[<< wikibooks] LMIs in Control/pages/dt mixed H2 Hinf optimal output feedback control
WIP, Description in progress
This part shows how to design dynamic outpur feedback control in mixed

H

2

{\displaystyle {\mathcal {H}}_{2}}
and

H

∞

{\displaystyle {\mathcal {H}}_{\infty }}
sense for the continuous time.

== Problem ==
Consider the discrete-time generalized LTI plant

P

{\displaystyle {\mathcal {P}}}
with minimal state-space realization

x
˙

=
A
x
+

[

B

1
,
1

B

1
,
2

]

[

w

1

w

2

]

+

B

2

u
,

{\displaystyle {\dot {x}}=Ax+{\begin{bmatrix}B_{1,1}&B_{1,2}\end{bmatrix}}{\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}+B_{2}u,}

[

z

1

z

2

]

=

[

C

1
,
1

D

1
,
2

]

x

k

+

[

D

11
,
11

D

11
,
12

D

11
,
21

D

11
,
22

]

[

w

1

w

2

]

+

[

D

12
,
1

D

12
,
2

]

u
,

{\displaystyle {\begin{bmatrix}z_{1}\\z_{2}\end{bmatrix}}={\begin{bmatrix}C_{1,1}\\D_{1,2}\end{bmatrix}}x_{k}+{\begin{bmatrix}D_{11,11}&D_{11,12}\\D_{11,21}&D_{11,22}\end{bmatrix}}{\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}+{\begin{bmatrix}D_{12,1}\\D_{12,2}\end{bmatrix}}u,}

y
=

C

d
2

x
+

[

D

21
,
1

D

21
,
2

]

[

w

1

w

2

]

+

D

d
22

u

{\displaystyle y=C_{d2}x+{\begin{bmatrix}D_{21,1}&D_{21,2}\end{bmatrix}}{\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}+D_{d22}u}

== Theorem ==
A continuous-time dynamic output feedback LTI controllerwith state-space realization

(

A

c

,

B

c

,

C

c

,

D

c

)

{\displaystyle (A_{c},B_{c},C_{c},D_{c})}

is to be designed to minimize the

H

2

{\displaystyle {\mathcal {H}}_{2}}
norm of the closed-loop transfer matrix

T

11

(
s
)

{\displaystyle T_{11}(s)}
from the exogenous
input

w

1

{\displaystyle w_{1}}
to the performance output

z

1

{\displaystyle z_{1}}
while ensuring the H∞ norm of the closed-loop
transfer matrix

T

22

(
s
)

{\displaystyle T_{22}(s)}
from the exogenous input

w

2

{\displaystyle w_{2}}
to the performance output

z

2

{\displaystyle z_{2}}
is less than

γ

d

{\displaystyle \gamma _{d}}
,
where

T

11

(
s
)
=

C

C
L
1
,
1

(
s
I
−

A

C
L

)

−
1

B

C
L
1
,
1

,

{\displaystyle T_{11}(s)=C_{CL1,1}(sI-A_{CL})^{-1}B_{CL1,1},}

T

22

(
s
)
=

C

C
L
1
,
2

(
s
I
−

A

C
L

)

−
1

B

C
L
1
,
2

+

D

C
L
11
,
22

,

{\displaystyle T_{22}(s)=C_{CL1,2}(sI-A_{CL})^{-1}B_{CL1,2}+D_{CL11,22},}

A

d

C
L

=

[

A
+

B

2

D

c

D
~

−
1

C

2

B

2

(
I
+

D

c

D
~

−
1

D

22

)

C

c

B

c

D
~

−
1

C

2

A

c

+

B

c

D
~

−
1

D

22

C

c

]

{\displaystyle A_{d_{CL}}={\begin{bmatrix}A+B_{2}D_{c}{\tilde {D}}^{-1}C_{2}&B_{2}(I+D_{c}{\tilde {D}}^{-1}D_{22})C_{c}\\B_{c}{\tilde {D}}^{-1}C_{2}&A_{c}+B_{c}{\tilde {D}}^{-1}D_{22}C_{c}\end{bmatrix}}}
,

B

C
L
1
,
1

=

[

B

1
,
1

+

B

2

D

c

D
~

−
1

D

21
,
1

B

c

D
~

−
1

D

21
,
1

]

{\displaystyle B_{CL1,1}={\begin{bmatrix}B_{1,1}+B_{2}D_{c}{\tilde {D}}^{-1}D_{21,1}\\B_{c}{\tilde {D}}^{-1}D_{21,1}\end{bmatrix}}}
,

B

C
L
1
,
2

=

[

B

1
,
2

+

B

2

D

c

D
~

−
1

D

21
,
2

B

c

D
~

−
1

D

21
,
2

]

{\displaystyle B_{CL1,2}={\begin{bmatrix}B_{1,2}+B_{2}D_{c}{\tilde {D}}^{-1}D_{21,2}\\B_{c}{\tilde {D}}^{-1}D_{21,2}\end{bmatrix}}}
,

C

C
L
1
,
1

=

[

C

1
,
1

+

D

12
,
1

D

c

D
~

−
1

C

2
,
1

D

12
,
1

(
I
+

D

c

D
~

−
1

D

22

)

C

c

]

{\displaystyle C_{CL1,1}={\begin{bmatrix}C_{1,1}+D_{12,1}D_{c}{\tilde {D}}^{-1}C_{2,1}&D_{12,1}(I+D_{c}{\tilde {D}}^{-1}D_{22})C_{c}\end{bmatrix}}}
,

C

C
L
1
,
2

=

[

C

1
,
2

+

D

12
,
2

D

c

D
~

−
1

C

2
,
2

D

12
,
2

(
I
+

D

c

D
~

−
1

D

22

)

C

c

]

{\displaystyle C_{CL1,2}={\begin{bmatrix}C_{1,2}+D_{12,2}D_{c}{\tilde {D}}^{-1}C_{2,2}&D_{12,2}(I+D_{c}{\tilde {D}}^{-1}D_{22})C_{c}\end{bmatrix}}}
,

D

C
L
11
,
22

=

D

11
,
22

+

D

12
,
2

D

c

D
~

−
1

D

21
,
2

{\displaystyle D_{CL11,22}=D_{11,22}+D_{12,2}D_{c}{\tilde {D}}^{-1}D_{21,2}}
,
and

D
~

=
I
−

D

22

D

c

{\displaystyle {\tilde {D}}=I-D_{22}D_{c}}
.

== Synthesis Method ==
Solve for

A

n

∈

R

n

x

×

n

x

,

B

n

∈

R

n

x

×

n

x

,

C

n

∈

R

n

u

×

n

x

,

D

n

∈

R

n

u

×

n

y

,

X

1

,

Y

1

∈

S

n

x

,
Z
∈

S

n

Z

1

,

{\displaystyle A_{n}\in \mathbb {R} ^{n_{x}\times n_{x}},B_{n}\in \mathbb {R} ^{n_{x}\times n_{x}},C_{n}\in \mathbb {R} ^{n_{u}\times n_{x}},D_{n}\in \mathbb {R} ^{n_{u}\times n_{y}},X_{1},Y_{1}\in \mathbb {S} ^{n_{x}},Z\in \mathbb {S} ^{n_{Z_{1}}},}
and

μ
∈

R

>
0

{\displaystyle \mu \in \mathbb {R} _{>0}}
that minimizes

J

(
μ
)
=
μ

{\displaystyle {\mathcal {J}}(\mu )=\mu }
subjects to

X

1

>
0
,

Y

1

>
0

Z
>
0
,

{\displaystyle X_{1}>0,\ Y_{1}>0\ Z>0,}

[

N

11

A
+

A

n

T

+

B

2

D

n

C

2

B

1
,
1

+

B

2

D

n

D

21
,
1

∗

X

1

A
+

A

T

X

1

+

B

n

C

2

+

C

2

T

B

n

T

X

1

B

1
,
1

+

B

n

D

21
,
1

∗

∗

−
I

]

<
0

{\displaystyle {\begin{bmatrix}N_{11}&A+A_{n}^{T}+B_{2}D_{n}C_{2}&B_{1,1}+B_{2}D_{n}D{21,1}\\*&X_{1}A+A^{T}X_{1}+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&X_{1}B_{1,1}+B_{n}D_{21,1}\\*&*&-I\end{bmatrix}}<0}
,

[

N

11

A
+

A

n

T

+

B

2

D

n

C

2

B

1
,
1

+

B

2

D

n

D

21
,
1

Y

1

T

C

1
,
2

T

+

C

n

T

D

12
,
2

T

∗

X

1

A
+

A

T

X

1

+

B

n

C

2

+

C

2

T

B

n

T

X

1

B

1
,
1

+

B

n

D

21
,
1

C

1
,
2

T

+

C

2

T

D

n

T

D

12
,
2

T

∗

∗

−

γ

d

I

D

11
,
22

T

+

D

21
,
2

T

D

n

T

D

12
,
2

T

∗

∗

∗

−

γ

d

I

]

<
0

{\displaystyle {\begin{bmatrix}N_{11}&A+A_{n}^{T}+B_{2}D_{n}C_{2}&B_{1,1}+B_{2}D_{n}D{21,1}&Y_{1}^{T}C_{1,2}^{T}+C_{n}^{T}D_{12,2}^{T}\\*&X_{1}A+A^{T}X_{1}+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&X_{1}B_{1,1}+B_{n}D_{21,1}&C_{1,2}^{T}+C_{2}^{T}D_{n}^{T}D_{12,2}^{T}\\*&*&-\gamma _{d}I&D_{11,22}^{T}+D_{21,2}^{T}D_{n}^{T}D_{12,2}^{T}\\*&*&*&-\gamma _{d}I\end{bmatrix}}<0}
,

[

Y

1

I

Y

1

C

1
,
1

T

+

C

n

T

D

12
,
1

T

∗

X

1

C

1
,
1

T

+

C

2

T

D

n

T

D

12
,
1

T

∗

∗

Z

]

>
0

{\displaystyle {\begin{bmatrix}Y_{1}IY_{1}C_{1,1}^{T}+C_{n}^{T}D_{12,1}^{T}\\*&X_{1}&C_{1,1}^{T}+C_{2}^{T}D_{n}^{T}D_{12,1}^{T}\\*&*&Z\end{bmatrix}}>0}
,

D

11
,
11

+

D

12
,
1

D

n

D

21
,
1

=
0
,

{\displaystyle D_{11,11}+D_{12,1}D_{n}D_{21,1}=0,}

[

X

1

I

∗

Y

1

]

>
0
,

{\displaystyle {\begin{bmatrix}X_{1}&I\\*&Y_{1}\end{bmatrix}}>0,}

tr

Z
<
μ
,

{\displaystyle Z<\mu ,}

where

N

11

=
A

Y

1

+

Y

1

A

T

+

B

2

C

n

+

C

n

T

B

2

T

{\displaystyle N_{11}=AY_{1}+Y_{1}A^{T}+B_{2}C_{n}+C_{n}^{T}B_{2}^{T}}
.
The controller is recovered by

A

c

=

A

K

−
B

c

(
I
−

D

22

D

c

)

−
1

D

22

C

c

,

{\displaystyle A_{c}=A_{K}-B{c}(I-D_{22}D_{c})^{-1}D_{22}C_{c},}

B

c

=

B

K

(
I
−

D

c

D

22

)
,

{\displaystyle B_{c}=B_{K}(I-D_{c}D_{22}),}

C

c

=
(
I
−

D

c

D

22

)

C

K

,

{\displaystyle C_{c}=(I-D_{c}D_{22})C_{K},}

D

c

=
(
I
+

D

K

D

22

)

−
1

D

K

,
w
h
e
r
e
<
m
a
t
h
>

[

A

K

B

K

C

K

D

K

]

=

[

X

2

X

1

B

2

0

I

]

−
1

(

[

A

n

B

n

C

n

D

n

]

−

[

X

1

A

d

Y

1

0

0

0

]

)

[

Y

2

T

0

C

2

Y

1

I

]

−
1

{\displaystyle D_{c}=(I+D_{K}D{22})^{-1}D_{K},where