Model Predictive Control System Design and Implementation Using ...

3.8 Closed-form Solution of Constrained Control for SISO Systems 135
is replaced by the estimated state ˆx(k i ), which leads to the question of the
effect of estimation error on the constraints.
After ranking the constraints, assuming that the most important constraints
are on the amplitude of the control, and the less important constraints
are on the difference of the control and the least important constraints are on
the output, the implementation is as follows:
1. The least important constraints on y if they are violated, will be treated
as an active constraint so as to obtain η that will satisfy this constraint
on y.
2. Check whether this η will lead to the satisfaction of the constraints on
Δu. Ifnot,Δu will be modified to satisfy the constraints on Δu. With
this new Δu, we need to check if the constraints on u are satisfied. If not,
u will be modified to satisfy the constraints on u.
3.8.1 MATLAB Tutorial: Constrained Control of DC Motor
A DC motor has a continuous-time transfer function as
K
G(s) =
s(Ts+1) , (3.70)
where the control signal is input voltage to the motor and the output is the
angular position. The discrete-time model with zero-order-hold is expressed
as
b 1 z + b 2
G(z) =
z 2 . (3.71)
+ a 1 z + a 2
Instead of using an observer in the implementation, we will choose the state
variables according to the input and output signals so that the state variables
are directly measurable. To do so, let us look at the difference equation that
relates the input to the output:
y(k +2)+a 1 y(k +1)+a 2 y(k) =b 1 u(k +1)+b 2 u(k). (3.72)
Choosing the state variable vector as
⎡
⎣
⎤ ⎡
y(k +1)
y(k) ⎦ =
u(k)
x m (k) = [ y(k) y(k − 1) u(k − 1) ] T
⎣ −a ⎤ ⎡
1 −a 2 b 2
1 0 0 ⎦ ⎣
y(k)
⎤ ⎡
y(k − 1) ⎦ + ⎣ b ⎤
1
0 ⎦ u(k)
0 0 0 u(k − 1) 1
y(k) = [ 100 ] x m (k). (3.73)
The augmented state-space model is obtained by choosing the state variable
vector as
x(k) = [ Δy(k) Δy(k − 1) Δu(k − 1) y(k) ] T
,