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1564 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
(21) is the solution to problem (20). Finally, certainty
equivalence between problems (20) and (19) completes
the proof.
Theorem 2: Let hypotheses 1-3 hold; then the solution
of K ∗ , to problem (12)-(17) is coincident with the
solution of appropriate weighted infinite horizon MPC
problem (19).
Proof. Problem (6)-(11) can be exactly restated as the
existence of Lagrange multiplier, , s.t.
1 1
s.t. ⎢ 0
T ⎥
⎣S1 I1⎦
⎡ T2 S2⎤ ⎢ 0
T ⎥ >
⎣S2 I2⎦
T
λ
i
, γ
A PA 1 P1 Q 0
j
⎧
p m ⎫
T1 ρ1I1
⎪
min { ∑qY
i i+ ∑ rU j j +
K, ∑x≥ 0, Yi, U
⎪
j i= 1 j=
1
⎪
⎪
⎪
p
m
2 2 ⎪
T2 ρ2I2
⎪∑λi( Y i− yi ) + ∑γ
j( U j −uj
)} ⎪
i= 1 j=
1
T
⎪
⎪ where, 1 ( ) ( )
max ⎨
T
s. t. ( A+ BK) ∑ ( A+
BK)
⎬. (22)
λi≥0, γ j≥0
x
T
T
⎪
⎪ S2
RK B PBK B PA
T
⎪ + E∑ w E =∑x
⎪ LQR inverse-optimal Control, ,
⎪
T T
⎪
⎪ ϕiC∑ x C ϕi = Yi
⎪
⎪
T T
⎪
∃P
≥ 0 , Q ≥ 0 , R > 0 , P 1 > 0 , s.t.
⎪⎩
ϕ jK∑ x K ϕ j = U j ⎪⎭
If rewrite the minimization objective function as:
p
m
T
T
min { ∑( qi + λi) Y i+ ∑ ( rj + γ j) U j}
RK B PBK B PA
K, ∑x≥ 0, Yi, U j i= 1 j=
1
,
p
m
T
2 2
A PA
− ∑λiyi − ∑γ
ju
1 P1
Q
j
i= 1 j=
1
λ i≥ γ j ≥ .This indicates
∗
λ i γ j dependent solutions, K ( λi, γ j)
,coincide
⎡ T1 S1⎤ ⎢ 0
T ⎥ >
⎣S1 I1⎦
i = qi + i i = rj + j).
T T T T
1
Because of the variance constraints (16) and (17),
T
problem cannot be equivalent to the infinite
1 ≤ ρ1I1
⎡ T1 S1⎤ and Lemma 2, ⎢ 0
Theorem 2 guarantees that the MVC 3 T ⎥ >
problem would
⎣S1 I1⎦
∗
T
T
K λi
γ j such that there to T 1 − S 1 S 1 > 0 , that is 1 I 1 SS 1 1
⎡ T2 S2⎤ λ , γ j . But, weighted matrix Q , R of infinite
⎢ 0
T ⎥ >
⎣S2 I2⎦
T
T
2 = + +
T2 ≤ ρ2I2
LQR inverse-optimal control can be described as:
If ∃P
≥ 0 , Q ≥ 0 , R > 0 , P 1 > 0 and symmetric
min ρ 1 + ρ 2
(23) PPT , , , T, QR ,
then, it is clear that the above three assumptions are
satisfied for all values of 0, 0
that all ,
with the solution to some infinite horizon MPC problem
( λ λ , γ γ
MVC 3
horizon MPC problem (19). Theorem 2 shows that
introducing variance constraints to MVC 3 problem (21),
is exactly the reason to adjust the MPC controller weight
matrix, Q , R .
generate a linear feedback ( , )
exists a feasible solution for infinite horizon MPC
problem. Unfortunately, the exact form of this infinite
horizon MPC problem is unclear, unless the MVC 3
solution procedure provides us with the optimal
Lagrangians i
horizon MPC can be solved by given feedback gain, K .
Then, it can be updated with the Riccati equation.
E. LQR Inverse-Optimal Control and Its LMI Method
matrices, T 1 , T 2 that
1 1 2
⎡ T S ⎤ >
(24)
(25)
− − < (26)
< (27)
< , (28)
S = A+ BK P A+ BK − P+ Q+ K RK ,
= + + .Then, through the solution of
QR,can be obtained.
LQR inverse-Optimal Control [27] is described as:
T T T T
A PA − P −K RK − K B PBK + Q = 0 (29)
+ + = 0 (30)
− < , (31)
where, (31) ensures that ( A, Q ) is detectable. As (29)
cannot be converted to the LMI form, it can be
constructed as,
S = A PA −P −K RK − K B PBK + Q ,
where, T 1 is symmetric matrix; ρ 1 is a scalar; I 1 is a
unit matrix of appropriate dimension. From LMI theory
is
T
equivalent
ρ > .Then, approximate
solution of equation (29), R , P , can be gotten through
choosing a small enough ρ 1 . Similarly,
S RK B PBK B PA
Equation (26) is the rewriting of (31). Then, the LMI
form of LQR inverse-Optimal Control can be gotten.
Equations (29)-(31) are constructed to LMI form to get
parameters QR. , However, Matrix Q here is nondiagonal
matrix, practical application is inconvenience.
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