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1566 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
• Mismatch model
In order to create performance degradation condition,
A disturbance, Δ A was added to the nominal model (2),
to construct a man-made mismatch model as,
⎛ 0 1 0 0 ⎞
⎜
0 0 1 0
⎟
A'
= A+Δ A= ⎜
⎟. (36)
⎜ 0 0 0 1 ⎟
⎜
⎟
⎝−600 −300 −100 −30⎠
• MPC tuning parameters
Based on the extended MVC 3 algorithm combined
with LQR inverse optimal, the MPC controller matrixes
of above mismatch model can be gotten.
R = 0.0388 , Q = diag(0.0706 0.0002 0.0357 0.000)
Through MVC 3 performance evaluation, controller
performance declining was detected. If it dropped below
threshold, ψ , (here, ψ is set to 0.8), then, weight
parameter of manipulated variable in MVC 3 , R ,is
updated by R'
= R+ ξ I ,(here,ψ is set to 0.5). Repeat the
MPC weight matrix calculation process to update Q,
R,
the new MPC controller parameters
as R = 0.0952
,
Q = diag(0.1703 0.0006 0.0552 0.0000) can be
gotten. Apply the new controller parameters to operation
to restore the desired operational performance.
V. SIMULATION AND ANALYSIS OF MPC CONTROLLER
A. Simulation and Comparison of MPC Controller and
the LQR Controller
• Simulation of LQR controller
LQR controlled system in the Simulink of Matlab was
shown in Fig. 3. Parameter of LQR block was set to
K = ( −31.623 − 20.151 72.718 13.155)
, which was
provided by Googol Technology LTD.
Figure 3. Simulation block of LQR control loop
• Simulation of MPC controller
MPC controlled system in the Simulink was shown in
Fig. 4, where, parameter of Acker block was set to
stabilizer feedback gain. Weighted matrixes of MPC
block were set the parameters calculated by the nominal
model.
Figure 4. Simulation block of MPC control loop
• Analysis of simulations results
Simulation curve charts of MPC controller and LQR
controller are shown in Fig. 5, where, u denotes
manipulative variable, which is cart angular velocity.
angle, pos denote outputs, they are pendulum angle and
cart position.
The maximum deviation of MPC controller and LQR
controller are shown in Table I and comparison bar chart
is shown in Fig. 6.
TABLE I.
MAXIMUM DEVIATION COMPARISON
u angle pos
MPC 0.8240 0.0096 0.0093
LQR 34.4589 0.3101 0.3533
Obviously, due to the introduction of steady state
manipulative variable and outputs covariance constraint,
MPC controller can make the maximum deviation
significantly reduced than LQR controller, which greatly
improved the system dynamic performance.
B. Simulation of MPC Controller Tuning
• Simulation of MPC controller tuning
MPC controller tuning dynamic curves was obtained
by replace original weight matrix Q , R of MPC controller
with mismatched and Controller-tuned parameters, shown
in Fig. 7, where, the legend (good, bad and tuned) means
controller running under, nominal model, mismatch
model and MPC controller tuned.
Through curves, if using original controller parameters
to control mismatch model, it would lead an increase on
manipulative variable and outputs deviation. After
adjusting controller parameters by controller tuning
system, the deviation reduced to some extent. This proves
the feasible of MPC controller tuning algorithm.
• Extended MVC 3 performance evaluation method
The extended MVC 3 performance evaluation curve is
shown in Fig. 8.
Set λ = 1 in the MVC 3 performance evolution object
function (18) and get
p m
i= 1
i i
j=
1
j j
−7
benchmark J 3 = ∑qY + ∑ rU by LMI method,
MVC
here, J 3 = 1.3217 × 10 .Then actual run-time
MVC
variance was compared with this benchmark, get η
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