Robustness in the Gap Metric and Coprime Factor Perturbations for LTV
Systems
Seddik M. Djouadi
Abstract— In this paper, we study the problem of robust
stabilization for linear time-varying (LTV) systems subject to
time-varying normalized coprime factor uncertainty. Operator
theoretic results which generalize similar results known to
hold for linear time-invariant (infinite-dimensional) systems are
developed. In particular, we compute a tight upper bound for
the maximal achievable stability margin under TV normalized
coprime factor uncertainty in terms of the norm of an operator
with a time-varying Hankel structure. We point to a necessary
and sufficient condition which guarantees compactness of the
TV Hankel operator, and in which case singular values and
vectors can be used to compute the time-varying stability
margin and TV controller.
D EFINITIONS AND N OTATION
•
B(E, F ) denotes the space of bounded linear operators
from a Banach space E to a Banach space F , endowed
with the operator norm
kAk :=
sup
kAxk, A ∈ B(E, F )
x∈E, kxk≤1
•
`2 denotes the usual Hilbert space of square summable
sequences with the standard norm
kxk22 :=
∞
X
¡
¢
|xj |2 , x := x0 , x1 , x2 , · · · ∈ `2
j=0
•
•
Pk the usual truncation operator for some integer k,
which sets all outputs after time k to zero.
An operator A ∈ B(E, F ) is said to be causal if it
satisfies the operator equation:
Pk APk = Pk A, ∀k positive integers
• tr(·) denotes the trace of its argument.
The subscript “c ” denotes the restriction of a subspace of
operators to its intersection with causal (see [19], [6] for the
definition) operators. “⊕” denotes for the direct sum of two
spaces. “? ” stands for the adjoint of an operator or the dual
space of a Banach space depending on the context [4], [5].
I. I NTRODUCTION
The gap metric was introduced to study stability
robustness of feedback systems. It induces the weakest
topology in which feedback stability is robust [1], [20], [18],
[22], [24]. In [20] Georgiou showed the relationship between
the gap metric and a particular two-block H ∞ problem.
In [18], the authors showed that feedback optimization
in the gap metric is equivalent to feedback optimization
S.M. Djouadi is with the Electrical Engineering & Computer Science Department, University of Tennessee, Knoxville, TN 37996-2100.
djouadi@ece.utk.edu
with respect to normalized factor perturbations. They
computed the largest possible uncertainty radius such that
robust stability is preserved. Extensions to time-varying
systems have been proposed in [26], [21] where a geometric
framework for robust stabilization of infinite-dimensional
time-varying systems was developed. The uncertainty was
described in terms of its graph and measured in the gap
metric. Several results on the gap metric and the gap
topology were established, in particular, the concept of a
graphable subspace was introduced. In [6], some of the
results obtained in [20] were generalized, in particular, the
gap metric for time-varying systems was generalized to
a two-block time varying optimization analogous to the
two-block H ∞ - optimization proposed in [20]. This was
achieved by introducing a metric which is the supremum
of the sequence of gaps between the plants measured at
every instant of time. The latter reduces to the standard gap
metric for linear time-invariant (LTI) systems.
In [7], [6], [8] using the time-varying gap metric it is
shown that the ball of uncertainty in the time-varying gap
metric of a given radius is equal to the ball of uncertainty
of the same radius defined by perturbations of a normalized
right coprime fraction, provided the radius is smaller than
a certain quantity. In [8] tight lower and upper bounds
are derived for computing the maximal stability margin
(quantified in the TV gap metric), in terms of using coprime
factorizations. These bounds are equal to the maximal
stability margin for LTI systems. In [17], the authors
showed that the time varying (TV) directed gap reduces
to the computation of an operator with a TV Hankel plus
Toeplitz structure. Computation of the norm of such an
operator can be carried out using an iterative scheme known
to hold for standard two-block H ∞ problems [17]. The
minimization in the TV directed gap formula was shown to
be a minimum using duality theory.
In this paper, we use the equivalence between uncertainty
quantified in TV gap metric balls and coprime factor
uncertainty for LTV systems. We study the problem of
robust stabilization for time-varying normalized coprime
factor perturbations and obtain operator theoretic results,
which generalize similar results in [11], [18], [22] known
to hold for LTI systems. In particular, we compute a tight
upper bound on the maximal achievable stability margin
under TV normalized coprime factor uncertainty in terms
of the norm of a TV Hankel operator. The upper bound
reduces to the maximal stability margin for LTI systems.
We point to a necessary and sufficient condition which
guarantees compactness of the TV Hankel operator, and
in which case singular values and vectors can be used to
compute the TV optimal stability margin and TV controller.
Therefore, generalizing similar results obtained in [22]
for LTI systems to TV ones. The technique developed to
compute the upper bound also applied to the lower bound
on the maximal achievable stability margin in the TV case.
To this end, it suffices to restrict the relevant operators to a
particular subspace.
The rest of the paper is organized as follows. In section
II the gap metric is introduced. In section III the relation
between the TV gap metric and coprime factorization is
discussed, and computations in terms of operator theory are
developed. In section IV the computation of the optimal
controller via the Youla parameter is presented. We conclude
with a summary of the paper contributions in section V.
II. T HE T IME -VARYING G AP M ETRIC
LTV systems may be regarded as causal linear (possibly
unbounded) operators acting on `2 as multiplication operators. To each plant P we associate its domain
D(P ) = {u ∈ `2 : P x ∈ `2 }
(1)
A LTV plant P is stabilizable if there exists a LTV controller
C such that the operator [27], [6]
µ
I
−P
−C
I
¶
: D(P ) ⊕ D(C) −→ `2 ⊕ `2
(2)
is invertible with bounded inverse. The inverse is given by
µ
−1
(I − CP )
P (I − CP )−1
−1
C(I − P C)
I(I − P C)−1
¶
(3)
It is readily seen from (3) that P is stabilizable if all its
entries belong to Bc (`2 , `2 ).
The LTV plant P has a right coprime factorization if there
exist operators M and N both in Bc (`2 , `2 ), such that,
P = N M −1 , and a left coprime factorization if there exist
M̂ , N̂ ∈ Bc (`2 , `2 ), such that P = M̂ −1 N̂ . In addition there
exist X, Y ∈ Bc (`2 , `2 ), and X̂, Ŷ ∈ Bc (`2 , `2 ), such that,
XN + Y M
N̂ X̂ + M̂ Ŷ
= I
= I
(4)
Such factorizations exist if and only if P is stabilizable
[23]. That is, The LTV plant P is stabilizable if and only if
it has left and right coprime factorizations
P = M̂ −1 N̂ = N M −1
(5)
There exist causal bounded linear operators U, V, Û , and V̂
such that
µ
¶µ
¶
V̂
−Û
M U
N V
−N̂ M̂
µ
¶µ
¶
M U
V̂
−Û
=
N V
−N̂ M̂
µ
¶
I 0
=
(6)
0 I
and the all P stabilizing LTV controllers C can be parameterized as
C
= (U + M Q)(V + N Q)−1
= (V̂ + QN̂ )−1 (Û + QM̂ ),
Q ∈ Bc (`2 , `2 ) (7)
Following [18], [6] we µ
are interested
in normalized coprime
¶
M
factorizations. That is,
is an isometry from `2 into
N
`2 ⊕ `2 , in this case, M ? M + N ? N = I as is in the LTI
case, where M ? , N ? are the adjoint operators of M and N ,
respectively [6].
Suppose that two LTV plants Gµ
1 and G
¶2 have normalized
µ
¶
M1
M2
right coprime factorizations
and
,
N1
N2
respectively. Denote by Π1 and Π2 the orthogonal
projections on their ranges. Then the gap between between
G1 and G2 is defined by [6]
δ(P1 , P2 ) = kΠ1 − Π2 k
and the directed gap
→
−
δ (G1 , G2 ) = k(I − Π2 )Π2 k
The gap is then [18], [21]
¡→
¢
−
−
→
δ(P1 , P2 ) = max δ (G1 , G2 ), δ (G2 , G1 ) 9)
(8)
(9)
(10)
It is well known that if δ(G1 , G2 ) < 1, then all these
quantities are equal.
µ
¶
µ
¶
M1
M2
Normalized coprime factorizations
and
N1
N2
imply that
µ
¶
µ
¶
M1
M2
(I − Pn ),
(I − Pn )
(11)
N1
N2
are isometries on (I − Pn )`2 with range in (I − Pn )`2 ⊕
(I − Pn )`2 [7], [6]. Following [7], [6] µ
Let Π1n¶denote the
M1
orthogonal projection on the range of
(I − Pn ),
N1
Π2n is defined similarly. Define
°½µ
°
¶
¾
°
°
−
→
I − Pn
−0
°
δn (G1 , G2 ) = °
− Π2n Π1n °
° (12)
0
I − Pn
and
δn (G1 , G2 )
= kΠ1n − Π2n k
¡→
¢
−
−
→
= max δn (G1 , G2 ), δn (G2 , G1 ) (13)
Lemma 1: [7]
°µ
µ
¶
¶
°
M2
?
?
°
δn (G1 , G2 ) = ° I −
(I − Pn )(M2 , N2 )
N2
°
µ
¶
°
M1
(I − Pn )°
° (14)
N1
The following result generalizes the time-invariant counterpart derived in [20].
Theorem 1: [6]
°µ
¶ µ
¶ °
° M1
°
−
→
M2
sup δn (G1 , G2 ) =
inf 2 2 °
−
Q°
°
° (15)
N1
N2
Q∈Bc (` ,` )
n≥0
The directed time varying gap between G1 and G2 is then
defined as
−
→
→
−
α (G1 , G2 ) := sup δn (G1 , G2 )
(16)
n≥0
and the time varying gap [6]
´
³
→
→
α(G1 , G2 ) = max −
α (G1 , G2 ), −
α (G2 , G1 )
(17)
The function α is a metric and for time-invariant systems
reduces to the standard gap metric δ [7]. Note that in the
time-invariant case Theorem 1 reduces to the two-block
H ∞ optimization derived for the gap metric in [20]. In [17]
−
→
α (G1 , G2 ) was solved in terms of a time-varying Hankel
operator, and in fact using duality theory the infimum in
(15) is shown to be attained by some Q in Bc (`2 , `2 ).
Denote by Bs (P, r) the set of all P1 with right coprime
factorization N1 M1−1 for which
°µ
¶°
¶ µ
° M
M1 °
°
°<r
−
(18)
° M
N1 °
and denote by B(P, r) the set of all plants P1 such that
α(P, P1 ) < r. then the following result in [8] holds.
Theorem 2: [8]
B(P, r) = Bs (P, r)
(19)
Theorem 2 relates coprime uncertainty balls to balls defined
in the time-varying gap metric. In particular, maximizing
the uncertainty radius for coprime uncertainty results in
maximizing the uncertainty radius quantified in the TV gap
metric.
In the next section, we study the problem of robust stabilization for time-varying normalized coprime factor perturbations and obtain operator theoretic results, which generalize
similar results in [11], [18], [22] known to hold for LTI
systems.
III. ROBUST S TABILIZATION U NDER C OPRIME FACTOR
U NCERTAINTY
In [9] the authors proved a robust stabilization result
under proper coprime factor uncertainty for LTI systems.
This result was generalized to LTV systems in [8], and is
summarized in the following theorem.
Theorem 3: Let C be a controller with right coprime
representation
C = (U + M Q)(V + N Q)−1
Then the following are equivalent
1) C stabilizes all plant P1 with normalized left coprime
factorization
¡
¢−1
(20)
P1 = M̂ + ∆M̂
(N̂ + ∆N̂ )
where ∆M̂ , ∆N̂ ∈ Bc (`2 , `2 ) and k(∆M̂ , ∆N̂ k < r,
for some r > 0.
2)
°µ
¶°
° U + MQ ° 1
°
°
(21)
° V + NQ ° ≤ r
A similar result holds for (TV) normalized right coprime
factorizations.
If ropt is the supremum over all r such that C stabilizes
Bs (P, r), then for TV systems the following inequality holds
[8]
°µ
°¾
¶
½
° U + MQ
°
°
(I − Pn )°
inf
inf2 2 °
°
V
+
N
Q
n≥0 Q∈Bc (` , ` )
°µ
¶°
° U + MQ °
1
°
°
≤
≤
inf
V + NQ °
ropt
Q∈Bc (`2 ,`2 ) °
(22)
For time-invariant systems these numbers are equal.
In the same vein as [18], we define ro as the supremum
over all r such that there exists a ¡fixed controller
¢−1 C which
stabilized all plants P1 with P1 = M̂ + ∆M̂
(N̂ + ∆N̂ ),
∆M̂ , ∆N̂ ∈ Bc (`2 , `2 ) and k(∆M̂ , ∆N̂ k < r. It follows
that
°µ
¶ µ
¶ °
° U
°
M
−1
°
inf
ro =
+
Q°
(23)
°
V
N
Q∈Bc (`2 ,`2 ) °
The infimum is achieved for some Q ∈ Bc (`2 , `2 ) [16], [17].
In the sequel we are concerned with solving the optimization
(23) which reciprocal is an upper bound for the optimal
robustness radius. The solution proposed applies as well to
the lower bound in (22) by a restriction to a subspace, that
is, by restricting the domain of definition of the operators
introduced in the sequel to the range of (I − Pn ). Now
observe that (as in the time-invariant case) the operator
µ
¶
M? N?
Ž :=
∈ B(`2 , `2 × `2 )
(24)
−N̂ M̂
and the operator induced norm
°µ
°µ
¶°
¶ °
° A °
° A
°
°
° :=
°
sup
f°
° B °
°
°
B
f ∈`2 , kf k2 ≤1
2
³
´ 12
=
kAf k22 + kBf k22
µ
¶
A
∈ B(`2 , `2 × `2 )
B
(25)
(26)
is unitarily invariant, that is, for any unitary operator U ∈
B(`2 , `2 × `2 ), we have
° µ
¶° °µ
¶°
°
°
°
A °
°U
°=° A °
°
B ° ° B °
In particular, for U = Ž, we have
°µ
° µ
¶ µ
¶ °
¶ µ
¶ °
° U
°
°
°
M
U
M
°
°
°
+
Q° = °Ž
+
Q°
° V
°
N
V
N
°µ
¶°
° R+Q °
°
= °
°
°
I
¡
¢1
= 1 + kR + Qk2 2
(27)
where R := M ? U + N ? V ∈ B(`2 , `2 × `2 ).
Next, we show that
¡
¢− 1
ro = 1 + kHR k2 2
(28)
where HR is the time-varying Hankel operator with symbol
R. Note that as in the LTI case 0 < ro ≤ 1.
To define the operator HR , we need some mathematical
preliminaries. Let
Qn := I − Pn , for n = −1, 0, 1, · · ·
where Pn is the standard truncation operator, which also an
orthogonal projection that sets all outputs after time n to
zero, and
P−1
and P∞
:= 0
:= I
The quantity ro is a tight bound on the maximal achievable
TV stability margin under coprime factor uncertainty for
LTV systems, and reduces to the maximal stability margin
for LTI system.
Proof. Let Π1 be
projection on the subspace
µ the orthogonal
¶
M
(A2 ⊕ A2 ) ª
A2 the orthogonal complement of
N
µ
¶
M
A2 in the operator Hilbert space A2 ⊕ A2 under
N
the inner product
(A, B) := tr(B ? A), A, B ∈ A2 ⊕ A2
Then Qn is a projection, and we associate to it the following
nest
N := {Qn `2 , n = −1, 0, 1, · · · }
The orthogonal projection is then defined on the spaces
µ
¶
M
Π1 : A2 ⊕ A2 7−→ (A2 ⊕ A2 ) ª
A2
N
Call
The space of causal bounded linear operators Bc (`2 , `2 ) can
be viewed as a triangular or nest algebra, which leaves
invariant every subspace N ∈ N , that is, T N ⊆ N ,
∀T ∈ Bc (`2 , `2 ). In fact, Bc (`2 , `2 ) can be written as
Bc (`2 , `2 ) =
=
where HR = (I − P)(M ? U + N ? V ).
Formula (32) is the time-varying analogue of the standard
Nehari problem.
The following expression for ro was obtained in the LTI case
was obtained in [11] using state space techniques, and in [18]
using operator theory. in [11]. We will give its time-varying
counterpart and give an operator theoretic proof along the
lines of [18].
Theorem 4:
°
°2
°
°
ro−2 = 1 − °H
(33)
°
M̂ ? 

N̂ ?
{A ∈ B(`2 , `2 ) : Pn A(I − Pn ) = 0, ∀ n}
{A ∈ B(`2 , `2 ) : (I − Qn )AQn = 0, ∀ n}
(29)
Now, call C2 the class of compact operators on `2 called the
Hilbert-Schmidt or Schatten 2-class [10], [3] under the norm,
³
´ 12
kAk2 := tr(A? A)
µ
S := (A2 ⊕ A2 ) ª
(30)
Then A2 is the space of causal Hilbert-Schmidt operators.
Define the orthogonal projection P of C2 onto A2 . P is the
lower triangular truncation or nest projection.
Following [15] an operator X in B(`2 , `2 ) determines a
Hankel operator HX on A2 if
HX A = (I − P)XA,
2
2
for A ∈ A2
(31)
2
For R ∈ B(` , ` × ` ) the TV Hankel operator solves the
minimization [16]
inf
Q∈Bc (`2 ,`2 )
kR + Qk = kHR k
(32)
¶
A2
Define the operator
Ξ
:
Ξ :=
A2 µ
7−→ A¶2 ⊕ A2
U
Π1
V
(34)
Using the Commutant Lifting Theorem [6], it follows
ro−1 = kΞk
(35)
Next, let Γ be the left multiplication operator
Γ
Γ
Define the space
A2 := C2 ∩ Bc (`2 , `2 )
M
N
: S 7−→ A2
:= (−N̂ , M̂ )|S
Then, for X ∈ A2 we have
µ
¶
µ
¶
U
M
ΞX =
X−
Y ∈S
V
N
for some Y ∈ A2 and
·µ
¶
µ
¶ ¸
U
M
Γ(ΞX) = (−N̂ , M̂ )
X−
Y =X
V
N
therefore ΓΞ = I on S. Moreover, for s ∈ S, we have
Γs =
=
(−N̂ , M̂ )s
·µ
¶ µ
¶ ¸
s1
M
(−N̂ , M̂ )
−
s3
s2
N
(36)
°
°
its norm °H
for some s1 , s2 and s3 in A2 . Now, apply Ξ to get
µ
¶
µ
¶
U
M
Ξ(Γs) =
(−N̂ , M̂ )s −
(−N̂ , M̂ )s
V
N
and thus
Ξ(Γs) = s
showing that ΞΓ = I on S, ΓΞ = I on A2 , that is Γ is the
inverse of Ξ, Γ = Ξ−1 . This implies
inf
s∈S, ksk2 ≤1
kΓsk−1 =: τ (Γ)−1
(37)
?
Now, note that the adjoint operator of Γ, Γ is defined by
Γ?
Γ? X
:
A2 −
7 →S
µ
¶
−N̂ ?
= Π1
X, X ∈ A2
M̂ ?
and the operator Υ defined by
is equal to the maximal singular
M̂ ? 
N̂ ?
value. A necessary and sufficient condition for any Hankel
operator HX to be compact is that X belongs to a triangular
algebra A plus the space of compact operator denoted K,
that is, X ∈ A + K [15]. In our case, if we define K
as the space of compact operator
from A°2 into C2 and
°
° 
°
A := B(A2 , A2 ⊕ A2 ), then °H
° is compact if
?
M̂


N̂ ?
and only if
µ
¶
M̂ ?
∈A+K
N̂ ?

which by (6) is equal to
µ
¶
µ
¶
U
M
Ξ(Γs) =
(−N̂ , M̂ )s −
(V̂ , Û )s
V
N
kΞk =
°
°
°
(38)
This implies that H
 attains its norm at some X ∈
M̂ ? 
N̂ ?
A2 . As in the LTI case [22], this has implications for the
operators Ξ and Γ, in particular Ξ attains its norm. This is
summarized in the following theorem.
Theorem 5: Let 0 < λ < 1. Then the following are
equivalent

1) λ is a singular value of Υ.
A2 7−→ C2 ª A2 =: A⊥
?
⊥
2
2) There exists X
µ
¶
µ 6= 0¶ ∈ A2 , Y 6= 0 ∈ A2 , W ∈
−N̂ ?
M
Υ = (I − P)
A2 such that
(A2 ⊕ A2 ) ª
M̂ ?
N
¶
µ
µ
¶
−N̂ ?
1
−N̂ ?
defined
Consider the left multiplication operator
X − (1 − λ2 ) 2 W = λY ?
M̂ ?
M̂ ?
from A2 into C2 which is isometric since we are using
(−N̂ , M̂ )Y ? = λX
(41)
normalized coprime factorizations. Then for any X ∈ A2
1
we have
3) (1 − λ2 ) 2 is a singular value of Γ.
µ
¶
µ
¶
µ
¶
?
?
?
?
⊥
−N̂
−N̂
−N̂
4) There exists X
µ 6= 0¶ ∈ A2 , Y 6= 0 ∈ A2 , W ∈
X = P
X + (I − P)
X
?
?
?
M
M̂
M̂
M̂
A2 such that
(A2 ⊕ A2 ) ª
µ
¶
µ
¶
N
?
−N̂
−N̂ ?
= Π1
X + (I − P)
X
1
M̂ ?
M̂ ?
(−N̂ , M̂ )W = (1 − λ2 ) 2 X (42)
?
µ
¶
= Γ X + ΥX
1
−N̂ ?
X − λY ? = (1 − λ2 ) 2 W (43)
M̂ ?
But since ΓΓ? + Υ? Υ = I, that is, the operator Γ? + Υ is
an isometry, and τ (Γ? )2 + kΥk2 = 1. Further, Ξ? Γ? = I
1
1
5) (1 − λ2 ) 2 is a singular value of Ξ.
and so kΓ? k ≥ kΞk
, i.e., Γ? is bounded below implying
Proof. Follows as in the LTI case [22].
kτ (Γ)k = τ (Γ? )k.
By definition
IV. O PTIMAL ROBUSTNESS IN THE TV G AP M ETRIC
°
°
°
°
(39)
kΥk = °H
°
?
M̂ 
In this section we give a necessary and sufficient condition

 and generalizes
for λ to be a singular value of H
N̂ ?
M̂ ? 

This yields
N̂ ?
its LTI counterpart.
°2
°
° 
°
(40)
° +kΞk−2 = 1
°H
Theorem 6: Let 0 < λ < 1. Then λ is a singular value
?
M̂ 

of Υ. if and only if there exists W 6= 0 ∈ A2 such that
?
N̂
³µ −N̂ ? ¶
´
1
which implies the result of the theorem (33).
(−N̂ , M̂ ) − (1 − λ2 ) 2 I W ∈ A2
(44)
?
M̂
 is compact, then
µ
¶
If the Hankel operator is H
M̂ ? 
M

Moreover, if (44) holds then W ∈ (A2 ⊕ A2 ) ª
A2
N̂ ?
N
Υ
:
and Theorem 5, 3) and 4) hold for
[9] M. Vidyasagar and H. Kimura, Robust stabilization for uncertain linear
multivariable systems, Automatica, vol. 22, pp. 85-94, 1986
1
[10] Schatten R. Norm Ideals of Completely Continuous Operators,
X =
(−
N̂
,
M̂
)W
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¡ ?
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¡ ?
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?
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The maximal stability margin under coprime factor uncertainty for LTV systems is characterized as the norm of these
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the case of LTI systems in [22], and in fact reduce to them
in that case.
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