8.1.  INTRODUCTION

State-space analysis was introduced in Chapter 2, and has been used in parallel with the classical frequency-domain analyses techniques presented in Chapters 3 through 7. It was shown that the state-space approach is applicable to a wider class of problems such as multiple-input/multiple-output (MIMO) control systems. Chapter 7 applied the frequency-domain approaches such as the Bode diagram, and the root locus to linear control-system design.

In the design of a control system, the question arises as to where to place the closed-loop roots. In Section 7.9 which presented the root-locus method, we could specify where to place the dominant-pair of complex-conjugate roots in order to obtain a desired transient response. However, we could not do so with great certainty because we were never sure what effect the higher-order poles would have on the second-order approximation.

The control-system design engineer desires to have design methods available which would enable the design to proceed by specifying all of the closed-loop poles of higher-order control systems. Unfortunately, the frequency-domain design methods presented in Chapter 7 do not permit the control-system engineer to specify all poles in control systems which are higher than two because they do not provide a sufficient number of unknowns for solving uniquely for the specified closed-loop poles. This problem is overcome using state-space methods which provide additional adjustable parameters, and methods for determining these parameters.

This chapter presents a modern control-system design method using state-space techniques known as pole placement or pole assignment. This design technique is similar to what we did in Section 7.9 where we placed two dominant complex-conjugate poles of the closed-loop transfer function in desired locations in order to obtain desirable transient responses. However, in this chapter, we will show how pole placement allows the control-system engineer to place all of the poles of the closed-loop transfer function in desirable locations. Ackermann’s formula is also presented for designs using pole placement for application in those control systems that require feedback from state variables which are not phase variables (where each subsequent state variable is defined as the derivative of the previous state variable). A practical problem arises with the pole placement method involving cost and the availability of determining (measuring) all of the system variables needed for obtaining a solution. In many pracical control systems, all of the system state variables may not be available due to cost considerations, environmental considerations (e.g., nuclear power plant control systems), and the availability of transducers to measure certain states. For these cases, it is necessary for the control-system engineer to estimate the state variables that cannot be measured from the state variables that can be measured. Therefore, in addition to pole placement, this chapter also presents the very important subjects of controllability, observability, and estimation.

This chapter on modern control-system design also presents the design of robust control systems. Robust control systems are concerned with determining a stabilizing controller that achieves feedback performance in terms of stability and accuracy requirements, but the controller must achieve performance that is robust (insensitive) to plant uncertainty, parameter variation, and external disturbances. The design of two-degrees-of-freedom compensation control systems exhibiting desirable robustness to plant uncertainty, parameter variation, and external disturbances is presented.

This chapter concludes with an introduction to H^∞ control concepts which is a new technique that emerged in the 1980s that combines both the frequency- and time-domain approaches to provide a unified design approach. The H^∞ approach has dominated the trend of control-system development in the 1980s and 1990s. The H^∞ control-system design approach expands on the concept of robustness presented in this chapter, sensitivity (presented in Chapter 5), together with the frequency and state-variable domain techniques presented in this book. The H^∞ approach is applied to determine the optimum sensitivity for control systems.