Robust analysis and synthesis for the gradient-like behaviour of uncertain pendulum-like systems

The problem of passivity analysis finds important applications in many signal processing systems such as digital quantizers, decision feedback equalizers and digital and analog filters. This paper considers the passivity analysis problem for a large class of systems which involve uncertain parameters, time delays, quantization errors, and unmodeled high order dynamics. By characterizing these and many other types of uncertainty using a general tool called integral quadratic constraints (IQCs), we present a solution to the problem of robust passivity analysis. More specifically, we determine if a given uncertain system is robustly passive. The solution is given in terms of the feasibility of a linear matrix inequality (LMI) which can be solved efficiently.

Dichotomy, or monostability, is one of the most important properties of nonlinear dynamic systems. For a dichotomous system, the solution of the system is either unbounded or convergent to a certain equilibrium, thus periodic or chaotic states cannot exist in the system. In this paper, a new methodology for the analysis of dichotomy of a class of nonlinear systems is proposed, and a linear matrix inequality (LMI)-based criterion is derived. The results are then extended to uncertain systems with real convex polytopic uncertainties in the linear part, and the LMI representation for robust dichotomy allows the use of parameter-dependent Lyapunov function. Based on the results, a dynamic output feedback controller guaranteeing robust dichotomy is designed, and the controller parameters are explicitly expressed by a set of feasible solutions of corresponding linear matrix inequalities. An extended Chua's circuit with two nonlinear resistors is given at the end of the paper to demonstrate the validity and applicability of the proposed approach. It is shown that by investigating the convergence of the bounded oscillating solutions of the system, our results suggests a viable and effective way for chaos control in nonlinear circuits.

In this paper, we investigate the problem of robust exponential stability analysis for uncertain linear systems with discrete interval time-varying delay, distributed time-varying delay and nonlinear perturbations. Based on constructing an augmented Lyapunov–Krasovskii functional with some parameter, decomposition technique of coefficient matrix, mixed model transformation with Leibniz–Newton formula and utilization of zero equations, new delay-range-dependent robust exponential stability criteria are derived in terms of linear matrix inequalities (LMIs). Numerical examples are given to show the superiority of our results to those in the literature.

The problem of passivity analysis finds important applications in many signal processing systems such as digital quantizers, decision feedback equalizers, and digital and analog filters. Equally important is the problem of passification, where a compensator needs to be designed for a given system to become passive. This paper considers these two problems for a large class of systems that involve uncertain parameters, time delays, quantization errors, and unmodeled high-order dynamics. By characterizing these and many other types of uncertainty using a general tool called integral quadratic constraints (IQCs), we present solutions to the problems of robust passivity analysis and robust passification. More specifically, for the analysis problem, we determine if a given uncertain system is passive for all admissible uncertainty satisfying the IQCs. Similarly, for the problem of robust passification, we are concerned with finding a loop transformation such that a particular part of the uncertain signal processing system becomes passive for all admissible uncertainty. The solutions are given in terms of the feasibility of one or more linear matrix inequalities (LMIs), which can be solved efficiently.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper addresses robust performance analysis problems of linear time-invariant (LTI) systems affected by real parametric uncertainties. These problems, known also as a special class of structured singular value computation problems, are inherently intractable (NP-hard problems). As such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions, where linear matrix inequality (LMI) plays an important. Nevertheless, since LMI-based conditions are expected to be conservative in general, it is often the case that we cannot conclude anything if the LMI at hand turns out to be infeasible. This motivates us to consider the dual of the LMI and examine the structure of the dual solution. By pursuing this direction, in this paper, we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained. In particular, a set of uncertain parameters that violates the specified performance can be computed. These results come from block-moment matrix structure of the dual variable, which is consistent with the recent results on polynomial optimization. This particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained. </para>

Robust analysis and control for a class of uncertain nonlinear discrete-time systems

The stability and performance robustness analysis for a class of uncertain nonlinear systems with bounded structured uncertainties are characterized in terms of various types of nonlinear matrix inequalities (NLMIs). As in the linear case, scalings or multipliers are used to find Lyapunov functions that give sufficient conditions, and the resulting NLMIs yield convex feasibility problem. For these problems, robustness analysis is essentially no harder than stability analysis of the system with no uncertainty. Sufficient conditions for the solvability of related robust synthesis problems are developed in terms of NLMIs as well. >

The problem of robust performance analysis for the delta operator formulated uncertain systems is considered. Sufficient conditions for D-stability robustness of the unified delta operator systems with structured uncertainty are presented in terms of matrix measure approach, which the desired closed-loop poles are constrained to locate within a prespecified D-region. The proposed results can unify related conclusions of continuous time and discrete time systems into the delta operator framework.

Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays

Robust passivity analysis for stochastic impulsive neural networks with leakage and additive time-varying delay components

The authors consider the problem of robust stability analysis and stabilization of uncertain two-dimensional (2-D) discrete systems described by the Fornasini-Marchesini second local state-space (LSS) model. Sufficient conditions for the uncertain 2-D discrete systems to be robustly stable are given in terms of linear matrix inequalities (LMI's). The LMI approach is also introduced to solve the robust stabilization problem of the uncertain 2-D systems via static-state feedback. Furthermore, sufficient conditions for the 2-D uncertain systems to be free of overflow oscillations under a saturation arithmetic are established.

This paper investigates the robust stability analysis and control synthesis of discrete-time switched systems with norm-bounded uncertainty under arbitrary switching by linear matrix inequality (LMI) approach together with switched Lyapunov function method. First, the equivalence between the existence of a switched Lyapunov function and two different matrix inequalities is established. These two matrix inequality conditions can be further proven to be equivalent to two LMI-based conditions. Then, some LMI-based conditions for stabilization via switched static state feedback as well as switched static output feedback are presented. In the case of state feedback, the conditions are necessary and sufficient, whereas for output feedback, the conditions are only sufficient as they depend on particular state-space representation used to describe the switched systems.

A delay decomposition approach to robust stability analysis of uncertain systems with time-varying delay

This paper investigates the problem of robust delay-dependent stability analysis of uncertain neutral delay systems. Based on a new approach which suggests very closed lower bound of Jensen's inequality by the use of Wirtinger inequality, an extended inequality is proposed. Moreover, by considering norm bounded uncertainties in system matrices, a robust stability criterion of uncertain neutral delay systems is studied. Two numerical examples are given to describe the less conservatism of the proposed methods.

Robust controller synthesis attracts reviving research interest, driven by\nthe rise of learning-based systems where uncertainty and perturbation are\nubiquitous. Facing an uncertain situation, a robustly stabilizing controller\nshould maintain stability while operating under a perturbed system deviating\nfrom its nominal specification. There have been numerous results for robust\ncontroller synthesis in multiple forms and with various goals, including\nmu-synthesis, robust primal-dual Youla, robust input-output, and robust system\nlevel parameterizations. However, their connections with one another are not\nclear, and we lack a general approach to robust controller analysis and\nsynthesis.\n To serve this purpose, we derive robust stability conditions for general\nsystems and formulate the general robust controller synthesis problem. The\nconditions hinge on the realization-stability lemma, a recent analysis tool\nthat unifies existing controller synthesis methods. Not only can the conditions\ninfer a wide range of existing robust results, but they also lead to easier\nderivations of new ones. Together, we demonstrate the effectiveness of the\nconditions and provide a unified approach to robust controller analysis and\nsynthesis.\n