Adaptive Identification Algorithm for Continuous-Time Linear Time-Invariant Systems in Frequency Domain⁎

July 9-11, 2018. Stockholm, Sweden on System Identification Proceedings,18th IFAC Symposium July 9-11, 2018. Stockholm, Sweden Available online at www.sciencedirect.com

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Adaptive Identification Algorithm Adaptive Identification Algorithm Adaptive Identification Algorithm for Continuous-Time Linear for Continuous-Time Linear for Continuous-Time Linear Time-Invariant Systems in Time-Invariant Systems in  in Time-Invariant Systems Frequency Domain Frequency Domain  Frequency Domain Wen Mi ∗,∗∗ Wei Xing Zheng ∗∗ ∗,∗∗ Wen Mi Wei Xing Zheng ∗∗ ∗,∗∗ ∗∗ Wen Mi Wei University Xing Zheng ∗ School of Mathematical Science, of Electronic and

∗ School Science of Mathematical Electronic Technology of China, Science, Chengdu,University Sichuan, of 611731, P. R.and China ∗ School of Mathematical Science, University of Electronic Technology Science(e-mail: of China, Chengdu, Sichuan, 611731, P. R.and China [email protected]) Technology Science(e-mail: of China, Chengdu, Sichuan, 611731, P. R. China ∗∗ [email protected]) School of Computing, Engineering and Mathematics, Western ∗∗ (e-mail: Engineering [email protected]) School Computing, Western SydneyofUniversity, Sydney, NSW and 2751,Mathematics, Australia (e-mail: ∗∗ School of Computing, Engineering and Mathematics, Western Sydney University, Sydney, NSW 2751, Australia (e-mail: [email protected]) Sydney University, Sydney, NSW 2751, Australia (e-mail: [email protected]) [email protected])

Abstract: This paper considers the problem of identifying continuous-time linear time-invariant Abstract: paper domain. considersAn theadaptive problem of identifying continuous-time linear time-invariant systems in This frequency identification is proposed in terms of continuous Abstract: This paper considers the problem of identifying continuous-time linear time-invariant systems in frequency domain. An adaptive identification is proposed in terms of rational orthogonal bases, where one-by-one selection of poles is performed forcontinuous the basis systems in frequencybases, domain. An adaptive identification proposed in terms of continuous rational where one-by-one selection ofis poles is performed forunder the these basis functions.orthogonal It is shown that the proposed algorithm achieves approximate optimality rational orthogonal bases, where one-by-one selection of poles is performed for the basis functions. is shown that of thethe proposed algorithm achieves these selections. It The adaptivity proposed algorithm lies in approximate that differentoptimality poles are under selected for functions. It is shown that the proposed algorithm achieves approximate optimality under these selections. The adaptivity of the proposed algorithm lies in that different poles are selected for basis functions for different systems. The effectiveness of the proposed adaptive algorithm is selections. The adaptivity of systems. the proposed algorithm lies in that different poles are selected for basis functions different demonstrated byforsimulation results. The effectiveness of the proposed adaptive algorithm is basis functions for different systems. demonstrated by simulation results. The effectiveness of the proposed adaptive algorithm is © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. demonstrated by simulation Keywords: Orthogonal basis,results. frequency domain identification, adaptive algorithm, rational Keywords: Orthogonal basis, frequency domain identification, adaptive algorithm, rational function. Keywords: function. Orthogonal basis, frequency domain identification, adaptive algorithm, rational function. 1. INTRODUCTION den Hof et al. (1995); Ninness et al. (1997); Ninness & 1. INTRODUCTION den Hof et (1997); al. (1995); Ninness et al.(1998, (1997); NinnessVan & Gustafsson Ak¸cay & Ninness 1999a,b); 1. INTRODUCTION den Hof et al. (1995); Ninness et al. (1997); Ninness & Gustafsson (1997);(2003); Ak¸cay Mi & Ninness (1998, 1999a,b); Van Gucht & Bultheel & Qian (2012). A summative The modeling of dynamical systems can be roughly catego- Gustafsson (1997); Ak¸cay & Ninness (1998, 1999a,b); Van Gucht & these Bultheel (2003); Mi & Qian (2012). A summative book on researches in this stage is Heuberger et al. The modeling of dynamical systems can be roughly rized as time domain identification (S¨oderstr¨ om &categoStoica Gucht & Bultheel (2003); Mi & Qian (2012). A summative book on these researches in this stage is Heuberger et al. The modeling of dynamical systems can be roughly catego(2005). rized asLjung time domain identification oderstr¨ om & Stoica (1989); (1999); Billings & Wei(S¨ (2008); Laurain et al. book on these researches in this stage is Heuberger et al. (2005). rized asLjung time domain identification (S¨ oderstr¨ om & Stoica (1989); (1999); Billings & Wei (2008); Laurain et al. (2015); Mu et al. (2017)) and frequency domain identifica(2005). (1989); Ljung Billings & Wei (2008); Laurain et al. Denote by Π the open right-half complex plane. The (2015); Mu et (1999); al. (2017)) andPintelon frequency domain identification (Pintelon et al. (1994); & Schoukens (2001); Denote by Π the open right-half complex plane. The continuous-time rational orthogonal basis functions are (2015); Mu et al. (2017)) andPintelon frequency Schoukens domain identification (2003)). (Pintelon et al. (1994); (2001); Bai The rational orthogonal& basis is known as Denote by Π the open right-half complex plane. The continuous-time rational orthogonal basis functions are defined in terms of a sequence of complex-valued numbers tion (Pintelon et al. (1994); Pintelon & Schoukens (2001); Bai (2003)). The rational orthogonal basis is known as an essential tool in identification of linear time-invariant continuous-time rational orthogonal basis functions are defined in terms of a sequence of complex-valued numbers Bai (2003)). tool Theinrational orthogonal basistime-invariant is known as {ξ k } ⊂ Π as an essential identification of linear (LTI) systems, including the popular Laguerre model defined in terms of a sequence of complex-valued numbers {ξ } ⊂ Π as an essential tool including in identification of linear time-invariant  (LTI) systems, the popular model {ξk } ⊂ Π as and two-parameter Kautz model (M¨ akil¨ a Laguerre (1990a,b, 1991); k 2{ξk } (LTI) systems, including the popular Laguerre model B{ξ1 ,...,ξk } (s) = 2{ξ } φk (s), k = 1, 2, . . . , (1) and two-parameter KautzLater model akil¨ a (1990a,b,case 1991); Wahlberg (1991, 1994)). on,(M¨ the generalized of  + ξkk φk (s), k = 1, 2, . . . , (1) B{ξ1 ,...,ξk } (s) = s2{ξ and two-parameter KautzLater model akil¨ a (1990a,b,case 1991); Wahlberg (1991, 1994)). on,(M¨ the generalized of rational orthogonal bases has been studied systematically s + ξ k} Wahlberg (1991, 1994)). Later on, the generalized case of where B{ξ1 ,...,ξk } (s) = s + ξk φk (s), k = 1, 2, . . . , (1) rational orthogonal bases has been studied systematically k using rational orthogonal basis functions, such as Van rational orthogonal bases has beenfunctions, studied systematically using rational orthogonal basis such as Van where k−1  s − ξl  using rational orthogonal basis functions, such asScience Van where This work was supported in part by the National Natural φ1 (s) ≡ 1, φk (s) = k−1  s − ξl , k = 2, . . .  This workofwas supported part by the National Natural Science Foundation China under in Grants 11401079 and 11571083 and in s+ξ φ1 (s) ≡ 1, φk (s) = k−1  l=1 s − ξl , k = 2, . . .  l This supported in part by the National Natural Science Foundation ofwas China under Grants 11401079 and 11571083 and in part by work the Australian Research Council under Grant DP120104986. φ1 (s) ≡ 1, φk (s) = l=1 s + ξl , k = 2, . . . Foundation of China under Grants 11401079 11571083 and in part by the Australian Research Council under and Grant DP120104986. s + ξl l=1

part by the©Australian Copyright 2018 IFACResearch Council under Grant DP120104986.239 Copyright © 2018 IFAC 239 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 239Control. Peer review©under of International Federation of Automatic

10.1016/j.ifacol.2018.09.141

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and {·} denotes the real part of a complex-valued number. For simplicity, we will use Bk instead of B{ξ1 ,...,ξk } in the remaining part of the paper. With respect to the usual inner product defined by  ∞ 1 Bm , Bn   Bm (jω)Bn (jω)djω 2πj −∞  1, m = n = 0, m = n, {Bk }k≥1 is an orthogonal system in the Hardy-2 space H2 (Π). Furthermore, this result can be extended to other function spaces with some extra conditions. Lemma 1. (Ak¸cay & Ninness (1999a)). The model set spanned by the basis functions {Bk }k≥1 is complete in all of the spaces Hp (Π), 1 < p < ∞, and a subset of A(Π) if and only if ∞  {ξk } = ∞, 1 + |ξk |2 k=1

where A(Π) is the set of functions that are analytic on Π and continuous on the boundary. In the discrete-time case, the practical use of the rational orthogonal systems is based on the a-priori-known poles. As the poles for the rational orthogonal bases are given a priori, the corresponding models mainly take advantage of the orthogonality. The works in e Silva (1996); Sabatini (2000); Casini et al. (2003); Mi & Qian (2012) were devoted to finding pole locations for basis functions. In Nara & Ando (2010); Mi & Qian (2014), poles locations of fixed-order systems were investigated. Nevertheless, estimation of poles of the underlying system is not an easy job for both discrete-time and continuous-time systems. In this paper, we study the system identification problem in frequency domain for continuous-time systems in a novel way by selecting the poles for the basis functions consecutively. An adaptive algorithm based on rational orthogonal bases (1) is developed to get efficient approximations through finding poles for the basis functions of (1) instead of finding true poles of the systems. Convergence of the proposed algorithm is analyzed. 2. PROBLEM STATEMENT In this paper, it is assumed that a set of frequency domain measurements {En } is available. Furthermore, it is assumed that these frequency domain measurements are obtained from a continuous LTI system and the transfer 240

function G(s) of this system is analytic on the right-half complex plane. The measurements {Ek }N k=1 are set to be Ek = G(jωk ) + vk ,

k = 1, 2, . . . , N,

(2)

where the measurement error {vk } may be a deterministic bounded sequence with |vk | < or a stochastic sequence with zero mean and variance σ 2 < ∞. Because of the symmetry of the frequency response, we only need the measurements in the positive interval with ωk = kh as usual, where h is the sampling size, and the rest measurements will be obtained as the complex conjugate of the data when ω < 0. Let Xn = span{B1 , B2 , . . . , Bn }. The identification problem to be considered can now be stated as follows. Frequency domain identification problem: Given measurements {(ωk , Ek )}N k=1 in (2) from G(s) ∈ H2 (Π) with all elements coming in complex conjugate pairs, find an approximation Gn (s) ∈ Xn such that {ξk }nk=1 = arg min{G(jω) − Gn (jω)2 , ξk ∈ Π}. The goal is to find a useful sequence {ξk }nk=1 so as to generate an infinite number of generalized rational orthogonal basis functions {Bk (s)}nk=1 instead of estimating the poles of the system by experience. In the meantime, these basis functions can lead to fast and efficient approximation to the original system.

3. ADAPTIVE DECOMPOSITION ALGORITHM In this section, we provide a brief introduction to the adaptive decomposition algorithm for H2 (Π) functions given in Qian (2010). This algorithm is based on the rational orthogonal system (1). For any functions f (s), g(s) ∈ H(Π), the inner product in H2 (Π) is defined by  ∞ 1 f, g = f (jω)g(jω)djω, 2πj −∞ and for g(s) ∈ H2 (Π), the Cauchy integral formula reads  g(z) 1 dz, (3) g(s) = 2πj Π s − z where Π denotes the boundary of Π. Define the kernel as  2{ξ} eξ (s) = . (4) s+ξ Then the inner product has a beautiful property: for any function f (s) ∈ H2 (Π), some simple computation with Cauchy integral formula (3) yields

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f, eξ  =

1 2π



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∞

f (jω)eξ (jω)dω −∞   2{ξ}  1 = 2{ξ}f (ξ). f (s) = 2πj Π −s + ξ

f (s) =

∞ 

241

hk , e{ξk } e{ξk } (s)

k=1

k−1  l=1

s − ξl , s + ξl

(7)

where the convergence is in the H2 norm sense.

The decomposition process for f (s) ∈ H2 (Π) given in Qian (2010) can be summarized as follows.

4. TWO-STAGE IDENTIFICATION ALGORITHM

First, let h1 (s) = f (s). Then f (s) can be written as       f (s) = f (s) − h1 , e{ξ1 } e{ξ1 } (s) + h1 , e{ξ1 } e{ξ1 } (s) s − ξ1 , = h1 , e{ξ1 } e{ξ1 } (s) + h2 (s) s + ξ1 and

Actually, the algorithm given in Section 3 is a greedytype approximation via (6), so it will not result in a best rational approximation to the original system. Though sacrificing a little optimality, it gives an easily-realized and efficient algorithm.

  s + ξ1 . h2 (s) = f (s) − h1 , e{ξ1 } e{ξ1 } (s) s − ξ1

(5)

Lemma 2. The analytic function in the bracket in (5) has a zero at ξ1 , and hence h2 (s) is a function in H2 (Π). Repeating this process n times, f (s) can be further expressed by f (s) = h1 , e{ξ1 } e{ξ1 } (s) + h2 , e{ξ2 } e{ξ2 } (s) + · · · + hn+1 (s)

s − ξ1 s + ξ1

n  s − ξk s + ξk k=1

= h1 , e{ξ1 } B1 (s) + h2 , e{ξ2 } B2 (s) + · · · + hn , e{ξn } Bn (s) + Rn (s). By Lemma 2, hk (s) belongs to H2 (Π) and is given by   2{ξk−1 }hk−1 (ξk−1 ) s + ξk−1 . hk = hk−1 (s) − s − ξk−1 s + ξk−1 ξk in each iteration is chosen by the following criterion ξk = arg max{|hk , eξ |2 , ξ ∈ Π},

(6)

and the sequence {ξk } under this selection criterion is a called greedy sequence since the process is a greedy type philosophy DeVore & Temlyakov (1996). The remainder Rn (s) is given by

For the transfer function G(s) to be approximated, a set of frequency domain measurements {(ωk , G(jωk ))}N k=1 is available. We intend to look for a fast and efficient approximation to G(s) by means of {(ωk , G(jωk ))}N k=1 . In this section, we propose a modified algorithm, which will use the adaptive decomposition method through constructing a new H2 function by measured data. In this two-stage algorithm, there are two issues to resolve. First, the adaptive decomposition method in Section 3 is based on a function, but here in our identification problem, only a set of measured data is available. Second, it is necessary to guarantee that the approximating functions have only realvalued coefficients both in the numerator and denominator because the finite approximations are rational functions. To resolve the first issue, we will combine the data set  {(ωk , G(jωk ))}N k=1 and a new function f (s) with the Cauchy integral formula as the first stage, viz,  1 G(z) dz. G(s) = 2πj Π s − z With G(jω) ∈ L2 (R), it can be substituted, in the L2 norm sense, by  G(jω)χ(·, ·)(ω) k

=

n  s − ξk Rn (s) = hn+1 (s) . s + ξk k=1

N 

G(jωk )χ(ωk−1 ,ωk ) (ω) +

k=1

where χ(ωk−1 ,ωk ) (ω) =

The following theorem in Qian (2010) summarizes the result of the adaptive decomposition. Theorem 1. (Qian (2010)). For any f (s) ∈ H2 (Π), repeating the shifting process consecutively, there holds 241

G(jωk )χ(ωk ,ωk+1 ) (ω)

k=−N

−→ G(jω),

Throughout all these processes, the convergence is guaranteed.

−1 



1, if ω ∈ (ωk−1 , ωk ), 0, if ω ∈ / (ωk−1 , ωk ).

Then the new constructed H2 (Π) function is given by N s − jωk−1 1  G(jωk ) ln f(s) = 2πj s − jωk k=1

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5. CONVERGENCE OF THE PROPOSED ALGORITHM

−1 s − jωk 1  , G(−jωk ) ln 2πj s − jωk+1 k=−N

where ωk = kh, k = 1, 2, . . . , N , and h is the sampling size. When the sampled data {Ek }N k=1 in (2) are used, G(jωk ) is substituted with Ek , which yields N s − jωk−1 1  Ek ln f(s) = 2πj s − jωk

k

k=1

+

−1 1  s − jωk . Ek ln 2πj s − jωk+1

In this section, we mainly make convergence analysis of the two-stage algorithm. When the data is corrupted by noise, say  vk χk (·, ·)(ω), v(ω) =

(8)

the first approximation of the constructed function f (s) can be decomposed into two parts

k=−N

Note that this setting means equally spaced sampling in frequency, which is not necessary. The function f(s) constructed in (8) belongs to H2 (Π) and has the property f(s) = f(s).

The identification problem is concerned not only about fast approximation but also the real-valued coefficients issue in the approximated rational functions. In Ak¸cay & Ninness (1999a), the conjugate points were used to get the approximation with real coefficients. In our two-stage algorithm, we present a new way in which the conjugate greedy sequence is utilized. The motivation is based on the next result. Theorem 2. Given function f (s) ∈ H2 (Π) with conjugation relationship f (s) = f (s), ∞ if {ξk }∞ k=1 is a greedy sequence of f (s), then {ξ k }k=1 is also a greedy sequence of f (s).

f(s) = F (s) + V (s),

where F (s) is the following discrete Cauchy integral with the true system responses:  ∞  1 k G(jωk )χ(·, ·)(ω) dω F (s) = 2π −∞ s − jω with respect to {ωk , G(ωk )}, and the noisy part V (s) is given by  ∞ 1 v(ω) dω. V (s) = 2π −∞ s − jω When the noise is deterministic and bounded by ε > 0, we have the following result. Theorem 3. Consider a continuous-time LTI system with transfer function G(s) ∈ H2 (Π). Given a set of frequency responses {Ek }N k=1 with deterministic and bounded noise v satisfying v2 ≤ ε, ε > 0, then the n-th approximating partial sum Gn (s) obtained by the two-stage algorithm converges to G(s) in the H2 norm sense as lim

N →∞,n→∞, →0

After the n-th process, there could be an approximating partial sum On (s) =

n 

(10)

Consider a continuous-time LTI system given below

hk , e{ξk } B{ξ1 ,...,ξk } (s)

with respect to the selected sequence {ξk }N k=1 , and another n (s) approximating partial sum O n (s) = O

G − Gn H2 = 0.

6. ILLUSTRATIVE EXAMPLE

k=1

n 

(9)

hk , e{ξk } B{ξ1 ,...,ξk } (s).

k=1

G(s) =

k=1

where a1 = 2.04,

Then we can construct Gn (s) as the approximating system, where  1 n (s) , On (s) + O Gn (s) = 2

n (s), Gn (s) are all rational functions. It is and On (s), O not difficult to notice that Gn (s) has only real-valued coefficients. 242

540.70748 × 1017 , 10  (s + ak )

a2 = 18.3,

a4 = 95.15, a7 = 257.21,

a3 = 250.13,

a5 = 148.85, a8 = 298.03,

a6 = 205.16, a9 = 320.97,

a10 = 404.16. The number of data used is N = 300. Some points of the greedy sequence under different numbers of samples found by the two-stage algorithm are listed in Table 1.

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G G

G G

0 −0.1

−0.1

−0.2

−0.2

−0.2 Imaginary

−0.1

−0.3

−0.3 −0.4

−0.4

−0.1

0

0.1

0.2

0.3 Real

0.4

0.5

0.6

0.7

−0.6 −0.1

0

0.1

(a) 3rd order 0.3

0.3 Real

0.4

0.5

0.6

0.7

−0.1

0.1

−0.1 Imaginary

0

−0.1

−0.3

−0.2 −0.3

−0.4

−0.5

−0.5

−0.5

−0.6

−0.6

−0.6

−0.7

−0.7 0.6

0.8

1

0.5

0.6

0.7

noised data G Gn

−0.3

−0.4

0.4 Real

0.4

−0.2

−0.4

0.2

0.3 Real

0.1

−0.1 −0.2

0.2

0.2

0

0

0.1

0.3

noised data G Gn

0.2

0

−0.2

0

(c) 5th order

0.3

Imaginary

Imaginary

0.1

0.2

(b) 4th order

noised data G Gn

0.2

−0.3

−0.5

−0.6

−0.6

n

−0.4

−0.5

−0.5

G G

0

n

n

Imaginary

Imaginary

0

243

−0.7 −0.2

0

(d) 3rd order

0.2

0.4 Real

0.6

0.8

1

−0.2

(e) 4th order

0

0.2

0.4 Real

0.6

0.8

1

(f) 5th order

Fig. 1. Frequency responses of the 3rd, 4th and 5th order approximation without noise and with noise (SNR= 20) by the two-stage algorithm, respectively.

Table 1. Poles found by the two-stage algorithm noise-free noisy

ξ1 1.2310 1.1510

ξ2 18.3310 17.4110

ξ3 1.6810 1.3910

ξ4 33.7110 38.1410

ξ5 0.7210 1.5110

Examples to clarify efficiency of using general rational orthogonal bases and comparison with other models are readily found in the literature, see, for example, Ak¸cay & Ninness (1999a); Ninness et al. (1997). It can also be seen that the maximum selection criterion theoretically guarantees that the approximations are better than no choice to the poles of basis functions. In Fig. 1, the frequency responses of the original system G(s) and the approximations Gn (s) by using the two-stage algorithm are plotted for illustration. The three sub-figures at the top of Fig. 1 depict the frequency responses of the true system (the solid lines) and the approximating systems by the two-stage algorithm with the 3rd, 4th, 5th order (the dash-dot lines) in the case of no noise. The three sub-figures at the bottom of Fig. 1 indicate the frequency responses of the true system (the solid lines) and the 243

ξ6 58.0910 95.2010

ξ7 1.3110 7.8310

ξ8 9.5810 0.0110

ξ9 0.7310 99.9910

ξ10 33.0110 34.1810

approximating systems by the two-stage algorithm with the 3rd, 4th, 5th order (the dash-dot lines) in the noisy case. It can be seen that as the number of basis functions increases, the approximation becomes better.

7. CONCLUSION A new efficient adaptive algorithm has been proposed in this paper for frequency domain identification of continuous-time linear systems. The identification algorithm has been built upon the rational orthogonal system. The novelty is the adaptivity in selecting poles for the basis functions instead of using the poles of the underlying system. The proposed algorithm is only approximately optimal, but it is easily implementable and work efficiently with sufficiently many measurements.

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