Abstract
Identification for closed-loop two-dimensional (2-D) causal, recursive, and separable-in-denominator (CRSD) systems in the Roesser form is discussed in this study. For closed-loop 2-D CRSD systems, under feedback control condition, there exists some correlation between the unknown disturbances and future inputs which offers the fundamental limitation for utilizing standard open-loop 2-D CRSD systems subspace identification methods. In other words, the existing open-loop subspace approaches will result in biased estimates of plant parameters from closed-loop data. In this study, based on orthogonal projection and principal component analysis, novel 2-D CRSD subspace identification methods are developed, which are applicable to both open-loop and closed-loop data. Additionally, the whiteness external excitation case is discussed and subsequently modified instrument variables are adopted to improve the proposed subspace algorithm. An illustrative example of the injection molding process and several numerical examples are used to validate consistency and efficiency of the proposed subspace approaches for 2-D CRSD systems.
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References
Bauer, D. (2005). Asymptotic properties of subspace estimators. Automatica, 41, 359–376.
Chiuso, A. (2007). The role of vector autoregressive modeling in predictor-based subspace identification. Automatica, 43, 1034–1048.
Chiuso, A., & Picci, G. (2005). Consistency analysis of some closed-loop subspace identification methods. Automatica, 41(3), 377–391.
Favoreel, W., De Moor, B., & Van Overschee, P. (2000). Subspace state space system identification for industrial processes. Journal of Process Control, 10(2), 149–155.
Gustafsson, T. (2002). Subspace-based system identification: Weighting and pre-filtering of instruments. Automatica, 38(3), 433–443.
Hinamoto, T. (1980). Realizations of a state-space model from two-dimensional input-output map. IEEE Transactions on Circuits and Systems, 27(1), 36–44.
Hong, X., Lai, X., & Zhao, R. (2016). A fast design algorithm for elliptic-error and phase-error constrained LS 2-D FIR filters. Multidimensional Systems and Signal Processing, 27(2), 477–491.
Huang, B., Ding, S. X., & Qin, S. J. (2005). Closed-loop subspace identification: An orthogonal projection approach. Journal of Process Control, 15(1), 53–66.
Huang, B., & Kadali, R. (2008). Dynamic modeling, predictive control and performance monitoring: A data-driven subspace approach. Berlin: Springer.
Jansson, M. (2003). Subspace identification and ARX modeling. In Proceedings of the 13th IFAC SYSID symposium.
Kaczorek, T. (1985). Two-dimensional linear systems. Berlin: Springer.
Katayama, T. (2006). Subspace methods for system identification. Berlin: Springer.
Katayama, T., & Tanaka, H. (2007). An approach to closed-loop subspace identification by orthogonal decomposition. Automatica, 43(9), 1623–1630.
Lashgari, B., Silverman, L. M., & Abramatic, J. F. (1983). Approximation of 2-D separable in denominator filters. IEEE Transactions on Circuits and Systems, 30(2), 107–121.
Lin, W. L., Qin, S. J., & Ljung, L. (2004). On consistency of closed-loop subspace identification with innovation estimation. In 2004 43rd IEEE conference on decision and control (pp. 2195–2200).
Liu, J. (2013). Observer-based higher order sliding mode control of power factor in three-phase AC/DC converter for hybrid electric vehicle applications. International Journal of Control, 87(6), 1117–1130.
Liu, J., Laghrouche, S., Harmouche, M., & Wack, M. (2014). Adaptive-gain second-order sliding mode observer design for switching power converters. Control Engineering Practice, 30, 124–131.
Liu, T. (2012). A bias-eliminated subspace identification method for errors-in-variables systems. In International symposium on advanced control of chemical processes (ADCHEM) (pp. 10–13).
Liu, T., Shao, C., & Wang, X. Z. (2013). Consistency analysis of orthogonal projection based closed-loop subspace identification methods. In European Control Conference (ECC) (pp.1428–1432).
Ljung, L., & McKelvey, T. (1996). Subspace identification from closed loop data. Signal Processing, 52(2), 209–215.
Nam, S. C., Abe, M., & Kawamata, M. (2007). GA-based design of 2D state-space digital filters with linear phase characteristics. Journal of Circuits, Systems, and Computers, 16(02), 287–303.
Pal, D., & Pillai, H. K. (2014). On restrictions of nD systems to 1-D subspaces. Multidimensional Systems and Signal Processing, 25(1), 115–144.
Qin, S. J. (2006). An overview of subspace identification. Computers & Chemical Engineering, 30(10), 1502–1513.
Qin, S. J., & Ljung, L. (2003). Closed-loop subspace identification with innovation estimation. In Proceedings of the 13th IFAC SYSID symposium (pp. 887–892).
Ramos, J. (1994). A subspace algorithm for identifying 2-D separable in denominator filters. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 41(1), 63–67.
Ramos, J. A., Alenany, A., Shang, H., & dos Santos, P. J. L. (2011). Subspace algorithms for identifying separable-in-denominator two-dimensional systems with deterministic inputs. IET Control Theory & Applications, 5(15), 1748–1765.
Roesser, R. P. (1975). A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, 20(1), 1–10.
Ruscio, D. (2009). Closed and open loop subspace system identification of the Kalman filter. Modeling, Identification and Control, 30(2), 71–86.
Shi, J., Gao, F., & Wu, T. J. (2005). Robust design of integrated feedback and iterative learning control of a batch process based on a 2D Roesser system. Journal of Process Control, 15(8), 907–924.
Shyu, J. J., Pei, S. C., Huang, Y. D., & Chen, Y. S. (2014). A new structure and design method for variable fractional-delay 2-D FIR digital filters. Multidimensional Systems and Signal Processing, 25(3), 511–529.
Söderström, T. (2007). Errors-in-variables methods in system identification. Automatica, 43(6), 939–958.
Treasure, R., Sreeram, V., & Ngan, K. N. (2004). Balanced identification and model reduction of a separable denominator 2-D system. In 5th Asian control conference (pp. 2048–2052).
van der Veen, G., van Wingerden, J. W., Bergamasco, M., Lovera, M., & Verhaegen, M. (2013). Closed-loop subspace identification methods: An overview. IET Control Theory & Applications, 7(10), 1339–1358.
Van Overschee, P., & De Moor, B. (1995). A unifying theorem for three subspace system identification algorithms. Automatica, 31(12), 1853–1864.
Wang, D., Zilouchian, A., & Bai, Y. (2005). An algorithm for balanced approximation and model reduction of 2-D separable-in-denominator filters. Multidimensional Systems and Signal Processing, 16(4), 439–461.
Wang, J., & Qin, S. J. (2002). A new subspace identification approach based on principal component analysis. Journal of Process Control, 12(8), 841–855.
Wang, J., & Qin, S. J. (2006). Closed-loop subspace identification using the parity space. Automatica, 42(2), 315–320.
Wang, L., Mo, S., Qu, H., Zhou, D., & Gao, F. (2013). H\(\infty \) design of 2D controller for batch processes with uncertainties and interval time-varying delays. Control Engineering Practice, 20(10), 1321–1333.
Wu, L., Shi, P., Gao, H., & Wang, C. (2008). H\(\infty \) filtering for 2D Markovian jump systems. Automatica, 44(7), 1849–1858.
Xiao, C., Sreeram, V., Liu, W.Q., & Venetsanopoulos, A.N. (1997). Identification and model reduction of 2-D systems via the extended impulse response Gramians. In American control conference (pp. 3567–3571).
Yang, R., Xie, L., & Zhang, C. (2006). H-2 and mixed H-2/H-infinity control of two-dimensional systems in Roesser model. Automatica, 42, 1507–1514.
Zhao, Y., Liebgott, H., & Cachard, C. (2015). Comparison of the existing tool localisation methods on two-dimensional ultrasound images and their tracking results. IET Control Theory & Applications, 9(7), 1124–1134.
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This study was supported by the National Natural Science Foundation of China under Grant 61374099 and the Program for New Century Excellent Talents in University under Grant NCET-13-0652.
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Cheng, J., Fang, M. & Wang, Y. Subspace identification for closed-loop 2-D separable-in-denominator systems. Multidim Syst Sign Process 28, 1499–1521 (2017). https://doi.org/10.1007/s11045-016-0427-y
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DOI: https://doi.org/10.1007/s11045-016-0427-y