Closed-Loop Systems

Stability of closed-loop feedback systems is a major control objective. Closed-loop system observability and controllability are prerequisites. If a plant is completely state controllable (observable), so is the closed-loop system. Observable eigen modes of the plant are also observable modes of the closed-loop system. As to the stability of a closed-loop system, it is not sufficient to consider input–output stability. A closed-loop system must be checked for internal stability which depends on both, the sensitivity and the complementary sensitivity function. The two functions are also important for the two major performance objectives, namely reference tracking and disturbance rejection. For input–output stability of closed-loop systems, applicability of the well-known Nyquist criterion extends from SISO to nominal MIMO systems. Instead of considering a single Nyquist plot of the determinant of the return difference matrix, the encirclement of all eigenloci around the critical point \(-1\) can be checked. In the case of plant uncertainties, the small-gain theorem is useful. This chapter finally considers the stability of feedback loops with a diagonal controller and a MIMO system.