17.1 Introduction

The time delays that usually occur in relay and sliding mode control systems must be considered in system analysis and design [1]. On the other hand, the presence of time delay does not allow the sliding mode control to be designed in the space of state variables. Even in the simplest one-dimensional delayed relay control system only oscillatory solutions can occur [2]. That is why the following directions in relay delayed control require investigation:

• research into time delay compensation;

• control of the amplitude of any oscillations.

A Pade approximation of the delay reduces the relay delay output tracking problem to a sliding mode control problem for a nonminimum-phase system. This approach was suggested in Reference 3. The sliding mode control [4] was designed in the space of predictor variables (see [5] also). This approach allowed the eigenvalue assignment problem to be solved without any restriction on the time delay and spectral properties of the open loop system. However, the sliding mode control design in the space of predictor variables [6, 7]:

• cannot compensate even matched uncertainties;

• in the simplest case of square systems, if the dimensions of the state space and control vector are the same, the sliding mode design in the space of predictor variables can remove uncertainties in the space of predictor variables but cannot guarantee that the effects of uncertainties in the space of state variables will be compensated.

Robustness properties of Smith predictors with respect to uncertainties...