806 Chapter 10 / Control Systems Design in State SpaceConcluding Comments on Optimal Regulator Systems
1.Given any initial state x(t 0 ),the optimal regulator problem is to find an allowable
control vector u(t)that transfers the state to the desired region of the state space
and for which the performance index is minimized. For the existence of an optimal
control vector u(t),the system must be completely state controllable.
2.The system that minimizes (or maximizes, as the case may be) the selected
performance index is, by definition, optimal. Although the controller may have
nothing to do with “optimality” in many practical applications, the important point
is that the design based on the quadratic performance index yields a stable control
system.
3.The characteristic of an optimal control law based on a quadratic performance
index is that it is a linear function of the state variables, which implies that we need
to feed back all state variables. This requires that all such variables be available for
feedback. If not all state variables are available for feedback, then we need to
employ a state observer to estimate unmeasurable state variables and use the es-
timated values to generate optimal control signals.
Note that the closed-loop poles of the system designed by the use of the
quadratic optimal regulator approach can be found from
Since these closed-loop poles correspond to the desired closed-loop poles in the
pole-placement approach, the transfer functions of the observer controllers can
be obtained from either Equation (10–74) if the observer is of full-order type or
Equation (10–108) if the observer is of minimum-order type.
4.When the optimal control system is designed in the time domain, it is desirable to
investigate the frequency-response characteristics to compensate for noise effects.
The system frequency-response characteristics must be such that the system at-
tenuates highly in the frequency range where noise and resonance of components
are expected. (To compensate for noise effects, we must in some cases either modify
the optimal configuration and accept suboptimal performance or modify the
performance index.)
5.If the upper limit of integration in the performance index Jgiven by Equation
(10–114) is finite, then it can be shown that the optimal control vector is still a
linear function of the state variables, but with time-varying coefficients. (Therefore,
the determination of the optimal control vector involves that of optimal time-
varying matrices.)
10–9 Robust Control Systems
Suppose that given a control object (i.e., a system with a flexible arm) we wish to de-
sign a control system. The first step in the design of a control system is to obtain a
mathematical model of the control object based on the physical law. Quite often the
model may be nonlinear and possibly with distributed parameters. Such a model may
be difficult to analyze. It is desirable to approximate it by a linear constant-coefficient
system that will approximate the actual object fairly well. Note that even though the
∑s I-A+BK∑= 0
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