Design of Control Systems in State Space

Abstract

In this chapter, design of control systems in state space is presented. The necessary and sufficient condition for arbitrary pole-placement is that the system is completely state controllable. The state variables are available for feedback. Control input is unconstrained. In pole-placement design, all the poles are located as desired. The appropriate feedback gain matrix has to be determined. We present three methods for the determination of the state feedback gain matrix K: (i) direct comparison method; (ii) using transformation matrix; and (iii) using Ackermann’s formula. In pole-placement design of control systems, access to the state variables is necessary. However, if they are not accessible then we are forced to estimate them. A device that estimates the state variables from the control and output variables is called the state observer. The same three methods are used to determine the observer gain matrix also. Obtaining the gain vector for state variable feedback can also be determined using optimization theory. All practical systems can be implemented with sampled-data with adequate accuracy. The digital implementation provides the advantages of low cost and high reliability.