93
_*
.*
The discrete-time matrices F^k) and G^k) corresponding to the plant
are obtained [26] by the relationships
F*(k) e^kT)T
(5-32)
0
(5-33)
Where T is again the sample period. By defining F^k) and G^k) in
this manner it is implicitly assumed that the dynamics of the linearized
time-varying system (5-13) do not change over any given sample period.
Thus, excluding the case when the reference and disturbance signals are
constant, equation (5-30) is indeed only an approximation.
Assuming that our discrete-time model is reasonably accurate, the
feedback gains K^k) and K2(k) are selected to give stability of the
discrete-time system (5-30). The actual mechanism for selecting the
feedback gains shall not be discussed, however, solving an algebraic
Riccati equation would be one approach [24]. In any event there is a
stabilizability (controllability) requirement for the discretized system
which must be met. In general, the controllability requirement will be
met whenever the continuous-time system is controllable [27] so that
controllability of the pair given in equation (5-14) is often
sufficient.