The Development of Neural Network Based System Identification ...

6.6 PREDICTION FROM THE STATE SPACE MODELS 169
Similarly, the predicted output variables are derived from the predicted state variables
by substitution leading to:
y(k + 1 | k) = CAx(k) + CB∆u(k) + CB d w(k)
y(k + 2 | k) = CAx(k + 1 | k) + CB∆u(k + 1) + CB d w(k + 1 | k)
= CA 2 x(k) + CAB∆u(k) + CB∆u(k + 1)
+ CAB d w(k) + CB d w(k + 1 | k)
.
y(k + N p | k) = CA Np x(k) + CA Np−1 B∆u(k) + CA Np−2 B∆u(k + 1)
+ CA Np−Nc B∆u(k + N c − 1) + CA Np−1 B d w(k)
+ CA Np−2 B d w(k + 1 | k) + · · · + CB d w(k + N p − 1 | k) (6.28)
Here, the w(k) is a zero mean white noise sequence and the future predicted value of
w(k + 1 | k) is assumed to be zero. Note that the predicted output variables (6.28) and
the predicted state variable (6.27) are presented using only the current state variable
information x(k) and sequence of current and future control movements ∆u(k + j),
where j = 0, 1, 2, · · · , N c − 1. For notation simplicity, the expectation operator can be
omitted. By defining the following vectors, (similar to definition in Equation (6.2)):
Ŷ = [y(k + 1 | k) y(k + 2 | k) y(k + 3 | k) · · · y(k + N p | k)] T
∆U = [∆u(k) ∆u(k + 1) ∆u(k + 2) · · · ∆u(k + N c − 1)] T
the formulation for the predicted output variables (6.28) can be represented in a compact
matrix form as:
Ŷ = Γx(k) + Φ∆U (6.29)