ELE7029: Optimal Digital Control System
Lecture IV
Discrete-time Control Design via LMI Optimization
Goals of this lecture
· LMIs for Stability, H2 and H∞ performances
· Stabilizing state feedback control
· H2 state feedback control
· H∞ state feedback control
· H2 output feedback control
· H∞ output feedback control
S.-H. Lee (shlee@ieee.org) Elec. Engr., Hanyang University Spring 2022
Outline
1. Fundamental LMIs
2. Stabilizing State Feedback Control
3. H2 State Feedback Control
4. H∞ State Feedback Control
5. H2 Output Feedback Control
6. H∞ Output Feedback Control
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 1
Fundamental LMIs
Consider a linear discrete-time system
Tzw :

x(k+ 1) = Φx(k) + Γw(k)
z(k) = Cx(k) +Dw(k)
Let us describe LMI conditions for stability, H2 and H∞ performances for
the system.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 2
■ Stability:
The system x(k+ 1) = Φx(k) is stable iff
i) ∃ X > 0 s.t.
ΦXΦT − X < 0 (4.1a)
i) ∃ X > 0 s.t. −X ΦX
XΦT −X

< 0 (4.1b)
ii) ∃ X > 0 and G s.t. −X ΦG
GTΦT X − (G+GT)

< 0 (4.1c)
iii) ∃ X > 0 and G and W s.t. −X −ΦW −WTΦT WT +ΦX
W + XΦT X − (G+GT)

< 0 (4.1d)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 3
Note: Introducing a slack variable G ∈ Rnx×nx
P ΦG
GTΦT G+GT − P

> 0
· Suff: G= GT = P recovers the LMI (4.1b)
· Nec: Find that G+GT − P> 0 and
(P−G)TP−1(P−G) = P−G−GT +GTP−1G≥ 0 assures
GTP−1G≥ G+GT − P
Therefore
P ΦG
GTΦT GTP−1G

> 0
Congruence transformation by T= diag(I,G−1P) recovers the standard
form.
TT

P ΦG
GTΦT GTP−1G

T=

P ΦP
PΦT P

> 0
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 4
Good to Note: H2 gain with D= 0
Recall
ΦPΦT − P+ Γ Γ T = 0 → tr(CPCT) = ∥Tzw∥22
⇒ ΦPΦT − P+ Γ Γ T < 0 → tr(CPCT)> ∥Tzw∥22
⇒
 P ΦP Γ(∗) P 0
(∗) (∗) I
> 0 → tr(CPCT)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 5
Find
tr(CPCT)< µ ⇐ CPCT ⇔

W CP
(∗) P

> 0, tr(W)< µ
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 6
■ H2 Performance (Discrete-time):
Theorem 4.1 (H2 performance in discrete-time)
For a system Tzw with D= 0, the following statements, involving matrix
variables X, G, W, Z are equivalent
i) ∥Tzw∥2 < ν
ii) ∃ X > 0 and Z such that X ΦX Γ∗ X 0
(∗) (∗) I
> 0, Z CX(∗) X

> 0, tr(Z)< ν2 (4.2a)
iii) ∃ X > 0, G and Z such that X ΦG Γ(∗) G+GT − X 0
(∗) (∗) I
> 0, Z CG(∗) G+GT − X

> 0, tr(Z)< ν2
(4.2b)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 7
iv) ∃ X > 0, G, W, and Z s.t.−X −ΦW −WTΦT WT +ΦX Γ(∗) X − (G+GT) 0
(∗) (∗) −I
< 0,

Z CG
(∗) G+GT − X

> 0, tr(Z)< ν2
(4.2c)
Exercise: Verify their equivalence given by (4.2).
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 8
■ Discrete-time Bounded Real Lemma:
For the system Tzw, the followings are equivalent:
i) ∥Tzw∥∞ = ∥C(zI−Φ)−1Γ +D∥∞ < γ
ii) ΦTXΦ− X + CTC
− (ΦTXΓ + CTD)(Γ TXΓ +DTD− γ2I)−1(ΦTXΓ + CTD)T < 0
with nonsingular Γ TXΓ +DTD− γ2I
iii) •
Φ Γ
C D
˜T •X 0
0 I
˜•
Φ Γ
C D
˜
−
•
X 0
0 γ2I
˜
< 0
iv) •
Φ Γ
γ−1C γ−1D
˜T •X 0
0 I
˜•
Φ Γ
γ−1C γ−1D
˜
−
•
X 0
0 I
˜
< 0
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 9
Verification:
Find
x(k)
w(k)
T ‚
Φ Γ
C D
T
X 0
0 I

Φ Γ
C D

−

X 0
0 γ2I
Œ
x(k)
w(k)

< 0
leads to
xT(k+ 1)Xx(k+ 1)− xT(k)Xx(k) + zT(k)z(k)− γ2wT(k)w(k)< 0
Let V(k) = xT(k)Xx(k). Then
V(k+ 1)− V(k) + zT(k)z(k)− γ2wT(k)w(k)< 0
→ zT(k)z(k)< γ2wT(k)w(k) → ∥Tzw∥∞ < γ
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 10
Find
Φ Γ
C D
T
X 0
0 I

Φ Γ
C D

−

X 0
0 γ2I

< 0
⇒

ΦTXΦ− X + CTC ΦTXΓ + CTD
Γ TXΦ+DTC Γ TXΓ +DTD− γ2I

< 0
⇒ ΦTXΦ− X + CTC
− (ΦTXΓ + CTD)(Γ TXΓ +DTD− γ2I)−1(ΦTXΓ + CTD)T < 0
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 11
Also, find
x(k)
w(k)
T ‚
Φ Γ
γ−1C γ−1D
T
X 0
0 I

Φ Γ
γ−1C γ−1D

−

X 0
0 I
Œ
x(k)
w(k)

< 0
leads to
xT(k+ 1)Xx(k+ 1)− xT(k)Xx(k) + γ−2zT(k)z(k)−wT(k)w(k)< 0
Let V(k) = xT(k)Xx(k). Then
V(k+ 1)− V(k) + γ−2zT(k)z(k)−wT(k)w(k)< 0
zT(k)z(k)< γ2wT(k)w(k) → ∥Tzw∥2∞ < γ2
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 12
Using
P> 0, GTPG − P′ < 0 ⇐⇒
−P′ GTP
PG −P

< 0 ⇐⇒
−P PG
GTP −P′

< 0,
for (iii)
Φ Γ
C D
T
X 0
0 I

Φ Γ
C D

−

X 0
0 γ2I

< 0,
we find its equivalent form−X 0 XΦ XΓ0 −I C D
ΦTX CT −X 0
Γ TX DT 0 −γ2I
< 0
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 13
Applying permutations leads to−X XΦ XΓ 0ΦTX −X 0 CT
Γ TX 0 −γ2I DT
0 C D −I

A congruence transformation by T= diag(γX−1,γX−1,γ−1,γ) leads to−γ
2X−1 Φ(γ2X−1) Γ 0
γ2X−1ΦT −γ2X−1 0 γ2X−1CT
Γ T 0 −I DT
0 C(γ2X−1) D −γ2I
< 0
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 14
Applying the change of variables X := γ2X−1 leads to−X ΦX Γ 0XΦT −X 0 XCT
Γ T 0 −I DT
0 CX D −µI
< 0 (4.3)
where µ= γ2.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 15
Exercise: Alternatively, let us consider
GTPG − P< 0 →
−P GTP
PG −P

< 0 →
−P PG
GTP −P

< 0
and
Φ Γ
γ−1C γ−1D
T
X 0
0 I

Φ Γ
γ−1C γ−1D

−

X 0
0 I

< 0
Again permutations followed by congruence transformations and the
change of variables exhibit the LMI−X ΦX Γ 0XΦT −X 0 XCT
Γ T 0 −I DT
0 CX D −µI
< 0
where µ= γ2.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 16
■ H∞ Performance:
Tzw :

x(k+ 1) = Φx(k) + Γw(k)
z(k) = Cx(k) +Dw(k)
Lemma 4.1 (H∞ performance in disscrete-time)
The inequality ∥Tzw∥2∞ < µ holds if and only if there exists P> 0 such that P ΦP Γ 0(∗) P 0 PCT(∗) (∗) I DT
(∗) (∗) (∗) µI
> 0 (4.4a)
is feasible.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 17
With a slack variable G P ΦG Γ 0(∗) G+GT − P 0 GTCT(∗) (∗) I DT
(∗) (∗) (∗) µI
> 0 (4.4b)
Exercise: Verify (4.4b).
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 18
Good to Note: The standard Problem
K
G, a generalized system
w ∈W, a set of disturbances
z, a controlled variable
Goal: Given W, synthesize K (feedback control) s.t. zTz is as small as
possible and Tzw is stable.
w
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 19
In the state space with
G :

x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k)
z(k) = C1x(k) +D11w(k) +D12u(k)
y(k) = C2x(k) +D21w(k) +D22u(k)
(4.5)
Applying state feedback control u(k) = Kx(k) leads to a clp
Tzw :

x(k+ 1) = (Φ+ Γ2K)x(k) + Γ1w(k)
z(k) = (C1 +D12K)x(k) +D11w(k)
(4.6)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 20
Stabilizing State Feedback Control
Let us consider a linear discrete-time system
x(k+ 1) = Φx(k) + Γu(k)
Applying state feedback control u(k) = Kx(k) leads to
x(k+ 1) = (Φ+ Γ2K)x(k)
The closed-loop system is asymptotically stable iff there exists a P ∈ Snx
such that
P (Φ+ Γ2K)P
(∗) P

> 0 (4.7)
is feasible.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 21
Applying the change of variables Z = KP leads to an LMI
P ΦP+ Γ2Z
(∗) P

> 0 (4.8)
With P> 0 and Z, the stabilizing control gain
K = ZP−1 (4.9)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 22
With a slack variable G ∈ Rnx×nx
P (Φ+ Γ2K)G
(∗) G+GT − P

> 0
Applying the change of variables Z = KG leads to an LMI
P ΦG+ Γ2Z
(∗) G+GT − P

> 0
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 23
The closed loop system is asymptotically stable iff there exists a P ∈ Snx ,
Z ∈ Rnu×nx , G ∈ Rnx×nx such that
P ΦG+ Γ2Z
(∗) G+GT − P

> 0 (4.10)
With P> 0, Z, and G, the stabilizing control gain
K = ZG−1 (4.11)
Exercise: Explain why P ∈ Snx satisfying the LMIs is positive definite. Also,
explain why G ∈ Rnx×nx satisfying the LMI becomes nonsingular such that
computation of stat feedback control gain is possible.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 24
H2 State Feedback Control
Consider a linear discrete-time system
P :

x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k)
z(k) = C1x(k) +D11w(k) +D12u(k)
y(k) = x(k)
(4.12)
Applying state feedback control u(k) = Kx(k) leads to a clp
Tzw :

x(k+ 1) = (Φ+ Γ2K)x(k) + Γ1w(k)
z(k) = (C1 +D12K)x(k) +D11w(k)
(4.13)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 25
Assuming D11 = 0, ∥Tzw∥22 < µ holds iff there exists P ∈ Snx , W ∈ Snz such
that
tr(W)< µ,

W (C1 +D12K)P
(∗) P

> 0 (4.14a) P (Φ+ Γ2K)P Γ1(∗) P 0
(∗) (∗) I
> 0 (4.14b)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 26
Applying the change of variables Z = KP ∈ Rnu×nx leads to
tr(W)< µ,

W C1P+D12Z
(∗) P

> 0 (4.15a) P ΦP+ Γ2Z Γ1(∗) P 0
(∗) (∗) I
> 0 (4.15b)
With P> 0, W, and Z, the H2 state feedback control gain
K = ZP−1. (4.16)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 27
Using the dilated LMI condition (4.2b)
Assuming D= 0, ∥Tzw∥22 < µ holds iff there exists G ∈ Rnx×nx and P ∈ Snx ,
W ∈ Snz such that
tr(W)< µ,

W (C1 +D12K)G
(∗) G+GT − P

> 0 (4.17a) P (Φ+ Γ2K)G Γ1(∗) G+GT − P 0
(∗) (∗) I
> 0 (4.17b)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 28
Applying the change of variables Z = KG ∈ Rnz×nx leads to
tr(W)< µ,

W C1G+D12Z
(∗) G+GT − P

> 0 (4.18a) P ΦG+ Γ2Z Γ1(∗) G+GT − P 0
(∗) (∗) I
> 0 (4.18b)
With P> 0, W, Z, and G, the H2 state feedback control gain
K = ZG−1. (4.19)
Exercise: Find the H2 state feedback control gain, using the dilated cond.
(4.2c).
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 29
H∞ State Feedback Control
Consider a linear discrete-time system
P :

x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k)
z(k) = C1x(k) +D11w(k) +D12u(k)
y(k) = x(k)
(4.20)
Applying state feedback control u(k) = Kx(k) leads to a clp
Tzw :

x(k+ 1) = (Φ+ Γ2K)x(k) + Γ1w(k)
z(k) = (C1 +D12K)x(k) +D11w(k)
(4.21)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 30
∥Tzw∥2∞ < µ holds, iff there exists P ∈ Snx such that P (Φ+ Γ2K)P Γ1 0(∗) P 0 P(C1 +D12K)T(∗) (∗) I DT11
(∗) (∗) (∗) µI
> 0 (4.22)
is feasible.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 31
Applying the change of variables Z = KP ∈ Rnu×nx leads to P ΦP+ Γ2Z Γ1 0(∗) P 0 PCT1 + ZTDT12(∗) (∗) I DT11
(∗) (∗) (∗) µI
> 0 (4.23)
With P> 0 and Z, the H∞ state feedback control gain
K = ZP−1 (4.24)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 32
Using the dilated cond. (4.4b)
∥Tzw∥2∞ < µ holds, iff there exist P ∈ Snx and G ∈ Rnx×nx such that P (Φ+ Γ2K)G Γ1 0(∗) G+GT − P 0 GT(C1 +D12K)T(∗) (∗) I DT11
(∗) (∗) (∗) µI
> 0 (4.25)
is feasible.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 33
Applying the change of variables Z = KG ∈ Rnu×nx leads to P ΦG+ Γ2Z Γ1 0(∗) G+GT − P 0 GTCT1 + ZTDT12(∗) (∗) I DT11
(∗) (∗) (∗) µI
> 0 (4.26)
With P> 0, Z, and G, the H∞ state feedback control gain
K = ZG−1 (4.27)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 34
H2 Output Feedback Control
Dynamic output feedback problem:
A generalized system
P :

x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k)
z(k) = C1x(k) +D11w(k) +D12u(k)
y(k) = C2x(k) +D21w(k)
Output feedback control
u(k) =

ΦK ΓK
CK DK

:

xˆ(k+ 1) = ΦK xˆ(k) + ΓKy(k)
u(k) = CK xˆ(k) +DKy(k)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 35
Closed-loop
Tzw :
x(k+ 1)xˆ(k+ 1)
z(k)
= Φcl Γcl
Ccl Dcl
x(k)xˆ(k)
w(k)

where
Φcl =

Φ+ Γ2DKC2 Γ2CK
ΓKC2 ΦK

, Γcl =

Γ1 + Γ2DKD21
ΓKD21

,
Ccl =

C1 +D12DKC2 D12CK

, Dcl =

D11 +D12DKD21

S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 36
The LMIs for the closed-loop H2 performance
tr(W)< µ,

W CclP
(∗) P

> 0,
 P ΦclP Γcl(∗) P 0
(∗) (∗) I
> 0
With a slackvariable G
tr(W)< µ,

W CclG
(∗) G+GT − P

> 0,
 P ΦclG Γcl(∗) G+GT − P 0
(∗) (∗) I
> 0
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 37
A congruence transformation by T= diag(T,T, I)
TT
 P ΦclG Γcl(∗) G+GT − P 0
(∗) (∗) I
T> 0
TTPT TTΦclGT TTΓcl(∗) TT(G+GT − P)T 0
(∗) (∗) I
> 0 (4.28)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 38
Define
K =

ΦK ΓK
CK DK

, G=

X ?1,2
U ?2,2

, G−1 =

YT ?¯1,2
VT ?¯2,2

and
T =

I YT
0 VT

Here, ? and ?¯ denote arbitrary matrices with compatible dimensions.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 39
Want to have
GG−1 =

X ?1,2
U ?2,2

YT ?¯1,2
VT ?¯2,2

=

XYT+?1,2VT X?¯1,2+?1,2?¯2,2
UYT+?2,2VT U?¯1,2+?2,2?¯2,2

=

I 0
0 I

with ?1,2V
T = 0, XYT = I, UYT+?2,2VT = 0 such that GT =

X I
U 0

We can find ?¯1,2, ?¯2,2, ?1,2, and ?2,2 satisfying the above condition.
However, the matrices are not to be literally used in our formulation.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 40
Introduce the changes of variables:
Q F
L R

=

V YΓ2
0 I

ΦK ΓK
CK DK

U 0
C2X I

+

Y
0

Φ

X 0

(4.29)
Po J
JT H

= TTPT (4.30)
S= YX + VU (4.31)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 41
Find
Q F
L R

=

VΦKU + YΓ2CKU + VΓKC2X + YΓ2DKC2X + YΦX VΓK + YΓ2DK
CKU +DKC2X DK

Thus
Q= VΦKU + YΓ2CKU + VΓKC2X + YΓ2DKC2X + YΦX
F = VΓK + YΓ2DK
L= CKU +DKC2X
R= DK
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 42
Each block in the LMI (4.28)
TTΦclGT =

I 0
Y V

Φ+ ΓDKC2 Γ2CK
ΓKC2 ΦK

GT
=
 ΦX + Γ2DKC2X + Γ2CKU Φ+ Γ2DKC2YΦX + YΓ2DKC2X + VΓKC2X
+ YΓ2CKU + VΦKU
YΦ+ YΓ2DKC2 + VΓKC2

=

ΦX + Γ2L Φ+ Γ2DKC2
Q YΦ+ FC2

Exercise: Verify the above formulation for TTΦclGT
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 43
TTΓcl =

I 0
Y V

Γ1 + Γ2DKD21
ΓKD21

=

Γ1 + Γ2DKD21
YΓ1 + YΓ2DKD21 + VΓKD21

=

Γ1 + Γ2DKD21
YΓ1 + FD21

CclGT =

C1 +D12DKC2 D12CK
X I
U 0

=

C1X +D12DKC2X +D12CKU C1 +D12DKC2

=

C1X +D12L C1 +D12DKC2

Dcl =

D11 +D12DKD21

S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 44
Also, find
TT(G+GT − P)T =

X + XT I+ ST
I+ S Y + YT

−

Po J
JT H

Exercise: Verify the above formulation for TT(G+GT − P)T.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 45
Thus, in the LMI (4.28)
(1, 1) = TTPT =

Po J
JT H

(1,2) = TTΦclGT =

ΦX + Γ2L Φ+ ΓRC2
Q YΦ+ FC2

(1, 3) = TTΓcl =

Γ1 + Γ2RD21
YΓ1 + FD21

(2, 2) = TT(G+GT − P)T =

X + XT I+ ST
I+ S Y + YT

−

Po J
JT H

S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 46
A congruence transformation by T= diag(I,T)
TT

W CclG
(∗) G+GT − P

T< 0

W CclGT
(∗) T(G+GT − P)TT

< 0
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 47
Find
CclGT =

C1X +D12L C1 +D12DKC2

TT(G+GT − P)T =

X + XT − Po I+ ST − J
I+ S− JT Y + YT −H

S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 48
Substituting each block leads to the LMIs for H2 output feedback control
synthesis:
tr(W)< µ,
W C1X +D12L C1 +D12RC2(∗) X + XT − Po I+ ST − J
(∗) (∗) Y + YT −H
> 0 (4.32a)

Po J ΦX + Γ2L Φ+ Γ2RC2 Γ1 + Γ2RD21
(∗) H Q YΦ+ FC2 YΓ1 + FD21
(∗) (∗) X + XT − Po I+ ST − J 0
(∗) (∗) (∗) Y + YT −H 0
(∗) (∗) (∗) (∗) I
> 0 (4.32b)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 49
Utilizing the solutions S, X, Y,

Q F
L R

, one can find K:
▶ Find U and V from (4.31).
▶ Compute from (4.29)
K =

V YΓ2
0 I
†
Q F
L R

−

Y
0

Φ

X 0
U 0
C2X I
†
(4.33)
with
V YΓ2
0 I
†
=
‚
V YΓ2
0 I
T
V YΓ2
0 I
Œ−1
V YΓ2
0 I
T

U 0
C2X I
†
=

U 0
C2X I
T ‚
U 0
C2X I

U 0
C2X I
TŒ−1
Then, we have ΦK, ΓK, CK, and DK.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 50
Discrete-time H2 output feedback control: with D22 ̸= 0
Generalized systemx(k+ 1)z(k)
y(k)
=
 Φ Γ1 Γ2C1 D11 D12
C2 D21 D22
x(k)w(k)
u(k)

Output feedback control
u(k) =

ΦK ΓK
CK DK

:

xˆ(k+ 1) = ΦK xˆ(k) + ΓKy(k)
u(k) = CK xˆ(k) +DKy(k)
Closed-loop system
Gcl(z) =

Φcl Γcl
Ccl Dcl

= Ccl (zI−Φcl)−1 Bcl +Dcl
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 51
with
Φcl =

Φ+ Γ2DKD˜−1C2 Γ2(I+DKD˜−1D22)Ck
BKD˜
−1C2 ΦK + ΓKD˜−1D22CK

,
Γcl =

Γ1 + Γ2DKD˜−1D21
BKD˜
−1D21

,
Ccl =

C1 +D12DKD˜−1C2 D12(I+DKD˜−1D22)CK

,
Dcl =

D11 +D12DKD˜−1D21

where D˜= I−D22DK
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 52
Now solve for Φˆ ∈ Rnx×nx , Γˆ ∈ Rnx×ny , Cˆ ∈ Rnu×nx , Dˆ ∈ Ruu×ny , X1 > 0,
Y1 > 0, Z > 0, and ν ∈ R>0 that minimize ν subject to the following LMIs:
X1 I X1Φ+ ΓˆC2 Φˆ X1Γ1 + ΓˆD21
(∗) Y1 Φ+ Γ2DˆC2 ΦY1 + Γ2Cˆ Γ1 + Γ2DˆD21
(∗) (∗) X1 I 0
(∗) (∗) (∗) Y1 0
(∗) (∗) (∗) (∗) I
> 0
 Z C1 +D12DˆC2 C1YT1 +D12Cˆ(∗) X1 I
(∗) (∗) Y1
> 0
D11 +D12DˆD21 = 0
X1 I
(∗) Y1

> 0
trZ < ν
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 53
The controller is recovered by
DK = (I+ D¯D22)
−1D¯
CK = (I−DKD22)C¯
ΓK = Γ¯ (I−DKD22)
ΦK = Φ¯− ΓK(I−D22DK)−1D22CK
where
Φ¯ Γ¯
C¯ D¯

=

X2 X1Γ2
0 I
−1
Φˆ Γˆ
Cˆ Dˆ

−

X1ΦY1 0
0 0

YT2 0
C2Y1 I
−1
and the matrices X2 and Y2 chosen to satisfy X2Y
T
2 = I− X1Y1.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 54
Note:
▶ If D22 = 0, then Φk = Φ¯, ΓK = Γ¯ , CK = C¯, and DK = D¯.
▶ If D11 = 0, D12 ̸= 0, and D21 ̸= 0, then it is open simplest to choose
Dˆ= 0 in order to satisfy the equality constraint D11 +D12DˆD21 = 0.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 55
H∞ Output Feedback Control
Discrete-time H∞ output feedback control:
A discrete-time linear generalized system
x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k)
z(k) = C1x(k) +D11w(k) +D12u(k)
y(k) = C2x(k) +D21w(k)
Output feedback control
u(k) =

ΦK ΓK
CK DK

:

xˆ(k+ 1) = ΦK xˆ(k) + ΓKy(k)
u(k) = CK xˆ(k) +DKy(k)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 56
Then the closed-loop system
Gcl(z) =

Φcl Γcl
Ccl Dcl

= Ccl (zI−Φcl)−1 Bcl +Dcl
with
Φcl =

Φ+ Γ2DKC2 Γ2CK
ΓKC2 ΦK

, Γcl =

Γ1 + Γ2DKD21
ΓKD21

,
Ccl =

C1 +D12DKC2 D12CK

, Dcl =

D11 +D12DKD21

S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 57
The LMI for the closed-loop H∞ performance: P ΦclP Γcl 0(∗) P 0 PCTcl(∗) (∗) I DTcl
(∗) (∗) (∗) µI
> 0 (4.34)
With a slack variable G P ΦclG Γcl 0(∗) G+GT − P 0 GTCTcl(∗) (∗) I DTcl
(∗) (∗) (∗) µI
> 0 (4.35)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 58
A congruence transformation by T= diag(T,T, I, I)
TT
 P ΦclG Γcl 0(∗) G+GT − P 0 GTCTcl(∗) (∗) I DTcl
(∗) (∗) (∗) µI
T> 0
T
TPT TTΦclGT T
TΓcl 0
(∗) TT(G+GT − P)T 0 TTGTCTcl
(∗) (∗) I DTcl
(∗) (∗) (∗) µI
> 0 (4.36)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 59
Thus, in the LMI (4.36)
(1, 1) = TTPT =

Po J
JT H

(1,2) = TTΦclGT =

ΦX + Γ2L Φ+ ΓRC2
Q YΦ+ FC2

(1, 3) = TTΓcl =

Γ1 + Γ2RD21
YΓ1 + FD21

(2, 2) = TT(G+GT − P)T =

X + XT I+ ST
I+ S Y + YT

−

Po J
JT H

(2,4) = TTGTCTcl =

XTCT1 + L
TDT12
CT1 + C
T
2R
TDT12

(3, 4) = DTcl =

DT11 +D
T
21R
TDT12

S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 60
Substituting each expression leads to an LMI for H∞ output feedback
control synthesis:
Po J ΦX + Γ2L Φ+ Γ2RC2 Γ1 + Γ2RD21 0
(∗) H Q YΦ+ FC2 YΓ1 + FD21 0
(∗) (∗) X + XT − Po I+ ST − J 0 XTCT1 + LTDT21
(∗) (∗) (∗) Y + YT −H 0 CT1 + CT2RTDT21
(∗) (∗) (∗) (∗) I DT11 +DT21RTDT11
(∗) (∗) (∗) (∗) (∗) µI
> 0
(4.37)
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 61
Utilizing the solutions S, X, Y,

Q F
L R

, one can compute K:
▶ Find S, X, Y → U, VT
▶ Compute
K =

V YΓ2
0 I
†
Q F
L R

−

Y
0

Φ

X 0
U 0
C2X I
†
(4.38)
with
V YΓ2
0 I
†
=
‚
V YΓ2
0 I
T
V YΓ2
0 I
Œ−1
V YΓ2
0 I
T

U 0
C2X I
†
=

U 0
C2X I
T ‚
U 0
C2X I

U 0
C2X I
TŒ−1
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 62
Discrete-time H∞ output feedback control: with D22 ̸= 0
Generalized systemx(k+ 1)z(k)
y(k)
=
 Φ Γ1 Γ2C1 D11 D12
C2 D21 D22
x(k)w(k)
u(k)

Output feedback control
u(k) =

ΦK ΓK
CK DK

:

xˆ(k+ 1) = ΦK xˆ(k) + ΓKy(k)
u(k) = CK xˆ(k) +DKy(k)
Closed-loop system
Gcl(z) =

Φcl Γcl
Ccl Dcl

= Ccl (zI−Φcl)−1 Bcl +Dcl
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 63
with
Φcl =

Φ+ Γ2DKD˜−1C2 Γ2(I+DKD˜−1D22)Ck
BKD˜
−1C2 ΦK + ΓKD˜−1D22CK

,
Γcl =

Γ1 + Γ2DKD˜−1D21
BKD˜
−1D21

,
Ccl =

C1 +D12DKD˜−1C2 D12(I+DKD˜−1D22)CK

,
Dcl =

D11 +D12DKD˜−1D21

where D˜= I−D22DK
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 64
Solve for Φˆ ∈ Rnx×nx , Γˆ ∈ Rnx×ny , Cˆ ∈ Rnu×nx , Dˆ ∈ Rnu×ny , X1 > 0, Y1 > 0,
and γ ∈ R>0 that minimizes γ subject to the following LMIs:

X1 I X1Φ+ ΓˆC2 Φˆ X1Γ1 + ΓˆD21 0
(∗) Y1 Φ+ Γ2DˆC2 ΦY1 + Γ2Cˆ Γ1 + Γ2DˆD21 0
(∗) (∗) X1 I 0 CT1 + CT2DT12
(∗) (∗) (∗) Y1 0 Y1CT1 + CˆTDT12
(∗) (∗) (∗) (∗) −γI DT11 +DT21DˆTDT12
(∗) (∗) (∗) (∗) (∗) −γI
> 0

X1 I
(∗) Y1

> 0
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 65
The controller is recovered by
DK = (I+ D¯D22)
−1D¯
CK = (I−DKD22)C¯
ΓK = Γ¯ (I−DKD22)
ΦK = Φ¯− ΓK(I−D22DK)−1D22CK
where
Φ¯ Γ¯
C¯ D¯

=

X2 X1Γ2
0 I
−1
Φˆ Γˆ
Cˆ Dˆ

−

X1ΦY1 0
0 0

YT2 0
C2Y1 I
−1
and the matrices X2 and Y2 chosen to satisfy X2Y
T
2 = I− X1Y1.
If D22 = 0, then Φk = Φ¯, ΓK = Γ¯ , CK = C¯, and DK = D¯.
S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 66