On invariance conditions in variable structure controllers

In practical systems, plant poles vary to some degree for reasons such as nonlinearity, ageing, wear, temperature dependence, etc., producing changes in the plant matrix. The range of such parameter variations can often be predetermined and account of this taken in the variable structure system control algorithms. Itkis (1978) and Utkin (1977, 1978) developed conditions guaranteeing reachability and sliding motion in such controllers, and suggest that the closed loop system response can be made invariant to certain classes of plant parameter variation and external disturbance. It is shown that this invariance of response only applies when the state used to construct the switching function model are derivatives of one another, i.e. in companion form. Conditions are developed that guarantee sliding motion and reachability for variable structure control of a SISO plant with uncertainties in its matrix using companion and noncompanion form states. The closed loop characteristic equations of both types of controller is developed. It is shown, in the case of companion form states, that the output dynamic response can be made invariant to plant parameter changes and conversely, in the case of noncompanion form states, the output dynamic response becomes a direct function of the changes in the open loop plant matrix. It is further shown, in the latter case, that the stability of the closed loop response may also be affected