Control of Piecewise Smooth Systems: Generalized Absolute Stability and Applications to Supercavitating Vehicles

Abstract

Many systems in engineering applications are modeled as piecewise smooth systems. The piecewise smoothness presents great challenges for stability analysis and control synthesis for these systems. Over the years, the theory of absolute stability has been one of the few tools developed by control theory researchers to meet these challenges. For systems in which the nonlinearity is known to be bounded within certain sectors, many stability and control problems can be addressed using results from absolute stability theory.

During the last few decades, many important advances have been made in the study of the absolute stability. In these studies, it is commonly assumed that the sector bound for the system nonlinearity is \textit{symmetric} with respect to the origin in state space. However, in many practical engineering systems, the nonlinearity does not satisfy such a symmetry assumption. To study stability and control problems for these systems, in this work the author studies generalized absolute stability problems involving \textit{asymmetric} sector bounds. Nonlinear systems with Lure' structure are considered. For second-order systems, conditions that are both necessary and sufficient for generalized absolute stability are obtained. These conditions can be easily tested in engineering applications. For general finite-order systems, sufficient conditions are provided for generalized absolute stability. The derived conditions may be easily tested by using numerical tools for linear programming. With the generalizations in this work, absolute stability theory becomes a more powerful tool in the sense that it applies to an extended class of piecewise smooth systems in which the nonlinearities can be asymmetric with respect to the state variables.

This work includes general theoretical questions as well as detailed investigations of an application to models of supercavitating vehicles. For these high-speed underwater vehicles, the dive-plane motion is naturally modeled as a piecewise smooth system with a dead zone. The strong nonlinear planing force plays an important role in determining the dive-plane dynamics. To design control laws that stabilize the dive-plane motion, the necessary and sufficient condition for generalized absolute stability of second-order systems is applied to a reduced-order model obtained through the backstepping control approach. The obtained sufficient conditions for generalized absolute stability of finite-order systems can also be successfully applied for stabilizing the dive-plane motion. In comparison with alternative control approaches, control designs with the aid of theoretical findings in generalized absolute stability lead to stability that is robust to the modeling errors in the nonlinearity such as the magnitude, local slope and the dead zone location.

The dissertation also includes basic results on bifurcation and bifurcation control of supercavitating vehicles. The presence of bifurcations in the dive-plane dynamics is demonstrated, and control techniques for modifying the bifurcation behavior to improve the vehicle dynamic performance are developed. These results complement the absolute stability results to give a more complete picture of the dynamics and control of supercavitating vehicles.