V.V. Grushkovskaya

University of Wuerzburg | JMU · Institute of Mathematics

Nonlinear control-affine systems with time-varying vector fields are considered in the paper. We propose a unified control design scheme with oscillating inputs for solving the trajectory tracking and stabilization problems. This methodology is based on the approximation of a gradient like dynamics by trajectories of the designed closed-loop system...

The paper presents a novel family of dynamic extremum seeking controls for nonholonomic systems with output described by a convex function. It is assumed that the vector fields of the system satisfy the controllability rank condition with first‐ and second‐order Lie brackets. The proposed algorithm consists of two components. The first one is the s...

In this article, we consider extremum seeking problems for a general class of nonlinear dynamic control systems. The main result of the article is a broad family of control laws which optimize the steady‐state performance of the system. We prove practical asymptotic stability of the optimal steady‐state and, moreover, propose sufficient conditions...

In this paper, a class of nonlinear driftless control-affine systems satisfying the bracket generating condition is considered. A gradient-free optimization algorithm is developed for the minimization of a cost function along the trajectories of the controlled system. The algorithm comprises an approximation scheme with fast oscillating controls fo...

This paper focuses on the problem of constructing time-varying feedback laws that asymptotically stabilize a given part of the state variables for nonlinear control-affine systems. It is assumed that the class of systems under consideration satisfies nonlinear controllability conditions with respect to the stabilizable variables. Under these assump...

In this paper, a class of nonlinear driftless control-affine systems satisfying the bracket generating condition is considered. A gradient-free optimization algorithm is developed for the minimization of a cost function along the trajectories of the controlled system. The algorithm comprises an approximation scheme with fast oscillating controls fo...

This paper focuses on the development of stability conditions for systems of nonlinear non‐autonomous ordinary differential equations and their applications to control problems. We present a novel approach for the study of asymptotic stability properties for nonlinear non‐autonomous systems based on considering a parameterized family of sets. The p...

A class of nonlinear control-affine systems with bounded time-varying drift is considered. It is assumed that the control vector fields together with their iterated Lie brackets satisfy Hormander's condition in a neighborhood of the origin. Then the problem of exponential stabilization is treated by exploiting periodic time-varying feedback control...

This paper deals with the stabilization problem for nonlinear control-affine systems with the use of oscillating feedback controls. We assume that the local controllability around the origin is guaranteed by the rank condition with Lie brackets of length up to 3. This class of systems includes, in particular, mathematical models of rotating rigid b...

The paper deals with the extremum seeking problem for a class of cost functions depending only on a part of state variables of a control system. This problem is related to the concept of partial asymptotic stability and analyzed by Lyapunov's direct method and averaging schemes. Sufficient conditions for the practical partial stability of a system...

This paper deals with the stabilization problem for nonlinear control-affine systems with the use of oscillating feedback controls. We assume that the local controllability around the origin is guaranteed by the rank condition with Lie brackets of length up to~3. This class of systems includes, in particular, mathematical models of rotating rigid b...

The problem of tracking an arbitrary curve in the state space is considered for underactuated driftless control-affine systems. This problem is formulated as the stabilization of a time-varying family of sets associated with a neighborhood of the reference curve. An explicit control design scheme is proposed for the class of controllable systems wh...

In this work, discrete-time extremum seeking algorithms for unconstrained optimization problems are developed. A general class of non-commutative maps and one- and two point function evaluation polices are presented to approximate a gradient-descent algorithm, suitable for extremum seeking problems. Moreover, adaptive step size rules are discussed...

A class of nonlinear control-affine systems with bounded time-varying drift is considered. It is assumed that the control vector fields together with their iterated Lie brackets satisfy Hörmander’s condition in a neighborhood of the origin. Then the problem of exponential stabilization is treated by exploiting periodic time-varying feedback control...

The paper deals with the extremum seeking problem for a class of cost functions depending only on a part of state variables of a control system. This problem is related to the concept of partial asymptotic stability and analyzed by Lyapunov’s direct method and averaging schemes. Sufficient conditions for the practical partial stability of a system...

In this paper, we consider the problem of extremum seeking in the presence of obstacles. The analytical expression of the cost function as well as locations and shapes of the obstacles are assumed to be partially or completely unknown. We describe a broad family of control algorithms for a unicycle type system, which provides a solution of the abov...

In this paper, we propose a new control design scheme for solving the obstacle avoidance problem for nonlinear driftless control-affine systems. The class of systems under consideration satisfies controllability conditions with iterated Lie brackets up to the second order. The time-varying control strategy is defined explicitly in terms of the grad...

In this paper, we propose and practically evaluate a class of gradient-free control functions ensuring the motion of a unicycle-type system towards the extremum point of a time-varying cost function. We prove that the unicycle is able to track the extremum point, and illustrate our results by numerical simulations and experiments that show that the...

This paper considers the obstacle avoidance problem for control-linear systems satisfying the Hörmander condition with the first-order Lie brackets. To solve the problem under consideration, we use a family of oscillating periodic control functions with state-dependent coefficients. The proposed approach is shown to be applicable for different shap...

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 s...

The paper focuses on the development of the navigation function approach for nonlinear systems with fast oscillating controls. This approach allows to solve the obstacle avoidance problem and generate reference trajectories on the state space with obstacles by using the gradient flow of a navigation function. In general, such gradient flow cannot b...

This work addresses the problem of solving distributed optimization problems over directed graphs. Using techniques from geometric control theory, a novel methodology is proposed to design continuous-time distributed algorithms that approximate the solutions to the saddle-point algorithms over directed graphs. Graph-theoretic conditions are establi...

The paper presents a control algorithm that steers a system to an extremum point of a time-varying function. The proposed extremum seeking law depends on values of the cost function only and can be implemented without knowing analytical expression of this function. By extending the Lie brackets approximation method, we prove the local and semi-glob...

In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes the 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 r...

We study a point-to-point control problem for driftless control-affine systems. The class of problems under consideration satisfies controllability conditions with the Lie brackets up to the second order. To solve the control problem, we use trigonometric polynomials whose coefficients are computed by expanding the solutions into the Volterra serie...

The paper is devoted to the study of a class of optimization problems with convex objective and constraint functions by using extremum seeking methods. Multi-agent systems with single integrator and unicycle dynamics are considered. The purpose is to develop a control algorithm that steers the agents to the set of saddle points of the associated La...

This paper is devoted to the study of the asymptotic behavior of an essentially nonlinear system with resonant frequencies. Namely, it is assumed that the matrix of linear approximation of the system has several subsets of multiple purely imaginary eigenvalues. For such systems, the paper presents sufficient conditions for the asymptotic stability...

The paper is devoted to the analysis of the decay rate of solutions of a nonlinear system with two pairs of purely imaginary eigenvalues. The main result is the power estimate for the norm of solutions. It is proven that the order of such estimate varies for cases of a diagonalizable matrix of linear approximation, and for a matrix containing a Jor...

In this paper, we propose a stabilization scheme for nonlinear control systems whose vector fields satisfy Hörmander's 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 Lyapunov's direct method, we reduce the stabilizat...

This paper is focused on the analysis of the asymptotic behavior of solutions for a nonlinear system with (Formula presented.) 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 st...

This paper is devoted to the study of attractive sets for dynamical systems in a metric space with a measure. It is assumed that the measure of a set of points in the phase space is increasing along the flow. We prove that an invariant set is an attractor for almost all initial conditions under some extra assumptions. For a system of autonomous ord...

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 Ly...

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...

This paper is devoted to the stability problem of a nonlinear system in the critical case of q pairs of purely imaginary eigenvalues. The center manifold theory and the normal form method are exploited in this study. The main result of the paper is a power estimate for the solutions in the case when stability is ensured by third order forms. A Lyap...