'(k) = -Kl(k)'(k) - K2(k)n(k) (5-28)
To be consistant with the time-varying feedback law required for the
quasi-state approach, the gains are shown to be functions of the sample integer k. From (5-28) and the assumption that a zero-order hold will be employed, the actual control input to the plant can be expressed as
"(t) = -Kl(k)^(k) - K2(k)n(k) , kT < t < (k+l)T (5-29)
A necessary requirement for the discrete-time control scheme to be satisfactory is that the control law of (5-29) must result in asymptotic stability of the closed-loop transient system.
One possible method [24], [25] of selecting the feedback is based on a discretized model for the nominal linearized system. This nominal linearized system is given in continuous-time form by equation (5-13). Since the internal model system has already been given in discrete-time form, only the part of the linearized system corresponding to the plant must be discretized. Assuming that the continuous-time system (5-13) is slowly time-varying, we may write the discrete-time equation approximating the dynamics of (5-13) as
xA(k+l) Fd(k)XA(k) (5-30)
where *F(k) - Gd(k)Kl(k) -Gd(k)K2(k)
TA*d(k) dI (5-31)
Ad-BdH Ad I