Alexander Zuyev

Partial asymptotic stabilization of nonlinear distributed parameter systems

The paper is devoted to stability and stabilization of a class of evolution equations arising from mathematical modeling of hybrid mechanical systems with flexible parts. A sufficient condition is obtained for partial strong asymptotic stability of nonlinear, infinite-dimensional dynamic systems in Banach spaces. This result is applied to deriving a control law that stabilizes a part of the variables describing a rotating rigid body endowed with a number of elastic beams. Results of numerical simulations are presented.

Exponential Stabilization of Nonholonomic Systems by Means of Oscillating Controls

This paper is devoted to the stabilization problem for nonlinear driftless control systems by means of a time-varying feedback control. It is assumed that the vector fields of the system together with their first order Lie brackets span the whole tangent space at the equilibrium. A family of trigonometric open-loop controls is constructed to approximate the gradient flow associated with a Lyapunov function. These controls are applied for the derivation of a time-varying feedback law under the sampling strategy. By using Lyapunovs direct method, we prove that the controller proposed ensures exponential stability of the equilibrium. As an example, this control design procedure is applied to stabilize the Brockett integrator.

Stabilization and Observability of a Rotating Timoshenko Beam Model

A control system describing the dynamics of a rotating Timoshenkonbeam is considered. We assume that the beam is driven by a controlntorque at one of its ends, and the other end carries a rigid body asna load. The model considered takes into account the longitudinal,nvertical, and shear motions of the beam. For this distributednparameter system, we construct a family of Galerkin approximationsnbased on solutions of the homogeneous Timoshenko beam equation. Wenderive sufficient conditions for stabilizability of such finitendimensional system. In addition, the equilibrium of the Galerkinnapproximation considered is proved to be stabilizable by annobserver-based feedback law, and an explicit control design isnproposed.

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.

Approximate controllability of a rotating Kirchhoff plate model

In this paper, we consider a mechanical system consisting of a rigid body with a thin elastic plate. The plate vibration is governed by the Kirchhoff plate theory. We derive the equations of motion as a system of ordinary and partial differential equations and consider the angular acceleration as a control parameter. The dynamical equations are transformed into an infinite set of ordinary differential equations with respect to modal coordinates. It is shown that such a system is not controllable in general. Hence, the approximate controllability problem is set out for the dynamics restricted to an invariant manifold. Then sufficient conditions for the approximate controllability are proposed. Simulation results are presented to illustrate the spillover effect.

Approximate controllability and spillover analysis of a class of distributed parameter systems

A linear control system in Hilbert space is considered. An estimate of solutions of the system is carried out for the class of optimal control corresponding to a subsystem with finite number of degrees of freedom. It is proved that the family of controllers considered solves the problem of approximate controllability for the infinite dimensional system.

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.

Application of the Return Method to the Steering of Nonlinear Systems

The controllability property plays a crucial role in mathematical control theory. For linear systems, necessary and sufficient conditions of controllability are given by the Kalman criterion. This criterion also allows to study the local controllability by linear approximation [5]. There is a number of necessary as well as sufficient conditions of controllability for nonlinear systems expressed in terms of properties of the corresponding Lie algebra (see, e.g., [2, 7, 9]). However, in general case, a constructive test of controllability of nonlinear systems is a difficult problem and, to our knowledge, there is no effective estimate of the number of iterated Lie brackets for the test of an appropriate rank condition [1]. In this paper, we use controllability conditions of nonlinear systems based on a modification of the return method (cf. [4, 6, 8]).

Stabilizability conditions in terms of critical Hamiltonians and symbols

Symmetric functions of critical Hamiltonians, called symbols, were used in such problems of nonlinear control as the characterization of symmetries and feedback invariants. We derive here a stabilizability condition in the class of almost continuous feedback controls based on symbols. The methodology proposed consists of defining a selector of the multivalued covector field satisfying an extra degree condition on a sphere close to the origin. Then the above selector is extended to some neighborhood in order to get an exact differential form. Integration of that form gives us a control Lyapunov function candidate (Theorem 3.1). The condition proposed is applied to the analysis of a planar control-affine system with polynomial homogeneous vector fields. Unlike known results in the literature, we consider the case when each vector field vanishes at the origin and the control is bounded. We give stabilizability conditions explicitly in terms of the control system parameters (Theorem 4.1).