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


                            (t) = Fx(t) + Gu(t)                    (4-41)


select the control u(t) to minimize the quadratic performance index



                   J = f [x'(t)Qx(t) + u'(t)Ru(t)]dt               (4-42)
                        0

Where    Q > 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