Page 101 - Biosystems Engineering

82 Chapter Two

thus needs to be rewritten in terms of these variables, and then linear-
ized. Using Eqs. (2.98) and (2.99), u can be written as follows:
k
⋅
u = α(PS ⋅ ) PS ES
k k k k
Δ
= α(PS ⋅ ) (PS + PS ⋅ ) (ES + ΔES )
k k−1 k k−1 1 k
β
α
≈ ( PS ) ⋅PS ⋅ES + ( PS ) ⋅PS ⋅ ΔES (2.102)
k − 1 k − 1 k − 1 k − 1 k −1 1 k
+ β( PS ) ⋅ES ⋅ ΔPS + β( PS ) ⋅ ΔPS ⋅ ΔEES
k − 1 k − 1 k k − 1 k k
where α is the global amplification and β is the local amplification.
Note that after linearizing the system, the local amplification, as
defined in Eq. (2.99) appears in the equations. By neglecting the sec-
ond-order term and defining a new input vector v containing the
k
change of the pump setting and the change of the engine speed at
time k, u can be written as
k
u = α(PS ⋅ ) PS ⋅ES + B v (2.103)
k k−1 k−1 k−1 u k
where

B = β(PS ⋅ ) [ES PS ]
u k−1 k−1 k−1
Δ
⎡ PS ⎤ (2.104)
v = ⎢ k ⎥
k Δ
⎣ ES k⎦
The system input thus becomes v , and Eq. (2.103) becomes an extra
k
system state with the state equation:
u = u + B v (2.105)
k k−1 u k
Note that B depends on the pump-setting state of the system. Using
u
a constant B across the prediction horizon thus implies linearization.
u
To keep the pump setting and the engine speed within their physical
range, constraints can be introduced. However, constraints can only
be placed on inputs, states, and outputs. Because the only inputs after
linearization are ΔPS and ΔES , integrator states need to be added to
k k
calculate PS and ES . This way, constraints can be imposed on the
k k
calculated PS and ES . The extended state–space description then
k k
becomes
⎧ x k+1 = A x + B v k
L k
L
⎨ (2.106)
⎩ y = C x + D v k
L k
L
k
×
×
×
where system matrices A ∈ 55 , B ∈ 52 , C ∈ 15 , and
L
L
L
×
D ∈ 12 . State vector x now contains five elements: the two model
L k
states of Eq. (2.100), the input state of Eq. (2.105) and two integrator
states for PS and ES .
k k