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.     

Robust Performance Controller Synthesis Based on Discrete‐Time Noncausal Linear Periodically Time‐Varying Scaling

This paper is concerned with the technique called discrete‐time noncausal linear periodically time‐varying (LPTV) scaling for robust stability analysis and synthesis. It is defined through the lifting treatment of discrete‐time systems, and naturally leads to a sort of noncausal operation of signals. In the robust stability analysis of linear time‐invariant (LTI) systems, it has been shown that even static noncausal LPTV scaling induces some frequency‐dependent scaling when it is interpreted in the context of lifting‐free treatment. This paper first discusses in detail different aspects of the effectiveness of noncausal LPTV scaling, with the aim of showing its effectiveness in controller synthesis. More precisely, we study the robust performance controller synthesis problem, where we allow the controllers to be LPTV. As in the LTI robust performance controller synthesis problem, we tackle our problem with an iterative method without guaranteed convergence to a globally optimal controller. Despite such a design procedure, the closed‐loop H_∞ performance is expected to improve as the period of the controller is increased, and we discuss how the frequency‐domain properties of noncausal LPTV scaling could contribute to such improvement. We demonstrate with a numerical example that an effective LPTV controller can be designed for a class of uncertainties for which the well‐known μ‐synthesis fails to derive even a robust stabilization controller.     

Noncausal linear periodically time-varying scaling for robust stability analysis of discrete-time systems: Frequency-dependent scaling induced by static separators

This article is concerned with robust stability analysis of discrete-time systems and introduces a novel and powerful technique that we call noncausal linear periodically time-varying (LPTV) scaling. Based on the discrete-time lifting together with the conventional but general scaling approach, we are led to the notion of noncausal LPTV scaling for LPTV systems, and its effectiveness is demonstrated with a numerical example. To separate the effect of noncausal and LPTV characteristics of noncausal LPTV scaling to see which is a more important source leading to the effectiveness, we then consider the case of LTI systems as a special case. Then, we show that even static noncausal LPTV scaling has an ability of inducing frequency-dependent scaling when viewed in the context of the conventional LTI scaling, while causal LPTV scaling fails to do so. It is further discussed that the effectiveness of noncausal characteristics leading to the frequency-domain interpretation can be exploited even for LPTV systems by considering the νN-lifted transfer matrices of N-periodic systems.     

Separator-type robust stability theorem of sampled-data systems allowing noncausal LPTV scaling

This paper derives a necessary and sufficient condition for robust stability of sampled-data systems, which is stated by using the notion of separators that are dealt with in an operator-theoretic framework. Such operator-theoretic treatment of separators provides a new perspective, which we call noncausal linear periodically time varying scaling and leads to reducing conservativeness in robust stability analysis. A numerical example is given to demonstrate the results.     

Robustness of discrete periodically time-varying control under LTIunstructured perturbations

Investigates robustness of linear periodically time varying (LPTV) control of discrete linear time invariant (LTI) plants subject to LTI unstructured perturbations. The note first derives a necessary and sufficient condition for robust stability of an LPTV system subject to LTI perturbations, which is less conservative than the well known small gain condition. It then presents a quantitative analysis on the robustness of LPTV control under LTI unstructured perturbations in comparison with that of LTI control. It is shown that under the normal value of the controller period suggested in the previous literature, the stability margin is deteriorated by LPTV control if LTI unstructured perturbations are considered. Hence LTI control is superior to LPTV control in this respect     

时滞切换不确定神经网络系统的指数稳定性

本文研究了具有无穷时滞切换不确定细胞神经网络(UCNNs)系统任意切换下的指数稳定性.利用同胚映射和M-矩阵理论,得到UCNNs系统平衡点存在性,唯一性和指数稳定性的充分条件;利用Lyapunov泛函方法,研究了时滞切换UCNNs系统任意切换下的鲁棒指数稳定性,并得到确保系统全局指数稳定的充分条件.     

Stable inversion of LPTV systems with application in position-dependent and non-equidistantly sampled systems

Many control applications, including feedforward and learning control, involve the inverse of a dynamical system. For nonminimum-phase systems, the response of the inverse system is unbounded. For linear time-invariant (LTI), nonminimum-phase systems, a bounded, noncausal inverse response can be obtained through an exponential dichotomy. For generic linear time-varying (LTV) systems, such a dichotomy does not exist in general. The aim of this paper is to develop an inversion approach for an important class of LTV systems, namely linear periodically time-varying (LPTV) systems, which occur in, e.g. position-dependent systems with periodic tasks and non-equidistantly sampled systems. The proposed methodology exploits the periodicity to determine a bounded inverse for general LPTV systems. Conditions for existence are provided. The method is successfully demonstrated in several application cases, including position-dependent and non-equidistantly sampled systems.     

Linear periodically time-varying discrete-time systems: Aliasing and LTI approximations

Linear periodically time-varying (LPTV) systems are abundant in control and signal processing; examples include multirate sampled-data control systems and multirate filter-bank systems. In this paper, several ways are proposed to quantify aliasing effect in discrete-time LPTV systems; these are associated with optimal time-invariant approximations of LPTV systems using operator norms.     

THE PERFORMANCE OF DISCRETE LINEAR TIME VARYING CONTROL OF LINEAR PERIODIC PLANTS

This paper analyzes the performance of discrete linear time varying (LTV) control of discrete linear periodically time varying (LPTV) plants for l₂ disturbance rejection. It extends the results of [11–13] for linear periodic controllers to general LTV control of LPTV plants. It is shown that LPTV control provides strictly better control performance than linear strictly time varying control for LPTV plants. The analysis is carried out in frequency domain. This approach provides not only new results on disturbance rejection of LTV control but also some new insight into properties of general LTV systems.     

Trajectory-based design of robust non-fragile controllers for a class of uncertain linear continuous-time systems

This paper deals with a design problem of robust non-fragile stabilizing controllers for a class of uncertain linear continuous-time systems. The proposed design approach of robust non-fragile controllers is based on computation of the trajectory for the uncertain linear system and differs from the existing methods based on quadratic stabilization via Lyapunov criterion. In this paper, we show that sufficient conditions for asymptotical stability of the linear system with uncertainties and controller gain variations, and a LMI-based design algorithm of a robust non-fragile controller. Furthermore, we extend the result to the design problem of decentralized robust non-fragile controllers for a class of uncertain large-scale interconnected systems. Finally, the effectiveness of the proposed design scheme of robust non-fragile controllers is shown through illustrative examples.     

Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming

A wide variety of stability and performance questions about linear dynamical systems can be reformulated as convex optimization problems involving linear matrix inequalities (LMIs). These techniques have been recently extended to nonlinear systems with polynomial or rational dynamics through the use of sum of squares (SOS) programming.In this paper we further extend the class of systems that can be analyzed with convexity-based methods. We show how to analyze the robust stability properties of uncertain nonlinear systems with polynomial or rational dynamics, via contraction analysis and SOS programming. Since the existence of a global contraction metric is a sufficient condition for global stability of an autonomous system, we develop an algorithm for finding such contraction metrics using SOS programming. The search process is made computationally tractable by relaxing matrix definiteness constraints, the feasibility of which indicates the existence of a contraction metric, to SOS constraints on polynomial matrices. We illustrate our results through examples from the literature and show how our contraction-based approach offers advantages when compared with traditional Lyapunov analysis.     

Passivity-based control of nonlinear flexible multibody systems

In this paper, global asymptotic stability of a class of nonlinear multibody flexible space structures under certain dissipative compensation is established. Furthermore, for an important subclass of such systems, the stability is shown to be robust to certain types of actuator and sensor nonlinearities. The results are applicable to robust stabilization of a wide class of systems, including flexible space structures and manipulators with articulated flexible appendages. The stability proofs use the Lyapunov approach and exploit the inherent passivity of such systems     

Semi‐global stabilization via linear sampled‐data output feedback for a class of uncertain nonlinear systems

This paper develops a systematic design scheme to construct a linear sampled‐data output feedback controller that semi‐globally asymptotically stabilizes a class of uncertain systems with both higher‐order and linear growth nonlinearities. To deal with the uncertain coefficients in the systems, a robust state feedback stabilizer and a reduced‐order sampled‐data observer, both in the linear form, are constructed and then integrated together. The semi‐global attractivity and local stability are delicately proved by carefully selecting a scaling gain using the output feedback domination approach and a sampling period sufficiently small to restrain the state growth under a zero‐order‐holder input. Copyright © 2014 John Wiley & Sons, Ltd.     

Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints

Stability analysis and control of linear impulsive systems is addressed in a hybrid framework, through the use of continuous-time time-varying discontinuous Lyapunov functions. Necessary and sufficient conditions for stability of impulsive systems with periodic impulses are first provided in order to set up the main ideas. Extensions to the stability of aperiodic systems under minimum, maximum and ranged dwell-times are then derived. By exploiting further the particular structure of the stability conditions, the results are non-conservatively extended to quadratic stability analysis of linear uncertain impulsive systems. These stability criteria are, in turn, losslessly extended to stabilization using a particular, yet broad enough, class of state-feedback controllers, providing then a convex solution to the open problem of robust dwell-time stabilization of impulsive systems using hybrid stability criteria. Relying finally on the representability of sampled-data systems as impulsive systems, the problems of robust stability analysis and robust stabilization of periodic and aperiodic uncertain sampled-data systems are straightforwardly solved using the same ideas. Several examples are discussed in order to show the effectiveness and reduced complexity of the proposed approach.     

Piecewise quadratic Lyapunov functions over conical partitions for robust stability analysis

In this paper, the robust stability problem for uncertain linear continuous‐time systems is faced making use of piecewise quadratic Lyapunov functions (PQLF). PQLF are obtained by partitioning the state space into polyhedral cones and by associating a quadratic form with each cone. The proposed formulation allows us to recover, as particular cases of PQLFs, not only the class of quadratic functions but also the class of polyhedral functions. In this way, we manage to show the universality of the class of PQLF for the robust stability problem. The main contribution of the paper is the formulation of a low‐computational cost procedure for the stability analysis of uncertain linear systems. Several numerical examples are included in the paper, where the proposed approach is tested on some benchmark cases taken from the literature. Comparisons with existing methods show that the proposed method performs better under several aspects. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust kernel Isomap

Isomap is one of widely used low-dimensional embedding methods, where geodesic distances on a weighted graph are incorporated with the classical scaling (metric multidimensional scaling). In this paper we pay our attention to two critical issues that were not considered in Isomap, such as: (1) generalization property (projection property); (2) topological stability. Then we present a robust kernel Isomap method, armed with such two properties. We present a method which relates the Isomap to Mercer kernel machines, so that the generalization property naturally emerges, through kernel principal component analysis. For topological stability, we investigate the network flow in a graph, providing a method for eliminating critical outliers. The useful behavior of the robust kernel Isomap is confirmed through numerical experiments with several data sets.     

Recursive scaling design for robust global nonlinear stabilization via output feedback

This paper proposes a novel approach to robust backstepping for global stabilization of uncertain nonlinear systems via output feedback. The design procedure developed in this paper is based on the concept of state‐dependent scaling, which handles output‐feedback stabilization problems of strict‐feedback systems with various structures of uncertainties in a unified way. The proposed method is suitable for numerical computation. The theory of the method employs the Schur complements formula instead of Young's inequality and completing the squares. This paper shows a condition of allowable uncertainty size under which an uncertain system is globally stabilized by output feedback. A class of systems is shown to be always globally stabilizable for arbitrarily large nonlinear size of uncertainties. A recursive procedure of robust observer design for such a class of uncertain systems is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Global finite-time stabilization for a class of switched nonlinear systems via output feedback

This paper addresses the problem of global finite-time stabilization for a class of uncertain switched nonlinear systems via output feedback under arbitrary switchings. Based on the adding a power integrator approach, we design a homogeneous observer and controller for the nominal switched system without the perturbing nonlinearities. Then, a scaling gain is introduced into the proposed output feedback stabilizer to implement global finite-time stability of the closed-loop system. In addition, the proposed approach can be also extended to a class of switched nonlinear systems with upper-triangular growth condition. Two examples are given to illustrate the effectiveness of the proposed method.     

Properties of linear switching time-varying discrete-time systems with applications

We study two discrete-time, linear switching time-varying (LSTV) structures, each of which consists of a periodic switch connected to several linear time-invariant (LTI) systems. Such structures can be used to represent any linear periodically time-varying (LPTV) systems. We give basic properties associated with the LSTV structures in terms of their LTI building blocks, and then apply the results to solve a general approximation problem: How to optimally approximate an LPTV system with period p by an LPTV system with period ? The optimality is measured using norms. The study is extended to general multirate periodic systems.