model to be used for design purposes may be a simplified one, it is necessary that such
a model must include any intrinsic character of the actual object. Assuming that we can
get a model that approximates the actual system quite well, we must get a simplified
model for the purpose of designing the control system that will require a compensator
of lowest order possible. Thus, a model of a control object (whatever it may be) will
probably include an error in the modeling process. Note that in the frequency-response
approach to control systems design, we use phase and gain margins to take care of
the modeling errors. However, in the state-space approach, which is based on the dif-
ferential equations of the plant dynamics, no such “margins” are involved in the
design process.
Since the actual plant differs from the model used in the design, a question arises
whether the controller designed using a model will work satisfactorily with the actu-
al plant. To ensure that it will do so, robust control theory has been developed since
around 1980.
Robust control theory uses the assumption that the models we use in designing con-
trol systems have modeling errors. We shall present an introduction to this theory in this
section. Basically, the theory assumes that there is an uncertainty or error between the
actual plant and its mathematical model and includes such uncertainty or error in the
design process of the control system.
Systems designed based on the robust control theory will possess the following
properties:
(1)Robust stability. The control system designed is stable in the presence of
perturbation.
(2)Robust performance. The control system exhibits predetermined response
characteristics in the presence of perturbation.
This theory requires considerations based on frequency-response analysis and time-
domain analysis. Because of the mathematical complications associated with robust con-
trol theory, detailed discussion of robust control theory is beyond the scope of the senior
engineering student. In this section, only introductory discussion of robust control the-
ory is presented.
Uncertain Elements in Plant Dynamics. The term uncertaintyrefers to the dif-
ferences or errors between the model of the plant and the actual plant.
Uncertain elements that may appear in practical systems may be classified as struc-
tureduncertainty and unstructureduncertainty. An example of structured uncertainty is
any parametric variation in the plant dynamics, such as variations in poles and zeros
of the plant transfer function. Examples of unstructured uncertainty include frequency-
dependent uncertainty, such as high-frequency modes that we normally neglect in mod-
eling plant dynamics. For example, in the modeling of a flexible-arm system, the model
may include a finite number of modes of oscillation. The modes of oscillation that are not
included in the modeling behave as uncertainty of the system. Another example of un-
certainty occurs in the linearization of a nonlinear plant. If the actual plant is nonlinear
and its model is linear, then the difference acts as unstructured uncertainty.
In this section we consider the case where the uncertainty is unstructured. In addi-
tion we assume that the plant involves only one uncertainty. (Some plants may involve
multiple uncertain elements.)
Section 10–9 / Robust Control Systems 807