Robust Model Predictive Control with input and output constraints
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Model Predictive Control
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Model Predictive Control is an open-loop control design procedure where at each sampling time k, plant measurements are obtained and a model of the process is used to predict future outputs of the system. Using these predictions,
m
{\displaystyle m}
control moves
u
(
k
+
i
|
k
)
,
i
=
0
,
1
,
.
.
.
,
m
−
1.
{\displaystyle u(k+i|k),i=0,1,...,m-1.}
are computed by minimizing a nominal cost
J
p
(
k
)
{\displaystyle J_{p}(k)}
over a prediction horizon
p
{\displaystyle p}
. The objective is to minimize the nominal cost function.
We consider the nominal cost function as:
min
u
(
k
+
i
)
,
i
=
0
,
1
,
.
.
.
,
m
−
1
J
p
(
k
)
{\displaystyle \min _{u(k+i),i=0,1,...,m-1}J_{p}(k)}
where,
J
p
(
k
)
=
Σ
i
=
0
p
[
x
(
k
+
i
|
k
)
T
Q
1
x
(
k
+
i
|
k
)
+
u
(
k
+
i
|
k
)
T
R
u
(
k
+
i
|
k
)
]
{\displaystyle J_{p}(k)=\Sigma _{i=0}^{p}[x(k+i|k)^{T}Q_{1}x(k+i|k)+u(k+i|k)^{T}Ru(k+i|k)]}
Q
1
>
0
{\displaystyle Q_{1}>0}
and
R
>
0
{\displaystyle R>0}
Q
1
{\displaystyle Q_{1}}
and
R
{\displaystyle R}
are positive definite weighting matrices.
In this case, we take
p
=
∞
{\displaystyle p=\infty }
. This is also called infinite horizon MPC.
Uncertainties
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Here, we consider system uncertainties that are modeled as polytopic uncertainties or structured uncertainties.
Polytopic Uncertainty
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The set
Ω
{\displaystyle \Omega }
is the polytope
Ω
=
C
o
[
A
1
B
1
]
,
[
A
2
B
2
]
,
.
.
.
.
,
[
A
L
B
L
]
{\displaystyle \Omega =Co{[A_{1}B_{1}],[A_{2}B_{2}],....,[A_{L}B_{L}]}}
Where,
C
o
{\displaystyle Co}
denotes the convex hull.
Structured Uncertainty
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The operator
Δ
{\displaystyle \Delta }
is a block-diagonal:
Δ
=
[
Δ
1
Δ
2
⋱
Δ
r
]
{\displaystyle \Delta ={\begin{bmatrix}\Delta _{1}&&&\\&\Delta _{2}&&&\\&&\ddots &\\&&&\Delta _{r}\end{bmatrix}}}
Each
Δ
{\displaystyle \Delta }
can be a repeated scalar block or a full block.
The System
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Consider a linear time-varying(LTV) system:
x
(
k
+
1
)
=
A
(
k
)
x
(
k
)
+
B
(
k
)
u
(
k
)
,
{\displaystyle x(k+1)=A(k)x(k)+B(k)u(k),}
y
(
k
)
=
C
X
(
k
)
,
{\displaystyle y(k)=CX(k),}
[
A
(
k
)
B
(
k
)
]
∈
Δ
{\displaystyle {\begin{bmatrix}A(k)&B(k)\end{bmatrix}}\in \Delta }
Here,
u
(
k
)
∈
R
n
{\displaystyle u(k)\in \mathbb {R} ^{n}}
is the control input,
x
(
k
)
∈
R
n
{\displaystyle x(k)\in \mathbb {R} ^{n}}
is the state of the plant and
y
(
k
)
∈
R
n
{\displaystyle y(k)\in \mathbb {R} ^{n}}
is the plant output and
Δ
{\displaystyle \Delta }
is uncertainty set that is either polytopic system or structured uncertainty.
We modify the minimization of the nominal cost function to a minimization of the worst-case objective function.
The modified objective function minimizes the robust performance objective as follows:
min
u
(
k
+
i
)
,
i
=
0
,
1
,
.
.
.
,
m
−
1
max
[
A
(
k
+
i
)
B
(
k
+
i
]
∈
Δ
,
i
≥
0
J
∞
(
k
)
{\displaystyle \min _{u(k+i),i=0,1,...,m-1}\max _{[A(k+i)B(k+i]\in \Delta ,i\geq 0}J_{\infty }(k)}
where,
J
∞
(
k
)
=
Σ
i
=
0
∞
[
x
(
k
+
i
|
k
)
T
Q
1
x
(
k
+
i
|
k
)
+
u
(
k
+
i
|
k
)
T
R
u
(
k
+
i
|
k
)
]
{\displaystyle J_{\infty }(k)=\Sigma _{i=0}^{\infty }[x(k+i|k)^{T}Q_{1}x(k+i|k)+u(k+i|k)^{T}Ru(k+i|k)]}
The Data
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The LMI: Robust Model Predictive Control with State Feedback and input and output constraints for polytopic uncertainty
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min
γ
,
Q
,
Y
γ
{\displaystyle \min _{\gamma ,Q,Y}\gamma }
subject to
[
1
x
(
k
|
k
)
T
x
(
k
|
k
)
Q
]
≥
0
{\displaystyle {\begin{bmatrix}1&x(k|k)^{T}\\x(k|k)&Q\end{bmatrix}}\geq 0}
and
[
Q
Q
A
j
T
+
Y
T
B
j
T
Q
Q
1
1
/
2
Y
T
R
1
/
2
A
j
Q
+
B
j
Y
Q
0
0
Q
q
1
/
2
0
γ
I
0
R
1
/
2
Y
0
0
γ
I
]
≥
0
{\displaystyle {\begin{bmatrix}Q&QA_{j}^{T}+Y^{T}B_{j}^{T}&QQ_{1}^{1/2}&Y^{T}R^{1/2}\\A_{j}Q+B_{j}Y&Q&0&0\\Q_{q}^{1/2}&0&\gamma I&0\\R^{1/2}Y&0&0&\gamma I\end{bmatrix}}\geq 0}
Input constraint is given by the following LMI:
[
u
m
a
x
2
I
Y
Y
T
Q
]
≥
0
{\displaystyle {\begin{bmatrix}u_{max}^{2}I&Y\\Y^{T}&Q\end{bmatrix}}\geq 0}
Output constraint is given by the following LMI:
[
Q
(
A
j
Q
+
B
j
Y
)
T
C
T
C
(
A
j
Q
+
B
j
Y
)
y
m
a
x
2
I
]
≥
0
{\displaystyle {\begin{bmatrix}Q&(A_{j}Q+B_{j}Y)^{T}C^{T}\\C(A_{j}Q+B_{j}Y)&y_{max}^{2}I\end{bmatrix}}\geq 0}
The LMI: Robust Model Predictive Control with input and output constraints with State Feedback for structured uncertainty
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min
γ
,
Q
,
Y
,
Λ
γ
{\displaystyle \min _{\gamma ,Q,Y,\Lambda }\gamma }
subject to
[
1
x
(
k
|
k
)
T
x
(
k
|
k
)
Q
]
≥
0
{\displaystyle {\begin{bmatrix}1&x(k|k)^{T}\\x(k|k)&Q\end{bmatrix}}\geq 0}
[
Q
Y
T
R
1
/
2
Q
Q
1
1
/
2
Q
C
q
T
+
Y
T
D
q
u
T
Q
A
T
+
Y
T
B
T
R
1
/
2
Y
γ
I
0
0
0
Q
1
1
/
2
Q
0
γ
I
0
0
C
q
Q
+
D
q
u
Y
0
0
Λ
0
A
Q
+
B
Y
0
0
0
Q
−
B
p
Λ
B
p
T
]
≥
0
{\displaystyle {\begin{bmatrix}Q&Y^{T}R^{1/2}&QQ_{1}^{1/2}&QC_{q}^{T}+Y^{T}D_{qu}^{T}&QA^{T}+Y^{T}B^{T}\\R^{1/2}Y&\gamma I&0&0&0\\Q_{1}^{1/2}Q&0&\gamma I&0&0\\C_{q}Q+D_{qu}Y&0&0&\Lambda &0\\AQ+BY&0&0&0&Q-B_{p}\Lambda B_{p}^{T}\end{bmatrix}}\geq 0}
where
Λ
=
[
λ
1
I
n
1
λ
2
I
n
2
⋱
λ
r
I
n
r
]
>
0
{\displaystyle \Lambda ={\begin{bmatrix}\lambda _{1}I_{n1}&&&\\&\lambda _{2}I_{n2}&&&\\&&\ddots &\\&&&\lambda _{r}I_{nr}\end{bmatrix}}>0}
Input constraint is given by the following LMI:
[
u
m
a
x
2
I
Y
Y
T
Q
]
≥
0
{\displaystyle {\begin{bmatrix}u_{max}^{2}I&Y\\Y^{T}&Q\end{bmatrix}}\geq 0}
Output constraint is given by the following LMI:
[
y
m
a
x
2
Q
(
C
q
Q
+
D
q
u
Y
)
T
(
A
Q
+
B
Y
)
T
C
T
C
q
Q
+
D
q
u
Y
T
−
1
0
C
(
A
Q
+
B
Y
)
0
I
−
C
B
p
T
−
1
B
p
T
C
T
]
≥
0
{\displaystyle {\begin{bmatrix}y_{max}^{2}Q&(C_{q}Q+D_{qu}Y)^{T}&(AQ+BY)^{T}C^{T}\\C_{q}Q+D_{qu}Y&T^{-1}&0\\C(AQ+BY)&0&I-CB_{p}T^{-1}B_{p}^{T}C^{T}\end{bmatrix}}\geq 0}
where,
T =
[
t
1
I
n
1
t
2
I
n
2
⋱
t
r
I
n
r
]
≥
0
{\displaystyle {\begin{bmatrix}t_{1}I_{n_{1}}&&&\\&t_{2}I_{n_{2}}&&\\&&\ddots &\\&&&t_{r}I_{n_{r}}\end{bmatrix}}\geq 0}
Conclusion:
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Implementation
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