New Control Methods for a Class of Nonlinear Systems with Constrained Input

Abstract

Conventional nonlinear feedback control tools include linearization, gain scheduling, integral control, feedback linearization, sliding mode control, Lyapunov redesign, back stepping, passivity based control etc. Each of these techniques is designed to deal with a specific nature of problem. None of these methods are universal in the sense that it can be applied to all classes of nonlinear control problems. The realm of nonlinear control systems encounters theoretical and practical problems that do not fit into existing frameworks. This demands development of novel and innovative methods that go beyond conventional philosophy of control systems. This thesis also deals with such class of problems that is difficult to deal due to usual nonlinear control techniques. The core issue is hard constraints on the input of the system, that restrict the freedom of a control designer to incorporate control methods based on continuous stabilization, cancellation, compensation and/or adjustment of control parameters. The thesis starts with a discussion on sampled data tracking problem for a class of multi-input multi-output (MIMO) nonlinear systems. The nature of system is generic enough to handle many theoretical and practical problems. However, the thesis broadly focuses on a challenging example of the two-axis orientation control of a gyroscopic system with constrained input. During a single sample period, only a fixed amplitude pulse of variable position and width can be applied as a single control input. The example also falls in the category of under actuated systems due to single control of two axes. Alternately, pulse width and position can be construed as two inputs of the system. The output is also assumed to be available at only the sampling instants. All these restrictions result in a complex problem whose exact solution is not possible and thus we have to resort to approximate methods. The thesis begins with exploration of classical techniques. Firstly, a more conventional pulse width modulation approach based on principle of equivalent areas is proposed. This is followed by an error minimized control technique which is based on optimal control. The solution minimizes a cost function so as to obtain optimal values of pulse width and position. The problems of local minima and non-causality have to be addressed in order to solve the problem. The main contribution of the thesis is a particle controller for the class of systems under discussion. The classical theory of particle filters is adapted in order to solve the global optimization problem. A deterministic problem is solved using stochastic tools. The idea is to associate the cost function to be minimized with a probability density function (pdf). Input samples are drawn according to this pdf which are subsequently assigned weights using simulations of the system. The process includes steps like generation, refinement, regeneration, resampling etc. some of which are familiar in the realm of particle filters. This unconventional control philosophy has the potential to address a variety of control problems that are difficult to handle using available tools. Extensive Monte Carlo simulations have been performed for each of the above techniques. Where applicable, performance comparisons have also been made. The suggested techniques are computationally heavy and require fast processing. However, they suit parallel computing and can thus be embedded using FPGAs or ASICs.