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Chapter 5: Pole Placement

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5.1 Introduction

In this chapter, I introduce the pole placement design method, which allows the designer to locate the closed-loop system poles at arbitrarily selected points in the complex plane. With an appropriate choice of pole locations, the closed-loop system can exhibit any response speed and damping characteristics desired. This assumes the plant and actuators are capable of producing an approximately linear response and some enabling conditions on the plant are satisfied.

However, this design freedom comes at a small cost. The control algorithm used in pole placement assumes that the full internal state of the plant is known at all times. Most often, this is not the case. In a typical design, perhaps only one of several internal state variables is measured. The values of the unmeasured states must be estimated during controller operation.

An observer (also called a state estimator) performs the task of estimating the complete set of state variables internal to the plant. The observer develops state estimates on the basis of the measured plant outputs and the known plant inputs.

Two restrictions on the structure of the plant model must be satisfied for the pole placement design method to succeed. The first requires that the actuators be capable of driving the system in a manner that allows control of all modes of behavior. This property is called controllability. The other restriction is that the sensors must measure sufficient system parameters to enable construction of a complete state estimate. This is called observability. Some straightforward tests of the plant model will determine whether it is controllable and observable.

This design approach is applicable to MIMO systems as well as to SISO systems. The same general state-space model form is used for SISO and MIMO systems, and the steps in the design method are applicable in both situations.

The pole placement design method uses some fairly complex matrix algorithms for controller and observer design. The MATLAB Control System Toolbox provides robust implementations of these algorithms. The algorithms themselves will not be discussed here. Instead, I cover the steps necessary to apply the commands in the Control System Toolbox for pole placement design.