Ch_ 6

6.8. Lhtear discrete-ti~nc opti~t~alolfrp~ftfeedback coflhol/ers of reduced cli,i~e~lsio~u

Consider the linear time-invariant discrete-time system

all for i 2 i,, where col [w,(i), i~'?(i)], 2 ill,forms a sequence of uncorrelated stochastic vectors uncorrelated with xn. Consider for this system the timeinvarianl controller

Assume that the interconnection of controller and plant is asymptotically stable.

(a) Develop matrix relations that can be used to compute expressions of Ihe form

Presuming that computer programs can be developed that determine the controller matrices L, K,, F, and K, such that 6-539is minimized while 6-540is constra~nedto a given value, outline a method for determining discrete-time optimal output feedback controllers of reduced dimensions. (Compare the continuous-time approach discussed in Section 5.7.)

(b) When gradient methods are used to solve numerically the optimization problem of (a), the follow~ngresult is useful. Let M, N , and R be given matrices of compatible dimensions, each depending upon a parameter y. Let Sbe the solution of the linear matrix equation

S = M S M+~N,

and consider the scalar

tr (SR)

as a function of y. Then the gradient of 6-542with respect to 11 is given by