Construction of Lyapunov functions for nonlinear planar systems by linear programming

In this paper, the structure of several convex cones that arise in the study of Lyapunov functions are investigated. In particular, the cones associated with quadratc Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases some technical results are presented on the structure of individual cones and it is shown how these insights can lead to new results on the problem of common Lyapunov function existence. 

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.

We present a novel numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. Our proposed approach constructs a continuous piecewise affine (CPA) function given a suitable partition of the state space, called a triangulation, and values at the vertices of the triangulation. The vertex values are obtained from a Lyapunov function in a classical converse Lyapunov theorem and verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Furthermore, by refining the triangulation, we show that it is always possible to construct a CPA Lyapunov function. Numerical examples are presented demonstrating the effectiveness of the proposed method.

On the equivalence between global recurrence and the existence of a smooth Lyapunov function for hybrid systems

ON THE EXISTENCE OF LYAPUNOV FUNCTIONS IN GENERAL SYSTEMS

This paper deals with the stability of continuous-time 2D nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are extended to 2D nonlinear systems. To check the stability a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what proposed in this paper are the first results of this kind on stability of continuous-time 2D nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems is proposed. The proposed method is based on the sum of squares decomposition, therefore it formulates the Lyapunov function search algorithmically. In this way polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.

In this paper, the global asymptotic stability and strict passivity of three types of nonlinear RLC circuits are investigated by utilizing the Lyapunov direct method. The stability conditions are obtained by constructing appropriate Lyapunov function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. The meaning of Lyapunov functions is not clear by many specialists whose studies based on Lyapunov theory. They construct Lyapunov functions by using some properties of Lyapunov functions with much trial and errors or for a system choose candidate Lyapunov functions. So, for a given system Lyapunov function is not unique. But we insist that Lyapunov (energy) function is unique for a given physical system. In this study we highly simplified Lyapunov’s direct method with suitable tools. Our approach constructing energy function based on power-energy relationship that also enable us to take the derivative of integration of energy function. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The results are provided within and are of central importance for the analysis of nonlinear electrical, mechanical, and neural systems which based on the system energy perspective. The simulation results given from Matlab successfully verifies the theoretical predictions.

The classical Matrosov theorem concludes uniform asymptotic stability of time-varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semi-definite derivative along solutions) and another auxiliary function with derivative that is strictly nonzero where the derivative of the Lyapunov function is zero (Mastrosov in J Appl Math Mech 26:1337–1353, 1962). Recently, several generalizations of the classical Matrosov theorem have been reported in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005). None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) which is a very useful analysis and controller design tool for nonlinear systems. Inspired by generalized Matrosov conditions in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005), 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 and use this result to determine an ISS controller.

The operation of dynamic systems is often accompanied by abrupt and random changes in their configurations, which will dramatically change the stationary probability density function of their response. In this article, an effective procedure is proposed to reshape the stationary probability density function of nonlinear stochastic systems against abrupt changes. Based on the Markov jump theory, such a system is formulated as a continuous system with discrete Markov jump parameters. The limiting averaging principle is then applied to suppress the rapidly varying Markov jump process to generate a probability-weighted system. Then, the approximate expression of the stationary probability density function of the system is obtained, based on which the reshaping control law can be designed, which has two parts: (i) the first part (conservative part) is designed to make the reshaped system and the undisturbed system have the same Hamiltonian; (ii) the second (dissipative part) is designed so that the stationary probability density function of the reshaped system is the same as that of undisturbed system. The proposed law is exactly analytical and no online measurement is required. The application and effectiveness of the proposed procedure are demonstrated by using an example of three degrees-of-freedom nonlinear stochastic system subjected to abrupt changes.

Converse Lyapunov function theory guarantees the existence of strict Lyapunov functions in many situations, but the functions it provides are often abstract and nonexplicit, and therefore may not lend themselves to engineering applications. Often, even when a system is known to be stable, one still needs explicit Lyapunov functions; however, once an appropriate strict Lyapunov function has been constructed, many robustness and stabilization problems can be solved through standard feedback designs or robustness arguments. Non-strict Lyapunov functions are often readily constructed. This book contains a broad repertoire of Lyapunov constructions for nonlinear systems, focusing on methods for transforming non-strict Lyapunov functions into strict ones. Their explicitness and simplicity make them suitable for feedback design, and for quantifying the effects of uncertainty. Readers will benefit from the authors mathematical rigor and unifying, design-oriented approach, as well as the numerous worked examples.

In this paper, Lyapunov functions are generated to determine the regions of asymptotic stability of power systems under transient disturbances. With suitable assumptions, the swing equations of the synchronous machines connected to a power system are second- or third-order nonlinear autonomous differential equations. The Lyapunov functions V, employed to determine the domains of asymptotic stability of these nonlinear differential equations, are simple quadratic forms, whose coefficients are chosen so that the Routh-Hurwitz criteria are satisfied for the corresponding linear differential equations. For a synchronous machine swinging against an infinite bus, three typical Lyapunov functions are generated, taking transient saliency and positive-sequence damping into account. The application of Aizerman's method to the same problem leads to a fourth Lyapunov function. The domains of stability given by these Lyapunov functions are compared with the actual stability region obtained by numerical integration. Consideration of one time constant of the prime-mover governor leads to a third-order differential equation. A Lyapunov function is generated, and the stability surface obtained is verified by computing the swing curves numerically for various initial conditions. Finally, it is explained how some of these techniques could be extended for generating Lyapunov functions for multimachine systems.

This paper studies the problem of safety-critical tracking control for uncertain nonlinear systems which are allowed to be with high relative degree constraints, parametric uncertainty in the control and drift vector fields. The key to our method is to construct an augmented stored matrix and an augmented estimation vector to simultaneously handle unknown parameters in the control and drift vector fields. By designing a parameter update law depending on stored data and instantaneous data, a concurrent learning control Lyapunov function (CL-CLF) is proposed to achieve exponential convergence of tracking errors and parameter estimation errors under a finite excitation condition. By developing a class <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{K}$</tex-math> </inline-formula> function with respect to the augmented stored matrix, a CL high order control barrier function (CL-HOCBF) is constructed to ensure the safety of uncertain nonlinear systems with arbitrary relative degree constraints. Based on the CL-CLF and CL-HOCBF, the safety-critical controller can be obtained by solving quadratic programming (QP) without adding additional constraints. Theoretical analysis shows that the proposed CLF-CBF-QP framework can complete the challenge of transforming stability and safety requirements into verifiable linear constraints on control inputs in the presence of uncertainty in both the control and drift vector fields. In addition, the effectiveness of the developed approach is also verified by addressing the adaptive cruise control problem with uncertain vehicle dynamics. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In many control applications, the relative degrees of safety-critical constraints typically are larger than one, such as constraints regarding the configuration of mechanical systems. In addition, uncertainties in parameters including friction, control coefficients and damping widely exist in real-world systems. Motivated by these practical factors, this paper develops a novel CLF-CBF-QP framework for more general safety-critical systems with high relative degree constraints, parametric uncertainty in the control and drift vector fields, which can overcome the difficulty that transforms stability and safety requirements into verifiable linear constraints on control inputs. In particular, the practicability of presented method is demonstrated by solving the adaptive cruise control problem for automobiles with safety constraints and vehicle limitations, unknown parameters in aerodynamic drag term and control effectiveness.

We propose a sampling-based approach to learn Lyapunov functions for a class of discrete-time autonomous hybrid systems that admit a mixed-integer representation. Such systems include autonomous piecewise affine systems, closed-loop dynamics of linear systems with model predictive controllers, piecewise affine/linear complementarity/mixed-logical dynamical systems in feedback with a ReLU neural network controller, etc. The proposed method comprises an alternation between a learner and a verifier to search for a Lyapunov function from a family of parameterized Lyapunov function candidates. In each iteration, the learner uses a collection of state samples to select a Lyapunov function candidate through a convex program in the parameter space. The verifier then solves a nonconvex mixed-integer quadratic program in the state space to either validate the proposed Lyapunov function candidate or reject it with a counterexample, i.e., a state where the Lyapunov condition fails. This counterexample is then added to the sample set of the learner to refine the set of Lyapunov function candidates in the next iteration. By designing the learner and the verifier according to the analytic center cutting-plane method from convex optimization, we show that when the set of Lyapunov functions is full-dimensional in the parameter space, our method finds a Lyapunov function in a finite number of steps. We demonstrate our stability analysis method on closed-loop MPC dynamical systems and a ReLU neural network controlled PWA system.

A sufficient condition for the existence of a common quadratic Lyapunov function (CQLF) for the linear systems x/spl dot/=A/sub i/x, A/sub i//spl isin/{A/sub 1/,A/sub 2/,...,A/sub m/}, A/sub i//spl isin/IR/sup n/spl times/n/ is that the matrices can be simultaneously triangularized using a non-singular transformation T. In this paper, we show that this result follows trivially from the structure of the matrices in the set A, and that the switching system, constructed by switching between the matrices in this set, is benign from a stability viewpoint. Finally, we then discuss several conditions under which a transformation T exists.

This paper deals with the question of the existence of weak and strong common&#13;\nquadratic Lyapunov functions (CQLFs) for stable discrete-time linear timeinvariant&#13;\n(LTI) systems. The main result of the paper provides a simple characterisation&#13;\nof pairs of such systems for which a weak CQLF of a given form exists&#13;\nbut for which no strong CQLF exists. An application of this result to second&#13;\norder discrete-time LTI systems is presented.