Model Predictive Control System Design and Implementation Using ...

214 6 Continuous-time MPC
through the computation of the controllability and observability matrices.
Here, the controllability matrix is calculated as
⎡
[ BABA 2 B ] = ⎣ 01 0 ⎤
10−ω0
2 ⎦ ,
00 1
and the observability matrix is calculated as
⎡
⎣ C ⎤ ⎡
CA ⎦ = ⎣ 001 ⎤
100⎦ .
CA 2 010
Both matrices have non-zero determinant. Therefore, the augmented model is
both controllable and observable.
Here, the desired closed-loop polynomial is selected as (s +3ω 0 ) 3 .The
feedback control law is defined as
⎡
˙u(t) =− [ ]
k 1 k 2 k 3
⎣ ẋ1(t)
⎤
ẋ 2 (t) ⎦ . (6.15)
x 3 (t)
The closed-loop characteristic polynomial is calculated via
⎡
⎤
s −1 0
det ⎣ ω0 2 + k 1 s + k 2 k 3
⎦ = s 3 + k 2 s 2 +(ω0 2 + k 1 )s + k 3 .
−1 0 s
By equating the closed-loop characteristic polynomial to the desired closedloop
polynomial, we find the coefficients for the controller as k 1 = 26ω0 2,
k 2 =9ω 0 and k 3 =27ω0.
3
To examine the behaviour of the closed-loop system, we note that the
derivative of the constant input disturbance is zero, i.e., d(t) ˙ =0.Theclosedloop
system is
⎡
⎣ ẍ1(t) ⎤ ⎡
⎤ ⎡
0 1 0
ẍ 2 (t) ⎦ = ⎣ −27ω0 2 −9ω0 2 −27ω 2 ⎦ ⎣ ẋ1(t) ⎤
0 ẋ 2 (t) ⎦ (6.16)
ẋ 3 (t)
1 0 0 x 3 (t)
⎡
˙u(t) =− [ ]
k 1 k 2 k 3
⎣ ẋ1(t) ⎤
ẋ 2 (t) ⎦ . (6.17)
x 3 (t)
Figure 6.1a shows that the derivative ˙u(t) exponentially decays to zero, and
Figure 6.1b shows the area under the plot ˙u(t) 2 is bounded for an arbitrarily
large t. Since the signal u(t) is an exponentially decay function that goes to
zero, ∫ ∞
˙u(t) 2 dt < ∞. In conjunction with the discussion given in Sections 5.2
0
and 5.3, this example demonstrated that the derivative of the control signal
is a good candidate to be modelled by using a set of Laguerre functions.