FOR traditional discrete-time sampled systems, the operation

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1 29 American Control Conference Hyatt Regency Riverfront St Louis MO USA June FrA132 Least-squares based iterative parameter estimation for two-input multirate sampled-data systems Jing Lu Xinggao Liu Feng Ding Abstract This paper studies identification problems for twoinput multirate systems with colored noises (The method in the paper can be easily extended to multi-input multirate systems) The state-space models are derived for the multirate systems with two different input sampling periods and furthermore the corresponding transfer functions are obtained To solve the difficulty of identification models with unmeasurable noises terms the least-squares based iterative algorithm is presented by replacing the unmeasurable variables with their iterative estimates Finally the simulation results indicate that the proposed algorithm has good performances I INTRODUCTION FOR traditional discrete-time sampled systems the operation frequencies of hold and sampler are equal and synchronous at time Systems existing two or over two operation frequencies are called multirate systems For a twoinput system if the updating periods of the two inputs T 1 and T 2 are not equal then we get a multirate sampled-data system For decades the studies of multirate systems have focused on not only petroleum chemical process control but also achieved a series of research results in theory such as adaptive control [1] [3] optimal control [4] [5] and so on In the multirate identification literature Li et al used multirate input-output data and system states to estimate the parameters of lifted state-space models for multirate systems [6]; Ding and Chen proposed a hierarchical identification method of lifted state-space models for general dual-rate systems [7] [8]; Li et al used the subspace approach to directly identifiy a residual model for fault detection and isolation for systems with non-uniformly sampled multirate data without any knowledge of the system [9]; Ding and Ding applied the extended least squares algorithm to identify the dual-rate systems directly from the available input-output data [1] This paper focuses on parameter identification for twoinput multirate sampled-data systems with colored noises The fundamental idea of the proposed methods is to use the iterative techniques to deal with the identification problem of the multirate systems by adopting the iterative estimation theory: when computing the parameter estimates the unknown variables in the information vector are replaced This work was supported by the National Natural Science Foundation of China J Lu and F Ding are with the Control Science and Engineering Research Center Jiangnan University Wuxi P R China ljjudith@yahoocomcn (J Lu); haibeizhang@163com (CX Zhang); fding@jiangnaneducn (F Ding) XG Liu is with the College of Information Science and Engineering Zhejiang University Hangzhou PR China 3127 lxg@zjueducn with their corresponding estimates at the current iteration and these estimates of the unknown variables are again computed by the preceding parameter estimates Based on this idea we present a least-squares based iterative identification algorithm The main advantage of such iterative algorithms is that they can produce more accurate parameter estimates than the recursive methods [11] see the example later In addition the iterative methods can be extended to other cases such as multi-input multi-output multirate systems This paper is organized as follows Section II derives the state-space models and obtains the corresponding transfer functions Section III presents a least-squares based iterative algorithm for two-input multirate systems and Section IV gives the recursive least-squares identification algorithm compared with least-squares based iterative algorithm Section V provides an illustrative example Finally concluding remarks are given in Section VI II PROBLEM FORMULATION The focus of this paper is a class of multirate systems two-input single-output multirate systems with colored noises as depicted in Figure 1 P c1 and P c2 are two continuous-time processes the inputs u 1 (t) and u 2 (t) to P c1 and P c2 are produced separately by zero-order holds H T1 and H T2 with periods T 1 and T 2 processing two discrete-time signals u 1 (kt 1 ) and u 2 (kt 2 ); the noise output e(t) is produced by the v (t) through the continuous-time process P N ; the output y(t) by the superposition of the unmeasurable outputs y 1 (t) and y 2 (t) which is corrupted by the noise e(t) are sampled by the sampler S T with period T yielding a discretetime signal y(kt) Without loss of generality we assume that T 1 = h T 2 = p 2 h and p 2 are two coprime integers T := p 2 h is f rame period (h is called base period) Since the two input updating periods are not equal to the output sampling period the system in Figure 1 is a multirate system u 1 (kt 1 ) u 2 (kt 2 ) Fig 1 HT1 HT2 u 1 (t) u 2 (t) Pc1 Pc2 v (t) PN y 1 (t) e(t) + + y(t) ST y 2 (t) y(kt) The two-input single-output multirate sampled-data systems Throughout the paper we assume that P ci i = 12 are liner time-invariant continuous-time processes with the following /9/$25 29 AACC 4379

2 state-space representation: P ci : {ẋi (t) = A ci x i (t)+b ci u i (t) i = 12 y i (t) = C i x i (t)+d i u i (t) x i (t) R n i is the state vector of the ith subsystem u i (t) is the control input of the ith subsystem y i (t) is the unmeasurable output of the ith subsystem A ci B ci C i D i are matrices of appropriate sizes For such multirate systems the input-output data available are {u 1 (kt + j 1 T 1 )u 2 (kt + j 2 T 2 )y(kt): j i = 12 1 k = 12 } ( = p 2 /p i ) Since the zero-hold technique is used the input u i (t) keeps invariant over interval [kt +(j 1)T i kt + jt i ) i = 12 j = 12 We discretize P ci with the sampling period T to get [7] x i ((k+ 1)T) = e A (k+1)t cit x i (kt)+ e A ci((k+1)t τ) B ci u i (τ)dτ kt = e AciT x i (kt) kt+ jti + e A ci(kt+ T i τ) B ci u i (τ)dτ kt+( j 1)T i = e A cit x i (kt) + e A Ti ci( j)t i e Acit dt B ci u i (kt +(j 1)T i ) = A i x i (kt)+ B i j u i (kt +(j 1)T i ) y i (kt) = C i x i (kt)+d i u i (kt) A i = e A cit R n i n i i = 12 B 1 j = e A T1 c1(p 2 j)t 1 e Ac1t dtb c1 R n 1 j = 12 p 2 B 2 j = e A T2 c2( j)t 2 e Ac2t dtb c2 R n 2 j = 12 The output equation is y(kt) = y 1 (kt)+y 2 (kt)+e(kt) Let z 1 be a unit backward shift operator ie z is a unit forward shift operator z 1 u i (kt + T i ) = u i (kt + T i T) zx(kt) = x(kt + T) we have y(kt) = C 1 (zi A 1 ) 1 p 2 B 1 j u 1 (kt +(j 1)T 1 ) +C 2 (zi A 2 ) 1 B 2 j u 2 (kt +(j 1)T 2 ) +D 1 u 1 (kt)+d 2 u 2 (kt)+e(kt) p 1 2 =: α 1 (z) β 1 j(z)u 1 (kt +(j 1)T 1 ) + 1 α 2 (z) (1) β 2 j(z)u 2 (kt +(j 1)T 2 )+e(kt) (2) α i (z) = z n i det [zi A i ] =: 1+α i1 z 1 + α i2 z 2 + +α ini z n i i = 12 β i1 (z) = z n i C i adj [zi A i ]B i1 + D i α i (z) =: β i1 ()+β i1 (1)z 1 + β i1 (2)z 2 + +β i1 (n i)z n i β 1 j (z) = z n 1 C 1 adj [zi A 1 ]B 1 j =: β 1 j (1)z 1 + β 1 j (2)z 2 + +β 1 j (n 1)z n 1 j = 23 p 2 β 2 j (z) = z n 2 C 2 adj [zi A 2 ]B 2 j =: β 2 j (1)z 1 + β 2 j (2)z 2 + +β 2 j (n 2)z n 2 j = 23 Similarly we discretize P N with the period T to get e(kt) = η(z) v(kt) (3) = 1+γ 1 z 1 + γ 2 z 2 + +γ nc z n c η(z) = 1+η 1 z 1 + η 2 z 2 + +η ne z n e Thus substituting (3) into (2) gives p 2 y(kt) = 1 α 1 (z) β 1 j (z)u 1(kT +(j 1)T 1 ) + 1 α 2 (z) β 2 j (z)u 2(kT +(j 1)T 2 )+ η(z) v(kt) (4) Multiplying both sides of (4) by α 1 (z)α 2 (z) and let α(z) := α 1 (z)α 2 (z) β 1 j (z) := α 2 (z)β 1 j (z) β 2 j(z) := α 1 (z)β 2 j (z) ζ(z) := α 1 (z)α 2 (z)η(z) then the above equation can be rewritten as α(z)y(kt) = + p 2 β 1 j (z)u 1 (kt +(j 1)T 1 ) β 2 j (z)u 2 (kt +(j 1)T 2 )+ ζ(z) v(kt) (5) α(z) = 1+α 1 (z)z 1 + α 2 z 2 + +α n z n β i1 (z) =: β i1 ()+β i1 (1)z 1 + β i1 (2)z 2 + +β i1 (n)z n i = 12 β 1 j (z) =: β 1 j (1)z 1 + β 1 j (2)z 2 + +β 1 j (n)z n j = 23 p 2 β 2 j (z) =: β 2 j (1)z 1 + β 2 j (2)z 2 + +β 2 j (n)z n j = 23 =: 1+γ 1 z 1 + γ 2 z 2 + +γ nc z n c ζ(z) = 1+ζ 1 z 1 + ζ 2 z 2 + +ζ nd z n d The objective of the paper is to present the least-squares based iterative identification algorithm to identify the parameters α j β i j (r) γ j and ζ j 438

3 III THE LEAST-SQUARES BASED ITERATIVE ALGORITHMS Define the information vector ψ(kt) and the parameter vector ϑ as ϕs (kt) ψ(kt) := R n n ϕ n (kt) := m+n c + n d m = ( + p 2 + 1)n+2 ϕ s (kt) := [ y(kt T) y(kt 2T) y(kt nt)φ T 1 (kt)φt 2 (kt)]t R m φ i (kt) := [u i (kt)u i (kt T)u i (kt 2T) u i (kt nt)u i (kt T + T i ) u i (kt 2T + T i ) u i (kt nt + T i ) u i (kt T +( 1)T i ) u i (kt 2T +( 1)T i ) u i (kt nt +( 1)T i )] T R n+1 ϕ n (kt) := [ e(kt T) e(kt 2T) e(kt n c T) v(kt T)v(kT 2T) v(kt n d T)] T R n c+n d θ ϑ := s R n θ n θ s := [α 1 α 2 α n β 11 ()β 11 (1)β 11 (2) β 11 (n)β 12 (1)β 12 (2) β 12 (n) β 1p2 (1)β 1p2 (2) β 1p2 (n)β 21 ()β 21 (1) β 21 (2) β 21 (n)β 22 (1)β 22 (2) β 22 (n) β 2p1 (1)β 2p1 (2) β 2p1 (n)] T R m θ n := [γ 1 γ 2 γ nc ζ 1 ζ 2 ζ nd ] T R n c+n d By the above definition we have Then (5) can be written as e(kt) = ϕ T n (kt)θ n + v(kt) (6) y(kt) = ϕ T s (kt)θ s + e(kt) (7) y(kt) = ψ T (kt)ϑ + v(kt) (8) The information vector ψ(kt) in (8) contains unmeasurable variables e(kt jt) and v(kt jt) To solve the difficulty these unmeasurable variables are replaced with their iterative estimates by the iterative method For convenience let t = kt ψ(t) := ψ(kt) Thus (8) can be written as y(t) = ψ T (t)ϑ + v(t) (9) Consider the newest q and define the stacked output vector Y(t) stacked information vector Ψ(t) and white noise vector V(t) as y(t) y(t T) Y(t) := Rq (1) y(t qt + T) ψ T (t) ψ T (t T) Ψ(t) := Rq n (11) ψ T (t qt + T) v(t) v(t T) V(t) := Rq (12) v(t qt + T) If we take q = L t = L (L is the data length) then Y(t) and Ψ(t) contain all the measured data From (9)-(12) we have Y(t) = Ψ(t)ϑ +V(t) (13) Notice that V(t) is a white noise vector with zero mean and define a quadratic criterion function [12] J(ϑ) = [Y(t) Ψ(t)ϑ] T [Y(t) Ψ(t)ϑ] (14) Provided that ψ(t) is persistently exciting Ψ T (t)ψ(t) is an invertible matrix Minimizing J(ϑ) in (14) gives the leastsquares estimate of ϑ: ˆϑ(t) = [Ψ T (t)ψ(t)] 1 Ψ T (t)y(t) (15) Since ψ(t jt) in Ψ(t) contains unmeasurable variables e(t jt) and v(t jt) (see the definitions of ϕ n (t) and Φ(t)) so the estimate of ϑ is impossible to compute by (15) Here a new iterative identification algorithm based on the hierarchical identification principle is developed to solve the difficulty The details are as follows [ Let ] l = 123 be ˆθ an iteration variable and ˆϑ l (t) := sl (t) be the estimate ˆθ nl (t) θ of ϑ := s R n the unknown variables e(t jt) and θ n v(t jt) in the information vector are replaced with their estimates ê l (t jt) and ˆv l (t jt) then the estimate of ϕ n (t) is ˆϕ nl (t) := [ ê l (kt T) ê l (kt 2T) ê l (kt n c T) ˆv l (tt T) ˆv l (t 2T) ˆv l (t n d T)] T R n c+n d (16) Replacing ϕ(t) in ψ(t) with ˆϕ nl (t) then we have the estimate of ψ(t) as ϕs (t) ˆψ l (t) = R ˆϕ nl (t) n (17) From (7) we have e(t jt) = y(t jt) ϕ T s (t)θ s 4381

4 If θ s in the above equation is replaced with ˆθ sl 1 (t) then the estimate of e(t jt) can be computed by ê l (t jt) = y(t jt) ϕ T s (t jt) ˆθ sl 1 (t) (18) From (9) we have v(t jt) = y(t jt) ψ T (t jt)ϑ Replacing ψ(t) and ϑ in the above equation with their estimates ˆψ l 1 (t) and ˆϑ l 1 respectively we can compute the estimate ˆv l (t jt) by Define ˆv l (t jt) = y(t jt) ˆψ T l 1 (t jt) ˆϑ l 1 (t) (19) ˆΨ l (t) = [ ˆψ l (t) ˆψ l (t T) ˆψ l (t qt + T)] T R q n (2) The least-squares estimate of the parameter vector ϑ is obtained by replacing Ψ(t) with Ψ l (t) in (15) ˆϑ l (t) = [ ˆΨ T l(t) ˆΨ l (t)] 1 ˆΨ T l(t)y(t) l = 123 (21) Equations (16)-(21) form the least-squares based iterative (LSI) algorithm for estimating ϑ: ˆϑ l (t) = [ ˆΨ T l(t) ˆΨ l (t)] 1 ˆΨ T l(t)y(t) (22) Y(t) = [y(t)y(t T) y(t qt + T)] T (23) ˆΨ l (t) = ˆψ l (t) ˆψ l (t T) ˆψ l (t qt + T)] T (24) ϕs (t) ˆψ l (t) = (25) ˆϕ nl (t) ϕ s (t) = [ y(t T) y(t 2T) y(t nt) φ T 1 (t)φt 2 (t)]t (26) φ i (kt) = [u i (kt)u i (kt T)u i (kt 2T) u i (kt nt)u i (kt T + T i ) u i (kt 2T + T i ) u i (kt nt + T i ) u i (kt T +( 1)T i ) u i (kt 2T +( 1)T i ) u i (kt nt +( 1)T i )] T (27) ϕ nl (t) = [ ê l (t T) ê l (t 2T) ê l (t n c T) ˆv l (t T) ˆv l (t 2T) ˆv l (t n d T)] T (28) ê l (t jt) = y(t jt) ϕ T s (t jt) ˆθ sl 1 (t) j = 12 n c (29) ˆv l (t jt) = y(t jt) ˆψ T l 1 (t jt) ˆϑ l 1 (t) j = 12 n d (3) The LSI algorithm adopts the idea of updating the estimate ϑ using a fixed data batch with the data length L at each iteration and thus has higher parameter estimation accuracy than the recursive least-squares (RLS) algorithm To initialize the LSI algorithm we take ˆϑ (t) = 1 n with 1 n being an n-dimensional column vector whose elements are all 1 ê (t j) and ˆv (t j) are random numbers To summarize we list the steps involved in the LSI algorithm to compute ˆϑ l (t) as l increases: 1) Choose the data window length q n collect the input and output data {u 1 (kt + j 1 T 1 )u 2 (kt + j 2 T 2 )y(t): j i = 12 1 k = 12 q 1} and given the estimation accuracy ε and let t = kt = qt and ˆϑ (t) = 1 n 2) Collect the input and output data to form output vector Y(t) by (23) form φ 1 (t) and φ 2 (t) by (27) and ϕ s (t) by (26) 3) To initialize let l = 1 ê (t jt) and ˆv (t jt) equal random numbers form ˆϕ n (t) by (28) and ˆψ (t) by (25) 4) Compute ê l (t) and ˆv l (t) by (29) and (3) 5) Form ˆψ nl (t) by (28) ˆψ l (t) by (25) and ˆΨ l (t) by (24) 6) Update the estimate ˆϑ l (t) by (22) 7) Compare ˆϑ l (t) with ˆϑ l 1 (t): if ˆϑ l (t) ˆϑ l 1 (t) < ε then terminate the procedure and obtain the iterative times l = l and estimate ˆϑ l (t) = ˆϑ l (kt) and increment k by 1 ie t := (k + 1)T set ˆϑ (t) = ˆϑ l (kt) and go to step 2; otherwise increment l by 1 and go to step 4 IV THE RECURSIVE LEAST SQUARES ALGORITHMS For comparison we give the recursive least squares algorithm to identify parameter vector ϑ to identify (5) ˆϑ(t) = ˆϑ(t T)+L(t)[y(t) ˆψ T (t) ˆϑ(t T)] P(t T) ˆψ(t) L(t) = 1+ ˆψ T (t)p(t T) ˆψ(t) P(t) = [I L(t) ˆψ T (t)]p(t T) ˆψ(t) = [ϕ T s (t) ê(t T) ê(t 2T) ê(t n ct) ˆv(tT T) ˆv(t 2T) ˆv(t n d T)] T ê(t) = y(t) ϕ T s (t) ˆθ s (t) ˆv(t) = y(t) ˆψ T (t) ˆϑ(t) To initialize the above algorithm we take P() = p I with p normally a large positive number eg p = 1 6 and ˆϑ() some small real vector eg ˆϑ() = 1 n /p with 1 n being an n-dimensional column vector whose elements are all 1 V EXAMPLE Example For the system depicted in Figure 1 take the process model to be P c1 (s) = 1 5s+1 P 1 c2(s) = 6s+1 and h = 1s = 2 p 2 = 3 hence T 1 = 2s T 2 = 3s and T = 6s the corresponding discrete-time state space models are x 1 (kt + T) = 88692x 1 (kt) +[ ] u 1(kT) u 1 (kt + 2) u 1 (kt + 4) y 1 (kt) = 2x 1 (kt) and x 2 (kt + T) = 9484x 2 (kt) u2 (kt) +[ ] u 2 (kt + 3) y 2 (kt) = 16667x 2 (kt) 4382

5 then the input and output representation is ( z z 2 )y(kt) = (36196z z 2 )u 1 (kt) +(37673z z 2 )u 1 (kt + 2) +(39211z z 2 )u 1 (kt + 4) +(46392z z 2 )u 2 (kt) +(48771z z 2 )u 2 (kt + 3) Introducing a noise model e(kt) = (1 7z 1 )v(kt) in the above equation we have ( z z 2 )y(kt) = (36196z z 2 )u 1 (kt) +(37673z z 2 )u 1 (kt + 2) +(39211z z 2 )u 1 (kt + 4) +(46392z z 2 )u 2 (kt) +(48771z z 2 )u 2 (kt + 3) +(1 7z 1 )v(kt) Here {u i (kt + j i T i )} is taken as a persistent excitation signal sequence with zero mean and unit variance and {v(kt)} as a white noise sequence with zero mean and variance σ 2 = 8 2 and σ 2 = 2 2 Applying the LSI algorithm and RLS algorithm to estimate the parameters of the above transfer function model the parameter estimation errors δ := ˆϑ l (kt) ϑ / ϑ (the LSI algorithm with l = 6 iterations) or δ := ˆϑ(kT) ϑ / ϑ (the RLS algorithm) versus t = kt are shown in Figures 2-3 From Figures 2-3 we can arrive at the following conclusions: For different noise variances the parameter estimation errors δ of both the LSI algorithm and RLS algorithm are becoming gradually smaller as t = kt increases; the parameter estimation errors of the LSI algorithm are smaller than those of the RLS algorithm for the given noise variances In other words under the same data length the parameter estimates given by the LSI algorithm have higher accuracy than those by RLS algorithm VI CONCLUSIONS This paper presents a least-squares based iterative algorithm for two-input multirate systems with colored noises The method can be extended to other cases for example general multi-input multi-output multirate systems with each of the input and output channels having different sampling periods and with colored noises REFERENCES [1] C Zhang RH Middleton and RJ Evans An algorithm for multirate sampling adaptive control IEEE Transactions on Automatic Control vol 34 no 7 pp [2] P Albertos J Salt and J Tormero Dual-rate adaptive control Automatica vol 32 no 7 pp δ LSI RLS t Fig 2 The parameter estimation errors δ versus t (σ 2 = 8 2 ) δ RLS 1 LSI t Fig 3 The parameter estimation errors δ versus t (σ 2 = 2 2 ) [3] F Ding and T Chen Least squares based self-tuning control of dualrate systems International Journal of Adaptive Control and Signal Processing vol 18 no 8 pp [4] HM Al-Rahmani and GF Franklin A new optimal multirate control of linear periodic and time-invariant systems IEEE Transactions on Automatic Control vol 35 no 4 pp [5] HM Al-Rahmani and GF Franklin Multirate control: A new approach Automatica vol 28 no 1 pp [6] D Li SL Shah and T Chen Identification of fast-rate models from multirate data International Journal of Control vol 74 no 6 pp [7] F Ding and T Chen State-space modeling and identification of general dual-rate stochastic systems Acta Automatica Sinica vol 3 no 5 pp [8] F Ding and T Chen Hierachical idetification of lifted state-space models for general dual-rate systems IEEE Transactions on Circuits and Systems I: Regular Papers vol 52 no 6 pp [9] W Li Z Han and S L ShahSubspace identification for FDI in systems with non-uniformly sampled multirate data Automatica vol 42 no 4 pp [1] J Ding and F Ding The residual based extended least squares identification method for dual-rate systems Computers and Mathematics with Applications vol 56 no 6 pp [11] F Ding and T Chen Bias compensation based recursive least squares identification algorithm for MISO systems IEEE Transactions on Circuits and Systems II: Express Briefs vol 53 no 5 pp [12] L Ljung System Identification: Theory for the User 2nd ed Englewood Cliffs New Jersey: Prentice-hall