Figures 5 shows the time responses obtained for
a step reference of 0.5 for
1
y , and 0.3 for
2
y , and
the disturbance rejection for a step disturbance of
0.05 applied to
1
u at 2=t min. Figure 6 shows the
control signals
1
u and
2
u .
For robustness under additive uncertainties at
high frequency, a high-pass filter is used for each
control signal, as described in Section 4.2 which
transfer form is
3.0/)7.01(
1
2
−
−= qIW
u
. Using
the optimization procedure based on LMIs gives a
multivariable Youla parameter as a
22 × matrix of
polynomials of order
20
Q
n .
Figure 7 shows the singular values analysis of
transfer from
b to control signals u (from
Figure 4). The greatest value of maximal singular
values represents the
∞
H norm. We can remark that
this
∞
H norm has been reduced. In this way the
stability robustness is improved with respect to high-
frequency additive unstructured uncertainties.
10
-1
10
0
10
1
10
2
-15
-10
-5
5
10
15
20
25
30
Frequency (rad/min)
Before robustification
After robustification
Singular Values (dB)
Figure 7: Singular values before and after robustification.
0
0
0.5
1
1.5
2
2.5 3
3.5
0.1
0.2
0.3
0.4
0.5
Time
minute
1
y
2
y
Setpoint
Before robustification
After robustification
Time Responses
Figure 8:
1
y and
2
y before and after robustification.
Figures 5 and 6 show that after robustification
the input/output behaviour is unchanged, but the
disturbance is rejected more slowly by the
robustified controller. In fact, the robustified
controller has a slower disturbance rejection, but a
higher robust stability. To support this, a high
frequency neglected dynamics of the actuator
1
u has
been considered. Thus the transfer between
11
/ uy
corresponds to
)07.01)(7.01/(1 ss ++ . Figure 8
illustrates that the initial controller behaviour is
destabilized by this uncertainty, but the robustified
controller remains stable; it also shows the influence
of the considered unstructured uncertainty to
2
y .
6 CONCLUSIONS
This paper has presented a new MIMO complete
methodology which enables robustifing an initial
multivariable MPC controller in state-space
formalism using the Youla parameter framework. In
order to improve robustness towards unstructured
uncertainties, a
∞
H convex optimization problem
was solved using the LMIs techniques. The major
advantage of the developed structure is the state-
space formulation of this MPC robustification
problem for MIMO systems with a reduced
computational effort compared to the transfer
function formalism. This method can also be applied
to non square systems, which otherwise are more
difficult to control. This technique enables also the
use of time-domain templates to manage the
compromise between stability robustness and
nominal performance.
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