Text 5. Stability and performance

Stability and performance are two of the fundamental issues in the design, analysis and evaluation of control systems. Stability means that, in the absence of external excitation, all signals in the system decay to zero. Stability of the closed loop system is an absolute requirement since its absence causes signals to grow without bound, eventually destroying and breaking down the plant. This is what happens when an aircraft crashes, or a satellite spins out of control or a nuclear reactor core heats up uncontrollably and melts down. In many interesting applications the open loop plant is unstable and the job of feedback control is to stabilize the system. While feedback is necessary to make the system track the reference input, its presence in control systems causes the potential for instability to be everpresent and very real. We shall make this notion more precise below in the context of servomechanisms. In engineering systems it is of fundamental importance that control systems be designed so that stability is preserved in the face of various classes of uncertainties. This property is known as robust stability.

The performance of a system usually refers to its ability to track reference signals closely and reject disturbances. A well designed control system or servomechanism should be capable of tracking all reference signals belonging to a class of signals, without excessive error, despite various types of uncertainties. In other words the worst case performance over the uncertainty set should be acceptable. This is, roughly speaking, referred to as robust performance.

In analysis and design it is customary to work with a nominal mathematical model. This is invariably assumed to be linear and time invariant, because this is the only class of systems for which there exists any reasonably general design theory. Nevertheless, such models are usually a gross oversimplification and it is therefore necessary to test the validity of any proposed design by testing its performance when the model is significantly different from the nominal.

In summary the requirements of robust stability and performance are meant to ensure that the control system functions reliably despite the presence of significant uncertainty regarding the model of the system and the precise description of the external signals to be tracked or rejected.

(from S.P.Bhattacharyya, H. Chapellat, L.H.Keel. Robust Control. The Parametric Approach)