Quasi-linear Control Theory: The Case of Disturbance Rejection

Professor Semyon M. Meerkov

Department of Electrical Engineering and Computer Science

University of Michigan

Quasi-linear control theory is a set of tools motivated by and similar to standard linear control techniques, but applicable to systems with nonlinear actuators and sensors. As examples of quasi-linear methods, the circle criterion and the method of harmonic balance can be mentioned: The former "expands" the critical point of the Nyquist stability criterion to a circle in order to test the stability of linear plants with nonlinear actuators. The latter extends the notion of transfer functions to nonlinear actuators (i.e., describing functions) and provides a linear-like method for oscillations analysis in nonlinear situations. To our knowledge, all the existing quasi-linear results are concerned with issues of stability and oscillations. The present work addresses issues of performance, i.e., disturbance rejection and reference tracking. In this talk, the former is discussed. Specifically, we provide an extension of the LQR/LQG methodology to systems with saturating actuators (referred to as SLQR/SLQG, where S stands for "saturating"). The development is based on the method of stochastic linearization. We show that the solution of the SLQR/SLQG problem is provided by standard Riccati and Lyapunov equations coupled with two transcendental equations, which account for the variance of the signal at the output of the controller and the Lagrange multiplier, associated with the optimization problem. We show that under the usual stabilizability and detectability conditions, these equations have a unique solution and provide a method for its determination using a simple bisection algorithm. Finally, we investigate the stability properties of the closed loop systems with SLQR/SLQG controllers and illustrate the technique developed by an application to a marine ship roll-damping problem.

This is joint work with Professor P. T. Kabamba and Dr. C. Gokcek

Friday, March 16, 2001

3:30 - 5:00 p.m.

1500 EECS