domain: spi math

text:
Ahlem Sassi
email: ahlem.sassi@esme.fr

Islam Boussaada
email: islam.boussaada@l2s.centralesupelec.fr

Silviu-Iulian Niculescu

Reduced-Order Observer Design for Time-Delay Systems using Partial Pole Placement

This paper develops a functional reduced-order observer for a class of Linear Time-Invariant (LTI) systems with time-delay in the state. The proposed observer provides exponentially stable estimation errors. The unbiasedness conditions on the nominal part of the error dynamics of the reducedorder observer were given by employing an algebraic framework. Under such conditions, we propose the use of a partial pole placement of the error dynamical equations to handle sufficient conditions for the existence of the proposed observer. Specifically, we make use of the multiplicity-induced-dominancy property of the characteristic function corresponding to the system's error. The performance and the effectiveness of the developed observer is highlighted through two examples: first, the estimation of the HIV-1 infection dynamics, and second, the improvement of the environmental performance through the study of a diesel engine system.

I. INTRODUCTION

Delay-differential equations are one of the simplest ways to represent time heterogeneity. Transportation, propagation and communication represent the simplest phenomena where the delay appears naturally. As expected, due to its infinite dimensional character, the presence of time delay in a system model renders the control/observer design more difficult. However, it should be mentioned that if such a delay is properly taken into account, it may bring advantages in controlling system dynamics as emphasized by [START_REF] Richard | Time-delay systems : An overview of some recent advances and open problems[END_REF] [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF].

Over the past decades, several significant advances have been done in stability analysis and feedback control of timedelay systems, [START_REF] Niculescu | Delay Effects on Stability, A Robust Control Approach[END_REF], [START_REF] Gu | Stability of Time-Delay Systems[END_REF] and the references therein. In the sequel, we are interested on the observation of time-delay systems under the assumption that the delay is known, and, in particular, reduced-order observers. Roughly speaking, since the number of state variables in a reduced-order observer is less than the order of the considered system, the reducedorder observer is parsimonious, often a desirable engineering quality. But, in addition, a reduced-order observer may have better properties than a full-order observer, especially with respect to robustness of a control system which uses an observer to implement the control algorithm in an "observerbased" control design. Reduced-order observers are well recognized for having faster convergence rates and lower computational burden, as the only state variables to be Silviu.Niculescu@l2s.centralesupelec.fr estimated are the ones which are actually not measured (see, for example, [START_REF] Trinh | Linear functional state observer for time-delay systems[END_REF], [START_REF] Darouach | Existence and design of functional observers for linear systems[END_REF], [START_REF]Linear functional observers for systems with delays in state variables[END_REF], [START_REF] Boutayeb | Observer design for linear time-delay systems[END_REF], [START_REF] Trinh | Functional Observers for Dynamical Systems[END_REF].

It is well known that the aim of observer design is to reconstruct the trajectory of states of a dynamic system. An observer is an auxiliary system such that the error between the observer state and the system state decreases asymptotically towards zero. As a result, stability analysis tools are crucial in the design of observers. Among the existing methods, eigenvalue-based methods have become popular in the stabilization of linear time-invariant (LTI) time-delay systems [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems[END_REF] and recently in observers design [START_REF] Sassi | Observer design in LTI time-delay systems using partial pole placement with applications[END_REF], [START_REF] Rojas-Ricca | Multiplicity-induced dominance in stabilization of state predictors for time-delay systems[END_REF]. A recent pole placement approach, called Partial Poles Placement (PPP), has been introduced in [START_REF] Mazanti | Multiplicity-induceddominancy for delay-differential equations of retarded type[END_REF]. It derives from two properties called respectively multiplicity-induced-dominancy (MID) and coexistent-real-roots-inducing-dominancy (CRRID), see for instance [START_REF] Boussaada | Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design[END_REF].

In a previous work [START_REF] Sassi | Observer design in LTI time-delay systems using partial pole placement with applications[END_REF], we propose the design of a fullorder observer using the pole placement approach based on the MID property. Here, we propose a more general observer with reduced order to estimate a function of the states or the unmeasured ones. For the design, we call the generic multiplicity-induced-dominancy (GMID) which is a more general result for Delay Differential Equations (DDE) of order n with delayed polynomial term of order n -1, which allows us to avoid assumption of the real rootedness of the non-delayed polynomial corresponding to the characteristic function. To the best of the authors' knowledge, such a construction represents a novelty in the open literature.

In this paper, we investigate the design of a reduced order observer for delay systems which employs the pole placement approach, for stabilization of the error systems involving a known time-delay. Some prerequisites and the problem formulation are presented in Section II. In Section III, the main results on the existence of the observer and stability analysis of the error system are given. In Section IV, the proposed observer is applied to the linear model of HIV-1 infection dynamics and the diesel engine which both involves known time-delay. Some concluding remarks end the paper. The notations are standard.

II. PREREQUISITES AND PROBLEM STATEMENT

Consider a generic dynamical system with a single delay described by the DDE:

ξ(t) = Aξ(t) + A d ξ(t -τ ), (1)
under appropriate initial conditions, where ξ ∈ R n is the state vector, τ is a positive constant delay. The matrices: A ∈ R n×n and A d ∈ R n×n are known constant matrices.

It is well known that the asymptotic behavior of the solutions of (1) is determined from its spectrum (see, e.g. [START_REF] Bellman | Differential-difference equations[END_REF]), that is the set of the (characteristic) roots of the associated characteristic function (denoted ∆(s, τ ) in the sequel).

The characteristic function ∆ : C × R + → C corresponding to (1) writes as follows:

∆(s, τ ) = det(sI n -A -A d e -τ s ) (2)
A generic result on the location of spectral values corresponding to (2) is given by the following: Proposition 1: [START_REF] Michiels | Stability and stabilization of time-dely systems[END_REF] If s is a characteristic root of system (1), then it satisfies

|s| ≤ ||A + A d e -τ s || 2
(3) This result combined with the triangular inequality provides a generic envelope curve around the characteristic roots corresponding to system (1).

In the sequel, we are interested in the study of a class of LTI delay systems characterized by the quasipolynomial function of the form:

∆(s, τ ) = P 0 (s) + P 1 (s)e -τ s (4)
where deg(P 0 ) > deg(P 1 ). The question of analytical characterization of its rightmost root will be a key in the resolution of our problem.

A. Partial pole placement in delay systems

One of the most natural and classical ways to stabilize a dynamical LTI delay system is to select the free parameters of the controller in order to choose the location of finitely many roots while also guaranteeing that the dominant root 1 is among the chosen ones. This has been the subject of several recent works, such as [START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay[END_REF], [START_REF] Boussaada | Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design[END_REF], [START_REF] Mazanti | Multiplicity-induceddominancy for delay-differential equations of retarded type[END_REF]. Contrary to the strategy of FPP used, e.g., in [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF], the controllers designed calling these approaches do not make the closedloop system finite-dimensional, instead, control its rightmost spectral value. These methods also extend to some partial differential equations, as detailed, for example, in [START_REF] Mazanti | Effects of roots of maximal multiplicity on the stability of some classes of delay differential-algebraic systems: the lossless propagation case[END_REF]. In particular, we shall use the MID property as described in [START_REF] Balogh | Conditions for stabilizability of time-delay systems withreal-rooted plant[END_REF]. For LTI systems including delays in their model representation, spectral methods can be used to understand the asymptotic behavior of solutions by considering the roots of some characteristic function (see, e.g., [START_REF] Hale | Introduction to functional differential equations[END_REF], [START_REF] Michiels | Stability, control, and computation for time-delay systems: An eigenvalue-based approach[END_REF], [START_REF] Bellman | Differential-difference equations[END_REF], [START_REF] Cooke | On zeroes of some transcendental equations[END_REF], [START_REF] Stépán | Retarded dynamical systems: stability and characteristic functions[END_REF], [START_REF] Wright | Stability criteria and the real roots of a transcendental equation[END_REF]) which, for (1), is the function

∆ : C → C defined for s ∈ C by ∆(s) = s n + n-1 k=0 a k s k + e -sτ m k=0 α k s k . ( 5
)
More precisely, the exponential behavior of solutions of ( 5) is given by the real number γ 0 = sup{ℜ(s)/s ∈ C, ∆(s) = 0}, called the spectral abscissa of ∆, in the sense that, for every ε > 0, there exists C > 0 such that, for every solution e of ( 10), one has

|e(t)| ≤ Ce (γ0+ε)t max θ∈[-τ,0] |e(θ)| [21].
Moreover, all solutions of (5) converge exponentially to 0 if 1 the rightmost characteristic root in the complex plane and only if γ 0 < 0. An important difficulty in the analysis of the asymptotic behavior of ( 5) is that, contrary to the delayfree case, the corresponding characteristic function ∆ has infinitely many roots. Theorem 1 ( [START_REF] Boussaada | The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions[END_REF]): Consider the qua-sipolynomial ∆ given by [START_REF] Trinh | Linear functional state observer for time-delay systems[END_REF]. Let s 0 ∈ R be a root of ∆ with maximal multiplicity i.e. M (s

0 ) = deg(∆), then, 1) (Retarded) If m < n, then s 0 is a strictly dominant root of ∆. 2) (Neutral) If m = n, then s 0 is a dominant root of
∆ and, for every other complex root s of ∆, one has ℜ(s) = s 0 . Notice that the GMID consists in forcing a root to reach its maximal multiplicity, which does not allow any degree of freedom in assigning s 0 . In order to allow for some additional freedom when assigning s 0 , one can relax such a contraint by forcing the root s 0 to have a multiplicity lower than the maximal.

B. Problem formulation 1) Studied Class of systems: The class of system investigated in this work is defined by the following state space model,

ẋ(t) = Ax(t) + A d x(t -τ ) + Bu(t) (6a) y(t) = Cx(t) (6b) x(t) = ϕ(t) t ∈ [0, τ ] (6c) z(t) = Lx(t) (6d)
where x ∈ R n , u ∈ R m , y ∈ R p are the states vector, the control input and the measurements vector respectively. z ∈ R r is the vector to be estimated where r ≤ n. τ is a known constant delay. Finally, ϕ(t) is a continuous function defining the corresponding initial condition. The matrices:

A ∈ R n×n , A d ∈ R n×n , B ∈ R n×m and C ∈ R p×n are known constant matrices of appropriate dimensions. It is assumed that: rank(L) = r (7)
In order to estimate the states z(t), we consider the following structure of functional observer

η(t) = N η(t) + N d η(t -τ ) + M y(t) + M d y(t -τ ) (8a) + F u(t) (8b) ẑ(t) = η(t) + Ey(t) (8c)
where η ∈ R r . ẑ ∈ R r is the estimate of z. The matrices N , M , N d , M d , F and E are unknown, which should be determined later such that the estimation of the states ẑ converge to its real values z.

2) The error equation: Let us note e(t) = z(t) -ẑ(t) = Ψx(t) -η(t), where Ψ = L -EC. The estimation error dynamics is described by the following system

ė(t) = N e(t) + N d e(t -τ ) + (ΨA -N Ψ -M C)x(t) + (ΨA d -N d Ψ -M d C)x(t -τ ) + (F -ΨB)u(t) (9)
Proposition 2: The dynamical system (8) represents a reduced funtional observer for system [START_REF] Darouach | Existence and design of functional observers for linear systems[END_REF], if and only if the dynamic error system given by

ė(t) = N e(t) + N d e(t -τ ) (10)
is asymptotically stable and the following equation hold

F -ΨB = 0 ΨA -N Ψ -M C = 0 ΨA d -N d Ψ -M d C = 0 Proof 1:
Let us compute the dynamics of the estimation error, as follows:

ė(t) = Ψ ẋ(t) -η(t) = ΨAx(t) + ΨA d x(t -τ ) + ΨBu(t) -N η(t) -N d η(t -τ ) -M y(t) -M d y(t -τ ) -F u(t)
By adding and substracting the term ±N Ψx(t) ± N d Ψx(tτ ), system (9) is obtained. Thus, in order to ensure the unbiasedness of the proposed observer, the dynamics of the error estimation must be independant of the state vector x and the input vector u. So, we need to put the terms which do not depend on these errors equal to zero. The convergence of the proposed observer require the stability of the error system [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems[END_REF].□

The first step for the stability analysis of the estimation error is to guarantee the unbiasedness of system [START_REF] Trinh | Functional Observers for Dynamical Systems[END_REF]. In other words,we have to put the terms which do not depend on the error equal to zero. Thus, the matrix F had to be chosen as

F = ΨB (11)
and the following Sylvester equations must hold

ΨA -N Ψ -M C = 0 (12a) ΨA d -N d Ψ -M d C = 0 (12b)
Then, for the second step, we need to ensure the stability of the error dynamics, i.e. to choose the matrices N and N d in order to guarantee the stability of the error dynamics.

III. MAIN RESULTS

A. Resolution of the Sylvester equations

To ensure the unbiasedness of the observer, the Sylvester equations described by ( 12) should be verified.

Using the expression Ψ = L-EC, the Sylvester equations are rewritten as

LA = N L + KC + ECA (13a) LA d = N d L + K d C + ECA d ( 13b
)
where K = M -N E, and

K d = M d -N d E.
It's clear that these Sylvester equations ( 13) can be written in the following compact form

V = XW (14)
where V and W contain the known matrices such as

V = [LA, LA d ] W T =       L 0 0 L C 0 0 C CA CA d      
and X is the matrix of the unknown observer's matrices to be designed, described as follows

X = [N, N d , K, K d , E]
Equation ( 14) admits a solution if and only if the following rank condition is satisfied

rank(W ) = rank([V T , W T ] T ) (15)
and, by defining ℓ = 2(r + p) + p, the general solution is given by

X = V W † + Z(I ℓ -W W † ) ( 16
)
where Z is an arbitrary matrix with appropriate dimension, which will be chosen in order to satisfy the convergence of the estimation errors and W † is any generalised inverse of W , which fulfils the following equation [START_REF] Rao | Generalized Inverse of Matrices and its Applications[END_REF]

W = W W † W (17)
Then, the observers matrices can be expressed through a single gain matrix Z as

      N N d K K d E       X =       N 1 N d1 K 1 K d1 E1       V W † +Z       N 2 N d2 K 2 K d2 E2       I ℓ -W W † (18)
In other words:

N = V W † R N + Z(I ℓ -W W † )R N (19a) N d = V W † R N d + Z(I ℓ -W W † )R N d (19b) K = V W † R K + Z(I ℓ -W W † )R K (19c) K d = V W † R K d + Z(I ℓ -W W † )R K d (19d) E = V W † R E + Z(I ℓ -W W † )R E (19e) with R T N = [I r 0 r×r 0 p×r 0 p×r 0 p×r ] R T N d = [0 r×r I r 0 p×r 0 p×r 0 p×r ] R T K = [0 r×p 0 r×p I p 0 p×p 0 p×p ] R T K d = [0 r×p 0 r×p 0 p×p I p 0 p×p ] R T E = [0 r×p 0 r×p 0 p×p 0 p×p I p ]

B. Stability and attenuation analysis of the estimation error

Since the unbiasedness conditions given in section III-A are satisfied, the dynamic of the estimation error is written as in equation [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems[END_REF]. Let us consider the estimation error dynamic written in the following form

ė(t) -N e(t) -N d e(t -τ ) = 0 (20)
So that the characteristic function associated ∆ :

C × R + → C is given by ∆(s, τ ) = |sI n -N -N d e -τ s | (21)
Hence, the error system is characterised by the quasipolynomial function with the form of equation ( 4) where the polynomial P 0 is given by:

P 0 (s) = |sI n -N | = n n s n + n n-1 s n-1 + . . . + n 1 s + n 0
In order to check the stability of the estimation error, the following assumption is necessary:

Assumption 1: rank(N d ) = 1.
In a previous work [START_REF] Sassi | Observer design in LTI time-delay systems using partial pole placement with applications[END_REF] the intermediate MID property has been exploited through the result of [START_REF] Balogh | Conditions for stabilizability of time-delay systems withreal-rooted plant[END_REF] restricting the observer design to the following additional assumption:

Assumption 2: The polynomial P 0 to be realrooted. The use of the GMID property allows to relax such an assumption by using the result from [START_REF] Boussaada | The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions[END_REF]. In the following theorem, we state sufficient condition that ensures the stability of the error estimation and thus the convergence of the proposed observer.

Theorem 2: Consider that Assumption 1 holds and let a negative s 0 be a multiple root of ( 21) with the maximal multiplicity. The error system ( 10) is exponentially stable with s 0 as decay rate and the estimated states vector converge to its real value.

Proof 2: Since condition 15 holds, the error dynamics is written under equation [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems[END_REF]. Under the assumption 1, the quasipolynomial ( 21) is a particular case of (5). Thus, following Theorem 1, a root s 0 of ∆ is necessarily dominant. So that, it corresponds to the exponential decay of [START_REF] Balogh | Conditions for stabilizability of time-delay systems withreal-rooted plant[END_REF]. □

IV. ILLUSTRATIVE EXAMPLES

In this section, two case studies are proposed and discussed.

A. HIV-1 Infection Model

In HIV-1 infection, the virus life cycle plays a crucial role in disease progression. The binding of a viral particle to a receptor on a target cell initiates a cascade of events that can ultimately lead to the target cell becoming productively infected, i.e. producing new virus. Some models [START_REF] Perelson | Hiv-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time[END_REF], [START_REF] Perelson | Mathematical analysis of hiv-1, dynamics in vivo[END_REF] assumed this process to occur instantaneously. In other words, it is assumed that as soon as virus contacts a target cell, the cell begins producing the virus. However, in reality there is a time delay between initial viral entry into a cell and subsequent viral production. This delay needs to be taken into account to accurately determine the half life of free virus from drug perturbation experiments. The intracellular delay is incorporated in the system model by assuming that the generation of virus producing cells at time t is due to the infection of target cells at time (t -τ ), where τ is a constant delay. The HIV-1 dynamics is given by the following equations

dT * (t) dt = kT 0 V I (t -τ ) -δT * (t) (22a)
dV

I (t) dt = (1 -n p )N δT * (t) -cV I (t) (22b)
dV N I (t) dt = n p N δT * (t) -cV N I (t) (22c)
where T * is the concentration of productively infected Tcells, V I and V N I represent the plasma concentrations of virions in the infectious pool (produced before the drug effect) and in the noninfectious pool (produced after the drug effect), respectively. For detailed study, refer to [START_REF] Perelson | Hiv-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time[END_REF] on the experiment and the data. A detailed list of variable is given below. To apply the proposed approach, it is convenient to transform system [START_REF] Michiels | Stability, control, and computation for time-delay systems: An eigenvalue-based approach[END_REF] into the form of system [START_REF] Darouach | Existence and design of functional observers for linear systems[END_REF], where x(t) = [T * (t) V I (t) V N I (t)] T , and the matrices A and A d are given by

A =   -δ 0 0 (1 -n p )N δ -c 0 n p N δ 0 -c   A d =   0 kT 0 0 0 0 0 0 0 0   (23)
The level of virions circulating in plasma for both infectious and non infectious pool can be mesaured. So, we assume that the vector of measurement is given by

y(t) = Cx(t) = 0 1 0 0 1 1 x(t) = V I (t) V I (t) + V N I (t)
and we propose to estimate the concentration of productively infected T-cells denoted T * . In other words, we propose to estimate z(t) = T * (t), with L = [1 0 0]. For this aim, we consider an observer in the form [START_REF] Boutayeb | Observer design for linear time-delay systems[END_REF]. So the error dynamics is given by [START_REF] Trinh | Functional Observers for Dynamical Systems[END_REF]. For this application, the rank condition [START_REF] Bellman | Differential-difference equations[END_REF] holds. As the observer order is r = 1, so assumption 1 is already considered and just need to tune the matrix Z to obtain a stable error system. The quasipolynomial function is with the following form

∆(s, τ ) = (s -N ) -e -τ s N d = (s -(N 1 + ZN 2 )) -e -τ s (N d1 + ZN d2 ) (24)
Where Z ∈ R r×ℓ . The spectrum of the quasipolynomial is computed for a matrix Z = z 1 z 2 0 0 0 0 0 0 and is shown in Figure 1. By setting the initial condition to x(0) = [1 180 0] T and η(0) = [0] and with a suitable choice of the matrix Z = -1.57 1 0 0 0 0 0 0 , the Fig. 1. Spectrum distribution of the quasipolynomials [START_REF] Stépán | Retarded dynamical systems: stability and characteristic functions[END_REF] estimation of the concentration of productively infected Tcells z(t) converges to its real values T * (t) (see Figure 2).

B. Diesel Engine Model

A diesel engine with an exhaust gas recirculation (EGR) valve and a turbo-compressor with a variable geometry turbine (VGT) was modeled in [START_REF] Jankovic | Delay Differential Equations : Recent Advances and New Directions[END_REF]. The system is subjected to intake-to-exhaust transport delay. A LTI time-delay model is obtained by linearizing the original non-linear model under the assumption that the engine operates at a constant speed N = 1500 RPM. This model is described as

ẋ(t) = A 0 x(t) + A 1 x(t -τ ) + Bu(t) (25a) y(t) = Cx(t) = [0 1 0]x(t) (25b



 and τ is given as 0.06s at this particular operating point. The state variables are defined as intake manifold pressure (x 1 ), exhaust manifold pressure (x 2 ), and compressor power (x 3 ). Furthermore, the system has two control inputs in which u 1 is an input for the EGR valve openning and u 2 is an input for the VGT mass flow rate.

The model includes intake-to-exhaust transport delay τ = 60ms when engine speed is 1500 RPM. Due to the timedelay assignable to the transport time of the gas from intake to exhaust manifold, the system is represented a DDE as in Equation [START_REF] Wright | Stability criteria and the real roots of a transcendental equation[END_REF]. Since the observability matrix is full rank matrix, system [START_REF] Wright | Stability criteria and the real roots of a transcendental equation[END_REF] where r = 2, P 0 (s) is with order 2 and P 1 (s) is with order 1. The matrix Z is tuned to get assumption 1.

For instance, choosing Z as :

Z =
x 0 0 0 0 0 0 0 0 y 0 0 0 z one can define the rightmost root s 0 of ( 26), with a maximal multiplicity 4, as a function of the delay τ as shown in Figure 3. To show the convergence of the states estimation to For the simulation, the initial states are chosen as x(0) = 0.3 0.2 -0.3 T and z(0) = 0 0 T . As shown in Figure 4, the error e(t) of the states and their estimation converge to zero in a short span of time. Thus, the observer reveals satisfactory performance. Evolution of the errors between the system's states and their estimations.

V. CONCLUDING REMARKS

In this paper, a reduced-order observer was developed for a class of LTI systems described by delay-differential equations, including a known delay in the system statespace representation. By using a pole placement approach derived from the GMID property, we present some sufficient conditions for the existence of the observer, which guarantee that the estimation error converges asymptotically towards zero. The efficiency of the proposed method is illustrated through two examples.

Fig. 2 .

 2 Fig. 2. Evolution of concentration of productively infected T-cells T * (t)(-) and its estimation z(t)(--).

  is observable and the corresponding quasipolynomial function is given by∆(s, τ ) = det(sI r -N -N d e -τ s ) = det(sI r -(N 1 + ZN 2 )) -(N d1 + ZN d2 )e -τ s(26)

Fig. 3 .

 3 Fig. 3. The spectral abcissa of (26) as a function of the delay τ .

  Fig.[START_REF] Gu | Stability of Time-Delay Systems[END_REF].Evolution of the errors between the system's states and their estimations.