Computing Lyapunov functions for strongly asymptotically stable differential inclusions (original) (raw)
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.
On practical stability of differential inclusions using Lyapunov functions
Discrete and Continuous Dynamical Systems - Series B, 2017
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.
Stability and robustness of homogeneous differential inclusions
2016 IEEE 55th Conference on Decision and Control (CDC), 2016
The known results on asymptotic stability of homogeneous differential inclusions with negative homogeneity degrees and their accuracy in the presence of noises and delays are extended to arbitrary homogeneity degrees. Discretization issues are considered, which include explicit and implicit Euler integration schemes. Computer simulation illustrates the theoretical results.
2014
The objective of the research is to develop a general method of constructing Lyapunov functions for non-linear non-autonomous differential inclusions described by ordinary differential equations with parameters. The goal has been attained through the following ideas and tools. First, three-point Poincare strategy of the investigation of differential equations and manifolds has been used. Second, the geometric-topological structure of the non-linear non-autonomous parametric differential inclusions has been presented and analyzed in the framework of hierarchical fiber bundles. Third, a special canonizing transformation of the differential inclusions that allows to present them in special canonical form, for which certain standard forms of Lyapunov functions exist, has been found. The conditions establishing the relation between the local asymptotical stability of two corresponding particular integral curves of a given differential inclusion in its initial and canonical forms are ascertained. The global asymptotical stability of the entire free dynamical systems as some restrictions of a given parametric differential inclusion and the whole latter one per se has been investigated in terms of the classificational stability of the typical fiber of the meta-bundle. There have discussed the prospects of development and modifications of the Lyapunov second method in the light of the discovery of the new features of Lyapunov functions.
Input / output stability of systems with switched dynamics and outputs
Systems & Control Letters, 2008
This paper provides the extension of results on input-to-output stability properties of switched systems to switched systems whose dynamics are described by forced differential inclusions and whose outputs are obtained via switched set-valued maps. Lyapunov characterizations of these input/output stability properties, obtained in terms of certain conceptual output functions, are also presented.
2014
The objective of the research is to develop a general method of constructing Lyapunov functions for non-linear non-autonomous differential inclusions described by ordinary differential equations with parameters. The goal has been attained through the following ideas and tools. First, three-point Poincare's strategy of the investigation of differential equations and manifolds has been used. Second, the geometric-topological structure of the non-linear non-autonomous parametric differential inclusions has been presented and analyzed in the framework of hierarchical fiber bundles. Third, a special canonizing transformation of the differential inclusions that allows to present them in special canonical form, for which certain standard forms of Lyapunov functions exist, has been found. The conditions establishing the relation between the local asymptotical stability of two corresponding particular integral curves of a given differential inclusion in its initial and canonical forms are ascertained. The global asymptotical stability of the entire free dynamical systems as some restrictions of a given parametric differential inclusion and the whole latter one per se has been investigated in terms of the classificational stability of the typical fiber of the metabundle. There have discussed the prospects of development and modifications of the Lyapunov second method in the light of the discovery of the new features of Lyapunov functions.
Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators
Journal of Mathematical Analysis and Applications, 2018
We give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state. These sets are not necessarily weakly closed as in [5, 6], while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for a-Lyapunov pairs of lower semi-continuous (not necessarily weakly-lsc) functions associated to these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.