A “mixed” small gain and passivity theorem in the frequency domain

We show that the negative feedback interconnection of two causal, stable, linear time-invariant systems, with a “mixed” small gain and passivity property, is guaranteed to be finite-gain stable. This “mixed” small gain and passivity property refers to the characteristic that, at a particular frequency, systems in the feedback interconnection are either both “input and output strictly passive”; or both have “gain less than one”; or are both “input and output strictly passive” and simultaneously both have “gain less than one”. The “mixed” small gain and passivity property is described mathematically using the notion of dissipativity of systems, and finite-gain stability of the interconnection is proven via a stability result for dissipative interconnected systems.     

Small-gain theorem for ISS systems and applications

We introduce a concept of input-to-output practical stability (IOpS) which is a natural generalization of input-to-state stability proposed by Sontag. It allows us to establish two important results. The first one states that the general interconnection of two IOpS systems is again an IOpS system if an appropriate composition of the gain functions is smaller than the identity function. The second one shows an example of gain function assignment by feedback. As an illustration of the interest of these results, we address the problem of global asymptotic stabilization via partial-state feedback for linear systems with nonlinear, stable dynamic perturbations and for systems which have a particular disturbed recurrent structure.The final version of this work was finished when Z.-P. Jiang held a visiting position at I.N.R.I.A., Sophia Antipolis.     

非线性系统的有界增益分解

本文对一类典型结构的反馈控制系统，讨论了对象的有界增益稳定的右互质分解和闭环系统有界增益稳定性的关系，对存在有界增益右互质分解的系统，给出了一种镇定方案。并且研究了这种的鲁棒性。     

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.     

Boundedness properties of nonlinear quasi-dissipative systems

Boundedness results for feedback interconnections of quasi-dissipative systems are presented. A classical result due to Hill and Moylan is generalized to the case of finite power gain stability of feedback interconnection of quasi-dissipative systems. It is also shown that under the assumption of strong finite-time detectability, such an interconnection has uniformly ultimately bounded trajectories for any uniformly bounded input.     

Passivity and stability of switched systems: A multiple storage function method

This paper presents a concept of passivity for switched systems using multiple storage functions. This passivity property is invariant under compatible feedback interconnection. Branicky's stability theorem of multiple Lyapunov functions is generalized by relaxing the non-increasing condition on values of Lyapunov-like functions. Using this result we show that a passive switched system is stable in the sense of Lyapunov. Moreover, asymptotic stability is reached if all subsystems are asymptotically detectable.     

On integral input-to-state stability for a feedback interconnection of parameterised discrete-time systems

This paper addresses integral input-to-state stability (iISS) for a feedback interconnection of parameterised discrete-time systems involving two subsystems. Particularly, we give a construction for a smooth iISS Lyapunov function for the whole system from the sum of nonlinearly weighted Lyapunov functions of individual subsystems. Motivations for such a construction are given. We consider two main cases. The first one investigates iISS for the whole system when both subsystems are iISS. The second one gives iISS for the interconnected system when one of subsystems is allowed to be input-to-state stable. The approach is also valid for both discrete-time cascades and a feedback interconnection of iISS and static systems. Examples are given to illustrate the effectiveness of the results.     

Contraction-based stabilisation of nonlinear singularly perturbed systems and application to high gain feedback

Recent development of contraction theory-based analysis has opened the door for inspecting differential behaviour of singularly perturbed systems. In this paper, a contraction theory-based framework is proposed for stabilisation of singularly perturbed systems. The primary objective is to design a feedback controller to achieve bounded tracking error for both standard and non-standard singularly perturbed systems. This framework provides relaxation over traditional quadratic Lyapunov-based method as there is no need to satisfy interconnection conditions during controller design algorithm. Moreover, the stability bound does not depend on smallness of singularly perturbed parameter and robust to additive bounded uncertainties. Combined with high gain scaling, the proposed technique is shown to assure contraction of approximate feedback linearisable systems. These findings extend the class of nonlinear systems which can be made contracting.     

Resilient decentralized stabilization of interconnected time‐delay systems with polytopic uncertainties

Decentralized delay‐dependent local stability and resilient feedback stabilization methods are developed for a class of linear interconnected continuous‐time systems. The subsystems are time‐delay plants which are subjected to convex‐bounded parametric uncertainties and additive feedback gain perturbations while allowing time‐varying delays to occur within the local subsystems and across the interconnections. The delay‐dependent local stability conditions are established at the subsystem level through the construction of appropriate Lyapunov–Krasovskii functional. We characterize decentralized linear matrix inequalities (LMIs)‐based delay‐dependent stability conditions by deploying an injection procedure such that every local subsystem is delay‐dependent robustly asymptotically stable with an γ‐level ??₂‐gain. Resilient decentralized state‐feedback stabilization schemes are designed, which takes into account additive gain perturbations such that the family of closed‐loop feedback subsystems enjoys the delay‐dependent asymptotic stability with a prescribed γ‐level ??₂‐gain for each subsystem. The decentralized feedback gains are determined by convex optimization over LMIs. All the developed results are tested on representative examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Strong iISS is preserved under cascade interconnection

In a recent paper, we have introduced the notion of Strong iISS as a compromise between the strength of input-to-state stability (ISS) and the generality of integral ISS (iISS). In this note, we continue the investigations around this property by studying its behavior in an interconnection context. In particular, we show that the cascade of Strongly iISS systems is itself Strongly iISS and we recall some useful tools to study Strongly iISS systems in feedback interconnection.     

Robust control of decomposable LPV systems

This paper considers distributed control of a class of interconnected systems, namely decomposable linear parameter-varying (LPV) systems, which include multi-agent systems with LPV agent models and switching communication topology as a special case. Sufficient conditions for stability are established for uncertain time-invariant as well as for time-varying interconnection topologies in a known set. Recent work on distributed state feedback controller synthesis is extended to robust output feedback controller synthesis. Here robustness refers to variations in the topology as well as the LPV dynamics of the subsystems.     

Necessary and sufficient conditions for strong stability of distributed parameter systems

In this paper we propose definitions for strong stabilizability and strong detectability of infinite-dimensional control systems. We show that these definitions are appropriate by showing that they can be used to give necessary and sufficient conditions for the strong stability of a system in terms of its input-output stability. As an application, we discuss the strong stability of the feedback interconnection of two strongly stabilizable and strongly detectable systems.     

On output feedback stabilization of Euler-Lagrange systems with nondissipative forces

This paper provides a solution to the problem of output feedback stabilization of systems described by Euler-Lagrange equations perturbed by nondissipative forces. This class of forces appears in some applications where we must take into account the interaction of the system with its environment. The nonlinear dependence on the unmeasurable part of the state and the loss of the fundamental passivity property render most of the existing results on stabilization of nonlinear systems unapplicable to this problem. The technique we use consists of finding a dynamic output feedback controller and a nonlinear change of coordinates such that the closed loop can be decomposed as a cascade of an asymptotically stable system and an input-to-state stable system. This should be contrasted with the well-known passivity-based technique that aims at a feedback interconnection of passive systems. We believe this design methodology to be of potential applicability to other stabilization problems where passivity arguments are unapplicable.     

一类线性离散时滞大系统的分散镇定

用一组线性矩阵不等式给出一类线性离散时滞大系统分散能镇定的一个充分条件,进而,通过建立和求解一个凸优化问题,提出了具有较反馈增益参数的分散稳定化状态反馈控制律的设计方法,所得到的控制器不仅使得闭环境系统是稳定的,而且还可以使得闭环系统状态具有给定的衰减度。     

Stability Robustness of a Feedback Interconnection of Systems With Negative Imaginary Frequency Response

A necessary and sufficient condition, expressed simply as the dc loop gain (i.e., the loop gain at zero frequency) being less than unity, is given in this note to guarantee the internal stability of a feedback interconnection of linear time-invariant (LTI) multiple-input multiple-output systems with negative imaginary frequency response. Systems with negative imaginary frequency response arise, for example, when considering transfer functions from force actuators to colocated position sensors, and are commonly important in, for example, lightly damped structures. The key result presented here has similar application to the small-gain theorem, which refers to the stability of feedback interconnections of contractive gain systems, and the passivity theorem, which refers to the stability of feedback interconnections of positive real (or passive) systems. A complete state-space characterization of systems with negative imaginary frequency response is also given in this note and also an example that demonstrates the application of the key result is provided.     

Observer-based output feedback control of networked control systems with non-uniform sampling and time-varying delay

This paper investigates the problem of observer-based output feedback control for networked control systems with non-uniform sampling and time-varying transmission delay. The sampling intervals are assumed to vary within a given interval. The transmission delay belongs to a known interval. A discrete-time model is first established, which contains time-varying delay and norm-bounded uncertainties coming from non-uniform sampling intervals. It is then converted to an interconnection of two subsystems in which the forward channel is delay-free. The scaled small gain theorem is used to derive the stability condition for the closed-loop system. Moreover, the observer-based output feedback controller design method is proposed by utilising a modified cone complementary linearisation algorithm. Finally, numerical examples illustrate the validity and superiority of the proposed method.     

Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity

The stability analysis for nonlinear differential-algebraic systems is addressed using tools from classical control theory. Sufficient stability conditions relying on matrix inequalities are established via Lyapunov Direct Method. In addition, a novel interpretation of differential-algebraic systems as feedback interconnection of a purely differential system and an algebraic system allows reducing the stability analysis to a small-gain-like condition. The study of stability properties for constrained mechanical systems, for a class of Lipschitz differential-algebraic systems and for an academic example is used to illustrate the theory.      

采用还原方法的不确定关联时滞系统的鲁棒分散镇定

针对一类具有多输入时滞项及互联时滞项的不确定关联系统，提出了系统可鲁棒分散镇定的充分条件，即一组线性矩阵不等式(LMI)有解．系统的不确定性是未知时变且范数有界的，基于还原方法及LMI技术给出系统设计状态反馈分散控制器的方法．该控制器保证闭环系统全局渐近稳定，且设计简单，计算量小，易于工程实现．最后通过仿真例子说明了该方法的有效性．     

An ISS small gain theorem for general networks

We provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled input-to-state stable systems. For this result the interconnection gains are described in a nonlinear gain matrix, and the small gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than 1. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated lower-dimensional discrete time dynamical system.     

Markov跳跃非线性系统逆最优增益设计

证明了一类严格反馈Markov跳跃系统是依概率输入–状态可稳定的.其次,证明了逆最优增益设计问题可解的一个充分条件是存在一组满足小控制量的依概率输入–状态稳定控制李雅普诺夫函数.最后,利用积分反推方法,给出了严格反馈Markov跳跃系统逆最优增益设计问题的一个构造性解.其中,为了克服由于Markov跳跃引起的耦合项所带来的困难,所设计的李雅普诺夫函数以及控制器是与模态无关的.