Robust Stability Analysis of Nonlinear Hybrid Systems

We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems

Robust stability of time-varying uncertain systems with rational dependence on the uncertainty

Robust stability of time-varying uncertain systems is a key problem in automatic control. This note considers the case of linear systems with rational dependence on an uncertain time-varying vector constrained in a polytope, which is typically addressed in the literature by using the linear fractional representation (LFR). A novel sufficient condition for robust stability is derived in terms of a linear matrix inequality (LMI) feasibility test by exploiting homogeneous polynomial Lyapunov functions, the square matrix representation and an extended version of Polya's theorem which considers structured matrix polynomials. It is shown that this condition is also necessary for second-order systems, and that this condition is less conservative than existing LMI conditions based on the LFR for any order. © 2010 IEEE.published_or_final_versio

Computing upper-bounds of the minimum dwell time of linear switched systems via homogenous polynomial lyapunov functions

Regular Session - Switched Systems IIThis paper investigates the minimum dwell time for switched linear systems. It is shown that a sequence of upper bounds of the minimum dwell time can be computed by exploiting homogeneous polynomial Lyapunov functions and convex optimization based on LMIs. This sequence is obtained by adopting two possible representations of homogeneous polynomials, one based on Kronecker products, and the other on the square matrix representation. Some examples illustrate the use and the potentialities of the proposed approach.published_or_final_versionThe 2010 American Control Conference (ACC), Baltimore, MD., 30 June-2 July 2010. In Proceedings of the American Control Conference, 2010, p. 2487-249

Sufficient and Necessary LMI Conditions for Robust Stability of Rationally Time-Varying Uncertain Systems

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Toward non-conservative stability conditions for equilibrium points of genetic networks with SUM regulatory functions

An important problem in systems biology consists of establishing whether an equilibrium point of a genetic regulatory network is stable. This paper investigates this problem for genetic networks with SUMregulatory functions. It is shown that a sufficient condition for global asymptotical stability of an equilibrium point of these networks can be derived in terms of convex optimizations with LMI constraints by exploiting polynomial Lyapunov functions and SOS techniques. This condition is interesting because does not introduce approximations of the nonlinearities present in the genetic regulatory network, and the conservatism can be decreased by increasing the degree of the involved polynomials. ©2009 IEEE.published_or_final_versionThe Joint 48th IEEE Conference on Decision and Control and the 28th Chinese Control Conference (CDC/CCC 2009), Shanghai, China, 16-18 December 2009. In Proceedings of the IEEE Conference on Decision and Control, 2009, p. 5631-563

Computing equilibrium points of genetic regulatory networks

Computing equilibrium points of genetic regulatory networks is a problem of primary importance for numerous investigations in these systems. This paper addresses this problem for differential equation models, with the regulation function expressed in a general form which includes both SUM form and PROD form for saturation functions of any type. Specifically, a recursive algorithm is proposed, which provides at each recursion a region guaranteed to contain all equilibrium points. This region progressively shrinks, and asymptotically converges to the sought set of equilibrium points. Moreover, the proposed algorithm can also allow one to delimit and find limit cycles. Some numerical examples are reported to illustrate and validate the proposed algorithm, including examples where standard mathematical tools fail to compute the sought equilibrium points. © 2009 Springer Berlin Heidelberg.postprin

LMI techniques for optimization over polynomials in control: A survey

Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Plya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers. © 2006 IEEE.published_or_final_versio

Model Reference Fuzzy Adaptive PID Control and Its Applications in Typical Industrial Processes

To improve the dynamic response, regulation precision and robustness of the closed-loop system, a novel two degree of freedom control method called model reference fuzzy adaptive PID (MRFA-PID) control is proposed for industrial processes. The proposed control law consists of two parts, PID controller and fuzzy logic controller. The PID controller, which is designed for the nominal plant, guarantees the basic requirement on stability and product quality. The fuzzy logic controller, as an extra degree of freedom, improves the system dynamic performance, regulation precision, and robustness to the uncertainty of the system. The effectiveness of MRFA-PID control is illustrated by its applications in some typical industrial processes. Since the proposed method need not identify the uncertain parameters of the plant, it has a very good real-time performance, and is easy to be implemented on-line

Composite control Lyapunov functions for robust stabilization of constrained uncertain dynamical systems

This work presents innovative scientific results on the robust stabilization of constrained uncertain dynamical systems via Lyapunov-based state feedback control. Given two control Lyapunov functions, a novel class of smooth composite control Lyapunov functions is presented. This class, which is based on the R-functions theory, is universal for the stabilizability of linear differential inclusions and has the following property. Once a desired controlled invariant set is fixed, the shape of the inner level sets can be made arbitrary close to any given ones, in a smooth and non-homothetic way. This procedure is an example of ``merging'' two control Lyapunov functions. In general, a merging function consists in a control Lyapunov function whose gradient is a continuous combination of the gradients of the two parents control Lyapunov functions. The problem of merging two control Lyapunov functions, for instance a global control Lyapunov function with a large controlled domain of attraction and a local one with a guaranteed local performance, is considered important for several control applications. The main reason is that when simultaneously concerning constraints, robustness and optimality, a single Lyapunov function is usually suitable for just one of these goals, but ineffective for the others. For nonlinear control-affine systems, both equations and inclusions, some equivalence properties are shown between the control-sharing property, namely the existence of a single control law which makes simultaneously negative the Lyapunov derivatives of the two given control Lyapunov functions, and the existence of merging control Lyapunov functions. Even for linear systems, the control-sharing property does not always hold, with the remarkable exception of planar systems. For the class of linear differential inclusions, linear programs and linear matrix inequalities conditions are given for the the control-sharing property to hold. The proposed Lyapunov-based control laws are illustrated and simulated on benchmark case studies, with positive numerical results