Matrosov's theorem using a family of auxiliary functions: an analysis tool to aid time-varying nonlinear control design (original) (raw)

Smooth time-varying stabilization of driftless systems over communication channels

Systems & Control Letters, 2006

We address the problem of smooth time-varying stabilization of port-interconnected driftless passive systems. The benchmark that we study is reminiscent of driftless systems interconnected over communication channels and constitutes a generalization of the well-known chained-form nonholonomic systems. Our contribution consists in proposing a smooth time-varying controller that guarantees uniform global asymptotic stability; moreover, a necessary condition is also stated. Both the sufficient and necessary conditions are formulated in terms of the so-called -persistency of excitation previously used in set-point control of nonholonomic systems. The proof of sufficiency relies on a recently reported extended Matrosov's theorem which may be regarded as an extension of Krasovskȋi-La Salle's invariance principle to the case of nonautonomous systems.

A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems

IEEE Transactions on Automatic Control, 2000

A new infinitesimal sufficient condition is given for uniform global asymptotic stability (UGAS) for time-varying nonlinear systems. It is used to show that a certain relaxed persistency of excitation condition, called uniform -persistency of excitation (U -PE), is sufficient for uniform global asymptotic stability in certain situations. U -PE of the right-hand side of a time-varying differential equation is also shown to be necessary under a uniform Lipschitz condition. The infinitesimal sufficient condition for UGAS involves the inner products of the flow field with the gradients of a finite number of possibly sign-indefinite, locally Lipschitz Lyapunov-like functions. These inner products are supposed to be bounded by functions that have a certain nested, or triangular, negative semidefinite structure. This idea is reminiscent of a previous idea of Matrosov who supplemented a Lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is "definitely nonzero" where the derivative of the Lyapunov function is zero. For this reason, we call the main result a nested Matrosov theorem. The utility of our results on stability analysis is illustrated through the well-known case-study of the nonholonomic integrator.

Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem

2007

The classical Matrosov theorem concludes uniform asymptotic stability of time varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semidefinite derivative along solutions) and another auxiliary function with derivative that is strictly non-zero where the derivative of the Lyapunov function is zero [M1]. Recently, several generalizations of the classical Matrosov theorem that use a finite number of Lyapunovlike functions have been reported in [LPPT2]. None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) that is a very useful analysis and controller design tool for nonlinear systems. We provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy inequalities that have a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function based controller redesign. We illustrate our results by constructing a strong Lyapunov function for a simple Euler-Lagrange system controlled by an adaptive controller.

Exponential stabilization of nonholonomic chained systems

IEEE Transactions on Automatic Control, 1995

This paper presents a feedback control scheme for the stabilization of two-input, driftless, chained nonholonomic systems, also called chained form. These systems are controllable but not asymptotically stabilizable by a smooth static-state feedback control law. In addition, exponential stability cannot be obtained with a smooth, time-varying feedback control law. Here, global, asymptotical stability with exponential convergence is achieved about any desired configuration by using a nonsmooth, time-varying feedback control law. The control law depends, in addition to the state and time, on a function which is constant except at predefined instants of time where the function is recomputed as a nonsmooth function of the state. The inputs are differentiable with respect to time and tend exponentially toward zero. For use in the analysis, a lemma on the exponential convergence of a stable time-varying nonlinear system perturbed by an exponentially decaying signal is presented. Simulation results are also shown.

Nonsmooth control-Lyapunov functions

Proceedings of 1995 34th IEEE Conference on Decision and Control, 1995

It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finitedimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative that has been studied in set-valued analysis and the theory of differential inclusions with various names such as "upper contingent derivative." This result generalizes to the non-smooth case the theorem of Artstein relating closed-loop feedback stabilization to smooth CLF's. It relies on viability theory as well as optimal control techniques. A "non-strict" version of the results, analogous to the LaSalle Invariance Principle, is also provided.

A strict Lyapunov function for non-holonomic systems under persistently-exciting controllers

IFAC-PapersOnLine

We study the stability of a non linear time-varying skew symmetric systemṡ x = A(t, x)x with particular structures that appear in the study problems of non holonomic systems in chained form as well as adaptive control systems. Roughly, under the condition that each non diagonal element of A(t, x) is persistently exciting or uniform δ persistently exciting with respect x. Although some stability results are known in this area, our main contribution lies in the construction of Lyapunov functions that allows a computation of convergence rate estimates for the class of non linear systems under study.

Control of nonholonomic systems using reference vector fields

IEEE Conference on Decision and Control and European Control Conference, 2011

This paper presents a control design methodology for n-dimensional nonholonomic systems. The main idea is that, given a nonholonomic system subject to κ Pfaffian constraints, one can define a smooth, N-dimensional reference vector field F, which is nonsingular everywhere except for a submanifold containing the origin. The dimension N ≤ n of F depends on the structure of the constraint equations, which induces a foliation of the configuration space. This foliation, together with the objective of having the system vector field aligned with F, suggests a choice of Lyapunov-like functions V . The proposed approach recasts the original nonholonomic control problem into a lower-dimensional output regulation problem, which although nontrivial, can more easily be tackled with existing design and analysis tools. The methodology applies to a wide class of nonholonomic systems, and its efficacy is demonstrated through numerical simulations for the cases of the unicycle and the n-dimensional chained systems, for n = 3, 4.

Lyapunov stability theory of nonsmooth systems

IEEE Transactions on Automatic Control, 1994

This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajec• tories. Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right-hand side such as in nonsmooth dynamic systems or variable structure control.