The Development of Neural Network Based System Identification ...

168 CHAPTER 6 NEURAL NETWORK BASED PREDICTIVE CONTROL SYSTEM
6.6 PREDICTION FROM THE STATE SPACE MODELS
After formulation of the state space model, the next step in MPC control design is
to calculate the predicted plant output variables with the future control signal as
the adjustable variable. In this section, the prediction from the state space model is
described within a single optimisation window with N p sample length. Assume that
at the current sampling time k, the state variable vector x(k) is available through
measurement. The future state variables predicted from the state space model over N p
sample length can be represented in a sequence as:
x(k + 1 | k), x(k + 2 | k), · · · , x(k + N p | k)
Similarly, the sequence of future control variables over the control horizon N u are
denoted by:
∆u(k), ∆u(k + 1), · · · , ∆u(k + N c − 1)
where the control horizon N c is chosen to be equal or less to the prediction horizon N p .
The future state variables can be determined sequentially using the set of the future
control variables. Based on matrix A, B and C from Equation (6.26), the future state
variables are updated as follows:
x(k + 1 | k) = Ax(k) + B∆u(k) + B d w(k)
x(k + 2 | k) = Ax(k + 1 | k) + B∆u(k + 1) + B d w(k + 1 | k)
= A 2 x(k) + AB∆u(k) + B∆u(k + 1)
+ AB d w(k) + B d w(k + 1 | k)
.
x(k + N p | k) = A Np x(k) + A Np−1 B∆u(k) + A Np−2 B∆u(k + 1)
+ A Np−Nc B∆u(k + N c − 1) + A Np−1 B d w(k)
+ A Np−2 B d w(k + 1 | k) + · · · + B d w(k + N p − 1 | k) (6.27)