Concept Of Control Lyapunov Function Research Articles

Stabilizing Dynamic Controllers for Hybrid Systems: A Hybrid Control Lyapunov Function Approach

This paper proposes a dynamic controller structure and a systematic design procedure for stabilizing discrete-time hybrid systems. The proposed approach is based on the concept of control Lyapunov functions (CLFs), which, when available, can be used to design a stabilizing state-feedback control law. In general, the construction of a CLF for hybrid dynamical systems involving both continuous and discrete states is extremely complicated, especially in the presence of non-trivial discrete dynamics. Therefore, we introduce the novel concept of a hybrid control Lyapunov function, which allows the compositional design of a discrete and a continuous part of the CLF, and we formally prove that the existence of a hybrid CLF guarantees the existence of a classical CLF. A constructive procedure is provided to synthesize a hybrid CLF, by expanding the dynamics of the hybrid system with a specific controller dynamics. We show that this synthesis procedure leads to a dynamic controller that can be implemented by a receding horizon control strategy, and that the associated optimization problem is numerically tractable for a fairly general class of hybrid systems, useful in real world applications. Compared to classical hybrid receding horizon control algorithms, the proposed approach typically requires a shorter prediction horizon to guarantee asymptotic stability of the closed-loop system, which yields a reduction of the computational burden, as illustrated through two examples.

Lyapunov Function for Nonuniform in Time Global Asymptotic Stability in Probability with Application to Feedback Stabilization

We extend the well-known Artstein-Sontag theorem by introducing the concept of control Lyapunov function for the notion of nonuniform in time global asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. The main results of our work enable us to derive the necessary and sufficient conditions for feedback stabilization for affine in the control systems.

Control Lyapunov Functions: New Framework for Nonlinear Controller Design

Control Lyapunov function (CLF) is a successful attempt to directly use of the Lyapunov function stability analysis technique of nonlinear systems in the synthesis problem. In this paper, on the basis of Freeman's work (1996), the concept of CLF are re-analyze through using the method of set-valued analysis. And then, a new CLF based nonlinear controller design framework, called generalized pointwise min-norm (GPMN), is proposed. Simultaneously, three robust GPMN controllers are introduced with respect to respectively parameter uncertainties, external disturbance, and the combining cases. Actually, the framework provides us a new idea of nonlinear controller design since within which many other controller design indexes can be combined without re-considering the closed loop stability. Finally, a simple simulation is conducted to show one of the typical applications.

Stabilization of nonlinear switched systems using control Lyapunov functions

In this paper, we study the stabilization of general nonlinear switched systems by using control Lyapunov functions. The concept of control Lyapunov function for nonlinear control systems is generalized to switched control systems. The first part of our contribution provides a necessary and sufficient condition of stabilization. The main idea is to use a common control Lyapunov function; this is achieved with the converse Lyapunov theorem dedicated to switched systems. In the second part, an explicit construction of a common control Lyapunov function is addressed with respect to a finite family of switched systems. The approach uses a family of control Lyapunov functions attached to the subsystems.

NON AUTONOMOUS AFFINE SYSTEMS: CONTROL LYAPUNOV FUNCTION AND THE STABILIZATION PROBLEM

Stabilization using the concept of control Lyapunov function is investigated for non linear non autonomous systems. First, necessary conditions for stabilization are given. Then, sufficient conditions ensure the existence of continuous stabilizing feedback for non autonomous affine systems. Lastly, a generalization of Sontag's formula is given.

Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation

We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed.

Universal controllers for robust control problems

In this paper we discuss the construction of “universal” controllers for a class of robust stabilization problems. We give a general theorem on the construction of these controllers, which requires that a certain nonlinear inequality is solvablepointwisely or, equivalently, that arobust control Lyapunov function does exist. The constructive procedure producesalmost smooth controllers. The robust control Lyapunov functions extend to uncertain systems the concept of control Lyapunov functions. If such a robust control Lyapunov function also satisfies a small control property, the resulting stabilizing controller is also continuous in the origin of the state space. Applications of our results range from optimal to robust control.