Input-Output Representation and Identifiability ofDelay Parameters for Nonlinear Systems withMultiple Time-Delays

Milena Anguelova1,2 and Bernt Wennberg1,2

1 Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg,Sweden milena@chalmers.se

2 Mathematical Sciences, Göteborg University, SE-412 96 Gothenburg, Sweden

Summary. We have analyzed the identifiability of time-lag parameters in nonlinear delaysystems using an algebraic framework. The identifiability is determined by the form of thesystem’s input-output representation. The values of the time lags can be found directly fromthe input-output equations, if these can be obtained explicitly. Linear-algebraic criteria areformulated to decide the identifiability of the delay parameters when explicit computation ofthe input-output relations is not possible.

1 Introduction

Observability and parameter identifiability are important properties of a system where ini-tial state or parameter estimation are concerned. These properties guarantee that the desiredquantities can be uniquely determined from the available data.

For nonlinear systems without time delays, these properties are well-characterized, seefor instance [7, 13, 14] and the references therein. The characterization of observability andidentifiability has now been extended to nonlinear systems with time delays in [16] and [17],using an algebraic approach introduced in [10], and developed in [9]. In these works the timedelays themselves are assumed to be known, or multiples of a unit delay. The identifiabilityof general unknown time-delays has been analyzed only for linear systems [11, 15, 12, 3].Recently, we used the mathematical setting of [10, 9] and [16] to analyze the identifiabil-ity of the time-delay parameter for nonlinear systems with a single unknown constant timedelay ([2]).

In this paper we analyze the identifiability of the time-delay parameters for nonlinearcontrol systems with several unknown constant time delays. We show that state eliminationproduces input-output relations for the system, the form of which decides the identifiabilityof the delay parameters. The values of the delay parameters can be found directly from theinput-output equations, if the latter can be obtained explicitly. We formulate linear-algebraiccriteria to check the identifiability of the delay parameters which eliminate the need for anexplicit calculation of the input-output relations. The identifiability of the delay parameterscan be a necessary but not sufficient condition for the observability of the state variables (andidentifiability of the regular parameters in the system). The already established methods for

J.J. Loiseau et al. (Eds.): Topics in Time Delay Systems, LNCIS 388, pp. 243–253.springerlink.com c© Springer-Verlag Berlin Heidelberg 2009

244 M. Anguelova, and B. Wennberg

testing weak observability and identifiability for nonlinear delay systems alone [16, 17] can-not be used to determine the identifiability of the time-delay parameters and a prior analysisis necessary for the latter.

2 Notation and preliminary definitions

Consider nonlinear time-delay systems of the form:⎧⎪⎪⎨⎪⎪⎩x(t) = f

(x(t), x(t− τ1), . . . , x(t− τ�), u(t), u(t− τ1), . . . , u(t− τ�)

)y(t) = h

(x(t), x(t− τ1), . . . , x(t− τ�)

)x(t) = ϕ(t), t ∈ [−maxiτi, 0]u(t) = u0(t), t ∈ [−T, 0]

, (1)

where x ∈ Rn denotes the state variables, u ∈ Rm is the input and y ∈ Rp is theoutput. The unknown constant time-delays are denoted by the vector τ = (τ1, . . . , τ�),τi ∈ [0, T ), T ∈ R. The entries of f and h are meromorphic in their arguments and itis reasonable to assume that none of the hi are constants. The unknown continuous functionof initial conditions is denoted by ϕ : [−maxi(τi), 0] → Rn. The set of initial functions forthe variables x is denoted by C := C([−maxi(τi), 0],R

n). A meromorphic input functionu(t) is called an admissible input if the differential equation above admits a unique solution.The set of all such input functions is denoted by CU .

One of the objectives of this work is to investigate the property of local identifiability ofthe delay parametersτi. Intuitively, τi are identifiable if any two sets can be distinguished bythe system’s input-output behaviour. A formal definition is as follows:

Definition 1. The delay parameters τ are said to be locally identifiable at τ0 ∈ (0, T )� ifthere exists an open set W ( τ0,W ⊂ [0, T )�, such that ∀τ1 ∈ W : τ1 �= τ0, ∀ϕ0, ϕ1 ∈ C,there exist t ≥ 0 and u ∈ CU such that y(t, ϕ1, u, τ1) �= y(t, ϕ0, u, τ0), where y(t, ϕ, u, τ )denotes the parameterized output for the initial function ϕ, the admissible input u anddelays τ .

Following the notations and algebraic setting of [10, 9, 16] and [17], let K be the field ofmeromorphic functions of a finite number of variables from

{x(t− iτ ), u(t− iτ ), . . . , u(l)(t− iτ ), i = (i1, . . . , i�), ij , l ∈ Z+} ,

where we have denoted i1τ1 + · · · + i�τ� by iτ . Let E be the vector space over K given by

E = spanK{dξ : ξ ∈ K} . (2)

Then E is the set of linear combinations of a finite number of one-forms from {dx(t −iτ ), du(t− iτ ), . . . , du(l)(t− iτ )} with row vector coefficients in K.

Let K(δ] denote the set of polynomials in δ1, . . . , δ� with coefficients from K. This setis a noncommutative ring, where addition is defined as usual, while multiplication is definedas follows. Let a(δ] be a polynomial in K(δ], a(δ] =

∑k akδk , where we have denoted

δk11 . . . δk�� by δk , k = (k1, . . . , k�). Order the different powers k according to the largestk1, then k2, etc. According to this order, let the highest degree of a(δ] be ra , where ra =(ra,1, . . . , ra,�) and analogously for another polynomial b(δ] in K(δ]. Multiplication of a(δ]and b(δ] is then given by

Identifiability of Delay Parameters 245

a(δ]b(δ] =

ra+rb∑k=0

i≤ra,j≤rb∑i+j=k

ai(t)bj(t− iτ )δk . (3)

The ring K(δ] is Noetherian (see [1] for a proof) and therefore also a (left) Ore domain byCorollary 8.10 in Chapter 0 of [4]. It thus has the invariant basis number (IBN) by Proposi-tions (1.8) and (1.13) in [8] - a free left module over K(δ] has a uniquely defined rank and allits bases have the same cardinality. Let M denote the module spanK(δ]{dξ : ξ ∈ K}. Theclosure of a submodule N in M is the submodule

N = {w ∈ M : ∃a(δ] = a(δ1, . . . , δ�] ∈ K(δ], a(δ]w ∈ N} .

Equivalently, N is the largest submodule of M containing N and having a rank equal torankK(δ]N [16].

Differentiation of functions φ(x(t−iτ ), u(t−jτ ), . . . , u(l)(t−jτ )) in K and one-formsω =

∑i κ

ixdx(t− iτ )+

∑j,r νjdu

(r)(t− jτ ) in M is defined in the natural way [16, 17]:

φ =∑

i

∂φ

∂x(t− iτ )δif +

∑j,r

∂φ

∂u(r)(t− jτ )u(r+1)(t− jτ ) (4)

ω =∑

i

κixdx(t− iτ ) +

∑j,r

νjdu(r)(t− jτ ) +

+∑

i

κixdδif +

∑j,r

νjdu(r+1)(t− jτ ) . (5)

A one-form ω ∈ M is also an element of E and Poincaré’s lemma holds (see Lemma 3in [9]):

Lemma 1 (Poincaré). Consider a one-form ω ∈ M. Then there exists a function ξ ∈ K suchthat (locally) ω = dξ if and only if dω = 0.

Define

X = spanK(δ]{dx} , (6)

Yk = spanK(δ]{dy, dy, . . . , dy(k−1)} , (7)

U = spanK(δ]{du, du, . . . } . (8)

Then, from [16],(Yk + U) ∩ X = (Yn + U) ∩ X

for k ≥ n andrankK(δ](Yn + U) ∩ X ≤ n .

3 State elimination

In this section we consider the problem of obtaining an input-output representation from thestate-space form of a time-delay control system. This problem has been treated for polyno-mial systems in [6]. We show that for a system of the form (1) there always exists, at leastlocally, a set of input-output delay-differential equations of the form:

246 M. Anguelova, and B. Wennberg

F (δ, y, . . . , y(k), u, . . . , u(J)) :=

= F (y(t− i0τ ), . . . , y(k)(t− ikτ ), u(t− j0τ ), . . . , u(J)(t− jlτ )) = 0 ,(9)

such that any pair (y(t), u(t)) which solves (1), also satisfies (9), for t such that all derivativesinvolved are continuous. The function F is meromorphic in its arguments.

Theorem 1. Given a system of the form (1), there exists an integer J ≥ 0 and an open densesubset V of C × CJ+1

U , such that in the neighborhood of any point of V , there exists aninput-output representation of the system of the form (9).

Proof : The proof is an adaptation of the proof of Theorem 2.2.1. in [5] for the analogousresult for ODE-systems.

Let f be an r-dimensional vector with entries fj ∈ K. Let ∂f∂x

denote the r × n matrixwith entries (

∂f

∂x

)j,i

=∑

k

∂fj∂xi(t− kτ )

δk ∈ K(δ] , (10)

where the sum is over finitely many terms, for fj is a function of finitely many delays of xi.Denote by s1 the least positive integer such that

rankK(δ]∂(h1, ..., h

(s1−1)1 )

∂x= rankK(δ]

∂(h1, ..., h(s1)1 )

∂x. (11)

Inductively, for 1 < i ≤ p, set si = 0 if

rankK(δ]

∂(h1, ..., h(s1−1)1 , . . . , hi−1, . . . , h

(si−1−1)

i−1 )

∂x=

= rankK(δ]

∂(h1, ..., h(s1−1)1 , . . . , hi−1, . . . , h

(si−1−1)

i−1 , hi)

∂x, (12)

and otherwise, let si be the least positive integer such that

rankK(δ]∂(h1, ..., h

(s1−1)1 , . . . , h

(si−1)i )

∂x= rankK(δ]

∂(h1, ..., h(s1−1)1 , . . . , h

(si)i )

∂x. (13)

Let S = (h1, ..., h(s1−1)1 , . . . , h

(sp−1)p ), where hi does not appear if si = 0. Then

rankK(δ]∂S

∂x= s1 + · · · + sp ≤ n . (14)

For simplicity, denote the elements in S by x1, . . . , xs1+···+sp . The definition of si (equation

(13)) implies that∂h

(si)i∂x

is in M, where

M = spanK(δ]

{∂(x1, . . . , xs1+···+si)

∂x

}. (15)

Thus, there exist nonzero polynomials ai(δ] ∈ K(δ], i = 1, . . . , p such that ai(δ]∂h

(si)i∂x

∈M. Therefore,

ai(δ]dh(si)i +

m∑r=1

J∑j=0

bi,j,r(δ]du(j)r −

s1+···+si∑j=1

ci,j(δ]dxj = 0 , (16)

Identifiability of Delay Parameters 247

for some J ≥ 0, where J is the highest derivative of u appearing in the functions in Sand bi,j,r(δ], ci,j(δ] ∈ K(δ]. Since all functions are assumed meromorphic and we havecontinuous dependence for the output on the input and initial function, the above equalityholds on an open dense set of C × CJ+1

U . The left hand side of equation (16), being equalto zero, is a closed one-form on M. Applying the Poincaré lemma, Lemma 1, we obtainfunctions ξi ∈ K such that

dξi = ai(δ]dh(si)i +

m∑r=1

J∑j=0

bi,j,r(δ]du(j)r −

s1+···+si∑j=1

ci,j(δ]dxj

andξi(δ, h

(si)i , x1, . . . , xs1+···+si , u, . . . , u

(J)) = 0 . (17)

This produces an input-output equation

ξi(δ, y(si)i , y1, . . . , y

(s1−1)1 , . . . , y(si−1)

i , u, . . . , u(J)) = 0 , (18)

if si ≥ 1 and

ξi(δ, yi, y1, . . . , y(s1−1)1 , . . . , y

(si−1−1)

i−1 , u, . . . , u(J)) = 0 , (19)

for si = 0, where only those yj for which sj ≥ 1 appear in equations (18) and (19).This is true for each i, 1 ≤ i ≤ p, resulting in p input-output equations of the form (9). �

4 Identifiability of the delay parameters

In this section we formulate criteria for determining the identifiability of the delay parametersτi from (1). The starting point is the linear form of the input-output equations (16), whosei-th equation is

ai(δ]dy(si)i +

m∑r=1

J∑j=0

bi,j,r(δ]du(j)r =

s1+···+si∑j=1

ci,j(δ]dxj . (20)

With no loss of generality, we assume that the polynomials ai(δ], bi,j,r(δ] and ci,j(δ] abovehave an element from K as greatest common divider. Let

ai(δ] =∑

k

aikδk , bi,j,r(δ] =∑

k

bi,j,rkδk and ci,j(δ] =

∑k

ci,jkδk .

Denote all the different monomials δk appearing above by Δi,1, . . . ,Δi,q . Each Δi,j can bewritten

Δi,j = δki,j,11 . . . δ

ki,j,�� , (21)

where ki,j,1, . . . , ki,j,� are nonnegative integers, and Δi,j thus represents the (integer) linearcombination Ti,j = ki,j,1τ1 + · · · + ki,j,�τ� of the different time delays. If any one of theterms in ai(δ], bi,j,r(δ] or ci,j(δ] is a polynomial in δ of degree zero, that is, the input-output equations contain un-delayed variables, then we set Δi0 equal to δ0, where δ0 denotesthe identity operator, and the corresponding Ti,0 is zero. There are Ai,s, Bi,j,r,s and Ci,j,s inK such that

248 M. Anguelova, and B. Wennberg

ai(δ] =

q∑s=0

Ai,sΔi,s, bi,j,r(δ] =

q∑s=0

Bi,j,r,sΔi,s, ci,j(δ] =

q∑s=0

Ci,j,sΔi,s. (22)

Let Δi = δki,11 . . . δ

ki,�� be the monomial δk in ai(δ] with smallest index k = ki,j,1, . . . , ki,j,�

(ordered after ki,j,1, . . . , ki,j,�) - it is either equal to Δi,0 or is among the Δi,1, . . . ,Δi,q . De-note its corresponding linear combination of time delays by Ti. Consider the matrix

M =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

k11,1 − k1,1 · · · k11,� − k1,�

......

...k1q,1 − k1,1 . . . k1q,� − k1,�

......

...kp1,1 − kp,1 · · · kp1,� − kp,�

......

...kp1,1 − kp,1 · · · kpq,� − kp,�

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (23)

for which we have the equality

[T11 − T1, . . . , T1q − T1, . . . , Tp1 − Tp, . . . , Tpq − Tp]tr = M [τ1, . . . , τ�]

tr , (24)

where vtr denotes the transpose of v.We can now formulate the identifiability criteria for τ1, . . . , τ� in the following proposi-

tion:

Proposition 1. If M is defined as above and y(si+j)i (t) is not identically equal to zero for

any 0 ≤ j < iq − 1, i = {1, . . . , p}, then τ1, . . . , τ� are locally identifiable generically, ifand only if rank(M) = ".

Proof : Using (22) in (20) and multiplying by Δ−1i from the left we obtain

Δ−1i

∑qs=0 Ai,sΔi,sdy

(si)i = Δ−1

i

∑s1+···+sij=1

∑qs=0 Ci,j,sΔi,sdxj−

− Δ−1i

∑mr=1

∑Jj=0

∑qs=0 Bi,j,r,sΔi,sdu

(j)r ,

(25)

which can be written (with no un-delayed terms dy(si)i on the right-hand side)

dy(si)i = Δ−1

i

∑s1+···+sij=1

∑qs=0 Ci,j,sΔi,sdxj−

− Δ−1i

∑mr=1

∑Jj=0

∑qs=0 Bi,j,r,sΔi,sdu

(j)r −

− Δ−1i

∑qs=1 Ai,sΔi,sdy

(si)i .

(26)

The Poincaré lemma, Lemma 1, then gives locally

y(si)i (t) = fi

(x1(t), . . . , xs1+···+si(t), u(t), . . . , u(J)(t),

x1(t− Ti1 + Ti), . . . , xs1+···+si(t− Ti1 + Ti), y(si)i (t− Ti1 + Ti),

u(t− Ti1 + Ti), . . . , u(J)(t− Ti1 + Ti),

. . . ,

x1(t− Tiq + Ti), . . . , xs1+···+si(t− Tiq + Ti), y(si)i (t− Tiq + Ti),

u(t− Tiq + Ti), . . . , u(J)(t− Tiq + Ti)

).

(27)

Let first rank(M) = ". Then, q ≥ " and we thus have at least " different Δiq :s. Considerthose of the equations (27), for which iq ≥ 1. Evaluated at a fixed time point t0 ≥ T , (27)gives an equation for Ti1 − Ti, . . . , Tiq − Ti:

Identifiability of Delay Parameters 249

y(si)i (t0) = fi

(x1(t0), . . . , xs1+···+si(t0), u(t0), . . . , u

(J)(t0),

x1(t0 − Ti1 + Ti), . . . , xs1+···+si(t0 − Ti1 + Ti), y(si)i (t0 − Ti1 + Ti),

u(t0 − Ti1 + Ti), . . . , u(J)(t0 − Ti1 + Ti),

. . . ,

x1(t0 − Tiq + Ti), . . . , xs1+···+si(t0 − Tiq + Ti), y(si)i (t0 − Tiq + Ti),

u(t0 − Tiq + Ti), . . . , u(J)(t0 − Tiq + Ti)

).

(28)If the time-point t0 is chosen large enough to ensure the existence of all time-derivativesinvolved (which can be achieved by choosing for example t0 ≥ (maxisi − 1)T ), then dif-ferentiating (28) with respect to time gives new equations for Ti1 − Ti, . . . , Tiq − Ti whichare independent, since dy(j)

i , j ≥ 0 are linearly independent over K due to iq ≥ 1:

y(si+j)i (t0) = dj

dtjfi(x1(t), . . . , xs1+···+si(t), u(t), . . . , u(J)(t),

x1(t− Ti1 + Ti), . . . , xs1+···+si(t− Ti1 + Ti), y(si)i (t− Ti1 + Ti),

u(t− Ti1 + Ti), . . . , u(J)(t− Ti1 + Ti),

. . . ,

x1(t− Tiq + Ti), . . . , xs1+···+si(t− Tiq + Ti), y(si)i (t− Tiq + Ti),

u(t− Tiq + Ti), . . . , u(J)(t− Tiq + Ti)

)|t0.

(29)Unless y(si+j)

i (t) is identically zero for some 0 ≤ j < q − 1, the first q of these equationsidentify Ti1 − Ti, . . . , Tiq − Ti locally (the rest can in some cases be used to analyze globalidentifiability). Since rank(M) = ", equation (24) implies that the time delays τi, i =1, . . . , " are generically defined uniquely by the locally identifiable linear combinations Ti1−Ti, . . . , Tiq − Ti (an obvious exception is the case of commensurate time-delays). Thus, allτi, i = 1, . . . , " are generically locally identifiable, which completes the proof.

Let now rank(M) < ", that is, there are infinitely many τi:s which give the same linearcombinations {Ti1 − Ti, . . . , Tiq − Ti}, i = 1, . . . , p and the input-output equations (27)cannot be used to identify the τi:s. We treat the case " ≥ 2 and refer to [2] for the case of asingle delay. If rank(M) = 0, the proof is analogous to the proof of Theorem 2 in [2] and istherefore left out.

We will show that τ1, . . . , τ� are not locally identifiable generically by showing that (1)can be represented locally as a neutral system with time lags τ = (τ1, . . . , τ�), where thelatter are the nonzero elements in {Ti1− Ti, . . . , Tiq− Ti}, i = 1, . . . , p. To see this, observethat by using equation (27), we obtain the following delay-differential equations for each iwith si ≥ 1:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

˙xs1+···+si−1(t) = xs1+···+si−1+1(t)...

˙xs1+···+si(t) = fi( ˙xs1+···+si(t− τ1), . . . , ˙xs1+···+si(t− τ�), x(t), x(t− τ1),

. . . , x(t− τ�), u(t), . . . , u(J)(t), u(t− τ1), . . . , u(t− τ�), . . . ,

u(J)(t− τ1), . . . , u(J)(t− τ�)) .

(30)We now combine these delay-differential equations for all i = 1, . . . , p with si ≥ 1 and setthe outputs y1, . . . , yp equal to y1, . . . , yp. The outputs yj for which sj ≥ 1 are amongstthe x, and the rest of the yj :s (for which sj = 0) are dependent on x as solutions to differ-ence equations according to (27). Using appropriate initial conditions x(t) = ϕ(t), t ∈[t0 − maxj τj , t0], we obtain a neutral system. Any input-output pair (y(t), u(t)) resultingfrom the original system (1) also satisfies the above neutral system for t0 ≥ maxiτj such

250 M. Anguelova, and B. Wennberg

that all derivatives are continuous. Thus, the input-output behaviour of the system does notdistinguish the infinitely many τi:s which give the same linear combinations τ and τ1, . . . , τ�are not locally identifiable generically. �

5 Examples

In this section we illustrate the theory by simple examples and show that weak observability(and/or parameter identifiability for the regular model parameters) does not necessarily implyidentifiability of the delay parameters, or vice versa. Thus, the already established methodsfor testing observability and identifiability for nonlinear delay systems [16, 17] cannot beused to determine the identifiability of the delay parameters.

Example 1: ⎧⎪⎪⎨⎪⎪⎩x1(t) = x2

2(t− τ1) + u(t)x2(t) = x1(t− τ2)x2(t)y(t) = x1(t)x(t) = ϕ(t), t ∈ [−τ, 0]

(31)

We havey = (δ1(x2))

2 + uy = 2δ1(x2)δ1δ2(x1)δ1(x2) + u = 2(δ1(x2))

2δ1δ2(x1) + u(32)

and ⎡⎣ dydy − dudy − du

⎤⎦ =∂(S,h

(s1)1 )

∂x

[dx1

dx2

]=

=

⎡⎣ 1 00 2δ1(x2)δ1

2(δ1(x2))2δ1δ2 4δ1δ2(x1)δ1(x2)δ1

⎤⎦ [ dx1

dx2

].

(33)

Clearly, the matrix∂(S,h

(s1)1 )

∂xhas rank 2 over K(δ] and so the system is weakly observ-

able according to the definition in [16], if τ1 and τ2 are known. However, τ1 and τ2 are notidentifiable. The input-output equation in linear form (eq. (20)) is:

dy − du + 2δ1δ2(x1)du = 2(δ1(x2))2δ1δ2dy + 2δ1δ2(x1)dy, (34)

and we see that there are two monomials, Δ1 = Δ10 = δ0, the identity operator and Δ11 =δ1δ2 (with corresponding combinations T1 = T10 = 0 and T11 = τ1 + τ2 of the twotime-delays). Thus, M =

[1 1]

with rank 1, and the time lags are not identifiable.Following the the first part of the proof of Corollary 1, we can use the change of variables

x1 = y = x1, x2 = y = (δ1(x2))2 + u to rewrite the system as⎧⎨⎩

˙x1(t) = x2(t)˙x2(t) = 2(x2(t) − u(t))x1(t− T11) + u(t)y(t) = x1(t)

. (35)

Example 2: ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩x1(t) = −x2(t− τ1)x2(t) = x1(t− τ2)y1(t) = x1(t)y2(t) = x2(t− τ2)x(t) = ϕ(t), t ∈ [−T, 0]

(36)

Identifiability of Delay Parameters 251

We have

y1 = −δ1x2

y1 = −δ1δ2x1and

∂(S, h(s1)1 , h

(s2)2 )

∂x=

⎡⎢⎢⎣1 00 δ1

δ1δ2 00 δ2

⎤⎥⎥⎦ . (37)

The input-output equations in linear form are

dy1 = −δ1δ2dy1

δ1dy2 = −δ2dy1 ⇔ dy2 = −δ−11 δ2dy1 .

(38)

Then, Δ10 is the identity δ0 operator, Δ11 = δ1δ2, Δ21 = δ1 and Δ22 = δ2 (with cor-responding combinations T10 = 0, T11 = τ1 + τ2, T21 = τ1 and T22 = τ2 of the twotime-delays). Thus, Δ1 = δ0 and Δ2 = Δ21 (with T1 = T10 = 0 and T2 = T21 = τ1) and

M =

[1 1−1 1

],

which is of rank 2. Thus τ1 and τ2 are identifiable.For this simple example, we can actually calculate the values of the two time lags from

the explicit input-output equations y1 = −δ1δ2y1, y2 = −δ−11 δ2y1. To illustrate, we have

carried out a numerical simulation using the parameters τ1 = 1, τ2 =√

2, ϕ1(t) = et andϕ2(t) = t+ 1. We then plot

μ1(T11 − T10)|t0 := y1(t0) + y1(t0 − T11 + T10)

andμ2(T22 − T21)|t0 := y2(t0) + y1(t0 − T22 + T21)

for t0 = 4. As expected, the functions are zero for T11 − T10 = 1 +√

2 and T22 − T21 =√2− 1 and locally these are the only roots, see Fig. 1. Globally, there are also other roots for

μ2(T22 − T21) that can be seen in Fig. 1. They can be discarded in this case by plotting

−2 0 2 4−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

A plot of the function μ1(T

11−T

10) for t

0=4

T11

−T10

μ 1

−1 −0.5 0 0.5 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

A plot of the function μ2(T

22−T

21) for t

0=4

T22

−T21

μ 2

Fig. 1. The functions μ1(T11 − T10)|4 and μ2(T22 − T21)|4.

252 M. Anguelova, and B. Wennberg

μ2(T22 − T21)|t0 :=d

dt(y2(t0) + y1(t0 − T22 + T21))|t0

(and other subsequent time-derivatives) since T22 − T21 must be a root for this, too, see Fig.2. The simulations were carried out using MATLAB.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

T22

−T21

d/dt

(μ2) |4

Fig. 2. The function μ2(T22 − T21)|4.

6 Conclusions

We have analyzed the identifiability of the time-lag parameters in nonlinear systems withmultiple constant time delays.

State elimination is shown to yield an external input-output representation of the system,the form of which decides the identifiability of the delay parameters. For simpler models withfew variables and parameters, the input-output equations can be used directly to identify thevalues of the time-lags from measured data.

We have formulated linear-algebraic criteria to check the identifiability of the delay pa-rameters which eliminate the need for an explicit calculation of the input-output relations.

7 Acknowledgements

This work was supported by the National Research School in Genomics and Bioinformat-ics, the Swedish Research Council, and the Swedish Foundation for Strategic Research viaGMMC and CMR.

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