On-line identification of continuous time-delay ... - Semantic Scholar

i n t . j . c o n t r o l , 1997, v o l . 66, n o . 1, 23± 42

On-line identi® cation of continuous time-delay systems combining least-squares techniques with a genetic algorithm ZI-JIANG YANG ‹ , TO M OHIRO HACHIN O ‹ and TERU O TSUJI ‹ This paper proposes a new approach to on-line identi® cation of continuous timedelay systems from sampled input± output data. In order to track the time-varying time-delay and system parameters, the linear recursive least-squares (RLS) method is combined in a bootstrap manner with the genetic algorithm (GA) which has a high potential for global optimization. The time-delay is coded into binary bit strings and searched by the GA, while the system parameters are updated by the RLS method. Since only the time delay is searched by the GA, a small population size for the GA is su cient and hence it is possible to implement the algorithm on line on the digital computers. Furthermore, this method (GALS method) is hybridized with the sequential nonlinear least-squares method which is eŒective in local search, to improve the speed of convergence. Simulation results show that both the GALS and the hybrid methods are quite e cient. It is also veri® ed that, since the hybrid method is eŒective in both global and local optimizations, it has superior tracking performance over the GALS method especially in the case where the system parameters and time delay vary continuously with time.

1.

Introduction

M any practical systems such as therm al processes, chem ical processes and biological systems, etc., have inherent time delay. If the time delay used in the system model for controller design does not coincide with the actual process time delay, a closed-loop system may be unstable or exhibit unacceptable transient response characteristics. The identi® cation of linear systems with unknown time-varying time delay is of practical importance for system analysis, prediction and control design, etc. Therefore, there has been continuing interest in m ethods of identi® cation of timedelay system s in the last two decades. Some oŒ-line identi® cation methods for time-delay system s have been reported in the literature (R ao and Sivakumar 1979 , Saha and Rao 1981, Pearson and W uu 1984 , Zheng and Feng 1991 , Y ang et al. 1994). However, in many practical systems, the time delay and the system parameters may vary with tim e. In these cases, oŒ-line identi® cation m ethods are not feasible. In order to track the time-varying system parameters and time delay, a practical on-line identi® cation method is strongly required. Several on-line identi® cation methods based on the discrete-time m odel have been reported, such as the correlation analysis approach (Isermann and Bauer 1974 , Zheng and Feng 1990), variable regression approach (Elnagga r et al. 1989 , Dumont et al. 1993), and som e other methods (Boker & Keviczky 1985 , Pupekis 1985, Ferretti et al. 1991). However, in the case of the discrete-time model, the sam pling period may

Received 7 February 1995. Revised 8 September 1995. Second revision 29 January 1996. ‹ Department of Electrical Engineering, Kyushu Institute of Technology, Sensui-cho, Tobataku, Kitakyushu 804, Japan. Tel : 81 93 884 3228 ; Fax : 81 93 884 0879 ; e-mail : yang ! cse.kyutech.ac.jp. 0020± 7179 } 97 $12.00 ’

1997 Taylor & Francis Ltd

24

Z.-J. Yang et al.

be required to be very small such that the tim e delay is an integral multiple of the sampling period whereas, if the sampling period is too sm all, the identi® cation becomes num erically di cult (U nbehauen and R ao 1990). M oreover, the param eters in the discrete-tim e model usually do not correspond to the physical parameters. Therefore the im portance of the identi® cation based on the continuous-time m odel has been recognized in recent years (Unbehauen and Rao 1990). In this paper, we also consider the identi® cation of time-delay systems based on the continuous-tim e model. One popular approach of the conventional on-line identi® cation methods using the continuous-time m odel is based on the approxim ation of the tim e delay in the frequency domain by a rational transfer function such as the polynomial approxim ation (Gaw throp and N ihtila$ 1985), and Pade! approxim ation (Agarwal and Canudas 1987 , Bai and Chyung 1993). This approach can be im plemented for on-line estimation, but it requires estimation of m ore parameters because the order of the approximated system model is increased, and an unacceptable approximation error occurs when the system has a large tim e delay. Furthermore, it is di cult to separate the parameters concerning the time delay from those concerning the system param eters (Agarwal and Canudas 1987). The other approach is based on a nonlinear estimation method such as the nonlinear least-squares method (Gaw throp et al. 1989, Zhao et al. 1991, Tuch et al. 1994). These nonlinear estim ation methods are in essence local search techniques that search for the optimum by using a gradient-following technique. Since the mean squares error function is usually m ultim odal with respect to the time delay even if the system parameters are known a priori (Pupeikis 1985 , Zheng and Feng 1991), in this approach, the estimates of the system param eters and time delay often converge to local optim a and cannot track the abruptly jum ping time delay and system parameters. The genetic algorithm (GA) is a parallel global probabilistic search procedure based on the mechanics of natural selection and natural genetics (G oldberg 1989 , Davis 1990). Because the GA sim ultaneously evaluates many points in the param eter space, it can in eŒect search many local optim a and thereby increases the likelihood of ® nding the global optimum. In recent years, the G A has received considerable attention in various ® elds, because it has a high potential for global optimization. Studies on the control and identi® cation problems using the GA were made for example by Lansberry et al. (1992), Kristinsson and D umont (1992) and Jeong and Lee (1994). For details of the GA, the readers are referred to the books by G oldberg (1989) and D avis (1990). The presence of the unknown time delay greatly complicates the param eter estimation problem, essentially because the parameters of the model are not linear with respect to the time delay. However, once the time delay is determined, the m odel becomes linear for the other parameters and hence the common recursive least-squares (RLS) method can be utilized directly (Elnagga r et al. 1989 , D umont et al. 1993, Yang et al. 1994). M otivated by this fact, in this paper, we proposed a novel approach to online identi® cation of tim e-delay systems from sampled input± output data, com bining the GA with the RLS method to give the GALS m ethod. That is, the tim e delay is determined by the GA, whereas the system parameters are estimated by the RLS method. In general, it is impossible to separate the estimation of the system param eters from that of the tim e delay except when speci® c conditions are assum ed. In the proposed m ethod, the system parameters and time delay are estimated reciprocally in a bootstrap manner. Although the G A has an excellent property for global optimization, em pirically it

On-line identi® cation of continuous tim e-delay systems

25

is inferior in local search (K itano 1993 ) to the least-squares or gradient-based identi® cation techniques which converge to the optimum rapidly from its neighbourhood. To im prove the convergence property, the hybrid method combining the GA LS m ethod with the sequential nonlinear least-squares (SN LS) m ethod described by G oodwin and Sin (1984) is also proposed. The main advantage of the hybrid method is that it yields a faster convergence speed while requiring a smaller population size than the G ALS method, because it is eŒective in both global and local optim izations. Usually, the system behaviour is more sensitive to the time delay than to the other linear system parameters (Chen and Loparo 1993), that is a small error in the timedelay estim ate m ay often cause a large error in the system parameter estimate. Therefore the on-line identi® cation algorithms for time-varying time-delay systems may not be robust to the measurem ent noise of high levels, even if the instrumental variable (IV) m ethod is em ployed. From the other studies reported in the literature (Agarwal and Canudas 1987 , Elnagga r et al. 1989 , Dumont et al. 1993), it should be em phasized here that the on-line identi® cation methods in this paper are limited to the only the cases of low m easurement noise. The oŒ-line identi® cation method com bining the IV method with the GA algorithm which yields accurate estimates of the timeinvariant tim e delay and system parameters in the case of measurement noise of considerably high levels, can be found in the paper by Yang et al. (1994). To avoid direct approxim ations of the signal derivatives, the identi® cation algorithm is based on the approximated discrete-time estimation model derived by the aid of a digital pre-® lter in which the system param eters remain in their original form (Sagara et al. 1991 , 1993 , Yang et al. 1994). As will be explained later, in our estimation model the time-delay need not be an integral multiple of the sampling period in contrast to most of the identi® cation methods of the time-delay systems. Various numerical examples show that the proposed m ethods are quite satisfactory for the system s with tim e-varying system parameters and time delay.

2.

Statement of the problem Consider the following single-input single-output continuous tim e-delay system : n

3

i= !

a p n Õ ix(t) ¯ i

m

3

i= "

b p mÕ i u(t® i

s )

(a ¯ !

1)

(1)

where p is a diŒerential operator, u(t) and x(t) are the real input and output respectively, and s and a (i ¯ 1, 2, ¼ , n), b (i ¯ 1, 2, ¼ , m) are the time delay and i i system param eters respectively. n and m are assumed to be known (n " m). It is assumed that a zero-order hold is utilized such that u(t) ¯

u` (k),

(k®

1) T %

t!

kT

(2)

where T is the sampling period. The measurement of the output variable is assumed to be corrupted by a low-level zero-mean stochastic measurem ent noise Š (k) : y(k) ¯

x(k)­

Š (k)

(3)

26

Z.-J. Yang et al.

Our goal is to identify the system parameters and time delay on line from sam pled data of the input and the noisy output.

3.

On-line identi® cation of the system parameters

In this section, on-line identi® cation of the system param eters via the RLS method in the case when the time delay has been determined by some other m ethods is brie¯ y reviewed. The identi® cation algorithm which estimates the system parameters and tim e delay by com bining the RLS method with the GA will be presented in the next section. Since diŒerential operations m ay accentuate the measurem ent noise as well as the round-oŒnoise, it is inappropriate to identify the parameters using direct approximations of diŒerentiations (Young 1981 , U nbehauen and Rao 1990). O ur objective here is to introduce a digital low-pass ® lter which would reduce the noise eŒects su ciently. Then we can obtain an approximated discrete-time estimation model in which the system parameters remain in their original form (Sagara et al. 1991 , 1993 , Yang et al. 1994). In this paper we introduce a low-pass pre-® lter Q( p) as Q( p) ¯

1 ( a p­

1) n

(4)

where a is the tim e constant which determines the passband of Q( p). The pre-® lter (4) was ® rst suggested by Young (1964) where it is suggested that the ® lter should be chosen so that it matches approximately the passband of the system under study. Subsequently, Young discussed the optim al choice of pre-® lters for discrete- and continuous-tim e models (Young 1976 , 1981 , Young and Jakeman 1980 ) and proposed recursive and iterative methods for implem enting this optimal approach. In the simplest case of white m easurement noise, for example, this optim al method adaptively selects the pre-® lter such that its denominator polynomial converges on the denom inator of the system m odel. In the present study, we con® ne our attention to identi® cation of time-varying parameters and time delay in the case of low measurement noise. W e have found empirically that, in the case of low measurement noise, the param eter estimates are not so sensitive to design of the passband of the pre® lter (Sagara et al. 1991). To achieve fast tracking to the parameter and tim e-delay variation, a pre-® lter with a slightly fast time constant is usually recommended, that is the passband of the pre-® lter may be slightly broader than that of the system under study. This approach is very similar to that proposed by Young (1964, 1965 , 1966 , 1970) and is m ost useful, as Young showed, in the case of tim e-varying param eter estimation and adaptive control, where an adaptive ® lter would add unnecessary com plication. In higher-m easurement-noise situations (i.e. the noise-to-signal ratio (NSR) is greater than 5 % ), the approach proposed in the present paper will yield biased estimates of the parameters and so some procedure for removing this bias will be required. In his papers, Young Proposed an IV m odi® cation to the RLS methods and, in the later papers, showed that the resulting re® ned instrum ental variable (RIV) algorithm , which incorporates both adaptive pre-® ltering and adaptive IV estimation, had optimal statistical properties. In addition, the simpli® ed re® ned instrumental variable (SR IV) (Young 1985 , Young et al. 1991 ) algorithm, which does not require

27

On-line identi® cation of continuous tim e-delay systems

sim ultaneous noise model estimation and is simpler to implement in practice than the RIV algorithm , yields consistent and relatively e cient estimates of the transfer function model parameters. Clearly these same modi® cations could be applied to the approach at present proposed in order to m ake it less vulnerable to higher measurement noise levels. In our previous work (Yang et al. 1994), we have veri® ed that the oŒ-line identi® cation method combining the IV method with the GA algorithm can yield accurate estimates of the time-invariant tim e delay and system parameters in the case of measurement noise of considerably high levels. M ultiplying both sides of system (1) by Q( p) and using the bilinear transformation based on the block-pulse functions (Jiang and Schaufelberger 1992), we obtain the following approximate discrete-tim e estimation model (Sagara et al. 1991 , 1993 , Yang et al. 1994) : n

3

n ya(k)­ !

i= "

where

Q (z Õ " )

0 21

sg)¯

Q (z Õ " )

0 21

r(k) ¯

!

!

n

3

i= !

Q (z Õ " ) ¯ !

s g )­

b i n (n Õ m+i)ua(k®

(5)

r(k)

i= "

n iya(k) ¯ n (nÕ m+i)ua(k®

m

3

a i n iya(k) ¯

T i

5

T n Õ m+i

(1­

z Õ " ) nÕ i y` (k)

z Õ " ) n Õ m+i(1®

z Õ " ) mÕ i u` (k®

l) (6) 6 7

!

0 1

z Õ " )­

(T } 2) (1­

a Q (z Õ " ) i

[a (1®

z Õ " )i (1®

(1­

T i (1­ 2

z Õ " )i (1® z Õ " )] Õ n

z Õ " ) nÕ i Š (k)

8

and y` (k) ¯ (1­ z Õ " ) y(k) } 2 is the block-pulse approxim ation of y(t) (Jiang and Schaufelberger 1992), s g is given by

sg ¯

s T

¯

l­

D T

(7)

where 0 % D ! T and l is a non-negative integer. So far in the literature, most of the methods for identifying time-delay system s by using the discrete-time model assum e that the time delay is an integral m ultiple of the sampling interval. This is generally not true, especially when the sampling interval is large, and in this case the error in the time-delay estim ate can cause instability of the closed-loop control system (Chen and Loparo 1993). However, in the discrete-time model identi® cation, too small a sam pling interval may cause some numerical problems (Unbehauen and Rao 1990). It should be noted here that in our estimation model it is not required that the time delay s is an integral m ultiple of the sam pling interval, that is it is allowable that D 1 0. In this case, n (k® s g ) can be obtained (n Õ m+i)ua by the linear interpolation of n (k® l ) and n (k® l® 1). Of course, if the a a (nÕ m+i)u (n Õ m+i)u input± output data are rapidly sampled, the time delay can then be expressed as an integral multiple of the sampling interval and the approximation by the linear interpolation need not be employed. Too sm all a sampling period, however, m ay be a heavily computational burden. Therefore a trade-oŒshould be made here.

28

Z.-J. Yang et al. Equation (5) can be written in vector form :

n ya(k) ¯ ! z T(k, s g ) ¯ h T¯

z T (k, s g ) ± h ­

,®

n ya(k), I "

[®

[a , I

n nya(k), n (n Õ m+ )ua(k® "

, an , b , I

"

5

r(k)

s g ), I

,n

s g )]

(k®

nu a

, b m]

"

(8) 7

6

8

From (8), the system parameters can be estimated by the following RLS method provided that the time delay has been determined by som e other m ethods :

h # (k) ¯

h # (k®

e (k) ¯

n ya(k)® !

L(k) ¯

P(k) ¯

z T(k, s g ) ± h # (k®

1

q (k)

9 P(k®

1)

1) z(k, s g )

P(k®

z T (k, s g ) ± P(k®

q (k)­

5

L(k) e (k)

1)­

P(k®

1)®

(10) 6

1) z(k, s g )

q (k)­

7

1) z(k, s g ) ± z T(k, s g ) P(k® z T(k, s g ) ± P(k®

1)

1) z(k, s g )

: 8

where q (k) is the forgetting factor. It is well known that the least-squares h # (k) is obtained in the sense of minimizing the following mean squares equation error : V( h , s ) ¯

1 ks +N 3 q (k) k s+N Õ k ( n ya(k)® N ! k=k + s

4.

z T (k, s g ) ± h ) #

(10)

"

Identi® cation algorithm combining the recursive linear least-squares method with the genetic algorithm

The G ALS method used for on-line identi® cation of the system param eters and the tim e delay is proposed in detail in this section. In general, it is impossible to separate the estimation of the system parameters from that of the time delay except when speci® c conditions are assumed. Therefore, in the proposed method, the bootstrap technique, by which the system parameters and time delay are estimated reciprocally, is adopted. At each tim e step, the system param eters are updated by the RLS method based on the estim ates of the system param eters and tim e delay obtained at the previous tim e step. Then the ® tness values for the candidates of the tim e delay coded in binary bit strings are calculated with the system param eter estimate updated by the RLS method at the current time step. The estimate of the time delay which is determined from the individual string with the best ® tness value am ong the current candidate population of the time delay is passed to the linear RLS estimator for the update of the system parameter estimate at the next time step. At the same tim e, genetic operations, that is reproduction based on the current ® tness values, cross-over, mutation, etc., are carried out for the binary bit strings of the current generation, such that the GA progresses to the next generation. Therefore the system parameters and time delay are estimated on line in a bootstrap manner, every time that the input± output data are measured. The bootstrap technique in time-delay system identi® cation can also be found in the papers by Elnagga r et al. (1989), Gaw throp et al. (1989) and Dumont et al. (1993).

29

On-line identi® cation of continuous tim e-delay systems

Figure 1.

Flow chart of the GALS method.

In the algorithm, the tim e delay s is coded into binary bit strings of L bits. Now assume that the search range of the tim e-delay is [s ,s ] and denote the decimal min max values of the binary representation as X ; then the tim e delay is decoded linearly as follows :

s ¯

s max ® 2 L®

s min X­ 1

s min

(11)

The algorithm of the proposed GALS method is described as follows. Step 1. Set the initial values h # (k s ) and s W (k s ). Let the time step index k ¯ generation index of the G A g ¯ 0. Step 2. Calculate the signal vector z(k, s gW (k®

1)) using s W (k®

k s­

1 and the

1).

Step 3. U pdate the system parameter estim ate h # (k) using the RLS method in (9). Step 4. (1) In the case of g ¯ 0, generate an initial populations of M strings ( s W i(k) (i ¯ 1, 2, I , M )) represented in binary bits as the candidates of s W (k) randomly. (2) In the case of g 1

0, the genetic operations (a)± (d ) are carried out.

(a) Reproduction. In this paper, reproduction is implem ented as a linear search through roulette wheel slots weighted in proportion to the ® tness value (see (12) in Step 5) of the individual string in the old generation. Each one of the old population is reproduced with the

30

Z.-J. Yang et al. probability of J ( h # (k® 1), s gW (k® 1)) } 3 M J ( h # (k® 1), s gW (k® 1)). Praci i j= " i j tically, the linear ® tness scaling (Goldberg 1989 ) is utilized. Additionally, the best string of the last generation is always reproduced. (b) Cross-o Š er. Pick up two strings random ly and decide whether or not to cross them over according to the cross-over probability P . If a c cross-over is required, exchange strings at a crossing position. The crossing position is chosen randomly. (c) M utation. Alter a bit of string (0 or 1) according to the mutation probability P . m

(d ) Refreshment. The refreshment operation (so called by the present authors) is not standard in the sim ple version of the GA. It is introduced here in order to keep the diversity of the candidate population of the tim e delay, such that the G A can track the timevarying tim e delay. By this operation, every sth generation, the best q % of individual strings are survived according to their ® tness values (see (12) in Step 5), whereas the rest of strings are replaced by random ly generated strings. Step 5. Calculate the ® tness value for each of the candidates of the time delay : J ( h # (k), s g W (k)) ¯ i i

(w 1

3

k

j=kÕ w+ "

q (k) kÕ j [n ya( j)® !

*

z T ( j, s g W (k)) h W (k)]# i

(i ¯

Õ "

1, 2, ¼

,M)

(12)

where w is the time window length. Usually, a large w makes the execution of the GA slow down, whereas a small w makes the estimates tend to oscillate. Therefore a trade-oŒshould be considered to choose a suitable w. Em pirically, it is recom mendable to take w to be 80± 120. Step 6. D etermine the time-delay estimate s W (k) of the current time step from the individual string with the best ® tness value am ong the current candidate population of the time delay. Step 7. Increase the time step index and the generation index respectively as k ¯ g ¯ g­ 1, and go to Step 2.

k­

1,

For clarity, the ¯ ow chart of the G ALS method is shown in Fig. 1. Some researchers such as Kristinsson and Dum ont (1992) and Jeong and Lee (1994) also challenged to apply the G A to system identi® cation problem . In their work, however, all the parameters including the tim e delay to be estimated are represented with binary bit strings and the identi® cation is carried out by only the G A. Since the GA has to search for the optimal solution in the com plicated m ultidim ensional search space, all the parameters including the time delay m ust be coded into binary strings with numerous bits. Therefore a large population size is required to avoid premature convergence. On the other hand, in our method, only the time delay of the system input is represented with binary bit strings, and hence the GA searches for only one single value. Therefore it is su cient to choose a sm all population size such that it is possible to im plement the algorithm on line on the digital com puters. Typically, a population size of 15± 20 is su cient.

On-line identi® cation of continuous tim e-delay systems

31

Compared with the conventional identi® cation methods of the tim e-delay systems, the proposed m ethod is relatively com putationally demanding because of the use of GA . However, in recent years, since the powerful high performance microprocessing units (M PUs) are availa ble easily at cheap prices, the computational burden should not be a bottleneck to our method. As will be shown by numerical studies, the high perform ance of the proposed method is quite attractive so that the computational cost seems trivial. M oreover, compared with the electrical serosystems, m any practical systems such as thermal processes, chemical processes and biological systems which have inherent time delay allow users to choose a relatively large sampling period, therefore it is possible to execute the GALS method within one single sam pling period. Additionally, since the GALS method is a parallel algorithm , it is considerable to im plem ent it in a multirate fashion, or with a parallel signal processing technique on multiple M PUs. This is outside the scope of the present paper.

5.

Hybrid method combining the genetic algorithm least-squares method with the sequential nonline ar least-squares method

System analysis, prediction and control design are sensitive to the time delay and the system param eters. Therefore it is required that the estim ates of the system parameters and tim e delay track the true values rapidly when the system changes with tim e. However, it has been pointed out that, although the GA has an excellent property for global optimization, it is inferior in local search to the conventional methods (K itano 1993). O n the other hand, although the gradient-based optim ization techniques are poor for global optim ization and often converge to local optima, they converge to the optimum relatively rapidly from its neighbourhood. It has been shown by many examples that hybrid m ethods combining the GA with other conventional optim ization techniques can improve the accuracy and convergence speed of search (Kitano 1993). Here, the G ALS method proposed in  4 and the SN LS method (Goodwin and Sin 1984) are hybridized in order to improve the speed of convergence. In the proposed hybrid m ethod, the G ALS method and the SNLS method run in a parallel fashion, while still comparing and exchanging the results with each other at every time step. The ® tness values of the estim ates by both the GA LS method and the SN LS method are compared, and the superior estimates are adopted as the current estim ation results and are used for the estimate updates at the next time step. If the G ALS m ethod yields better estimates, the SNLS method is reset by the GALS estimates while, if the SNLS method yields better estimates, the GALS method is reset by the SNLS estimates. In the latter case, the time-delay estimate obtained by the SN LS method is coded into a binary string and replaces one of the strings of the GA . Compared with the identi® cation by only the GALS method, the hybrid m ethod has better convergence owing to the support of the SN LS m ethod. In the hybrid method, the task of the global search is carried out by the GALS method, and the SNLS method is devoted to the local search. Therefore, for the hybrid method, it is allowable to choose a less ® ne resolution (a shorter length of the binary bit string) and hence a smaller population size than the GALS method. This leads to a reduction in the com putational burden.

32

Z.-J. Yang et al.

5.1. Identi® cation by using the sequential nonline ar least-squares method The approximate discrete-time estimation model described in (8) is rewritten as follows :

} T(k, s g ) ± H ­

n ya(k) ¯ ! } T (k, s g ) ¯ H T¯

,®

n ya(k), I "

[®

[a , I

n nya(k), n (n Õ m+ )ua(k® " ,b ,s ] m

,a ,b ,I n

"

5

r(k)

"

s g ), I

,n

(k®

nu a

s g ), 0]

(13) 6 7

8

By m inimizing the m ean squares error 1 ks +N

3

V(H ) ¯

} T (k, s g ) ± H ]#

q (k) k s+N Õ k [n ya(k)® !

N k=k + s "

(14)

we can identify the system parameters and tim e delay by the SNLS method (Goodwin and Sin 1984 ) as

H = (k) ¯

e (k) ¯ R(k) ¯

H = (k®

1)­

n ya(k)® ! 1

q (k)

R(k) f (k, H = (k®

} T (k, s g ) ± H = (k®

9 R(k®

5

1)) e (k)

1) (15) 6

1)®

R(k®

q (k)­

1) f (k, H = (k®

1)) ± f T (k, H = (k®

f T(k, H = (k®

1)) ± R(k®

1)) R(k®

1) f (k, H = (k®

1) 1))

:

7

8

where A

®

C

n ya(k) "

]

®

f (k, H ) ¯

®

¥ e (k) ¯ ¥ H

n nya(k)

n (n Õ m+ )ua(k® "

sg)

(16)

]

® B

3

m i= "

n nu a(k®

sg)

b i n (n Õ m+iÕ )ua(k® "

sg ) D

A purely continuous-time version of this method can be found in the paper by Tuch et al. (1994). W e have observed that, by the SN LS method, the estimates converge to the true values rapidly, provided that the initial values are near the true values. However, in general, it is very di cult to ® nd such initial values, since the system to be identi® ed is unknown, and hence the SN LS method m ay often converge to local optima. In our experience, even if the SN LS m ethod will fortunately reach the true values, the convergence m ay take much tim e.

5.2. Hybrid m ethod The proposed hybrid method combining the GALS m ethod with the SNLS method is given as follows.

33

On-line identi® cation of continuous tim e-delay systems

Figure 2.

Flow chart of the hybrid method.

Step 1. Set the initial values h # (k ) and s W (k ) for the G ALS method, and GALS s GALS s H = T(k s ) ¯ [h # TSNLS (k s ), s W SNLS (k s )] for the SN LS method such that h # GALS (k s ) ¯ h # SNLS (k s), s W GALS (k s ) ¯ s W SNLS (k s ). Let the time step index k ¯ k s ­ 1 and the generation index of the G A g ¯ 0. Step 2. Perform the G ALS method from Step 2 to Step 6. Step 3. U pdate the estim ates of the parameters including the tim e delay H = (k) using the SN LS m ethod in (15). Step 4. Calculate the ® tness value in the SN LS method as follows : J

SNLS

( H = (k)) ¯

(w 1

3

k

i=kÕ w+ "

q (k) kÕ i [n ya(i)® !

} T(i, s g W SNLS (k)) ± H = (k)]#

*

Õ "

(17)

Step 5. If J & J SNLS (J GALS denotes the best ® tness value in the GALS method), GALS then let h # (k) and s W (k) be the results of the estimation, and set H = T (k) ¯ GALS GALS [ h # T (k), s W (k)]. GALS

GALS

34

Z.-J. Yang et al.

Step 6. If J ! J SNLS , then let H = (k) be the result of the estimation. Only if the time GALS delay estimated by the SN LS method s W (k) ` [s ,s ], set the GALS SNLS min max estimates as h # (k) ¯ h # (k), s W (k) ¯ s W (k) and code s W (k) into a GALS SNLS SNLS SNLS SNLS binary bit string. From (11), we have X¯

N INT

9

( s W SNLS (k)®

s min ) (2 L ® s max ® s min

1)

:

(18)

where NIN T[ [ ] denotes the integer num ber obtained by rounding a real num ber. Then X is transform ed into a binary bit string, which replaces one of the strings in the GA. Step 7. Increase the time step index and the generation index respectively as k ¯ g ¯ g­ 1, and go to Step 2.

k­

1,

The ¯ ow chart of the hybrid method is shown in Fig. 2.

6.

Illustrative exam ples

6.1. Exam ple 1 Consider the following continuous time-delay system described by xX (t)­

a xc (t)­ "

a x(t) ¯ #

s )

b u(t® "

(19)

where the system parameters and time delay change piecewise constantly as shown in Table 1. The input signal is output of a zero-order hold driven by a white signal ® ltered by a second-order Butterworth ® lter L( p) ¯

² ( p } x c )# ­

2 " /# ( p } x c)­

1´ Õ "

(x c ¯

4± 0)

(20)

The sam pling period is taken to be T ¯ 0± 05 and the identi® cation is carried out from t ¯ 150 s to t ¯ 650 s. The time constant a in the low-pass pre-® lter Q( p) is designed as 0± 4. W e have found em pirically that, in the case of low measurement noise, the estim ates are not so sensitive to design of the passband of the pre-® lter (Sagara et al. 1991). The NSR of the output signal is assumed to be 5 %. The design param eters of the G A are given as follows : (1) population size : GALS m ethod, M ¯ 20 ; hybrid m ethod, M ¯ 16 ; (2) string length : GALS m ethod, L ¯ 14 : hybrid m ethod, L ¯ 7 ; (3) cross-over probability P ¯ 0± 8 ; c (4) mutation probability P ¯ 0± 03 ; m (5) search range of the time delay : GALS m ethod, [0, (2 " % ® 1)¬ 0± 001] ¯ [0, 16± 383] ; hybrid m ethod, [0, (2 ( ® 1)¬ 0± 129] ¯ [0, 16± 383]. Note here that the resolution of the time-delay coding in the hybrid method is 0± 129, which is much coarser than that of the GALS m ethod, 0± 001. It should be commented here that, in the hybrid method, the task of the global search is carried out by the GALS method, and the SNLS method is devoted to the

On-line identi® cation of continuous tim e-delay systems

Table 1.

Figure 3.

Time (s)

a

[0± 300] [300± 450] [450± 650]

3± 0 2± 0 2± 0

a "

b #

4± 0 4± 0 5± 0

"

2± 0 3± 0 4± 0

35

s 9± 130 11 ± 610 10 ± 350

True values of the system parameters and time delay.

Estimates by the SNLS method (Example 1) : (a) time delay s W ; (b) system parameter aW ; (c) system parameter aW ; (d) system parameter bW . "

#

"

local search. Therefore, for the hybrid method, it is allowable to choose a less ® ne resolution (a shorter length of the binary bit string) and hence a smaller population size than for the GA LS method. Every tenth generation (s ¯ 10 in Step 2 of the G ALS method), the best 25 % of individual strings are survived, and the rest of the strings are replaced by randomly generated strings. This operation is introduced to keep the diversity of the candidate population, such that the GA can track the time-varying time delay. The tim e window length w of the ® tness value is taken to be 100. The initial values for both the GALS and hybrid m ethods are chosen as follows :

h # (k ) ¯ #

[1, 1, 1] T ,

s W (k s ) ¯

P(k ) ¯

10 & I,

s

H = (k s ) ¯ [1, 1, 1, 1] T R(k ) ¯ 10 & I s 1,

where I is the unit matrix. The forgetting factor is given as

q (k) ¯

(1®

0± 001) q (k®

1)­

0± 001

( q (k ) ¯ 0± 95 ; if q (k) " 0± 99, then q (k) ¯ 0± 99). s The results by the com mon SN LS method are shown in Fig. 3, where the broken lines denote the true values and the solid lines denote the estimates. It is clear that the SNLS method fails to track the time-varying system parameters and tim e delay

36

Z.-J. Yang et al.

Figure 4. Estimates by the GALS method (Example 1) : (a) time delay s W ; (b) zoomed-up trajectory of s W ; (c) system parameter aW ; (d) system parameter aW ; (e) system parameter " # bW . "

because this method is only eŒective in local optimization. The results obtained by the GA LS method and the hybrid m ethod are shown in Fig. 4 and Fig. 5 respectively. From Figs 4 and 5, it is clear that the estim ates obtained by both the proposed methods can track the abruptly jumping system parameters and time delay rapidly. Furtherm ore, from the zoomed-up trajectories of the time-delay estim ate, it can be found that the estimates obtained by the hybrid method converge to the true values slightly more rapidly than those determined by the GALS m ethod, that is the hybrid method im proves the convergence speed. The execution tim e (im plemented on Sun W orkstation Sparc 10, m odel 40) of the three algorithms coded in Fortran 90 are compared in Table 2. Although the hybrid method seems more sophisticated than the GALS method, it requires a small population size and hence requires a shorter execution time than the G ALS method. Additionally, it is observed that the tim e-delay estimate converges m uch m ore rapidly than the system parameter estimate. This is because usually the system

On-line identi® cation of continuous tim e-delay systems

37

Figure 5. Estimates by the hybrid method (Example 1) : (a) time delay s W ; (b) zoomed-up trajectory of s W ; (c) system parameter aW ; (d) system parameter aW ; (e) system parameter " # bW . "

Table 2.

Method

Time (s)

SNLS GALS Hybrid

30 ± 8 112 ± 8 100 ± 7

Central processing unit time of the identi® cation algorithms.

behaviour is more sensitive to the time delay than to the other linear system parameters (Chen and Loparo 1993), that is a sm all bias in the time-delay estimate may often cause a large bias in the system parameter estimate. This em phasizes the im portance of the accurate estim ation of the time delay.

38

Z.-J. Yang et al.

Figure 6.

Estimates by the SNLS method (Example 2) : (a) time delay s W ; (b) zoomed-up trajectory of s W ; (c) system parameters aW , aW and bW . "

#

"

6.2. Exam ple 2 Consider the time-varying tim e-delay system described by xX (t)­

a (t) xc (t)­ "

a x(t) ¯ #

b u[t® "

a (t) ¯

3± 0­

a ¯

4± 0

b ¯

2± 0

s (t) ¯

9± 0­

"

# "

5

s (t)]

0

0± 5 cos

2 p (t®

150± 0)

500± 0

1 (21) 6 7

0± 5 sin

0

2 p (t® 150± 0) 500± 0

1 8

The sampling period, the pre-® lter Q( p), the N SR, the initial setting, the forgetting factor, the design param eters of the GA, etc., are the same as those in Example 1. The results obtained by the comm on SNLS m ethod are shown in Fig. 6. It can be found that it takes a long time for the estimates to catch up with the true values, although the estimates ® nally track the slowly changing parameters and time delay. It should be noted here that this is a lucky case. W ith some other initial settings, the estim ates by the SN LS method converge to the local minima and never reach the true values. Therefore, from the viewpoint of on-line use, this method is not satisfactory. The result obtained by the GALS method are shown in Fig. 7. Although the G ALS method tracks the tim e delay with a sm all bias owing to its relatively slow speed in local search (Kitano 1993), as mentioned previously, since a small bias of the time delay may in¯ uence the system parameter estim ate greatly, the estim ated system param eters

On-line identi® cation of continuous tim e-delay systems

Figure 7.

39

Estimates by the GALS method (Example 2) : (a) time delay s W ; (b) zoomed-up trajectory of s W ; (c) system parameters aW , aW and bW . "

#

"

determined by the GALS method are not acceptable. The results obtained by the hybrid method are shown in Fig. 8. It can be veri® ed that the hybrid method tracks the continuously time-varying system param eters and tim e delay accurately, because of its high ability in both global and local optimizations. Comparison of the zoomed-up trajectories of the tim e delay estimated by the three methods shows that the hybrid method yields a m uch more accurate estimate than do the other two methods. In this sim ulation study, we have veri® ed that the hybrid method is preferable. Several studies on the identi® cation of time-varying time-delay systems have been reported in the literature (Agarwal and Canudas 1987 , Elnagga r et al. 1989 , Dumont et al. 1993). In their work, only the systems where the system parameters and timedelay are piecewise constant as in Example 1 of this paper are considered. H owever, strictly speaking, such system s can be viewed as piecewise time invariant, that is they are tim e invariant except for the tim e instants of jumps. To our knowledge, identi® cation of the system s where the parameters and time delay vary continuously with time as in Exam ple 2 has not been studied widely in the literature. W e have found that it is uneasy to identify such a system by the conventional time-delay system identi® cation methods. This example shows the attractive applicability of the hybrid method. It should be commented here that the on-line identi® cation m ethods proposed in this paper are lim ited to only the cases of low measurement noise. As the other works reported in the literature (Agarwal and Canudas 1987 , Elnagga r et al. 1989 , Dumont et al. 1993), the on-line identi® cation algorithms for time-varying time-delay systems may not be robust to the measurement noise of high levels. A nd we have veri® ed that in the case of m easurement noise of high levels the estimates of the tim e-varying timedelay and param eters are very sensitive and thus not acceptable even if the IV method instead of the least-squares m ethod is em ployed.

40

Z.-J. Yang et al.

Figure 8.

Estimates by the hybrid method (Example 2) : (a) time delay s W ; (b) zoomed-up trajectory of s W ; (c) system parameters aW , aW and bW . "

7.

#

"

Conclusions

In this paper, a new approach to on-line identi® cation of continuous tim e-delay systems from sampled input± output data has been studied. In order to track the timevarying time delay and system parameters, we have proposed a new method (GALS method) combining the R LS m ethod in a bootstrap manner with the G A which has a high potential for global optimization. Furthermore, in order to im prove the convergence property, the hybrid method com bining the GALS and SN LS methods has also been proposed. Simulation results show that the proposed methods are quite eŒective in the case where the system changes abruptly, and that the hybrid method has a superior convergence property to the GA LS method especially in the case where the system changes with tim e continuously. It should be emphasized, here especially, that the hybrid method requires a smaller GA population size and hence, entails a moderate computational burden, since it is eŒective in both global and local optim izations. Therefore the hybrid method is highly expected to be applicable to the on-line identi® cation of real systems.

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