The Development of Neural Network Based System Identification ...

164 CHAPTER 6 NEURAL NETWORK BASED PREDICTIVE CONTROL SYSTEM
6.4 NON-MINIMAL STATE SPACE MODEL REALISATION
There are generally three types of models used in MPC design such as finite impulse
response (FIR)/step response models, transfer function models and state space models
[Rossiter, 2003]. The state space model is preferred over other model types in MPC
controller design due to its simplicity and effective handling of multi-variable problems.
The transfer function models obtained from instantaneous linearisation of the NN model
can also be represented in terms of state space model realisation. This can be done
by reformulating the state variables of the state space model to be identical to the
feedback variables that have been used in the ARX model [Wang, 2009b, Ordys and
Clarke, 1993]. Consider the general discrete time ARX model that describes the input
and output relationship as:
A(q −1 )(1 − q −1 )y(k) = B(q −1 )∆u(k) + q −1 ɛ(t) (6.16)
where ɛ(t) is the input disturbance which is assumed to be a sequence of integrated
white noise, A(q −1 ) and B(q −1 ) are polynomials in the time shift operator given by the
following forms:
A(q −1 ) = 1 + a 1 q −1 + a 2 q −2 · · · + a n q −n
B(q −1 ) = b 1 q −1 + b 2 q −2 · · · + b n q −n (6.17)
Let the polynomial A(q −1 )(1 − q −1 ) be referred as:
A(q −1 )(1 − q −1 ) = 1 + ā 1 q −1 + ā 2 q −2 + · · · + ā n+1 q −(n+1) (6.18)
Next, by choosing the state variable vector as:
x(k) = [y(k) y(k − 1) · · · y(k − n − 1) ∆u(k − 1) ∆u(k − 2) · · · ∆u(k − n)] T (6.19)