Control of Linear Plants in the Presence of Disturbances

Abstract

Consider a control system operating in the presence of disturbances, and given reference inputs which define the desired profiles of controlled variables. Accordingly, the behaviour of control system in the case of linear controlled plant is described by linear non-homogeneous differential equation

$$\dot X = Ax + Bu + Qf(t),$$

((10.1))

where f (t) ∈ℝ¹ is the vector characterizing external actions. If the error coordinates are substituted in the state vector for the controlled variables, reference inputs will play a part of disturbances in the new space. Assume that (10.1) is written after such a substitution, vector f (t) then is just a disturbing action, and it is desirable to reduce its effect upon the system behaviour or eliminate it at all. This problem is pivotal in the invariance theory. Let us discuss in short the ideas that may underlie the design of invariant systems.