Volume1.pdf

Notes of robust control
• Zeros (continuous-time systems)
• Parametrization (continuous-time systems)
• H2 control (continuous-time systems)
• H∞ control (continuous-time systems)
• H∞ filtering in discrete-time (discrete-time systems)

.
ZEROS AND POLES OF
MULTIVARIABLE SYSTEMS
• Transmission zeros
SUMMARY
Rank property and time-domain characterization
• Invariant zeros
Rank property and time-domain characterization
• Decoupling zeros
• Infinite zero structure

.
(Transmission) Zeros and poles of a rational
matrix
Basic assumption:
−1
G ( s)
= C(
sI − A)
B + D,
p outputs,
[ G(
s)
] = min( p,
m)
r
normal rank
=
Smith-McMillan form of G(s)
⎡ f1(
s)
⎢
⎢
0
M ( s)
= ⎢ M
⎢
⎢ 0
⎢
⎣ 0
εi(
s)
fi
( s)
= , i = 1,
2,
L,
r
ψ ( s)
εi divides εi+1
ψi+1 divides ψi
i
f
2
0
( s)
M
0
0
L
L
O
L
L
f
r
0
0
M
( s)
0
0⎤
0
⎥
⎥
M⎥
⎥
0⎥
0⎥
⎦
• The transmission zeros are the roots of e i(s), i=1,2,…,r
m inputs
• The (transmission) poles are the roots of y i(s), i=1,2,…,r

.
Example
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡−
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
=
1
1
,
0
1
0
0
0
0
5
13
,
0
2
1
0
,
0
1
1
1
0
5
0
0
0
0
7
10
0
0
1
0
D
C
B
A
Hence
)
5
)(
2
(
1
)
2
)(
3
(
)
1
)(
3
(
)
(
−
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−
+
−
=
s
s
s
s
s
s
s
G
)
5
)(
2
(
1
0
3
)
(
,
1
−
−
⎥
⎦
⎤
⎢
⎣
⎡ −
=
=
s
s
s
s
M
r
There is one transmission zero at s=3 and two poles in s=2 and s=5.

Theorem 1
.
Rank property of transmission zeros
The complex number λ is a transmission zero of G(s) iff there exists
polynomial vector z(s) such that z(λ)≠0 and
⎧ lim G ( s)
z(
s)
= 0
⎪
s → λ
⎨
lim G(
s)'
z(
s)
= 0
⎪
⎪⎩
s → λ
p ≥ m
p ≤ m
Remark
In the square case (p=m), it follows
det
[ G ( s)
]
∏
= r
r
i = 1
∏
i = 1
ε
ψ
i
i
( s)
( s)
if
if
so that it is possible to catch all zeros and poles if and only if there are not
cancellations.
Proof of the theorem
Assume that p≥ m and let M(s) be the Smith-Mc Millan form of G(s), i.e. G(s)=L(s)M(s)R(s) for suitable polynomial
and unimodular matrices L(s) (of dimension p) and R(s) (of dimensions m). It is possible to write:
where E(s)=diag{εi(s)} and Ψ(s)=diag{ψi(s)}. Now assume that λ is a zero of the polynomial eκ(s). Recall that R(s) is
unimodular and take the polynomial vector z(s)=[R -1 (s)]k, i.e. the k-th column of the inverse of R(s). Since ψk(λ)≠0, it
follows that G(s)z(s)=L(s)E(s)Ψ −1 (s)R(s)z(s)=L(s)E(s) [Ψ −1 (s)]k= L(s)[E(s)]k/ψ k (s)=[L(s)]kε k (s) /ψ k (s). This quantity
goes to zero as s tends to λ. Vice-versa, if z(s) is such that G(s)z(s) goes to zero as s tends to λ, then the limit as s tends
to λ of E(λ)Y -1 −1
⎡E( s)
Ψ ( s)
⎤
M
( s)
= ⎢ ⎥
⎣ 0 ⎦
(s)R(λ)z(λ) being zero means that at least one polynomial εk(s) should vanish at s= λ.
The proof in the opposite case (p≤m) follows the same lines and therefore is omitted.

.
Example
G
⎡1
s)
= ⎢
⎣0
( s −1)(
s + 2)
⎤ 1
( s −1)
⎥
⎦ ( s −1)(
s + 2)
( 2
Hence,
⎡1
G(
s)
= ⎢
⎣0
so that, with
⎡ 1
0⎤⎢
( s −1)(
s +
⎢
1
⎥
⎦⎢
0
⎢⎣
⎡ − ( s −1)(
s + 2)
⎤
z(
s)
= ⎢
⎥
⎣ 1 ⎦
it follows
lim G(
s)
z(
s)
= 0
s →1
2)
Notice that it is not true that G(s)z(1) tends to zero as s tends to one. .
⎤
0
⎥ 1
⎥
s −1
⎥ ( s −1)(
s + 2)
s + 2⎥⎦

.
Time-domain characterization of transmission
zeros and poles
Let Σ=(Α,Β, C ,D) be the state-space system with transfer function
G(s)=C(sI-A) -1 B+D. Let Σ′=(A′,C′,B′,D′) the “transponse” system whose
transfer function is G(s) ′=B′(sI-A′) -1 C′+D′. Now, let
Σˆ
⎧ Σ
= ⎨
⎩ Σ '
Finally, let δ i (t) denote the i-th derivative of the impulsive “function”.
Theorem 2
p
p
≥
≤
m
m
Gˆ
( s )
⎧ G ( s )
⎨
⎩ G ( s )'
(a) The complex number λ is a pole of Σ iff the exists an impulsive input
such that the forced output yf(t) of Σ is
(b) The complex number λ is a zero of Σ iff the exists an
exponential/impulsive input
such that the forced output yf(t) of Σ is
=
u ( t )
y
f
y f
Zeros blocking property. The proof of the theorem is not reported here for space limitations.
∑
i = 0
= ν
α
p
p
i
δ
≥
≤
i
m
m
( t )
λt
( t ) = y 0e , ∀ t ≥
λ t
i
u ( t ) u 0 e + α
iδ
( t ) , u 0
= ∑
i = 0
ν
( t ) = 0 , ∀ t ≥
0
0
≠
0

.
Invariant zeros
Given a system (A,B,C,D) with transfer function G(s)=C(sI-A) -1 B+D, the
polynomial matrix
P ( s )
=
⎡ sI − A −
⎢
⎣ C D
is called the system matrix.
Consider the Smith form S(s) of P(s):
⎡α1(
s)
⎢
⎢
0
S(
s)
= ⎢ M
⎢
⎢ 0
⎢
⎣ 0
αi divides αi+1.
α
2
0
( s)
M
0
0
L
L
O
L
L
αν
0
0
M
( s)
0
0⎤
0
⎥
⎥
M⎥
⎥
0⎥
0⎥
⎦
The invariant zeros are the roots of αi(s), i=1,2,…,ν
−−−−−−−−−−−−−−−−−
Notice that
P ( s)
=
⎡sI −
⎢
⎣ C
A
so that the rank of P(s) is ϖ=n+rank(G(s))
B
⎤
⎥
⎦
0⎤
⎡( sI − A )
I
⎥ ⎢
⎦ ⎣ 0
−1
0 ⎤ ⎡sI
−
⎥
G ( s)
⎢
⎦ ⎣ 0
A
− B ⎤
I
⎥
⎦

Theorem 3
.
Rank property of invariant zeros
The complex number λ is an invariant zero of (A,B,C,D) iff P(s) looses
rank in s=λ, i.e. iff there exists a nonzero vector z such that
⎧ P ( λ ) z = 0
⎪
⎨
P
⎪
⎪⎩
( λ )' z = 0
if
if
Notice that if T is the transformation for a change of basis in the state-space, then
P ( s ) =
⎡ sI − A − B
⎢
⎣ C D
⎤
⎥
⎦
=
⎡T
⎢
⎣ 0
Therefore, the invariant zeros are invariant with respect to a change of basis.
p
p
≥
≤
m
m
0⎤
⎡sI − A −
I
⎥ ⎢
⎦ ⎣ C D
⎤ ⎡T
⎥ ⎢
⎦ ⎣
Proof of Theorem 3
Consider the case p≥m and let P(s)=L(s)S(s)R(s), where R(s) and R(s) are suitable polynomial
unimodular matrices. Hence, if P(λ)z=0 for some z≠0, then S(λ )R(λ)z=0 so that S(λ)y=0, with y≠0.
Hence, λ must be the root of at least one polynomial αi(s). Vice-versa, if αk(λ) for some k, then
z=[R -1 (λ)]k is such that P(λ)z=L(λ)S(λ)R(λ)z= L(λ)S(λ)R(λ)[R -1 (λ)]k=L(λ)[S(λ)]k=0.
B
−1
0
0 ⎤
⎥
I ⎦

.
Time-domain characterization of invariant zeros
Let Σ=(Α,Β, C ,D) be the state-space system with transfer function
G(s)=C(sI-A) -1 B+D. Let Σ′=(A′,C′,B′,D′) the “transponse” system whose
transfer function is G(s) ′=B′(sI-A′) -1 C′+D′. Now, let
⎧ Σ
Σˆ
= ⎨
⎩Σ
'
Theorem 4
p ≥
p ≤
m
m
G s
Gˆ
⎧ ( )
( s)
= ⎨
⎩G
( s)'
The complex number λ is an invariant zero Σ iff at leat one of the two
following conditions holds:
(i) λ is an eigenvalue of the unobservable part of Σ
(ii) there exist two vectors x0 and u0≠0 such that the forced output of Σ
corresponding to the input u(t)=u0e λt , t≥0 and the initial state
x(0)=x0 is identically zero for t≥0.
p ≥
p ≤
Zeros blocking property. The proof of the theorem is not reported here for space limitations.
m
m

.
Trasmission vs invariant zeros
Theorem 5
(i) A transmission zero is also an invariant zero.
(ii) Invariant and transmission zeros of a system in minimal form do
coincide.
Proof of point (i). Let p≥m and let λ be a transmission zero. Then, there exists an
exponential/impulsive input
λt
ν
∑
i=
0
( i)
u ( t)
= u e + α δ ( t)
such that the forced output is zero, i.e.
y
f
0
t
i
ν
A(
t−τ)
⎡ λτ
( i)
⎤ λt
( t)
= C∫
e B⎢u
0e
+ ∑ α iδ
( τ)
⎥dτ
+ Du0e
= 0,
t > 0 . Letting
⎣
i=
0 ⎦
0
⎡α0
⎤
⎢ ⎥
ν
[ ] ⎢
α1
x =
⎥
0 B AB L A B , it follows
⎢ M ⎥
⎢ ⎥
⎣αν
⎦
Ce
t
At
x0
C
A(
t−τ
) λτ
λt
e Bu0e
dτ
+ Du0e
+ ∫
0
=
0,
t > 0
so that λ is an invariant zero. The proof in the case p≤m is the same, considering the transponse
system.
Proof of point (ii). Let p≥m. We have already proven that a trasmission zero is an invariant zero.
Then, consider a minimum system and an invariant zero λ. Hence Av+Bw=λw, Cv+Dw=0. Notice
that w≠0 since the system is observable. Moreover, thanks to the reachability condition there exists
a vector
⎡α
0 ⎤
⎢ ⎥
ν
[ ] ⎢
α1
v = B AB L A B ⎥
⎢ M ⎥
⎢ ⎥
⎣αν
⎦
Since λ is an invariant zero there exists an initial state x(0)=v and an input u(t)=we λt , such that
Ce
t
At A
λ
v + C∫
e
By noticing that
C
0
( t−τ
) λτ
t
Bwe dτ
+ Dwe = 0,
t > 0
Ce
At
ν
∑
t
At i
A(
t−τ)
( i)
v = Ce A Bαi
= Ce B αiδ
( τ)
dτ
, it follows,
i=
0
t
ν
A( t−τ)
⎡ λτ
λ
∫ e B⎢u
0e
+ ∑
0 ⎣
i=
0
trransmission zero.
∫
0
ν
∑
i=
0
( i)
⎤
t
α iδ
( τ)
⎥dτ
+ Du0e
= 0,
t > 0,
which means that λ is a
⎦

Associated Smith forms
.
S
P C
C
Decoupling zeros
⎡ sI − A ⎤
( s ) = ⎢ ⎥ PB ( s)
= [ sI − A − B]
⎣ C ⎦
( s )
=
⎡
⎢
⎣
diag
C { α ( s ) } ⎤
B
i
⎥⎦ S B ( s ) =
diag { α i ( s ) }
0
[ 0 ]
The output decoupling zeros are the roots of α1 C (s) α2 C (s)… αn C (s), the
input decoupling zeros are the roots of αi B (s) α2 B (s)…αn B (s) and the inputoutput
decoupling zeros are the common roots of both.
Lemma
(i) A complex number λ is an output decoupling zero iff there exists
z≠0 such that PC(λ)z=0.
(ii) A complex number λ is an input decoupling zero iff there exists
w≠0 such that P B(λ)′w=0.
(iii) A complex number λ is an input-output decoupling zero iff there
exist z≠0 and w≠0 such that PC(λ)z=0, PB(λ)′w=0.
(iv) For square systems the set of decoupling zeros is a subset of the
set of invariant zeros.
(v) For square invertible systems the transmission zeros of the inverse
system coincide with the poles of the system and the invariant
zeros of the inverse system coincide with the eigenvalues of the
system.

.
Infinite zero structure
Mc Millan form over the ring of causal rational matrices
Given the transfer function G(s) there exists two unimodular matrices in
that ring (bi-proper rational functions) such that
⎡M
( s)
0⎤
G(
s)
= V ( s)
⎢ ⎥W
( s),
M ( s)
= diag
,
⎣ 0 0⎦
where δ1, δ2, …, δr are integers such that δ1≤ δ2≤…≤ δr. They describes the
structure of the infinity zeros.
• Inversion
• Model matching
• ……
Example: compute the finite and infinite zeros of:
⎡s
+ 1
⎢
⎢s
− 1
G(
s)
= ⎢ 1
⎢
⎢ 1
⎢
⎣ s
0
1
s
0
s
3
s + 1 ⎤
⎥
s − 1 ⎥
s + 1 ⎥
s ⎥
2
− 3s
+ s + 5⎥
3 ⎥
s ( s − 3)
⎦
{ } r
−δ1
−δ2
−δ
s , s , L s

PARAMETRIZATION OF STABILIZING
CONTROLLERS
Summary
• BIBO stability and internal stability of feedback systems
• Stabilization of SISO systems. Parameterization and
characterization of stabilizing controllers
• Coprime factorization
• Stabilization of MIMO systems. Parameterization and
characterization of stabilizing controllers
• Strong stabilization

Stability of feedback systems
d
r e u y
C(s) P(s)
The problem consists in the analysis of the asymptotic stability
(internal stability) of closed-loop system from some characteristic
transfer functions. This result is useful also for the design.
THEOREM 1
The closed loop system is asymptotically stable if and only if the
transfer matrix G(s) from the input to the output
is stable.
G(
s)
⎡r
⎤
⎢ ⎥
⎣d
⎦
−1
⎡ ( I + P(
s)
C(
s))
⎢
−1
⎣(
I + C(
s)
P(
s))
C(
s)
⎡e⎤
⎢ ⎥
⎣u⎦
−1
− ( I + P(
s)
C(
s))
P(
s)
⎤
⎥
( I + C(
s)
P(
s))
⎦
= −1

Stability of interconnected SISO systems
The closed loop system is asymptotically stable if and only if the
three transfer functions
S(
s)
=
1
are stable.
+
1
, R(
s)
C(
s)
P(
s)
=
1
+
C(
s)
, H ( s)
C(
s)
P(
s)
P(
s)
C(
s)
P(
s)
• Notice that if S(s), R(s), H(s) are stable, then also the
complementary sensitivity
C(
s)
P(
s)
T ( s)
=
= 1 − S(
s)
1+
C(
s)
P(
s)
is stable.
• The stability of the three transfer functions are made to avoid
prohibited (unstable) cancellations between C(s) and P(s).
Example
Let C(s)=(s-1)/(s+1), P(s)=1/(s-1) . Then S(s)=(s+1)/(s+2),
R(s)=(s-1)/(s+2) e H(s)=(s+1)/(s 2 +s-2)
=
1
+

Comments
The proof of Theorem 1 can be carried out pursuing different
paths. A simple way is as follows. Let assume that C(s) and P(s)
are described by means of minimal realizations C(s)=(A,B,C,D),
P(s)=(F,G,H,E). It is possible to write a realization of the closedloop
system as Σcl=(Acl,Bcl,Ccl,Dcl). Obviously, if Acl is
asymptotically stable, then G(s) is stable. Vice-versa, if G(s) is
stable, asymptotic stability of Acl follows form being
Σcl=(Acl,Bcl,Ccl,Dcl) a minimal realization. This can be seen
through the well known PBH test.
In our example, it follows
A
cl
G(
s)
=
0
⎡
⎢
⎣−
⎡
⎢
= ⎢
⎢
⎢⎣
1
s
s
s
s
+
+
−
+
−
−
1
2
1
2
2
1
⎤
⎥
⎦
,
( s
−
−
B
cl
=
⎡
⎢
⎣
1
1
( s + 1)
1)(
s + 2)
s + 1
s + 2
⎤
⎥
⎥
⎥
⎥⎦
1
0
⎤
⎥
⎦
,
SMM
>
⎡−
= ⎢
⎣−
Hence, Acl is unstable and the closed-loop system is reachable and
observable. This means that G(s) is unstable as well.
C
cl
⎡
⎢
⎢
⎣
( s
−
1
1
−
1
1)(
s
0
0
2
+
⎤
⎥
⎦
,
2)
D
cl
( s
=
+
⎡
⎢
⎣
1
1
0
1)(
s
−
0
1
⎤
⎥
⎦
1)
⎤
⎥
⎥
⎦

Parametrization of stabilizing controllers
1° case: SISO systems and P(s) stable
d
r e u
C(s) P(s)
THEOREM 2
The family of all controllers C(s) such that the closed-loop system
is asymptotically stable is:
C(
s)
=
1
−
Q(
s)
P(
s)
Q(
s)
where Q(s) is proper and stable and Q(∞)P(∞) ≠1.

Proof of Theorem 2
Let C(s) be a stabilizing controller and define
Q(s)=C(s)/(1+C(s)P(s)). Notice that Q(s) is stable since it is the
transfer function from r to u. Hence C(s) can be written as
=Q(s)/(1-P(s)Q(s)), with Q(s) stable and, obviously, the stability
of both Q(s) and P(s) implies that Q(∞)P(∞)=C(∞)/(1+C(∞)P(∞))
≠1.
Vice-versa, assume that Q(s) is stable and Q(∞)P(∞)≠1. Define
C(s)=Q(s)/(1-P(s)Q(s)). It results that
S(s)=1/(1+C(s)P(s))=1-P(s)Q(s),
R(s)=C(s)/(1+C(s)P(s))=Q(s),
H(s)=P(s)/(1+C(s)P(s))=P(s)(1-P(s)Q(s))
are stable. This means that the closed-loop system is
asymptotically stable.

Parametrization of stabilizing controllers
2° case: SISO systems and generic P(s)
It is always possible to write (coprime factorization)
P ( s)
=
N(
s)
M ( s)
where N(s) and M(s) are stable rational coprime functions, i.e.
such that there exist two stable rational functions X(s) e Y(s)
satisfying (equation of Diofanto, Aryabatta, Bezout)
N(
s)
X ( s)
+ M ( s)
Y ( s)
= 1
THEOREM 3
The family of all controllers C(s) such that the closed-loop system
is well-posed and asymptotically stable is:
C(
s)
=
X ( s)
Y(
s)
+
−
M ( s)
Q(
s)
N(
s)
Q(
s)
where Q(s) is proper, stable and such that Q(∞)N(∞) ≠ Y(∞).

Comments
The proof of the previous theorem can be carried out following
different ways. However, it requires a preliminary discussion on
the concept of coprime factorization and on the stability of
factorized interconnected systems.
d
r e z1 u z2 y
Mc(s) -1
LEMMA 1
Let P(s)=N(s)/M(s) e C(s)=Nc(s)/Mc(s) stable coprime
factorizations. Then the closed-loop system is asymptotically
stable if and only if the transfer matrix K(s) from [r′ d′]′ to [z1′
z2′]′ is stable.
K(
s)
M(s) -1
Nc(s) N(s)
⎡
⎢
⎣
r
d
=
⎤
⎥
⎦
⎡
⎢
⎣−
M
c
N
( s)
c
( s)
⎡
⎢
⎣
z
z
1
2
⎤
⎥
⎦
N(
s)
M ( s)
⎤
⎥
⎦
−
1

Proof of Lemma 1
Define four stable functions X(s),Y(s),Xc(s),Yc(s) such that
X ( s)
N(
s)
+ Y ( s)
M ( s)
= 1
X c c c c
( s)
N ( s)
+ Y ( s)
M ( s)
= 1
To proof the Lemma it is enough to resort to Theorem 1, by
noticing that the transfer functions K(s) and G(s) (from [r′ d′]′ to
[e′ u′]′) are related as follows:
K(
s)
=
⎡
⎢
⎣−
Y ( s)
c
X ( s)
X
c
( s)
Y ( s)
⎤
⎥
⎦
G(
s)
⎡
− ⎢
⎣−
⎡I
0⎤
⎡ 0 − N ⎤
G(
s)
=
⎢ ⎥ + ⎢ ⎥ K(
s)
0 I N 0
⎣
⎦
⎣
c
⎦
0
X
X
0
c
⎤
⎥
⎦

Proof of Theorem 3
Assume that Q(s) is stable and Q(∞)N(∞) ≠ Y(∞). Moreover,
define C(s)=(X(s)+M(s)Q(s))/(Y(s)-N(s)Q(s)). It follows that
1=N(s)X(s)+M(s)Y(s)=N(s)(X(s)+M(s)Q(s))+M(s)(Y(s)-
N(s)Q(s))
so that the functions X(s)+M(s)Q(s) e Y(s)-N(s)Q(s) defining C(s)
are coprime (besides being both stable). Hence, the three
characterist transfer functions are:
S(s)=1/(1+C(s)P(s))=M(s)(Y(s)-N(s)Q(s)),
R(s)=C(s)/(1+C(s)P(s))=M(s)(X(s)+M(s)Q(s)),
H(s)=P(s)/(1+C(s)P(s))=N(s) (Y(s)-N(s)Q(s))
Since they are all stable, the closed-loop system is asymptotically
stable as well (Theorem 1).
Vice-versa, assume that C(s)=Nc(s)/Mc(s) (stable coprime
factorization) is such that the closed-loop system is well-posed
and asymptotically stable. Define Q(s)=(Y(s)Nc(s)-
X(s)Mc(s))/(M(s)Mc(s)+N(s)Nc(s)). Since the closed-loop system
is asymptotically stable and (Nc,Mc) are coprime, then, in view of
Lemma 1 it follows that
Q(s) = (Y(s)Nc(s)-X(s)Mc(s))/(M(s)Mc(s)+N(s)Nc(s)) =
= [0 I]K(s)[Y(s)′-X(s)′]′
is stable. This leads to
C(s)=(X(s)+M(s)Q(s))/(Y(s)-N(s)Q(s)).
Finally, Q(∞)N(∞) ≠ Y(∞), as can be easily be verified.

Coprime Factorization
1° case: SISO systems
Lemma 1 makes reference to a factorized description of a transfer
function. Indeed, it is easy to see that it is always possible to write
a transfer function as the ratio of two stable transfer functions
without common divisors (coprime). For example
with
It results
with
s + 1
G(
s)
= = N(
s)
M ( s)
( s − 1)(
s + 10)(
s − 2)
N(
s)
=
( s
+
1
10)(
s
+
1)
N(
s)
X(
s)
+ M(
s)
X ( s)
=
X ( s)
=
Euclide’s Algorithm
4(
16s
− 5)
,
( s + 1)
,
Y ( s)
M ( s)
1
=
( s
( s
=
+
+
−
( s − 1)(
s
( s + 1)
15)
10)
1
−
2
2)

Coprime Factorization
1° case: MIMO systems
In the MIMO case, we need to distinguish between right and left
factorizations.
Right and left factorization
Given P(s), find four proper and stable transfer matrices such that:
− 1
−1
P( s)
= Nd
( s)
M d ( s)
= M s(
s)
N s(
s)
Nd ( s),
M d ( s)
right coprime X d ( s)
Nd
( s)
+ Yd
( s)
M d(
s)
Ns ( s),
M s(
s)
left coprime N s(
s)
X s(
s)
+ M s(
s)
Ys
( s)
Double coprime factorization
Given P(s), find eight proper and stable transfer functions such
that:
− 1
−1
( i) P(
s)
= Nd
( s)
M d(
s)
= M s(
s)
Ns
( s)
( ii)
⎡
⎢
⎣
Y
d
( s)
X
( s)
M
( s)
( s)
− N ( s)
M ( s)
N ( s)
Y ( s)
s
Hermite’s algorithm
Observer and control law
d
s
⎤ ⎡ d −
⎥ ⎢
⎦ ⎣ d
X
s
s
⎤
⎥
⎦
=
I
=
I
=
I

Construction of a stable double coprime
factorization
Let (A,B,C,D) a stabilizable and detectable realization of P(s) and
conventionally write P=(A,B,C,D) .
THEOREM 4
Let F e L two matrices such that A+BF e A+LC are stable. Then
there exists a stable double coprime factorization, given by:
Md=(A+BF,B, F,I), Nd=(A+BF,B,C+DF,D)
Ys=(A+BF,L,-C-DF,I), Xs=(A+BF,L,F,0)
Ms=(A+ LC,L,C,I), Ns=(A+LC,B+LD,C,D)
Yd=(A+LC,B+LD,- F,I), Xd=(A+LC,L,F,0)

Proof of Theorem 4
The existence of stabilizing F e L is ensured by the assumptions of
stabilizability and detectability of the system. To check that the 8
transfer functions form a stable double coprime factorization
notice first that they are all stable and that Md e Ms are biproper
systems. Finally one can use matrix calculus to verify the theorem.
Alternatively, suitable state variables can be introduced. For
example, from
x&
y
u
=
=
=
Ax
Cx
Fx
+
+
+
Bu
Du
v
=
=
( A
( C
+
+
control
BF ) x
DF)
x
law
+
+
Bv
Dv
one obtains y=P(s)u=Nd(s)v, u=Md(s)v, so that P(s)Md(s)=Nd(s).
Similarly,
x&
y
=
=
Ax
Cx
⎧ =
⎨
⎩ η =
θ &
A
C
+
+
θ
θ
Bu
Du
+
+
Bu
Du
+
−
L
η
y
observer
implies η=Ns(s)u-Ms(s)y=Ns(s)u-Ms(s)P(s)y=0 (stable autonomous
dynamic) so that Ns(s)=Ms(s)P(s). Analoguously one can proceed
for the rest of the proof.

Parametrization of stabilizing controllers
3° case: MIMO systems and generic P(s)
− 1
−1
( i) P(
s)
= Nd
( s)
M d ( s)
= M s(
s)
Ns
( s)
( ii)
⎡ Yd
( s)
X d ( s)
⎤ ⎡M d ( s)
− X s(
s)
⎤
⎢
=
N s s M s s
⎥ ⎢
Nd
s Ys
s
⎥
⎣−
( ) ( ) ⎦ ⎣ ( ) ( ) ⎦
THEOREM 5
The family of all proper transfer matrices C(s) such that the
closed-loop system is well-posed and asymptotically stable is:
C(
s)
= ( X
= ( Y
d
s
C(s) P(s)
( s)
+ M
( s)
Q
( s)
− Q ( s)
N ( s))
s
d
s
d
( X
I
( s))(
Y ( s)
− N
−1
s
d
d
( s)
Q
( s)
+ Q ( s)
M
s
d
( s))
s
−1
( s))
where Qd(s) [Qs(s)] is stable and such that Nd(∞)Qd(∞)≠Ys(∞)
[Qs(∞)Ns(∞)≠Yd(∞)].

Comments
The proof of Theorem 5 follows the same lines as that of Theorem
3. Thowever, the generalization of Lemma 1 to MIMO is required.
d
r e z1 u z2 y
Mc(s) -1
LEMMA 2
Let P(s)=Nd(s)Md(s) -1 and C(s)=Nc(s)Mc(s) -1 stable right coprime
factorizations. Then the closed-loop system is asymptotically
stable if and only if the transfer matrix K(s) from [r′ d′]′ to [z1′
z2′]′ is stable.
⎡ M c(
s)
K(
s)
= ⎢
⎣−
Nc
( s)
Md(s) -1
Nc(s) Nd(s)
N
M
d
d
( s)
⎤
( s)
⎥
⎦
−1

Proof of Lemma 2
It is completely similar to the proof of Lemma 1.
Proof of Theorem 5
Assume that Q(s) is stable and define
C(s)=(Xs(s)+Md(s)Q(s))(Ys(s)-Nd(s)Q(s)) -1
It results that
I=Ns(s)Xs(s)+Ms(s)Ys(s)
=Ns(s)(Xs(s)+Md(s)Q(s))+Ms(s)(Y(s)s-Nd(s)Q(s))
so that the functions Xs(s)+Md(s)Q(s) e Ys(s)-Nd(s)Q(s) defining
C(s) are right coprime (besides being both stable). The four
transfer matrices characterizing the closed-loop are:
(1+PC) -1 = (Ys-NdQ)Ms
-(1+PC) -1 P= (Ys-NdQ)Ns
(1+CP) -1 C=C(I+PC) -1 = (Xs+MdQ)Ms
(1+CP) -1 =I-C(I+PC) -1 C= I-(Xs+MdQ)Ns
They are all stable so that the closed-loop system is asymptotically
stable.
Vice-versa, assume that C(s)=Nc(s)Mc(s) -1
(right coprime
factorization) is stabilizing. Notice that the matrices
⎡ Yd
( s)
X d(
s)
⎤ ⎡M
d ( s)
Nc
( s)
⎤
⎢
and
≈
N s s M s s
⎥ ⎢
Nd
s M c s
⎥
⎣−
( ) ( ) ⎦ ⎣ ( ) ( ) ⎦
have stable inverse. Hence,
⎡ Yd
( s)
X d ( s)
⎤ ⎡M
d ( s)
Nc
( s)
⎤ ⎡I
Yd
( s)
N c(
s)
X d ( s)
Nc
( s)
⎤
⎢
⎥ ⎢
⎥ = ⎢
⎥
⎣−
N s ( s)
M s ( s)
⎦ ⎣ N d ( s)
M c ( s)
⎦ ⎣0
N s ( s)
N c(
s)
+ M s ( s)
M c ( s)
⎦
(*)
has stable inverse as well. Then,
K

Q(s)=(Yd(s)Nc(s)-Xd(s)Mc(s))(Ms(s)Mc(s)+Ns(s)Nc(s)) -1
is well–posed and stable. Premultiplying equation (*) by
[Md(s) –Xs(s);Nd(s) Ys(s)]
it follows
Nc(s)Mc(s) -1 =(Xs(s)+Md(s)Qd(s))(Ys(s)-Nd(s)Qd(s)) -1 .

Observation
Taking Q(s)=0 we have the so-called central controller
C0(s) =Xs(s)Ys(s) -1
which coincides with the controller designed with the pole
assignment technique.
Taking r=d=0 one has:
ξ&
= Aξ
+ Bu + L(
Cξ
+ Du − y)
u = Fξ
LEMMA 3
−1
−1
−1
[ ( I − Y ( s)
N ( s)
Q(
s))
] Y ( )
−1
C( s)
= C0
( s)
+ Yd
( s)
Q(
s)
s d
s s
u y
P
Yd -1
C0
Nd
Q
Ys -1

Strong Stabilization
The problem is that of finding, if possible, a stable controller
which stabilizes the closed-loop system.
Example
s −1
P(
s)
=
( s − 2)(
s + 2)
is not stabilizable with a stable controller. Why?
s − 2
P(
s)
=
( s −1)(
s + 2)
is stabilizable with a stable controller. Why?
Stabilizazion of many plants
Two-step stabilization
C(s) P(s)

Interpolation
Let consider a SISO control system with P(s)=N(s)/M(s) e
C(s)=Nc(s)/Mc(s) (stable coprime stabilization). Then the closedloop
system is asymptotically stable if and only if
U(s)=N(s)Nc(s)+M(s)Mc(s) is an unity (stable with stable inverse).
Since we want that C(s) be stable, we can choice Nc(s)=C(s) and
Dc(s)=1. Then,
U ( s)
− M ( s)
C(
s)
=
N ( s)
Of course, we must require that C(s) be stable. This fact depends
on the role of the right zeros of P(s). Indeed, if N(b)=0, with
Re(b)≥0 (b can be infinity), then the interpolation condition must
hold true:
U ( b)
= M ( b)
Consider the first example and take M(s)=(s-2)(s+2)/(s+1) 2 ,
N(s)=(s-1)/(s+1) 2 . Then, it must be: U(1)=-0.75, U(∞)=1.
Obviously this is impossible.
Consider the second example and take M(s)=(s-1)/(s+2) e N(s)=(s-
2)/(s+2) 2 . It must be U(1)=0.25, U(∞)=1, which is indeed possible.

THEOREM 6
Parity interlacing property (PIP)
• P(s) è strongly stabilizable if and only if the number of poles of
P(s) between any pair of real right zeros of P(s) (including
infinity) is an even number.
• We have seen that the PIP is equivalent to the existence of the
existence of a unity which interpolates the right zeros of P(s).
Hence, if the PIP holds, the problem boils down to the
computation of this interpolant.
• In the MIMO case, Theorem 6 holds unchanged. However it is
worth pointing out that the poles must be counted accordingly
with their Mc Millan degree and the zeros to be considered are
the so-called blocking zeros.

REFERENECES
J.G.Truxal, Control Systems Synthesis, Mc-Graw Hill, NY, 1955.
J.R. Ragazzini, G.E. Franklin, Sampled-data Control Systems,
Mc-Graw Hill, NY, 1958.
M. Vidyasagar, Control System Synthesis: a Factorization
Approach, MIT Press, Cambridge, MA, 1985.
J.C. Doyle, B.A. Franklin, A.R. Tannenbaum, Feedback Control
Theory, Mc Millan, NY, 1992.
D.C. Youla, J.J. Bongiorno, N.N. Lu, Single-loop feedback
stabilization of linear multivariable dynamical plants”,
Automatica, 10, p. 159-175, 1974.
G. Zames, “Feedback and optimal sensitivity in model reference
transformation, multiplicative seminorms, approximate inverses,
IEEE Trans. AC-26, p. 301-320, 1981.
V. Kucera, Discrete Linear Control: The Polynomial Equation
Approach, Wiley, NY, 1979.

H2 CONTROL
SUMMARY
• Definition and characterization of the H2 norm
• State-feedback control (LQ)
• Parametrization, mixed problem, duality
• Disturbance rejection, Estimation (Kalman filter), Partial
Information
• Conclusions and generalizations

The L2 space
L2 space: The set of (rational) functions G(s) such that
1
2π
∞
∫
−∞
[ G(
− jω)'G(
jω
] dω
< ∞
trace )
(strictly proper – no poles on the imaginary axis!)
With the inner product of two functions G(s) and F(s):
∞
∫
−∞
1
G ( s),
F(
s)
= trace [ G(
− jω
)' F(
jω)
] dω
2π
The space L2 is a pre-Hilbert space. Since it is also complete, it is indeed a
Hilbert space. The norm, induced by the inner product, of a function G(s)
is
G(
s)
2
⎡ ∞
⎢ 1
= ∫ ⎢ trace
2π
⎢
⎣ −∞
[ G(
− jω
)'G(
jω)
]
dω
⎤
⎥
⎥
⎥
⎦
1/
2

The subspaces H2 and H2 ^
The subspace H2 is constituted by the functions of L2 which are analytic in
the right half plane. (strictly proper – stable !)
The subspace H2 ⊥ is constituted by the functions of L2 which are analytic
in the left half plane. (stricty proper – untistable !)
Note: A rational function in L2 is a strictly proper function without poles
on the imaginary axis. A rational function in H2 is a strictly proper
function without poles in the closed right half plane. A rational function in
H2 ⊥ is a strictly proper function without poles in the closed left half plane.
The functions in H2 are related to the square integrable functions of the
real variable t in (0,∞]. The functions in H2 ⊥ are related to the square
integrable functions of the real variable t in (-∞,0].
A function in L2 can be written in an unique way as the sum of a function
in H2 and a function in H2: G(s)=G1(s)+G2(s). Of course, G1(s) and G2(s)
are orthogonal, i.e. =0, so that
2
2
G s)
=
G
2 1(
s)
+
2
( G ( s)
2
2
2

Consider the system
x& ( t)
= Ax(
t)
+ Bw(
t)
z(
t)
= Cx(
t)
x(
0)
= 0
System theoretic interpretation
of the H2 norm
And let G(s) be its transfer function. Also consider the quantity to be
evaluated:
J
1
=
m
∑∫
i=
1 0
∞
z
( i)
'(
t)
z
( i)
( t)
dt
where z (i) represents the output of the system forced by an impulse input at
the i-th component of the input.
J
1
1
2π
=
∞
m
i=
1 0
m
− ∞ i=
1
∞
∑∫
∫∑
z
( i)
'(
t)
z
( i)
1
( t)
dt =
2π
− ∞ i=
1
Z
1
ei'G(
− jω)'G(
jω)
eidω
=
2π
∞
m
∫∑
( i)
( − jω)'Z
∞
∫
−∞
trace
( i)
( jω)
dω
=
[ G(
−jω)'G(
jω)
]
= G(
s)
2
2

Computation of the norm
Lyapunov equations
“Control-type Lyapunov equation”
“Filter-type Lyapunov equation”
G(
s)
A o o
' P + P A + C'C
= 0
Pr r
2
2
A'
+ P A + BB'
= 0
=
J
1
=
m
∑∫
∞
∞
z
( i)
'(
t)
z
( i)
( t)
dt =
i=
1 0
0 i=
1
∞ ⎡
m ⎤
= trace ⎢e
A't
C'Ce
At
Beiei'
B'⎥dt
∫ ⎢ ∑ ⎥
0 ⎢⎣
i=
1 ⎥⎦
⎡ ∞
⎤
⎢ A't
At ⎥
= trace ⎢B'
e C'
Ce dt B⎥
= trace ∫ ⎢
⎣ 0
⎥
⎦
⎡ ∞
⎤
⎢
trace C e
At
BB'
e
A't
⎥
= ⎢
dt C'⎥
= trace ∫ ⎢
⎣ 0
⎥
⎦
m
∫∑
trace
[ B'
P B]
[ CP C']
r
[ e 'B
'e
A't
C'Ce
At
i
Bei]
0
dt
=

Other interpretations
Assume now that w is a white noise with identity intensity and consider
the quantity:
J
2
= lim E ( z'(
t)
z(
t))
t→∞ It follows:
J
2
t
t
⎡ ( )
( ) ( )' '
'(
)
lim trace E
⎡
Ce
A t−τ
Bw d w B e
A t−
C'd
⎤⎤
= ⎢
⎥
E =
0
0
t ⎣
⎢∫
τ τ ∫ σ
σ
σ
⎣
⎥
→∞
⎦⎦
⎡
= lim trace ⎢
t→∞
⎣
⎡
= lim trace ⎢
t→∞
⎣
t t
∫∫
0 0
Finally, consider the quantity
J
3
and let again w(.) be a white noise with identity intensity. It follows:
Notice that J3=(||G(s)||2) 1/2 since
t
∫
0
Ce
1
= lim
T →∞ T
Ce
A(
t−τ)
A(
t−τ)
E
T
∫
0
BE
BB'e
[ w(
τ)
w(
σ )']
A'(
t−τ
)
z'(
t)
z(
t)
dt
⎤
C'dτ
⎥
⎦
B'e
=
A'(
t−σ
)
G(
s)
Aτ A'τ
[ CP(
t)
C']
, P(
t)
= e BB'
e dτ
E(
z'(
t)
z(
t))
= trace
∫
t
0
⎤
C'dτ
dσ
⎥
⎦
P&
1
( t)
= AP ( t)
+ P(
t)
A'+
BB'
, AY + YA'+
BB'
= 0,
Y = lim ∫ P(
τ ) dτ
T →∞ T
2
2
=
T
0

x&
Optimal Linear Quadratic Control (LQ)
versus Full-Information H2 control
= Ax + Bu
Assume that
And notice that these conditions are equivalent to
Find (if any) u(⋅) minimizing J
The above matrix result is the well known Schur Lemma. The proof is as follows. If H≥0, then
Vice-versa, assume that H≥0 and R>0, and suppose by contradiction that W is not positive
semidefinite, then Wx=νx, for some ν
0,
W
0
= Q − SR
[ ] 0
1
'≥
0
− S

Optimal Linear Quadratic Control (LQ)
versus Full-Information H2 control
Let C11 a factorization of W=C11′C11 and define
C
u
1
=
R
Then, it is easy to verify that
∞
J = ∫ ( x'Qx
+ 2u'
Sx + u'
Ru
) dt = ∫ z'zdt
= z
0
Moreover, the free motion of the state can be considered as a state motion
caused by an impulsive input. Hence, with w(t)=imp(t), let
The (LQ) problem with stability is that of finding a controller
fed by x and w and yielding u that minimizes the H2 norm of the transfer
function from w to z.
• LQS problem
⎡ C
⎢ 1/
⎣R
11
= − 2
1/
2
u,
⎤
,
S'
⎥
⎦
D
z = C x + D
1
12
x(
0)
⎡0⎤
= ⎢
I
⎥
⎣ ⎦
1
= 0
12
u
x& = Ax + B u + B w
2
z = C x + D
u
12
1
u
ξ = Fξ
+ G x + G
&
1
1
21
2
w
= Hξ
+ E x + E w
∞
0
2
2

x&
= Ax + B w + B u
z = C x + D
1
⎡x
⎤
y = ⎢ ⎥
⎣w⎦
1
12
u
Full information
w z
u x
2
Problem: Find the minimun value of ||Tzw||2 attainable by an
admissible controller. Find an admissible controller minimizing
||Tzw||2. Find a set of all controllers generating all ||Tzw||20
(Ac,C1c) detectable
Ac = A-B2(D12′D12) -1 D12′C1 stable invariant zeros of (A,B2,C1,D12)
C1c = (I-D12 (D12′D12) -1 D12′)C1
P
K

Solution of the Full Information Problem
Theorem 1
There exists an admissible controller FI minimizing ||Tzw||2 . It is
given by
F
2
-1 ( 'D
) ( B ' + D 'C
)
= −
P
D12 12 2 12 1
where P is the positive semidefinite and stabilizing solution of the
Riccati equation
A'
P+
PA−(
PB + C 'D
)( D 'D
)
A
= A−B
( D
2
'D
1
')
12
( B 'P+
D
−1
12
'C
) = A+
B F
−1
cc 2 12 12 2 12 1
2 2
The minimum norm is:
2 2
Tzw α , α = P
2
cB1
= trace(
B
2
1'
PB1
) , Pc
12
( B 'P+
D 'C
) + C 'C
= 0
2
12
1
1
1
stable
= ( s)
= ( C + D F )( sI − A−B
F )
The set is given by
Q(s)
u w
where Q(s) is a stable strictly proper system, satisfying
Q
2 2 2
( s)
< γ −α
2
F2
x
1
12
2
2
2
−1

Proof of Theorem 1
The assumptions guarantee the existence of the stabilizing solution to the
Riccati equation P. Let v=u-F2x so that u=F2x+v, where v is a new input.
Hence, z(s)=Pc(s)B1w(s)+U(s)v, where U(s)=Pc(s)B2+D12. It follows that
Tzw(s)=Pc(s)B1+U(s)Tvw(s). The problem is recast as to find a controller
minimizing the norm from w to z of the following system
where Pv is given by:
x&
= Ax + B w + B u
v = −F
x + u
2
⎡x
⎤
y = ⎢ ⎥
⎣w⎦
1
2
w v
Pv
u x
K
Notice that Tzw(s) is strictly proper iff Tvw(s) is such. Exploiting the
Riccati equality it is simple to verify that U(s) ~ U(s)=D12′D12 and that
U(s) ~ 2 2 2
Pc(s) is antistable. Hence, ||Tzw(s)|| =||Pc(s)B1|| + ||Tvw(s)|| . Hence
2
2
2
the optimal control is v=0, i.e. u=F2x.
2 2
Finally, take a controller K(s) such that ||Tzw(s)|| < γ . From this controller
2
and the system it is possible to form the transfer function Q(s)=Tvw(s). Of
2 2 2
course, it is ||Q(s)|| < γ -α . It is enough to show that the controller
2
yielding u(s)=F2x(s)+v(s)=F2x(s)+Q(s)w(s) generates the same transfer
function Tzw(s). This computation is left to the reader.
w

x&
= Ax + B w + B u
z = C x + D
1
y = C
2
x + w
Disturbance Feedforward
1
12
u
Same assumptions as FI+stability of A-B1C2
C2=0 → direct compensation
C2≠0 → indirect compensation
A-B1C2 B1 B2 y u
R(s) = 0 0 I
I 0 0
-C2 I 0
2
KDF
R(s)
KFI(s)
The proof of the DF theorem consist in verifying that the transfer function from w to z achieved by
the FI regulator for the system A,B1,B2,C1, D12 is the same as the transfer function from w to z
obtained by using the regulator KDF shown in the figure.

x&
= Ax + B w + ( B u)
z = C x + u
1
1
y = C2x
+ D21w
w y
Output Estimation
w z
u y
2
P11(s)
P21(s)
P
K(s)
K(s)
u =
−zˆ
Assumptions
(A,C2) detectable
D21D21′> 0 Af = A-B1 D21′(D21D21′) -1 C2
B1f = B1(I-D21′ (D21D21′) -1 D21)
(Af,Bf) stabilizable
(A-B2C1) stable
z

Solution of the
Output Estimation problem
Theorem 2
There exists an admissible controller (filter) minimizing ||Tzw||2. It
is given by
ξ&
= Aξ
+ B
u
where
L = −(
PC + B
2
2
2
D'
21
)
( ) -1
D D
21
21
and P is the positive semidefinite and stabilizing solution of the
Riccati equation
AP+
PA'−(
PC '+
BD
A
= −C
ξ
1
2
= A−(
PC '+
B D
2
u + L
1
2
21
( C
')(
D
')(
D
D
D
−1
( C P+
D B ')
+ B B '=
0
= A+
L C
−1
ff 2 1 21 21 21
2 2
2
ξ
21
− y)
21
')
')
2
21
1
1
stable
1

Solution of the
Output Estimation problem (cont.)
The optimal norm is
2
2
Tzw = 2
f
2
−1
C1Pf
( s)
= trace(
C1PC1'),
P ( s)
= ( sI−
A−L2C2
) ( B1
+ L2D21)
The set of all controllers generating all ||Tzw||2 < γ is given by
M(s)=
Aff -B2C1 L∞ -B2
C1 0 I
C2 I 0
u y
M
Where Q(s) is proper, stable and such that ||Q(s)|| 2 2 <
γ 2 −||C1Pf(s)|| 2 2
Q

Proof of Theorem 2
It is enough to observe that the “transpose system”
λ&
= A'λ
+ C'
μ
ϑ
= B 'λ
+ D
=
B
1
2
'λ
+ ς
1
ς
21
+ C
'ν
2
'ν
has the same structure as the system defining the DF problem.
Hence the solution is the “transpose” solution of the DF problem.
It is worth noticing that the H2 solution does not depend on the
particular linear combination of the state that one wants to
estimate.

Given the system
x& = Ax + ς + B
1
y = Cx + ς
2
22
u
Filtering
where [z′1 z′2]′ are zero mean white Gaussian noises with intensity
W
⎡W11
W12
⎤
= ⎢ ,
12'
⎥ W
⎣W
W22⎦
22
> 0
find an estimate of the linear combination Sx of the state such as
to minimize
J
= lim
t →∞
Letting
C
B
1
11
= −S,
B
11
y = W
E
'=
W
[ ( Sx(
t)
− u(
t))'(
Sx(
t)
− u(
t))
]
C
−1/
2
22
11
2
y,
= W
−W
12
−1
22
W
C,
−1
22
W
D
z = C x + u,
1
12
21
',
=
⎡ς
⎢
⎣ς
[ 0
1
2
B
1
I],
= [ B
⎤ ⎡B
⎥ = ⎢
⎦ ⎣ 0
11
11
W
12
W
W
−1/
2
22
]
−1/
2
12W22
1 / 2
W22
⎤
⎥w
⎦
the problem is recast to find a controller that minimizes the H2
norm from w to z.

z = C x + D
y = C x + D
The partial information problem
(LQG)
x&
= Ax + B w + B u
1
2
12
21
u
w
Assumptions: FI + DF.
1
w z
u y
2
P
K(s)

Theorem 3
There exists an admissible controller minimizing ||Tzw||2. It is
given by
ξ&
= Aξ
+ B u + L ( C ξ − y)
u = −F
ξ
The optimal norm is
2
2
2
2
Tzw P 2 c s)
L2
+ C 2 1Pf
( s)
= P ( s)
B
2
c 1 + F 2 2Pf
= ( ( s)
= γ
The set of all controllers generating all ||Tzw||2 < γ is given by
S(s)=
2
2
2
A +B2F2+L2C2 L2 -B2
-F2 0 I
C2 I 0
2
2
2
2
0
u y
S
Where Q(s) is proper, stable and such that ||Q(s)|| 2 2 < γ 2 −γο 2
Proof
Separation principle: The problem is reformulated as an Output Estimation problem for the estimate
of -F2 x.
Q

Important topics to discuss
• Robustness of the LQ regulator (FI)
• Robustness of the Kalman filter (OE)
• Loss of robustness of the LQG regulator (Partial Information)
• LTR technique

.
THE H¥
CONTROL PROBLEM
SUMMARY
• Definition and characterization of the H∞ norm
• Description of the uncertainty: standard problem
• State-feedback control
• Parametrization, mixed problem, duality
• Disturbance rejection, Estimation, Partial Information
• Conclusions and generalizations

.
G ( s)
20
10
0
-10
-20
50
0
-50
-100
-150
10 0
What the H¥ norm is?
rational
stable
proper
G (s)
= G(
s)
=
∞
sup Re( s ) ≥ 0
Bode Diagrams
From: U(1)
Frequency (rad/sec)
max G(
jω)
Ω = σ ( Ω)
= λmax(
Ω * Ω)
= λmax(
ΩΩ*)
For the frequency-domain definition we can consider a transfer function without poles on the
imaginary axis. It is easy to understand that the norm of a function in L∞ can be recast to the norm
of its stable part given through the so-called inner-outer factorization. The time-domain definition,
given in the next page, requires, for obvious reasons, the stability.
10 1
ω
10 2

G(s) = D+C(sI-A) -1 B
x&
= Ax + Bw
.
Time-domain characterization
z = Cx + Dw
A = asymptotically stable
G(s)
∞
=
sup
w∈L
There does not exist a direct procedure to compute the infinity
norm. However, it is easy to establish whether the norm is
bounded from above by a given positive number γ.
The symbol indifferently the space of square integrable or the space of the strictly proper rational
function without poles on the imaginary axis. If w(.) is a white noise, the infinity norm represents
the square root of the maximal intensity of the output spectrum.
2
z
w
2
2

.
BOUNDED REAL LEMMA
Let γ be a positive number and let A be asymptotically stable.
THEOREM 1
The three following conditions are equivalent each other:
(i) ||G(s)||∞ < γ
(ii) ||D||

Comments
Notice that as γ tends to infinity, the Riccati equation becomes a
Lyapunov equation that admits a (unique) positive semidefinite
solution, thanks to the system stability. This is obvious if one
notices that the infinity norm of a stable system is finite.
Proof of Theorem 1
For simplicity, let consider the case where the system is strictly proper, i.e. D=0. The equation and
the inequality are
Also denote: G(s) ~ =G(-s)′.
.
PBB'P
A'
P + PA + 2 + C'C
= ( < ) 0
γ
Points (ii)→(i) and (iii) →(i).
Assume that there exists a positive semidefinite (definite) solution P of the equation (inequality).
We can write:
PBB'
P
( sI + A')
P − P(
sI − A)
+ + 2 C'C
= ( < ) 0
γ
Premultiply to the left by B′(sI+A′) -1 e to the right by (sI-A) -1 B, it follows
G
so that ||G(s)||∞ < γ.
~
( s)
G(
s)
= γ
T ( s)
= γI
−γ
−1
2
I − T
~
( s)
T(
s)
B'
P(
sI − A)
Points (i)→(ii)
Assume that ||G(s)||∞ < γ. We now proof that the Hamiltonian matrix
⎡ A
H = ⎢
⎣−
C'C
−1
−2
γ BB'⎤
⎥
− A'
⎦
does not have eigenvalues on the imaginary axis. Indeed, if, by contradiction jω is one such
eigenvalue, then

( jω
− A)
x − γ
( jω
+ A')
y + C'Cx
= 0
.
−2
BB'
y = 0
Hence Cx = -γ -2 C(jωI-A) -1 BB′(jωI+A) -1 C′Cx, so that G(-jω)′G(jω)Cx=0 and Cx=0. Consequently,
y=0 and x=0, that is a contradiction. Then, since the Hamiltonian matrix does not have imaginary
eigenvalues, it must have 2n eigenvalues, n of them having negative real parts. The remaining on
eigenvalues are the complex conjugate of the previous ones. This fact follows from the matrix being
Hamiltonian, i.e. satisfying JH+H′J=0, where J=[0 I;-I 0]. Let take the n-dimensional subspace
generated by the (generalized) eigenvectors associated with the stable eigenvalues and let choice a
matrix S=[X* Y*]* whose range coincides with such a subspace. For a certain asymptotically stable
matrix T (restriction of H) it follows HS=ST. We now proof that X*Y=Y*X. Indeed let define
V=X*Y-Y*X=S*JS and notice that VT=S*JST=S*JHS=-S*H′JS=-T*S*JS=-T*V, so that
VT+T*V=0. The stability of T yields (by the well known Lyapunov Lemma) that the unique
solution is V=0, i.e. X*Y=Y*X. We now proof that X is invertible. Indeed from HS=ST it follows
that AX+γ -2 BB′Y=XT and –A′Y-C′CX=YT. Premultiplying the first equation by Y* yields
Y*AX+γ -2 Y*BB′Y =Y*XT=X*YT. Hence if, by contradiction, v∈Ker(X) then v*Y*BB′Yx=0 so
that B′Yv=0. From the first equation we have XTv=0 and -A′Yv=YTv. In conclusion, if v∈Ker(X)
then Tv∈Ker(X). By induction it follows T k v∈Ker(X), and again
-A′YT k v =YT k+1 v, k nonnegative. Take the monic polynomial of minimum degree h(T) such that
h(T)v=0 (notice that it always exists) and write h(T)=(λI-T)m(T). Since T è asymptotically stable, it
results Re(λ)

.
Worst Case
The “worst case” interpretation of the H∞ norm is given by the
following result:
THEOREM 2
Let A be stable, ||G(s)||∞ < γ and let x0 be the initial state of the
system. Then,
where P is the solution of the BRL Riccati equation.
Proof of Theorem 2
2 2 2
sup w ∈L
z −γ
w = x
2
2 0'
Px
2 0
Consider the function V(x)=x′Px and its derivative along the trajectories of the system. Letting
Δ=(γ 2 I-D′D) -1 we have:
V&
= x'(
A'
P + PA+
PBw + B'PB)
x
= −x'C'
Cx − x'
( PB + C'D)
Δ(
B'P
+ D'
C)
x = −z'
z
+ w'(
D'C
+ B'
P)
x + x'(
C'D
+ PB)
w+
w'D'
Dx +
− x'
( PB + C'
D)
Δ(
B'P
+ D'C)
x =
2
−1
= −z'
z + γ w'
w − ( w − w )'Δ
( w−
w )
ws
where wws = Δ(B’P+D’C)x is the worst disturbance. Recalling that the system is asymptotically
stable and taking the integral of both hands, the conclusion follows.
ws

.
Observation: LMI
Schur Lemma
0
'
'
'
'
'
'
2
>
⎥
⎦
⎤
⎢
⎣
⎡
−
+
+
−
−
−
D
D
I
C
D
P
B
D
C
PB
C
C
PA
P
A
γ
( ) 0
'
)
'
'
(
'
)
'
(
'
0
1
2
<
+
+
−
+
+
+
>
−
C
C
DC
P
B
D
D
I
D
C
PB
PA
P
A
P
γ

.
x& = ( A + LΔN)
x ,
⎧ x&
⎪
⎨ z
⎪
⎩w
=
=
=
Complex Stability Radius and
quadratic stability
Ax + Lw
Nx
Δx
Δ
≤ α
The system is said to be quadratically stable if there exists a
solution (Lyapunov function) to the following inequality, for
every Δ in the set
THEOREM 3
The system is quadratically stable if and only if ||N(sI-A) -1 L||∞

Proof of Theorem 3
First observe that the following inequality holds:
( A + LΔN)'P
+ P(
A + LΔN)
=
.
A'
P + PA + N'
Δ'ΔN
+ PLL'
P − ( N'
Δ'−PL)(
ΔN
− L'
P)
≤ A'P
+ PA + PLL'
P + N'
Nα
2
= α
2
( A'
X
XLL'
X
+ XA + −2
α
where Pα 2 =X. Hence, if there exists X>0 satisfying
+ N'
N)
then A+LΔN is asymptotically stable for every Δ, ||Δ|| ≤α, with the
same Lyapunov function (quadratic stability). This happens if
||N(sI-A) -1 L||∞0, we have that there exists
s=jω that violates (*), a contradiction.

Real stability radius
A= stable (eigenvalues with strictly negative real part)
r ( A,
B,
C)
= inf
r
Linear algebra problem: compute
μr
Proposition
= inf
s∈
jω
= inf
s∈
jω
m×
p
{ Δ : Δ ∈ R and A + BΔC
is unstable}
m×
p
inf{
Δ : Δ ∈ R and det(
sI − A − BΔC)
= 0}
m×
p
inf{
Δ : Δ ∈ R
−1
and det(
I − ΔC(
sI − A)
B)
= 0}
[ ] 1 −
inf
m×
p
{ Δ : Δ ∈ R and det(
I − Δ ) = 0}
( M ) = M
μ
⎛ ⎡ Re M − γ Im M ⎤⎞
⎜
⎟
⎜ ⎢
⎥⎟
⎝ ⎣ Im M Re ⎦⎠
r ( M ) = inf σ 2
γ∈(
0,
1]
−1
γ
M
Taking M=G(s)=C(sI-A) -1 B it follows
⎛ ⎡ Re G(
jω)
− γ Im G(
jω)
⎤ ⎞
sup inf σ ⎜
⎟
2
∈ ⎜ ⎢
⎥ ⎟
ω γ ( 0,
1]
⎝ ⎣γ
Im G(
jω)
Re G(
jω)
⎦ ⎠
−1
rr ( A,
B,
C)
= −1

Example
0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−
−
−
−
−
=
4175
.
0
3834
.
0
6711
.
0
0535
.
0
0668
.
0
0077
.
0
5297
.
0
0346
.
0
8310
.
0
6793
.
0
5194
.
0
6789
.
0
3835
.
0
0470
.
0
9347
.
0
2190
.
0
,
11
5
.
14
9
5
.
33
38
60
40
167
13
17
12
41
20
30
20
79
C
B
A
⎥
⎦
⎤
⎢
⎣
⎡−
=
Δ
4996
.
0
1214
.
0
1214
.
0
4996
.
0
worst

.
Entropy
Consider a stable system G(s) with state space realization
(A,B,C,0), and assume that ||G(s)||∞

Comments
It is easy to check that the γ-entropy measure is not a norm, but
can be considered as a generalization of the square of the H2 norm
Indeed the γ-entropy can be written as
Graph of f(x 2 ) parametrized in γ>1. For large γ the function f(x 2 )
gets closer to x 2 (red line).
.
I
G(
s)
γ
m
1
( G)
= f ( σ
2π
∫∑
f ( x
2
2
2
) = −γ
f
1
=
2π
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
− ∞ i=
1
2
+∞
ln
+∞
( 1
m
∫∑
− ∞ i=
1
σ
x
−
γ
2
i
2
i
2
2
[ G(
jω)
] dω
[ G(
jω)
]
)
) dω
0
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
x

Comments
An interpretation for the γ-entropy for SISO systems is as follows.
Consider the feedback configuration:
.
w z
G(s)
Δ(s)
where w(.) is a white noise and Δ(s) is a random transfer function
with Δ(jω1) independent on Δ(jω2) and uniformly distributed on
the disk of radius γ -1 in the complex plane.
Hence the expectation value over the random feedback transfer
function is
2
⎛ G(
s)
⎞
⎜
⎟
E = Iγ
( G)
Δ
⎜ 1 G(
s)
( s)
⎟
⎝
− Δ 2 ⎠

Proof of the Proposition
First notice that f(x 2 ) ≥ x 2 so that the conclusion that Iγ(G) ≥||G(s)|| 2 2 follows immediately.
Now, let
Of course it is ri≥β>1. Then
so that
which is the conclusion. In order to prove that the entropy can be computed from the stabilizing
solution of the Riccati equation, recall (Theorem 1) that, since ||G(s)||∞

.
z = Cx + Du
ACTUATOR DISTURBANCE
x&
= Ax + B(
w + u)
The optimal H2 state-feedback controller is
u
= F x,
2
F
2
= −(
D'D)
Proposition
The optimal H2 control law ensures an H∞ norm of the closed loop
system (from w to z) lower than 2||D||.
Proof. Consider the equation of the Bounded Real Lemma for the closed-loop system, i.e.
−2
( A + BF2
)'Q
+ Q(
A + BF2
) + γ QBB'Q
+ ( C + DF2
)'(
C + DF2)
< 0
Now take Q=P/α and consider both equations in P. It follows:
( 1
This condition is verified by choosing α≤1 and the minimum γ such that R≤0, i.e. the minimum γ
such that
α-α 2
The term α-α 2 is maximized by α=0.5, for which α-α 2 2 −2
2 −2
2
( α − α ) I − γ D'
D ≥ α −α
− γ σ ( D)
= 0
=0.25, so that the minimum γ is
−1
0 = A'
P + PA + C'C
− F D'
DF
2
( B'
P + D'C)
−1
−1
−1
−1
−2
( I − D(
D'D)
D')
C + PBRB'
P < 0,
R = ( D'
D)
( 1−
α ) + α
−1
−α ) C'
I γ
γ =
2σ( D)
2

The Small Gain Theorem
THEOREM 4
Assume that G1(s) is stable. Then:
(i) The interconnected system is stable for each stable G2(s) with
||G2(s)||∞α −1 then there exists a stable G2(s) with
||G2(s)||∞

(iii) Proof of Theorem 4
Point (i).
If ||G2(s)||∞0, and write the
singular value decomposition of G1(jω), i.e.
G1(jω)=U(jω)Σ(jω)V ~ (jω) where Σ(jω)=[S(jω)′ 0]′ and S(jω) is
square with dimension m. Moreover, take a stable G2(s) such that
G2(jω)=ρV(jω)[I 0]U ~ (jω). Notice that G2 ~ (jω)G2(jω)=ρ 2

The H¥
design problem
du dc
c 0 y u up K(s) G(s)
c
Find K(s) in such a way to guarantee:
• Stability
• Satisfactory performances
Design in nominal conditions
Uncertainties description
Design under uncertaint conditions
dr

G(s) = Gn(s)
Nominal Design
The performances are expressed in terms of requirements on some
transfer functions
du dc
c 0 y K(s) u up G(s)
c
Characteristic functions:
• Sensitivity Sn(s)=(I+Gn(s)K(s)) -1
dc → c, c 0 →y, -dr→y
• Complementary sensitivity
Tn(s)=Gn(s)K(s)(I+Gn(s)K(s)) -1
c 0 →c, -dr→c
• Control sensitivity
Vn(s)=K(s)(I+Gn(s)K(s)) -1
c 0 →up, -dr→up, -dc→up
dr

Shaping Functions
In the SISO case, the requirement to have a “small” transfer
function φ(s) can be well expressed by saying that the absolute
value |φ(jω)|, for each frequency ω, must be smaller than a
given function θ(ω) (generally depending on the frequency):
φ( jω)
< θ ( ω),
∀ω
Analogously, in the MIMO case, one can write
This can be done by choosing a matrix transfer function W(s),
stable with stable inverse (“shaping” function) , such that
∀ω
σ
[ φ ( jω)
] < θ ( ω),
ω
σ ∀
−1
[ W ( jω)
] = θ ( ω)
← W ( s)
φ ( s)
< 1
A general requirement is that the sensitivity function is small at
low frequency (tracking) whereas the complementary sensitivity
function is small at high frequiency (feedback disturbances
attenuation). Mixed sensitivity:
⎡W1
( s)
Sn
( s)
⎤
⎢
2(
) ( )
⎥
⎣W
s Tn
s ⎦
∞
< 1
∞

Description of the uncertainties
The nominal transfer function Gn(s) of the process G(s) belongs
to a set G which can be parametrized through a transfer
function Δ(s) included in the set
D
α
For instance,
{ Δ(
s)
Δ(
s)
∈ H , Δ(
s)
< α }
= ∞
∞
• G = {G(s) | G(s) = Gn(s)+Δ(s)}
• G = {G(s) | G(s) = Gn(s)(I+Δ(s))}
• G = {G(s) | G(s) = (I+Δ(s))Gn(s)}
• G = {G(s) | G(s) = (I-Δ(s)) -1 Gn(s)}
• G = {G(s) | G(s) = (I-Gn(s)Δ(s)) -1 Gn(s)}

Examples
• G = {G(s) | G(s) = Gn(s)+Δ(s)}
Uncertaint unstable zeros
s − 2 ε
G(
s)
=
+
( s + 2)(
s + 1)
( s + 2)(
s + 1)
• G = {G(s) | G(s) = Gn(s)(I+Δ(s))}
Unmodelled high frequency poles or unstable zeros
G(
s)
1 ⎛ εs
⎞
= ⎜1−
⎟,
( s + 1)
⎝ ( 1+
εs)
⎠
• G = {G(s) | G(s) = (I-Δ(s)) -1 Gn(s)}
Unmodelled unstable poles
⎛ 10 ⎞
G(
s)
= ⎜1−
⎟
⎝ ( 1+
s)
⎠
• G = {G(s) | G(s) = (I-Gn(s)Δ(s)) -1 Gn(s)}
Uncertain unstable poles
−1
⎛ 1 ⎞
G( s)
= ⎜1−
ε
⎟
⎝ s −1
⎠
G(
s)
1
( s + 10)
−1
1
( s −1)
1 ⎛ 2 ⎞
= ⎜1−
⎟
( s + 1)
⎝ ( 1+
s)
⎠

Design in nominal conditions
Example of mixed performances
z1
W 1
du dc
w = c 0 y K(s) u up Gn(s) c z2
W2 w z
P
u y
K
⎡ z1
⎤
z =
⎢ , w = c
z
⎥
⎣ 2 ⎦
W5=1
0
⎡W
=
⎢
⎢
0
⎢⎣
I
1
dr
−W
G
P 2
1
W G
− G
n
n
n
⎤
⎥
⎥
⎥⎦

Design under uncertain condition
Robust stability and nominal performances
z1
W 1
z2
K(s) Gn(s) dc
c 0 y u c
w z
P
u y
K
⎡ z1⎤
z = ⎢ w =
z
⎥,
⎣ 2⎦
W4
dc
Δ
W5
⎡−W
=
⎢
⎢
0
⎢⎣
− I
dc=w
dr
−W
G
P 4
1
1
W G
− G
n
n
n
⎤
⎥
⎥
⎥⎦

Design under uncertain condition
Robust stability and robust performances
z1
W 1
z2
K(s) Gn(s) dc
c 0 y u c
1) Robust sensitivity performance
2) Robust stability
dc=w
Result: (take W5=1)
1) and 2) are both achieved if the controller is such that
W4
Δ
W5
−1
[ I −W
( s)
Δ(
s)
W ( s)
V ( s)
] , ∀ Δ(
s)
< 1
W1 ( s)
S(
s)
= W1(
s)
Sn(
s)
5
4 n
W4 n
∞
( s)
V ( s)
W5(
s)
∞
< 1
⎡W1
( s)
Sn
( s)
⎤
⎢
4(
) ( )
⎥
⎣W
s Vn
s ⎦
∞
< 1
dr
∞

The proof of the result follows from 1) and 2) and on the following
Lemma
Let X(s) and Y(s) in L∞ and Ψ a generic L∞ function such that ||Y∞||

The Standard Problem
w z
u y
All the situations studied so far can be recast in the so-called
standard problem: Find K(s) in such a way that:
• The closed-loop system is asymptotically stable
•
T(
z,
w,
s)
∞
< γ
P
K
Existence of a feasible K(s)and parametrization of all controllers
such that the closed-loop norm between w and z be less than g.

x&
= Ax+
B w + B u
z = C x + D
1
⎡x
⎤
y = ⎢ ⎥
⎣w⎦
1
12
u
Ac = A-B2(D12′D12) -1 D12′C1
C1c = (I-D12 (D12′D12) -1 D12′)C1
Full information
w z
u x
2
P
K
w
Assumptions
(A,B2) stabilizzabile
D12′D12>0
(Ac,C1c) rivelabile

Solution of the Full Information Problem
THEOREM 5
There exists a controller FI, feasible and such that ||T(z,w,s)||∞ < γ
if and only if there exists a positive semidefinite and stabilizing
solution of the Riccati equation
−1
B1B
1'
A'
P+
PA−(
PB2
+ C1'D12)(
D12'D12)
( B2'
P+
D12'C1)
+ P P+
C1'C
2
1 = 0
γ
−1
B1B
1'
A−B2
( D12'D12')
( B2'P+
D12'C1)
+ P stable
2
γ
F
∞
=
−
F∞
-γ
u w
Q(s)
-2 B1′P
-1 ( 'D
) ( B 'P
+ D 'C
)
D12 12 2 12 1
Q(s) is a stable system, proper and satisfying ||Q(s)||∞ < γ.
x

Comments
• As (γ→∞) the central controller (Q(s)=0) coincides with the H2
optimal controller
• The parametrized family directly includes the only static
controller u=F∞x.
• The proof of the previous result, in its general fashion, would
require too much time. Indeed, the necessary part requires a
digression on the Hankel-Toeplitz operator. If we drop off the
result on the parametrization and limit ourselves to the static
state-feedback problem, i.e. u=Kx, the following simple result
holds:
THEOREM 6
There exists a stabilizing control law u=Kx such that the norm of
the system (A+B2K,B1,C1+D12K) is less than γ if and only if there
exists a positive definite solution of the inequality:
A'
P + PA − ( PB
2
+ C'
D
B1B1
'
+ P P + C1'
C1
< 0
2
γ
12
)( D
12
'D
12
)
−1
( B
2
' P + D
12
'C
1
)

Observation: LMI
P > 0
A'P
+ PA − ( PB
Ac = A-B2(D12′D12) -1 D12′C1
+ C'D
B1B1'
+ P P + C1'C
1 < 0
2
γ
X
> 0
C1c = (I-D12 (D12′D12) -1 D12′)C1
2
12
⎡ − XAc
'−Ac
X − B1B1'+
γ
⎢
⎣ C1c
X
)( D
2
12
'D
12
)
−1
( B
2
Schur Lemma
X= γ 2 P -1
B
2
B
2
'
'P
+ D
XC '⎤
1c
> 0
2 ⎥
γ I ⎦
12
'C
1
)

Proof of Theorem 6
Assume that there exists K such that A+B2K is stable and the
norm of the system (A+B2K,B1,C1+D12K) is less than γ. Then,
from the we know that there exists a positive definite solution of
the inequality
( A+ B K)'P
+ P(
A + B K)
+ γ
+ ( C
1
2
+ D
12
Now, defining
F
K)'(
C
1
+
D
12
2
K)
< 0
−2
-1 ( 'D
) ( B ' + D 'C
)
= −
P
D12 12 2 12 1
PB B 'P
+
1
1
(*)
the Riccati inequality can be equivalently rewritten as
A'
P + PA − ( PB
P
B
1 1
2
γ
B
'
1
2
1
+ C'
D
12
)( D
' D
) +
so concluding the proof. Vice-versa, assume that there exists a
positive definite solution of the inequality. Then, inequality (*)
is satisfied with K=F, so that with such a K, the norm of the
closed-loop transfer function is less than γ (BRL).
12
12
)
−1
P + C 'C
+ ( K − F)'(
K − F)
< 0
( B ' P + D
2
12
'C
1

Define:
A
C
W
R
R
Parametrization of all algebraic
“state-feedback” controllers
THEOREM 7
The set of all controllers u=Kx such that the H∞ norm of the
closed-loop system is less than γ is given by:
K
W
1
=
=
=
[ A B2
]
[ C1
D12
]
[ W W ]
= W
> 0
2
1
'W
−1
1
⎡− ARW
−WAR'−
B1B
⎢
⎣ CRW
'
2
1
'
WCR
'⎤
2 ⎥ > 0
γ I ⎦

Proof of Theorem 7
We prove the theorem in the simple case in which C1′D12=0 and
D12′D12=I. The LMI is equivalent to
K = W 'W
W
1
AW
> 0
1
2
1
−1
1
+ W A'+
B B '+
W B
1
1
2
2
'+
B W
2
2
'+
γ
If there exists W satisfying such an inequality, it follows that the
inequality
Is satisfied with K=W2′W1 -1 and P=γ 2 W1 -1 . The result follows from
the BRL. Vice-versa, assume that there exists P>0 satisfying this
last inequality. The conclusion directly follows by letting
W1==γ 2 P -1 e and W2=W1K′.
−2
WC
'CW
+ γ
1
1
1
1
−2
W W '<
0
−2
P( A + B2
K)
+ ( A + B2K)'P
+ γ
PB1B
1'P
+ C1'C1
+ KK'<
0
2
2

Example
⎡ 0
x&
= ⎢
⎣−1
1 ⎤ ⎡0⎤
⎥x
+ ⎢ ⎥(
w + u)
−1⎦
⎣1⎦
⎡1
z = ⎢
⎣0
0⎤
⎡0⎤
⎥x
+ ⎢ ⎥u
0⎦
⎣1⎦
γinf = 0.75, γ = 0.76
||T(z1,w)||∞
2.5
2
1.5
1
0.5
robustness
⎡0
ΔA
= ⎢
⎣Ω
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
H2
H
Ω
0⎤
0
⎥
⎦

x&
=
z
ξ
=
Ax + B w + B
C
1
x +
1
D
12
= Lx + Mu
u
Mixed Problem H2/H¥
2
u
The problem consists in finding u=Kx in such a way that
min T ( ξ, w,
s,
K)
: T ( z,
w,
s,
K)
≤ γ
2
K
• For instance, take the case z=ξ. The problem makes sense since
a blind minimization of the H∞ infinity norm may bring to a
serious deterioration of the H2 (mean squares) performances.
• Obviously, the problem is non trivial only if the value of γ is
included in the interval (γinf γ2), where γing is the infimun
achievable H∞ norm for T(z,w,s,K) as a function of K, whereas
γ2 is the H∞ norm of T(z,w,s,KOTT) where KOTT is the optimal
unconstrained H2 controller.
• This problem has been tackled in different ways, but an analytic
solution is not available yet. In the literature, many sub-optimal
solutions can be found (Nash-game approach, convex
optimization, inverse problem, …)
∞

x&
=
z
ξ
=
Ax + B w + B
C
1
x +
D
= Lx + Mu
Post-Optimization procedure
1
12
u
2
u
simplifying assumptions L′M=0, M′M=I
Let Ksub a controller such that ||T(z,w,s,Ksub)||∞

THEOREM 8
Post-Opt Algorithm
Each controller Kα of the family (α-con) is a stabilizing controller
and is such that
T ( ξ,
w,
s,
K
( 1
d
−α
)
dα
α
)
2
= T ( ξ,
w,
s,
K2−
T ( ξ,
w,
s,
K
α
)
2
≤ 0
Interpretation: For α=0, the controller is K0=Ksub. Hence,
||T(z,w,s,K0)||∞

Proof of Theorem 8
First notice that the equation (α-RIC) canm be rewritten as:
( 2 α α α 2
A + B K )'P
+ P ( A + B K ) + K 'K
+ L'
L = 0 ( α − RIC)
so that
α
||T(ξ,w,s,Kα)||2 =(Trace(B1′PαB1)) 1/2
The fact that the norm ||T(ξ,w,s,Kα)||2 is symmetric with respect to
α=1 follows directly (α-RIC) by inspection. Taking the derivative
with respect to α one obtains
F
α
' Xα
+ XαFα
− 2(
1−α
) Λα'Λα
= 0
where Xα is the derivate of Pα with respect to α and
Fα=Aα-B2αB2α′Pα, Λα=Ksub+PαB2
Matrix Fα is stable since Pα is the stabilizing solution of (α-RIC).
Hence, for the Lyapunov Lemma the conclusion follows.
α
α

Example
⎡ 0
x&
= ⎢
⎣−1
1 ⎤ ⎡0⎤
⎥x
+ ⎢ ⎥(
w + u)
−1⎦
⎣1⎦
⎡1
z = ⎢
⎣0
0⎤
⎡0⎤
⎥x
+ ⎢ ⎥u
0⎦
⎣1⎦
γinf = 0.75, γ2 = 0.8415, γ=0.8
Ksub=Kcen=[-0.6019 -0.7676]
robustness
a*=0.56, Ka*=[-0.4981 -0.5424]
Performances with respect to W
1.1
1
0.9
0.8
0.7
0.6
0.5
0.64
0.63
0.62
0.61
0.6
0.59
0.58
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Ka
Kcen
K2
0.4
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
1
0.95
0.9
0.85
0.8
⎡0
ΔA
= ⎢
⎣Ω
0⎤
0
⎥
⎦
0.75
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
3
2.5
2
1.5
1
K2
Kcen
0.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Ka

x&
= Ax + B w + B u
z = C x + D
1
y = C
2
x + w
Disturbance Feedforward
1
12
u
Same assumptions as FI+stability of A-B1C2
C2=0 → direct compensation
C2≠0 → indirect compensation
A-B1C2 B1 B2 y u
R(s) = 0 0 I
I 0 0
-C2 I 0
2
KDF
R(s)
KFI(s)
The proof of the DF theorem consist in verifying that the transfer function from w to z achieved by
the FI regulator for the system A,B1,B2,C1, D12 is the same as the transfer function from w to z
obtained by using the regulator KDF shown in the figure.

x&
= Ax + B w + ( B u)
z = C x + u
1
1
y = C2x
+ D21w
w y
Output Estimation
w z
u y
2
P11(s)
P21(s)
P
K(s)
K(s)
ξ
Assumptions
(A,C2) detectable
D21D21′> 0 Af = A-B1 D21′(D21D21′) -1 C2
B1f = B1(I-D21′ (D21D21′) -1 D21)
(Af,B1f) stabilizable
(A-B2C1) stable
u
ξˆ = −
z

Solution of the
Output Estimation problem
THEOREM 9
There exists a feasible controllor (Filter) such that ||T(z,w,s)||∞ < γ
if and only if there exists the stabilizing positive semidefinite
solution of the Riccati equation
M(s)=
A
C
I
L
AΠ
+ ΠA′
− ( ΠC
A − ( ΠC
2 f
f
ff
∞
=
=
=
Aff L∞ -B2 γ -2 ΠC1′
C1 0 I
C2f If 0
( )
( )
( )
( ) 1 -
-1
A − ΠC2'
D21D21'
C2
-1
D21D21'
C2
-1
D21D21'
−
−(
ΠC
'+
B D ')
D D '
=
2
'+
B D
1
2
21
2
'+
B
')(
D
1
1
21
D
21
21
D
')(
D
21
')
−1
21
C1'C
1
C2
Π 2
γ
21
B
2
C
1
stable
u y
M
Q(s) is a proper stable system satisfying ||Q(s)||∞ < γ.
21
D
21
')
−1
( C Π +
2
D
21
C1'C
1
B1
) + Π Π + B1
B
2
1'
= 0
γ
Q

Proof of Theorem 9
It is enough to observe that the “transpose system”
λ&
= A'λ
+ C'
μ
ϑ
= B 'λ
+ D
=
B
1
2
'λ
+ ς
1
ς
21
+ C
'ν
2
'ν
Has the same structure as the system defining the DF problem.
Hence the solution is the “transpose” solution of the DF problem.
It is worth noticing that the solution depends on the particular
linear combination of the state that one wants to estimate.

Example
⎡ 0
x&
= ⎢
⎣−1
y =
1 ⎤ ⎡0⎤
x + w
−1
+ Ω
⎥ ⎢
1
⎥
⎦ ⎣ ⎦
[ 1 0]
x + w2
1
damping ξ = ( 1−
Ω)
/ 2
Let consider six filters:
Filter K1: Kalman filter. The noises are assumed to be white
uncorrelated gaussian noises with identity intensities, namely
W1=1, W2=1.
Filter K2: Kalman filter. The noises are assumed to be white
uncorrelated gaussian noises with W1=1, W2=0.5.
Filter K3: Kalman filter. The noises are assumed to be white
uncorrelated gaussian noises with W1=0.5, W2=1.
Filters H1,H2,H3: H∞ filters with γ=1.1, γ=1.01, γ=1.005 (notice
that with γ=1 the stabilizing solution of the Riccati equation does
not exist).
Other techniques of Robust filtering are possible
Norm H2
2.2
2
1.8
1.6
1.4
1.2
1
0.8
H3
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
H2
H1
Norm H∞
7
6
5
4
3
2
1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Assumptions: FI + DF.
Partial Information
w z
u y
Theorem 10
There exists a feasible such that ||T(z,w,s)||∞ < γ if and only if
• There exists the stabilizing positive semidefinite solution of
−1
B1B
1'
A'
P+
PA−(
PB2
+ C1'D12)(
D12'
D12)
( B2'
P + D12'C1
) + P 2 P+
C1'C1
= 0
γ
• There exists the stabilizing positive semidefinite solution of
•
x&
= Ax + B w + B u
z = C x + D
y = C x + D
−1
C1'C1
AΠ+
ΠA'−(
ΠC2
'+
B1D21')(
D21D21')
( C2Π+
D21B1')
+ Π Π+
B1B
1'=
0
2
γ
max λ ( PΠ)
< γ
i
1
2
i
1
12
21
u
w
2
2
P
K(s)

Structure of the regulator
u y
S∞
Q
Afin B1fin B2fin
S∞(s) = F∞ 0 (D21D21′) -1/2
C2fin (D21D21′) -1/2 0
Afin = A−B2(D12′D12) -1 D12C1−B2(D12′D12) -1 B2′P+γ -2 B1B1′P+
+ (I−γ -2 ΠP) -1 L∞(C2+γ -2 D21B1′P)
B1fin = −(I−γ -2 ΠP) -1 L∞
B2fin = (I−γ -2 ΠP) -1 (B2+γ -2 ΠC1′D12)(D12′D12) -1/2
C2fin = −(D21D21′) -1 (C2+γ -2 D21B1′P)

Comments
It is easy to check that the central controller (Q(s)=0) is described
by:
ξ&
= A ξ + B u + Z L C ξ − y + D w*)
+ B w*
u
=
F
∞
ξ
2
∞
∞ ( 2
21 1
This closely resembles the structure of the optimal H2 controller.
Notice, however, that the well-known separation principle does
not hold for the presence of the worst disturbance w*=B1′Pξ.
Proof of Theorem 10 (sketch)
Define the variables r e q as:
w = r + γ -2 B1′Px
q = u − F∞x
In this way, the system becomes
x&
= ( A + γ
q
y
= −F
= ( C
∞
2
−2
x + u
+ γ
B B 'P)
x + B r + B
−2
1
D
1
21
B ')
x + D
1
1
21
w
2
u
This system has the same structure as the one of the OE problem.
Hence, the solution can be obtained from the solution of the OE
problem, by recognizing that the solution Πt of the relevant
Riccati equation is related to the solutions P e Π above in the
following way:
Π = Π(
I −γ
P
t
−2 Π
)
−1

⎡P
P = ⎢
⎣P
11
21
P
P
12
22
⎤
⎥
⎦
T ( z,
w,
s)
= P
11
T ( z,
w,
s)
= T ( s)
−T
1
Operatorial Approach
w z
P
u y
( s)
+ P
2
12
( s)
K
( I − K(
s)
P ( s)
)
( s)
Q(
s)
T
( s)
K(
s)
P
( s)
The functions T1, T2, T3 depend on a double coprime factorization
of P22. The function Q(s) is the Youla-Kucera parameter (stable
and proper function).
• Inner-outer factorization and reduction of the problem to a
distance Nehari problem
3
22
−1
21

Laurent and Hankel operators
• Let F(s) ∈ L∞ . The map ΛF: L2 → L2 defined as
ΛF : G(s) → ΛFG(s) = F(s)G(s)
is called the Laurent operator (with symbol F).
• Consider G(s)∈L2 and let G(s) =Gs(s)+Ga(s), with Gs(s) ∈H2 and Ga(s)
∈ H2 ⊥ . The map Π s: L2 → H2 defined as
Πs : G(s) → Π sG(s) = Gs(s)
is called the stable projection.
• Consider G(s)∈L2 and let G(s) =Gs(s)+Ga(s), with Gs(s) ∈H2 and Ga(s)
∈ H2 ⊥ . The map Π a: L2 → H2 ⊥ defined as
Π a : G(s) → Π aG(s) = Ga(s)
is called the untistable projection.
• Let F(s)∈L∞. The map ΓF: H2 ⊥ → H2 defined as
ΓF : G(s) → ΓFG(s) = Π sΛFG(s)
is called the Hankel operator (with symbol F).
Notice that ΛF is linear and ||ΛF||=||F(s)||∞ so that is a bounded operator. It is immediate to check that
the projections Πs and Πa are indeed linear operators. The Hankel operator is the result of the
composition of the Laurent operator and the stable projection, i.e. ΓF=ΠsΛF . Notice also that only
the stable part of F(s) contributes to ΓFG(s). Indeed, since G(s) since G(s)∈Η2 ⊥ it follows
ΓFG(s)=ΠsΛFG(s)= ΠsF(s)G(s)= Πs[(Fs(s)+Fa(s))G(s)]= ΠsFs(s)G(s)=ΓFsG(s)

The adjoint operators
• The adjoint Laurent operator (with symbol F) is the adjoint operator
with symbol F ~ , i.e.
Λ*F = ΛF ~
• The adjoint Hankel operator (with symbol F) is
Γ*F = Π aΛ ∗ F
Laurent: G1(s)∈ L2 and G2(s)∈ L2 . Then
G
1
G , F
1
,
*
1
Λ FG
2 = Λ FG1,
G 2 =
2 ∫
+∞
π −∞
~
G
2
=
G , Λ
1
F
~
G
2
trace
[ G ( − jω)'
F(
− jω)'G
( jω)
]
Hankel: H(s) ∈ H2 and G(s)∈ H2 ⊥ . Recall that =0 if a∈H2 and b∈
H2 ⊥ . Then
G, Γ
G, F
*
F
~
H
H
=
=
Γ G, H
F
G, Π
a
F
~
=
H
Π FG, H
=
s
G, Π Λ
a
=
F
~
H
1
FG, H
=
2
dω
=

Time-domain interpretation of the Hankel
operator
Let F(s)=C(sI-A) -1 B, with A stable, (A,C) observable and (A,B) reachable.
Hence, the inverse Laplace transform of F(s) is
⎧ 0
( t)
= ⎨
⎩Ce
f At
t ≤ 0
t > 0
Now, consider the map Γf : L2(−∞,0) → L2(0, +∞)
Defined by
⎧ 0
⎪ 0
Γf
: u( t)
→ Γf
u(
t)
= y(
t)
= ⎨ At −Aτ
Ce ( )
⎪ ∫ e Bu τ dτ
⎩ −∞
B
t ≤ 0
t > 0
This operator maps past input to future outputs. Indeed, the past inputs (with x(−∞)=0) contribute
to form the state x(0) from which the free output can be calculated. Γf is the time-domain
counterpart of ΓF. Γf can be seen as the composition of two operators Ψo (the observability
operator) and Ψr (the reachability operator)
Γ
f
Ψ
Ψ
r
o
= Ψ
r
Ψ
o
,
Ψ
: u(
t)
→ Ψ u(
t)
: = x =
r
r
n
: L ( −∞,
0)
→ R ,
⎧ 0
: x → Ψox
: = y(
t)
= ⎨ At
⎩Ce
x
2
t < 0
t ≥ 0
→ L
The operator Ψo is injective thanks to the observability of (A,C). Indeed, from Ψo x=0 it follows
x=0.
Conversely, the operator Ψr is surjective thanks to the reachability of (A,B). Indeed, given any point
x of R n it follows that Ψru(t)=x, where u(t)=0, for t>0 and u(t)=B′e -A′t Pr -1 for t≤0, where Pr is the
unique solution of the Lyapunov equation A′Pr+PrA+BB′=0
0
∫ ∞
−
e
− Aτ
Ψ
o
Bu(
τ)
dτ
: R
n
2
( 0,
∞)

Time-domain interpretation of the adjoint
Hankel operator
The adjoint reachability and observability operators are as follows:
Ψ
Ψ
*
r
*
o
− ⎧
* B'e
: x → Ψr
x = u(
t)
= ⎨
⎩ 0
: y(
t)
→ Ψ
*
o
y(
t)
= x =
C'
y(
τ ) dτ
Therefore the adjoint Hankel operator in time-domain is
Γ
Γ
*
f
*
f
: L
2
[ 0,
∞)
→ L ( −∞,
0]
: y(
t)
→ Γ
*
f
2
The rank of Ψr and Ψo is n = McMillan degree of C(sI-A) -1 B. Moreover,
since Ψo is injective and Ψr surjective, it turns out that Γf (and ΓF) has rank
n. Therefore the self adjoint operatopr Γ * FΓF has rank n, so that it admits
eigenvalues.
∞
∫
0
e
⎧
⎪B'e
y(
t)
= u(
t)
= ⎨
⎪
⎩
A't
x
−A'τ
∞
A't
∫
0
e
t ≤ 0
t > 0
A'τ
0
C'
y(
τ)
dτ
t ≤ 0
t > 0

Reachability and observability grammians
Notice that Pr and Po are the unique (positive definite) solutions of the
Lyapunov equations:
APr+PrA′+BB′=0
PoA+A′Po+C′C=0
The operator Γ * FΓF and the matrix PrPo share the same nonzero
eigenvalues. To see this notice that Γ*fΓf = Ψ * rΨ * oΨoΨr=PrPo
The Hankel norm of the system is
______________
Example:
Consider the stable part and take a minimal realization:
It results
Γ
Ψ
Ψ
F
r
*
o
Ψ
Ψ
=
*
r
o
Γ
=
=
f
∞
∫
0
∞
∫
0
e
e
=
Aτ
A'τ
BB'e
C'Ce
λ
A'τ
Aτ
dτ
= P
r
dτ
= P
( P P )
max r o
5(
s −1)(
s + 5)(
s − 3)
22.
5 2.
5(
3s
− 5)
F(
s)
=
= 5 + +
2
2
( s + 1)
( s − 7)
s − 7 ( s + 1)
⎡ 0 1 ⎤ ⎡0⎤
A = ⎢ , = , =
1 2
⎥ B ⎢
1
⎥ C
⎣−
− ⎦ ⎣ ⎦
[ −12.
5 7.
5]
⎡0.
25 0 ⎤ ⎡303.
125 78.
125⎤
Pr
= ⎢ ,
,
0 0.
25
⎥ Pr
= ⎢
78.
125 53.
125
⎥ λ
⎣ ⎦ ⎣
⎦
max
(P P ) = 81.3827, Γ
r
o
o
F
=
9.
0212

Formulae for the minimization problem via
the operatorial approach
All stable functions
)
(
)
(
ˆ
)
(
)
(
)
(
)
(
)
(
)
(
ˆ
)
(
)
(
)
(
)
(
21
3
12
2
21
12
11
1
s
P
s
M
s
T
s
M
s
P
s
T
s
P
s
Y
s
M
s
P
s
P
s
T
=
=
+
=
I
s
X
s
N
s
Y
s
M
s
M
s
N
s
Y
s
X
s
N
s
M
s
M
s
N
s
P
=
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
=
−
−
)
(
)
(
)
(
)
(
)
(
ˆ
)
(
ˆ
)
(
ˆ
)
(
ˆ
)
(
ˆ
)
(
ˆ
)
(
)
(
)
(
1
1
22
[ ] [ ]
)
(
ˆ
)
(
)
(
ˆ
)
(
ˆ
)
(
)
(
ˆ
)
(
1
s
M
s
Q
s
Y
s
N
s
Q
s
X
s
K −
−
=
−
K
P

Scalar case
It is possible to perform an inner-outer factorization of T4=T2T3=T4iT4o so
that, letting F=T4i -1 T1 and X=T4oQ we have:
T ( s)
− T ( s)
Q(
s)
T ( s)
T
1
4i
The problem is then reduced to a Nehari problem.
Now, let
F
( s)
Theorem 11
The function
is such that
−1
2
T ( s)
− T
1
F(
s)
= F
a
PQβ
= λ
max
40
( s)
= C(
sI − A)
A'
P + PA = BB',
a
3
( PQ)
β,
∞
=
( s)
Q(
s)
( s)
+ F ( s)
+ D,
−1
f ( s)
= B'(
sI − A')
X
0
λ
( s)
= F ( s)
−
s
−1
max
B
β,
∞
T ( s)
− T ( s)
Q(
s)
1
=
F
AQ + QA'
= C'C
χ = [ λ
a
4
F ( s)
− X ( s)
max
( s)
∈ H
( PQ)]
⊥
2
−1
Pβ
g(
s)
= C(
sI − A)
( PQ)
g(
s)
f ( s)
0
F( s)
X ( s)
= infx
( s ) ∈H
− ∞
F(
s)
− X ( s)
,
−1
∞
∞
χ
=
F ( s)
∈ H
s
2

Proof of Theorem 11
From F(s) construct f(s), g(s) and λmax(PQ). Let be X o (s) be the optimal
solution and define h(s)=(F(s)-X o (s))f(s). Notice that h(s) ∈L2, since F(s)-
X o (s) ∈L∞ and f(s) ∈ H2. It follows:
h(
s)
− Γ
〈 h(
s),
h(
s)
〉 + 〈 Γ
Now, taking into account that Γ * F ~ f(s) ∈H2 ⊥ and X o f(s)∈H2, it follows
〈 h(
s),
Γ
〈 Π
a
So that
*
~
F
*
~
F
f ( s)
2
2
*
~
F
f ( s)
〉
F(
s)
f ( s),
Γ
h(
s)
− Γ
*
~
( F(
s)
− X
= 〈 h(
s)
− Γ
f ( s),
Γ
= 〈 Π
*
~
2
⎡ o
( F(
s)
− X ( s))
−
∞
⎢⎣
F
F
f ( s)
o
2
2
*
~
F
*
~
F
f ( s)
〉 = 〈 Γ
f ( s),
h(
s)
− Γ
f ( s)
〉 − 2〈
h(
s),
Γ
[ F(
s)
− X
a
( s))
f ( s)
2
2
− λ
max
max
*
~
F
o
( PQ)
⎤
⎥⎦
f ( s)
〉
The Nehari theorem says that the minimum of ||F(s)-X(s)||∞ is equal to
||ΓF~||=λmax(PQ) [the proof is omitted]. Hence, it follows ||F(s)-X o (s)||∞ =
λmax(PQ) so that h(s)=Γ * F~f(s). On the other hand, as it is easy to check,
Γ * F~f(s)=g(s). Therefore,
*
~
*
~
F
( s)]
f ( s),
Γ
f ( s),
Γ
= 〈 h(
s),
h(
s)
〉 − 〈 Γ
λ
( PQ)
o
X F
( s)
=
F(
s)
− Γ ~
*
~
F
*
~
F
f ( s)
f ( s)
2
2
F
*
~
F
f ( s)
〉
f ( s),
Γ
2
2
g(
s)
f ( s)
≤
f ( s)
〉 =
f ( s)
〉
*
~
F
=
=
f ( s)
〉 =

⎡a
⎢
⎣a
1
2
⎤
⎥
⎦
Chain Scattering Representation
⎡b
= Σ⎢
⎣b
1
2
⎤
⎥
⎦
a1 b1
If Σ21 is invertible, then b1=Σ21 -1 (a2−Σ22b2) and
⎡a
⎢
⎣b
1
1
⎤
⎥
⎦
⎡a
= Chain(
Σ)
⎢
⎣a
1
2
⎤
⎥
⎦
• Algebra of chain scattering representation
a2
Chain(
a1
b1
Σ
⎡Σ
) = ⎢
⎣
b2
−1
−1
12 − Σ11Σ21
Σ22
Σ11Σ21
Σ −1
−1
− Σ21
Σ22
Σ21
chain(Σ)
b2
a2
⎤
⎥
⎦

Homographic Trasformation
a1
b1
a2
b2
a1 b2
b1
a2
Y is J-unitary
Σ
S
Θ =
chain(Σ)
S
1
22
21
12
11
21
1
22
12
11
1
1
)
)(
(
)
;
(
)
(
)
;
(
,
−
−
Θ
+
Θ
Θ
+
Θ
=
Θ
=
Φ
Σ
Σ
−
Σ
+
Σ
=
Σ
=
Φ
Φ
=
S
S
S
HM
S
I
S
S
LF
b
a
⎥
⎦
⎤
⎢
⎣
⎡
−
=
−
Ψ
=
Ψ
=
Ψ
Ψ
I
I
J
s
s
J
s
J
s
0
0
,
)'
(
)
(
,
)
(
)
(
~
~

The Standard Problem via chain-scattering
representation
w z
P
u y
We say that G(s) admits a J-unitary factorization if
G(s)=Θ(s)Λ(s)
Where Θ(s) is J-unitary and Λ(s) is unimodular (stable with stable
inverse) .
Theorem12
Assume that P has a chain scattering representation G=chain(P)
such that G is left invertible and does not have poles or zeros on
the imaginary axis. Then, the H∞ problem is solvable if and only if
G has a J-unitary factorization G=ΘΛ. The controllers K can be
written as K=HM(Λ -1 ;S) for each proper and stable S.
• J-spectral Factorization in control and filtering
K

A robust stability problem
k
z = w,
1 − gk
Z δ
w
k
q =
1 − gk
We want to maximize ||d||∞ (minimize ||q|∞). For the stability of the closed
loop it must be
(1) q stable
(2) gq stable
(3) (1+gq)g =stable
Hence, if pi are the unstable poles of g (assumed to be simple and with
Re(pi)>0) q must be such that
(0) q minimum norm
(1) q stable
(2) q(pi)=0
(3) (1+gq)(pi)=0
Letting
w
s + pi
q = qa = q∏i
pi
− s
it follows that q w must be such that
(0) q W minimim norm
(1) q W stable
(2) q W (pi)=-ag -1 (pi)
g
k

A simple example
where
z
δ
w
u y
g
s + 2
g =
( s + 1)(
s −1)
Find k in such a way that ||d||¥ is maximized.
Take
s + 1
a =
1−
s
and find q w stable of minimum norm such that such that
q w (1)=-ag -1 (1)=4/3. It follows q w =4/3 so that ||d||∞max =3/4 and
w
q 4(
s + 1)
= = −
1 + q g ( 3s
+ 5)
k w
k

In state-space
A=[0 1;1 0], B1=[0;0], B2=[0;1], C1=[0 0], D12=1, C2=[2 1], D21=1.
Stabilizing solution of:
A′P+PA-PB2B2′P+PB1B1′P/γ 2 +C1′C1=0
Stabilizing solution of:
AQ+QA′-PC2′C2Q+PC1′C1Q/ γ 2 +B1B1′=0
rs(PQ)

Operatorial approach
P
⎡0
s)
= ⎢
⎣1
Bezout
1⎤
g
⎥
⎦
Model Matching
Inner-outer factorization of t4=t2t3
Unstable part fa of f=t4i -1 t1
Hence, X 0 =0, Q=0 and
s + 2 n
= 2
s −1
m
s + 2
( s + 1)
( g =
n = 2
s + 5/
3
− ny + mx = 1, x = , y = −4
s + 1
t
t
1
2
t
=
p
= t
4
3
11
=
+
p
12
s −1
s + 1
myp
21
− 4 / 3(
s −1)
=
s + 1
2
( s −1)
= t4it4o
, t4i
= t4
= , t 2 4o
( s + 1)
/ 3
= 1
− 4/
3(
s + 1)
8/
3 8/
3
f =
= −4
/ 3−
, f a = −
s −1
s −1
s −1
K(
s)
=
y
x
− 4/
3(
s + 1)
4(
s + 1)
=
= −
s + 5/
3 3s
+ 5
,
m =
s −1
s + 1

REFERENCES
V.M. Adamjan, D.Z. Arov and M.G. Krein. Infinite block Hankel
matrices and related extension problem. American Mathematical Society
Translation, 111: 133-156, 1978.
Bala Subrahmanyam, Finite Horizon H∞ and Related Control Problems,
Birkhäuser, Boston, 1995.
Basar, T., and Bernhard, P., H∞ Optimal Control and Related Miinimax
Design Prooblems: A Dynamic Game Approach, Birkhäuser, Boston,
1995.
Chen, B.M., Robust and Optimal Control, Springer-Verlag, 2000.
Colaneri, P., Geromel, J.C., and Locatelli, A., Control Theory & Design:
An H2 and H∞ Viewpoint, Academic Press, San Diego, 1997.
Dahleh, M.A., and Diaz-Bobillo, I., Control of Uncertain Systems, A
Linear Programming Approach, Prentice Hall, Englewood Cliffs, 1995.
Davis, C., Kahan W.M., Weinberger, H.F., "Norm.preserving dilations
and their applications to optimal error bounds", SIAM Journal of Control
and Optimization, vol. 20, pp. 445-469, 1982.
Doyle J.C., "Analysis of feedback systems with structured
uncertainties", Proceedings of the IEEE, Pard D, vol. 129, pp. 242-250,
1982Hazony, D., "Zero cancellation synthesis using impedence
operators", IRE Trans. on Circuit Theory, vool. 8, pp. 114-120, 1961.

Doyle J.C., Lecture Notes in Advanced Multivariable Control, Lecture
Note ONR/Honeywell Workshop, Minneapolis, 1984.
Doyle, J.C., Francis, B.A., and Tannenbaum, A.R., Feedback Control
Theory, MacMillan Publishing Co., New York, 1992.
J.C. Doyle, K. Glover, P.P. Khargonekar, B. Francis, “State-space
solution to standard H2-H∞ control problems”, IEEE Trans. Autom.
Control, AC-34, pp. 831-847, 1989.
D. Fragopoulos, H∞ synthesis theory using polynomial system
representations”, PhD Thesis, University of Strathclyde, 1994.
Francis B.A., and Zames, G., "On L∞- optimal sensitivity theory for
SISO feedback systems", IEEE Trans. Automat. Contr., vol. AC-29, pp.
9-16, 1984.
B.A. Francis, “A course in H∞ control theory”, in Lecture Notes in
Control and Information Science, Vol. 88, Springer-Verlag, Berlin,
1987.
Glover, K., "All optimal Hankel-norm approximations of linear
multivariable systems and their L∞-error bounds, International Journal
of Control, vol. 39, pp. 1115-1193, 1984.
M. Green, K. Glover, D.J.N. Limebeer, J.C. Doyle, “A J-spectral
factorization approach to H∞ control”, SIAM J. Contr. And
Optimization, 28, pp. 1350-1371, 1990.
M.J. Grimble and V. Kucera, “Polynomial methods for control systems
design”, Springer Verlag 1996.
Helton, J.W:, and Merino, O., Classical Control Using H∞ Methods:
Theory, Optimization and Design, SIAM, Philadelphia, 1988.

Heyde, R.A., H∞ Aerospace Control Design: A VSTOL Flight
Application, Springer-Verlag, New-York, 1995.
Kimura, H., Okinishi, F., "Chain scattering approach to control system
design", In Trends in Control, An European Perspective, A.Isidori Ed.,
pp. 151-171, Springer Verlag,, Berlin, 1995.
Kimura, H., "Conjugation, interpolation and model-matching in H∞,
International Journal of Control, vol. 49, pp. 269-307, 1989.
H. Kwakernaak, M. Sebek, “Polynomial J-spectral factorization”, IEEE
Trans. Autom. Control, AC-39, pp. 315-328, 1994.
McFarlane D., and Glover, K, "A loop shaping design procedure using
H∞ synthesis", IEEE Trans. Automat. Control, vol. 37, pp. 759-769,
1992.
G. Mensma, “Frequency-domain methods in H∞ Control”, PhD Thesis,
University of Twente, 1993.
Mustafa, D. and Glover, K., Minimum Entropy H∞ Control, Lecure
Notes in Control and Information Sciences, Vol. 16, Springer-Verlag,
1990.
R. Nevanlinna,, "Über beschränkte Funktionen, die in gegeben Punkten
vorgeschriebenne Werte annehmen", Ann. Acad, Sci. Fenn., Vol. 13, pp.
27-43, 1919.
G. Pick, "Über Beschränkuungen analytischer Funktionen, welche durch
vorgegebene Funktions wert bewirkt sind", Mathematische Ann., Vol.
77, pp. 7-23, 1916.

Stoorgvogel, A., The H∞ Control Problem: A State Space Approach,
Prentice Hall, Englewwod Cliffs, New Jersey, 1992.
Van Keulen, B., H∞ Control for Distribuited Parameter Systems: A
State Space Approach, Birkhäuser, Boston, 1993.
W.M. Vidyasagar, “Control system synthesis: a factorization approach”,
MIT Press, Cambridge, 1985.
I. Yaesh, “Linear Optimal Control and Estimation in the Minimum H∞
norm sense”, Phd thesis, Tel Aviv University, 1992.
Zames, G., "Feedback and optimal sensitivity: Model refernce
transformations, multiplicity seminorms, and approximation inverses",
IEEE Trans. Automat. Contr., vol. AC-26, pp. 301-320, 1981.

H¥ FILTERING IN DISCRETE-TIME
P.Colaneri
Dipartimento di Elettronica e Informazione del Politecnico di
Milano

• H¥ analisys
SUMMARY
H∞ norm. Small gain theorem. Algebraic and difference analysis Riccati
equation
• H¥ filtering (infinite horizon)
Stochastic filtering. Basic prediction problem. J-spectral factorization.
Duality. Krein spaces. Risk sensitivity.
• Mixed H 2 /H¥ filtering
Convex optimization. α procedure
• H¥ filtering (finite horizon)
Feasibility. Convergence. Gamma Switching
• Conclusions

x(
t + 1)
= Ax ( t)
+ Bw(
t)
z(
t)
= Cx(
t)
+ Dw(
t)
H¥ NORM
−1
G ( z)
= C(
zI − A)
B + D
2
jϑ − jϑ
yF
G( z ) ∞ = sup λ max ( G( e ) G( e )' ) = sup = sup
ϑ∈ 0 , π
w l
2
w ( )
z − 0. 5
G( z)
= 2
z + 0. 5z + 0. 7
2
2
∈ 2 w z ∈l2
2
G( z ) w( z )
w( z )
If w( t)
is a white noise, the norm represents the square-root of the peak
value of the spectrum of the output
______________________________________________________________
For the frequency-domain definition it is enough to require that G(z) don’t have poles on the unit circle (L ∞ ). It is easy
to reckognize that the H ∞ norm of a function in L ∞ coicides with the norm of its stable part G o (z) in the inner-outer
factorization G(z)=G o (z)G i (z). The time-domain definition requires the external stability of the system. The symbol l 2
indicates the set of the square-summable discrete-time funtions in Z + . Noice that if w belongs to l 2 and the system is
asymptotically stable, the forced y F is in l 2 as well.
2
2
2
2

SMALL GAIN THEOREM: robust stability
Let G(z) and Δ(z) stable. Then the closed-loop system (Δ,G) is stable for
each ||Δ(z)|∞≤ α iff ||G(z)||∞

Descriptor simplectic system:
Theorem 2
RICCATI EQUATION
FOR THE H¥ ANALISYS
A is stable and G( z)
< γ iff there exists Q ≥ 0 satisfying
∞
• (i)
2 −1
Q = AQA '+ BB ' + ( AQC ' + BD ')( γ I − DD '− CQC ') ( AQC '+ BD ')'
• (ii) γ 2
I − DD' − CQC'
> 0
feasibility
2 −1
• (iii) A + ( AQC '+ BD ')( γ I − DD' −CQC')
C
stability

Notes:
The Riccati equation (or its dual) is associated with the simplectic system written above and described by equations
below.
⎡I
⎢
⎣0
2
−1
2
−1
− C'(
γ I − DD'
) C ⎤ ⎡A'+
C'
( γ I − DD')
DB'
η(
t + 1)
=
2
−1
⎥ ⎢
−2
A + BD'
( γ I − DD'
) C⎦
⎣ − B(
I − γ D'D)
B'
0⎤
⎥η(
t)
I ⎦
For the existence of the stabilizing solution of the Riccati equation it is necessary that such a system does not have
characteristic values with unitary modulus.
From the equation it readily follows that:
γ
2
I − G(
z)
G(
z
−1
)'=
Ψ(
z)
Ψ(
z)
= I − C(
zI − A)
−1
2 [ γ I − DD'−CQC']
( BD'+
AQC'
)( γ
2
Ψ(
z
−1
)',
I − DD'−CQC'
)
This explains the fact that γ2I − DD ' −CQC
' must be positive. As will be clear from the proof, the theorem can be
formulated for the inequality
− Q + AQA' + ( AQC ' + BD' )( γ I − DD ' − CQC ' ) ( AQC ' + BD'
)' ≤
2 1 0
without the stability condition (iii).
−
Notice that as γ→∞ (the loop in the figure opens) the Riccati equation boils down to the Lyapunov equation
associated with the H 2 norm.
Proof of Theorem 2
Assume that there exists a solution Q of the Riccati equation. Such an equation can be transformed in the following
way (the computations are left to the reader):
γ
2
I − G ( z ) G ( z
−1 )' = Ψ( z ) γ
2
I − DD ' −CQC
' Ψ ( z
−1
) ',
where
Ψ( z) = I−C( zI− A) −1 ( BD' + AQC' )( γ
2
I −DD' −CQC'
)
−1
.
Hence, on the basis of the property γ 2
I − DD ' − CQC ' > 0 and invertibiliy of Ψ ( e )
jϑ we have that
2 jϑ − jϑ jϑ 2
− jϑ
γ I − G ( e ) G ( e )' = Ψ ( e ) γ I − DD' − CQC ' Ψ ( e )' > 0 , ∀ϑ
,
i.e. the thesis.
Vice-versa assume that G ( z ) ∞ < γ . We prove that the symplectic system
⎡I
⎢
⎣0
2
−1
2
−1
− C'(
γ I − DD'
) C ⎤ ⎡A'+
C'
( γ I − DD')
DB'
η(
t + 1)
=
2
−1
⎥ ⎢
−2
A + BD'
( γ I − DD'
) C⎦
⎣ − B(
I − γ D'D)
B'
−1
0⎤
⎥η(
t)
I ⎦
does not have characteristic values with unitary modulus. Indeed, assume, by contradiction, λ is such, i.e.

2 −1 2 −1
λx − λC ' ( γ I − DD ' ) Cy = ( A' + C' ( γ I − DD ' ) DB ' ) x
2 −1 −2
λ( A + BD ' ( γ I − DD ' ) C ) y = y − B( I − γ D' D ) B' x
for a certain [x 'y']'≠0 and define p=(γ 2 I-DD') -1 (DB'x+λCy). It results G(λ -1 )G(λ)'=γ 2 I, which is a contradiction unless
p=0. On the other hand, p=0 implies A'x=λx, and, since A is stable, x=0. Finally, p=0 and x=0 imply (λA-I)y=0 so
that y=0 ( A is stable). Hence, x=0 and y=0 is a contradiction.
Since the symplectic system does not have characteristic values on the unit circle, there are n characteristic values
inside (possible in zero) and n outside (the reciprocals) the unit circle (possibly at infinity). Grouping the 2ndimensional
vectors associated with the n characteristic values inside the unit circle, one can construct a matrix [X'
Y']', whose range is a ln-dimensional subspace. It follows
⎡I
⎢
⎣0
2
−1
2
−1
− C'
( γ I − DD'
) C ⎤⎡X
⎤ ⎡A'+
C'
( γ I − DD')
DB'
⎥⎢
⎥T
=
2
−1
⎢
−2
A + BD'
( γ I − DD'
) C⎦
⎣Y
⎦ ⎣ − B(
I −γ
D'D)
B'
0⎤⎡
X ⎤
⎥⎢
⎥
I ⎦⎣Y
⎦
where T is a matrix with eigenvalues inside the unit circle (stable matrix). Consider now the solutions of the system
r(t+1)=Tr(t), i.e. r(t)=T t r(0) and left multiply by r(t) the two matrix equations above. Next, define p(t)=B'Xr(t) and
q(t)=CYr(t). It follows
p( t + 1)
= A' p( t) + C ' v1( t ), v2 ( t ) = B' p( t ) + D ' v1( t )
Aq ( t + 1)
= q ( t ) − Bv2 ( t) ,
1
v1( t ) = ( Cq( t + 1)
+ Dv ( t ))
2 2
γ
from which, passing to the Z-transform
−1 2 −1
2
G ( z ) ' G ( z ) v1( z ) = γ v1( z) , G ( z ) G ( z ) ' v2( z ) = γ v2 ( z ) .
From the assumption it is v1 ( t ) = 0, v2 ( t ) = 0 , so that B'X =0 and C'Y=0. Hence, from the symplectic system it
follows A'X=XT, AYT=Y. Assume that there exists a vector d such that Xd=0 and take the minimal degrre polynomial
ϖ(T) of T such that ϖ(T)d=0. Such a polynomial can be written as ϖ(T)=(λI-T)μ(T). For the minimality of ϖ(T) it
results μ(T)d≠0. Then, AYTd=λAYd=Yd. Since λ has modulus less than 1, λ-1 has modulus greater than one so that
this equation contradicts the stability of A', unless Yd=0. On the other hand, this last conclusion, together with Xd=0
contradicts the n-dimensionality of the range of [X 'Y']'.

THE DIFFERENCE EQUATION
x( t + 1)
= Ax( t ) + Bw( t )
y( t ) = Cx( t ) + Dw( t )
Q( t + 1)
= AQ( t ) A' + ( AQ( t ) C' + BD' )( γ I − DD' − CQ( t ) C' ) ( AQ( t ) C' + BD' )' + BB'
Q( 0 ) = Q > 0
0
Theorem 3
N
N
2 −1
J = ∑ y( t )' y( t ) − γ ∑w( t )' w( t ) − γ x( 0 )' Q x( 0 ) < 0, ∀( w(.), x(
0 )) ≠ 0
t=
0
t=
0
2
2
−
0 1
iff there exists Q(t)≥0 in [0,N] such that γ 2
I − DD' − CQ( t ) C'
>
0

Proof of Theorem 3
Consider the cost.
N
N
~
2
J ( N)
= z(
t)'
z(
t)
−γ
w(
t)'
w(
t)
+ x(
N + 1)'
PN
+ 1x(
N + 1)
< 0,
x(
0)
= 0,
∀w(.)
≠ 0
∑
t=
0
~
∑
t=
0
We proof the dual result: J ( N)
< 0,
x(
0)
= 0,
∀w(.)
≠ 0 iff there exists the solution of the equation
P( t ) = A' P( t + 1)
A +
2 −1
+ ( A' P( t + 1) B + C' D )( γ I − D' D − B' P ( t + 1 ) B ) ( A' P( t + 1 ) B + C' D )' + C' C ,
P( N + 1) = PN + 1 > 0
with γ 2
I − D'D − B' P( t + 1) B>
0 , in [0,N+1]. Sufficient condition: if there exists the solution with the indicated
properties, then pre-premultiplying the equation by x(t)' and post-multiplying by x(t), and summing all terms, it results
N
∑
t=
0
(*)
where
z(
t)'
z(
t)
−γ
2
N
∑
t=
0
w(
t)'w(
t)
+ x(
N + 1)'
P
~ 1
N + 1
x(
N + 1)
= x(
0)'P
( 0)
x(
0)
−
2
−
w ( t)
= ( I − D'
D − B'
P(
t + 1)
B)
( A'
P(
t + 1)
B + C'
D)'
x(
t)
γ .
Hence , being x(0)=0 we have J ( N)
< 0,
∀w(.)
≠ 0 .
~
~
N
∑
t=
0
( w(
t)
−w~
( t))'(
w(
t)
− w~
( t))
Vice-versa, assume that J ( N)
< 0,
∀w(.)
≠ 0 and take w(.)=0 in [0,N-1]. Being x(0)=0, with such a choice we have
0 > w( N)' − γ 2I+
D'D + BPN + 1B'
w( N), ∀w( N)
≠0
so that
− γ + + + >
2
I D' D BPN 1B'
0.
Furthermore, from
~ ~
J ( N)
= J ( N −1)
− ( w(
N)
− w~
( N ))'(
w(
N)
− w~
( N))
< 0,
∀w(.)
∈ ,
it turns out taht
~
( N −1) > 0,
∀w(.)
∈ N
[ 0,
−1]
J .
By iterating this reasoning we have
γ 2
I − D'D − B' P( t + 1) B>
0
in [0,N]. Theorem 3 is proven from (*) by noticing that
−1 2
P( t ) = Q( t ) γ .
[ 0 N ]

STOCHASTIC FILTERING
⎧ x(
t + 1)
= Ax(
t)
+ v1(
t)
⎪
⎨ z(
t)
= Lx(
t)
⎪
⎩ y(
t)
= Cx(
t)
+ v2(
t)
where v1( t), v2 ( t)
are zero-mean white gaussian noises . Moreover,
⎡v1
( t)
⎤ ⎡ V1
E ⎢ =
2(
)
⎥ ⎢
⎣v
t ⎦ ⎣V12
'
V
V
12
2
⎤
⎥,
⎦
V
2
> 0
The problem is to find an estimate z ˆ( t)
in H∞ of a linear combination
z ( t)
= Lx(
t)
of the state, i.e. one wants to find an estimator based on the
observations of { y( i),
i ≤ t − N}
) in such a way that the maximum of the
output spectrum of the error ε = z − zˆ
due to [ v 1'
v2']'
is minimized (or
bounded from above by a certain positive scalar γ).
Letting
u( t)
= −zˆ
( t),
ε ( t)
= Lx(
t)
+ u(
t),
D =
B =
1/
2 [ 0 V2
] , C = C
−1
1/
2
−1/
2
[ ( V −V
V V ')
V V ]
1
12
2
12
12
2
y(
t)
= V
−1/
2
2
y(
t)
and defining with w t
( ) a zero mean white gaussian noise with identity
covariance, it follows:

BASIC FILTERING PROBLEM
S
⎧ x(
t + 1)
= Ax(
t)
+ Bw(
t)
⎪
S : ⎨ ε(
t)
= Lx(
t)
+ u(
t)
⎪
⎩ y(
t)
= Cx(
t)
+ Dw(
t)
⎧ xˆ
( t + 1)
= A
F : ⎨
⎩ u(
t)
= C
F
F
xˆ
( t)
+ B
xˆ
( t)
+ D
Basic problem: Find A F , BF
, CF
, DF
such that
F
F
e
y(
t − N)
y(
t − N)
• (S,F) is asymptotically stable
T < γ
•
ε w
∞
________________________________________________________________
If z(t)=Lx(t) is the variable to be estimated and z ˆ( t)
the estimated variable, it follows u( t)
= −zˆ
( t)
. Hence ε( t )
represents the filtering error z( t)
= z(
t)
− zˆ
( t)
= ε(
t)
. If w(t) is a white noise, the problem is that of minimizing
the peak value of the error spectrum If w(t) is a generic signal in l2 , the problem is that of minimizing the l2 “gain”
between the error and the disturbance.

SOLUTION OF THE FILTERING PROBLEM
DD'> 0
Ipotesi: A = stable
( A , B,
C,
D)
does not have unstable invariant zeros
Theorem 4
There exists a filter F solving the basic problem iff ∃ Q≥0 solution of
⎡ CQA'+
DB'⎤
⎡DD'+
CQC'
i)
Q = AQA'+
BB'−⎢
⎥ ⎢
⎣ LQA'
⎦ ⎣ LQC'
ii) V = γ
2
I − LQL'+
LQC'(
DD'+
CQC ')
⎡ CQA'+
DB'⎤
⎡DD'+
CQC'
iii)
Aˆ
= A − ⎢ ⎥ ⎢
⎣ LQA'
⎦ ⎣ LQC'
'
2
'
−1 CQL
CQL'
⎤
2
− γ I + LQL'
⎥
⎦
'>
0
CQL'
⎤
2
−γ
I + LQL'
⎥
⎦
CENTRAL FILTER
ξ ( t + 1)
= Aξ(
t)
+ F ( y(
t)
−Cξ
( t)),
zˆ(
t)
= Lξ
( t)
+ F ( y(
t)
− Cξ
( t)),
F
1
2
F
1
−1
−1
⎡ CQA+
⎢
⎣
filter feasibility
⎡C⎤
⎢ ⎥
⎣L⎦
= LQC'(
CQC'+
DD')
−1
stable
= ( AQC'+
BD')(
CQC'+
DD')
−1
' DB'⎤
LQA'
⎥
⎦
_________________________________________________________________________
The assumption on the zeros is equivalent to the stabilizability of the pair
~ ~ ~
−1
~
−1
( A,
B),
A = A − BD'
( DD'
) C,
B = B(
I − D'(
DD'
) D)
. The assumption of stability of A can be weaked to
the assumption of detectability of (A,C) if one is only interested in the stability of the filter.

NOTES
• If G≠0, one obtains the central filter
ξ ( t + 1)
= Aξ(
t)
+ F ( y(
t)
− Cξ(
t))
+ Gu(
t)
zˆ
( t)
= Lξ(
t)
+ F ( y(
t)
−Cξ
( t))
2
1
• As γ → ∞ the H 2 filtering is obtained, where F 1 e F 2 depend on the
stabilizing solution of the standard Riccati equation:
Q = AQA '+ BB' − ( CQA' + DB')'( DD' + CQC) ( CQA' + DB')
• Notice that in the H ∞, filter the solution depends on L, i.e. on the
particular linear combination of the state to be estimated. This is a
difference between H2, and H ∞, design.
• The “gains” F 1 and F 2 are related by:
with
F
F
2
1
=
=
LK
−1
−1
[ A − BD'(
DD')
C]
K − BD'(
DD')
K = QC'(
CQC'+
DD')
In the uncorrelated case (DB'=0) these relations become
Hence
F = LK , F = AK
2 1
ξ ( t +
1)
= Aξ
( t)
+
rˆ
( t)
= Lξ
( t)
+ LK
−1
AK[
y(
t)
−Cξ
( t)
]
[ y(
t)
− Cξ
( t)
]
−1

BASIC ONE-STEP PREDICTION PROBLEM
Assume that we want to estimate z ( t)
= Lx(
t)
on the basis of the data
y (i)
, i≤ t.
S
P
⎧ x(
t + 1)
= Ax(
t)
+ Bw(
t)
+ Gu(
t)
⎪
S : ⎨ ε(
t)
= Lx(
t)
+ u(
t)
⎪
⎩ y(
t)
= Cx(
t)
+ Dw(
t)
⎧
~
x(
t + 1)
= A
~
P x(
t)
+ BP
y(
t)
P : ⎨
u(
t)
C
~
⎩ = Px
( t)
Basic problem: Find AP , BP , CP
such that:
• (S,F) asymptotically stable
• T rw ∞

SOLUTION OF THE ONE-STEP PREDICTION
PROBLEM
Assumption:
A = stable, DD'> 0, ( A, B, C, D ) without unstable zeros
Theorem 5
There exists a one-step predictor P solving the problem iff ∃ Q≥0
satisfying
'
⎡ CQA'+
DB'⎤
⎡DD'+
CQC'
i)
Q = AQA'+
BB'−⎢
⎥ ⎢
⎣ LQA'
⎦ ⎣ LQC'
CQL'
⎤
2 −γ
I + LQL'
⎥
⎦
⎡ CQA'+
DB'⎤
⎢ ⎥
⎣ LQA'
⎦
2
ii) γ I − LQL'>
0
predictor feasibility
iii )
⎡ CQA + DB ⎤ ⎡DD
+ CQC
Aˆ
' ' ' '
= A − ⎢
⎥ ⎢
⎣ LQA ' ⎦ ⎣ LQC '
'
− γ
CENTRAL PREDICTOR
CQL ' ⎤
⎥
I + LQL '⎦
η(
t + 1)
= Aη(
t)
+ F3(
y(
t)
−Cη(
t))
~ z(
t)
= Lη(
t)
2
−1
−1
⎡ C ⎤
⎢ ⎥
⎣−
L⎦
F = ( AQC' + BD' )( CQC' + DD' ) + A QL'V LQC' ( CQC' + DD'
)
3
A = A − ( AQC' + BD' )( CQC' + DD' ) C
X
2 −1
V = γ I − LQL' + LQC' ( DD' + CQC' ) CQL'
stable
−1 X
−1
−1 −1
_________________________________________________________________________
The assumption on the zeros is equivalent to the stabilizability of the pair
~ ~ ~
−1
~
−1
( A,
B),
A = A − BD'
( DD'
) C,
B = B(
I − D'(
DD'
) D)
. The assumption of stability of A can be weakened
to that of detectability of (A,C), if one is only interested to the stability of the predictor.

NOTES
• If G≠0, the so-called central predictor is obtained
μ t + 1)
= Aη(
t)
+ F
~ z(
t)
= Lη(
t)
( 3
( y(
t)
− Cη(
t))
+ Gu(
t)
3 → 1,
i.e.
the gain of the predictor and the gain of the filter coincide and depend
on the stabilizing solution of the standard Riccati equation:
• When γ → ∞ the H 2 predictor is recovered. Moreover, F F
Q = AQA '+ BB' − ( CQA' + DB')'( DD' + CQC) ( CQA' + DB')
• Notice that the H ∞ predictor depends on L, i.e. on the particular linear
combination of the state to be estimated. This is a difference between
H2, and H ∞, design.
• The “gain” F3 of the predictor and the “gains” F1 and F2 of the filter
are related by:
F = F +
−
A − F C QL'V F
3 1 1
________________________________________________________________
In the uncorrelated case (DB'=0) and under the assumption of reachability of (A,B), the Ricacti equation of
Theorem 3 can be rewritten in the following way:
⎡ −1
−1
1 ⎤
Q = A⎢Q
+ C'
( DD'
) C − L'
L A'+
BB'
2 ⎥
⎣
γ ⎦
Feasibility condition for the predictor γ 2
I − LQL'>
0
Feasibility condition for the filter
Filter/Predictor
−1 −2
Q + C' C − L' Lγ
> 0
F
1
−1
1 2
−1
−1
−1
−1
−1
1
= A(
Q + C'C
) C',
F2
= L(
Q + C'C)
C',
F3
= A(
Q + C'C
− L'L)
C'
2
γ
−1

PREDICTOR VS FILTER
The Riccati equation is the same, but the feasibility conditions are
different
System for the prediction error
μ( t + ) = ( A − F C) μ(
t) + ( F C − B) w( t)
1 3 3
e( t) = Lμ( t)
feasibility (analysis) : γ 2
I − LQL'
>
System for the filtering error
η(
t + 1)
= ( A − F C)
η(
t)
+ ( F D − B)
w(
t)
e(
t)
= ( L − F C)
η(
t)
+ F Dw(
t)
feasibility (analysis) :
γ
2
1
2 −1
2
1
0
I − LQL' + LQC '( DD' + CQC') CQL' −( F − F )( DD' + CQC')( F − F )' >
F = LQC'( CQC' + DD')
2
−1
2 2 2 2 0
________________________________________________________________
1) At γ fixed, the fesibility condition of the predictor is more stringent than that of the filter. They coincide as γ
tends to infinity, to witness the fact that the Kalman predictor and filter share the same existence conditions.
2) Assuming the existence of the stabilizing solution of the Riccati equation satisfying the feasibility condition, the
sufficient part of Theorem 3 is proved from the analysis result. Indeed, the analysis Riccati equation coincides
with the synthesis Riccati equation once the gains F 1 , F 2 (for the filter) or F 3 (for the predictor) are substituted
there.

Proof of Theorem 4/5
For simplicity we study the case where DB'=0, DD'=I, (A,B) reachable. Moreover, we limit the proof to filters
(predictors) in observer form. This is a restriction, but the analysis of all filters (predictors) ensuring the norm
bounded is far from being trivial and is not within our scope. Notice that if F 1 =AK e F 2 =LK are the characteristic
gains of a generic filter, the transfer function from the disturbance to the error is
T εw =(L-LKC)(zI-A+AKC) -1 (AKD-B)+LKD
So that the analysis Riccati equation is
−1 −2 −1
Q = A( X − L' Lγ ) A' + BB' ,
with the feasibility condition
X = ( I − KC ) Q( I − KC )' + KK'
(1)
γ 2
I − LXL'
> 0
and asymptotic stability of
(2)
2 −1
A( I − KC ) + AXL' ( γ I − LXL' ) ( I − KC )' L'
On the other hand, if F3 is the gain of a generic one-step predictor, the transfer function is
(3)
T εw =L(zI-A+F 3 C) -1 (F 3 D-B),
and then the analysis Riccati equation is:
−1
−2
−1
Q = A(
Q + C'C
− L'
Lγ
) + BB'+
( F
Y = ( Q
−1
− L'
Lγ
−2
)
3
− AYC'(
I + CYC')
−1
)( I + CYC')(
F
3
− AYC'(
I + CYC')
with the feasibility condition
γ 2
I − LQL ' > 0
(5)
and asymptotic stability of
2 −1
A − F3C + ( A + F3C ) QL' ( γ I − LQL' ) L'
(6)
Now, assume that there exists Q≥0 satisfying Q=A[Q -1 +C'C-L'Lγ -2 ] -1 +BB', and such that Q -1 +C'C-L'Lγ -2 >0 and
A(Q -1 +C'C-L'Lγ -2 ) -1 Q -1 stable. Take the filter characterized by F 1 =AK, F 2 =LK, with K=(Q -1 +C'C) -1 C'. From (1) it
results X=(Q-1 +C'C) -1 and hence
Q = AXA' + BB' + AXL' ( γ2I − LXL' ) −1 LXA' = A( X− 1 − L' Lγ− 2 ) −1 A' + BB' = A( Q−1 + C' C− L' Lγ− 2 ) −1
A' + BB'
Therefore, the analysis Riccati equation admits a positive semidefinite solution with
2 2 −1
γ I − LXL' = γ I − LQL ' LQC ' ( I + CQC ' ) CQL'
> 0,
so that the feasibility condition is satisfied. Also the stability condition (3) is satisfied.
Hence the norm is less than γ.
Assume now that there exists Q≥0 satisfying Q=A[Q -1 +C'C-L'Lγ -2 ] -1 +BB', and such that Q -1 -L'Lγ -2 >0 and A(Q -
1 +C'C-L'Lγ -2 ) -1 Q -1 stable. Take a predictor defined by F3 =A(Q -1 +C'C-L'Lγ -2 ) -1 C'. From (4) it results that
F 3 =AYC'(I+CYC') -1 and hence Q=AYA'+BB'-AYC'(I+CYC') -1 CYA'=A(Q -1 +C'C-L'Lγ -2 ) -1 A'+BB'. The analysis
Riccati equation admits a positive semidefinite solution and Q -1 -L'Lγ -2 >0 coincides with the feasibility condition
(5) for the predictor. Also the stability condition (6) is satisfied .
Hence the norm is less than γ.
Vice-versa, assume that there exists a filter such that the norm is less than γ. Then, there exists Q solving (1)-(3). On
the other hand, X=(Q -1 +C'C) -1 +(K-(Q -1 +C'C) -1 C')(I+CQC')(K-(Q -1 +C'C) -1 C')'≥(Q -1 +C'C) -1 and hence X -1 ≤Q -1 +C'C
implies that Q solves the inequality Q≥A(Q -1 '+C'C-L'Lγ -2 ) -1 A’+BB' whereas γ 2 I-LXL'>0 with X>0 implies that
Q -1 +C'C-L'Lγ -2 >0. The existence of Q solving Q=A(Q -1 +C'C-L'Lγ -2 ) -1 A'+BB' with Q -1 A'+C'C-L'Lγ -2 >0 and
satisfying the stability condition follows form standard monotonicity results of the solutions of the Riccati equation.
Finally, assume that there exists a predictor such that the norm is less than Then, there exists Q solving (4)-(6).
Hence Q≥A(Q -1 A'+C'C-L'Lγ -2 ) -1 A+BB' with γ 2 I-LQL'>0 The existence of Q solving Q=A(Q -1 A'+C'C-L'Lγ -2 ) -
1 A'+BB' with γ 2 I-LQL'>0 and satisfying the stability condition follows form standard monotonicity results of the
solutions of the Riccati equation
−1
(4)
)'

J-SPECTRAL FACTORIZATION
The Riccati equation underlying the problem solution can be compactly
written as:
~ ~ ~ ~ −1
~ ~
[ AQC
'+
BD'][
R + CQC
']
[ CQA'
DB']
Q = AQA'+
BB'−
+
~ ⎡C⎤
~ ⎡D⎤
⎡DD'
= ⎢ ⎥,
D = ⎢ ⎥,
R = ⎢
⎣ L⎦
⎣ 0 ⎦ ⎣ 0
0 ⎤
− γ I
⎥
⎦
C 2
Let define:
H
H
1
2
( z)
= D + C(
zI − A)
−
( z)
= L(
zI − A)
1
B
−1
B
⎡H1(
z)
H(
z)
= ⎢
⎣H
2(
z)
0⎤
I
⎥
⎦
⎡I
= ⎢
⎣0
0 ⎤
− γ I
⎥
⎦
J 2
Theorem 6
Assume that A is stable. There exists a stable filter F(z) such that
H2(z)-F(z)H1(z) is stable and
H ( z)
− F(
z)
H ( z)
2
H ( z)
− F(
z)
H ( z)
2
1
1
∞
stable
< γ
if and only if there exists a square Ω(z), stable with stable inverse, with
stable [Ω(z) -1 ]11, solving the factorization problem
H ( z)
JH ( z
)'=
Ω(
z)
JΩ(
z
−1
−1
)'
All feasible filters can be parametrized in the following way:
−1
F(
z)
= ( Ω21(
z)
− Ω22
( z)
U(
z))(
Ω11(
z)
− Ω12
( z)
U ( z))
, U ( z)
∈ H ∞,
U ( z)
< γ ∞

Notes:
From z = H2
( z)
w,
y = H1(
z)
w,
zˆ
= F(
z)
y , it results z − zˆ
= ( H1(
z)
− F ( z)
H2
( z))
w and then the
filtering problem can be equivalently rewritten as a model matching problem.
Proof of Theorem 6
Assume that the exists a feasible filter. We limit ourselves to the case where the filter is in the observer form. From
Theorem 4 we know that the exists a positive semidefinite solution Q of the Riccati equation. Moreover, such a
solution is stabilizing and feasible. It is a matter of cumbersome computations to show that
Ω(
z)
=
V
= γ
2
~
−1
~ ~ ~ ~ −1
[ I + C(
zI − A)
( BD'+
AQC
')(
R + CQC')
]
I − LQL'+
LQC'
( DD'+
CQC'
)
−1
CQL'
1/
2
⎡ ( DD'+
CQC'
)
⎢
⎢LQC
'(
DD'+
CQC'
)
⎣
is a J-spectral factor. The stability of Ω(z) follows from stability of A. Stability of Ω(z) -1 and [Ω(z) -1 ]11 follows from
Q being a stabilizing solution. Existence of V 1/2 follows from the feasibility condition.
Vice-versa, assume that Ω(z) with the given properties exists and take the formula defining all filters F(z). It follows
−1
−1
−1
−1
[ ( z)
− I]
= −(
Ω ( z)
− Ω ( z)
Ω ( z)
Ω ( z)
)( I −U
( z)
Ω ( z)
Ω ( z)
) [ U ( z)
I]
Ω(
z)
F (*)
22
Notice that, thanks to the small gain theorem,
( ) 1
−1 −
I
− U(
z)
Ω ( z)
Ω ( z)
21
11
12
is stable. Indeed, U(z) is stable and ||U(z)||∞

DUALITY'
The prediction problem is dual with respect to the full-state control
problem (x is measurable).
The filtering problem is dual with respect to the full-information
control problem (x and w are measurable).
⎧ x(
t + 1)
= Ax(
t)
+ Bw(
t)
+
⎪
S : ⎨ ε ( t)
= Lx(
t)
+ u(
t)
⎪
⎩ y(
t)
= Cx(
t)
+ Dw(
t)
( ( ( (
⎧ x(
t + 1)
= A'
x(
t)
+ L'ε
( t)
+ C'
y(
t)
S' R : ⎨ ( ( (
⎩ w(
t)
= B'x(
t)
+ D'
y(
t)
Theorem 7
y t)
K x(
t)
K ε ( t)
( ( (
= − − stabilizes S'R and T ( (
wε
< γ iff
( 1
2
ξ( t + ) = Aξ( t) + K '( y( t) − Cξ( t))
1 1
− u( t) = Lξ( t) + K '( y( t) − Cξ( t))
stabilizes S and Tεw ∞
2
< γ .
∞
OE
FI-SF
_________________________________________________________________
Notice that if K2 is zero (state-feedback), the filter is indeed a predictor. The proof of Theorem 7 is immediate. It
is enough to verify that for any K1 and K2 it results T ( (
εw
= Twε
' . Actually, an extended version of this result
allows to link any full-information controller K(z) for system S R ' to a filter F(z) for system S.

KREIN SPACES
The filtering (prediction) problem can be interpreted as a Kalman
filtering (prediction) problem in the Krein space (instead of Hilbert
space).
with
~ ⎡C⎤
C = ⎢ ⎥,
⎣L⎦
( )
~
( ) ( )
) (
⎧ x(
t + 1)
= Ax(
t)
+ v1
t
⎪
⎨ z(
t)
= Lx t
⎪
⎩ y(
t)
= Cx
t + v2
t
⎡⎡v1
⎤
E⎢⎢
⎥
⎣⎣v2⎦
[ v ' v ']
1
2
⎡BB'
⎤
=
⎢
⎥ ⎢
DB'
⎦ ⎢⎣
0
BD'
DD'
0
0 ⎤
0
⎥
⎥
2
−γ
I⎥⎦

RISK SENSITIVITY
x( t + ) = Ax ( t) + v ( t)
1 1
y( t) = Cx( t) + v ( t)
where the noises are white gaussian noises with zero mean.
The problem of the risk sensitive filter is to minimize
⎡ 2 ⎛ ⎛
minμ
( ϑ)
= min⎢−
log⎜
E⎜
zˆ
( y)
zˆ
( y)
⎢
⎜ ⎜
e
⎣
ϑ
⎝ ⎝
The parameter θ is the risk parameter
θ=0 risk-neutral Kalman
θ>0 risk-seeking modified Kalman
θ

MIXED H2/H¥ FILTERING
It makes sense to distinguish between two classes of disturbances:
those whose statistics is known and those which are only requested to
be bounded. For instance, in general, the spectral characteristics of the
transducer errors are known whereas those related to modelling errors
are unknown. Consequently, we can write a mixed filtering problem
associated to a variable z(t):
x( t + ) = Ax ( t)
+ Bw( t)
= Ax ( t)
+ B w ( t)
+ B w ( t)
z( t)
y( t)
1 1 1
= Lx( t)
= Cx( t)
+ Dw( t)
= Cx( t)
+ D w ( t)
+ D w ( t)
ε ( t) = z(
t)
− zˆ
( t)
Mixed Problem
Find a filter solving:
1 1
{ Tε , w ( F)
s.
t.
Tε
, w ( F)
≤ γ }
inf 1
2
2
∞
2 2
2 2
The solution to this problem is non trivial if γ < Tε, w ( Kalman)
2 ∞ .
Up to now there not exist a closed-form solution. It can be proven that
the optimal mixed solution Fo is on the “boundary”, i.e. such that
o
Tε, w ( F )
2 ∞ =γ.
___________________________________________________________________
Obviously, the mixed problem makes sense since there exist an infinite number of filters F (a parametrized family)
such that ( F)
≤ γ
Tε , w
2
∞

OBSERVATIONS
The gain F3 of the predictor ensuring the norm less than γ can be
-1
parametrized in the following way: F3=W1 W2, where (W1,W2) satisfyies an elliptic equation.
Convex Optimization
min tr(XW) ≥ min T ( W W )
ε w
1 1 −
2
2
2
⎡W1
W2
⎤
, W = ⎢ ⎥
⎣W2
' W3
⎦
where X depends from the data. If w 1=w 2, then one obtains the
central predictor. In this way one simply minimizes an upper bound
of the real cost. The solution may be too conservative.
Search for boundary values
The boundary values are (norm equal to γ) on the tangent sets between
the ellipsoid and the planes passing through the origin.
− γ2
W
2
γ2
W
1

a - PROCEDURE
We can parametrize with a scalar variable a family of predictor
(defined by the only gain K) which ensure the norm less than γ with
minimal two norm.
Idea
||T εw(K)|| 2
α
γ ||T εw(K)|| ∞
K( α) = ( 1−
α) K + α AQ( α) C' + BD' ( CQ( α)
C' + DD'
)
c
2 −1
Q = AQA' + BB' −( 2α
− α )( AQC' + BD' )( CQC' + DD' ) ( AQC' + BD'
)'
A = A − KcC, B = B − KcD where Kc is such that ||Tεw(Kc)|| ∞

⎧ x(
t + 1)
= Ax(
t)
+ Bw(
t)
⎪
S : ⎨ z(
t)
= Lx(
t)
⎪
⎩ y(
t)
= Cx(
t)
+ Dw(
t)
Finite horizon problem in 1,T :
FINITE HORIZON
Find a linear causal filter based on su y( 1),
y(
2),
L , y(
t)
, t ≤ T
providing an estimate z ˆ( t)
guaranteeing a level of attenuation γ>0,
i.e. such that J F 0 is a weighting matrix (it playes the role played by the
initial covariance matrix in the Kalman filter design). Putting the two equations together we get the difference
Riccati equation: Q( t+ ) = A Q( t) −
−
1
1
+ C'C− L' L/ 2
1
−1
γ + BB'
, with Q( 1)
= AΨ A' + BB'
.
.

zˆ
( t)
= Lη(
t)
SOLUTION OF THE FINITE HORIZON
PROBLEM
η(
t + 1)
= Aη(
t)
+ K ( t + 1)
The gain is given by:
K( t ) = Q( t ) C' I + CQ( t ) C'
[ y(
t + 1)
− CAη(
t)
]
−1
where Q( t ) is the solution of the difference Riccati equation
QUESTIONS
Under which conditions it is possible to guarantee the existence of the
time-varying filter for an arbitrarily long interval?
Under which conditions it is possible to guarantee the convergence for
t→∞ to the H∞ filter?
____________________________________________________________
The time-varying filter can be written as
ξ ( t + 1)
= Aξ
( t)
+ AK
zˆ
( t)
= Lξ
( t)
+ LK(
t)
( t)
[ y(
t)
− Cξ
( t)
]
[ y(
t)
− Cξ
( t)
]
The gains F1 ( t ) = AK ( t ), F2 ( t ) = LK ( t ) have the same structure as in the stationary case (uncorrelated noises).

1) A invertible
FEASIBILITY AND CONVERGENCE
2) There exists the stabilizing solution Qs of
−1
~ −1
~ −1
Q = A Q + C'
R C A'+
BB
3) The pair (A,B) is reachable
[ ] '
Thanks to these assumptions, there exists the unique (positive
semidefinite) solution of the Lyapunov equation
−1
~ −1
~ −1
−1
~ −1
~ −1
~ ~ ~ −1
~
[ Qs
+ C'
R C]
A'YA[
Qs
+ C'
R C]
− [ C'
( R + CQ
C')
] −
Y s
= C
Theorem 10
(i) The solution Q(t) of the difference Riccati equation is feasible, ∀
t>0, if, for some ε :
{ [ [ ] ] } 1
~ ~ ~ −1
~
−
Y − C'(
R + CQ
C')
C +
0 < ( 1)
< Q +
I
Q s s ε + +
(ii) The solution Q(t) of the difference Riccati equation is feasible and
converges to the stabilizing solution Qs if
0 < Q( 1)
< Q + Y + ε I
s
−1
____________________________________________________________
It is possible to see that the conditions introduced in the theorem are also necessary in two significant cases: L=0
(γ→∞) and C=0. In the first case Y=0 and therefore one recovers the condition of convergence for the Kalman
filter (Q(1)>0), whereas, in the second case the condition of convergence is equivalent to
S(1) = Q(1) -1 +C'C- L'Lγ -2 >S a ,
where S a is the antistabilizing solution of
S=(AS -1 A'+BB') -1 +C'C-L'Lγ -2

g-SWITCHING
If Q(1) is "too large", nothing can be said on the convergence of
Q(t,γs) towards the stabilizing solution Qs(γ) associated with a given
value of γ=γs.
The convergence condition is: 0 < Q( 1)
< Q ( γ ) + Y( γ ) + ε I .
Theorem 11
(i) The solution Y(γ) of the Lyapunov equation goes to zero for γ→∞
(ii) The stabilizingsolution Qs(γ) of the Riccati equation increases as γ
increases
Thanks to point (i) of the theorem, if Q(1) does not satisfy the
convergence condition, one can find a value of γ=γ 1>γ s such that the
convergence condition is satisfied for γ=γ1. The solution of the Riccati
equation with γ=γ1 converges to Q s(γ1). On the other hand, thanks to
point (ii) ot the theorem, one can say that
Qs(γ1)≤ Qs(γs)< Qs(γs)+(Y(γs)-1+εI)-1
So that ε>0 and t ε>0 are such that
Q(t ε,γ1)< Qs(γs)+(Y(γs)-1+εI)-1..
Finally, imposing
⎧γ
1 , 1 ≤ t < tε
γ ( t)
= ⎨
⎩γ
s , t ≥ tε
the solution Q(t,γ(t)) tends a Qs(γs) as t→∞.
_______________________________________________________
The γ-switching strategy modifies the cost:
2
T −
T
~ r(
t)
rˆ
( t)
⎡
2
⎤
J F = ∑ − ⎢∑
w(
t)
+ ( x(
0)
− xˆ
0 )'Ψ(
x(
0)
− xˆ
2
0 ) ⎥
t = 1 γ
( t)
⎣ t=
0
⎦
s
−1

EXAMPLE
A=0.5, B=[1 0], C=3, D=[0 1], L=2, T=1/12
Q( t + 1) = 0. 25
Q( t )
Feasibility condition
2
γ
Q( t ) <
2
4 − 9γ
Algebraic Riccati equation
Q = 0. 25
Q
Choice: γs=0.66
1
−1 −2
+ 9 − 4γ
1
−1 −2
+ 9 − 4γ
+ 1
+ 1 , Q(
1) = 4
Hence, Q s (γs)=1.5318 is the stabilizing solution
Q( 1)
< 5.4724 feasibility condition
Q(1) 0.6576
Integrating the equation with γ=γs it results Q( t )=4, 4.7167, 9.5394, -
2.2089, 0.6066, ..... and then the feasibility is lost after three steps.
Taking γ1 in such a way that 4

REFERENCES
T.Basar, P.Bernhard, "H∞ -Optimal Control and related minimax design
problems- a dynamic game approach", Birkhauser, 1991.
P.Bolzern, P.Colaneri, G.De Nicolao, "Transient and asymptotic analysis
of discrete-time H∞ filters", European Journal of Control, to appear.
J.C.Geromel, P.L.D.Peres, S.R.Souza, "A convex approach to the mixed
H2/H∞ control problem for discrete-time uncertain systems", SIAM J.
Control and Optimization, Vol. 33, 6, pp. 1816-1833, 1995.
M.Green, K.Glover, D.Limebeer, J.Doyle, "A J-spectral factorization
approach to H∞ control", SIAM J. Control and Optimization, 25,6, pp.
1350-1371.
M.J.Grimble, A. El Sayed, "Solution of the H∞ optimal linear filtering
problem for discrete-time systems", IEEE Trans. ASSP, 38, 7, pp. 1092-
1104, 1990.
W.Haddad, D.Bernstein, D.Mustafa, "Mixed H2/H∞ - regulation and
estimation", Systems and Control Letters, 16, 1991.
B.Hassibi, T.Kailath, "H∞ bounds for recursive-least square algorithms",
33rd Conference on Decision and Control, Lake Buena Vista, pp. 3927-
3938, 1994.
B.Hassibi, T.Kailath, "H∞ optimality of the LMS algorithms", IEEE
Trans. on Automatic Control, 44, 2, pp. 267-280, 1996.
B.Hassibi, A.H.Sayed, T.Kailath, "Linear estimation in Krein spaces -
Part II: Applications", IEEE Trans. AC, 41, pp. 34-49, 1996.
B.Hassibi, A.H.Sayed, T.Kailath, "Linear estimation in Krein spaces -
Part I: Theory", IEEE Trans. AC, 41, pp. 18-33, 1996.
B.Hassibi, A.H.Sayed, T.Kailath, "H∞ Optimality of the LMS
algorithm", IEEE Trans. ASSP, 44, pp. 267-1022, 1996.
D.H.Jacobson, "Optimal stochastic linear systems with exponential
performance criteria and their relation to deterministic games", IEEE
Trans. Automatic Control, AC-18, 2, 1973.
P.Lancaster, L.Rodman, "Algebraic Riccati Equations", Oxford Series
Pub., 1995.
D.J.Limebeer, M.Green, D.Walker, "Discrete-time H∞ control", Proc.
28th Conference on Decision and Control, Tampa (USA), pp. 392-396,
1989.

H.Rotstein, M.Sznaier, M.Idan, "H2/H∞ filtering theory and an
aerospace application", Int. J. Robust and Nonlinbear Control, 6, pp.
347-366, 1996.
J.Speyer, J.Deyst, D.H.Jacobson, "Optimization of stochastic linear
systems with additive measurement and process noise using exponential
performance criteria", IEEE Trans. Automatic Coontrol, AC-19, pp.358-
366, 1974.
J.L.Speyer, C.Fan, R.N.Banavar, "Optimal stochastic estimation with
exponential cost criteria", Proc. Conference on Decision and Control, pp.
2293-2298, 1992.
A.A.Stoorvogel, A.Saberi, B.M.Chen, "The discrete-time H∞ control
problem with measurementt feedback", Int. J. Robust and Nonlinear
Contr., 4, pp. 457-479, 1994.
Y.Theodor, U.Shaked, N.Berman, "Time-domain H∞ identification",
IEEE Trans. on Aut. Control, 41, pp. 1019-1022, 1996.
M.Vidyasagar, "Control Sysem Synthesis: a facorization approach", MIT
Press Series, 1985.
P.Whittle, "Risk Sensitive Optimal Control", New York: Wiley, 1990.
I.Yaesh, U.Shaked, "Game theory approach to optimal linear estimation
in the minimum H∞ norm sense", 28th Conference on Decision and
Control, Tampa, pp. 421-425, 1989.
I.Yaesh, U.Shaked, "A transfer function approach to the problem of
discrete-time systems: H∞-optimal linear control and filtering", IEEE
Trans. on Automatic Control, 36, 11, pp. 1264-1271, 1991.
I.Yaesh, U.Shaked, "Game theory approach to state-estimation of linear
discrete-time processes and its relation to H∞-optimal estimation", Int. J.
on Control, 55, 6, pp. 1443-1452, 1992.