System Dynamics Introduction to the Design and Simulation of Controlled Systems Introduction

System Dynamics (24.509)

VII. Introduction to the Design and Simulation of Controlled Systems

Introduction

There are two general approaches to developing controller designs - the classical method and the modern method. Some of the Case Studies in Section VI have already introduced the classical approach which includes PID controllers, lead-lag compensators, etc.. The classical design method relies on the root locus technique (Case Study E for example) and/or the various frequency response representations of LTI systems (Bode, Nyquist, and Nichols plots) to assist in the selection of the free variables introduced within the controller transfer function. The tuning of these control variables is usually performed via an educated or guided trial-and-error approach, and the creativity and experience of the designer plays an important role in the overall design process. The classical design method is usually restricted to single-input-single-output (SISO) systems.

Modern control theory employs a more formal mathematical approach for the design of control systems. Matrix methods are usually applied which allows the treatment of multiple-input-multiple-output (MIMO) systems. The objective of the design can often be stated precisely in quantitative terms in the form of a performance index, and the control variables are determined via application of a rigorous set of mathematical procedures. The trial-and-error aspect of the classical design method is considerably reduced and, in some cases, eliminated completely.

This section of notes introduces the subject of controller design using modern control methods. In particular, state feedback control, with and without a full state observer, is introduced and illustrated with some numerical examples. The idea of a simple classical proportional controller is also revisited and the combination of classical and modern control is used to help explain what is really happening with state control.

The development of these subjects is broken into two parts. The first challenge deals with the design of the control system. The root locus method is used to obtain the appropriate controller gain for the simple proportional controller and the pole placement method is used to obtain the gain matrix with the state control formulation (a similar method is used to obtain the observer gains). Once the design parameters are known, our focus then turns to addressing the actual simulation of the system with preset control parameters. In the simulation mode we address both linear and nonlinear systems (note that the controller design step is usually performed with a linear model of the system).

Finally, a sequence of Matlab examples is given for the so-called ‘inverted pendulum’ problem. This system is highly unstable and a robust control system is required for stable performance. Additionally, since the dynamics of the inverted pendulum is governed by a set of nonlinear equations, we can also explore the impact of using linear models to design a controller for a nonlinear system.

These three major subject areas are broken into a series of subtopics, as follows: