ABSTRACT
For the control design we choose the criterion 6.4, i.e., to minimize a trade-off between the size of the control error e = z - r and the size of the control input u:
min(||e||2Ql + ||ti||J,) = mm J'eT(t)Qie(t)+uT(t)Q^u(t)dt (9.3)
Recall that the left hand side can also be interpreted as the time average ("power") of the integrand in the right hand side, as well as mathematical expectations (variances) of stationary stochastic processes, see (5.26). (It might be a good idea to go back and check Sections 5.2 and 5.3 at this point.) We seek the optimal linear controller, so the minimization should be carried out over all linear transfer functions Fy and FT. The formal solution to this problem is
240 Chapter 9 Minimization of Quadratic Criteria: LQG
most easily found if the system (9.1) is represented in state space form. We shall compute this solution in the next section. Before that, some general reflections are in order.
