Robust control of a class of sampled-data systems with LTI uncertainties

The robust stability and robust performance in sampled-data control is studied for systems with LTI uncertainties. A class of problems is considered, which allows an exact characterization of robust stability of the sampled-data system in terms of the _H norm of a finite-dimensional discrete transfer function. In the problem class certain assumptions on the product of the uncertainty weight and the anti-aliasing prefilter are imposed. In return, the robustness conditions can be stated in a particularly compact form, in which the robust stability and performance problems for LTI perturbations reduce to standard finite-dimensional discrete robust stability and performance problems. The robust stability condition for LTI uncertainties is also compared to the robustness condition based on the small gain theorem. The analysis provides a method to estimate the conservatism of the small gain robustness condition directly in terms of the anti-aliasing prefilter and frequency weights. As a by-product of this analysis a non-trivial class of sampled-data systems is identified for which the small gain theorem gives a non-conservative condition for robust stability to norm-bounded LTI uncertainties.     

Robust stability and control of uncertain linear discrete-time periodic systems with time-delay

This paper deals with the problems of robust stability analysis and robust control of linear discrete-time periodic systems with a delayed state and subject to polytopic-type parameter uncertainty in the state-space matrices. A robust stability criterion independent of the time-delay length as well as a delay-dependent criterion is proposed, where the former applies to the case of a constant time-delay and the latter allows for a time-varying delay lying in a given interval. The developed robust stability criteria are based on affinely uncertainty-dependent Lyapunov–Krasovskii functionals and are given in terms of linear matrix inequalities. These stability conditions are then applied to solve the problems of robust stabilization and robust _H∞$H_{\infty }$  control via static periodic state feedback. Numerical examples illustrate the potentials of the proposed robust stability and control methods.     

Small-gain stability conditions for linear systems with time-varying delays

In this paper we show that a variety of stability conditions, both existing and new, can be derived for linear systems subject to time-varying delays in a unified manner in the form of scaled small-gain conditions. From a robust control perspective, our development seeks to cast the stability problem as one of robust stability analysis, and the resulting stability conditions are also reminiscent of robust stability bounds typically found in robust control theory. The development is built on the well-known conventional robust stability analysis, requiring essentially no more than a straightforward application of the small gain theorem. The derived conditions have conceptual appeal, and they can be checked using standard robust control toolboxes.     

Some new stability conditions for two-dimensional difference systems

This paper considers robust stability of two-dimensional (2D) difference systems in the Fornasini-Marchesini (F-M) state space setting. An approach based on the MacL aurine series expansion is presented. First, a new and less conservative sufficient condition for asymptotic stability of nominal 2D discrete systems is derived. Then sufficient conditions for robust stability of 2D discrete systems subjected to both structured and unstructured perturbations are given. Finally, new robust stability conditions for 2D discrete-delay systems are derived. Numerical examples are given to illustrate the results.     

Robust disturbance rejection in l optimal control systems

Given a class of plants formed by perturbing a nominal discrete-time linear shift-invariant plant with norm bounded unstructured perturbation, the problem of finding a single compensator that will stabilize all plants in this class and at the same time minimize the worst case norm of a certain performance measure is addressed. Assuming the performance measure of interest is the sensitivity function, first an expression for the supremum, over all plants in this class, of such a measure is derived for a system with a robustly stabilizing compensator. This is achieved by first showing that the proposed expression is an upper bound for the worst case sensitivity norm, and then by constructing a plant in the allowable class that will achieve this upper bound. Combined with already known necessary and sufficient conditions for robust stability, the derived expression provides an effective way of combining both robust stability and performance in one, easy to compute, measure. Secondly, a scheme for finding a controller that minimizes the worst case norm of the sensitivity function while achieving robust stability, is provided. This makes possible the incorporation of stability and performance robustness requirements in one design procedure.     

Robust stability analysis with cycling-based LPTV scaling: part I. Fundamental results on its relationship with lifting-based LPTV scaling

This paper is concerned with robust stability analysis of discrete-time linear periodically time-varying (LPTV) systems using the cycling-based LPTV scaling approach. To study the properties of this approach in comparison with the lifting-based LPTV scaling approach, we consider exploiting the framework of representing the associated robust stability conditions with infinite matrices. Since it serves as a common framework for comparing the two different LPTV scaling approaches, it provides us with new insights into the relationship between the cycling-based and lifting-based scaling approaches. In particular, we derive fundamental results that enable us to reduce the comparison, with respect to conservativeness in robust stability analysis, of the two scaling approaches with restricted and tractable classes of separators to a modified comparison of the associated classes of what we call infinite-dimensional separators arising in the above infinite matrix framework.     

Performance robustness of discrete-time systems with structureduncertainty

Given an interconnection of a nominal discrete-time plant and a stabilizing controller together with structured, norm-bounded, nonlinear/time-varying perturbations, necessary and sufficient conditions for robust stability and performance of the system are provided. It is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem. The resulting stability robustness problem is shown to be equivalent to a simple algebraic one, the solution of which provides the desired necessary and sufficient conditions for performance/stability robustness. These conditions provide an effective tool for robustness analysis and can be applied to a large class of problems. In particular, it is shown that some known results can be obtained immediately as special cases of these conditions     

Stability of a class of discrete-time nonlinear recursive observers

This work provides a framework for nominal and robust stability analysis for a class of discrete-time nonlinear recursive observers (DNRO). Given that the system has linear output mapping, local observability and Jacobian matrices satisfying certain conditions, the nominal and robust stability of the DNRO is defined by the property of estimation error dynamics and is analyzed using Lyapunov theory. Moreover, a simultaneous state and parameter estimation scheme is shown to be Input-to-State Stable (ISS), and adaptively reduce plant-model mismatch on-line. Three design strategies of the DNRO that satisfy the stability results are given as examples, including the widely used extended Kalman filter, extended Luenberger observer, and the fixed gain observer.     

Robust performance analysis with LMI-based methods for realparametric uncertainty via parameter-dependent Lyapunov functions

Robust performance analysis for linear time-invariant systems with linear fractional transformation real parametric uncertainty is considered. New conditions of robust stability/performance based on parameter-dependent Lyapunov functions are proposed. The robust stability/performance measures are: robust pole location, robust H_∞ performance and robust H₂ performance. Linear matrix inequality (LMI)-based sufficient conditions for the existence of parameter-dependent Lyapunov functions are derived by using the quadratic separation concept. The performances of the proposed conditions are compared with existing tests     

Robust stability analysis and stabilization for uncertain linear neutral delay systems

This paper considers the problem of robust stability analysis and robust stabilization for uncertain neutral delay systems. The system under consideration is subject to norm-bounded parameter uncertainty appearing in all the matrices of the state-space model. Both delay-dependent and -independent robust stability conditions are obtained in terms of linear matrix inequalities. Based on these, the corresponding conditions for the exsitence of robust stabilizing state feedback controllers are developed. When these conditions are feasible, the desired state feedback controller gain matrices can be obtained via convex optimization.     

IMPROVED CONDITIONS FOR DELAY‐DEPENDENT ROBUST STABILITY AND STABILIZATION OF UNCERTAIN DISCRETE TIME‐DELAY SYSTEMS

This paper provides improved delay‐dependent conditions for the robust stability and robust stabilization of discrete time‐delay systems with norm‐bounded parameter uncertainties. It is theoretically established that the proposed conditions are less conservative than those discussed in the literature. The new approach proposed in this paper in a derivation of delay‐dependent conditions and involves the use of neither model transformation nor bounding techniques for some cross terms. A numerical example is provided to demonstrate the reduced conservatism of the proposed conditions.     

Design of robust stable controls for nonlinear objects

We consider a problem of discrete control for a class of nonlinear time-varying objects. Only set estimations for object parameters are available. The aim is to design controls that ensure robust stability of closed-loop systems in a given domain of state space. Since the considered class of objects is large enough not to have a stabilizing control, the proposed design method has to verify at the last step if the obtained conditions of robust stability are satisfied for a nonlinear system “in a given domain.”     

一类不确定系统的鲁棒稳定性分析

研究了一类不确定系统的稳定性问题, 以线性矩阵不等式(LMI)的形式给出了该类系统的鲁棒稳定条件, 并将该结果推广到区间系统的稳定性分析, 提出了研究区间系统稳定性问题的新方法, 给出了线性矩阵不等式形式的稳定判据, 该判据将区间系统的稳定性问题转化为一类线性矩阵不等式的可解问题. 最后通过仿真算例验证了本文结果具有更小的保守性.     

Estimating the conservatism of Popov''s criterion for real parametric uncertainties

We consider the problems of robust stability and performance analysis of linear systems subject to parametric uncertainties. The Popov criterion provides lower bounds of stability margins and upper bounds on robust performance. In this paper we propose a method for obtaining upper bounds on stability margin or lower bounds on robust H₂ performance based on the outcome of the lower bound computation for the stability problem or the upper bound computation for the H₂ performance problem. We make several numerical tests.     

脉冲随机混合系统的稳定性与鲁棒稳定性分析

针对一类在切换时刻具有脉冲行为的Markov切换随机系统,首先,利用多Lyapunov函数的方法研究系统的稳定性,得到系统依概率稳定的充分条件,该条件以线性矩阵不等式(LMI)的形式给出;然后,进一步对系统的稳定化以及鲁棒稳定性问题进行分析与设计,给出相应的状态反馈增益矩阵和脉冲增益矩阵的求解方法;最后,数值算例说明了所设计方法的有效性.     

Stability analysis and gain-scheduled state feedback control for continuous-time systems with bounded parameter variations

The problems of robust stability analysis and state feedback control based on gain-scheduling for continuous-time systems with time-varying parameters that have bounded rates of variation and lie inside a polytope are addressed in this article. With respect to previous results in the literature, two main contributions of the article are: (i) the robust stability analysis conditions are less conservative and demand less computational effort than the existing ones; (ii) the conditions can be extended to cope with the problem of control design for this class of system. Parameter-dependent linear matrix inequality (LMI) conditions are given for the existence of a parameter-dependent Lyapunov function quadratic in the state and homogeneous polynomially of arbitrary degree in the parameter assuring robust stability. Two convex procedures based on LMIs exhibiting distinct complexities are proposed to solve the problem of robust stability. An extension to deal with the computation of a stabilising parameter-dependent state feedback gain for this class of time-varying systems is also provided, as a sequence of LMIs of increasing precision. Examples illustrate the results, including comparisons with other techniques from the literature.     

一类不确定性系统鲁棒H∞控制器

研究一类具有匹配不确定性系统鲁棒H∞控制问题,第一,基于矩阵不等式给出了二次稳定的条件并且对系统的H∞性能进行了分析;第二,给出了系统的鲁棒H∞控制器,该控制器不仅满足系统二次稳定的条件,而且也满足H∞性能约束条件;最后,数值算例说明了控制器的有效性和可行性。     

Fixed-architecture controller synthesis for systems with input–output time-varying nonlinearities

In this paper we develop a fixed-architecture controller analysis and synthesis framework that addresses the problem of multivariable linear time-invariant systems subject to plant input and plant output time-varying nonlinearities while accounting for robust stability and robust performance over the allowable class of nonlinearities. The proposed framework is based on the classical Luré problem and the related Aizerman conjecture concerning the stability of a feedback loop involving a sector-bounded nonlinearity. Specifically, we extend the classical notions of absolute stability theory to guarantee closed-loop stability of multivariable systems in the presence of input nonlinearities. In order to capture closed-loop system performance we also consider the minimization of a quadratic performance criterion over the allowable class of input nonlinearities. Our approach is directly applicable to systems with saturating actuators and provides full and reduced-order dynamic compensators with a guaranteed domain of attraction. The principal result is a set of constructive sufficient conditions for absolute stabilization characterized via a coupled system of algebraic Riccati and Lyapunov equations. The effectiveness of design approach is illustrated by several numerical examples. © 1997 John Wiley & Sons, Ltd.     

Analysis and Synthesis of Robust Control Systems Using Linear Parameter Dependent Lyapunov Functions

This note provides sufficient robust stability conditions for continuous time polytopic systems. They are obtained from the Frobenius-Perron Theorem applied to the time derivative of a linear parameter dependent Lyapunov function and are expressed in terms of linear matrix inequalities (LMI). They contain as special cases, various sufficient stability conditions available in the literature to date. As a natural generalization, the determination of a guaranteed H₂ cost is addressed. A new gain parametrization is introduced in order to make possible the state feedback robust control synthesis using parameter dependent Lyapunov functions through linear matrix inequalities. Numerical examples are included for illustration