Stabilization problem with oscillating inputs under nonlinear controllability conditions
Speaker: Prof. Dr. Alexander Zuyev
Affiliation: Max-Planck-Institut für Dynamik Komplexer Technischer Systeme, Germany
Abstact: This talk is devoted to the development of potential function approach for solving the problems of stabilization and collision avoidance for underactuated nonlinear control systems. In the first part of the talk, we propose a stabilization scheme for driftless control-affine systems whose vector fields satisfy controllability rank condition with iterated Lie brackets. Our control design scheme is based on the use of trigonometric time-varying feedback laws with bounded frequencies. By using the Chen-Fliess series expansion and a modification of Lyapunov’s direct method, we reduce the stabilization problem to a system of algebraic 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. In the second part of this talk, we discuss the problem of generating reference trajectories on the state space with obstacles by using the gradient flow of a navigation function. In general, such gradient flow cannot be implemented for underactuated control systems, and the approximation of non-admissible velocities is required for the control design. We present here approximation results under nonlinear controllability assumptions.