This chapter is devoted to discrete-time control system design. The problem of forming desired output transients for a discrete-time system described by a difference equation is discussed. The insensitivity condition for the output transients with respect to varying parameters of the system and external disturbances is introduced, and a discrete-time control law is constructed. Desired output behavior with prescribed dynamics is achieved by inducing two-time-scale motions in the closed loop system, despite uncertainty in the system description. The singular perturbation method is used to analyze fast and slow motions in the discrete-time closed-loop control system. The approach may be considered as the discrete-time counterpart of the above design methodology for continuous-time control systems with the highest derivative in feedback. The chapter opens with explanations in simplified form, while various peculiarities associated with the sampling process will be discussed in later sections.

11.1 SISO Two-Time-Scale Discrete-Time Control Systems

11.1.1 Discrete-Time Systems

Let us consider a discrete-time control system given by a difference equation of the form

(11.1)

where

• k is the discrete time variable, k = 0, 1, ...;

• y _k is the output, available for measurement;

• u _k is the control;

• w _k is the external disturbance, unavailable for measurement.

Assumption 11.1:

For the unforced system (11.1) we have

(11.2)

In other words, only small variations of y _k occur during the settling time of the nth-order discrete-time system with deadbeat response.

Note that the requirement (11.2) can be easily provided for a sampled-data...