Alexandre Seuret

Matthew M Peet

Stability analysis of sampled-data systems using Sum of Squares

Keywords: Sampled-Data systems, Lyapunov function, Sum of Squares

This article proposes a new approach to stability analysis of linear systems with sampled-data inputs or channels. The method, based on a variation of the discrete-time Lyapunov approach, provides stability conditions using functional variables subject to convex constraints. These stability conditions can be solved using the sum of squares methodology with little or no conservatism in both the case of synchronous and asynchronous sampling. Numerical examples are included to show convergence.

I. INTRODUCTION

In recent years, much attention has been paid to Networked Control Systems (NCS) (see [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], [START_REF] Zampieri | A survey of recent results in Networked Control Systems[END_REF]). These systems contain several distributed controllers and plants which are connected through a communication network. In this model, the inputs to the plants are updated as often as the network capacity allows (sampling times) -ideally once per round-trip-time. In such applications, however, a heavy transient load on the network or on a processor can change the sampling period of the controller. To compensate for this effect, the sampling period is sometimes explicitly included in the controller design. However, this approach is problematic in that the sampling period is difficult to predict and variation of this sampling period may destabilize the closed-loop system. The result is that a significant effort has been made to develop stability conditions for sampleddata systems which are robust with respect to variations of sampling period.

Sampled-data systems have been extensively studied in the literature [START_REF] Chen | Optimal sampled-data control systems[END_REF]- [START_REF] Zhang | Stability of networked control systems[END_REF] and the references therein. It is now possible to design controllers which guarantee the robustness of the solutions of the closed-loop system under periodic samplings. However in the case of asynchronous sampling, there are still several open problems. For example, consider the practical situation where the difference between two successive sampling instants is not constant but timevarying. Recently, several articles have addressed the problem of time-varying periods based on a discrete-time approach, [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF]- [START_REF] Hetel | Stabilization of arbitrary switched linear systems with unknown time-varying delays[END_REF]. Recent papers have considered the modeling of continuous-time systems with sampled-data control in the form of continuous-time systems with delayed control input. In [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], a Lyapunov-Krasovskii approach was introduced. Improvements were provided in [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-andhold devices[END_REF], [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF], using the small gain theorem, and in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], based on the analysis of impulsive systems. These approaches dealt with time-varying sampling periods as well as with uncertain systems (see [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF] and [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF]). Despite much progress, however, all these conditions are conservative. This means that the sufficient conditions obtained using continuous time approaches are not able to guarantee asymptotic stability for certain stable systems. Recently several authors [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF]- [START_REF] Seuret | Stability analysis for sampled-data systems with a timevarying period[END_REF] refined those approaches and obtained tighter conditions.

The key insight of this paper is that once we have developed the discrete-continuous Lyapunov conditions sufficient for stability, then these conditions can be verified computationally using recently developed algorithms for the optimization of polynomial functions. In particular, we use the tools developed in [START_REF] Peet | Positive forms and stability of linear time-delay systems[END_REF] to reformulate the stability question as a convex optimization problem with polynomial variables. We then use the software package SOSTOOLS [START_REF] Prajna | Introducing SOS-TOOLS: a general purpose sum of squares programming solver[END_REF] to solve the optimization problem. As can be seen in the numerical examples, the result is a sequence of stability tests of increasing accuracy. Furthermore, in the numerical examples, the accuracy of the stability test approaches the analytical limit exponentially fast as the degree of the polynomial variables increases.

This article is based on a Lyapunov approach introduced in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. This previous result was based on the discrete-time Lyapunov theorem and was expressed using a continuous-time model of sampleddata systems. More precisely, this article analyzed the link between the discrete-time Lyapunov theorem employed, for instance in [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF]- [START_REF] Hetel | Stabilization of arbitrary switched linear systems with unknown time-varying delays[END_REF], and the continuous-time approach proposed in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF], [START_REF] Seuret | Stability analysis for sampled-data systems with a timevarying period[END_REF]. In previous work, asymptotic stability criteria can be found for both synchronous and asynchronous samplings. Those criteria were expressed in terms of linear matrix inequalities. The main contribution of this paper is the use of sum of squares tools to construct more sophisticated Lyapunov functions and consequently prove larger upper-bounds for the maximum allowable sampling period than bounds existing in the literature (based on the continuoustime modeling).

This article is organized as follows. The next section formulates the problem. Section III presents a theorem on exponential stability of sampled-data systems expressed in terms of sum of squares. Section IV provides some details on the implementation of the stability conditions using SOSTOOLS. Some examples and simulations are provided in Section V and show the efficiency of the method. Section VI concludes the article.

Notation : Throughout the article, the sets N, R + , R n , R n×n and S n denote respectively the set of natural numbers, nonnegative real numbers, the set of n-dimensional real-valued vectors, the set of n ×n real valued matrices and the subspace of R n×n of symmetric matrices. The superscript 'T ' stands for the matrix transposition. The notation P ≻ 0 for P ∈ S n means that P is positive definite. The symbols I and 0 represent the identity and the zero matrices of appropriate dimension. For any matrix A ∈ R n×n , the notation He(A) stands for A + A T . We define a functional as any map whose domain includes a function space.

II. PROBLEM FORMULATION

Consider the following sampled-data system

ẋ(t) = Ax(t) + Bu(t k ) ∀t ∈ [t k , t k+1 ), (1)
where x ∈ R n and u ∈ R m represent the state and the input vectors. Define the sampling times {t k } k∈N to be an increasing sequence of positive scalars such that k∈N [t k , t k+1 ) = [0, + ∞). Suppose that the sampling intervals, T k , are bounded so that there exist positive scalars T min ≤ T max such that

T k := t k+1 -t k ∈ [T min , T max ], ∀k ∈ N. (2)
The sequence {t k } k∈N represents the sampling instants of the controller. The matrices A and B are constant, known, and of appropriate dimension. The control law is chosen to be linear state feedback,

u = Kx (3)
with a gain K ∈ R m×n . Then the closed-loop system is governed by

ẋ(t) = Ax(t) + BKx(t k ) ∀t ∈ [t k , t k+1 ). (4)
The differential equation ( 4) with the control law (3) can be integrated over a sampling period. If we define the function

Γ(s) = e As + s 0 e A(s-θ ) dθ BK ∀s ∈ [0, T max ],
we have the following discrete-time system

x(t k+1 ) = Γ(T k )x(t k ), (5)
where

x(t) = Γ(t -t k )x(t k ) for t ∈ [t k ,t k+1 ).
Notation: Taking a cue from time-delay systems theory, we denote the segment of solution on t ∈ [t k ,t k+1 ) by x T k , so that

x T k (s) = Γ(s)x(t k ) ∀s ∈ [0, T k ).
We use K n to denote the space of continuous maps from [0, T max ] → R n , where recall T max is the upper-bound on the T k . This function-based model for sampled-data systems was introduced in [START_REF] Yamamoto | A function space approach to sampled data control systems and tracking problems[END_REF].

If T k is constant, the discrete dynamics become x(t k+1 ) = Γ(T )x(t k ), where T is the sampling period. A simple method to check stability of the system is to ensure that Γ(T ) has all eigenvalues inside the unit circle. If T k is time-varying, then it is not sufficient to verify that Γ(T k ) has all eigenvalues inside the unit circle for all T k ∈ [T min , T max ] to prove stability. However, it is still possible to apply the Lyapunov Theorem for discrete-time systems, i.e. find a positive definite matrix P such that Γ T (T k )PΓ(T k ) -P ≺ 0 for all T k ∈ [T min , T max ]. This is an infinite dimensional problem in that we must perform an eigenvalue test at an infinite number of points. It is well-known that verifying such infinite-dimensional stability conditions can prove difficult. If the system contains uncertainty in system parameters such as the sampling period, the difficulty increases. Nonetheless, several authors have successfully investigated this approach to stability analysis [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF], [START_REF] Oishi | Stability and stabilization of aperiodic sampled-data control systems: An approach using robust linear matrix inequalities[END_REF], [START_REF] Hetel | Robust stability and control of switched linear systems[END_REF].

Noting that sampled-data systems are a special case of timedelay systems with time-varying delay, many authors treat sampleddata systems in a manner similar to time-delay systems and use similar approaches to the question of stability. For example, sufficient conditions for stability of sampled-data systems were derived in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF] by analyzing stability of a class of systems with time-varying delay. However, these results were somewhat conservative in that they did not account for the unique structure of the variation in delay in a sampled-data system. In [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], the authors introduce a new type of Lyapunov-Krasovskii functional which depends more explicitly on the delay function. In particular, they use the fact the derivative of the delay function representing the effect of the sampling is equal to 1 almost everywhere in their formulation. This led to improvement in the accuracy of the stability conditions. In the present article , we take a different approach which does not model the hold as a delay, but rather uses a new type of sampled-data Lyapunov Theorem introduced in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] and inspired by [START_REF] Peet | Positive forms and stability of linear time-delay systems[END_REF]. The conditions are enforced using sum of squares optimization.

III. STABILITY OF SAMPLED-DATA SYSTEMS

A. Main Theorem

In this section we introduce a new Lyapunov theorem which applies to general linear sampled-data systems. This theorem accounts for the interaction between the continuous and discrete states of a sampleddata system. A version of this result was introduced in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] and was partially inspired by the concept of spacing functions introduced in [START_REF] Peet | Positive forms and stability of linear time-delay systems[END_REF]. Essentially, the theorem says that if there exists a Lyapunov function which has a net decrease over every sampling interval, then there exists a storage function which is continuously decreasing for all time. We assume global existence and continuity of solutions.

Theorem 1: [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] For given positive scalars α, 0 < T min ≤ T max , suppose V : R n → R + satisfies the following for 0

< µ 1 < µ 2 µ 1 |x| 2 ≤ V (x) ≤ µ 2 |x| 2 , for all x ∈ R n . ( 6
)
Assume that the sampling interval, T k , satisfies (2), then the following two statements are equivalent.

(i) There exists a positive constant ε, such that for all solutions x of (4), and for all k ≥ 0,

V (x(t k+1 )) -e -2αT k V (x(t k )) < -ε|x(t k )| 2 .
(ii) There exist a positive constant δ , continuous functionals

Q k : R×K n → R, differentiable over [t k t k+1 ] which satisfy, for all k ≥ 0 and for all z ∈ K n Q k (T k , z(•)) = e -2αT k Q k (0, z(•)), (7)
and such that for all solutions of system (4), and for all t

∈ [t k ,t k+1 ] d dt [V (x(t)) + Q k (t -t k , x T k )] +2αV (x(t)) + 2αQ k (t -t k , x T k ) < -δ x T k . (8)
where

x T k = max τ∈[0, T max ] |x T k (τ)|.
Moreover, if either of these statements is satisfied, then system (4) is exponentially stable about the origin with a guaranteed decay rate α.

Proof: The proof of this theorem can be found in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. There are several articles in the literature which use related approaches (see for instance [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF]). Typically, these results are derived from a Lyapunov-Krasovskii-type theorem which requires positive definiteness of the functional. In the above result, positivity is relaxed through the use of the spacing functional Q.

The following sections show how the conditions of Theorem 1 can be enforced using sum of squares optimization for exponential stability in both the synchronous and asynchronous case. This is similar to the approach taken in [START_REF] Peet | Positive forms and stability of linear time-delay systems[END_REF].

B. Stability under asynchronous sampling

The case of asynchronous sampling, where T k is unknown but bounded in some range, is clearly more realistic than the synchronous case in a networked control scenario.

ẋ(t) = Ax(t) + BKx(t k ), for t ∈ [t k ,t k + T k ), k ≥ 0, (9)
where T k , for all k ≥ 0 is a time-varying positive parameter in [T min , T max ]. The following theorem gives conditions for stability. The conditions of the theorem can be enforced using sum of squares, as will be described shortly. Theorem 2: Consider System (4). For given α > 0 and 0 < T min < T max < ∞, if there exist P ∈ S n , positive definite and a bi-

polynomial matrix M : [0, T max ] × [T min , T max ] → S 2n such that for all T ∈ [T min , T max ], P ≻ 0, I n I n T M(0, T ) I n I n = 0, M(T, T ) = 0, (10)
and such that for all τ ∈ [0, T ] and for all T ∈ [T min , T max ], the following inequality holds

Ψ α (τ, T ) ≺ 0,
where

Ψ α (τ, T ) = Ψ(τ, T ) + 2α 0
I n P 0 I n T + 2αM(τ, T ) Ψ(τ, T ) = He   0 I n P (BK) T A T T   +He M(τ, T ) 0 0 BK A + ∂ ∂ τ M(τ, T ) (11)
Then if T k ∈ [T min , T max ] for all k ≥ 0, the closed loop system is exponentially stable about the origin with the guaranteed decay rate α.

τ ∈ [0, T k ] and z ∈ K n Q(τ, T k , z) = z(0) z(τ) T M(τ, T k ) z(0) z(τ) .
First, from [START_REF] Hetel | Stabilization of arbitrary switched linear systems with unknown time-varying delays[END_REF], we note that

Q(0, T k , z) = z(0) z(0) T M(0, T k ) z(0) z(0) = z(0) T I n I n T M(0, T k ) I n I n z(0) = 0. Furthermore, Q(T k , T k , z) = z(0) z(T ) T M(T k , T k ) z(0) z(T ) = 0. Therefore, we have Q(T k , T k , z) = Q(0, T k , z) = 0 and hence condition (7) is satisfied.
Computing the derivative term (8), we get

d dt [V (x(t)) + Q(t -t k , T k , x T k )] = ẋT (t)Px(t) + x(t) T P ẋ(t) + d dt   x T k (0) x T k (t -t k ) T M(t -t k , T k ) x T k (0) x T k (t -t k )   = x T (t)P ẋ(t) + ẋT (t)Px(t) +2 0 ẋ(t) T M(t -t k , T k ) x(t k ) x(t) + x(t k ) x(t) T d dt M(t -t k , T k ) x(t k )
x(t) .

Recalling that ẋ(t) = Ax(t) + BKx(t k ), we get

d dt [V (x(t)) + Q(t -t k , T k , x T k )] = 2 x(t k ) x(t) T 0 I n P BK A x(t k ) x(t) +2 x(t k ) x(t) T 0 0 BK A T M(t -t k , T k ) x(t k ) x(t) + x(t k ) x(t) T Ṁ(t -t k , T k ) x(t k ) x(t) = x(t k ) x(t) T Ψ(t -t k , T k ) x(t k ) x(t) , for all t ∈ [t k ,t k + T k ). Introducing the matrix Ψ α in the previous equality yields d dt [V (x(t)) + Q(t -t k , T k , x T k )] +2α [V (x(t)) + Q(t -t k , T k , x T k )] = x(t k ) x(t) T Ψ α (t -t k , T k ) x(t k ) x(t) (12)
Thus if there exists a solution of inequality [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF], it implies that the left hand side of the previous equality is strictly negative definite. Then integrating this inequality yields, for all t

∈ [t k , t k + T ) V (x(t)) + Q(t -t k , T k , x T k ) < e -2α(t-t k ) V (x(t k )).
In particular, taking t = t k + T = t k+1 , we get

V (x(t k+1 )) < e -2αT k V (x(t k )).
This ensures exponential stability of the discrete-time system for the constant sampling period T . Finally Theorem 1 allows us to conclude the proof.

We emphasize that Theorem 2 only guarantees exponential stability of the solutions of system (9) for any asynchronous sampling which belongs to the interval [T min , T max ]. If T min = T max = T , the conditions of Theorem 2 address the problem of exponential stability under constant sampling period T .

If α is chosen equal to zero, then the conditions of Theorem 2 concerns now the problem of asymptotic stability under asynchronous samplings characterized by T min and T max .

As mentioned in the introduction, the case of asymptotic stability of sampled-data systems with a constant sampling period can also be verified numerically by checking if the eigenvalues of the matrix Γ(T ) lie within the unit circle. However, this method fails if the system matrices (A, B) are uncertain. For example, if matrices (A, B) lie in a polytope, it is difficult to investigate the eigenvalues of the matrix Γ(T ). By contrast, in Theorem 2, the stability condition depends linearly on the matrices A and B. Therefore, it is relatively simple to extend the previous results to the case of systems with polytopic uncertainty. This is a significant advantage of the proposed methodology.

Another remark concerns the choice of the functional Q introduced in Theorem 2. Note that, in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] in the proof of Theorem 2, the condition Ψ α (τ, T ) < 0, for all τ in [0, T ] and for all T ∈ [T min , T max ], is equivalent to the negativity of the derivative of the function V (x(t))+Q(t -t k , x T k ). The stability condition Ψ α (τ, T ) < 0 is equivalent to the negativity of the derivative of the function V (x(t))+Q(t -t k , T k , x T k ). This is not the case in [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF] or [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. Indeed, in these articles, the authors introduce an integral term of the form

Q(t -t k , T k , x KT ) = (t k+1 -t) t t k ẋT (s)R ẋ(s)ds, so that V + Q + Q.
Although this term has an important role in reducing the conservatism of the stability conditions in [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF] and [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], it unavoidably leads to the use of the Jensen's inequality [START_REF] Gu | Stability of time-delay systems[END_REF] to compute an upper bound of the derivative of V + Q + Q. It is well known that the Jensen's inequality introduces conservatism in the stability conditions [START_REF] Briat | Convergence and equivalence results for the Jensen's inequality -application to time-delay and sampled-data systems[END_REF]. In the present paper, this term is unnecessary, yielding an exact condition for the derivative of V + Q. This implies that the stability conditions from Theorem 2 should be less conservative than the ones from [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF] or [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. The conservatism of the previous Theorem 2 only depends on the degree, N, of the polynomial matrix M.

IV. SUM OF SQUARES AS ALGORITHMIC TOOL

The conditions associated with Theorem 2 require us to find a matrix P and a function M(τ) which satisfy linear matrix inequality conditions at an infinite number of values of τ. To test these conditions we must use algorithms for the optimization of functions subject to positivity/negativity constraints. One such algorithm is based on the sum of squares (SOS) decomposition of positive polynomials.

To apply this methodology we assume that M is polynomial, can be approximated by a polynomial, or there is a change of coordinates that renders it polynomial.

Denote by R[y] the ring of polynomials in y = (y 1 , . . . , y n ) with real coefficients. Denote by Σ s the cone of polynomials that admit a SOS decomposition, i.e., those p ∈ R[y] for which there exist

h i ∈ R[y], i = 1, . . . , M so that p(y) = M ∑ i=1 h 2 i (y).
If p ∈ Σ s , then clearly p(y) ≥ 0 for all y. The converse is not always true, although the converse does hold for univariate matrix-valued polynomials. The advantage of SOS is that while the problem of testing if p(y) ≥ 0 is known to be NP-hard, testing if p(y) ∈ Σ s is an SDP, [START_REF] Parrilo | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[END_REF], and hence is worst-case polynomial-time. The SDPs related to SOS can be formulated and solved efficiently the Matlab toolbox SOSTOOLS [START_REF] Prajna | Introducing SOS-TOOLS: a general purpose sum of squares programming solver[END_REF], which interfaces with semidefinite solvers such as SeDuMi [START_REF] Sturm | Using SeDuMi 1.02, a Matlab Toolbox for optimization over symmetric cones[END_REF]. Consider now the conditions in Theorem 2. These can be expressed as L 1 (0) = L 2 (T ) = 0 with positivity condition

L 3 (s) ≥ 0, s ∈ S . (13)
where L 1 , L 2 and L 3 are linear functions of M and P and S = [0, T ]. The equality constraints and global positivity can be easily implemented using SOSTOOLS. However, because we only desire positivity on an interval, global positivity may be overly restrictive.

To eliminate this conservatism we represent S = [0, T ] as a semialgebraic set:

S = {s ∈ R | g i (s) ≥ 0, i = 1, . . . , M},
where g i (s) are polynomial functions. For S = [0, T ], we use g 1 (s, T ) = -(Ts)s. In order to test condition [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF], we apply S-procedure-style results which allow us to test positivity on a semialgebraic set using SOS. Specifically, condition (13) holds if there exists SOS polynomials P i ∈ Σ s , subject to the constraint

L 3 (s) + M ∑ i=1 g i (s)P i (s, y) = P 0 (s).
Intuitively, the above condition guarantees that when s ∈ S , we have L(s) ≤ -∑ M i=1 g i (s)p i (s, y) ≤ 0 since g i ≥ 0 and p i ≥ 0, and therefore L(s) ≤ 0 for those s. While these conditions are relatively simple, they can, in fact, be made non-conservative through the use of Positivstellensatz results [START_REF] Putinar | Positive polynomials on compact semi-algebraic sets[END_REF].

Theorems

T max for Ex.1 T max for Ex.2 [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF] 0.869 0.99 [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] 1.113 1.99 [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF] 1.695 2.03 [START_REF] Liu | Discontinuous Lyapunov functionals for linear systems with sawtooth delays[END_REF] 1.695 2.53 [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] 1 Given these g i , we apply the same procedure as for Theorem 2.

V. NUMERICAL EXAMPLES

To illustrate the methodology, we consider control of system (1) with controller u(t) = Kx(t) using a network connection where u(t) is updated at times t k where t k+1t k = T k for all k = 1, 2, • • • , ∞.

• Example 1 from [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF]:

A = 0 1 0 -0.1 , BK = 0 0 -0.375 -1.15 , • Example 2 from [13]: A = -2 0 0 -0.9 , BK = -1 0 -1 -1 ,
• and Example 3 from [START_REF] Gu | Stability of time-delay systems[END_REF], [START_REF] Michiels | Using delays and timevarying gains to improve the static output feedback stabilizability of linear systems : a comparison[END_REF]:

A = 0 1 -2 0.1 , BK = 0 0 1 0 .
Tables I, II and III summarize the results obtained in the literature and using the theorems provided in the present paper for examples 1, 2 and 3. One can see that the obtained results are less conservative then existing ones.

In Figures 1, 2 and3, we use our algorithm to examine the effect of samplings on the decay rate of the system based on the conditions from Theorem 2 applied to synchronous samplings, asynchronous samplings with T min = 0 and T min = 0.9T max ( = 0), respectively. Our result shows that for synchronous samplings, the use of an appropriate sampling period can increase the convergence rate of the same system with a continuous-time controller. This behavior can also be appreciated when one considers asynchronous samplings whose lower bound is strictly greater than 0 as it is shown in Figure 2.

Another important remark deals with Example 3. This system is well known in the time-delay literature because the delay has a stabilizing effect. This means that its solutions are not stable for sufficiently small delay but become stable for sufficiently large delay. The method proposed in this article is able to take into account this phenomena and is also able to isolate several intervals of possible values for the length of the sampling interval where the system is stable for asynchronous and synchronous sampling. Figure 4 illustrates the intervals of stability and exponential decay rate for synchronous sampling.

Note that our analysis of Example 3 indicates that Theorem 2 (with N = 5) can be used to prove stability for asynchronous sampling in [0.4, 1.828] OR [2.520, 3.550], but not over both simultaneously. This means that stability is not guaranteed if the sampling switches from one interval to the other. This recalls the classical behavior of switched systems: A system which switches between two stable subsystems is not necessarily stable. In this article, a novel analysis of continuous linear systems under asynchronous sampling is provided. This approach is based on the discrete-time Lyapunov Theorem applied to the continuoustime model of the sampled-data systems. Numerical results compare favorably with results in the literature. Perhaps the most important feature of the method presented in this paper is that it is expressed using the sum-of-squares framework and is thus easily extended to nonlinear systems and systems with parametric uncertainty.

  Moreover the condition Γ T (T )PΓ(T )e -2αT P ≺ 0 is satisfied for all T ∈ [T min , T max ] Proof: Let k be a positive integer and T k ∈ [T min , T max ]. Consider the classical quadratic Lyapunov function for linear continuous-time systems. Define V : R n → R + as V (x) = x T Px, where P ≻ 0 is in S. This function V satisfies condition (6) from Theorem 1. Now define the following function for all

- 6 .

 6 a) Application to Theorem 2: For Theorem 2, M(s, T ) is a function of two variables s and T , where T ∈ [T min , T max ] and s ∈ [0, T ]. To describe the relevant semialgebraic set we use g 1 (s, T ) = -(Ts)s and g 2 (T ) = -(T max -T )(T -T min ).

Fig. 1 :Fig. 2 :Fig. 3 :

 123 Fig. 1: Bound on Decay Rate vs. Synchronous Sampling Period, T min = T max for Example 2 with N = 3.

Fig. 4 :

 4 Fig. 4: Bound on Decay Rate vs. Synchronous Sampling Period, T for Example 3 with N = 3

TABLE I :

 I Maximum allowable sampling period T max for examples 1, 2, with T min = 10 -6 .

	Theorems for Ex.3	T = T min = T max ∈ [ Tmin , Tmax ]
	[18]	[0.201, 1.623]
	Th.2 N = 1	∅
	Th.2 N = 3	[0.2007, 2.016] [2.606, 3.055]
	Th.2 N = 5	[0.2007, 2.020] [2.470, 3.694]

TABLE II :

 II Interval of allowable constant sampling period T for example 3 and α = 10

TABLE III

 III

: Interval of allowable asynchronous samplings of the form [T min , T max ] for example 3 and α = 10 -6 .


A Schur algorithm for symmetric inner functions.

Martine Olivi, Bernard Hanzon, Ralf L.M. Peeters Abstract. Symmetric inner rational functions naturally arise in the description of physical systems which satisfy the conservation and reciprocity laws. Inner matrix functions can be parametrized by a sequence of interpolation vectors obtained from a tangential Schur algorithm. In this paper, we present a Schur type algorithm which allows to describe symmetric inner functions, based on a two-sided Nudelman interpolation problem. This Schur algorithm gives rise to an interesting interpretation in the context of surface acoustic wave filters.

I. INTRODUCTION.

Symmetric inner functions naturally arise in the description of physical systems which satisfy the conservation and reciprocity laws. This study was initially motivated by an application to the synthesis of SAW (surface acoustic wave) filters. A SAW filter can be viewed as composed by a finite number of cells, each cell containing a reflection center with reflection coefficient r i , r i ∈] -1, 1[, and an electroacoustic center [START_REF] Baratchart | Parameter determination for surface acoustic wave filters[END_REF]. The acoustic matrix which relates ingoing waves to outgoing waves can be computed from the sequence of reflection coefficients r 1 , r 2 , . . . , r N . Put

t n = 1 -r 2
n , P n = t 1 t 2 . . . t n , and define Schur polynomials φ n and ψ n of degree (n -1) satisfying the Levinson recursions

φ n (z) = zφ n-1 (z) + r n ψ n-1 (z) ψ n (z) = zr n φ n-1 (z) + ψ n-1 (z), with φ 1 (z) = 1 ψ 1 (z) = r 1 .
Note that these polynomials have real coefficients. Also define the reciprocal polynomial of a polynomial p(z) of formal degree n as

p(z) = z n p(1/z) = z n p(1/z). (1)
The acoustic matrix associated with the first n-cells of the filter is thus given by

Q n (z) = z -jψ n (z 2 ) P n z n-1 P n z n-1 -j ψ n (z 2 ) / φ n (z 2 ), (2)
INRIA, BP 93, 06902 Sophia-Antipolis Cedex, FRANCE, {olivi}@sophia.inria.fr, phone: 33-4-92387877, fax: 33-4-92387858 School of Mathematical Sciences, University College, Cork, Ireland, Tel.: +353-21-4902376, Fax +353-21-4903784, E-mail: b.hanzon@ucc.ie Dept. Mathematics, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht, The Netherlands, Tel.: +31-43-3883365, Fax: +31-43-3884910, Email: ralf.peeters@math.unimaas.nl in which j denotes the square root of -1. Note that,

φ n (z 2 ) = z 2(n-1) φ n (1/z 2 ), ψ n (z 2 ) = z 2(n-1) ψ n (1/z 2 ).
The matrices Q n (z) are symmetric inner matrices, of McMillan degree 2n, with complex coefficients. In this example, the complex structure is very particular, but it can be much more complicated if one considers a combination of filters. Rational complex matrices which are both inner and symmetric seem to be relevant objects in physics and electrical engineering. However, while there exists an important literature on inner functions and some papers on real symmetric rational matrix functions [START_REF] Fuhrmann | On symmetric rational matrix functions[END_REF], [START_REF] Helmke | Tangent spaces of rational functions[END_REF], to our knowledge symmetric inner matrices have not been studied.

Tangential Schur algorithms provide interesting tools to parametrize inner functions by means of interpolation values [START_REF] Alpay | On the differential structure of matrix-valued rational inner functions[END_REF], [START_REF] Fulcheri | Matrix Rational H 2 -Approximation: a Gradient Algorithm Based on Schur Analysis[END_REF], [START_REF] Kimura | Directional interpolation approach to H ∞ -optimization and robust stabilization[END_REF]. The object of this paper is to present a Schur algorithm which allows to describe symmetric inner functions. In order to take into account symmetry, the usual Nevanlinna-Pick interpolation problem underlying the Schur algorithm will be replaced by a two-sided Nudelman interpolation problem.

II. SCHUR ALGORITHMS, INTERPOLATION AND PARAMETRIZATION.

The Schur algorithm was originally a nice recursive test for checking the boundedness of an analytic function S(z) in the disk: define a sequence of functions by S 0 = S and

S i+1 (z) = S i (z) -S i (0) z(1 -S i (0) * S i (z)) ; then, |S(z)| < 1 for |z| < 1 if and only if |S i (0)| < 1
for all i ≥ 0. Such an analytic function is called a Schur function and it is completely characterize by the sequence of numbers γ i = S i (0). Then, Schur type algorithms were used to approach interpolation problems with metric constraints (Nevanlinna-Pick problems): find a Schur function which takes certain specified values at certain points inside the unit disk.

Generalizations of Nevanlinna-Pick problems to the matrix case were considered, often motivated by some sample problems in control and system theory: sensitivity minimization, robust stabilization and model reduction. For these questions we refer the reader to [START_REF] Ball | Interpolation of rational matrix functions[END_REF] and the bibliography therein. An interpolation condition for a rational matrix function can take several forms, namely

S(w)u = v, (3) x T S(w) = y T , (4) x T S(w)u = ρ, (5) S(w) = M, (6)
in which w is a complex scalar, u, v, x, y, are vectors and M is a complex matrix. The solutions of such interpolation problems can be usually parametrized via a linear fractional transformation, that is a transformation of the form

T Θ (R) = (Θ 1 R + Θ 2 )(Θ 3 R + Θ 4 ) -1 , (7)
associated with some rational matrix function Θ(z) with block decomposition

Θ = Θ 1 Θ 2 Θ 3 Θ 4 .
The most popular interpolation problem, also called Nevanlinna-Pick problem consists of a set of directional interpolation conditions of the form (3). Schur algorithms to solve this problem were presented among others in [START_REF] Alpay | On the differential structure of matrix-valued rational inner functions[END_REF], [START_REF] Kimura | Directional interpolation approach to H ∞ -optimization and robust stabilization[END_REF].

In [START_REF] Alpay | On the differential structure of matrix-valued rational inner functions[END_REF] a Schur algorithm is used to provide parameters for the set of rational inner functions of fixed McMillan degree. We briefly recall this result. Let

J = I p 0 0 -I p , K = 0 I p I p 0 . (8)
For any matrix function F (z), we define

F ♯ (z) = F (z) * , F (z) = F (z). (9)
A 2p × 2p rational matrix function Θ(z) is called J-inner if, at every point of analyticity z of Θ it satisfies

Θ(z)JΘ(z) * ≤ J, |z| < 1, (10)
Θ(z)JΘ(z) * = J, |z| = 1. (11)
The linear fractional transformation [START_REF] Helmke | Tangent spaces of rational functions[END_REF] possesses the following useful properties: if Q = T Θ (R) where R and Q are invertible, then

Q -1 = T KΘK (R -1 ), (12)
Q T = T K Θ(1/z)K (R T ). (13)
A p × p rational matrix function S(z) is Schur if it is analytic and contractive in the disk

S(z)S(z) * ≤ I p , |z| < 1 , (14)
and inner if in addition it takes unitary values on the circle

S(z)S(z) * = I p , |z| = 1 . (15)
A linear fractional transformation (7) associated with a Jinner matrix function Θ transforms inner functions into inner functions.

Let Q(z) be a p × p inner function of McMillan degree n, then (i) w ∈ C, |w| < 1 being given, we can always find some direction u ∈ C p , u = 1 such that v given by the interpolation condition

Q(w)u = v,
has norm strictly less than 1.

(ii) in that case ( v < 1), a J-inner matrix function can be built from the interpolation data w, u, v, namely

Θ w,u,v (z) = I 2p + (z -1) 1 -|w| 2 1 -v 2 v u v u * (z -w)(1 -w) J (16)
such that the function Q(z) can be represented by the linear fractional transformation (7) associated with Θ w,u,v , for some inner function Q n-1 (z) of degree (n -1).

The tangential Schur algorithm consists of repeating this process, and thus provides a sequence of inner functions,

Q n = Q, Q n-1 , . . . , Q 1 , Q 0 , in which for k = n, n -1, . . . , 1, Q k has degree k and satisfies the interpolation condition Q k (w k )u k = v k , v k < 1,
and Q 0 is a constant unitary matrix.

A local parametrization (or chart) for the set of inner functions of McMillan degree n can be associated with a sequence of interpolation points (w 1 , w 2 , . . . , w n ) and interpolation directions (u 1 , u 2 , . . . , u n ). An inner function of McMillan degree n can be parametrized in this chart, if the Schur algorithm can be run until Q 0 (i.e. for k = n, . . . , 1, the interpolation values

v k = Q k (w k )u k satisfy v k < 1)
. Note that, by (i), for any Q(z), we can find a sequence of interpolation values and interpolation directions such that this condition is satisfied. The interpolation values v n = v, v n-1 , . . . , v 1 , are then the parameters of Q in the chart, together with the unitary matrix Q 0 .

Recall that a linear fractional transformation T H associated with such a J-unitary matrix H is a bijection on the set of inner functions which preserves the McMillan degree [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF]. Moreover, if Θ is a J-inner function, we have that

T ΘH (R) = T Θ (T H (R)) .
This leads to the following remark.

Remark 1: The function Θ w,u,v satisfies the condition Θ w,u,v (1) = I 2p . This implies that the constant unitary matrix Q 0 in the Schur algorithm is such that Q 0 = Q(1). This choice naturally arises in a function space approach [START_REF] Dym | J-contractive matrix functions, reproducing kernel spaces and interpolation[END_REF]. The matrix Θ w,u,v also has the nice factorization (obtained from [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF]Lemma 5.1]

applied to KΘ ♯ w,u,v K) Θ w,u,v (z) = H(vu * ) -1 I p 0 0 I p + (β ♯ w (z) -1)
uu * H(vu * ), in which β w and H(vu * ) are defined by ( 18) and (21) respectively. However, assertion (ii) is still true if we replace Θ w,u,v in the linear fractional transformation by any Θ w,u,v,H = Θ w,u,v H, where H is a constant J-unitary matrix. This matrix H may depend (smoothly) on the interpolation data. The corresponding Schur algorithm will provide from Q a different sequence of inner matrices Q k . Such a choice may be more convenient in some context. In [START_REF] Kimura | Directional interpolation approach to H ∞ -optimization and robust stabilization[END_REF], a similar Schur algorithm is presented which gives rise to a circuit theoretical interpretation. It is described in the case of continuous-time transfer functions, but it corresponds in our discrete-time setting to the choice of Θ w,u,v H(vu * ) -1 . In [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF], the freedom in the choice of the matrix H has been used to associate with the Schur algorithm a nice recursive construction of balanced realizations. This is possible with the J-inner matrix Θw,u,v = H(wvu * ) I p 0 0 I p + ( 1-wz z-w -1)uu * H(vu * ). Now, we turn to the case of symmetric functions. In view of (13) we have the following preliminary result:

Lemma 1: Let Θ(z) be a J-unitary function such that

Θ (1/z) = KΘ(z)K.
Then, the linear fractional transformation T Θ preserves symmetry.

III. ELEMENTARY SYMMETRIC INNER FUNCTIONS.

The Schur algorithm we shall describe in the next section produces a sequence of symmetric inner functions whose McMillan degree decreases by 2 at each step. Thus, it ends either on a symmetric unitary matrix or on an inner function of McMillan degree 1. In this section, we shall characterize these elementary symmetric inner functions. As mentioned previously, constant J-unitary functions also play an important role in these questions. We shall also characterize J-unitary matrices H which satisfies the condition H = KHK, so that T H preserves symmetry. Symmetric complex matrices have been studied in [START_REF] Horn | Matrix Analysis[END_REF]. We recall the following result.

Lemma 2: A complex matrix X is both symmetric and unitary if and only if it can be written as X = OΛO T where O is a real orthogonal matrix and

Λ = diag(λ 1 , λ 2 , . . . , λ n ), |λ k | = 1.
Every inner function of McMillan degree 1 can be written in the form (see [4, th.1.4.])

B(z) = (I p + (β ζ (z) -1)xx * ) X, (17)
where β ζ (z) is the Blaschke factor

β ζ (z) = (z -ζ)(1 -ζ) (1 -ζz)(1 -ζ) , |ζ| < 1 (18)
X is a unitary matrix uniquely determined by X = B(1) and x * is a left kernel vector of B(ζ) such that x = 1.

If B(z) is symmetric, then the matrix X = B(1) is symmetric too. Since it is unitary, there exists a unitary matrix Y such that (see Lemma 2)

X = Y Y T .
Consider the matrix

B(z) = Y * B(z) Ȳ = I p + (β ζ (z) -1)yy * , (19)
where y = Y * x. The matrix B(z) will be symmetric if and only if ǫy is real, for some unit complex number ǫ. Lemma 3: The inner function B(z) given by ( 17) is symmetric if and only if it can be written in the form

B(z) = Y I p + (β ζ (z) -1)yy T Y T ,
in which y is a unit real p-vector and Y a unitary matrix.

It is known [4, Th.1.2] that every 2p×2p J-unitary matrix can be expressed in the form

H = H(E) P 0 0 Q , (20)
where P and Q are p × p unitary matrices, and H(E) the Halmos extension of a strictly contractive p × p matrix E, that is

H(E) = (I p -EE * ) -1/2 E(I p -E * E) -1/2 E * (I p -EE * ) -1/2 (I p -E * E) -1/2 . (21) Lemma 4: The J-unitary matrix H(E) satisfies H(E) = KH(E)K if and only if E is symmetric. IV. A SCHUR ALGORITHM FOR SYMMETRIC INNER FUNCTIONS.
Let Q be a p × p symmetric inner function of McMillan degree n ≥ 2 and assume it satisfies the Nevanlinna-Pick interpolation condition

Q(w)u = v, (22)
where w belongs to the open unit disk, u and v are complex p-vectors, u = 1 and v < 1. In order to take into account the symmetry of Q(z), we shall consider simultaneously the symmetric interpolation condition

u T Q(w) = v T . (23)
The interpolation problem: find Q which satisfies ( 22) and (23), is a special case of the two-sided Nevanlinna-Pick problem in which the interpolation points coincide. It can be viewed as a two-sided Carathéodory-Fejér problem [4, th.6.3.] or as a particular case of a the two-sided Nudelman interpolation problem considered in [2, ex.18.5.3]. For this problem to be well-posed, an extra interpolation condition of the form

u T Q ′ (w)u = ρ, (24)
in which Q ′ denotes the derivative of Q, must be added to the previous ones.

Put σ = 1 -v 2 1 -|w| 2 , (25)
and associate with the set of interpolation data

δ = (w, u, v, ρ), (26)
the Pick matrix

Λ δ = σ ρ ρ σ , (27)
and the J-inner matrix function

Θ δ (z) = I 2p + (z -1)C (z -w) -1 0 0 (1 -z w) -1 Λ -1 δ (1 -w) -1 0 0 (1 -w) -1 C * J (28) with C = v -ū u -v . ( 29
)
Remark 2: Note that Θ δ (1) = I 2p and that Θδ (1/z) = KΘ δ (z)K, so that by Lemma 1 the linear fractional transformation T Θ δ preserves symmetry. It is proved in [START_REF] Ball | Interpolation of rational matrix functions[END_REF] that there exists a Schur function satisfying the interpolation conditions ( 22), ( 23) and ( 24) if and only if the Pick matrix Λ δ is positive definite. In the case of a symmetric inner function we have the following representation theorem.

Proposition 1: Let Q(z) be a p × p symmetric inner function of McMillan degree n ≥ 2 which satisfies ( 22), ( 23) and ( 24) and such that Λ δ is positive definite. Then, Q(z) can be represented as

Q = T Θ δ (R),
where Θ δ is given by (28) and R is a p × p symmetric inner function of McMillan degree (n -2).

Proof. We give a constructive proof of this result which follows the approach developed by Potapov to describe the multiplicative structure of J-inner functions [START_REF] Potapov | The multiplicative structure of J-contractive matrix functions[END_REF]. We first consider the case where the interpolation data are of the form δ 0 = (0, u, 0, ρ), so that (22) takes the simpler form

Q(0)u = 0.
From δ 0 we construct the positive definite matrix Λ δ0 given by

Λ δ0 = 1 ρ ρ 1 , |ρ| < 1,
and the matrix Θ δ0 (z) can be easily computed as

Θ δ0 (z) = I p + z-1 1-|ρ| 2 ūu T -ρ(z-1) 1-|ρ| 2 ūu * ρ(z-1) (1-|ρ| 2 )z uu T I p -z-1 (1-|ρ| 2 )z uu * .
Then, we write the Taylor series of Q(z) about 0:

Q(z) = Q(0) + zQ ′ (0) + . . . .
and the singular value decomposition of Q(0)

Q(0) = V diag(0, . . . , 0, λ 1 , . . . , λ r )U * , 0 < λ 1 ≤ . . . ≤ λ r ,
for some unitary matrices U and V . From (22) we may choose the vector u as first column vector of U . Now, let

B(z) = I p + (z -1)uu * = U diag(z, 1, . . . , 1)U * , then Q(z) = Q 1 (z)B(z) for some inner function Q 1 (z) of McMillan degree (n -1), and
Q 1 (z) = Q(z)B(z) -1 = Q ′ (0)(zI p + (1 -z)uu * ) + z × . . . .
We then have the following interpolation condition for Q 1 (z)

u T Q 1 (0) = u T Q ′ (0)uu * = ρu * .
This interpolation condition differs from (3) but can be approached in a similar way [START_REF] Fulcheri | Matrix Rational H 2 -Approximation: a Gradient Algorithm Based on Schur Analysis[END_REF]. Since ρu = |ρ| < 1, its solutions can be represented by a linear fractional transformation Q 1 = T Θ1 (R 1 ) for some inner matrix function R , which is precisely Q = T Θ δ 0 (R 1 ). Moreover, by Lemma 1 R 1 is symmetric which proves the result in this case. Then, we assume that the interpolation data are of the form δ 1 = (w, u, 0, ρ), so that (22) has the form Q(w)u = 0, and that the associated Pick matrix The Pick matrix associated with these data

Λ δ1 = 1 1-|w| 2 ρ ρ
Λ δ ′ 1 = 1 ρ(1 -|w| 2 ) 1- w 1-w ρ(1 -|w| 2 ) 1-w 1-w

1 ofQ 1 1 andQ

 111 McMillan degree (n -2) (since Q 1 has McMillan degree (n -1)), and where Θ 1 is the J-inner functionΘ 1 (z) = I 2p + (z -(z) = (I p + (z-1) (1-|ρ| 2 ) ūū * )R 1 (z) -(z-1) (1-|ρ| 2 ) ρūu * (z-1) (1-|ρ| 2 ) ρuū * R 1 (z) + I p -(z-1) (1-|ρ| 2 ) |ρ| 2 uu * -(z) = (I p + (z-1) (1-|ρ| 2 ) ūū * )R 1 (z) -(z-1) (1-|ρ| 2 ) ρūu * (z-1) z(1-|ρ| 2 ) ρuū * R 1 (z) + I p -(z-1) z(1-|ρ| 2 ) uu * -1

1 1 -

 1 |w| 2 is positive definite. Consider the inner function Q w defined by Q w (β w (z)) = Q(z),where β w (z) is the Blaschke factor defined in (18). The matrix Q w is symmetric, has same McMillan degree as Q(z) and satisfies the interpolation conditionsQ w (0)u = 0, u T Q ′ w (0)u = ρ(1 -|w| 2 ) 1 -w 1 -w ,associated with the interpolation data δ ′ 1 = 0, u, 0, ρ(1 -|w| 2 ) 1 -w 1 -w .

is positive definite since Λ δ1 is. The previous result thus applies to the matrix Q w , so that

for some symmetric inner function R w of McMillan degree (n -2). It is then easily verified that

in which R(z) = R w (β w (z)) is symmetric and has degree (n -2). Now, we come to some general data δ = (w, u, v, ρ) for which v is a nonzero vector. We first assume that E = Q(w) is a strictly contractive matrix and we consider the function

The Pick matrix Λ δ associated with the interpolation data δ = (w, û, 0, ρ 1-v 2 ) is again strictly positive since Λ δ is. Thus, by what precedes,

for some symmetric inner function R of McMillan degree (n -2). We then have (note that

This follows immediately from the fact that

In the case where Q(w) is not strictly contractive, we can write the Takagi's factorization [START_REF] Horn | Matrix Analysis[END_REF]Cor.4.4.4] for Q(w)

where U is a unitary matrix and the σ k 's are real nonnegative numbers of modulus less than or equal to one, since

and choose

It is not difficult to prove that Eu = v and the previous proof can be adapted to this case. 2 Remark 3: If both v and ρ are zero in the interpolation conditions ( 22), ( 23) and ( 24), the linear fractional representation

The Schur algorithm: let Q(z) be an inner function of McMillan degree n ≥ 2. Then, (i) we can find a set of interpolation data δ = (w, u, v, ρ) such that Q(z) satisfies ( 22), ( 23), (24) and the Pick matrix Λ δ given by ( 27) is positive definite.

Indeed, we can take for example a zero w of Q(z) and an associated direction u, so that

Then, let ρ = u T Q ′ (w)u. By Schwartz lemma [12, chap.12] applied to the Schur analytic function u T Q(z)u, we have

The equality is strict and thus the Pick matrix Λ δ positive definite unless if u T Q(z)u = ξβ w , ξ unit complex number. In this case, the matrix Q(z) is of the form

and it is not difficult to find a unit vector u such that u T Q ′ (ω)u = ρ = 0 and Q(ω)u = v, with v < 1. The Pick matrix Λ δ is again positive definite.

(ii) given a set of interpolation data δ such that the associated Pick matrix Λ δ is positive definite, any matrix Q satisfying (22), ( 23), (24) can be represented by a linear fractional transformation of the form

for some symmetric inner function R of McMillan degree (n -2).

Repeating this process we construct a sequence of symmetric inner function

The Schur algorithm stops when the degree is 1 or 0. If n is odd, then Q f is a symmetric inner function of McMillan degree 1 while if n is even, then Q f is a symmetric unitary matrix. These elementary symmetric inner functions were described in section III.

V. SCHUR PARAMETERS FOR A SAW FILTER.

NORMALIZATION ISSUES.

We return to the example proposed in the introduction. Let M n (z) = Q n (z)/z where Q n (z) is the acoustic matrix given by [START_REF] Ball | Interpolation of rational matrix functions[END_REF]. It satisfies the interpolation condition ( 22) with

It can be verified that u T Q ′ n (0)u = 0, so that the associated Pick matrix Λ = t 2 n I 2 is clearly positive definite. The associated J-inner matrix can be computed as

We may also run two steps of the classical tangential Schur algorithm described in section II, with

We get

Note that this matrix doesn't satisfy the condition Φ n (1) = I 2p that was imposed to the J-inner functions Θ δ in the symmetric Schur algorithm. The matrix Φ n (1) is in fact equal to H(vu * ). Note that, in this particular case,

is symmetric, so that by Lemma 4 and Lemma 1 the linear fractional transformation T H(vu * ) preserves symmetry. Moreover, we have that

It appears that the sequence of acoustic matrices M n corresponds to a symmetric Schur algorithm in which the J-inner function Θ δ H(vu * ) has been chosen instead of Θ δ . This leads us to some considerations on the choice of the J-inner function in a Schur algorithm. Concerning the tangential Schur algorithm described in section II, we have already mentioned that several choices have been made in the literature (see remark 1) and in particular, that of [START_REF] Kimura | Directional interpolation approach to H ∞ -optimization and robust stabilization[END_REF] which allows for a nice circuit interpretation, and that of [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF] which leads to a recursive construction of balanced realizations. In the context of SAW filters, the Schur algorithm associated with the matrices Φ n has a nice physical interpretation: it gives the sequence of acoustic matrices M n corresponding to the first n-cells of the filter. Moreover, it can be proved that it allows for a recursive construction of balanced realizations as described in [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF]. This is not very surprising since, when the interpolation point w is zero, the choices in [START_REF] Kimura | Directional interpolation approach to H ∞ -optimization and robust stabilization[END_REF] and [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF] for the matrix Θ agree. In general, the matrix vu * is not symmetric. It is symmetric only if v = λū for some complex number λ. This raises the question of the choice of a symmetric contractive matrix E, such that the Schur algorithm associated with Θ δ H(E) allows for a recursive construction of balanced realizations.


Martine Olivi

Bernard Hanzon

Ralf L M Peeters

Lossless scalar functions: boundary interpolation, Schur algorithm and Ober's canonical form

, this structure is also derived from a LC ladder. In this paper, a connection is established between Ober's canonical form and a Schur algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober's form as interpolation values at infinity and a recursive construction of the balanced realization.

I. INTRODUCTION

There is a rich literature on the construction of canonical forms for various classes of linear systems. It may be noticed that for balanced state-space canonical forms, the class of lossless systems lies at the heart of many such constructions ( [START_REF] Ober | Balanced parametrization of classes of linear systems[END_REF], [START_REF] Hanzon | Overlapping block-balanced canonical forms for various classes of linear systems[END_REF]). Lossless systems are important in their own right too because of their usefulness for several other purposes: (i) to study H 2 -model approximation problems; (ii) to perform system identification by the method of separable least squares (see [START_REF] Bruls | Linear and nonlinear system identification using separable least-squares[END_REF]) using output normal forms; (iii) to generate polyphase representations of orthogonal filter banks when implementing orthogonal wavelet decomposition schemes.

In the continuous-time case, a balanced canonical form for lossless systems was first presented in the work of Ober ([6], [START_REF] Ober | Asymptotically stable allpass transfer functions: canonical form, parametrization and realization[END_REF]). This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. To deal with the discrete-time case one may apply the bilinear transform, which preserves balancedness but destroys upper triangularity of K. In [START_REF] Hanzon | Balanced parametrizations of stable SISO all-pass systems in discrete-time[END_REF] another balanced canonical form for discrete-time lossless systems was developed directly starting from upper triangularity of K. This canonical form is conveniently parameterized with Schur parameters. In [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential schur algorithm[END_REF] this result is generalized to the multivariable case by exploiting the connection with the tangential Schur algorithm. In the discrete-time setting, the construction of such balanced canonical forms is achieved in a recursive way with unitary matrix multiplications. Each step of the recursion involves an interpolation condition at a point located either inside or outside the unit circle.

To transfer these discrete-time results back to continuoustime, the bilinear transform can again be used. As a result, unitary matrix multiplications are replaced by more general In the present paper we note that the canonical form of Ober involves interpolation conditions at infinity. Using the results of [START_REF] Ball | Interpolation of rational matrix functions[END_REF] on boundary interpolation problems we give a parametrization of continuous-time lossless functions. By making appropriate choices for the interpolation conditions, we show how the balanced state-space canonical form of Ober can now be recovered.

II. PRELIMINARIES.

We denote by Π + := {s ∈ C; Re s > 0} and Π -:= {s ∈ C; Re s < 0} the right and left half-planes, by D the open unit disk and by T the unit circle. For any matrix function R(s), we define the matrix functions R * (s) and R ♯ (s) by

R * (s) := R(s) * , and R ♯ (s) := R * (-s). (1)
Note that if s lies on the imaginary axis, then R ♯ (s) = R(s) * . Let

J = I p 0 0 -I p .
A 2p × 2p rational matrix function Θ(s) is called J-lossless if, at every point of analyticity s of Θ it satisfies

Θ(s) * JΘ(s) ≤ J, Re s > 0, (2)
Θ(s) * JΘ(s) = J, Re s = 0, (3)
Θ(s) * JΘ(s) ≥ J, Re s < 0. (4)
It can be shown that this definition contains redundancy in the sense that either one of these three defining properties (2)-( 4) is implied by the other two (cf., e.g., [START_REF] Dym | J-contractive matrix functions, reproducing kernel spaces and interpolation[END_REF], [START_REF] Genin | On σ-lossless transfer functions and related questions[END_REF]). A p × p rational matrix function F (s) is called lossless, if and only if

F (s)F (s) * ≤ I p , Re s > 0, (5)
with equality on the imaginary axis. By analytic continuation, the identity on the imaginary axis extends almost everywhere, so that any rational lossless function F (s) is invertible, its inverse being (-I)-lossless and given by F (s) -1 = F ♯ (s).

Along with a 2p × 2p rational function Θ(s) blockpartitioned as follows

Θ(s) = Θ 11 (s) Θ 12 (s) Θ 21 (s) Θ 22 (s) , (6)
with each block of size p × p, we associate the linear fractional transformation T Θ which acts on p × p rational functions F (s) :

T Θ (F ) = [Θ 11 F + Θ 12 ][Θ 21 F + Θ 22 ] -1 . (7)
Linear fractional transformations occur extensively in representation formulas for the solution of various interpolation problems ( [START_REF] Ball | Interpolation of rational matrix functions[END_REF]). If Θ(s) is a J-lossless matrix function, then the map T Θ sends every lossless function onto a lossless function.

If (A, B, C, D) is a (minimal) balanced realization of a lossless function G(s), then it holds that In the sequel, since we deal with scalar functions, only the case p = 1 is needed. The natural framework for interpolation theory is that of complex functions. However, systems are often real-valued and their transfer function real, that is, they satisfy the relation F (s) = F (s). In this paper, we are mainly interested in this case.

D is unitary (of size p × p), (8)
C = -DB * (of size p × n), (9)
A + A * = -BB * . ( 10

III. THE CANONICAL FORM OF OBER

A balanced canonical form for real lossless systems was first presented in the work of Ober ([6], [START_REF] Ober | Asymptotically stable allpass transfer functions: canonical form, parametrization and realization[END_REF]): any lossless function G(s) of McMillan degree n has a unique balanced realization G(s) = D + C(sI n -A) -1 B parametrized by s 1 = ±1, and b 1 , α 1 , α 2 , . . . , α n-1 positive real numbers such that

B T = b 1 0 • • • 0 , C T = s 1 B, A =          - b 2 1 2 α 1 -α 1 0 α 2 0 -α 2 0 . . . 0 0 α n-1 -α n-1 0          , D = -s 1 . (11)
The connection between the matrix G(s) and the parameters in the canonical form is better understood with the help of the following family of polynomials (see [START_REF] Ober | Asymptotically stable allpass transfer functions: canonical form, parametrization and realization[END_REF]

): for k = 0, . . . n -2, let ∆ n-k (s) = det(sI n-k -A n-k ), (12)
A n-k =          0 α k+1 0 -α k+1 0 α k+2 -α k+2 . . . . . . 0 0 α n-1 -α n-1 0          .
These polynomials satisfy the recursion

∆ n-k (s) = s∆ n-k-1 (s) + α 2 k+1 ∆ n-k-2 (s), (13)
∆ j is monic and is an even (resp. odd) function for j even (resp. odd). We have that

G(s) = -s 1 ∆ n (s) - b 2 1 2 ∆ n-1 (s) ∆ n (s) + b 2 1 2 ∆ n-1 (s) . ( 14
)
From a coprime fraction representation of G(s), with monic denominator, the parameters b 1 , α 1 , . . . , α n-1 can be easily recovered by separating the odd and even part of the numerator and using the recursion [START_REF] Marmorat | Matrix rational H2-approximation: a state-space approach using Schur parameters[END_REF]. If g(s) = G(1/s), it is easily verified that the parameter b 1 satisfies the interpolation condition

g ′ (0) = s 1 b 2 1 .

IV. BOUNDARY INTERPOLATION AND PARAMETRIZATION OF LOSSLESS FUNCTIONS

In this section we adress the complex case again. Interpolation problems in which an interpolation condition is prescribed at the boundary of the stability domain have been studied in [9, chap.21].

Let f (s) be a lossless function; let σ ∈ iR, and consider the boundary interpolation condition f (σ) = ξ with |ξ| = 1. For the interpolation problem to be well-posed, some extra condition must be prescribed on the angular derivative f ′ (σ) of f at σ. Since |f (σ)| = 1, it can be proved that the quantity

ρ = -f (σ)f ′ (σ)
is a positive real number. This suggests the following angular derivative interpolation problem: (ADI) Given σ ∈ iR, ξ ∈ T and ρ > 0, find all the lossless functions f such that

f (σ) = ξ f ′ (σ) = -ξρ (15)
With these interpolation data, we associate the J-lossless function

θ σ,ρ,ξ (s) = J - 1 (s -σ)ρ ξ 1 ξ 1 * . ( 16
)
Any J-lossless function with a pole at σ can be written in this form up to a right constant J-unitary factor H. The matrix θ σ,ρ,ξ satisfies θ σ,ρ,ξ (∞) = J. The reason for this particular choice will become clear in section V. Theorem 1: The problem (ADI) always has a solution. The set of all solutions is given by

f = T θ σ,ρ,ξ (g), (17)
where θ σ,ρ,ξ (s) is given by ( 16) and g(s) is an arbitrary lossless function such that g(σ) = -ξ. Moreover, we have that

deg f = deg g + 1.
Proof. This result is a particular case of Theorem 21.1.5. in [START_REF] Ball | Interpolation of rational matrix functions[END_REF], which more generally describes all the Schur functions which satisfy a set of interpolation conditions of the form (15). Here the choice of the J-lossless matrix θ σ,ρ,ξ is also different. We give the main lines of the proof. The linear fractional transformation (17) can be computed as

f (s) = ξ + (1 -ρ(s -σ))g(s) ξg(s) + 1 + ρ(s -σ) .
Assume that g(s) is a lossless function and that it satisfies g(σ) = -ξ. Then θ σ,ρ,ξ being J-lossless, f (s) is lossless too and we have

f (σ) = ξ ξ + g(σ) ξ + g(σ) , f ′ (σ) = -ξρ (ξ + g(σ)) 2 (ξ + g(σ)) 2 ,
so that, with the assumption g(σ) = -ξ, the interpolation conditions (15) are satisfied. The assumption g(σ) = -ξ also ensures that deg f = deg g + 1 since then, no pole-zero cancellation can occur.

Conversely assume that f (s) is a lossless function of McMillan degree n satisfying (15), and define

g(s) = ξ -(1 + ρ(s -σ))f (s) ξf (s) -1 + ρ(s -σ) .
The function g is again lossless and has degree at most n+1. However, the interpolations conditions ensure that a double pole-zero cancellation occurs, so that the degree of g is in fact n -1. Applying l'Hôpital's rule, we can compute the value of g at σ upon differentiating twice the numerator and denominator expressions, and then evaluating at σ. We get

g(σ) = -ξ -2ρξ f ′ (σ) f ′′ (σ) ,
which proves that g(σ) = -ξ. 2 Let f = f n be a lossless function of McMillan degree n, then a Schur algorithm can be performed using angular derivative interpolation conditions. It consists of the construction of a sequence of lossless functions of decreasing degree f = f n , f n-1 , . . . , f j . . . f 0 . Assume that f j has been constructed and let

f j (σ j ) = ξ j (18) f ′ j (σ j ) = -ξ j ρ j , (19)
then the function f j-1 is given by

f j-1 = T -1 θ σ j ,ρ j ,ξ j (f j ),
where θ σj ,ρj ,ξj is given by ( 16). It has degree j -1 and satisfies the condition f j-1 (σ j ) = -ξ j . The function f 0 is a constant ξ 0 of modulus 1.

The interpolation points σ j being fixed, the interpolation values

(ξ 0 , ξ 1 , . . . , ξ n , ρ 1 , ρ 2 , ρ n )
can be taken as parameters as in the classical Schur algorithm. We have the right number of real parameters: 2n + 1 in the complex case and n in the real case. Using the reverse Schur algorithm, we define a map

(ξ 0 , ξ 1 , . . . , ξ n , ρ 1 , ρ 2 , ρ n ) → f,
where f is a lossless function of degree n, which is clearly one-to-one on its domain of definition: the parameters (ξ 0 , ξ 1 , . . . , ξ n , ρ 1 , ρ 2 , ρ n ) such that, for j = 1, . . . , n, ρ j > 0 and ξ j is a unit complex number such that f j-1 (σ j ) = -ξ j , where the f j are built recursively, so that f j only depends on the first 2j + 1 parameters (ξ 0 , ξ 1 , . . . , ξ j , ρ 1 , ρ 2 , ρ j ). When all interpolation points are equal, say σ j = σ, j = 1 . . . , n, the condition f j-1 (σ j ) = -ξ j simplifies to

ξ j = -ξ j-1 , = 1 . . . , n, (20)
and the domain of definition becomes simpler.

V. CONNECTION WITH OBER'S CANONICAL FORM

We shall study the real case with all the interpolation points at zero σ j = 0, |ξ j | = 1, j = 1, . . . n. Condition (20) implies that ξ 0 = ξ 1 = . . . = ξ n = ±1, so that the parameters are now ξ 0 = ±1 and the ρ j , ρ j > 0, j = 1, . . . n.

Remark. The choice of the matrix θ σ,ρ,ξ in (16) satisfying θ σ,ρ,ξ (∞) = J allows to take all the interpolations values ξ j equal. The more usual choice θ σ,ρ,ξ (∞) = I will correspond to a sequence ξ 0 , ξ 1 , . . . , ξ n alternating between the values 1 and -1.

To handle Ober's form which requires interpolation conditions at infinity, we shall consider the functions F j (s) = f j (1/s), j = 1, . . . , n.

From F 0 = ξ 0 , for j = 1, . . . , n, the Schur recursion gives

F j (s) = (s -ρ j )F j-1 (s) + ξ 0 s (s + ρ j ) + ξ 0 sF j-1 (s) , that is F j (s) = T θj (1/s) (F j-1 (s))
, where θ j is the J-lossless matrix

θ j (s) = J - 1 ρ j s ξ 0 1 ξ 0 1 T . ( 21
)
Putting a j = ρ j /2 and ξ 0 = 1, we get:

F 0 (s) = 1 F 1 (s) = s -a 1 s + a 1 F 2 (s) = s 2 -a 2 s + a 1 a 2 s 2 + a 2 s + a 1 a 2 F 3 (s) = s 3 -a 3 s 2 + (a 1 a 2 + a 2 a 3 )s -a 1 a 2 a 3 s 3 + a 3 s 2 + (a 1 a 2 + a 2 a 3 )s + a 1 a 2 a 3
If ξ 0 = -1, then each F j is just multiplied by -1. Theorem 2: Let ρ 1 , ρ 2 , . . . , ρ n be strictly positive real numbers; let F 0 = ξ 0 and for j = 1, . . . , n where the θ j are given by (21). Then, F n = F is a lossless function of McMillan degree n, and the parameters b 1 , α 1 , . . . , α n in the canonical form [START_REF] Genin | On σ-lossless transfer functions and related questions[END_REF] of F are related to the a j = ρ j /2, j = 1, . . . , n as follows:

s 1 = -ξ 0 b 2 1 = ρ n α 2 1 = a n a n-1 α 2 2 = a n-1 a n-2
. . .

α 2 n-1 = a 2 a 1 (22)
Moreover,

F j (s) = ξ 0 ∆ j (s) -a j ∆ j-1 (s) ∆ j (s) + a j ∆ j-1 (s) ,
where ∆ j is the polynomial defined by [START_REF] Peeters | Linear fractional transformations and balanced realization of discrete-time stable all-pass systems[END_REF].

Proof. For simplicity, we assume in the proof that ξ 0 = 1. We first prove that for j = 1, . . . , n

F j (s) = P j (s) -a j Q j (s) P j (s) + a j Q j (s) ,
where P j and Q j are polynomials, P j is monic and has same parity as j while Q j has the opposite one. It holds for j = 1.

We have that P 1 (s) = s and Q 1 (s) = 1. Assume it holds for j, then

F j+1 = (s -ρ j+1 )(P j -a j Q j ) + s(P j + a j Q j ) (s + ρ j+1 )(P j + a j Q j ) + s(P j -a j Q j )
= sP j -a j+1 P j + a j a j+1 Q j sP j + a j+1 P j + a j a j+1 Q j , so that the relation is also true at j + 1 with

P j+1 = sP j + a j a j+1 Q j Q j+1 = P j . Now it is clear from (14) that ∆ n = P n , ∆ n-1 = Q n = P n-1 and ρ n = -b 2 1
. Using (13), we have that

α 2 j+1 ∆ n-j-2 = ∆ n-j -s∆ n-j-1 = P n-j -sP n-j-1 ,
by the induction hypothesis, and therefore

α 2 j+1 ∆ n-j-2 = a n-j a n-j-1 P n-j-2 ,
so that for j = 0, . . . n -2,

P n-j-2 = ∆ n-j-2 , α 2 j+1 = a n-j a n-j-1 .
2 For j = 1, . . . , n, the lossless functions F j (s) satisfy the interpolation conditions

F j (∞) = ξ 0 lim s→∞ s 2 F ′ j (s) = ξ 0 ρ j .
It is interesting to notice that the parameters a i above have a direct electrical interpretation. In [START_REF] Johns | Orthonormal ladder filters[END_REF], the state-space structure [START_REF] Genin | On σ-lossless transfer functions and related questions[END_REF] is derived from a singly-terminated LC ladder, the parameters a i in (22) being the reciprocal values of capacitor and inductor values, alternatively.

VI. BALANCED REALIZATIONS

In this section, a recursive construction of balanced realizations is associated with the Schur algorithm described in section IV. We study the general case of complex transfer functions, but focus on the interpolation points all at infinity, which gives a nice structure. In the real case, it leads to Ober's canonical form. This construction is based on the results of [START_REF] Peeters | Linear fractional transformations and balanced realization of discrete-time stable all-pass systems[END_REF] in which linear fractional transformations on transfer functions are represented by corresponding linear fractional transformations on state-space realization matrices. Namely, there we have the following result.

Theorem 3: Suppose that θ(s) is proper, has McMillan degree 1 and admits a state-space realization

θ(s) = D 11 D 12 D 21 D 22 + C 1 C 2 (s -σ) -1 B 1 B 2 . Let G(s) = D + C(s I n -A) -1 B be a realization of G(s), then a realization of F (s) = T θ(1/s) (G(s)) = D + C(s I n+1 -Ã) -1
B is given by the linear fractional transformation:

D C B Ã = T Φ     D 0 C 0 1 0 B 0 A     where Φ =         D 11 0 0 D 12 C 1 0 0 1 0 0 0 0 0 0 I n 0 0 0 D 21 0 0 D 22 C 2 0 B 1 0 0 B 2 σ 0 0 0 0 0 0 I n        
.

We now apply this result to the J-inner matrix

θ(s) = J - 1 ρs ξ 1 ξ 1 *
, considered in section V, and consider the linear fractional transformation

F (s) = T θ(1/s) (G(s)),
which provides the solution to an interpolation problem at ∞, namely

F (∞) = ξ lim s→∞ s 2 F ′ (s) = ξρ.
The matrix Φ in theorem 3 is then

Φ =         -1 0 0 0 ξ/ √ ρ 0 0 1 0 0 0 0 0 0 I n 0 0 0 0 0 0 1 1/ √ ρ 0 ξ/ √ ρ 0 0 1/ √ ρ 0 0 0 0 0 0 0 I n        
, and the transformation on balanced realizations becomes

D C B Ã =        ξ - √ ρ 0 ξ√ ρ -ρ 1+ξD √ ρ 1+ξD C 0 ξ √ ρ 1+ξD B A -ξ 1+ξD BC        .
In the real case, with ξ 0 = ξ 1 = . . . = ξ n = 1, the linear fractional fractional transformation

F j (s) = T θj (1/s) (F j-1 (s)),
corresponds to the following recursion on state space realizations:

A j =    -ρ j /2 √ ρj 2 C j-1 √ ρj 2 B j-1 A j-1 - Bj-1Cj-1 2    , B j = √ ρ j 0 , C j = - √ ρ j 0 , D j = 1,
which leads to the balanced canonical form of Ober, with s 1 = -1. Note that subdiagonal entries are all positive while they are all negative in Ober's canonical form. An example: consider the ladder filter in [2]: F (s) = (-1) n q(-s) q(s) where q(s) = s 5 + .9287s 4 + 1.7726s 3 + 1.0557s 2 + .6917s + .1739. To compute its first parameter a 5 = ρ 5 /2, we separate the odd and even part of q(s), which gives

∆ 5 = s 5 + 1.7726s 3 + .6917s ∆ 4 = s 4 + .18725 + 1.1367s 2 a 5 = .9287
Then we use the recursion formula ( 13) and (22) to compute the a i = ρ i /2. We get

∆ 3 = s 3 + .7933s ∆ 2 = s 2 + .5452 ∆ 1 = s ∆ 0 = 1 a 4 = .6847 a 3 = .5016 a 2 = .4946 a 1 = 1.1025
Then, a balanced realization can be computed by the recursions below:

A =       -.9287 -.7974 0 0 0 .7974 0 -.586 0 0 0 .586 0 -.498 0 0 0 .498 0 -.73843 0 0 0 .73843 0       B T = -C = 1.3629 0 0 0 0 VII. CONCLUSION
In this paper, a connection has been established between Ober's canonical form for continuous-time lossless systems and a Schur algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober's form in terms of boundary interpolation values and a recursive construction of the balanced realization. We thus have an analog in continuous-time of the connection between Hessenberg canonical form and Schur parameters in discrete-time ( [START_REF] Hanzon | Balanced parametrizations of stable SISO all-pass systems in discrete-time[END_REF]).

Schur parameters present interesting properties from an optimization view point. Stability and the order of the system are a built-in property, computations can be performed in state-space using balanced realizations and often exhibit a nice behavior. In discrete-time, Schur parameters have been successfully used as optimization parameters, mainly in rational L 2 approximation ( [START_REF] Marmorat | Matrix rational H2-approximation: a state-space approach using Schur parameters[END_REF]). Algorithm in discretetime can be used for continuous-time identification problems. However, the approach can now be developed directly in continuous-time for a better adaptation.

Applications to the design of hyperfrequency filters are expected but they would require the multivariable case since these filters are 2 × 2 systems. In this problem, as often, the physical parameters are not suitable for optimization. It is therefore natural to split the design into two steps: first, solve a rational approximation problem using dedicated parameters and then compute all possible physical realizations of the filter. These physical realizations have a particular structure connected with the geometry of the filter, made of coupled resonant cavities. They are balanced realizations of the form (A, C T , C, I) where

A = - 1 2 C T C -j M C = j √ 2R 1 0 • • • 0 0 • • • 0 j √ 2R 2 , R 1 , R 2 > 0.
The coupling matrix M is symmetric and real, it indicates the way resonators are coupled to one an other. An approach based on computer algebra is presently used to find all the possible physical realizations corresponding to the rational approximation computed at the first step. The use of optimization parameters coming from the MIMO Ober's form would provide an approximant with a realization closer to the desired structure and would facilitate the second step.

But we now expect that boundary interpolation problems for MIMO transfer functions would provide such parameters. This generalization is under study.

  However, it can be established that the original balanced canonical form of Ober is not recovered in this way.

INRIA, BP 93, 06902 Sophia-Antipolis Cedex, FRANCE, {olivi}@sophia.inria.fr, phone: 33-4-92387877, fax: 33-4-92387858 School of Mathematical Sciences, University College, Cork, Ireland, Tel.: +353-21-4902376, Fax +353-21-4903784, E-mail: b.hanzon@ucc.ie Dept. Mathematics, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht, The Netherlands, Tel.: +31-43-3883365, Fax: +31-43-3884910, Email: ralf.peeters@math.unimaas.nl linear fractional transformations.