LMIs in Control/pages/Multi-Criterion LQG
This is a WIP based on the template
The System [ edit ]
The system is a linear time-invariant system, that can be represented in state space as shown below:
x
˙
=
A
x
+
B
u
+
w
,
y
=
C
x
+
v
,
z
=
[
Q
1
/
2
0
0
R
1
/
2
]
[
x
u
]
{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+Bu+w,\\y&=Cx+v,\\z&={\begin{bmatrix}Q^{1/2}&0\\0&R^{1/2}\end{bmatrix}}{\begin{bmatrix}x\\u\end{bmatrix}}\end{aligned}}}
where u is the input; y is the measured output; z is the output of interest; w and v are white noise signals with constant positive-definite spectral density matrices W and V, respectively.
Q
≥
0
{\displaystyle Q\geq 0}
and
R
>
0
{\displaystyle R>0}
, and the system is controllable and observable.
The Data [ edit ]
The matrices
A
,
B
,
C
,
Q
,
R
,
W
,
V
{\displaystyle A,B,C,Q,R,W,V}
and the noise signals
w
,
v
{\displaystyle w,v}
.
The Optimization Problem [ edit ]
In the Linear Quadratic Gaussian (LQG) control problem, the goal is to minimize a quadratic cost function while the plant has random initial conditions and suffers white noise disturbance on the input and measurement.
There are multiple outputs of interest for this problem. They are defined by
z
=
[
Q
1
/
2
0
0
R
1
/
2
]
[
x
u
]
,
Q
i
≥
0
,
R
i
>
0
,
i
=
0
,
.
.
.
,
p
.
{\displaystyle {\begin{aligned}z&={\begin{bmatrix}Q^{1/2}&0\\0&R^{1/2}\end{bmatrix}}{\begin{bmatrix}x\\u\end{bmatrix}},Q_{i}\geq 0,R_{i}>0,i=0,...,p.\end{aligned}}}
For each of these outputs of interest, we associate a cost function:
J
L
Q
G
i
=
lim
t
→
∞
E
z
i
(
t
)
T
z
i
(
t
)
,
i
=
0
,
.
.
.
,
p
.
{\displaystyle {\begin{aligned}J_{LQG}^{i}=\lim _{t\to \infty }Ez_{i}(t)^{T}z_{i}(t),i=0,...,p.\end{aligned}}}
Additionally, the matrices
X
L
Q
G
{\displaystyle X_{LQG}}
and
Y
L
Q
G
{\displaystyle Y_{LQG}}
must be found as the solutions to the following Riccati equations:
A
T
X
L
Q
G
+
X
L
Q
G
A
=
X
L
Q
G
B
R
−
1
B
T
X
L
Q
G
+
Q
=
0
A
Y
L
Q
G
+
Y
L
Q
G
A
T
−
Y
L
Q
G
C
T
V
−
1
C
Y
L
Q
G
+
W
=
0
{\displaystyle {\begin{aligned}A^{T}X_{LQG}+X_{LQG}A=X_{LQG}BR^{-1}B^{T}X_{LQG}+Q&=0\\AY_{LQG}+Y_{LQG}A^{T}-Y_{LQG}C^{T}V^{-1}CY_{LQG}+W&=0\\\end{aligned}}}
The optimization problem is to minimize
J
L
Q
G
0
{\displaystyle J_{LQG}^{0}}
over u subject to the measurability condition and the constraints
J
L
Q
G
i
<
γ
i
,
i
=
0
,
.
.
.
,
p
.
{\displaystyle J_{LQG}^{i}<\gamma _{i},i=0,...,p.}
. This optimization problem can be formulated as:
max
t
r
a
c
e
(
X
L
Q
G
U
+
Q
Y
L
Q
G
)
−
∑
i
=
1
p
γ
i
τ
i
,
{\displaystyle {\begin{aligned}\max trace(X_{LQG}U+QY_{LQG})-\sum _{i=1}^{p}\gamma _{i}\tau _{i},\end{aligned}}}
over
τ
1
,
.
.
.
,
τ
p
{\displaystyle \tau _{1},...,\tau _{p}}
, with:
Q
=
Q
0
+
∑
i
=
1
p
τ
i
Q
i
,
R
=
R
0
+
∑
i
=
1
p
τ
i
R
i
.
{\displaystyle {\begin{aligned}Q&=Q_{0}+\sum _{i=1}^{p}\tau _{i}Q_{i},\\R&=R_{0}+\sum _{i=1}^{p}\tau _{i}R_{i}.\end{aligned}}}
The LMI: Multi-Criterion LQG[ edit ]
max
:
t
r
a
c
e
(
X
U
+
(
Q
0
+
∑
i
=
1
p
τ
i
Q
i
)
Y
L
Q
G
)
−
∑
i
=
1
p
γ
i
τ
i
,
{\displaystyle {\begin{aligned}\max :trace(XU+(Q_{0}+\sum _{i=1}^{p}\tau _{i}Q_{i})Y_{LQG})-\sum _{i=1}^{p}\gamma _{i}\tau _{i},\end{aligned}}}
over
X
,
τ
1
,
.
.
.
τ
p
{\displaystyle X,\tau _{1},...\tau _{p}}
, subject to the following constraints:
X
>
0
,
τ
1
≥
0
,
.
.
.
,
τ
p
≥
0
,
A
T
X
+
X
A
−
X
B
(
R
0
+
∑
i
=
1
p
τ
i
R
i
)
−
1
B
T
X
+
Q
0
+
∑
i
=
1
p
τ
i
Q
i
≥
0.
{\displaystyle {\begin{aligned}X&>0,\\\tau _{1}\geq 0,...,\tau _{p}&\geq 0,\\A^{T}X+XA-XB(R_{0}+\sum _{i=1}^{p}\tau _{i}R_{i})^{-1}B^{T}X+Q_{0}+\sum _{i=1}^{p}\tau _{i}Q_{i}&\geq 0.\\\end{aligned}}}
Conclusion: [ edit ]
The result of this LMI is the optimal cost achievable, the result of the
m
a
x
{\displaystyle max}
optimization.
Implementation [ edit ]
A link to CodeOcean or other online implementation of the LMI
Related LMIs [ edit ]
Links to other closely-related LMIs
External Links [ edit ]
A list of references documenting and validating the LMI.
Return to Main Page: [ edit ]