Computing Lyapunov functions for strongly asymptotically stable differential inclusions (original) (raw)
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Linear programming based Lyapunov function computation for differential inclusions
Discrete & Continuous Dynamical Systems - B, 2011
We present a numerical algorithm for computing Lyapunov functions for a class of strongly asymptotically stable nonlinear differential inclusions which includes spatially switched systems and systems with uncertain parameters. The method relies on techniques from nonsmooth analysis and linear programming and constructs a piecewise affine Lyapunov function. We provide necessary background material from nonsmooth analysis and a thorough analysis of the method which in particular shows that whenever a Lyapunov function exists then the algorithm is in principle able to compute it. Two numerical examples illustrate our method.
CPA Lyapunov Functions: Switched Systems vs. Differential Inclusions
Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics, 2020
We present an algorithm that uses linear programming to parameterize continuous and piecewise affine Lyapunov functions for switched systems. The novel feature of the algorithm is, that it can compute Lyapunov functions for switched system with a strongly asymptotically stable equilibrium, for which the equilibrium of the corresponding differential inclusion is merely weakly asymptotically stable. For the differential inclusion no such Lyapunov function exists. This is achieved by removing constraints from a linear programming problem of an earlier algorithm to compute Lyapunov functions, that are not necessary to assert strong stability for the switched system. We demonstrate the benefits of this new algorithm sing Artstein's circles as an example.
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In this paper we consider the problem of practical stability for differential inclusions. We prove the necessary and sufficient conditions using Lyapunov functions. Then we solve the practical stability problem of linear differential inclusion with ellipsoidal righthand part and ellipsoidal initial data set. In the last section we apply the main result of this paper to the problem of practical stabilization.
Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions
Systems & Control Letters, 2006
We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in . To prove the result, we consider an optimal control problem which consists in finding the "most unstable" trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. This property is obtained as a special case of a reachability result by bang-bang controls which is of independent interest. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.
General Solution of Stability Problem for Plane Linear Switched Systems and Differential Inclusions
IEEE Transactions on Automatic Control, 2000
Characterization and control of stability of switched dynamical systems and differential inclusions have attracted significant attention in the recent past. The most of the current results for this problem are obtained by application of the Lyapunov function method which provides sufficient but frequently over conservative stability conditions. For planar systems, practically verifiable necessary and sufficient conditions are found only for switched systems with two subsystems. This paper provides explicit necessary and sufficient conditions for asymptotic stability of switched systems and differential inclusions with arbitrary number of subsystems; these conditions turned out to be identical for the both classes of systems. A precise upper bound for the number of switching points in a periodic solution, corresponding to the break of stability, is found. It is shown that, for a switched system, the break of stability may also occur on a solution with infinitely fast switching (chattering) between some two subsystems.
Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions
Nonlinear Analysis: Theory, Methods & Applications, 2012
The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by the means of the proximal subdifferential of the nominal functions while primal conditions are described in terms of the Dini directional derivative. We also propose a unifying review of many other criteria given in the literature. Our approach is based on advanced tools of variational analysis and generalized differentiation.
R-composition of quadratic Lyapunov functions for stabilizability of linear differential inclusions
2010
A novel state feedback control technique to stabilise linear differential inclusions through composite quadratic Lyapunov functions is presented. By using a gradient-based control technique, the minimum effort control is composed through intersection and union operations, derived from the theory of R-functions. While conventional min and max compositions are recovered as a special case, it is shown that smoother sublevel sets and everywhere differentiability are obtained tuning the composition parameter. Examples of both intersection and union compositions are provided to show that intermediate control performances in terms of convergence time are obtained, while improved performances in the control signal can be achieved.
Local nonsmooth Lyapunov pairs for first-order evolution differential inclusions
The general theory of Lyapunov's stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in a previous work . This new contribution focuses on the natural case when the maximally monotone operator governing the given inclusion has a domain with nonempty interior. This setting permits to have nonincreasing Lyapunov functions on the whole trajectory of the solution to the given differential inclusion. It also allows some more explicit criteria for Lyapunov's pairs. Some consequences to the viability of closed sets are given, as well as some useful cases relying on the continuity or/and convexity of the involved functions. Our analysis makes use of standard tools from convex and variational analysis.
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1996
New constructive criteria of asymptotic stability of selector-linear differential inclusions are established. The wellknown absolute stability problem is also considered as a particular case of the above problem. Asymptotically stable inclusions are very similar in properties to linear stable time-invariant systems. This similarity concerns the wide range of dynamic properties. In particular, asymptotic stability of selector-linear differential inclusions has an exponential type and a response of the system to a bounded action is bounded. It turns out that there exist periodic motions at the boundary of asymptotic stability region for two-and three-dimensional systems. In the general case of 71-dimensional systems the periodic motions proved to exist out of the closure of the asymptotic stability region. This property is the basis for the new criteria having the form of algebraic conditions. It is necessary to note that differential inclusions are particularly attractive for adequate description of dynamic systems with incomplete information.
Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions
Symmetry
In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system.