Victoria Grushkovskaya

Time-varying stabilization of a class of driftless systems satisfying second-order controllability conditions

In this paper, we propose a stabilization scheme for nonlinear control systems whose vector fields satisfy Hormanders condition with the second-order Lie brackets. This scheme is based on the use of trigonometric controls with bounded frequencies. By using the Volterra series and a modification of Lyapunovs direct method, we reduce the stabilization problem to a system of cubic equations and prove its local solvability. Our approach ensures exponential stability of the equilibrium and gives explicit formulas for the coefficients of the control functions. The proposed methodology is illustrated by a rolling disc example.

On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties

Abstract In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems. It provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability. In contrast to many known results, we also prove asymptotic and exponential stability in the sense of Lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields.

Optimal Stabilization Problem With Minimax Cost in a Critical Case

This work addresses the optimal stabilization problem of a nonlinear control system by using a smooth output feedback. The optimality criterion is the maximization of the decay rate of solutions in a neighborhood of the origin. We formulate this criterion as a minimax problem with respect to non-integral functional. An explicit construction of a Lyapunov function is proposed to evaluate the optimal cost. This design methodology is justified for nonlinear systems in a critical case of stability with a pair of purely imaginary eigenvalues. As an example, a minimax optimal controller is obtained for a spring-pendulum system with partial measurements of the state vector.

Motion planning for control-affine systems satisfying low-order controllability conditions

ABSTRACT This paper is devoted to the motion planning problem for control-affine systems by using trigonometric polynomials as control functions. The class of systems under consideration satisfies the controllability rank condition with the Lie brackets up to the second order. The approach proposed here allows to reduce a point-to-point control problem to solving a system of algebraic equations. The local solvability of that system is proved, and formulas for the parameters of control functions are presented. Our local and global control design schemes are illustrated by several examples.

Asymptotic decay of solutions to an essentially nonlinear system with two-frequency resonances

This paper is focused on the analysis of the asymptotic behavior of solutions for a nonlinear system with pairs of purely imaginary roots under the presence of two-frequency resonances of the fourth order. The main purpose of the paper is to estimate the norm of solutions, provided that the trivial solution is asymptotically stable regardless of forms higher than the third order. For this study, asymptotic stability conditions are presented and a Lyapunov function is constructed. These results are applied for estimating the decay rate of oscillations of a spring-pendulum system with partial dissipation.

Optimal stabilization of nonlinear systems by an output feedback law in a critical case

This paper is devoted to the stability analysis of nonlinear systems whose linear approximation exhibits a pair of purely imaginary eigenvalues. By using the center manifold approach and normalization procedure, we estimate the decay rate of solutions in the critical case considered. Such an estimate is applied for computing the cost of an optimal stabilization problem. As an example, the optimal stabilization problem by means of a smooth output feedback law is considered for a mechanical system with two degrees of freedoms.