The Development of Neural Network Based System Identification ...

6.5 GENERAL FORMULATION OF AUGMENTED MODEL 167
state space model equation is given:
⎡
⎤ ⎡
⎤ ⎡ ⎤ ⎡ ⎤
⎢
⎣ ∆x m(k + 1) ⎥ ⎢
⎦ = ⎣ A m Om
T ⎥ ⎢
⎦ ⎣ ∆x m(k) ⎥ ⎢
⎦ + ⎣ B m ⎥
⎦ ∆u(k)
y(k + 1) C m A m I q×q y(k) C m B m
⎡ ⎤
⎢
+ ⎣ B d ⎥
⎦ ∆w(k)
C m B d
[
y(k) =
O m
] ⎡ ⎤
⎢∆x m (k) ⎥
I q×q ⎣ ⎦ (6.25)
y(k)
where O m is a q × n 1 zero matrix and I q×q is an identity matrix with q × q dimensions.
Then, for notation simplicity, Equation (6.25) is denoted as:
x(k + 1) = Ax(k) + B∆u(k) + B d w(k)
y(k) = Cx(k) (6.26)
Since the original state space model is augmented with integrators and MPC
controller is based on the augmented model, it is important to ensure that the augmented
system plant meets controllable and observable conditions, especially with respect to
unstable system [Wang, 2009c]. Controllability is a pre-requisite for the predictive
control system to achieve the desired closed-loop control performance, and observability
is a pre-requisite for the successful design of an observer.