Section 10–8 / Quadratic Optimal Regulator Systems 793frequency-response method. If the system designed has poor stability margins, it
is possible that the designed system may become unstable if the mathematical
model involves uncertainties.
7.Note that for nth-order systems, classical design methods (root-locus and
frequency-response methods) yield low-order compensators (first or second order).
Since the observer-based controllers are nth-orderCor(N-m)th-order if the
minimum-order observer is usedDfor an nth-order system, the designed system
will become 2nth order Cor(2n-m)th orderD. Since lower-order compensators are
cheaper than higher-order ones, the designer should first apply classical methods
and, if no suitable compensators can be determined, then try the pole-placement-
with-observer design approach presented in this chapter.
10–8 Quadratic Optimal Regulator Systems
An advantage of the quadratic optimal control method over the pole-placement method
is that the former provides a systematic way of computing the state feedback control gain
matrix.
Quadratic Optimal Regulator Problems. We shall now consider the optimal
regulator problem that, given the system equation
(10–112)
determines the matrix Kof the optimal control vector
(10–113)
so as to minimize the performance index
(10–114)
whereQis a positive-definite (or positive-semidefinite) Hermitian or real symmetric
matrix and Ris a positive-definite Hermitian or real symmetric matrix. Note that the
second term on the right-hand side of Equation (10–114) accounts for the expenditure
of the energy of the control signals. The matrices QandRdetermine the relative
importance of the error and the expenditure of this energy. In this problem, we assume
that the control vector u(t)is unconstrained.
As will be seen later, the linear control law given by Equation (10–113) is the optimal
control law. Therefore, if the unknown elements of the matrix Kare determined so as
to minimize the performance index, then u(t)=–Kx(t)is optimal for any initial state
x(0). The block diagram showing the optimal configuration is shown in Figure 10–35.
J=
3
q0(xQx+u Ru)dt
u(t)=-Kx(t)
x# =Ax+Bu
x.=Ax+ Buu x- K
Figure 10–35
Optimal regulator
system.