L a g u err e d o m a in id en ti ca tio n o f co n tinu o u s lin ea r ...

[Pages:6]Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Laguerre domain identification of continuous linear time delay systems from

impulse response data

Egi Hidayat Alexander Medvedev

Uppsala University, Sweden (e-mail: egi.hidayat@it.uu.se, alexander.medvedev@it.uu.se).

Abstract: A method for Laguerre domain identification of continuous time delay systems from impulse response data is proposed. Linear time-invariant systems resulting from cascading finitedimensional dynamics with pure time delays are considered. The estimation of the combined dynamics is resolved by an iterative procedure relying on linearity of the system and its cascade structure. Subspace identification is utilized for estimation of both finite- and infinitedimensional dynamics. An application to blind identification of a mathematical model of an endocrine system with pulsatile regulation is also provided.

Keywords: time-delay systems; continuous time systems; impulse responses; Dirac functions; subspace methods; parameter estimation; system identification.

1. INTRODUCTION

Time delay estimation has for a long time been an active research field in signal processing and system identification. However, mostly discrete systems have been addressed as they obey finite-dimensional dynamics. On the contrary, continuous time delay systems possess infinitedimensional dynamics and thus require more advanced estimation techniques. Another important distinction in the continuous time framework is between pure time delay estimation versus estimation of time-delay systems incorporating both finite- and infinite-dimensional subsystems. The former problem is investigated more often while the latter one, usually termed as time delay system identification, is considered in only a few papers such as M?akil?a [1990] and more recently in Ahmed et al. [2006], Gomez et al. [2007], Belkoura et al. [2009], Najafi et al. [2010]. In Ahmed et al. [2006], a linear filter-based approach to identify continuous time delay models is proposed. A four-step iterative algorithm utilizing the least squares and instrumental variable methods is devised to estimate the model parameters and initial conditions of the finitedimensional part as well as the time delay. An on-line identification algorithm is suggested in Gomez et al. [2007] for single-input single-output continuous-time linear time delay systems from only output measurements. This algorithm utilizes an adaptive identifier with sliding mode solutions that treats the time delay differential equations similar to finite-dimensional dynamics. Belkoura et al. [2009] provide an extension of an earlier identification method introduced in Belkoura et al. [2006] for systems with structured entries. This on-line estimation technique applies the iterated convolution product concept. Najafi et al. [2010] propose a system identification method for systems with input time delay based on a wavelet approach. Partitioning of the system is used there to estimate the finite-dimensional dynamics and the time delay separately.

The authors were in part financed by the European Research

Council, Advanced Grant 247035 (SysTEAM)

The finite-dimensional system dynamics are identified using recursive least squares method. System identification is conventionally performed from data in time or, via the Fourier transform, in frequency domain. However, orthonormal functional bases such as Laguerre functions and Kautz functions, have been utilized in system modeling and identification for many years, Nurges and Yaaksoo [1981]. The notion of shift operator plays a key role in a systematic exposition of such an approach. The advantage of using e.g. the Laguerre shift is its similarity, in a certain sense, to the regular discrete time shift operator with the possibility for tuning the identification performance by means of the Laguerre parameter. Laguerre domain is particularly suitable for exponential L2 functions, which makes it attractive for the identification of stable time delay systems, M?akil?a [1990]. Time delay estimation from Laguerre spectra of the input and output signal was introduced by Fischer and Medvedev [1999]. As pointed out in a survey by Bj?orklund and Ljung [2003], this subspace estimation approach has remarkable robustness properties against finitedimensional perturbations. However, Laguerre domain descriptions and estimation algorithms of more general continuous time delay systems are not readily available, even for the case of impulse response. The need for system identification of a continuous linear time delay system driven by an impulse signal or a train of impulses arise in several fields. In endocrinology, such a mathematical model is utilized for representing pulsatile secretion of hormones, Churilov et al. [2009]. A radar is a good example of an application where the travel time of a signal is considered as time delay, Xu et al. [2008]. The signal in question is usually a pulse and attenuated by the channel media. The paper is organized as follows. Necessary background on Laguerre functions and Laguerre domain system models is briefly presented in Section 2. Section 3 provides a mathematical formulation of the considered identification

Copyright by the International Federation of Automatic Control (IFAC)

9064

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

problem which is consequently solved by subspace-based techniques in Section 4. Simulation and a numerical example in Section 5 are intended to illustrate the efficacy of the proposed method.

2. PRELIMINARIES 2.1 Laguerre functions

The Laplace transform of k-th continuous Laguerre func-

tion is given by

Lk(s)

=

s

2p +p

s-p k s+p

where k is a positive number and p > 0 represents the La-

guerre parameter. In terms of the Laguerre shift operator

T (s) a normalizing function T~(s), one has Lk(s) = T~T k.

where

T (s) =

s-p s + p;

T~ =

1 (1 - T (s)) = 2p

2p

s + p.

The functions Lk(s), k = {0, } constitute an orthonor-

mal complete basis in H2 with respect to the inner product

W, Lk

=1 2j

W (s)Lk(-s)ds.

-

(1)

Further, k-th Laguerre coefficient of W (s) H2 is evaluated as a projection of W (s) onto Lk(s)

wk = W, Lk ,

while the set wk, k = {0, } is referred to as the Laguerre

spectrum of W (s).

According to Riemann-Lebesgue lemma, the integral over

an infinite half arc 2 in Fig. 1 is

1 2j

W (s)Lk(-s)ds = 0

2

and line integral in (1) can be conveniently evaluated as a

contour integral

W, Lk

=

1 2j

W (s)Lk(-s)ds

(2)

over a clockwise contour on the whole right half part of

complex plane .

Fig. 1. Contour for inner product evaluation.

The time domain representations of the Laguerre functions are obtained by means of inverse Laplace transform,

lk(t) = L-1 {Lk(s)} with lk(t), k = {0, } yielding an orthonormal basis in L2.

2.2 Laguerre domain

Consider the continuous single-input single-output timedelay system

x (t) = Ax(t) + Bu(t - )

(3)

y(t) = Cx(t) + Du(t - )

where A, B, C, D are constant real matrices of suitable dimensions and > 0 is the time delay. The matrix A is

assumed to be (Hurwitz) stable and the initial conditions on (3) are x(0) = x0 and u() 0, [-, 0]. System description (3) stipulates the relationship between u(?) and y(?) in time domain. A corresponding description in

Laguerre domain renders the dependence between uk and yk for k = [0, ). Two important special cases of system (3) have been

previously treated in Laguerre domain and come in handy

in this study.

Pure time delay Consider the continuous delay system

y(t) = u(t - ).

(4)

It represents a special case of (3) with the matrices A, B, C

equal to zero and D = I.

Lemma 1. (Fischer and Medvedev [1999]). For system (4),

the following regression equation holds between the La-

guerre coefficients of the input {uk}Nk=0 and those of the

output {yk}Nk=0:

yk = Tk

(5)

with the elements of the regression vector of dimension

N +1

Tk = [k(1), . . . , k(N + 1)]

k(j + 1) = uk,

j=0

k (j

+

1)

=

(-2)j j!(j - 1)!

k-j

(k (k

- -

i i

- -

1)! j)!

ui,

kj>0

i=0

k(j + 1) = 0,

j>k

and the parameter vector

= 1 p ? ? ? (p )N T e-p .

Thus, the Laguerre coefficients of the output signal are given by

k

yk

=

e 2

j j!(j -

1)!

k-j

(k (k

- -

i i

- -

1)! j)!

ui

+

uk

,

(6)

j=1

i=0

with = -2p.

Finite-dimensional system Consider the continuous timeinvariant system

x (t) = Ax(t) + Bu(t)

(7)

y(t) = Cx(t)

representing a specialization of (3) with = 0 and D = 0. Assume that the Laguerre coefficient p does not belong to the spectrum of A, i.e. det(pI - A) = 0. From this point on, the resolvent matrix of A is denoted as

RA(s) = (sI - A)-1.

Let Uk and Yk be the vectors of Laguerre coefficients of the input and output signal, respectively, i.e.

Uk = [u0 . . . uk]T , Yk = [y0 . . . yk]T .

Lemma 2. (Fischer and Medvedev [1998]). For system (7),

the following relationships hold between the Laguerre coefficients of the input and those of the output:

Yk = kx0 + kUk

where

H

J

HF

HG

k

=

...

,

k

=

...

0 ? ? ? 0

J ??? 0

...

...

...

HFk

HF k-1G HF k-2G ? ? ? J

and the Laguerre domain system matrices F, G, H are defined in Table 1.

9065

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Table 1. System matrices in Laguerre domain.

F

T (A)

G

-T~(A)B

H

C T~(A)

J C(pI - A)-1B

In a convolution form, the result of Lemma 2 reads:

k-1

yk = HF k-1-iGui + J uk.

i=0

With a single impulse as input, ui = 2p, i = [0, k] and

k-1

yk =

2p(-2p)(-1)iCRAi+2(p)(pI + A)iB

i=0

+ 2pCRA(p)B.

(8)

It should be noted that for impulse response of (7), the

Laguerre coefficients of the output signal can also be

derived by calculating the linear integral of the scalar

product via counter-clockwise contour integration over the

left half plane of the complex plane, namely c in Fig. 1.

This procedure actually produces a simpler expression:

1

+j

2p(-s

-

p)k

yk = 2j -j CRA(s)B (-s + p)k+1 ds

(-1)k(p + s)k

=

2pC Res RA(s) (p - s)k+1

B

s=A

= (-1)k 2pCRAk+1(p)(pI + A)kB.

(9)

3. SYSTEM MODEL IN LAGUERRE DOMAIN

Time delay system (3) has not been previously studied in Laguerre domain. In this paper, a specialization of it to the case of D = 0 is considered

x (t) = Ax(t) + Bu(t - )

(10)

y(t) = Cx(t)

to keep the system impulse response within L2.

Proposition 3. The Laguerre spectrum of the output sig-

nal response y(t) of (10) to a Dirac delta function (i.e.

u(t) = (t)) under zero initial conditions is given by:

yk =

k k-i

k j

2pk-j

e

2

i(k - i - j)!

C

RAi+1

(p)B

,

i=0 j=0

(11)

for k = {0, }.

Proof. In Laplace domain, under the conditions of the proposition, the output signal y(t) in (3) is

Y (s) = CRA(s)e-sB.

Thus, by a straightforward calculation according to (2), one gets

yk

=

1 2j

CRA(s)e- sB

2p

(s + p)k (s - p)k+1

ds

=C

2p Res

RA

(s)e-

s

(s + p)k (s - p)k+1

B.

s=p

Evaluating the residues will provide an analytical expression of the Laguerre spectrum that can, after some algebra, be brought to the more concise representation of (11). 2

Corollary 4. With = 0, the expression for yk in (11) is equivalent to that provided by Lemma 2 for the impulse response of (7) in the LTI case as well as to the expression

given by (9).

Proof. This can be shown by manipulating (11) into an

equivalent form

k k-i

yk =

k j

2p(-2p)k-j k-i-j e (k - i - j)!

2

C RAi+1 (p)B .

i=0 j=0

Then consider only the values that are nonzero for = 0, i.e. when k - i - j = 0, which gives

kk

yk =

i

i=0

2p(-2p)i C RAi+1 (p)B .

Partial fraction decomposition is then applied to both

expressions. 2

Under zero initial conditions, system (10) is characterized by the transfer function

W (s) = CRA(s)B e-s = W0(s) e-s

(12)

and constitutes a cascade coupling of a linear finitedimensional system W0(s) similar to (7) and a pure time delay operator, as in (4). Due to linearity of both blocks, the order in which they follow can be interchanged without altering the input-output properties of the overall system, i.e. the time delay can be alternatively attributed to the output signal. Thus, provided either constituting block of W (s) is known in advance, the problem of identifying (10) from its impulse response in Laguerre domain is readily solved by existing techniques. Indeed, the impulse response of (12) can be written as

y(t) = w0(t - )

with w0(t) representing the impulse response of W0(s), i.e. w0(t) = L-1{W0(s)}. Assume now that the delayfree part of (10) is known and the Laguerre spectrum of w(t) can therefore be evaluated according to Lemma 2. Then the problem of estimating is just the problem of estimating the time delay for the case of known Laguerre spectra of input and output already solved in Fischer and Medvedev [1999]. If, on the other hand, the value of timedelay is assumed to be known, it can be attributed to the input signal. The problem at hand turns then into the estimation of a linear time-invariant finite-dimensional system in Laguerre domain, the one that is readily solved by means of e.g. subspace identification in Fischer and Medvedev [1998]. The input signal in the case of impulse response is a delayed -function. Notice here that one does not have to evaluate the Laguerre coefficients of the function and simply shifting the time axis for the delay value will suffice.

Fig. 2. Two equivalent input-output realizations of system (12) under non-zero initial conditions

For a system with zero initial conditions, switching the order of the blocks would not have any influence on the input-output properties. However, for a general situation, initial conditions cannot be disregarded. In Fig. 2, the effect of the initial conditions on the output signal is singled out. For configuration A, the finite-dimensional dynamic block produces system output in response to a delayed input signal. Conversely, in configuration B,

9066

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

the delay element is excited by the output of the finite-

dimensional block while input-output equivalence between the configurations is preserved. The real polynomials Pj(?), j = 1, 2, . . . defined as

j-1 j - 1 j-i

Pj () =

i (j - i)!

i=0

(13)

play an important role in the Laguerre representation of time delay system (10). Due to the presence of binomial

coefficients in the expression for (13), it is desirable to evaluate Pj (?) in a recursive manner to avoid possible numerical problems. The following result is instrumental in this respect.

Lemma 5. (Three term relation). The polynomials Pj() obey the following three-term recurrence relation

Pj () = aj Pj+1() + bjPj () + cj Pj-1(),

with

aj = j + 1, bj = -2j, cj = j - 1.

Proof. The lemma can be proved straightforwardly by

expanding the polynomial sums and applying binomial coefficients identities. 2

Proposition 6. The following relationships hold for the Laguerre coefficients of the impulse response of system (10):

Yk

=

k x0

+

k

k

e

2

(14)

where

HB

0 ??? 0

1

HFB

k

=

...

HB ...

??? ...

0 ...

,

k

=

P1

()

...

.

HF kB HF k-1B ? ? ? HB

Pk ()

Proof. By Lemma 1, the Laguerre coefficients of the output signal of the time delay block, with the adjustment

for the initial conditions, are

Yk

=

k x0

+

e

2

(15)

where

HB

1,1 1,2 ? ? ? 1,k+1

HFB

=

...

,

=

2,1 ...

2,2 ...

??? ...

2,k+1 ...

,

HF kB

k+1,1 k+1,2 ? ? ? k+1,k+1

with i,j =

0 1 Pi-j ()

for i < j, for i = j, for i > j.

For individual Laguerre coefficients of output signal it then

applies:

y0

=

H x0

+

e 2

HB,

(16)

k

yk

=

HF kx0

+

e

2

H

F k-jPj () + I B,

j=1

for k 1. Now define

H 0 ??? 0

=

HF ...

H ...

... ...

0 ...

,

HF k HF k-1 ? ? ? H

then

Yk

=

k x0

+

(k

I

)B

e

2

.

The output is given by three separate terms, i.e. the

finite-dimensional dynamics, the time-delay, and the initial

state. Since the terms due to the time delay are scalar values, the equation could be rewritten as (14). 2

4. ESTIMATION ALGORITHM

The estimation algorithm is based on the idea of iteratively swapping the time delay block and the finite-dimensional block and using the knowledge acquired about one of the constituting blocks to improve an estimate of the another one, in a "bootstrapping" manner. A similar idea can e.g. be found in Nihtil?a et al. [1997], though not implemented in Laguerre domain. The estimation algorithm consists of four major steps, namely initialization, finite-dimensional dynamics estimation, time delay estimation, and loss function calculation. The second and third ones are performed in Laguerre domain. These two steps could use tuning of the Laguerre parameter to achieve better estimation results. Lower values of the Laguerre parameter would typically lead to faster convergence of Laguerre spectra, which is suitable for finite-dimensional dynamics estimation. On the contrary, relatively high values of the Laguerre parameter are needed for time delay estimation, to make sure that system dynamics is captured. The algorithm is worked out for the case of the impulse response of a continuous time delay system. For more general input, such as impulse trains, some modifications are required on each step. The case of double impulse is treated in the next section with regard to a biomedical application. The algorithm below is an outline of the estimation procedure, and followed up by a more detailed explanation of each step. (1) Based on measured data y(t) and system model,

provide initial estimation A^, B^, C^, x^0 and ^. (2) Choose appropriate values of the Laguerre parameter

pd for estimation of time delay and pf for finitedimensional dynamics. (3) Perform time delay estimation as follows: (a) Compute the Laguerre spectrum of the output

signal Yk with the chosen pd. (b) Given A^, B^, and C^ compute F and H. (c) Given B^, F , and H compute k and d. (d) Evaluate ^ d = +d (Y - kx^0). (e) Compute the time delay estimate.

^

=

-

1 2pd

^ dT ^ dT

^ d ^ d

(17)

(4) Perform subspace identification of finite-dimensional dynamics as follows:

(a) Given ^ compute k. (b) Evaluate Uk = ke-pf ^. (c) Compute the Laguerre spectrum of the output

signal Yk with Laguerre parameter pf . (d) Perform n4sid function with input Uk and out-

put Yk to obtain estimation of the state space representation in Laguerre domain (F^, F^B^, H^ , H^ B^)

and x^0. (e) Compute

A^ = -pf (I - F^)-1(I + F^)

(18)

C^ = 2pf H^ (I - F^)-1.

9067

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

(5) Compute the loss function f (i) and terminate the algorithm if stopping condition is satisfied, otherwise go back to step 3.

Initialization Initial guesses are provided for the system matrices A^, B^, C^, as well for x^0 and ^ in (10). In the case of heavily undersampled data, measured data y(t) have to

be interpolated and resampled.

Time delay estimation The estimates A^, B^, and C^ are utilized to produce an estimate of the time delay. The matrix representation in Laguerre domain is then calculated through the relations in Table 1. Configuration B in Fig. 2 is considered, i.e. a pure time delay with an input signal in the form of the impulse response of (7). Taking advantage of the proof of Lemma 1, the Laguerre coefficients for output signal (16) can identically be expressed as

k

yk

=

e

2

H

j k-j j!

k-i-1 k-i-j

Fi + Fk B

j=1 i=0

+ HF kx0, k 1.

In a matrix form representation, the shift property becomes more apparent:

Yk = kx0 + dd.

(19)

where

HB

0 ??? 0

H

F

B

...

...

d

=

...

...

...

;

k-1

HF kB HF iB ? ? ?

1

H

B

i=0

k!

1

d

=

e 2

...

k

Given the information about the finite-dimensional dynamics of the system, (19) can be solved yielding

^ d = +d (Yk - kx0). where (?)+ denotes pseudoinverse. As = -2pd^, a subspace-type estimate of ^ is obtained as (17) with d

representing d without the last element and d and d denoting d without the first element.

Identification of finite-dimensional dynamics Configura-

tion A in Fig. 2 is considered, i.e. a finite-dimensional

dynamics excited by a delayed impulse. Matrix represen-

tation (14) is the basis for the implementation of sub-

space identification in Laguerre domain. The estimated

time delay from previous step ^ is used to produce the

input

sequence

in

Laguerre

domain

such

that

Uk

=

k

e

2

,

with k defined in Proposition 6. The output data are

the Laguerre coefficients of output signal Yk. Further, the

finite-dimensional dynamics can be identified e.g. by using

the n4sid algorithm, Van Overschee and Moore [1994].

Execution of n4sid command results in the following state

space Laguerre representation

xk+1 = F^xk + F^B^uk yk = H^ xk + H^ B^uk

By consulting Table 1, the time domain system matrices can be obtained as (18).

Loss function minimization Given the true impulse re-

sponse with initial conditions

y(t) = CeAtx0 + CeA(t- )B

and the corresponding output of the identified model

y^(t) = C^eA^tx^0 + C^eA^(t-^)B^,

the loss function is defined as

f (^)

=

1 2

(y(t) - y^(t))2 dt.

0

The algorithm is terminated when the minimum of the loss

function is achieved.

5. APPLICATION

A pulse-modulated mathematical model describing an endocrine closed-loop system comprised of testosterone (Te), luteinizing hormone (LH), and gonadotropin releasing hormone (GnRH) is studied in Churilov et al. [2009]. An algorithm based on a nonlinear least squares approach has been developed in Hidayat and Medvedev [2010] to estimate the parameters of the system and the pulsatile input in order to study the dynamic relation between LH and GnRH.

In this section, the feasibility of the proposed algorithm for identification of time-delay systems is tested on simulated GnRH-LH data. The goal is to estimate the biological parameters of the model and the delay between two consecutive pulsatile GnRH secretions. With the concentration of GnRH denoted as R(t) and the concentration of the hormone it releases, i.e. LH, denoted as H(t), model (7) assumes the following values

A=

-b1 0 g1 -b2

,B=

1 0

, y(t) = H(t), x(t) =

R(t) H (t)

.

System model The pulse-modulated hormone regulation

system exhibits sustained nonlinear oscillations with one

or two -functions fired in each period. Assume that two

pulses of GnRH occur at the firing times t = 0 and t =

within the least period of the closed system solution T :

u(t) = 0(t) + 1(t - ). Then the concentration of GnRH in response to these two

secretion events is described by

R(t) = (R(0) + 0)e-b1t,

0 t < ,

R(t) = (b1)e-b1t, (x) = R(0) + 0 + 1ex , t < T.

For the measured LH concentration, it applies

H (t)

=

(R(0) b2

+ -

0 b1

)g1

(e-b1

t

-

e-b2t),

H (t)

=

b2

g1 -

b1

((b1)e-b1

t

-

(b2)e-b2t),

0 t < , t1 t < T.

Modified algorithm For this application, the estimation

algorithm has to be adjusted to take into account the

impulse fired in the beginning of the interval, i.e. without

delay. The output signal in Laplace domain is

Y (s) = CRA(s) x0 + B + Be-s

(20)

assuming 0 = 1 and 1 = to recover identifiability.

Time delay estimation For the second step of the estimation algorithm, expression (19) has to be augmented with a term reflecting the effect of the first impulse at t = 0

Yk = kx0 + d(0 + d)

where 0 is a vector with all elements equal to zero except for the first one since = 0 for = 0.

9068

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Finite dimensional dynamics estimation The third step

of the estimation algorithm has also to be modified to

account for the first impulse. From (20) and (14), the

Laguerre coefficients of the output signal can be defined

as following

Yk

=

k x0

+

k B

+

k

k

e

2

=

k x~0

+

k k e

2

where x~0 = x0 + B. Then, in Laguerre domain, it applies

x~k+1 = F x~k + F Bu~k

yk = Hx~k + HBu~k

where u~k = uk.

Simulation Identification data are generated by simulation performed with assumed values of hormones' half-life times. The releasing hormone has the decay rate of 0.07 per minute while the released hormone has 0.033 per minute. For this example, the time delay of the secondary pulse relative to the primary pulse is chosen equal to 52 minutes. White measurement noise is added to the simulated output data sampled each 10 min. The identification result is given in time domain on Fig. 3 and in Laguerre domain on Fig. 4.

2.5 Disturbed sampled data Undisturbed continuous signal Estimated output signal

2

1.5

1

0.5

0

-0.5 0

50

100

150

200

250

300

minutes

Fig. 3. Estimation of response with two consecutive pulses

Laguerre spectra 8

Interpolated disturbed signal Estimated output signal 6

4

2

0

-2

-4

-6

-8

-2

0

2

4

6

8

10

12

14

16

k

Fig. 4. Laguerre spectrum of simulated and estimated output signal

6. CONCLUSION

An algorithm for identification of continuous time-delay systems in Laguerre domain from impulse response data is suggested. The algorithm is based on an iterative "bootstrapping"-type technique where the numerically reliable tools of subspace identification are employed for the estimation of the finite- and infinite-dimensional dynamics. By separately tailoring the Laguerre parameter to the finite-dimensional dynamics of the identified system and

the time delay operator, better convergence of the estimation algorithms can be achieved. An application of the identification algorithm to the estimation of an endocrine system with pulsatile secretion is also provided.

REFERENCES

S. Ahmed, B. Huang, and S.L. Shah. Parameter and delay estimation of continuous-time models using a linear filter. Journal of Process Control, 16:323?331, 2006.

L. Belkoura, J.P. Richard, and M. Fliess. On-line identification of systems with delayed inputs. Proceedings of 16th conference mathematical theory of networks and systems, 2006.

L. Belkoura, J.P. Richard, and M. Fliess. Parameters estimation of systems with delayed and structured entries. Automatica, 45:1117?1125, 2009.

S. Bj?orklund and L. Ljung. A review of time-delay estimation techniques. Proceedings of the 42nd IEEE Conference on Decision and Control, 3:2502?2507, 2003.

A. Churilov, A. Medvedev, and A. Shepeljavyi. Mathematical model of non-basal testosterone regulation in the male by pulse modulated feedback. Automatica, 45: 78?85, 2009.

B.R. Fischer and A. Medvedev. Laguerre shift identification of a pressurized process. Proceedings of the 1998 American Control Conference, 3:1933?1937, 1998.

B.R. Fischer and A. Medvedev. L2 time delay estimation by means of Laguerre functions. Proceedings of the 1999 American Control Conference, 1:455?459, 1999.

O. Gomez, Y. Orlov, and I.V. Kolmanovsky. On-line identification of SISO linear time-invariant delay system from output measurements. Automatica, 43:2060?2069, 2007.

E. Hidayat and A. Medvedev. Parameter estimation in a pulsatile hormone secretion model. Technical report, no. 2010-007, 2010.

P.M. M?akil?a. Laguerre series approximation of finite dimensional systems. Automatica, 26(6):985?995, 1990.

M. Najafi, M. Zekri, and J. Askari. Input time delay systems identification via wavelet approach. Proceedings of the 2010 IEEE International Conference on Control and Automation, 8:2013?2017, 2010.

M. Nihtil?a, T. Damak, and J.P. Babary. On-line estimation of the time delay via orthogonal collocation. Simulation Practice and Theory, 5(2):101?120, 1997.

Y. Nurges and Y. Yaaksoo. Laguerre state equations for multivariable discrete systems. Automation and Remote Control, 42:1601?1603, 1981.

P. Van Overschee and B.D. Moore. N4SID: Subspace algorithms for the identification of combined deterministicstochastic systems. Automatica, 30:75?93, 1994.

Luzhou Xu, Jian Li, and P. Stoica. Target detection and parameter estimation for mimo radar systems. IEEE Transactions on Aerospace and Electronic Systems, 44: 927?939, 2008.

9069

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download