On stabilization of nonlinear systems with drift by time-varying feedback laws

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 bodies. We propose an explicit control design scheme with time-varying trigonometric polynomials whose coefficients depend on the state of the system. The above coefficients are computed in terms of the inversion of the matrix appearing in the controllability condition. It is shown that the proposed controllers can be used to solve the stabilization problem by exploiting the Chen-Fliess expansion of solutions of the closed-loop system. We also present results of numerical simulations for controlled Euler's equations and a mathematical model of underwater vehicle to illustrate the efficiency of the obtained controllers.Comment: This is the author's version of the manuscript accepted for publication in the Proceedings of the 12th International Workshop on Robot Motion Control (RoMoCo'19

Obstacle Avoidance Problem for Second Degree Nonholonomic Systems

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 gradient of a potential function. It is shown that the limit behavior of the closed-loop system is characterized by the set of critical points of the potential function. The proposed control design method can be used under rather general assumptions on potential functions, and particular applications with navigation functions are illustrated by numerical examples.Comment: This is the author's accepted version of the paper to appear in: 2018 IEEE Conference on Decision and Control (CDC), (c) IEE

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

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

Motion planning and stabilization of nonholonomic systems using gradient flow approximations

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. As an intermediate outcome, we characterize the asymptotic behavior of solutions of the considered class of nonlinear control systems with oscillating inputs under rather general assumptions on the generating potential function. These results are applied to examples of nonholonomic trajectory tracking and obstacle avoidance.Comment: submitte

Modeling and stabilization of a rotating mechanical system with elastic plates

A mechanical system consisting of a rigid body and attached Kirchhoff plates under the action of three independent controls torques is considered. The equations of motion of such model are derived in the form of a system of coupled nonlinear ordinary and partial differential equations. The operator form of this system is represented as an abstract differential equation in a Hilbert space. A feedback control law is constructed such that the corresponding infinitesimal generator is dissipative.Comment: A preprint version of the manuscript accepted for MATHMOD 2018 (9th Vienna International Conference on Mathematical Modelling, February 21-23, 2018

A Dynamic Observer for a Class of Infinite-Dimensional Vibrating Flexible Structures

Infinite-dimensional control systems with outputs are considered in the Hamiltonian formulation with generalized coordinates. An explicit scheme for constructing a dynamic observer for this class of systems is proposed with arbitrary gain coefficients. Sufficient conditions for the convergence of the constructed observer are obtained on the basis of the invariance principle. This result is applied to a flexible beam model attached to a mass-spring system with lumped and distributed actuators. The estimation error decay is illustrated with numerical simulations of finite-dimensional approximations of the observer dynamics.Comment: This is a preprint version of the paper accepted to the 2023 European Control Conferenc

Extremum Seeking Approach for Nonholonomic Systems with Multiple Time Scale Dynamics

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 for the nonholonomic dynamics and a model-free extremum seeking component with respect to the output measurements. Exponential convergence of the trajectories to an arbitrary neighborhood of the optimal point is established under suitable assumptions on time scale parameters of the extended system. The proposed algorithm is tested numerically with the Brockett integrator for different choices of generating functions.Comment: This is the author's version of the manuscript accepted for publication in the Proceedings of The 21st IFAC World Congress 2020 (IFAC 2020