Closed Loop Identification of Multi-Rate System by ‎Expectation-Maximization

محورهای موضوعی : Electrical Engineering

somayeh Jokar ¹ (Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran)
Amin Ramezani ² (Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran)
Mehdi Siahi ³ (Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran)

تاریخ دریافت : 1401/09/19 تاریخ پذیرش : 1402/04/06 تاریخ انتشار : 1402/03/11

کلید واژه: identification, Closed loop system, Multi-rate system, Expectation-maximization,

چکیده مقاله :

Closed-loop identification of multi-rate system with unknown parameters, that including prevalent non-uniform sampling data, is considered. The purpose is to identify a multi-rate closed loop model to approximate the parameters varying system. As far as the research has been done, the identification of multi-rate closed loop model with unknown parameters by using the expectation-maximization algorithm has not been done. To address this challenge, the two-stage method and expectation-maximization algorithm are applied in this paper to identify unknown parameters of system. In this case, by introducing the hidden variable, an EM is utilized to estimate the unknown model parameters. And also, it will be demonstrated that, to estimate of system parameters, Instead of the point estimate of the time variable, the full probability distribution of the time variable estimate is required. The performance of this procedure represents by simulation, and obtain consequences affirm that method has high precision and also has a high convergence speed. These simulations express that the performance of this algorithm is good, as the identified parameters accede the true parameters after several iterations. The Monte Carlo simulation with different noise realizations at SNR=26dB and SNR=46 dB are performed for showing the effectiveness of mentioned algorithm.

منابع و مأخذ:

[1] X.L. Feng, C.L. Wen, J.H. Park, "Sequential fusion h-infinity filtering for multi-rate multi-sensor time-varying systems - a krein-space approach", IET Control Theory Appl. Vol.11, No.3, p. 369–381, 2017.

[2] Y. Zhang, Z.D. Wang, L. Zou, " Fault detection filter design for networked multi-rate systems with fading measurements and randomly occurring faults", IET Control Theory Appl. Vol.10, No.5, p. 573–581, 2016.

[3] J. Chen, J. Li, Y.J. Liu,"Gradient iterative algorithm for dual-rate nonlinear systems based on a novel particle filter", J. Frankl. Inst. Vol.354,No.11, p. 4425–4437, 2017.

[4] F. Ding, T. Chen, "Combined parameter and output estimation of dual-rate systems using an auxiliary model", Automatica , Vol. 40, No. 10, p. 1739–1748,2004.

[5]. F. Ding, T. Chen, "Parameter estimation of dual-rate stochatic systems by using an output error method", IEEE Transactions on Automatic Control, Vol. 50, No. 9, p.1436–1441, 2005.

[6]. D. Barcelli, A. Bemporad, G. Ripaccioli, Decentralized hierarchical multi-rate control of constrained linear systems", in: 18th World Congress, The International Federation of Automatic Control, Milano, Italy, 2011, pp. 277–283.

[7]. L. Xie, H. Yang, F. Ding, "Inferential adaptive control for non-uniformly sampled–data systems", in: 2011 American Control Conference, San Francisco, CA, USA, 2011, pp. 4177–4182.

[8]. J. Ma, H. Ge, "Modified multi-rate detection for frequency selective Rayleigh fading CDMA channels", in: IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, 1998, pp. 1304–1308.

[9]. H. Geng, Y. Liang, X. Zhang, F. Yang, "Fast-rate residual generator based on multiple slow-rate sensors", IET Signal Proc. Vol. 8, No. 8, p.878–884, 2014.

[10]. R.B. Gopaluni, H. Raghavan, S.L. Shah, "System identification from multi-rate data", in: IFAC Advanced Control of Chemical Processes, Hong Kong. China, 2003, pp. 155–160.

[11]. F. Ding, G. Liu, X.P. Liu, "Partially coupled stochastic gradient identification methods for non-uniformly sampled systems", IEEE Trans. Autom Control, Vol.55,No. 8, p.1976–1981, 2010.

[12]. W.L. Yan, C.L. Du, C.K. Pang, "A general mult-irate approach for direct closed-loop identification to the nyquist frequency and beyond", Automatica, Vol. 53 p.164–170,2015.

[13] Q.M. Shao, A. Cinar, "System identification and distributed control for multi-rate sampled systems", J. ProcessControl, Vol. 34, p.1–12, 2015.

[14] M. Zamani, G. Bottegal, B.D.O. Anderson, "On the zero-freeness of tall multirate linear systems", IEEE Trans.Autom. Control, Vol.61, No.11, p. 3606–3611, 2016.

[15] Y. Zhang, Z.D. Wang, L. Zou, H.J. Fang, "Event-based finite-time filtering for multi-rate systems with fading
measurements", IEEE Trans. Aerosp. Electron. Syst, Vol.53, No.3, p. 1431–14412017.

 [16] F. Ding, D.D. Meng, J.Y. Dai, Q.S. Li, A. Alsaedi, T. Hayat, "Least squares based iterative parameter estimation algorithm for stochastic dynamical systems with ARMA noise using the model equivalence", Int. J. Control Autom.  Syst, Vol.16, No.2,p.630–639, 2018.

[17] Y.J. Wang, F. Ding, L. Xu, "Some new results of designing an IIR filter with colored noise for signal processing, Digit". Signal Process,Vol.72, p. 44–58, 2018.

[18] P. Ma, F. Ding, Q.M. Zhu, "Decomposition-based recursive least squares identification methods for multivariate pseudolinear systems using the multi-innovation", Int. J. Syst. Sci. Vol.49, No.5, p. 920–928, 2018.

[19] X. Zhang, L. Xu, "Combined state and parameter estimation for a bilinear state space system with moving average noise", J. Frankl. Inst. Vol.355, No.6, p.3079–3103, 2018.

[20] X. Zhang, F. Ding, F.E. Alsaadi, T. Hayat, "Recursive parameter identification of the dynamical models for
bilinear state space systems", Nonlin. Dyn. Vol.89, No.4, p.2415–2429, 2017.

[21] M.T. Chen, F. Ding, L. Xu, T. Hayat, A. Alsaedi, "Iterative identification algorithms for bilinear-in-parameter systems with autoregressive moving average noise", J. Frankl. Inst. Vol.354, No.17, p. 7885–7898, 2017.

[22].  A.P. Dempster, N.M. Laird, D.B. Rubin, "Maximum likelihood estimation from incomplete data via the EM
algorithm", J. R. Stat. Soc. B, Vol. 39, No. 1,p.1–38, 1977.

[23].  X. Wang, B. Quost, J.D. Chazot, J. Antoni, "Estimation of multiple sound sources with data and model uncertainties using the EM and evidential EM algorithms", Mech. Syst. Signal Process, Vol. 66, No. 67, p.159–177,2016.

[24]. Y. Chung, B.G. Lindsay, "Convergence of the EM algorithm for continuous mixing distributions", Stat. Probab.Lett, Vol. 96, p.190–195, 2015.

[25]. J.X. Ma, J. Chen, W.L. Xiong, "Expectation-maximization estimation algorithm for hammerstein models with non-gaussian noise and random time delay from dual-rate sampled-data", Digital Signal Process, Vol. 73, p.135–144, 2018.

[26]. MinhPhan, Richard W. Longman, "Identification of observer/Kalman filter Markov parameters - Theory and experiments", Journal of Guidance and Dynamics, Vol. 17, No.4, p.661-669, 1994.

[27].U. Forssell, L. Ljung, "Closed-loop identification revisited", Automatica, Vol. 35, No. 7, p.1215–1241,1999.

[28]. P. Van den Hof, R. Schrama, "An indirect method for transfer function estimation from closed loop data", Automatica, Vol. 29, No. 6, p.1523–1527, 1993.

[29]. B. Huang, S. Shah, "Closed-loop identification: a two-step approach", Journal of Process Control, Vol. 7, No. 6, p.425–438, 1997.

[30]. J.F. MacGregor, D. Fogal, "Closed loop identification: the role of the noise model and pre-filters", Journal of Process Control, Vol. 5, p.163–171, 1995.

[31] Y. Cao, P. Li, Y. Zhang, "Parallel processing algorithm for railway signal fault diagnosis data based on cloud
computing", Futur. Gener. Comput. Syst.Vol.88, p.279–283, 2018.

[32] Y.Z. Zhang, Y. Cao, Y.H. Wen, L. Liang, F. Zou, "Optimization of information interaction protocols in cooperative vehicle-infra structure systems", Chin. J. Electron. Vol.27, No.2, p.439–444, 2018,

[33] P. Li, R.X. Li, Y. Cao, G. Xie, "Multi-objective sizing optimization for island microgrids using triangular aggregation model and levy-harmony algorithm", IEEE Trans. Ind. Inform. Vol.14, No.8, p. 3495–3505, 2018.

[34] P. Li, R. Dargaville, Y. Cao, et al., "Storage aided system property enhancing and hybrid robust smoothing for
large-scale PV systems", IEEE Trans. Smart Grid,Vol. 8 ,No.6,p. 2871–2879,2017.

[35] Y. Cao, L.C. Ma, S. Xiao, et al., "Standard analysis for transfer delay in CTCS-3", Chin. J. Electron. Vol.26, No.5, p.1057–1063, 2017

[36] C.F.J. Wu, "On the convergence properties of the EM algorithm", Ann. Stat. Vol.11, No.1, p.95–103, 1983.

[37] Y. Gu, F. Ding, J.H. Li, "States based iterative parameter estimation for a state space model with multi-statedelays using decomposition", Signal Process. Vol.106, p.230–294, 2015.

[38] G. Zheng, J.P. Barbot, D. Boutat, "Identification of the delay parameter for nonlinear time-delay systems with
unknown inputs", Automatica, Vol. 4 ,No.6,p.1755–1760,2013.

[39]. G.Ya, L.Jicheng, L.Xiangli, C.Yongxin and J.Yan, "State space model identification of multirate process with time-delay using the expection maximaization", J of the Franklin Institute, 2018.

[40] F. Liu, "A note on marcinkiewicz integrals associated to surfaces of revolution", J. Aust. Math. Soc. Vol.104, No.3, p.380–402, 2018.

[41] F. Liu, H.X. Wu, "A note on the endpoint regularity of the discrete maximal operator", P. Am. Math. Soc. Vol.147, No.2, p. 583–596, 2019.

[42] F. Liu, "A note on Marcinkiewicz integrals associated to surfaces of revolution", J. Aust. Math. Soc. Vol.104, No.3, p.380–402, 2018.

[43] F. Liu, H.X. Wu, "Singular integrals related to homogeneous mappings in triebel-lizorkin spaces", J. Math. Inequal.Vol.11, No.4, p. 1075–1097, 2017.

[44] F.Z. Geng, S.P. Qian, "An optimal reproducing kernel method for linear nonlocal boundary value problems", Appl.Math. Lett.Vol.77, p.49–56, 2018.

[45] X.Y. Li, B.Y. Wu, "A new reproducing kernel collocation method for nonlocal fractional boundary value problems with non-smooth solutions", Appl. Math. Lett. Vol.86, p.194–199, 2018.

[46] F. Liu, "On the triebel-lizorkin space boundedness of marcinkiewicz integrals along compound surfaces", Math. Inequal. Appl. Vol.20, No.2, p. 515–535, 2017.

[47] C.C. Yin, J.S. Zhao, "Non exponential asymptotics for the solutions of renewal equations, with applications", J.Appl. Probab. Vol.43, No.3, p. 815–824, 2008.

[48] H.L. Gao, C.C. Yin, "The perturbed sparreandersen model with a threshold dividend strategy", J. Comput. Appl.Math. Vol.220, No. (1–2), p.394–408, 2008.

[49] C.C. Yin, C.W. Wang, "The perturbed compound poisson risk process with investment and debit interest,
Methodol". Comput. Appl. Probab. Vol.12, No.3, p.391–413, 2010.

[50] C.C. Yin, K.C. Yuen, "Optimality of the threshold dividend strategy for the compound poisson model", Stat.Probab. Lett. Vol.81, No.12, p. 1841–1846, 2011.

[51] C.C. Yin, Y.Z. Wen, "Exit problems for jump processes with applications to dividend problems", J. Comput. Appl.Math. Vol.245, p.30–52, 2013.

[52] C.C. Yin, Y.Z. Wen, "Optimal dividend problem with a terminal value for spectrally positive levy processes",
Insur. Math. Econ.Vol.53, No.3, p. 769–773, 2013.

[53] Y. Ji, F. Ding, "Multiperiodicity and exponential attractivity of neural networks with mixed delays, Circ. Syst". Signal Process. Vol.36, No.6, p. 2558–2573. 2017.

[54] N. Zhao, Y. Chen, R. Liu, M.H. Wu, W. Xiong, "Monitoring strategy for relay incentive mechanism in cooperative communication networks", Comput. Electr. Eng. Vol.60, p.14–29, 2017.

[55] P.C. Gong, W.Q. Wang, F.C. Li, H. Cheung, "Sparsity-aware transmit beam space design for FDA-MIMO radar", Signal Process. Vol.144, p.99–103, 2018.

[56] Z.H. Rao, C.Y. Zeng, M.H. Wu, et al.," Research on a handwritten character recognition algorithm based on an extended nonlinear kernel residual network", KSII Trans. Internet Inf. Syst. Vol.12, No.1,p. 413–435,2018.

[57] N. Zhao, R. Liu, Y. Chen, M. Wu, Y. Jiang, W. Xiong, C. Liu, "Contract design for relay incentive mechanism under dual asymmetric information in cooperative networks", Wirel. Netw.Vol.24, No.8, p. 3029–3044, 2018.

[58] J. Pan, W. Li, H.P. Zhang, "Control algorithms of magnetic suspension systems based on the improved double exponential reaching law of sliding mode control", Int. J. Control Autom. Syst. Vol16, No.6, p.2878–2887, 2018.

[59] X. Li, D.Q. Zhu, "An improved SOM neural network method to adaptive leader-follower formation control of AUVs", IEEE T. Ind. Electron. Vol.65, No.10, p. 8260–8270, 2018.

[60] J. Pan, H. Ma, X. Jiang, et al., "Adaptive gradient-based iterative algorithm for multivariate controlled autoregressive moving average systems using the data filtering technique, Complexity",2018. Article ID 9598307.doi: 10.1155/2018/9598307.

[61] T. Liu, F. Gao, "Closed-loop step response identification of integrating and unstable processes", Chem. Eng. Sci.65 (10) (2010) 2884–2895, doi:10.1016/j.ces.2010.01.013.