Subspace-based identification of discrete time-delay system

Abstract

We investigate the identification problems of a class of linear stochastic time-delay systems with unknown delayed states in this study. A time-delay system is expressed as a delay differential equation with a single delay in the state vector. We first derive an equivalent linear time-invariant (LTI) system for the time-delay system using a state augmentation technique. Then a conventional subspace identification method is used to estimate augmented system matrices and Kalman state sequences up to a similarity transformation. To obtain a state-space model for the time-delay system, an alternate convex search (ACS) algorithm is presented to find a similarity transformation that takes the identified augmented system back to a form so that the time-delay system can be recovered. Finally, we reconstruct the Kalman state sequences based on the similarity transformation. The time-delay system matrices under the same state-space basis can be recovered from the Kalman state sequences and input-output data by solving two least squares problems. Numerical examples are to show the effectiveness of the proposed method.

Keywords

Identification problems / Time-delay systems / Subspace identification method / Alternate convex search / Least squares

References

[1] Bayrak, A., Tatlicioglu, E., 2016. A novel online adaptive time delay identification technique. Int. J. Syst. Sci., 47(7):1574–1585.

[2] Belkoura, L., Orlov, Y., 2002. Identifiability analysis of linear delay-differential systems. IMA J. Math. Contr. Inform., 19(1-2):73–81.

[3] Ding, S.X., Zhang, P., Naik, A., , 2009. Subspace method aided data-driven design of fault detection and isolation systems. J. Process Contr., 19(9):1496–1510.

[4] Drakunov, S.V., Perruquetti, W., Richard, J.P., , 2006. Delay identification in time-delay systems using variable structure observers. Ann. Rev. Contr., 30(2):143–158.

[5] Gao, H., Chen, T., 2007. New results on stability of discretetime systems with time-varying state delay. IEEE Trans. Autom. Contr., 52(2):328–334.

[6] Gorski, J., Pfeuffer, F., Klamroth, K., 2007. Biconvex sets and optimization with biconvex functions: a survey and extensions. Math. Methods Oper. Res., 66(3):373–407.

[7] Hachicha, S., Kharrat, M., Chaari, A., 2014. N4SID and MOESP algorithms to highlight the ill-conditioning into subspace identification. Int. J. Autom. Comput., 11(1): 30–38.

[8] Huang, B., Kadali, R., 2008. Dynamic Modeling, Predictive Control and Performance Monitoring: a Data-Driven Subspace Approach. Springer, London, UK.

[9] Huang, B., Ding, S.X., Qin, S.J., 2005. Closed-loop subspace identification: an orthogonal projection approach. J. Process Contr., 15(1):53–66.

[10] Kailath, T., 1980. Linear Systems. Prentice-Hall, New Jersey, USA.

[11] Kolmanovskii, V., Myshkis, A., 1999. Introduction to the Theory and Applications of Functional Differential Equations. Springer, the Netherlands.

[12] Kudva, P., Narendra, K.S., 1973. An Identification Procedure for Discrete Multivariable Systems. Technical Report No. AD0768992, Yale University, USA.

[13] Lima, R.B.C., Barros, P.R., 2015. Identification of time-delay systems: a state-space realization approach. Proc. 9th IFAC Symp. on Advanced Control of Chemical Processes, p.254–259.

[14] Ljung, L., 1987. System Identification: Theory for the User. PTR Prentice Hall, New Jersey, USA.

[15] Lunel, S.M.V., 2001. Parameter identifiability of differential delay equations. Int. J. Adapt. Contr. Signal Process., 15(6):655–678.

[16] Lyzell, C., Enqvist, M., Ljung, L., 2009. Handling Certain Structure Information in Subspace Identification. Report, Linköping University Electronic Press, Sweden.

[17] Nakagiri, S., Yamamoto, M., 1995. Unique identification of coefficient matrices, time delays and initial functions of functional differential equations. J. Math. Syst. Estimat. Contr., 5(3):323–344.

[18] Niculescu, S.I., 2001. Delay Effects on Stability: a Robust Control Approach. Springer-Verlag, London, UK.

[19] Orlov, Y., Belkoura, L., Richard, J.P., , 2002. On identifiability of linear time-delay systems.IEEE Trans. Autom. Contr., 47(8):1319–1324.

[20] Orlov, Y., Belkoura, L., Richard, J.P., , 2003. Adaptive identification of linear time-delay systems. Int. J. Robust Nonl. Contr., 13(9):857–872.

[21] Park, J.H., Han, S., Kwon, B., 2013. On-line model parameter estimations for time-delay systems. IEICE Trans. Inform. Syst., 96(8):1867–1870.

[22] Pourboghrat, F., Chyung, D.H., 1989. Parameter identification of linear delay systems. Int. J. Contr., 49(2):595–627.

[23] Prot, O., Mercère, G., 2011. Initialization of gradient-based optimization algorithms for the identification of structured state-space models. Proc. 18th IFAC World Congress, p.10782–10787.

[24] Qin, P., Kanae, S., Yang, Z.J., , 2007. Identification of lifted models for general dual-rate sampled-data systems based on input-output data. Proc. Asian Modelling and Simulation, p.7–12.

[25] Qin, S.J., 2006. An overview of subspace identification. Comput. Chem. Eng., 30(10-12):1502–1513.

[26] Richard, J.P., 2003. Time-delay systems: an overview of some recent advances and open problems. Automatica, 39(10): 1667–1694.

[27] Wang, J., Qin, S.J., 2006. Closed-loop subspace identification using the parity space. Automatica, 42(2):315–320.

[28] Xie, L., Ljung, L., 2002. Estimate physical parameters by black box modeling. Proc. 21st Chinese Control Conf., p.673–677.

[29] Yang, X., Gao, H., 2014. Multiple model approach to linear parameter varying time-delay system identification with EM algorithm. J. Franklin Inst., 351(12):5565–5581.

[30] Yang, X., Lu, Y., Yan, Z., 2015. Robust global identification of linear parameter varying systems with generalised expectation-maximisation algorithm. IET Contr. Theory Appl., 9(7):1103–1110.

RIGHTS & PERMISSIONS

Zhejiang University and Springer-Verlag Berlin Heidelberg