Control Lyapunov functions for adaptive nonlinear stabilization

For the problem of stabilization of nonlinear systems linear in unknown constant parameters, we introduce the concept of an adaptive control Lyapunov function (aclf) and use Sontag's constructive proof of Artstein's theorem to design an adaptive controller. In this framework the problem of adaptive stabilization of a nonlinear system is reduced to the problem of nonadaptive stabilization of a modified system. To illustrate the construction of aclf's we give an adaptive backstepping lemma which recovers our earlier design.     

Asymptotic feedback stabilization: A sufficient condition for the existence of control Lyapunov functions

In this paper we study the feedback stabilization problem for a wide class of nonlinear systems that are affine in the control. We offer sufficient conditions for the existence of ‘Control Lyapunov functions’ that according to [3,23] and [28–30] guarantee stabilization at a specified equilibrium by means of a feedback law, which is smooth except possibly at the equilibrium. We note that the results of the paper present a local nature.     

Topological necessary conditions of smooth stabilization in the large

Topological necessary conditions of smooth stabilization in the large are obtained. In particular, if a smooth single-input nonlinear system is smoothly stabilizable in the large at some point of a connected component of an equilibrium set, then the connected component is an unbounded curve.     

Adding an integrator for the stabilization problem

We study the relationship between the following two properties: P1: The system is locally asymptotically stabilizable; and P2: The system is locally asymptotically stabilizable; where . Dayawansa, Martin and Knowles have proved that these properties are equivalent if the dimension n = 1. Here, using the so called Control Lyapunov function approach, (a) we propose another more constructive and somewhat simpler proof of Dayawansa, Martin and Knowles's result; (b) we show that, in general, P1 does not imply P2 for dimensions n larger than 1; (c) we prove that P2 implies P1 if some extra assumptions are added like homogeneity of the system. By using the latter result recursively, we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form.     

Constrained stabilization via smooth Lyapunov functions

We consider the problem of stabilizing a dynamic system by means of bounded controls. We show that the largest domain of attraction can be arbitrarily closely approximated by a “smooth” domain of attraction for which we provide an analytic expression. Such an expression allows for the determination of a (non-linear) control law in explicit form.     

Nonsmooth continuous-time optimization problems: necessary conditions

We consider Lipschitz continuous-time nonlinear optimization problems and provide first-order necessary optimality conditions of both Fritz John and Karush-Kuhn-Tucker types.     

Stabilization of Nonlinear Systems with Filtered Lyapunov Functions and Feedback Passivation

In this paper a generalized class of filtered Lyapunov functions is introduced, which are Lyapunov functions with time‐varying parameters satisfying certain differential equations. Filtered Lyapunov functions have the same stability properties as Lyapunov functions. Tools are given for designing composite filtered Lyapunov functions for cascaded systems. These functions are used to design globally stabilizing dynamic feedback laws for block‐feedforward systems with stabilizable linear approximation.     

Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems

We study the stability properties of a class of time-varying non-linear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body.     

Construction of control Lyapunov functions for damping stabilization of control affine systems

This paper considers the stabilization of nonlinear control affine systems that satisfy Jurdjevic–Quinn conditions. We first obtain a differential one-form associated to the system by taking the interior product of a non vanishing two-form with respect to the drift vector field. We then construct a homotopy operator on a star-shaped region centered at a desired equilibrium point that decomposes the system into an exact part and an anti-exact one. Integrating the exact one-form, we obtain a locally-defined dissipative potential that is used to generate the damping feedback controller. Applying the same decomposition approach on the entire control affine system under damping feedback, we compute a control Lyapunov function for the closed-loop system. Under Jurdjevic–Quinn conditions, it is shown that the obtained damping feedback is locally stabilizing the system to the desired equilibrium point provided that it is the maximal invariant set for the controlled dynamics. The technique is also applied to construct damping feedback controllers for the stabilization of periodic orbits. Examples are presented to illustrate the proposed method.     

Homogeneous Feedback Control of Nonlinear Systems Based on Control Lyapunov Functions

This paper is concerned with homogeneous feedback control for a class of nonlinear systems. By using control Lyapunov functions, homogeneous controllers of homogeneous and nonhomogeneous systems are respectively constructed, under which the stabilization of the systems under considerations are guaranteed. Then, the design method is extended for uncertain systems by means of homogeneous domination theory. Compared with the traditional design method based on robust control Lyapunov functions, the present design reduces the difficulty of constructing controllers. Finally, several simulation examples are given to illustrate the correctness of the design.     

Strict Lyapunov functions for time-varying systems

Uniformly asymptotically stable periodic time-varying systems for which is known a Lyapunov function with a derivative along the trajectories non-positive and negative definite in the state variable on non-empty open intervals of the time are considered. For these systems, strict Lyapunov functions are constructed.     

Lyapunov functions for the multivariable Popov criterion with indefinite multipliers

We demonstrate the existence of Lur’e-Postnikov type Lyapunov functions for a reasonably general form of the multivariable Popov criterion. In particular, the multipliers need not be positive semi-definite. We discuss the application to the special cases where the feedback element is either diagonal, unstructured or linear.     

Homogeneous Lyapunov functions for systems with structured uncertainties

The problem addressed in this paper is the construction of homogeneous polynomial Lyapunov functions (HPLFs) for linear systems with time-varying structured uncertainties. A sufficient condition for the existence of an HPLF of given degree is formulated in terms of a linear matrix inequalities (LMI) feasibility problem. This condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the Lyapunov function. The maximum ?_∞ norm of the parametric uncertainty for which there exists a homogeneous polynomial Lyapunov function is computed by solving a generalized eigenvalue problem. The construction of such Lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in LMI form. Comparisons with other classes of Lyapunov functions through numerical examples taken from the literature show that HPLFs are a powerful tool for robustness analysis.     

Global stabilization of MIMO minimum-phase nonlinear systems without strict triangular conditions

An extension of backstepping to a class of multivariable minimum-phase nonlinear systems is proposed. The systems are assumed to be in a special interlaced form which includes a lower triangular form as a special case. The extension involves the recursive application of backstepping and augmentation.     

Robust output feedback stabilization via a small gain theorem

In this paper, we give sufficient conditions for designing robust globally stabilizing controllers for a class of uncertain systems, consisting of ‘nominal’ nonlinear minimum phase systems perturbed by uncertainties which may affect the equilibrium point of the nominal system (‘biased’ systems). The constructive proof combines a systematic step-by-step procedure, based on H_∞ arguments, with a small gain theorem, recently proved for nonliner systems. At each step, one finds two Lyapunov functions, one for a state-feedback problem and the other one for an output injection problem. Combining these two functions, one derives at each step a Lyapunov function candidate for solving an ouptut feedback stabilization problem. This approach allows one to put into a unified framework many existing results on robust output feedback stabilization. © 1998 John Wiley & Sons, Ltd.     

Stabilization via dynamic output feedback for systems with output nonlinearities

The problem of asmptotically stabilizing a class of systems by means of continuous output feedback is considered. These systems are characterized by nonlinear terms, depending only on the ouputs. It is shown that for these systems stabilization via continuous state-feedback plus stabilization via output injection imply stabilization via continuous dynamic output-feedback. This generalizes a well-knwon result for linear systems.     

LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions

The robust stability of uncertain linear systems in polytopic domains is investigated in this paper. The main contribution is to provide a systematic procedure for generating sufficient robust stability linear matrix inequality conditions based on homogeneous polynomially parameter-dependent Lyapunov matrix functions of arbitrary degree on the uncertain parameters. The conditions exploit the positivity of the uncertain parameters, being constructed in such a way that: as the degree of the polynomial increases, the number of linear matrix inequalities and free variables increases and the test becomes less conservative; if a feasible solution exists for a certain degree, the conditions will also be verified for larger degrees. For any given degree, the feasibility of a set of linear matrix inequalities defined at the vertices of the polytope assures the robust stability. Both continuous and discrete-time uncertain systems are addressed, as illustrated by numerical examples.     

Conjugate Lyapunov functions for saturated linear systems

Based on a recent duality theory for linear differential inclusions (LDIs), the condition for stability of an LDI in terms of one Lyapunov function can be easily derived from that in terms of its conjugate function. This paper uses a particular pair of conjugate functions, the convex hull of quadratics and the maximum of quadratics, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a parametertized LDI system with an effective approach which enables optimization on the parameter of the LDI along with the optimization of the Lyapunov functions. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates the effectiveness of this paper's methods.     

Controller design for nonlinear affine systems by control Lyapunov functions

This paper addresses the stabilization problems for nonlinear affine systems. First of all, the explicit feedback controller is developed for a nonlinear multiple-input affine system by assuming that there exists a control Lyapunov function. Next, based upon the homogeneous property, sufficient conditions for the continuity of the derived controller are developed. And then the developed control design methodology is applied to stabilize a class of nonlinear affine cascaded systems. It is shown that under some homogeneous assumptions on control Lyapunov functions and the interconnection term, the cascaded system can be globally stabilized. Finally, some interesting results of finite-time stabilization for nonlinear affine systems are also obtained.