Research Article The Hierarchical Iterative Identification Algorithm for Multi-Input-Output-Error Systems with Autoregressive Noise

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1 Hindawi Complexity Volume 2017 Article ID pages Research Article The Hierarchical Iterative Identification Algorithm for Multi-Input-Output-Error Systems with Autoregressive Noise Jiling Ding Department of Mathematics Jining University Qufu China Correspondence should be addressed to Jiling Ding; Received 15 July 2017; Accepted 18 September 2017; Published 29 October 2017 Academic Editor: Guang Li Copyright 2017 Jiling Ding. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. This paper considers the identification problem of multi-input-output-error autoregressive systems. A hierarchical gradient based iterative (H-GI) algorithm and a hierarchical least squares based iterative (H-LSI) algorithm are presented by using the hierarchical identification principle. A gradient based iterative (GI) algorithm and a least squares based iterative (LSI) algorithm are presented for comparison. The simulation results indicate that the H-LSI algorithm can obtain more accurate parameter estimates than the LSI algorithm and the H-GI algorithm converges faster than the GI algorithm. 1. Introduction System identification studies mathematical models of dynamic systems by fitting experimental data to a suitable model structure 1 2. Many practical systems have multiple inputs and multiple outputs such as chemical processes 3 4 automation devices 5 7 and network communication engineering For decades much research has been performed on the multivariable systems and some typical approaches for the parameter estimation of the multivariable systems have been reported 13 such as the canonical approach 14 the iterative methods and the least squares methods 17. Recently Panda and Vijayaraghavan adopted the sequential relay feedback test to estimate the parameter of the linear multivariable systems 18. Jafari et al. presented an iterative least squares algorithm to identify the multivariable nonlinear systems with colored noises 19. The multivariable systems contain both parameter vectors and parameter matrices and the systems inputs and system outputs are relevant and coupled For the sake of reducing the computational complexity the hierarchical identification principle is utilized to transform a complex system into several subsystems and then to estimate the parameter vector of each subsystem respectively. In this literature Schranz et al. proposed a feasible hierarchical identification process for identifying the viscoelastic model of respiratory mechanics 25. Xu et al. developed the parameter estimation for dynamical response signals The iterative methods have been widely applied in identifying the parameters of linear or nonlinear systems Many iterative algorithms for system identification are based on the gradient method 31and the least squares method The basic idea of iterative methods is to update the parameter estimates using batch data. This paper focuses on the parameter estimation for output-error autoregressive (OEAR) systems using the hierarchical identification principle and the iterative identification principle and presents a hierarchical gradient based iterative (H-GI) algorithm and a hierarchical least squares based iterative (H-LSI) algorithm. The key is to decompose a multi-input OEAR system into two subsystems and then to identify each subsystem. The work in discussed the single-input single-output systems but many practical systems have multiple inputs and multiple outputs with the development of industrial technology. Compared with the work in this paper discusses the parameter estimation for multi-input OEAR systems and the presented H- LSI algorithm can achieve higher estimation accuracy than the LSI algorithm and the H-GI algorithm also can achieve higher estimation accuracy than the GI algorithm.

2 2 Complexity The rest of this paper is organized as follows. Section 2 gives some definitions and the identification model of multiinput OEAR systems. Section 3 presents a gradient based iterative algorithm and a least squares based iterative algorithm for multi-input OEAR systems. Section 4 derives a hierarchical gradient based iterative algorithm. Section 5 derives a hierarchical least squares based iterative algorithm. Section 6 provides two illustrative examples to demonstrate the effectiveness of the proposed algorithms. Finally concluding remarks are given in Section The Problem Formulation Let us define some notation. Symbols: meaning 1 n :anndimensional column vector whose entries are all 1 p 0 : a large positive constant for example p 0 =10 6 X T : the transpose of the vector or matrix X X 2 : X 2 = trxx T X fl A: A defined as X λ max X: the maximum eigenvalue of the symmetric real matrix X. Consider the following multi-input-output-error type models: r B j (z) y () = A j (z) u j () +w() (1) j=1 where y() R is the system output u j () R j = 12...rarethesysteminputsandw() R is the colored noise with zero mean. A j (z) and B j (z) are polynomials in the unit backward shift operator z and A j (z) fl 1+a j1 z +a j2 z 2 + +a jnj z n j B j (z) fl b j1 z +b j2 z 2 + +b jnj z n j b ji R. a ji R Assume that the orders n j are known y() = 0 u j () = 0 and w() = 0 as 0.Thecolorednoisew() can be fitted by a moving average process (2) w () =D(z) V () (3) where V() is the white noise with zero mean and C(z) D(z) arepolynomialsintheunitbackwardshiftoperatorz : C (z) fl 1+c 1 z +c 2 z 2 + +c nc z n c c j R D (z) fl 1+d 1 z +d 2 z 2 + +d nd z n d d j R. This paper considers the colored noise to be an autoregressive process so the models in (1) can be taken as the multi-input OEAR systems. Define the intermediate variables: (6) x j () fl B j (z) A j (z) u j () j=12...r. (7) From (4) and (7) we have w () = 1 C(z) w () + V () n c = i=1 c i w ( i) + V () x j () =1 A j (z)x j () +B j (z) u j () = n j i=1 a ji x j ( i) +b ji u j ( i). The output y() in (1) can be written as r n j y () = a ji x j ( i) +b ji u j ( i) j=1i=1 n c i=1 c i w ( i) + V (). Define the parameter vectors as θ fl θ T c T T R n n fl n 0 +n c n 0 fl r j=1 2n j (8) (9) or an autoregressive process w () = 1 V () (4) C (z) or an autoregressive moving average process w () = D (z) V () (5) C (z) θ fl θ T 1 θt 2...θT r T R n 0 θ j fl a j1 a j2...a jnj b j1 b j2...b jnj T R 2n j j=12...r c fl c 1 c 2...c nc T R n c (10)

3 Complexity 3 and the information vectors as φ () fl φ T () ψ T () T R n φ () fl φ T 1 () φt 2 ()...φt r ()T R n 0 φ j () fl x j () x j ( 2)... x j ( n j )u j ()u j ( 2)...u j ( n j ) T R 2n j ψ () fl w () w( 2)... w( n c ) T R n c. (11) (GI) algorithm for estimating the parameter vector θ of the multi-input OEAR systems: θ k () = θ k () +μ k θ k () i= p+1 k=12... φ k () = φ T k () ψt k ()T φ k (i) y (i) φ T k (i) φ k () = φ T 1k () φ T 2k ()... φ T rk ()T φ jk () = x jk ()... x jk ( n j ) According to the above definitions (8) and (9) can be written as w () = ψ T () c + V () (12) x j () = φ T j () θ j (13) y () = r j=1 φ T j () θ j + ψ T () c + V () = φ T () θ + ψ T () c + V () = φ T () θ + V (). (14) Equation (14) is the identification model of the multi-input OEAR system. 3. The Gradient Based and Least Squares Based Iterative Algorithm Consider the data {u 1 (i) u 2 (i)...u r (i) y(i)} from i= p+ 1 to i=and define quadratic criterion function as J (θ) fl y (i) φ T (i) θ 2. (15) i= p+1 u j ()...u j ( n j ) T ψ k () = w k () w k ( 2)... w k ( n c ) T x jk ( i) = φ T jk ( i) θ jk () w k ( i) =y( i) φ T k ( i) θ k () μ k 2λ max i= p+1 θ k () = θ T k () ĉt k ()T φ k (i) φ T k (i) θ k () = θ T 1k () θ T 2k ()... θ T rk ()T θ jk () = a j1k () a j2k ()... a jnj k () b j1k () bj2k ()... b jnj k () T ĉ k () = c 1k () c 2k ()... c nc k () T. (17) Let k = be an iteration variable and θ k () be the estimate of θ at iteration k. Minimizing J(θ) by using the negative gradient search we can obtain θ k () = θ k () μ k () grad J ( θ 2 k ()) = θ k () +μ k () i= p+1 φ (i) y (i) φ T (i) θ k () (16) where μ k () > 0 is an iterative step-size. Because the information vector φ() contains the unknown variables x j ( i) and w( i) weusetheestimates x jk ( i) and w k ( i) at iteration kto replace the unknown variables x j ( i) and w( i);wecanobtainthegradientbasediterative The convergence rate of the GI algorithm is slow. To improve the convergence speed we derive a least squares based iterative (LSI) identification algorithm. Minimizing J(θ) and letting the derivative of J(θ) with respect to θ be zero give the LSI identification algorithm for the multi-input OEAR systems: θ k () = i= p+1 φ k (i) φ T k (i) φ k () = φ T k () ψt k ()T i= p+1 φ k () = φ T 1k () φ T 2k ()... φ T rk ()T φ k (i) y (i) k=123...

4 4 Complexity φ jk () = x jk ()... x jk ( n j ) u j ()...u j ( n j ) T ψ k () = w k () w k ( 2)... w k ( n c ) T θ k () ĉ k () = θ k () +μ 1k () i= p+1 φ (i) y 1 (i) φ T (i) θ k () (24) x jk ( i) = φ T jk ( i) θ jk () w k ( i) =y( i) φ T k ( i) θ k () θ k () = θ T k () ĉt k ()T θ k () = θ T 1k () θ T 2k ()... θ T rk ()T θ jk () = a j1k () a j2k ()... a jnj k () b j1k () = ĉ k () +μ 2k () i= p+1 ψ (i) y 2 (i) ψ T (i) ĉ k (). (25) Here μ 1k () and μ 2k () are the iterative step-sizes or convergence factors. Substituting (19) into (24) and (20) into (25) we can obtain θ k () = θ k () +μ 1k () bj2k ()... b jnj k () T i= p+1 φ (i) y (i) ψ T (i) c φ T (i) θ k () (26) ĉ k () = c 1k () c 2k ()... c nc k () T. 4. The Hierarchical Gradient Based Iterative Algorithm Define intermediate variables: (18) y 1 () fl y () ψ T () c (19) y 2 () fl y () φ T () θ. (20) Using the hierarchical identification principle the multiinput OEAR system in (14) can be decomposed into two fictitious subsystems: y 1 () = φ T () θ + V () (21) ĉ k () = ĉ k () +μ 2k () i= p+1 ψ (i) y (i) φ T (i) θ ψ T (i) ĉ k (). (27) The parameter estimates θ k () and ĉ k () cannot be computed by (26) and (27) because the information vectors φ() and ψ() contain unknown variables x j ( i) and w( i)andthe parameter vectors θ and c in (26) and (27) are unknown. We solve this problem by replacing the unknown variables x j ( i) and w( i) with their corresponding estimates x jk ( i) and w k ( i) at iteration kand define the estimates φ k () and ψ k () at iteration k as φ k () fl φ T 1k () φ T 2k ()... φ T rk ()T R n 0 y 2 () = ψ T () c + V (). (22) Next we identify the parameters θ and c of each subsystem in (21) and (22) respectively. Define quadratic criterion functions as φ jk () fl x jk ()... x jk ( n j )u j ()...u j ( n j ) T R 2n j (28) J 1 (θ c) fl J 2 (θ c) fl y 1 (i) φ T (i) θ 2 i= p+1 y 2 (i) ψ T (i) c 2. i= p+1 (23) ψ k () fl w k () w k ( 2)... w k ( n c ) T R n c. From (12) and (14) we have Let θ k () and ĉ k () be the estimates of θ and c at iteration k. Using the negative gradient search and minimizing J 1 (θ c) and J 2 (θ c)wecanobtain x j ( i) = φ T j ( i) θ j w ( i) =y( i) φ T ( i) θ. (29)

5 Complexity 5 Substituting φ j ( i) and θ j with their estimates φ jk ( i) and θ jk () wecangettheestimates x jk ( i) and w k ( i) at iteration k: φ jk () = x jk ()... x jk ( n j )u j ( )...u j ( n j ) T (36) x jk ( i) = φ T jk ( i) θ jk () w k ( i) =y( i) φ T k ( i) θ k (). (30) ψ k () = w k () w k ( 2)... (37) w k ( n c ) T x jk ( i) = φ T jk ( i) θ jk () (38) Replacing φ() ψ() in(26)and(27)withtheirestimates φ k () and ψ k ()replacingc in (26) with its estimate ĉ k () and replacing θ in (27) with its estimate θ k ()wehave θ k () = θ k () +μ 1k () i= p+1 φ k (i) y(i) φ T k (i) θ k () ψ T k (i) ĉ k () ĉ k () = ĉ k () +μ 2k () i= p+1 ψ k (i) (31) w k ( i) =y( i) φ T k ( i) θ k () (39) μ k () max 2λ max i= p+1 2λ max i= p+1 ψ k (i) ψ T k (i) φ k (i) φ T k (i) (40) θ k () = θ T 1k () θ T 2k ()... θ T rk ()T (41) θ jk () = a j1k () a j2k ()... a jnj k () b j1k () y(i) φ T k (i) θ k () ψ T k () ĉ k (). bj2k ()... b jnj k () T (42) In order to guarantee the convergence of θ k () and ĉ k () a conservative choice is μ 1k () =μ 2k () = μ k () max 2λ max i= p+1 2λ max i= p+1 ψ k (i) ψ T k (i). φ k (i) φ T k (i) (32) At last we can summarize the hierarchical gradient based iterative parameter estimation (H-GI) algorithm for estimating θ and c of the multi-input OEAR systems: ĉ k () = c 1k () c 2k ()... c nc k () T. (43) The steps of computing the parameter estimates θ k () and ĉ k () for the multi-input OEAR systems are as follows. (1) Set the data length p let = pandcollectthe input-output data {u 1 (i) u 2 (i)...u r (i) y(i) : i = 01...p}. (2) Collect the input-output data {u 1 () u 2 ()...u r ()} and y(). (3) To initialize let k=1 θ 0 () = 1 n0 /p 0 ĉ 0 () = 1 nc / p 0 x j0 ( i) = 1/p 0 w 0 ( i) = 1/p 0 for i=12...maxn j n c. (4) Form φ jk () and ψ k () by (36) and (37) and form φ k () by (35). (5) Choose a μ k () satisfying (40) and update the estimate θ k () using (33) and ĉ k () using (34). θ k () = θ k () + μ k () i= p+1 θ k () ψ T k (i) ĉ k () ĉ k () = ĉ k () + μ k () i= p+1 θ k () ψ T k (i) ĉ k () φ k (i) y (i) φ T k (i) ψ k (i) y (i) φ T k (i) (33) (34) (6) Read θ k () and θ jk () using (41) and (42) and compute x jk ( i) using (38) and w k ( i) using (39). (7) Give a small positive ε > 0.If θ k () θ k () + ĉ k () ĉ k () > εincreasek by 1 and go to Step (4); otherwise obtain the parameters θ k () and ĉ k () and increase s by 1 and go to Step (2). 5. The Hierarchical Least Squares Based Iterative Algorithm φ k () = φ T 1k () φ T 2k ()... φ T rk ()T (35) The H-GI algorithm can produce higher parameter estimation accuracy compared with the GI algorithm but it

6 6 Complexity converges slowly. In order to solve this short board we derive a hierarchical least squares based iterative algorithm for the multi-input OEAR systems. Minimizing J 1 (θ c) and letting the partial derivative of J 1 (θ c) with respect to θ be zero and minimizing J 2 (θ c) and letting the partial derivative of J 2 (θ c) with respect to c be zero respectively we can obtain the least squares estimate θ(): θ () = ĉ () = i= p+1 i= p+1 φ (i) φ T (i) ψ (i) ψ T (i) i= p+1 i= p+1 Inserting (19) into (44) and (20) into (45) gives θ () = i= p+1 ĉ () = i= p+1 φ (i) φ T (i) φ (i) y 1 (i) (44) φ (i) y (i) ψ T (i) c i= p+1 i= p+1 ψ (i) ψ T (i) ψ (i) y (i) φ T (i) θ. ψ (i) y 2 (i). (45) (46) (47) The above estimates θ() and ĉ() are impossible to compute since the right-hand side of (46) contains the unknown parameter vector c and the unknown information vectors φ(i) and ψ(i) and the right-hand side of (47) also contains the unknown parameter vector θ and the unknown information vectors φ(i) and ψ(i).we solve this difficulty by replacing φ(i) ψ(i) with their estimates φ k (i) ψ k (i) and replacing c in (46) and θ in (47) with their estimates ĉ k () and θ k ().Thenwe can summarize the hierarchical least squares based iterative (LSI) algorithm of estimating the parameter vectors θ and c as follows: θ k () = i= p+1 ĉ k () = i= p+1 i= p+1 φ k (i) φ T k (i) φ k (i) y (i) ψ T k (i) ĉ k () i= p+1 ψ k (i) ψ T k (i) ψ k (i) y (i) φ T k (i) θ k () (48) (49) φ k () = φ T 1k () φ T 2k ()... φ T rk ()T (50) φ jk () = x jk ()... x jk ( n j )u j ()...u j ( n j ) T ψ k () = w k () w k ( 2)... w k ( n c ) T (51) (52) x jk ( i) = φ T jk ( i) θ jk () (53) w k ( i) =y( i) φ T k ( i) θ k () (54) θ k () = θ T 1k () θ T 2k ()... θ T rk ()T. (55) The procedure for computing the parameter estimation θ k () and ĉ k () is as follows. (1) Give the data length p let = p collect the input-output data {u 1 (i) u 2 (i)...u r (i) y(i) : i = 1...p} and give a small positive ε>0. (2) Collect the input-output data {u 1 () u 2 ()...u r ()} and y(). (3) To initialize let k=1 x j0 ( i) = 1/p 0 w 0 ( i) = 1/p 0 for i=12...maxn j n c. (4) Form φ jk () ψ k ()and φ k () by (51) (52) and (50) respectively. (5) Update the estimates θ k () and ĉ k () by (48) and (49) and read θ k () by (55). (6) Compute x jk () by (53) and w k () by (54). (7) If θ k () θ k () + ĉ k () ĉ k () > ε increase k by 1 and go to Step (4); otherwise obtain the parameters θ k () and ĉ k () and increase s by 1 and go to Step (2). 6. Example Example 1. Consider the following two-input OEAR system: y () = B 1 (z) A 1 (z) u 1 () + B 2 (z) A 2 (z) u 2 () + 1 C (z) V () A 1 (z) =1+a 11 z +a 12 z 2 = z z 2 B 1 (z) =b 11 z +b 12 z 2 =0.78z 0.40z 2

7 Complexity 7 Table 1: The LSI parameter estimates and errors for Example 1. k a 11 a 12 b 11 b 12 a 21 a 22 b 21 b 22 c 1 δ(%) True values Table 2: The H-LSI parameter estimates and errors for Example 1. k a 11 a 12 b 11 b 12 a 21 a 22 b 21 b 22 c 1 δ(%) True values A 2 (z) =1+a 21 z +a 22 z 2 = z z 2 B 2 (z) =b 21 z +b 22 z 2 = 0.56z z 2 C (z) =1+c 1 z = z θ =a 11 a 12 b 11 b 12 a 21 a 22 b 21 b 22 c 1 T = T. (56) H-LSI k Figure 1: The GI estimation errors δ versus k with different σ 2. LSI The inputs {u 1 () u 2 ()} are taken as two persistent excitation signal sequences with zero mean and unit variance and {V()} as a white noise sequence with zero mean and variance σ 2 = Take the data length L = 2000 applying the LSI algorithm and the H-LSI algorithm to estimate the parameters of this example system. The parameter estimates and their errors of the LSI algorithm are shown in Table 1 the parameter estimates and their errors of H-LSI algorithm are shown in Table 2 and the parameter estimation errors of the LSI and H-LSI algorithms versus k are shown in Figure 1. From the simulation results in Tables 1 and 2 and Figure 1 we can draw the following conclusions. (i) The estimation errors given by the LSI algorithm andh-lsialgorithmbecomesmallerandsmalleras iteration variable k increases. (ii) Under the same noise variance the estimation errors given by the H-LSI algorithm are lower than that given by the LSI algorithm.

8 8 Complexity Table3:TheH-LSIparameterestimatesanderrorsforExample2. k a 11 a 12 b 11 b 12 a 21 a 22 b 21 b 22 c 1 δ(%) True values Table 4: The H-GI parameter estimates and errors for Example 2. k a 11 a 12 b 11 b 12 a 21 a 22 b 21 b 22 c 1 δ(%) True values Table 5: The H-GI parameter estimates and errors for Example 2. k a 11 a 12 b 11 b 12 a 21 a 22 b 21 b 22 c 1 δ(%) True values (iii) The estimation accuracy of the H-LSI algorithm is close to their true values; this indicates that the proposed algorithm can effectively identify the multiinput OEAR systems. Example 2. Consider the following another two-input OEAR system: y () = B 1 (z) A 1 (z) u 1 () + B 2 (z) A 2 (z) u 2 () + 1 C (z) V () A 1 (z) =1+a 11 z +a 12 z 2 = z 0.20z 2 B 1 (z) =b 11 z +b 12 z 2 = 0.55z 0.80z 2 A 2 (z) =1+a 21 z +a 22 z 2 = z 0.10z 2 B 2 (z) =b 21 z +b 22 z 2 =0.40z 0.90z 2 C (z) =1+c 1 z = z θ =a 11 a 12 b 11 b 12 a 21 a 22 b 21 b 22 c 1 T = T. (57) The simulation conditions are the same as that of Example 1 and the noise variance σ 2 = Take the data length L = Applying the GI algorithm and the H-GI algorithm to estimate the parameters of this example system the simulation results are shown in Tables 3 5 and Figure 2. From the simulation results in Tables 3 5 and Figure 2 we can draw the following conclusions.

9 Complexity GI 0.1 H-GI Figure 2: The GI and H-GI estimation errors δ versus k. k (i) Under the same noise variance and data length the H-GI algorithm has less estimation errors than the GI algorithm. This shows that the H-GI estimation algorithm can obtain more accurate estimates than the GI algorithm. (ii) As the iterationvariable k increases the H-GI parameter estimates are very close to their true values. (iii) The proposed H-GI algorithm requires more iterations than the H-LSI algorithm to achieve almost same estimation accuracy. 7. Conclusions Combining the iterative technique and the hierarchical identification principle a H-GI algorithm and a H-LSI algorithm are derived for identifying the multi-input OEAR systems. Compared with the GI algorithm the H-GI algorithm can generate more accurate parameter estimates. Compared with the H-GI algorithm the H-LSI algorithm has faster convergence speed. The proposed methods can be extended to discuss the parameter estimation of the multi-input-output systems with colored noise and time-delay systems such as network and signal processing Conflicts of Interest The author declares that there are no conflicts of interest regarding the publication of this paper. Acknowledgments The author is grateful to her supervisor Professor Feng Ding at the Jiangnan University for his helpful suggestions and the main idea of this work comes from him and his book Multi- Innovation Identification Theory and Methods Beijing: Science Press This work was supported by the Natural Science Foundation of Shandong Province (China ZR2016FL08) and the Science Foundation of Jining University (China 2016QNKJ01). References 1 L. Xu A proportional differential control method for a time-delay system using the Taylor expansion approximation Applied Mathematics and Computation vol.236pp L. Xu L. Chen and W. Xiong Parameter estimation and controller design for dynamic systems from the step responses based on the Newton iteration Nonlinear Dynamicsvol.79no. 3 pp K. Rajarathinam J. B. Gomm D.-L. Yu and A. S. Abdelhadi PID controller tuning for a multivariable glass furnace process by genetic algorithm International Automation and Computingvol.13no.1pp D. Zumoffen M. Gonzalo and M. Basualdo Improvements on multivariable control strategies tested on the Petlyuk distillation column Chemical Engineering Science vol. 93 pp J. Na M. N. Mahyuddin G. Herrmann X. Ren and P. Barber Robust adaptive finite-time parameter estimation and control for robotic systems International Robust and Nonlinear Controlvol.25no.16pp H. Li Y. Shi and W. Yan On Neighbor Information Utilization in Distributed Receding Horizon Control for Consensus- Seeking IEEE Transactions on Cybernetics vol. 46 no. 9 pp L.FengM.WuQ.Lietal. ArrayFactorFormingforImage Reconstruction of One-Dimensional Nonuniform Aperture Synthesis Radiometers IEEE Geoscience and Remote Sensing Lettersvol.13no.2pp G. Gu S. Wan and L. Qiu Networked stabilization for multiinput systems over quantized fading channels Automatica vol. 61 pp A. Cristofaro and S. Pettinari Fault accommodation for multiinput linear sampled-data systems International Adaptive Control and Signal Processing vol.29no.7pp Y. Ji and X. Liu Unified synchronization criteria for hybrid switching-impulsive dynamical networks Circuits Systems and Signal Processingvol.34no.5pp L. Jia X. Li and M.-S. Chiu Correlation analysis based MIMO neuro-fuzzy Hammerstein model with noises Journal of Process Controlvol.41pp F.DingF.WangL.XuandM.Wu Decompositionbased least squares iterative identification algorithm for multivariate pseudo-linear ARMA systems using the data filtering Journal of The Franklin Institutevol.354no.3pp C. Yu and M. Verhaegen Blind multivariable ARMA subspace identification Automaticavol.66pp G. Mercère and L. Bako Parameterization and identification of multivariable state-space systems: a canonical approach Automatica vol. 47 no. 8 pp M. Li X. Liu and F. Ding Least-squares-based iterative and gradient-based iterative estimation algorithms for bilinear systems Nonlinear Dynamics vol. 89 no. 1 pp M.LiX.LiuandF.Ding Themaximumlikelihoodleast squares based iterative estimation algorithm for bilinear systems with autoregressive moving average noise The Franklin Institutevol.354no.12pp Y. Wang and F. Ding Recursive least squares algorithm and gradient algorithm for Hammerstein-Wiener systems using the

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Ding Highly efficient identification methods for dual-rate hammerstein systems IEEE Transactions on Control Systems Technologyvol.23no.5pp D. Wang and F. Ding Parameter estimation algorithms for multivariable Hammerstein CARMA systems Information Sciencesvol pp D. Wang Hierarchical parameter estimation for a class of MIMO Hammerstein systems based on the reframed models Applied Mathematics Lettersvol.57pp F. Ding X. Wang L. Mao and L. Xu Joint state and multiinnovation parameter estimation for time-delay linear systems and its convergence based on the Kalman filtering Digital Signal Processingvol.62pp L. Xu F. Ding Y. Gu A. Alsaedi and T. 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