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JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1565
From LMI and ARE equivalence relation [28], S1
is equal
to:
T T T T
S1 = A PA + A PBK + ( BK) PA + BK P( BK)
T
T
− P+ Q+
K RK
= ( A+ BK) P( A+ BK)
− P+ Q+
K RK
Therefore, LMI form of (23)-(28) can be gotten and the
solution of Q is diagonal matrix.
F. MPC Tuning Based on Extended MVC 3
Consider system (3), outputs and manipulative variable
constraints are as y ( k)
< y and u ( k)
< u , respectively,
i
i
where, yi ( k)
is i th element of y( k ) and u j ( k)
is j th
element of u ( k ) . When the MPC controller is put into
operation, extended MVC 3 performance evaluation
criteria is used to monitor controller performance. If the
performance evolution indexη is below the thresholdψ ,
weighted parameter R can be updated with extended
MVC 3 to improve the robustness of the controlled system.
The block diagram of MPC tuning is shown in Fig. 2.
Figure 2. Block diagram of MPC controller-tuning
IV. MPC CONTROLLER PERFORMANCE EVALUATION
AND TUNING SYSTEM IN SINGLE INVERTED PENDULUM
CONTROL
A. Model Preprocessing
• Stabilizer design
The single inverted pendulum is unstable system,
while, tuning system of infinite MPC require a stable
controlled object. Hence a stabilizer u =− Kx+ v is
needed. For the inverted pendulum system (2), Use
command K=acker(A,B,P) in Matlab, to configure
closed-loop pole to:
( 8 8 2 2 2 2 )
j
P = − − − + i − − i . (32)
j
T
.
The feedback gain is obtained as:
K = ( −17.4150 − 13.0612 60.9383 11.0204)
. (33)
The generalized system matrix after stabilization is as
follows:
⎛x
⎞ ⎛ 0 1 0 0 ⎞⎛x
⎞ ⎛0⎞
⎜
x
⎟ ⎜
0 0 1 0
⎟⎜ x
⎟ ⎜
1
⎟
⎜ ⎟
= ⎜ ⎟⎜ ⎟+
⎜ ⎟u
⎜ θ ⎟ ⎜ 0 0 0 1 ⎟⎜θ
⎟ ⎜0⎟
⎜ θ ⎟ ⎜ ⎟
512 384 136 20 ⎜θ
⎟ ⎜ ⎟
⎝− − − − ⎠
⎝ ⎠
⎝ ⎠ ⎝3⎠
. (34)
⎛x
⎞
⎜
x 1 0 0 0 x
⎟
⎛ ⎞ ⎛ ⎞ ⎛0⎞
y = = ⎜ ⎟
⎜ + u
θ
⎟ ⎜
0 0 1 0
⎟⎜θ
⎟ ⎜
0
⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎜
θ ⎟
⎝ ⎠
The following MPC controller performance evaluation,
tuning system based on extended MVC 3 was built on the
stabilized generalized system (34).
• Discretization
Since the derivation of the above extended MVC 3
algorithm is based on discrete state space model (3),
discretization of system (34) and constructing a suitable
noise are needed. Use command sys=c2d(A,B,Ts) in
Matlab, here Ts = 1s
, and consider the noise to be
stationary, Gaussian White-noise processes, the following
system can be obtained as:
⎛ xk ( + 1) ⎞ ⎛ 0.2584 0.1707 0.0306 0.0017 ⎞
⎜
xk ( 2)
⎟ ⎜
0.8519 0.3805 0.0556 0.0027
⎟
⎜
+
⎟
− − − −
= ⎜ ⎟
⎜θ
( k + 1) ⎟ ⎜ 1.3646 0.1716 −0.0181 −0.0023⎟
⎜ ⎟ ⎜ ⎟
⎝θ
( k + 2) ⎠ ⎝ 1.1850 2.2534 0.4863 0.0282 ⎠
⎛ xk ( ) ⎞ ⎛ 0.2319 ⎞ ⎛0⎞
⎜
xk ( + 1)
⎟ ⎜
0.1757
⎟ ⎜
⎜ ⎟
1
⎟
× + ⎜ ⎟uk
( ) + ⎜ ⎟wk
( )
⎜ θ ( k) ⎟ ⎜−1.
3885⎟ ⎜0⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝θ
( k + 1) ⎠ ⎝ 0.1646 ⎠ ⎝1⎠
where, ∑ w = 0.01 .
⎛ xk ( ) ⎞
xk ( ) 1 0 0 0
⎜
xk ( 1)
⎟
⎛ ⎞ ⎛ ⎞ +
yk ( ) = ⎜ ⎟
⎜
θ( k) ⎟=
⎜
0 0 1 0
⎟ , (35)
⎝ ⎠ ⎝ ⎠ ⎜ θ( k)
⎟
⎜ ⎟
⎝θ
( k + 1) ⎠
B. MPC Tuning Design
• MPC controller
For system (35), choose the initial objective function
of extended MVC 3 as
J = Y + Y + U ,where R = 1, Q = diag(1,1)
,output
1 2
variance bounds as yi
= 0.3, i = 1,2 , manipulative
variable variance bounds to be u = 0.8 .Using the
command of LMI toolbox in Matlab ,the MPC controller
weight matrix of the nominal model (35), R = 0.0167 ,
Q = diag(0.0204,0.1234,0.0007,0.0160) can be gotten.
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