured) time-varying perturbations of an asymptotically stable linear time-invariant discrete-time system. Lyapunov theory and singular value decomposition techniques are employed in deriving these bounds. Extensions of these results to linear stochastic systems with the Kalman filter as the stale estimator (LQG regulators) and to reduced-order dynamic compensator feedback are described. A state feedback control design method is presented for LQ regulators, using a quantitative measure called the Stability Robustness Index. Simple examples illustrate these new results.     

A riccati equation approach to the stabilization of uncertain linear systems

This paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems. The systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set. Furthermore, these uncertain parameters may be time-varying. The method used to establish asymptotic stability of the closed loop system (obtained when the feedback control is applied) involves the use of a quadratic Lyapunov function. The main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic Lyapunov function. Once the Lyapunov function has been obtained, it is used to construct the stabilizing feedback control law. The fundamental idea behind the algorithm presented involves constructing an upper bound for the Lyapunov derivative corresponding to the closed loop system. This upper bound is a quadratic form. By using this upper bounding procedure, a suitable Lyapunov function can be found by solving a certain matrix Riccati equation. Another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic Lyapunov function.     

Robust Output Feedback Stabilization of Switched Nonlinear Systems with Average Dwell Time

For some switched nonlinear systems, stabilization can be achieved under arbitrary switching with state feedback control. Due to switching zero dynamics, output feedback stabilization for some switched nonlinear systems needs dwell time between switching to guarantee system stability. In this paper, we consider a class of switched nonlinear systems with unknown parameters and unknown switching signals. We design a robust output feedback controller that stabilizes the system under a class of switching signals with average dwell time (ADT) where the value of ADT can be reduced by adjusting the control gain. For some special cases, common quadratic Lyapunov functions of the closed‐loop systems can be found and the value of ADT is further relaxed. Some examples and simulations are provided to validate the results.     

切换状态反馈控制系统的稳定性分析

研究了一类切换控制系统的稳定性分析问题,切换控制器由多个线性状态反馈控制器组成。运用Lyapunov方法得出了稳定性的时域判据,通过求解一个线性矩阵不等式的方法分析切换控制系统的稳定性,利用KYP引理得到了等价的频域判据;并且研究了一类含有参数不确定性的切换控制系统的鲁棒稳定性分析问题,给出了稳定性时域判据和频域判据。稳定性分析的主要结果是基于公共二次Lyapunov函数的,适用于切换系统任意切换的情形,该方法在工程实例中的应用表明了其有效性。     

State feedback design for input-saturating quadratic systems

This paper proposes a method to design stabilizing state feedback control laws for nonlinear quadratic systems subject to input saturation. Based on a quadratic Lyapunov function, a modified sector condition and a particular representation for the quadratic terms, synthesis conditions in a “quasi”-LMI form are stated in a regional (local) context. An LMI-based optimization problem is then derived for computing the state feedback gains maximizing the estimate of the stability region of the closed-loop system.     

Static Output Feedback Control for Interval Type‐2 T‐S Fuzzy Systems Based on Fuzzy Lyapunov Functions

This study aims to design an interval type‐2 (IT2) fuzzy static output feedback controller to stabilize the IT2 Takagi‐Sugeno (T‐S) fuzzy system. Conservative results may be obtained when a common quadratic Lyapunov function is utilized to investigate the stability of T‐S fuzzy systems. A fuzzy Lyapunov function is employed in this study to analyze the stability of the IT2 fuzzy closed‐loop system formed by the IT2 T‐S fuzzy model and the IT2 fuzzy static output feedback controller. Stability conditions in the form of linear matrix inequalities are derived. Several slack matrices are introduced to further reduce the conservativeness of stability analysis. The membership‐function shape‐dependent analysis approach is also employed to relax the stability results. The numerical examples illustrate the effectiveness of the proposed conditions.     

离散时滞切换系统的无记忆状态反馈镇定

针对一类子系统为离散时滞系统的切换系统,研究了稳定性与无记忆状态反馈镇定问题．采用多李雅普诺夫函数法,首先以线性矩阵不等式形式给出了在任意切换信号作用下离散时滞切换系统渐进稳定的一个充分性条件;然后给出了系统无记忆状态反馈镇定的控制器设计方案,并将结果推广到不确定离散时滞切换系统;最后用仿真算例验证了所提出设计方案的可行性。     

A training rule which guarantees finite-region stability for aclass of closed-loop neural-network control systems

A training method for a class of neural network controllers is presented which guarantees closed-loop system stability. The controllers are assumed to be nonlinear, feedforward, sampled-data, full-state regulators implemented as single hidden-layer neural networks. The controlled systems must be locally hermitian and observable. Stability of the closed-loop system is demonstrated by determining a Lyapunov function, which can be used to identify a finite stability region about the regulator point.     

An approach to stability criteria of neural-network control systems

This paper discusses stability of neural network (NN)-based control systems using Lyapunov approach. First, it is pointed out that the dynamics of NN systems can be represented by a class of nonlinear systems treated as linear differential inclusions (LDI). Next, stability conditions for the class of nonlinear systems are derived and applied to the stability analysis of single NN systems and feedback NN control systems. Furthermore, a method of parameter region (PR) representation, which graphically shows the location of parameters of nonlinear systems, is proposed by introducing new concepts of vertex point and minimum representation. From these concepts, an important theorem, which is useful for effectively finding a Lyapunov function, is derived. Stability criteria of single NN systems are illustrated in terms of PR representation. Finally, stability of feedback NN control systems, which consist of a plant represented by an NN and an NN controller, is analyzed.     

一类多输入级联非线性切换系统的全局镇定

研究一类带有部分线性系统的多输入级联非线性切换系统的全局镇定问题. 首先, 给出保证线性部分有一致规范型的充分条件. 其次, 利用一致规范型及其零动态的共同二次Lyapunov函数设计状态反馈使得线性部分在任意切换律下镇定. 最后, 通过构造共同Lyapunov函数能实现闭环系统在任意切换律下的全局渐近稳定性.     

Stability studies of control systems using a non-Luré type Lyapunov function

Stability of control systems with multiple nonlinearities is discussed. A non-Luré type Lyapunov function is presented, which surpasses the Luré-type function from the point of view of the stability region guaranteed. This function is used to establish a stability criterion for the system. The superiority of the function proposed is indicated by a numerical example, comparing the stability boundary to that obtained by a Luré-type Lyapunov function.     

Optimal infinite-horizon control and the stabilization of linear discrete-time systems: State-control constraints and nonquadratic cost functions

Stability results are given for a class of feedback systems arising from the regulation of time-invariant, discrete-time linear systems using optimal infinite-horizon control laws. The class is characterized by joint constraints on the state and the control and a general nonlinear cost function. It is shown that weak conditions on the cost function and the constraints are sufficient to guarantee asymptotic stability of the optimal feedback systems. Prior results, which concern the linear quadratic regulator problem, are included as a special case. The proofs make no use of discrete-time Riccati equations and linearity of the feedback law, hence, they are intrinsically different from past proofs.     

A dissipative dynamical systems approach to stability analysis of time delay systems

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite‐dimensional linear dynamical system and an infinite‐dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov–Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov–Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay. Copyright © 2004 John Wiley & Sons, Ltd.     

Conditions for stabilizability of linear switched systems

This paper investigates some conditions that can provide stabilizability for linear switched systems with polytopic uncertainties via their closed loop linear quadratic state feedback regulator. The closed loop switched systems can stabilize unstable open loop systems or stable open loop systems but in which there is no solution for a common Lyapunov matrix. For continuous time switched linear systems, we show that if there exists solution in an associated Riccati equation for the closed loop systems sharing one common Lyapunov matrix, the switched linear systems are stable. For the discrete time switched systems, we derive a Linear Matrix Inequality (LMI) to calculate a common Lyapunov matrix and solution for the stable closed loop feedback systems. These closed loop linear quadratic state feedback regulators guarantee the global asymptotical stability for any switched linear systems with any switching signal sequence.     

Suboptimal mean controllers for bounded and dynamic stochastic distributions

Based on the recently developed algorithms for the modelling and control of bounded dynamic stochastic systems (H. Wang, J. Zhang, Bounded stochastic distributions control for pseudo ARMAX stochastic systems, IEEE Transactions on Automatic control, 486–490), this paper presents the design of a subotpimal nonlinear mean controller for bounded dynamic stochastic systems with guaranteed stability. The B-spline functional expansion based square root model is used to represent the output probability density function of the system. This is then followed by the design of a mean controller of the output distribution of the system using nonlinear output tracking concept. A nonlinear quadratic optimization is performed using the well known Hamilton–Jacobi–Bellman equation. This leads to a controller which consists of a static unit, a state feedback part and an equivalent output feedback loop. In order to achieve high precision for the output tracking, the output feedback gain is determined by a learning process, where the Lyapunov stability analysis is performed to show the asymptotic stability of the closed loop system under some conditions. A simulation example is included to demonstrate the use of the algorithm and encouraging results have been obtained.     

The Exponential Stability for a Class of Hybrid Systems

The exponential stability with a nonsmooth Lyapunov function for a class of hybrid systems is studied in this paper. First, a sufficient condition is derived that has to be satisfied by the feedback control for the hybrid systems. Then, for the special case where the Lyapunov function involved is a kind of nonsmooth function, the maximum of finitely many smooth functions (for short, max‐type function), the stability of a hybrid system is considered, and a convenient criterion to determine the stability of the system is established. Finally, a numerical method of determining the control input value is developed.     

Static output feedback for discrete-time switched linear systems under arbitrary switching

This paper considers the problem of static output feedback (SOF) control for discrete-time switched linear systems under arbitrary switching. By the aid of switched quadratic Lyapunov functions combined with Finsler’s lemma, new sufficient conditions for SOF stabilization are obtained which guarantee a γ-performance of the closed-loop switched systems subject to input disturbances. The proposed method can work successfully in situations where the existing ones fail. Three examples are given to illustrate its effectiveness.     

Global quadratic stabilization of a class of nonlinear systems

The problem of quadratic stabilization for a class of nonlinear systems is examined in this paper. By employing a well-known Riccati approach, we develop a technique for designing a state feedback control law which quadratically stabilizes the system for all admissible uncertainties. This state feedback control law consists of linear and nonlinear feedback control terms. The linear feedback control term is generalized from a well-known H_∞ result, while the nonlinear term can be viewed as a correcting term for the presence of nonlinear bounded uncertainty. This stabilization result is extended to static output feedback and to systems for which the nonlinear uncertainty satisfies generalized matching conditions. Furthermore, we point out that in the presence of nonlinear uncertainty the global quadratic stability may be destroyed by some arbitrary small mismatched uncertainty in the matrix, and proceed to establish the region of semi-global quadratic stability of the controlled system. © 1998 John Wiley & Sons, Ltd.     

LMI-based model predictive control for switched nonlinear systems

This paper proposes an LMI approach to model predictive control of nonlinear systems with switching between multiple modes. In this approach, at each mode, the nonlinear system is divided to a linearized model in addition to a nonlinear term. A sum of squares (SOS) optimization problem is presented to find a quadratic bound for the nonlinear part. The stability condition of the switching system is obtained by using a discrete Lyapunov function and then the sufficient state feedback control law is achieved so that guarantees the stability of the system and also minimizes an infinite prediction horizon performance index. Moreover, two other LMI optimization problems are solved at each mode in order to find the maximum area region of convergence of the nonlinear system inscribed in the region of stability. The performance and effectiveness of the proposed MPC approach are illustrated by two case studies.     

反步法在三相电压型脉冲调宽逆变器控制中的应用

针对三相电压型脉冲调宽(pulse-width-modulation,PWM)逆变器这个时变的、多模态的非线性系统,应用开关函数建立了系统数学模型,引入开关周期平均算子将离散的系统化为连续系统,得到具有参数严格反馈形式的二阶非线性多输入多输出系统数学模型.并对此模型应用反向递推方法,通过逐步选择虚拟控制量和构造李雅普诺夫函数,使每个状态分量具有适当的渐近特性,实现整个系统在大扰动下的全局渐近稳定性,进而得到反馈控制律的一般表达式.将三相电压型PWM逆变器系统参数代入后得到其反馈控制规律.最后通过仿真实验验证了该方法的有效性和正确性.