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Optimal Feedback for the Linear Servomechanism Problem
In this section we discuss the consequences of using optimal
control theory as a means of determining the stabilizing feedback gains
for the linear servomechanism problem. Only the linear problem shall be
treated since the interpretation of the results for the nonlinear
problem is not clear. It is true, however, that the actual method of
feedback gain selection discussed in both this and the succeeding
section can be applied to the nonlinear problem when linearization
techniques are used.
Now consider the well known linear optimal control problem [16],
[20]. That is, given the linear time-invariant system
x(t) = Fx(t) + Gu(t) (4-41)
select the control u(t) to minimize the quadratic performance index
oo
J = / [x'(t)Qx(t) + u'(t)Ru(t)]dt (4-42)
o
Where Q 5 0 and R > 0 are symmetric matrices of appropriate
dimension. The optimal control law is found to be of the form
u(t) = -Kx(t) (4-43)
where K is a time-invariant feedback gain determined by solving an
algebraic Riccati equation.
The question answered here concerns the interpretation of applying
such a control to the linear servomechanism system. The reason that an
interpretation is considered necessary is that the optimal control is