5.11 Summary

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5.11 Summary

In this chapter, I discussed the pole placement design approach, which permits the positioning of closed-loop system poles at arbitrarily selected locations in the complex plane. This provides the designer with a great deal of freedom in determining the dynamic behavior of the system.

The feedback control algorithm used in the pole placement method is a gain matrix K multiplied by the system state vector x. The full state vector is required for this computation, including any state variables that are not directly measured with sensors. Consequently, it is necessary to estimate any unknown states for use in the control algorithm. An observer, also called a state estimator, performs this function.

Some restrictions on the structure of the state-space model must be satisfied for the pole placement design approach to succeed. First, the actuators must be capable of driving the system in a manner that produces the desired response. Second, the sensors must measure sufficient system parameters to enable construction of a complete state estimate. The ability of the actuators to fully control the system is called controllability, and the ability of the sensor measurements to enable complete state estimation is called observability. Performing some straightforward tests on the linear plant model determines whether it is controllable and observable.

Given a controllable and observable state-space plant model along with a set of desired system and observer pole locations, the design of the controller and observer requires only a few MATLAB Control System Toolbox commands. The most difficult part of the design process is the selection of closed-loop system and observer pole locations. The select_poles() function, as described in this chapter (provided on the accompanying CD-ROM), determines a suitable set of pole locations from the closed-loop settling time and damping ratio requirements.

The controller resulting from this design approach is a generalization of the PD controller described in Chapter 2. The addition of an error integral to the controller structure enables the elimination of steady-state errors resulting from modeling inaccuracies. Pole placement control with an error integral provides a generalization of the PID control technique discussed in Chapter 2.

The pole placement design approach described in this chapter is applicable to MIMO systems as well as to SISO systems. The same MATLAB commands are used to design the estimator and controller gain in both cases.