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Identifiability of a Nonlinear Delayed-Differential Aerospace Model
holds. From Assumption A2, it follows that
(
x;
)
2
2 2 ( ( ) ( 0 ))dl 0 1 W ( (l))dl: 0
H l ; l
(57)
Cauchy–Schwartz inequality implies that
(
x;
)
2
jH ((l); (l 0 ))j2 dl 0 1
Assumption H2’ implies that (x; ) is satisfied.
2 0
( ( ))
W l dl: (58)
0. Therefore, Assumption H2
Lilianne Denis-Vidal, Carine Jauberthie, and Ghislaine Joly-Blanchard Abstract—This technical note shows the identifiability of a nonlinear delayed-differential model describing aircraft dynamics. This result is obtained by an original combination: An extension of the results of Grewal and Glover to delay systems and an application of linear delayed-differential model identifiability of Orlov et al. . It requires initial conditions corresponding to an equilibrium state, which is implemented during the experimentation. Index Terms—Aerospace application, identifiability, nonlinear delayeddifferential systems.
I. INTRODUCTION ACKNOWLEDGMENT The authors would like to thank I. Karafyllis for his helpful suggestions and valuable comments.
REFERENCES [1] P.-A. Bliman, “Lyapunov equation for the stability of linear delay systems of retarded and neutral type,” IEEE Trans. Autom. Control, vol. 47, no. 2, pp. 327–335, Feb. 2002. [2] J. M. Coron and L. Praly, “Adding an integrator for the stabilization problem,” Syst. Control Lett., vol. 17, pp. 89–104, 1991. [3] R. Freeman and P. Kokotovic, Robust Nonlinear Control Design. Boston, MA: Birkhäuser, 1996. [4] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993, vol. 99, Applied Math, Sciences. [5] D. Ivanescu, S. I. Niculescu, J. M. Dion, and L. Dugard, “Control of distributed delay systems with uncertainties: A generalized Popov theory approach,” Kybernetika, vol. 37, no. 3, pp. 325–343, 2002. [6] M. Jankovic, “Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems,” IEEE Trans. Autom. Control, vol. 46, no. 7, pp. 1048–1060, Jul. 2001. [7] N. N. Krasovskii, Stability of Motion. Stanford, CA: Stanford Univ. Press, 1963. [8] F. Mazenc, S. Mondié, and R. Francisco, “Global asymptotic stabilization of feedforward systems with delay in the input,” IEEE Trans. Autom. Control, vol. 49, no. 5, pp. 844–850, May 2004. [9] F. Mazenc, S. Mondié, and S. I. Niculescu, “Global asymptotic stabilization for chains of integrators with a delay in the input,” IEEE Trans. Autom. Control, vol. 48, no. 1, pp. 57–63, Jan. 2003. [10] , “Global stabilization of oscillators with bounded input delayed,” Syst. Control Lett., vol. 53, pp. 415–422, 2004. [11] W. Michiels and D. Roose, “Global stabilization of multiple integrators with time-delay and input constraints,” in Proc. 3rd IFAC Workshop on Time Delay Systems TDS 2001, Santa Fe, NM, Dec. 2001. [12] W. Michiels, R. Sepulchre, and D. Roose, “Stability of perturbed delay differential equations and stabilization of nonlinear cascade systems,” SIAM J. Control Optim., vol. 40, no. 3, pp. 661–680, 2002. [13] A. R. Teel, “Semi-global stabilization of minimum phase nonlinear systems in special normal forms,” Syst. Control Lett., vol. 19, pp. 187–192, 1992. [14] , “Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorems,” IEEE Trans. Autom. Control, vol. 43, no. 7, pp. 960–964, Jul. 1998. [15] J. Tsinias, “Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation,” ESAIM: Control, Optim., Calc. Variat., vol. 2, pp. 57–85, Mar. 1997.
The results presented here have been motivated by experimental requirements during a collaboration with an industrial research center [2]. The aim of this collaboration was the determination of experiments leading to the estimation of some unknown parameters. More precisely, in the laboratory a scale model is catapulted in free flight conditions. Then it flies across a turbulence generated by a wind tunnel. The data, required for determining the paths and attitudes of the model, are obtained from on-board instrumentation (accelerometers, rate gyros, and an anemoclinometric probe). The involved model comes from the projection of motion equations onto the aerodynamic reference frame of an aircraft. It is given by the nonlinear retarded equations shown in (1) at the bottom of the next page. The different physical quantities and the meaning of the parameters will be precised in Section II. The final aim of the study is the numerical estimation of the aerodynamic coefficients, which requires a preliminary model identifiability analysis. It is why the note is concerned with the parameter identifiability of systems described by the nonlinear and retarded-differential equations shown in (2) at the bottom of the next page. Here, x(t; ) 2 IRn and y (t; ) 2 IRm denote the state variables and the measured outputs, respectively. The single input u is piecewise continuous, and the initial function x0 is continuous in t 2 [0 1 ; 0]. The parameters 1 and 2 denote delays, ( 1 ; 2 ) 2 U (U IR2 ). The unknown constant quantities f1 ; . . . ; p g to be estimated are in Up (Up IRp ). The parameter vector = (1 ; . . . ; p ; 1 ; 2 ) 2 U = Up 2 U (U IRp+2 ). The function f (:; :; :; :; 1 ; . . . ; p ) is real and twice continuously differentiable for every 2 U on M (a connected open subset of IRn 2IRn 2IR2IR such that (x(t; ); x(t0 1 ; ); u(t); u(t0 2 )) 2 M for every 2 U and every t 2 [0; T ]). The output is a linear function of the state (C is a matrix). Finally, in order to avoid nonessential technical difficulties, one delay in state and one delay in input are considered only. Now, let us recall the definition of model identifiability considered here. Definition 1.1: The model (2) is globally identifiable at 2 U if there exists a control u such that, for any ~ 2 U , the equality = ~ follows from y (t; ~) = y (t; ) 8t 2 [0; T ].
Manuscript received April 15, 2004; revised June 4, 2004 and May 17, 2005. Recommended by Associate Editor J. M. A. Scherpen. L. Denis-Vidal is with the University of Sciences and Technology, 59655 Villeneuve d’Ascq, France. C. Jauberthie and G. Joly-Blanchard are with the University of Technology, 60205 Compiègne, France (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2005.861700
0018-9286/$20.00 © 2006 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 1, JANUARY 2006
The identifiability of nonlinear dynamical systems has been extensively researched by different approaches [12]: linearization at an operating point [4], power series [9], similarity transformation [11], differential algebra [5], and system equivalence [3]. On the other hand, the identifiability of linear delayed-differential equations has been analyzed with restrictive conditions on the structure of the system [6], [10]. In [7] and [8], the identifiability of the transfer function is shown with a sufficient nonsmooth control input. But, to our knowledge, there is no existing method for solving the identifiability problem for general nonlinear and retarded systems with a priori unknown delays. Therefore, the idea of this note consists in extending [4, Th. 4] to some nonlinear delay systems in order that the results of [7] could be used. The aircraft application falls in with this argument because all the states are available to measurements and experimentation is done with initial conditions corresponding to an equilibrium point of the system. a constant input, let xe the corresponding equiMore precisely, let u librium state of (1), then the experimentation is led with initial condi + ( t) . tions xe and with a non constant input of the form u(t) = u Moreover, the equilibrium state xe is known by experimenters. Indeed, the experimentation in question is done in a wind tunnel. It begins by the free flight of a scale model, which leads to a measured equilibrium state; then the input is modified by the introduction of a turbulence generator [1]. The note is organized as follows. Section II provides the context of the application and the associated assumptions. In Section III, the extension of Theorem 4 by Grewal and Glover [4] is given and results of linear delayed-differential model identifiability [7] are recalled in order to prove the identifiability of the model (1) with specified initial conditions and input.
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II. APPLICATION CONTEXT The application is concerned with the glider longitudinal motion of an aircraft. The projection of the general equations of motion onto the aerodynamic reference frame of the aircraft gives the system shown in (3) at the bottom of the page, where V denotes the speed of the aircraft, the angle of attack, the pitch angle, q the pitch rate. In order to improve the model accuracy, the input u(t), which consists here in the turbulence, is supposed to act at three different points of the fuselage. With regard to the constants, denotes the air density, g the acceleration due to gravity, l a reference length, S a reference surface. The aero0 ; Cmq g are assumed to dynamic coefficients fCx0 ; Cx ; Cz0 ; Czq ; Cm 1 2 3 1 2 3 ; Cm g constitute the be known, but fCz ; Cz ; Cz ; ; Cm ; Cm unknown parameters to be estimated. Experiments are based on free flights of scale models in laboratory. In order to simplify the system writing, let us introduce km = (1=2)S=m, kl = (1=2Iyy )Sl, the known constants
k0 = 0 g k1 = 0km Cx0 k2 = 0km Cx k3 = km lCzq k4 = km Cz0 k5 = kl lCmq k6 = kl Cm0
(4)
and the unknown parameters 1 2 3 1 = km Cz 2 = km Cz 3 = km Cz 4 = 0 3 1 2 5 = kl Cm : 6 = kl Cm 7 = kl Cm
(5)
With these notations, system (3) leads to (1), introduced at the beginning of the note, which takes the form shown in (6) at the bottom of the next page, where f is twice continuously differentiable.
V_ (t) = k0 sin((t) 0 (t)) + V 2 (t)(k1 + k2 (t)) V (t)(_ (t) 0 _ (t)) = k0 cos((t) 0 (t)) + k3 q(t)V (t) + V 2 (t)[k4 + 1 ( (t) + u(t))+ 2 ( (t) + u(t 0 1 )) + 3 ( (t) + u(t 0 2 )) + 3 4 ( (t 0 3 ) + u(t 0 2 ))] q_(t) = k5 q(t)V (t) + V 2 (t)[k6 + 5 ( (t) + u(t)) + 6 ( (t) + u(t 0 1 )) +7 ( (t) + u(t 0 2 )) + 4 7 ( (t 0 3 ) + u(t 0 2 ))] _ (t) = q(t)
(1)
x_ (t; ) = f (x(t; ); x(t 0 1 ; ); u(t); u(t 0 2 ); 1 ; . . . ; p ); t 2 [0; T ] x(t; ) = x0 (t; ) ; t 2 [0 1 ; 0] y(t; ) = Cx(t; ):
(2)
V_ (t) _ (t)
0mg sin((t) 0 (t)) 0 SV (t)(Cx + Cx (t)) t = mV t mV (t)q (t) + mg cos((t) 0 (t)) 0 SV (t)(Cz + Czq lq V t 1 = m
1 2
1
( )
2
0
1 2
2
0
( ) ( )
1 + Cz ( (t) + u(t))
2 3 +Cz ( (t) + u(t 0 1 )) + Cz [ (t) + u(t 0 2 ) 0 ( (t 0 3 ) + u(t 0 2 ))]
q_(t)
= 2I1
1 SlV 2 (t) Cm0 + Cmq lqV ((tt)) + Cm ( (t) + u(t))
2 3 +Cm ( (t) + u(t 0 1 )) + Cm [ (t) + u(t 0 2 ) 0 ( (t 0 3 ) + u(t 0 2 ))]
_ (t)
= q ( t)
(3)
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The parameter vector to be estimated is given by
= f1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 1 ; 2 ; 3 g:
(7)
and a given parameter vector Now, let us consider a constant input u 2 U , where U will be precised subsequently. The corresponding equilibrium state xe = ( e ; qe ; e ; Ve ) is considered as an operating ; u ; 1 ; . . . ; 7 ) = 0, let point of (1) and is given by f (xe ; xe ; u
0 = k0 sin(e 0 e ) + Ve2 (k1 + k2 e ) 0 = k0 cos(e 0 e )+ Ve2 (k4 +(1 + 2 + 3 + 3 4 )( e + u )) (8) 0 = Ve2 (k6 + (5 + 6 + 7 + 4 7 )( e + u )) 0 = qe : Indeed, from a practical point of view, the equilibrium state xe is known (experiments in wind tunnel), the angles e and e are small and Ve is positive. Then, the solution of (8) is given by
e = 0 + +k + 0 u ; qe = 0 e = arctan k +( + k++ k+ )( +u) + e Ve =
p(k +k ) +( +k + + )( +u)
1=2
hfill
:
(9)
It is important to precise that the knowledge of xe is not sufficient to determine f1 ; . . . ; 7 g, the uniqueness being not assured by (9).
x_ (t; ) x(t; )
In this context, the experimental model, used for the parameter esti as input and initial conditions corremation, takes a perturbation of u sponding to xe
(t) = e ; q(t) = 0 ; (t) = e ; V (t) = Ve ; 0 3 t 0:
(10) The following step consists in linearizing (6) about xe , which gives with straightforward notations shown in (11) at the bottom of the page. In the precise case of the aircraft model (1) and with e = e 0 e , the matrices A1 , A2 , B1 , B2 , and B3 are given by (12) and (13) as shown at the bottom of the page. III. IDENTIFIABILITY RESULTS The identifiability of the model (1) and (10) is based on two arguments. A. Extension of [4, Th. 4] to Nonlinear Delayed-Differential Systems It will be explicated by the following proposition whose the proof is given in the Appendix. Proposition 3.1: Let the system shown in (14) at the bottom of the page hold, where xe is solution of f (xe ; xe ; u ; u ; 1 ; . . . ; p ) = 0. Let (15), as shown at the bottom of the page, hold the linearization of (14) about xe . If the model (15) is globally identifiable at 2 U , then as the model (14) is globally identifiable at , with a perturbation of u input.
= f (x(t; ); x(t 0 3 ; ); u(t); u(t 0 1 ); u(t 0 2 ); 1 ; . . . ; p ) = x0 (t) ; 0 3 t 0
(6)
_(t) = A1 (t) + A2 (t 0 3 ) + B1 (t) + B2 (t 0 1 ) + B3 (t 0 2 ) (s) = 0 ; s 2 [0 3 ; 0]
A1 =
A2 =
2V e (k 1 + k 2 e ) 2 Vk cos( e ) 0 0 0 0 0 0 0 Ve 3 4 0 0 Ve2 4 7 0 0 0 0
0k0 cos( e ) + Ve2 k2 0 k 0 V sin( e ) 0 Ve (1 + 2 + 3 ) 1 0 k3 k5 Ve Ve2 (5 + 6 + 7 ) 0 1 0 0 0 0 0 Ve 1 B 2 = 0 Ve 2 ; B1 = 0 Ve2 5 Ve2 6 0 0 0
(11)
k0 cos( e ) k V sin( e ) 0 0
B3 =
(12)
0
0 Ve ( 3 + 3 4 ) Ve2 (7 + 4 7 )
(13)
0
x_ (t; ) = f (x(t; ); x(t 0 1 ; ); u(t); u(t 0 2 ); 1 ; . . . ; p ) ; t 2 [0; T ] x(t; ) = xe ; t 2 [0 1 ; 0] y(t; ) = Cx(t; ) :
(14)
_(t; ) = A1 () (t; ) + A2 () (t 0 1 ; ) + B1 () (t) + B2 () (t 0 2 ) (t; ) = 0 ; t 2 [0 1 ; 0] (t; ) = C (t; ) :
(15)
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det[B1 + B2 + B3 ; (A1 + A2 )(B1 + B2 + B3 ); (A1 + A2 )2 (B1 + B2 + B3 ); (A1 + A2 )3 (B1 + B2 + B3 )] = 0
x (t; ) =
t
0
[f (xe + x (s; ); xe + x (s 0 1 ; ); u + (s); u + (s 0 2 ); 1 ; . . . ; p ) 0 f (xe ; xe ; u ; u ; 1 ; . . . ; p )]ds
kx (t; )kIR
t
0
(L1 kx (s; )kIR + L2 kx (s 0 1 ; )kIR + L3 j(s)j + L4 j(s 0 2 )j)ds
Indeed, it is shown that, if is a control providing the identifiability of (15), then the identifiability of the model (14) can be obtained by the =kk, with " small enough. control u + " Let us notice that Proposition 3.1 can be applied not only to the aircraft model (1) but also to any system of the form (14), and even with any fixed number of delays. B. An Application of Theorem 1 of [7, Th. 1] to (11), Which is the Linearization of (1) About the Equilibrium State (9) Proposition 3.2: For s functions
A(s) = A1 + A2 e0
s
2 C, let us define the entire matrix-valued
; B(s) = B1 + B2 e0
s
+ B3 e0 s : (16)
The Markov parameters Aj 01 (s)B(s), j = 1; . . . ; 4 of the time delay system (11) are identifiable and their identifiability is particularly enforced by an arbitrary sufficiently discontinuous control input (t). We are now in a position to state the identifiability of the aircraft model. Proposition 3.3: The model (1)–(10) is globally identifiable at 2 U (U = IRp 2 (IR+ )3 ), except at the points of a subset of measure zero given by the polynomial (with respect to 8 = 1 + 2 + 3 + 3 4 and 9 = 5 + 6 + 7 + 7 4 ); see (17), as shown at the top of the page. Proof: The linearization of (1) about the equilibrium point xe (9) is given by the model (11). By applying Proposition 3.2, the Markov parameters Aj 01 B , j = 1; . . . ; 4 are identifiable. On the other hand, the matrices fA1 ; A2 ; B1 ; B2 ; B3 g and the delays f 1 ; 2 ; 3 g are shown to be identifiable [7], if all the state variables are measured and (11) is weakly controllable. Therefore, it remains to prove the weak controllability, 2 C such that which is given by the existence of s rank[B(s); A(s)B(s); . . . ; A3 (s)B(s)] = 4. By taking s = 0, the determinant of the matrix [B(s); A(s)B(s); . . . ; A3 (s)B(s)] is given by the left-hand side member of (17). It is not difficult to show that it is a polynomial with respect to 8 and 9 , which is not identically equal to zero. Indeed, it can be checked that the coefficient of (9 )4 is equal to
Ve4 [k0 k3 cos( e ) + k2 (1 0 k3 )Ve2 ][2(1 0 k3 )k0 k2 cos( e ) k0 cos( e ) Ve2 which is not identically equal to zero and which is not equal to zero by taking experimental values for the known different physical quantities.
0k3 k7 sin( e )] 0 k32 k02 sin( e )
k2 0
(17)
(24)
(25)
Therefore, there exists a control (t) such that the identity (t) = ~ results in i = ~i (i = 12; 3, ), Ai = A~i (i = 1; 2), Bi = B ~i (t) (i = 1; 2; 3), with obvious notations. Now, it is clear that the identity of the previous matrices implies the identity of the parameters i = ~i (i = 1; . . . ; 7). Then, the model (11) is globally identifiable at 2 U except at the points of a subset of measure zero, which is usually called the structural identifiability. Then the application of Proposition 3.1 implies the global identifiability of the model (1)–(10) at 2 U except at the points of a subset of measure zero. IV. CONCLUSION A result about the identifiability of nonlinear delayed-differential system is proved in Proposition 3.1 by linearization about an operating point. It is restricted to models with initial conditions corresponding to an equilibrium point. By adding known results about the identifiability of linear delayed-differential systems, it has allowed the identifiability proof of a glider longitudinal motion model of an aircraft. However, it is obviously a sufficient condition only. It would be more interesting to find general necessary and sufficient conditions for identifiability of nonlinear retarded systems directly from relations between inputs, outputs and parameters as it is done in the case of nonlinear differential systems. APPENDIX Proof of Proposition 3.1 Let ~ 6= , ~ 2 U (resp. ~ 2 W ). Let us consider the equilibrium states corresponding respectively to ~ and
0 ye 0 ~ ye
= f (xe ; xe ; u ; u ; 1 ; . . . ; p ) = Cxe ~ ~ ; u; ~ ; . . . ; ~ ) = f (xe ; xe ; u 1 p ~ = Cxe :
(18)
~
For notational simplicity, let us introduce xe = xe , ye = ye , x ~e = xe , ~ y~e = ye . Two cases are possible: ye 6= y~e , then the system (14) is globally identifiable at – (1 ; . . . ; p ) since there exists an input u such that the outputs of the system (14) ye and y~e are distinct. ye = y~e . Let us evaluate the difference between the outputs – of the system (14)
~ = (t; ) 0 (t; ) ~ y(t; ) 0 y(t; ) ~ 0 (t; ))) ~ (19) +C((x (t; ) 0 (t; )) 0 (x (t; )
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where x (t; ) = x(t; ) 0 xe . By assumption, the linear system (15) is globally identifiable at , there exists an input (t) consequently such that the corresponding outputs ~ are distinct. Now, take the input (t; ) and (t; )
where
kE (t)k L (kx (t; )k +kx (0 ; )k IR
2 IR
8
2 IR
1
(t) = "
(t) kk
(kk = kkL
(0;T )
)
(20) and, by applying (27),
where " will be chosen judiciously subsequently. From linearity of the system (15) and zero initial conditions, the output corresponding to the input (t) is given by
~ k = " k(:; ) 0 (:; ) ~ k = "K k(:; ) 0 (:; ) kk
~ k "K ky(:; ) 0 y(:; ) ~ 0 (:; ) ~k 0kC k kx (:; ) 0 (:; )k + kx (:; )
: (23)
Estimation of kx (t; )kIR ; see (24), as shown at the top of the previous page. The assumption of the smoothness of the function f gives (25), as shown at the top of the previous page, where L1 ; L2 ; . . . designate real constants, hence t 0
kx (s; )k
IR
ds + L6 " :
(26)
This last estimation is due to x (s; ) = 0 for s 2 [0 1 ; 0], (s) = 0 for s 2 [0 2 ; 0] and k k = ". The application of Gronwall’s lemma gives
kx (t; )k L " : IR
•
7
9
2
:
(30)
t 10
0
kx (s; ) 0 (s; )k
IR
ds + L11 "2 (31)
then (Gronwall’s lemma)
kx (t; ) 0 (t; )k L IR
12
"2 :
(32)
~ k – The application of (32) Minimization of ky(:; ) 0 y(:; ) to (23) leads to ~ k "K 0 L" ky(:; ) 0 y(:; )
2
:
(33)
Now, there exists " > 0 such that "K 0 L"2 > 0. Therefore, ~ are distinct and the system (14) the outputs y(:; ) and y(:; ) is globally identifiable at .
which, by introducing (22), gives
5
IR
•
~ k k(:; ) 0 (:; ) ~k ky(:; ) 0 y(:; ) ~ 0 (:; ))) ~ k 0kC((x (:; ) 0 (:; )) 0 (x (:; )
IR
kx (t; ) 0 (t; )k L
(22)
where the constant K is strictly positive. The L2 -norm of (19) leads to
kx (t; )k L
IR
(21)
Consequently, the L2 -norm of the difference between the outputs of the system (15) is
•
kE (t)k L "
The same approach as in the previous estimation yields
" kk (t; ):
(t; ) =
+ j(t)j2 + j(t 0 2 )j2 ) (29)
(27)
Estimation of kx (t; ) 0 (t; )kIR – The assumption of the smoothness of the function f gives
d (x (t; ) 0 (t; )) = A1 ()(x (t; ) 0 (t; )) dt +A2 ()(x (t 0 1 ; ) 0 (t 0 1 ; )) + E (t) (28)
REFERENCES [1] P. Coton, “Validation de modèles de représentation du comportement des aérodynes en rafales verticales,” Rapport Tech. IMFL, vol. 81/20, 1981. [2] L. Denis-Vidal, G. Joly-Blanchard, C. Jauberthie, and P. Coton, “Aircraft parameter estimation: Successive steps,” in Proc. 5th IFAC NOLCOS’2001, St Petersburg, Russia, 2001, pp. 322–327. [3] L. Denis-Vidal and G. Joly-Blanchard, “Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems,” Automatica, vol. 40, pp. 287–292, 2004. [4] M. S. Grewal and K. Glover, “Identifiability of linear and nonlinear dynamical systems,” IEEE Trans. Autom. Control , vol. AC-21, pp. 833–837, 1976. [5] L. Ljung and T. Glad, “On global identifiability for arbitrary model parametrizations,” Automatica, vol. 30, no. 2, pp. 265–276, 1994. [6] S. I. Nakagiri and M. Yamamoto, “Unique identification of coefficient matrices, time delays and initial functions of functional differential equations,” J. Math. Syst. Estimat. Control, vol. 5, pp. 323–344, 1995. [7] Y. Orlov, L. Belkoura, J. P. Richard, and M. Dambrine, “On identifiability of linear time-delay systems,” IEEE Trans. Autom. Control, vol. 47, pp. 1319–1324, 2002. [8] , “Adaptative identification of linear time-delay systems,” Int. J. Robust Nonlinear Control, vol. 13, pp. 857–872, 2003. [9] H. Pohjanpalo, “System identifiability based on the power series expansion of the solution,” Math. Biosci., vol. 41, pp. 21–33, 1978. [10] S. M. Verduyn Lunel, “Parameter identifiability of differential delay equations,” in Int. J. Adapt. Control Signal Process., vol. 15, 2001, pp. 655–678. [11] S. Vajda, K. R. Godfrey, and H. Rabitz, “Similarity transformation approach to structural identifiability of nonlinear models,” Math. Biosci., vol. 93, pp. 217–248, 1989. [12] E. Walter, Identifiability of State Space Models, ser. Lecture Notes BioMathematics. Berlin, Germany: Springer-Verlag, 1982, vol. 46.
