Optimal robustness of passiv e discrete time systems
V. Mehrmann ∗
, P . V an Do oren †
Septem b er 15, 2019
Abstract
W e construct optimally robust realizations of a giv en rational transfer function that
represen ts a passiv e discrete-time system. W e link it to the solution set of linear matrix
inequalities defining passiv e transfer functions. W e also consider the problem of finding
the nearest passiv e system to a giv en non-passiv e one.
Keyw ords: Linear matrix inequalit y , passivit y , robustness, discrete-time system, port-Ha-
miltonian system
AMS Sub ject Classification : 93D09, 93C05, 49M15, 37J25
1 In tro duction
W e consider realizations of linear discrete-time dynamical systems for whic h the asso ciated
transfer function is p assive . Suc h transfer functions pla y a fundamen tal role in systems and
con trol theory: they represen t e. g., sp ectral densit y functions of sto c hastic pro cesses, sho w
up in sp ectral factorizations and are also related to discrete-time algebraic Riccati equations.
P assive transfer functions can b e described using conv ex sets, and this prop ert y has lead to
the extensiv e use of con vex optimization tec hniques in this area [5].
In this pap er w e show that in the set of p ossible realizations of a giv en passiv e transfer
function, there is a subset that maximizes robustness, in the sense that their so-called p assivity
r adius is nearly optimal. Related results for con tin uous-time systems w ere already obtained
in a companion pap er [15]. Here w e consider the discrete-time system
x k +1 = Ax k + B u k , x 0 = 0 ,
y k = C x k + D u k , (1)
where u k ∈ C m , x k ∈ C n , and y k ∈ C m are vector-v alued sequences denoting, resp ectiv ely , the
input , state , and output of the system. Denoting real and complex n -v ectors ( n × m matrices)
b y R n , C n ( R n × m , C n × m ), resp ectiv ely , the co efficien t matrices satisfy A ∈ C n × n , B ∈ C n × m ,
C ∈ C m × n , and D ∈ C m × m .
W e restrict ourselv es to systems whic h are minimal , i. e. the pair ( A, B ) is c ontr ol lable
(for all z ∈ C , rank [ z I − A B ] = n ), and the pair ( A, C ) is observable (i. e. ( A H , C H ) is
con trollable). Here, the Hermitian (or conjugate) transp ose (transp ose) of a v ector or matrix
1 Institut f ¨ ur Mathematik MA 4-5, TU Berlin, Str. des 17. Juni 136, D-10623 Berlin, German y . mehrmann@
math.tu- berlin.de .
2 Departmen t of Mathematical Engineering, Univ ersit´ e catholique de Louv ain, Louv ain-La-Neuv e, Belgium.
[email protected] .
1

V is denoted b y V H ( V T ) and the identit y matrix is denoted b y I n or I if the dimension is
clear. W e furthermore require that input and output dimensions are equal to m .
Passive systems are w ell studied in the contin uous-time case, starting with the w orks [23,
24]. Here we consider the equiv alen t definition in the discrete-time case and deriv e so-called
normalize d p assive r e alizations that could b e considered as “discrete-time p ort-Hamiltonian
systems”. Similar attempts w e re already made in the literature [11],[19],[20].
The pap er is organized as follo ws. After going o ver some preliminaries in Section 2, we
c haracterize in Section 3 what we called normalized passiv e realizations of a discrete-time
passiv e system. W e then sho w in Section 4 their relev ance in estimating the passivit y radius
of sicrete-time passiv e systems and construct in Section 5 realizations with nearly optimal
robustness margin for passivit y . In Section 7 w e describ e an algorithm to compute this
robustness margin. In Section 8 w e show ho w to use these ideas to estimate the distance to
the set of discrete-time passiv e systems.
2 P assiv e systems
Throughout this article w e will use the following notation. W e denote the set of Hermitian
matrices in C n × n b y H n . P ositive definiteness (semi-definiteness) of A ∈ H n is denoted b y
A > 0 ( A ≥ 0). The real and imaginary parts of a complex matrix Z are written as < ( Z )
and = ( Z ), resp ectiv ely , and ı is the imaginary unit. W e consider functions o v er H n , whic h is
a v ector space if considered as a r e al subspace of R n × n + ı R n × n .
The concept of p assivity is w ell studied. W e briefly recall some imp ortan t prop erties
follo wing [24], and refer to the literature for pro ofs and for a more detailed surv ey . Consider
a discrete-time system (1) with minimal state-space mo del
M := { A, B , C , D }
and transfer function T ( z ) := C ( z I n − A ) − 1 B + D and define the complex analytic function
of z ∈ C :
Φ( z ) := T H ( z − 1 ) + T ( z ) ,
whic h coin cides with the Hermitian part of T ( z ) on the unit circle:
Φ( e ıω )=[ T ( e ıω )] H + T ( e ıω ) .
The transfer function T ( z ) is called strictly p ositive-r e al if Φ( e ıω ) > 0 for all ω ∈ [ − π , π ]
and it is called p ositive-r e al if Φ( e ıω ) ≥ 0 for all ω ∈ [ − π , π ]; T ( z ) is called asymptotic al ly
stable if the eigen v alues of A are in the op en unit disc, and it is called stable if the eigen v alues
of A are in the closed unit disc, with an y eigen v alues o ccurring on the unit circle b eing
semi-simple. With these t w o prop erties, then T ( z ) is called strictly p assive if it is strictly
p ositiv e-real and asymptotically stable and it is called p assive if it is p ositiv e real and stable.
The transfer function T ( z ) is the Sc h ur complemen t of the so-called system p encil
S ( z ) := 

0 A − z I n B
z A H − I n 0 C H
z B H C D H + D

 (2)
and if the mo del M is minimal, then the finite generalized eigen v alues of S ( z ) are the finite
zeros of Φ( z ). The follo wing equiv alence transformation, using an arbitrary matrix X ∈ H n ,
2

lea ves the Sc h ur complemen t, and hence also the transfer function Φ( z ), unc hanged


0 A − z I n B
z A H − I n X − A H X A C H − A H X B
z B H C − B H X A D H + D − B H X B

 = 

I n 0 0
− A H X I n 0
− B H X 0 I m

 S ( z ) 
 I n − X 0
0 I n 0
0 0 I m

 .
(3)
Let us define the submatrix of (3), giv en b y
W ( X , M ) :=  X − A H X A C H − A H X B
C − B H X A D H + D − B H X B  , (4)
whic h we will also denote as W ( X ) when the underlying mo del M is ob vious from the con text.
Then it follo ws b y simple algebraic manipulation that
Φ( z ) =  B H ( z − 1 I n − A H ) − 1 I m  W ( X , M )  ( z I n − A ) − 1 B
I m  ,
and that T ( z ) is p ositiv e real if and only if there exists X ∈ H n suc h that the Linear Matrix
Inequalit y (LMI)
W ( X , M ) ≥ 0 (5)
holds. Moreo v er, T ( z ) is stable if and only if the matrix X in this LMI is also p ositiv e definite.
W e will therefore mak e frequen t use of the follo wing sets
X > := { X ∈ H n | W ( X , M ) ≥ 0 , X > 0 } , (6a)
X  := { X ∈ H n | W ( X , M ) > 0 , X > 0 } . (6b)
An imp ortan t subset of X > are those solutions to (5) for whic h the rank r of W ( X ) is minimal
(i. e. for which r = rank Φ( z )). If D H + D − B H X B is in vertible, then the minim um rank
solutions in X > are those for whic h rank W ( X ) = ran k( D H + D − B H X B ) = m , which in turn
is the case if and only if the Sc h ur complemen t of D H + D − B H X B in W ( X ) is zero. This
Sc hur complemen t is asso ciated with the discrete-time algebr aic R ic c ati e quation (ARE)
Ricc ( X ) := X − A H X A − ( C H − A H X B )( D H + D − B H X B ) − 1 ( C − B H X A ) = 0 . (7)
Solutions X to (7) pro duce a sp ectral factorization of Φ( z ), and eac h solution corresp onds to
a invariant subsp ac e spanned b y the columns of U :=  I n − X T  T that remains in v arian t
under the m ultiplication with the matrix
S :=  I n B ( D H + D ) − 1 B H
0 ( A − B ( D H + D ) − 1 C ) H  − 1  A − B ( D H + D ) − 1 C 0
C H ( D H + D ) − 1 C I n  , (8)
i. e. U satisfies S U = U A F where the so-called close d lo op matrix is defined as A F = A − B F
with F := ( D H + D − B H X B ) − 1 ( C − B H X A ). Suc h a subspace is called a L agr angian invariant
subsp ac e and the matrix S has a symple ctic structur e (see e.g., [14],[7]). Each solution X of (7)
can also b e asso ciated with an extende d L agr angian invariant subsp ac e for the p encil S ( z ),
spanned b y the columns of b
U :=  − X T I n − F T  T . In particular, b
U satisfies


0 A B
− I n 0 C H
0 C D H + D

 b
U = 

0 − I n 0
A H 0 0
B H 0 0

 b
U A F .
3

If D H + D − B H X B is singular then more complicated constructions are necessary , see [14].
In the con tin uous-time case, the definition of a passive systems has its origin in net w ork
theory , but its formal definition is asso ciated with the existence of a storage function and
a particular dissipation inequalit y . The equiv alen t concept for the discrete-time case again
follo ws from the LMI (5). If w e define the v ector z k as the stac k ed vector of th e state x k ab o v e
the input u k , and construct the inner pro duct z H
k W ( X ) z k , then w e obtain the inequalit y
x H
k X x k − x H
k +1 X x k +1 + y H
k u k + u H
k y k = z H
k W ( X ) z k ≥ 0 . (9)
Using the quadratic storage function H ( x i ) := 1
2 x H
i X x i this yields a dissip ation ine quality
H ( x k ) − H ( x 0 ) ≤
k − 1
X
i =0 < ( y H
i u i )
that is similar to the one of the con tin uous-time form ulation. It follo ws from the con tinuous-
time literature [24] and the bilinear transformation b et ween con tin uous-time and discrete-time
systems [1] that if the system M of (2) is minimal, then the LMI (5) has a solution X ≥ 0
if and only if M is a passiv e system. Moreo v er, the solutions of (5) also satisfy the matrix
inequalities
0 < X − ≤ X ≤ X + . (10)
The matrices X satisfying the matrix inequalities (10) also form a con v ex set, whic h w e call
X ± . W e th us ha v e the follo wing inclusions
X  ⊂ X > ⊂ X ±
whic h implies that all matrices in the sets X  and X > are b ounded. Notice also that the
(1 , 1) blo c k in the LMI (4),(6) is a discrete-time Ly apuno v equation with X > 0. This implies
that A is asymptotically stable if W ( X ) > 0 and is stable if W ( X ) ≥ 0, see also [13]. It is
also kno wn that if the system is strictly passive, meaning that Φ( e ıω ) > 0 for the whole unit
circle, then X − < X + .
Remark 2.1. The bilinear transformation b et w een con tin uous-time and discrete-time sys-
tems preserv es the solution sets X  and X > as w ell as the solutions X − and X + of the Riccati
equation. It w as shown, see e. g., [15], that the set X ± has a nonempt y in terior if and only if
X − < X + . Since X > is a subset of X ± it also follo ws X > has an empt y in terior when X + − X −
is singular.
3 Normalized passiv e realizations
A sp ecial class of realizations of discrete-time passiv e systems, are the ones asso ciated to a
normalized storage function H ( x k ) = 1
2 k x k k 2
2 .
Definition 3.1. A normalized passiv e system has the state-sp ac e form (1) wher e the system
matric es satisfy the matrix ine quality
 I n C H
C D H + D  −  A H
B H   A B  ≥ 0 . (11)
4

W e no w sho w that ev ery passiv e system has an equiv alen t normalized passiv e realization.
Consider a minimal state-space mo del M := { A, B , C , D } of a passive linear time-in v ariant
system and let X ∈ X > b e a solution of the LMI (5). W e then use a (Cholesky lik e) factor-
ization X = T H T whic h implies det T 6 = 0 and define a new realization
M T := { A T , B T , C T , D T } := { T AT − 1 , T B , C T − 1 , D }
so that
 T − H 0
0 I m   X − A H X A C H − A H X B
C − B H X A D H + D − B H X B   T − 1 0
0 I m  (12)
=  I n C H
T
C T D H
T + D T  −  A H
T
B H
T   A T B T  ≥ 0 , (13)
whic h expresses that the transformed realization M T is now normalized. Notice that the
factor T is unique up to a unitary factor U since T H U H U T = T H T . This unitary factor
do es not affect the normalization constrain t, but w e can c ho ose it to put A T in a sp ecial
co ordinate system. Notice that the inequalit y I n − A H
T A T ≥ 0 implies that A T is con tractive
and has a singular v alue decomp osition A T = U Σ V H where 0 ≤ Σ ≤ I n . The additional
unitary similarit y transformation { U H A T U , U H B T , C T U, D T } will then yield a new normalized
co ordinate system { A ˆ
T , B ˆ
T , C ˆ
T , D ˆ
T } where, in addition, A ˆ
T = Σ( V H U ), whic h is a p olar
decomp osition with a p ositiv e semidefinite Hermitian factor Σ that is diagonal and satisfies
0 ≤ Σ ≤ I n [10].
Ev en after the normalization, there is t ypically still a lot of freedom in the represen tation
of the system, since w e could ha v e used any matrix X from the set X > to normalize our
realization. In the remainder of this pap er, w e will fo cus on normalized passiv e realizations.
The freedom remaining is th us the choice of the matrix X from X > , whic h, as we will see,
can b e used to mak e the represen tation more robust, i.e., less sensitiv e to p erturbations. The
remainder of this pap er will deal with the question of ho w to mak e use of this freedom in the
state space transformation to determine a ’go o d’ or ‘nearly optimal’ normalized realization.
4 The passivit y radius
Our goal is to ac hiev e ‘go o d’ or ‘nearly optimal’ normalized realizations of a passiv e system.
A natural measure for this is a large p assivity r adius ρ M , whic h is the smallest p erturbation
(in an appropriate norm) to the co efficien ts of a mo del M that causes the p erturb ed system
to lo ose this prop ert y .
Once w e hav e determined a solution X ∈ X > to the LMI (5), w e can determine the
normalized represen tations as discussed in Section 3. F or each suc h represen tation w e can
determine the passivit y radius and then c ho ose the solution X ∈ X > whic h is most robust
under p erturbations ∆ M of the mo del parameters M := { A, B , C , D } . This is a suitable
approac h for p erturbation analysis, since as so on as w e fix X ∈ X  , w e will see that w e can
solv e for the smallest p erturbation ∆ M to our mo del M that mak es det W ( X, M + ∆ M ) = 0.
T o measure the size of the p erturbation ∆ M of a state space mo del M w e will use the
F rob enius norm or the 2-norm of the matrix ∆ S defined as
∆ S :=  ∆ A ∆ B
∆ C ∆ D  (14)
5

and w e use also the notion of X -p assivity r adius , whic h w as in tro duced in [2], and giv es a
b ound for the usual passivit y radius.
Definition 4.1. F or X ∈ X  the X -passivit y radius is define d as
ρ M ( X ) := inf
∆ S ∈ C n + m,n + m {k ∆ S k | det W ( X , M + ∆ M )=0 } .
Note that in order to compute ρ M ( X ) for the mo del M , w e m ust hav e a p oin t X ∈ X  ,
since W ( X , M ) m ust b e p ositiv e definite to start with and also X should b e p ositiv e definite
to obtain a state-space transformation from it. The follo wing relation b et w een the X -passivit y
radius and the usual passivit y radius w as already presented in [2].
Lemma 4.2. The p assivity r adius for a given mo del M satisfies
ρ M := sup
X ∈ X 
inf
∆ S ∈ C n + m,n + m {k ∆ S k| det W ( X , M + ∆ M ) = 0 } = sup
X ∈ X 
ρ M ( X ) .
W e no w pro vide an exact form ula for the X -passivit y radius based on a one parameter
optimization problem. F or this, w e p oin t out that the condition W ( X , M + ∆ M ) > 0 is
equiv alent to the condition
c
W ( X , M + ∆ M ) := 

X − 1 A + ∆ A B + ∆ B
A H + ∆ H
A X C H + ∆ H
C
B H + ∆ H
B C + ∆ C D H + ∆ H
D + D + ∆ D

 > 0 , (15)
whic h is no w an LMI in the unkno wn parameters of ∆ M (for a fixed X ). Setting
c
W := c
W ( X , M ) = 

X − 1 A B
A H X C H
B H C D H + D

 , E :=  E 1 E 2  

I n 0 0 0
0 0 I n 0
0 I m 0 I m

 ,
(16)
and using the matrix ∆ S in (14), this inequalit y can b e written as the structured LMI
c
W + E  0 ∆ S
∆ H
S 0  E T > 0 (17)
as long as the system is still passiv e. In order to violate this condition, w e need to find the
smallest ∆ S suc h that the determinan t of (17) b ecomes 0. Since c
W is p ositiv e definite, w e
can then construct its Cholesky factorization c
W := R H R . The matrix in (17) will b ecome
singular when the matrix
I 2 n + m + R − H E  0 ∆ S
∆ H
S 0  E T R − 1 (18)
b ecomes singular. The follo wing theorem, is analogous to results obtained for con tin uous-time
systems [2, 15, 18], and w e therefore omit the pro of. It giv es for this kind of problem the
minim um norm p erturbation ∆ S b oth in F rob enius norm and in 2-norm.
Theorem 4.3. Consider the matric es ˆ
X , c
W = R H R in (16) and the p ointwise p ositive
semidefinite matrix function
M ( γ ) :=  γ F H
1
γ − 1 F H
2   γ F 1 γ − 1 F 2  , F 1 := R − H E 1 , F 2 := R − H E 2 (19)
6

in the r e al p ar ameter γ ∈ (0 , ∞ ) . Then the lar gest eigenvalue λ max ( M ( γ )) is a unimo dal
function of γ (i.e. it is first monotonic al ly de cr e asing and then monotonic al ly incr e asing with
gr owing γ ). A t the minimizing value γ , M ( γ ) has an eigenve ctor z , i.e.
M ( γ ) z = λ max z , z :=  u
v  ,
wher e k u k 2
2 = k v k 2
2 = 1 . The minimum norm p erturb ation ∆ S is of r ank 1 and is given by
∆ S = uv H /λ max . It has norm 1 /λ max b oth in 2-norm and in F r ob enius norm.
A simple b ound for λ max can also b e obtained, as p oin ted out in [2] for the con tinuous-time
case. The pro of is essen tially the same and is therefore omitted.
Corollary 4.4. Consider the matric es c
W , F 1 , F 2 and M ( γ ) in The or em 4.3, and define
α := k F 1 k 2 and β := k F 2 k 2 . Then the norm of M ( γ ) is also the norm of γ 2 F 1 F H
1 + γ − 2 F 2 F H
2 ,
and
λ max = k M ( γ ) k 2 = min
γ > 0 k M ( γ ) k 2 = min
γ > 0 k γ 2 F 1 F H
1 + γ − 2 F 2 F H
2 k 2 ≤ 2 k F 1 k 2 k F 2 k 2 = 2 αβ .
This upp er b ound is r e ache d if and only if the matric es F 1 F H
1 and F 2 F H
2 have a c ommon
eigenve ctor asso cite d with the maximal eigenvalue.
The follo wing theorem is a v arian t of a result pro v en in [2], and constructs a rank one
p erturbation whic h makes the matrix W M +∆ M singular and therefore giv es an upp er b ound
for ρ M ( X ).
Theorem 4.5. L et M = { A, B , C, D } b e a given minimal p assive discr ete-time mo del and
assume that we ar e given a matrix X ∈ X  , then the X -p assivity r adius ρ M ( X ) is b ounde d
by
1 / (2 αβ ) ≤ ρ M ( X ) ≤ 1 / [(1 + | ˆ v H ˆ u | )( αβ )] ≤ 1 / ( αβ ) ,
wher e ˆ u, u and ˆ v , v ar e normalize d dominant singular ve ctor p airs of F 1 and F 2 , r esp e ctively :
F 1 u = α ˆ u, F H
1 ˆ u = αu, F 2 v = β ˆ v , F H
2 ˆ v = β v .
Mor e over, if ˆ u and ˆ v ar e line ar dep endent, then ρ M ( X )=1 / (2 αβ ) .
Pr o of. The pro of is analogous to the con tin uous-time case, see [15].
Finally , we point out here that in order to maximize the passivit y radius of a system
mo del M , one should maximize the smallest eigen v alue of the scaled matrix c
W ( X , M ). Let
D s = diag( I n , I n , I m / √ 2) and let us scale the inequality (17) with the matrix D s giv en b y
D s c
W ( X , M ) D s + D s E  0 ∆ S
∆ H
S 0  E T D s (20)
where no w D s E is an isometry . It then follo ws that in order to ha v e a p erturbation ∆ S of
norm ρ M ( X ) that mak es (20) singular, w e must ha v e
λ min ( D s c
W ( X , M ) D s ) ≤ ρ M ( X ) . (21)
This b ound expresses that if w e wan t to maximize ρ M ( X ) o v er all X ∈ X > , w e should
try to maximize λ min ( D s c
W ( X , M ) D s ) . The follo wing result sho ws that normalized passiv e
realizations can b e exp ected to ha ve a larger minimal eigen v alue in the matrix D s c
W ( I , M T ) D s
than the corresp onding minimal eigen v alue of the non-normalized matrix D s c
W ( X , M ) D s .
7

Lemma 4.6. L et X ∈ X  then the tr ac e of the matrix
min
det T 6 =0 trace[diag( T , T − H , I m )( D s c
W ( X , M ) D s ) diag( T H , T − 1 , I m )] = trace( D s c
W ( I , M T ) D s )
is minimize d by the matric es T such that X = T H T , while the determinant r emains invariant
det[diag( T , T − H , I m )( D s c
W ( X , M ) D s ) diag( T H , T − 1 , I m )] = det( D s c
W ( I , M T ) D s ) .
Pr o of. Note that transformation applied to D s c
W ( X , M ) D s is a congruence transformation
whic h preserv es the nonnegativit y of its eigen v alues and that the trace of the resulting matrix
is trace Z + trace Z − 1 + 1
2 trace( D H + D ), where Z := T X − 1 T H . It is w ell kno wn that
this is minimized when Z = I . The fact that the congruence transformation preserv es the
determinan t identit y is ob vious.
This lemma suggests that the smallest eigen v alue should increase as the pro duct of all
the eigen v alues remains constan t and their sum is b eing minimized, but this is of course not
guaran teed in general.
5 Maximizing the passivit y radius
In this section w e discuss another LMI in the matrices X > 0 with the same domain as
W ( X , M ) ≥ 0, giv en b y
f
W ( X , M ) := 

X X A X B
A H X X C H
B H X C D H + D

 ≥ 0 .
It is clear that f
W ( X , M ) is congruen t to diag ( X , W ( X , M )) and since X > 0, it has the
same solution set X > as W ( X , M ) ≥ 0. The LMI for the normalized passiv e realization
M T = { T AT − 1 , T B , C T − 1 , D } corresp onding to X = T H T , can b e obtained via a congruence
transformation as w ell
f
W ( I , M T ) := 
 I n A T B T
A H
T I n C H
T
B H
T C T D H
T + D T

 = 
 T − H 0 0
0 T − H 0
0 0 I m

 f
W ( X , M ) 
 T − 1 0 0
0 T − 1 0
0 0 I m

 ≥ 0 .
Let us no w consider the follo wing constrained LMI
f
W ( X , M ) ≥ ξ diag( X , X , 2 I m ) . (22)
Then the follo wing Theorem giv es a b ound on how large w e can c ho ose ξ in this LMI.
Theorem 5.1. L et M := { A, B , C , D } b e a minimal r e alization of a discr ete-time p assive
system, and let X b e any matrix in X > . Then ther e is a unique ξ ∗ ( X ) which is maximal for
the matrix ine quality (22) to hold, and which is strictly smal ler that 1. Mor e over, ξ ∗ ( X ) =
λ min ( D s f
W ( I , M T ) D s ) .
Pr o of. It follo ws from (10) that ev ery X ∈ X > is p ositiv e definite. Therefore it can b e fac-
torized as X = T H T with det T 6 = 0, and w e can consider the normalized system M T =
8

{ T AT − 1 , T B , C T − 1 , D } . It is easy to see that the condition (22) is equiv alen t to the corre-
sp onding LMI condition for the transformed system M T , whic h is giv en by
f
W ( I , M T ) ≥ ξ diag ( I n , I n , 2 I m ) .
The largest v alue ξ ∗ ( X ) of ξ for which this holds is clearly equal to
ξ ∗ ( X ) = max
ξ h ξ | D s f
W ( I , M T ) D s ≥ ξ I 2 n + m i = λ min ( D s f
W ( I , M T ) D s ) . (23)
Since D s f
W ( I , M T ) D s − ξ ∗ I 2 n + m is p ositive semi-definite, its diagonal m ust b e non-negativ e,
and thefore ξ ∗ can not b e larger than 1. Moreov er, ξ ∗ = 1 w ould imply then that A T , B T
and C T w ould b e zero.
Remark 5.2. Note that f
W ( I , M T ) = c
W ( I , M T ). F rom (21) one then obtains the inequalit y
λ min ( D s f
W ( I , M T ) D s ) ≤ ρ M T
whic h sho ws the relev ance of f
W ( I , M T ) in the maximization of the passivit y radius.
The use of the c haracterization ξ ∗ ( X ) := λ min D s f
W ( I , M T ) D s in terms of the LMI (22)
is crucial for the rest of this section. W e also p oin t out that Theorem 5.1 applies to all p oin ts
of X > , and therefore also of X  . But w e can distinguish b et ween both.
Corollary 5.3. The maximal value ξ ∗ ( X ) of a matrix X ∈ X > for a given mo del M e quals
0 if X is a b oundary p oint of X > and is strictly p ositive if and only if X is in X  .
Pr o of. If X is a b oundary p oin t of X > then det W ( X , M ) = 0 and also det f
W ( X , M ) = 0
and for those X , w e th us ha ve ξ ∗ ( X ) = 0. If X b elongs to X  , then f
W ( X , M ) > 0 and
diag( X , X , 2 I m ) > 0. Therefore there exists an ξ > 0 suc h that f
W ( X , M ) > ξ diag ( X , X, 2 I m ),
and hence ξ ∗ ( X ) > 0. Con v ersely , if ξ ∗ ( X ) > 0 then f
W ( X , M ) > 0 and W ( X , M ) > 0 whic h
implies that X ∈ X  .
In order to maximize ξ ∗ ( X ), w e consider for a giv en X in X > the matrix
f
W ( X , M ξ ) := 

X X A ξ X B ξ
A H
ξ X X C H
ξ
B H
ξ X C ξ D H
ξ + D ξ


corresp onding to the mo dified mo del M ξ := { A ξ , B ξ , C ξ , D ξ } := { A
(1 − ξ ) , B
(1 − ξ ) , C
(1 − ξ ) , D − ξ I m
(1 − ξ ) } .
It turns out that this matrix satisfies the iden tit y
(1 − ξ ) f
W ( X , M ξ ) = f
W ( X , M ) − ξ 

X 0 0
0 X 0
0 0 2 I m

 (24)
whic h is crucial for the follo wing Lemma.
Lemma 5.4. F or every X > 0 in X  and any 0 ≤ ξ − < ξ + ≤ ξ ∗ ( X ) , the p assivity LMIs for
the systems M ξ − and M ξ + ar e satisfie d. Mor e over, the solution set of f
W ( X , M ξ + ) ≥ 0 is
include d in the solution set of f
W ( X , M ξ − ) > 0 .
9

Pr o of. The LMIs for t w o differen t v alues of ξ are related as
(1 − ξ 2 ) f
W ( X , M ξ 2 ) = (1 − ξ 1 ) f
W ( X , M ξ 1 ) − ( ξ 2 − ξ 1 ) diag( X , X , 2 I m ) .
Since X ∈ X  , w e ha ve that ξ ∗ ( X ) > 0 and diag( X , X , 2 I m ) > 0. F or that X , it then follo ws
that
f
W ( X , M ) ≥ (1 − ξ − ) f
W ( X , M ξ − ) > (1 − ξ + ) f
W ( X , M ξ + ) ≥ (1 − ξ ∗ ( X )) f
W ( X , M ξ ∗ ( x ) ) ≥ 0 .
(25)
The systems M ξ − and M ξ + are th us passiv e, since their asso ciated LMIs ha v e a nonempty
solution set. No w consider any X for whic h f
W ( X , M ξ + ) ≥ 0. Since ξ + is strictly p ositiv e,
so is ξ ∗ ( X ) and hence X ∈ X  . It then follo ws from (25) that f
W ( X , ξ − ) > 0. Hence, the
solution set of f
W ( X , M ξ + ) ≥ 0 is included in the solution set of f
W ( X , M ξ − ) > 0.
Lemma 5.4 implies that for a giv en X ∈ X  , the solution sets of f
W ( X , M ξ ) ≥ 0 are
shrinking with increasing ξ . But w e still need to find the matrix X ∈ X > that maximizes
ξ ∗ ( X ). W e can answer this question b y relating this to the passivit y of the transfer function
of the mo dified system M ξ ,
T ξ ( z ) := C ξ ( z I n − A ξ ) − 1 B ξ + D ξ ,
whic h is minimal since M was assumed to be minimal. It follo ws from the discussion of
Section 2 that this transfer function corresp onds to a strictly passiv e system if and only if the
conditions (i) the transfer function T ξ ( z ) is asymptotically stable, and (ii) the matrix function
Φ ξ ( z ) := T H
ξ ( z − 1 ) + T ξ ( z ) is strictly p ositiv e on the unit circle e ıω , ω ∈ [ − π , π ], are satisfied.
It has b een sho wn in Section 2 that the zeros of Φ ξ ( z ) are the eigen v alues of the symplectic
matrix
S ξ :=  I n B ξ ( D H
ξ + D ξ ) − 1 B H
ξ
0 ( A ξ − B ξ ( D H
ξ + D ξ ) − 1 C ξ ) H  − 1  A ξ − B ξ ( D H
ξ + D ξ ) − 1 C ξ 0
C H
ξ ( D H
ξ + D ξ ) − 1 C ξ I n  , (26)
whic h are also the finite eigen v alues of the p encil
z 

0 − I n 0
A H
ξ 0 0
B H
ξ 0 0

 + 

0 A ξ B ξ
− I n 0 C H
ξ
0 C ξ D H
ξ + D ξ


or equiv alently , those of the p encil
z 

0 ( ξ − 1) I n 0
A H 0 0
B H 0 0

 + 

0 A B
( ξ − 1) I n 0 C H
0 C D H + D − 2 ξ I m

 (27)
and that the realization of M ξ is minimal. The algebraic conditions corresp onding to strict
passivit y of T ξ ( z ) are therefore
A1. A ξ has all its eigen v alues inside the unit disc (stability),
A2. the p encil (27) has no eigenv alues on the unit circle (p ositiv e realness).
10

These conditions are phrased in terms of eigen v alues of certain matrices that dep end on
the parameter ξ . Since eigen v alues are con tin uous functions of the matrix elemen ts, one can
consider limiting cases for the ab o ve conditions. As explained in Section 2 the passiv e transfer
functions are limiting cases of strictly passiv e ones. Those limiting cases corresp ond to the
v alue of ξ where one of the conditions A1. or A2. do es not hold an ymore.
Theorem 5.5. L et M b e a strictly p assive and minimal system. Then ther e is a b ounde d
supr emum Ξ := sup ξ { ξ | T ξ ( z ) is strictly passiv e } for which the fol lowing pr op erties hold
1. T Ξ ( z ) is p assive,
2. the solution set of f
W ( X , M Ξ ) ≥ 0 is not empty,
3. the solution set of f
W ( X , M Ξ ) > 0 is empty,
4. for any ξ < Ξ the solution set of f
W ( X , M ξ ) > 0 is non-empty,
5. Ξ := sup X ξ ∗ ( X ) for al l X ∈ X > .
Pr o of. The existence of a b ounded suprem um follo ws from the fact that T ξ ( z ) is strictly
passiv e only if ξ is smaller than 1 (see Theorem 5.1). Prop ert y 1. holds b ecause T Ξ ( z ) is
the limit of T ξ ( z ) for ξ → Ξ. Prop ert y 2. is a direct consequence of the previous prop ert y .
Prop ert y 3. follo ws b y con tradiction : if f
W ( X , M Ξ ) > 0 w ould not b e empt y , then ξ ∗ ( X )
for X in the domain of f
W ( X , M Ξ ) > 0, w ould b e larger that Ξ. Propert y 4. follo ws from
Lemma 5.4 where w e use any X in the domain of f
W ( X , M Ξ ) ≥ 0 and c ho ose ξ + = (Ξ + ξ ) / 2
and ξ − = ξ to sho w that X also lies in the domain of f
W ( X , M ξ ) > 0. Prop ert y 5. follo ws
from ξ ∗ ( X ) = max { ξ | f
W ( X , M ξ ) ≥ 0 } , whic h expresses that T ξ ( z ) is passiv e.
The follo wing theorem discusses the optimal passivity radius o v er all realizations of T ( z ).
Theorem 5.6. L et M := { A, B , C, D } b e a minimal r e alization of a strictly p assive tr ansfer
function T ( z ) := C ( z I − A ) − 1 B + D . Then
Ξ := sup
ξ { ξ | T ξ ( z ) is strictly passiv e }
is a lower b ound for the lar gest p ossible p assivity r adius within the set of al l r e alizations of
T ( z ) . Mor e over, normalize d r e alizations M T := { T − 1 AT , B T , T − 1 C , D } , wher e X := T H T
c orr esp onds to a solution X of f
W ( X , M Ξ ) ≥ 0 , have a p assivity r adius ρ M T lar ger than or
e qual to Ξ .
Pr o of. Consider realizations M T := { T − 1 AT , B T , T − 1 C , D } with X := T H T and X ∈
f
W ( X , M ) ≥ 0. It w as sho wn in Theorem 5.1 that for the corresp onding realization M T ,
w e ha v e that ξ ∗ ( X ) = λ min ( D s f
W ( I , M T ) D s ). Theorem 5.5 then sho ws that for a solu-
tion X of f
W ( X , M Ξ ) ≥ 0 corresp onding to the suprem um of all ξ ∗ ( X ), w e ha v e Ξ =
λ min ( D s f
W ( I , M T ) D s ). The lo w er b ound Ξ ≤ ρ M T then follows from Remark 5.2 and Lemma
4.2.
In Figure 1, we generated ran dom normalized passiv e systems and computed the follo wing
quan tities (using γ g m := p β /α as defined in App endix B):
11

Figure 1: Relativ e accuracies of four estimates of the passivit y radius of a random system :
λ min W ( I , M T ), λ min f
W ( I , M T ), λ min D s f
W ( I , M T ) D s , and E st = k [ γ g m N 1 | N 2 /γ g m ] k 2
2
1. The passivity radius ρ M T , computed to 4 digits of accuracy ,
2. λ min W ( I , M T ).
3. λ min f
W ( I , M T ),
4. λ min D s f
W ( I , M T ) D s which is a lo w er b ound for ρ M T ,
5. E st := k [ γ g m N 1 | N 2 /γ g m ] k 2
2 whic h is also a lo w er b ound for ρ M T .
In Figure 1 w e depict the quan tities (2.-5.) divided b y ρ M T to indicate their relative
b ounds. It can b e seen that the eigen v alues are in the in terv al
1
2 ρ M T ≤ λ min W ( I , M T ) , λ min f
W ( I , M T ) ≤ 2 ρ M T
and that
1
2 ρ M T ≤ λ min D s f
W ( I , M T ) D s ≤ ρ M T , k [ γ g m N 1 | N 2 /γ g m ] k 2
2 ≈ ρ M T .
Figure 1 indicates that 1 /g ( γ g m ) ≤ ρ M ( X ) is a v ery go o d estimate of the passivit y radius
(within 1% of the correct v alue) and that the b ound λ min ( D s f
W D s ) ≤ ρ M ( X ) holds.
6 A scalar example
In this section w e analyze a simple first order discrete-time scalar system. Its transfer function
T ( z ) = d + cb
z − a is asymptotically stable if a 2 < 1. Then
W ( x ) =  x − a 2 x c − abx
c − abx 2 d − b 2 x 
12

and the ro ots x − , x + of the quadratic p olynomial det W ( x ) = (1 − a 2 ) x (2 d − b 2 x ) − ( c − abx ) 2
happ en to b e the extremal solutions of the asso ciated Riccati equations. The set X > where
W ( x ) ≥ 0 is th us just the in terv al [ x − , x + ], pro vided these t w o ro ots are real. This p olynomial
can b e rewritten as
det W ( x ) = − b 2 x 2 + 2 β x − c 2 , where β := (1 − a 2 ) d + abc
and it has t w o real ro ots iff β 2 ≥ ( bc ) 2 or | β / ( bc ) | = | (1 − a 2 ) d
bc + a | ≥ 1.
The normalized passiv e realizations are those where w e normalize x to 1 b y the transfor-
mation that scales { a, b, c, d } to { a, b.t, c/t, d } , where x = t 2 ∈ [ x − , x + ]. In Figure 2 we sho w
a plot of the passivit y radius of the realizations M t := { a, b.t, c/t, d } as a function of t , and
also the follo wing quan tities:
• The true passivit y radius ρ M t := 1 /λ max M ( γ ∗ ) defined in Section 4,
• λ min W ( I , M t ) which is giv en in Section 2,
• λ min D s f
W ( I , M t ) D s which is a lo w er b ound for ρ M t ,
• the v alues of b t := b.t and c t := c/t .
Figure 2: Estimates λ min W ( I , M t ) and λ min D s f
W ( I , M t ) D s for the passivity radius ρ M t of
a scalar normalized system M t := { a, b.t, c/t, d } as function of t .
It is in teresting to see that the lo w er b ound λ min D s f
W ( I , M t ) D s is almost identical to
ρ M t for the scalar case and that the optim um is reac hed when b t = c t , so that
f
W = 

1 a b t
a 1 b t
b t b t 2 d − b 2
t .


13

W e sho w in App endix C that in this case no other realization has a b etter passivit y radius.
It is also w orth p oin ting out that λ min W ( I , M t ) reac hes is optim um v alue at another v alue
of t , but that ρ M t is nearly optimal at that p oin t.
7 Computing the largest v alue of ξ ∗ ( X )
In this section w e describ e an algorithm that computes, within a given tolerance τ , an ap-
pro ximation of the suprem um Ξ (see Theorem 5.5) of a given minimal realization M :=
{ A, B , C , D } that is passiv e.
First of all, if M is passiv e but not strictly passiv e then Ξ = 0. If M is strictly passiv e,
then a simple upp er b ound for Ξ follo ws by the stabilit y b ound
Ξ up = 1 − max
j | λ j ( A ) | .
The pro cedure to compute Ξ is then to v erify for 0 ≤ ξ ≤ Ξ up the second condition, namely
that the p encil (27) has no unit circle eigen v alues. The smallest v alue of ξ in this in terv al
where this condition fails, equals Ξ. (Note that this could b e equal to Ξ up .) One can then
apply a bisection metho d to this in terv al and c hec k the presence of unit circle eigen v alues in
the giv en in terv al. Setting Ξ lo = 0, w e then ha v e the follo wing pro cedure.
Bisection pro cedure for computing Ξ
ξ := (Ξ lo + Ξ up ) / 2 , if S ξ has unit circle eigen v alues then Ξ up := ξ , else Ξ l o := ξ .
Since the in terv al con taining Ξ shrinks b y a factor 2 in eac h step of this iteration, in
k = d log 2 (Ξ up /τ ) e steps, the in terv al [Ξ lo , Ξ up ] will b e of length less than or equal to τ .
One can also mak e use of the computed eigen v alue decomp ositions to construct an algo-
rithm with faster con v ergence. F or this, w e consider the generalized eigen v alue problem
Γ( ξ , ω ) := 

0 e ıω ( ξ − 1) I n + A B
A H + e − ıω ( ξ − 1) I n 0 C H
B H C D H + D − ξ I m

 ,
whic h is He rmitian for all real v alues of ω and ξ < 1. F or a giv en v alue of ˆ
ξ one can c heck
if Γ( ˆ
ξ , ω ) has real eigen v alues ω i (they corresp ond to unit circle eigen v alues of S ˆ
ξ ), and for
a giv en v alue of ˆ ω one can find the smallest real ro ot ξ i of Γ( ξ , ˆ ω ). These t w o ideas can b e
com bined in an algorithm for computing Ξ that is very similar to the computation of the H ∞
norm of a transfer function.
W e first recall some basic prop erties of the scalar function γ ( ξ , ω ) := λ min Γ( ξ , ω ), which
can b e deriv ed from the results describ ed in [4] and from the prop erties of eigen v alues of
Hermitian matrices.
1. γ ( ξ , ω ) is a real con tin uous function of the real v ariables ξ and ω ,
2. if γ ( ˆ
ξ , ω ) > 0 for all ω then ˆ
ξ < Ξ,
3. for ˆ
ξ < Ξ up the real zeros ω k of γ ( ˆ
ξ , ω ) corresp ond to a subset of the unit circle eigen-
v alues e ıω k of Γ( ˆ
ξ , ω ),
4. for a given v alue of ˆ
ξ , γ ( ˆ
ξ , ω ) is a quadratic function of ω in the neigh b orho o d of its
lo cal minima,
14

5. if ω 1 < ω 2 are tw o consecutiv e zeros of γ ( ˆ
ξ , ω ) then at the midp oin t ˆ ω := ( ω 1 + ω 2 ) / 2
the smallest real ro ot e
ξ of Γ( ξ , ˆ ω ) lies b et w een 0 and ˆ
ξ and is an impro v ed upp er b ound
for Ξ.
These ideas lead to the follo wing impro ved algorithm for the computation of Ξ.
Eigen v alue based pro cedure for computing Ξ
1. ˆ
ξ := Ξ up − τ ;
2. Compute the unit circle eigen v alues e ıω k of Γ( ˆ
ξ , ω ) and select those corresp onding to
real zeros ω k of γ ( ˆ
ξ , ω );
3. if γ ( ˆ
ξ , ω ) has no real zeros, then Ξ lo = ˆ
ξ , stop;
else tak e the midp oin t ˆ ω := ( ω 1 + ω 2 ) / 2 of the largest in terv al [ ω 1 , ω 2 ] of these ro ots
compute the real ro ots ξ i of Γ( ξ , ˆ ω ) and up date Ξ up := min i ξ i
mak e a guess for ˆ
ξ := Ξ up − τ and go to 2.
This algorithm is v ery similar to the metho ds prop osed in the literature for computing
the H ∞ norm of a transfer function (see e. g., [4]) and can therefore b e exp ected to require
only a few iterations to stop with an in terv al [Ξ l o , Ξ up ] of size τ .
Note that eac h step of b oth algorithms has a complexit y that is cubic in the matrix
dimensions. F or large scale problems, this complexit y b ecomes a problem, but there are
tec hniques that exploit sparsit y to reduce the complexit y , see e. g., [3, 12].
8 The distance to passivit y
In this section, w e consider the con v erse problem of computing the smallest p erturbation that
mak es a system passive. Supp ose that w e are giv en a minimal system M := { A, B , C , D }
that is not passiv e. Then w e study the problem of computing the smallest p erturbation ∆ M
of the mo del M that mak es the system M + ∆ M passive. It is clear that this is equiv alen t
to asking whic h is the smallest p erturbation ∆ M , measured via the matrix ∆ S in (14), such
that the LMI W ( X , M + ∆ M ) ≥ 0 has a Hermitian and p ositiv e semi-definite solution X .
Moreo ver X > 0 if the p erturb ed system remains minimal.
Definition 8.1. The distance to passivit y of a minimal mo del M := { A, B , C, D } is the
minimum norm k ∆ S k 2 or k ∆ S k F such that ther e exists a matrix X > 0 satisfying
c
W + E  0 ∆ S
∆ H
S 0  E T ≥ 0 , where c
W := 

X − 1 A B
A H X C H
B H C D H + D

 , (28)
and E is define d in (16) .
Note that (28) is an LMI in the parameters of ∆ M , but it is not linear in X . W e will
need the follo wing extension of Lemma 5.4, for whic h w e consider the LMI for the mo dified
mo del M − ξ := { A − ξ , B − ξ , C − ξ , D − ξ } := { A
(1+ ξ ) , B
(1+ ξ ) , C
(1+ ξ ) , D + ξ I m
(1+ ξ ) } with the corresp onding
transfer function
T − ξ ( z ) := C − ξ ( z I n − A − ξ ) − 1 B − ξ + D − ξ ,
15

and corresp onding LMI
f
W ( X , M − ξ ) := 

X X A − ξ X B − ξ
A H
− ξ X X C H
− ξ
B H
− ξ X C − ξ D H
− ξ + D − ξ

 ≥ 0 . (29)
Lemma 8.2. L et M := { A, B , C , D } b e a minimal non-p assive system. Then for every
X > 0 in H n ther e exists a ξ ∗ ( X ) > 0 such that the LMI (29) for the system M − ξ ∗ ( X ) holds.
Mor e over, for every value ξ > ξ ∗ ( X ) , the system M − ξ is p assive.
Pr o of. W e hav e the relation (1 + ξ ) f
W ( X , M − ξ ) = f
W ( X , M ) + ξ diag( X , X , 2 I m ), and since
f
W ( X , M ) is b ounded, the inequalit y f
W ( X , M − ξ ) ≥ 0 holds for a sufficien tly large v alue of
ξ . Let ξ ∗ ( X ) b e the smallest v alue for whic h the passivit y condition (29) holds, then
(1 + ξ ) f
W ( X , M − ξ ) = (1 + ξ ∗ ( X )) f
W ( X , M − ξ ∗ ( X ) )+( ξ − ξ ∗ ( X )) diag( X , X , 2 I m ) ,
whic h implies that the passivit y condition holds for all ξ > ξ ∗ ( X ).
T o determine the distance to passivit y , w e first restrict ourselv es to a p erturbation ∆ S
that has a particular structure.
Theorem 8.3. The minimum norm p erturb ation of the typ e
S + ∆ S = 1
(1 + ξ )  S +  0 0
0 ξ I m  (30)
that makes the system M p assive, c orr esp onds to the minimal value of ξ such that the mo del
M − ξ := { A
(1+ ξ ) , B
(1+ ξ ) , C
(1+ ξ ) , D + ξ I m
(1+ ξ ) } with tr ansfer function T − ξ ( z ) , is p assive.
Pr o of. It follo ws from (28) that ξ m ust satisfy the LMI (29) for some X > 0. By Lemma
8.2 there exists a b ounded minimal solution, whic h w e call Ξ. The mo del corresp onding to
S + ∆ S is M − ξ with transfer function (8). Therefore Ξ is the smallest v alue of ξ that mak es
the mo del M − ξ with transfer function T − ξ ( z ) b ecome passiv e. W e can then choose X > 0
from the domain of f
W ( X , M − Ξ ) ≥ 0 to satisfy (29).
The minimal v alue Ξ in Theorem 8.3 can b e computed with the algorithms describ ed in the
last section. It thus determines that passivit y radius for the constrained class of p erturbations
(30).
Since w e most lik ely made some of the eigen v alues of the LMI (29) strictly p ositiv e, rather
than nonnegativ e, w e can probably reduce the norm of the p erturbation ∆ S when remo ving
the constrain t (30). In order to do that, w e use a matrix X from the set f
W ( X , M − Ξ ) ≥ 0,
where Ξ w as obtained from the constrained problem. But once X is fixed, condition (28)
b eomes an LMI in the unkno w n perturbation ∆ S . W e can then minimize its 2-norm σ b y
solving the optimization problem
min
∆ S
σ , s.t.  σ I n + m ∆ S
∆ H
S σ I n + m  ≥ 0 , c
W + E  0 ∆ S
∆ H
S 0  E T ≥ 0 ,
or its F rob enius norm ˆ σ b y solving
min
∆ S
ˆ σ , s.t.  ˆ σ I ( n + m ) 2 v ec(∆ S )
v ec(∆ S ) H ˆ σ  ≥ 0 , c
W + E  0 ∆ S
∆ H
S 0  E T ≥ 0 .
16

Notice that the constrained problem of Theorem 30 pro vided a feasible starting v alue ∆ S for
these optimization problems. W e could also use another matrix X that is not in the solution
set of f
W ( X , M − Ξ ) ≥ 0, but then the norm of the starting p oin t ∆ S constructed from the
constrained problem w ould b e larger since ξ ∗ ( X ) > Ξ.
Remark 8.4. The same reasoning on ho w to compute the distance to the nearest passiv e
system can b e applied to estimate the distance of a system x k +1 = Ax k that is unstable to
the nearest stable system, see also [8, 9] for the con tin uous-time case. A result analogous to
Theorem 8.3 w ould giv e that a solution of the type
A + ∆ A = A/ (1 + ξ )
has a relativ e error A − 1 ∆ A with 2-norm Ξ
(1+Ξ) and F rob enius norm Ξ √ n
(1+Ξ) , where Ξ is the
minim um v alue of ξ suc h that the matrix A − ξ := A/ (1 + Ξ) is stable, and this can b e used
to find an appropriate matrix X for an LMI in ∆ A .
9 Conclusion
In this pap er w e ha v e in tro duced the notion of normalized passiv e realizations of a discrete-
time system and sho wn that they share prop erties with the normalized p ort-Hamiltonian
realizations of a con tinuous-time system in tro duced in [15]. W e also sho w ed that the normal-
ized passiv e realizations typically ha v e a b etter passivit y radius than non-normalized ones.
W e ha v e deriv ed metho ds to maximize a lo w er b ound on the passivit y radius and to construct
a nearly optimal ly r obust normalize d r e alization . The tec hniques dev elop ed in this pap er can
also b e applied to compute a nearb y passive system to a giv en non-passiv e one.
Ac kno wledgmen ts
This w ork w as supp orted b y the Deutsche F orschungsgemeinschaft , through TRR 154 ’Math-
ematical Mo delling, Sim ulation and Optimization using the Example of Gas Netw orks’. The
first author w as also supp orted b y the German F e der al Ministry of Educ ation and R ese ar ch
BMBF within the pr oje ct EiF er . The second author w as also supp orted b y the Belgian net-
w ork D YSCO, funded b y the In teruniv ersity A ttraction P oles Programme.
10 App endix A
Consider the unimo dal optimization problem
g ( γ ∗ ) := min
0 <γ < ∞ g ( γ ) , where g ( γ ) := k [ γ F 1 , F 2 /γ ] k 2
2 , F i ∈ C (2 n + m ) × ( n + m ) . (31)
If w e define α := k F 1 k 2 and β := k F 2 k 2 then it was already sho wn in Theorem 4.5 that
αβ ≤ g ( γ ∗ ) ≤ 2 αβ . W e can then deriv e the following result.
Lemma 10.1. The infinite se ar ch interval for γ in the minimization pr oblem (31) c an b e
r eplac e d by the close d interval γ ∈ [ γ lo , γ up ] :=  q β
2 α , q 2 β
α  . Mor e over, the function value
g ( γ g m ) at the ge ometric me an γ g m := q β
α is an upp er b ound for the minimum.
17

Pr o of. It is easy to see that g ( γ ) > 2 αβ outside the interv al γ ∈ [ γ lo , γ up ] and, since g ( γ ∗ ) ≤
2 αβ , the minimum m ust lie in the in terv al γ ∈ [ γ l o , γ up ]. An y function v alue in this in terv al
is of course an upp er b ound for the minim um.
11 App endix B
In this app endix w e describ e another c haracterization of ρ M ( X ). F or this, w e consider the
iden tity
M ( Q ) :=  F 1 F 2   0 Q
Q H 0   F H
1
F H
2  =  γ F 1 F 2 /γ   0 Q
Q H 0   γ F H
1
F H
2 /γ  ,
whic h holds for every real γ > 0 and ev ery nonsingular matrix Q . If w e constrain Q to b e
unitary , i. e. QQ H = Q H Q = I , then it follo ws that
h ( Q ) := k M ( Q ) k 2 ≤ g ( γ ∗ ) := min
0 <γ < ∞ g ( γ ) , g ( γ ) := σ 2
max [ γ F 1 , F 2 /γ ] = k [ γ F 1 , F 2 /γ ] k 2
2 .
W e no w pro v e that w e also ha v e
g ( γ ∗ ) = h ( Q ∗ ) := max
QQ H = Q H Q = I h ( Q ) = max
QQ H = Q H Q = I k M ( Q ) k 2 , (32)
whic h we pro v e b y constructing a matrix Q so that (32) holds. It follows from Theorem 4.3
that the minimizing righ t singular v ector z :=  u
v  satisfies
 γ F 1 F 2 /γ   u
v  = σ max w ,  γ F H
1
F H
2 /γ  w = σ max  u
v  , k u k 2 = k v k 2 .
It is then easy to v erify then that for a unitary Q satisfying Qv = u and Q H u = v , then
M ( Q ) z = σ 2
max z .
W e can no w use this construction to sho w that normalized realizations ha v e a b etter
passivit y radius than non-normalized ones. Let c
W = R H R b e the Cholesky factorization of
an arbitrary mo del M . The Cholesky factorization of the corresp onding matrix
c
W n = diag( T − 1 , T H , I m ) c
W diag( T − H , T , I m )
of the normalized mo del M T := { T − 1 AT , T − 1 B , C T , D } is then giv en b y
c
W n := R H
n R n = diag( T − 1 , T H , I m ) R H R diag ( T − H , T , I m )
and the relation R − H = R − H
n diag( T − 1 , T H , I m ) then yields
F 1 := R − H E 1 = R − H
n E 1 diag( T − 1 , I m ) = N 1 diag ( T − 1 , I m ) ,
F 2 := R − H E 2 = R − H
n E 2 diag( T H , I m ) = N 2 diag ( T H , I m ) .
It then follo ws from (32) that
ρ − 1
M ( X ) = min
0 <γ < ∞ k  γ F 1 F 2 /γ   γ F H
1
F H
2 /γ  k 2
≥ k  γ F 1 F 2 /γ   0 I n
I n 0   γ F H
1
F H
2 /γ  k 2
= k  N 1 N 2   0 I n
I n 0   N H
1
N H
2  k 2
= k N 1 N H
2 + N 2 N H
1 k 2 .
18

Note that
h ( I ) = k N 1 N H
2 + N 2 N H
1 k 2 = k R − H 

0 I n 0
I n 0 0
0 0 2 I m

 R − 1 k 2 ≤ 2 k N 1 k 2 k N 2 k 2
but w e need a lo wer bound for ρ − 1
M T ( X ). If c
W n comm utes with J := 

0 I n 0
I n 0 0
0 0 I m

 , whic h
implies that  A T B T  =  A H
T C H
T  and hence that M T is its o wn dual system, then





 ( R − H 

0 I n 0
I n 0 0
0 0 2 I m

 R − 1 ) 2 




 2
= 




 ( R − H 

I n 0 0
0 I n 0
0 0 2 I m

 R − 1 ) 2 




 2
= k ( D − 1
s c
W − 1
n D − 1
s ) 2 k 2 .
In this case it follo ws that
ρ M ( X ) ≤ λ min ( D s c
W n D s ) ≤ ρ M T ( X ) ,
whic h implies that suc h a normalized realization has a b etter passivit y radius than the cor-
resp onding non-normalized realization.
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