28

Usinga reduced set of state variables greatly reduces the

part of state in our control system.

computational effort required to construct a model of the discrete system. We will use Qto denote

the vector of RVs and define g(x) to be the projection function such that

Q= g (x).

(3.5)

Note that choosing to work strictly in the reduced state space carries the imlicit assumption that

controlling the reduced state is sufficient to control the complete system state in a desireable way.

For this to hold, g (x) and [!]Umust be chosen appropriately. However, there is no guarantee that

such an appropriate choice exists.

Replacing the full state in Eq. 3.3 with the reduced

state,

Q,

and

substituting

the

control

formulation of Eq. 3.4 yields the reduced-order system which we will control directly:

Q_i+1=h₍Q
i^,Unom+Ki^{* DU)}

(3.6)

For a given cycle, Q_i, U^nom, and [!]Uare treated as apriori information. We will use Q_i+1= h(K)

as a convinient short form of Eq. 3.6 when we are interested in discussing only the effect of Kon

the system.

Eq. 3.6 can be restated as

Q_i+1=Q^nom+ DQ_i+1=h₍Q
i^,Unom+Ki^{* DU)}

(3.7)

(3.8)

Q^nom=h₍Q_i,U^nom₎

where

We choose to approximate the response of this system about the nominal operating point Q^nom

(where K=0) using the following linear predictive model:

DQ
i+1^{=J K}i

(3.9)