A geometric approach to structural model matching by output feedback in linear impulsive systems

Zattoni, E.

International Journal of Applied Mathematics and Computer Science, Tome 28 (2018) no. 1, pp. 25-38.

Résumé

This paper provides a complete characterization of solvability of the problem of structural model matching by output feedback in linear impulsive systems with nonuniformly spaced state jumps. Namely, given a linear impulsive plant and a linear impulsive model, both subject to sequences of state jumps which are assumed to be simultaneous and measurable, the problem consists in finding a linear impulsive compensator that achieves exact matching between the respective forced responses of the linear impulsive plant and of the linear impulsive model, by means of a dynamic feedback of the plant output, for all the admissible input functions and for all the admissible sequences of jump times. The solution of the stated problem is achieved by reducing it to an equivalent problem of structural disturbance decoupling by dynamic feedforward. Indeed, this latter problem is formulated for the so-called extended linear impulsive system, which consists of a suitable connection between the given plant and a modified model. A necessary and sufficient condition for the solution of the structural disturbance decoupling problem is first shown. The proof of sufficiency is constructive, since it is based on the synthesis of the compensator that solves the problem. The proof of necessity is based on the definition and the geometric properties of the unobservable subspace of a linear impulsive system subject to unequally spaced state jumps. Finally, the equivalence between the two structural problems is formally established and proven.

Détail

Keywords: linear impulsive system, hybrid system, model matching, disturbance decoupling, geometric approach, structural properties
Mots-clés : układ liniowy impulsowy, układ hybrydowy, dopasowanie modelu, właściwości strukturalne

Bibliographie

[1] Basile, G. and Marro, G. (1992). Controlled and Conditioned Invariants in Linear System Theory, Prentice Hall, Englewood Cliffs, NJ.

[2] Carnevale, D., Galeani, S., Menini, L. and Sassano, M. (2016). Hybrid output regulation for linear systems with periodic jumps: Solvability conditions, structural implications and semi-classical solutions, IEEE Transactions on Automatic Control 61(9): 2416–2431.

[3] Carnevale, D., Galeani, S. and Sassano, M. (2014a). Francis equations vs invariant subspace algorithm for hybrid output regulation, 53rd IEEE Conference on Decision and Control, Los Angeles, CA, USA, pp. 4697–4702.

[4] Carnevale, D., Galeani, S. and Sassano, M. (2014b). A linear quadratic approach to linear time invariant stabilization for a class of hybrid systems, 22nd Mediterranean Conference on Control and Automation, Palermo, Italy, pp. 545–550.

[5] Choe, D.-G. and Kim, J.-H. (2002). Pitch autopilot design using model-following adaptive sliding mode control, Journal of Guidance, Control, and Dynamics 25(4): 826–829.

[6] Colaneri, P. and Kučera, V. (1997). The model matching problem for periodic discrete-time systems, IEEE Transactions on Automatic Control 42(10): 1472–1476.

[7] Conte, G. and Perdon, A.M. (1995). The disturbance decoupling problem for systems over a ring, SIAM Journal on Control and Optimization 33(3): 750–764.

[8] Conte, G., Perdon, A.M. and Zattoni, E. (2013). A geometric approach to output regulation for discrete-time switched linear systems, 21st Mediterranean Conference on Control and Automation, Platanias-Chania, Crete, Greece, pp. 852–857.

[9] Conte, G., Perdon, A.M. and Zattoni, E. (2014). Model matching problems for switching linear systems, IFAC Proceedings Volumes 47(3): 1501–1506.

[10] Conte, G., Perdon, A.M. and Zattoni, E. (2015a). The disturbance decoupling problem for jumping hybrid systems, 54th IEEE Conference on Decision and Control, Osaka, Japan, pp. 1589–1594.

[11] Conte, G., Perdon, A.M. and Zattoni, E. (2015b). The disturbance decoupling problem with quadratic stability for LPV systems, IFAC-PapersOnLine 48(26): 1–6.

[12] Conte, G., Perdon, A.M. and Zattoni, E. (2017). Unknown input observers for hybrid linear systems with state jumps, IFACPapersOnLine 50(1): 6458–6464.

[13] Djemai, M. and Deefort, M. (2015). Hybrid Dynamical Systems, Lecture Notes in Control and Information Sciences, Vol. 457, Springer, Berlin/Heidelberg.

[14] Engell, S., Frehse, G. and Schnieder, E. (2002). Modeling, Analysis and Design of Hybrid Systems, Lecture Notes in Control and Information Sciences, Vol. 279, Springer, Berlin/Heidelberg.

[15] Goebel, R., Sanfelice, R.G. and Teel, A.R. (2012). Hybrid Dynamical Systems: Modeling, Stability, and Robustness, Princeton University Press, Princeton, NJ.

[16] Haddad, W.M., Chellaboina, V.S. and Nersesov, S.G. (2006). Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton Series in Applied Mathematics, Vol. 6, Princeton University Press, Princeton, NJ.

[17] Hua, C.-C. and Ding, S. (2011). Model following controller design for large-scale systems with time-delay interconnections and multiple dead-zone inputs, IEEE Transactions on Automatic Control 56(4): 962–968.

[18] Kaczorek, T. (1982). Polynomial equation approach to exact model matching problem in multivariable linear systems, International Journal of Control 36(3): 531–539.

[19] Kouhi, Y., Bajcinca, N. and Sanfelice, R.G. (2013). Suboptimality bounds for linear quadratic problems in hybrid linear systems, 2013 European Control Conference, Zürich, Switzerland, pp. 2663–2668.

[20] Kučera, V. (1992). Model matching of descriptor systems by proportional state feedback, Automatica 28(2): 423–425.

[21] Lawrence, D.A. (2014). Controlled invariant subspaces for linear impulsive systems, 2014 American Control Conference, Portland, OR, USA, pp. 2336–2341.

[22] Lawrence, D.A. (2015). Conditioned invariant subspaces for linear impulsive systems, 2015 American Control Conference, Chicago, IL, USA, pp. 4850–4855.

[23] Li, Z., Soh, Y. and Wen, C. (2005). Switched and Impulsive Systems: Analysis, Design and Applications, Lecture Notes in Control and Information Sciences, Vol. 313, Springer-Verlag, Berlin/Heidelberg.

[24] Lin,W.-S. and Tsai, C.-H. (1999). Neurofuzzy-model-following control of MIMO nonlinear systems, IEE Proceedings: Control Theory and Applications 146(2): 157–164.

[25] Marro, G. and Zattoni, E. (2002). A novel geometric insight into the model matching problem with stability, 41st IEEE Conference on Decision and Control, Las Vegas, NV, USA, pp. 2137–2142.

[26] Marro, G. and Zattoni, E. (2005). Self-bounded controlled invariant subspaces in model following by output feedback: Minimal-order solution for nonminimum-phase systems, Journal of Optimization Theory and Applications 125(2): 409–429.

[27] Medina, E.A. and Lawrence, D.A. (2006). Controlled and conditioned invariants for linear impulsive systems, 45th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 2753–2758.

[28] Medina, E.A. and Lawrence, D.A. (2009). State estimation for linear impulsive systems, 2009 American Control Conference, St. Louis, MO, USA, pp. 1183–1188.

[29] Moog, C.H., Conte, G. and Perdon, A.M. (1991). Model matching and factorization for nonlinear systems: A structural approach, SIAM Journal on Control and Optimization 29(4): 769–785.

[30] Morse, A.S. (1973). Structure and design of linear model following systems, IEEE Transactions on Automatic Control 18(4): 346–354.

[31] Ni, M., Er, M., Leithead, W. and Leith, D. (2001). New approach to the design of robust tracking and model following controllers for uncertain delay systems, IEE Proceedings: Control Theory and Applications 148(6): 472–477.

[32] Noldus, E. (1987). Non-linear model following, Automatica 23(3): 387–391.

[33] Pascual, M., Garcerá, G., Figueres, E. and González-Espín, F. (2008). Robust model-following control of parallel UPS single-phase inverters, IEEE Transactions on Industrial Electronics 55(8): 2870–2883.

[34] Perdon, A.M., Conte, G. and Zattoni, E. (2016a). Necessary and sufficient conditions for asymptotic model matching of switching linear systems, Automatica 64(2): 294–304.

[35] Perdon, A.M., Zattoni, E. and Conte, G. (2016b). Disturbance decoupling with stability for linear impulsive systems, IFAC-PapersOnLine 49(9): 1–6.

[36] Perdon, A.M., Zattoni, E. and Conte, G. (2016c). Model matching with strong stability in switched linear systems, Systems Control Letters 97(11): 98–107.

[37] Rajasekaran, J., Chunodkar, A. and Padhi, R. (2009). Structured model-following neuro-adaptive design for attitude maneuver of rigid bodies, Control Engineering Practice 17(6): 676–689.

[38] Savkin, A.V. and Evans, R.J. (2002). Hybrid Dynamical Systems. Controller and Sensor Switching Problems, Control Engineering Series, Birkhauser, Boston, MA.

[39] Shioemaru, S. and Wu, H. (2001). Decentralized robust tracking and model following for uncertain large-scale interconnected systems, Journal of Optimization Theory and Applications 110(1): 35–52.

[40] Shyu, K.-K. and Chen, Y.-C. (1995). Robust tracking and model following for uncertain time-delay systems, International Journal of Control 62(3): 589–600.

[41] Van der Schaft, A.J. and Schumacher, H. (2000). An Introduction to Hybrid Dynamical Systems, Lecture Notes in Control and Information Sciences, Vol. 251, Springer, Berlin/Heidelberg.

[42] Wang, D., Wu, S. and Okubo, S. (2009). Design of the state predictive model following control system with time-delay, International Journal of Applied Mathematics and Computer Science 19(2): 247–254, DOI: 10.2478/v10006-009-0020-8.

[43] Wang, L., Zhu, J. and Park, J. (2012). A probabilistic approach for model following of Markovian jump linear systems subject to actuator saturation, International Journal of Control, Automation and Systems 10(5): 1042–1048.

[44] Wolovich, W.A. (1972). The use of state feedback for exact model matching, SIAM Journal on Control 10(3): 512–523.

[45] Wonham, W.M. (1985). Linear Multivariable Control: A Geometric Approach, 3rd Edn., Springer-Verlag, New York, NY.

[46] Wu, H. (2004). Adaptive robust tracking and model following of uncertain dynamical systems with multiple time delays, IEEE Transactions on Automatic Control 49(4): 611–616.

[47] Zakharov, A., Zattoni, E., Xie, L., Garcia, O. and Jämsä-Jounela, S.-L. (2013). An autonomous valve stiction detection system based on data characterization, Control Engineering Practice 21(11): 1507–1518.

[48] Zakharov, A., Zattoni, E., Yu, M. and Jämsä-Jounela, S.-L. (2015). A performance optimization algorithm for controller reconfiguration in fault tolerant distributed model predictive control, Journal of Process Control 34(8): 56–69.

[49] Zattoni, E. (2007). Decoupling of measurable signals via self-bounded controlled invariant subspaces: Minimal unassignable dynamics of feedforward units for prestabilized systems, IEEE Transactions on Automatic Control 52(1): 140–143.

[50] Zattoni, E. (2014). Measurable disturbance rejection with stability in continuous-time switched linear systems under dwell-time switching, European Control Conference 2014, Strasbourg, France, pp. 2242–2247.

[51] Zattoni, E. (2016a). Disturbance decoupling with stability in discrete-time switching linear systems: Arbitrary switching, International Journal of Pure and Applied Mathematics 110(1): 227–250.

[52] Zattoni, E. (2016b). Output feedback model matching in linear impulsive systems with control feedthrough: A structural approach, IOP Journal of Physics: Conference Series 783: 1–9.

[53] Zattoni, E. and Marro, G. (2013a). A constructive condition for inaccessible signal rejection with quadratic stability in discrete-time linear switching systems, 52nd IEEE Conference on Decision and Control, Florence, Italy, pp. 4650–4655.

[54] Zattoni, E. and Marro, G. (2013b). Measurable disturbance rejection with quadratic stability in continuous-time linear switching systems, European Control Conference 2013, Zürich, Switzerland, pp. 2157–2162.

[55] Zattoni, E., Perdon, A.M. and Conte, G. (2013a). A geometric approach to output regulation for linear switching systems, IFAC Proceedings Volumes 46(2): 172–177.

[56] Zattoni, E., Perdon, A.M. and Conte, G. (2013b). The output regulation problem with stability for linear switching systems: A geometric approach, Automatica 49(10): 2953–2962.

[57] Zattoni, E., Perdon, A.M. and Conte, G. (2014a). Disturbance decoupling with stability in continuous-time switched linear systems under dwell-time switching, IFAC Proceedings Volumes 47(3): 164–169.

[58] Zattoni, E., Perdon, A.M. and Conte, G. (2014b). Output feedback model matching with strong stability in continuous-time switched linear systems, 22nd Mediterranean Conference on Control and Automation, Palermo, Italy, pp. 525–530.

[59] Zattoni, E., Perdon, A.M. and Conte, G. (2015). Output regulation by error dynamic feedback in linear time-invariant hybrid dynamical systems, 14th European Control Conference, Linz, Austria, pp. 1438–1443.

[60] Zattoni, E., Perdon, A.M. and Conte, G. (2016). Disturbance decoupling with closed-loop modes stability in switched linear systems, IEEE Transactions on Automatic Control 61(10): 3115–3121.

[61] Zattoni, E., Perdon, A.M. and Conte, G. (2017a). Output regulation by error dynamic feedback in hybrid linear systems with state jumps, IFAC-PapersOnLine 50(1): 10808–10815.

[62] Zattoni, E., Perdon, A.M. and Conte, G. (2017b). Output regulation by error dynamic feedback in hybrid systems with periodic state jumps, Automatica 81(7): 322–334.