A simple algorithm for stable order reduction of z-domain Laguerre models

International audienceDiscrete-time Laguerre series are a well known and efficient tool in system identification and modeling. This paper presents a simple solution for stable and accurate order reduction of systems described by a Laguerre model

An Introduction To Models Based On Laguerre, Kautz And Other Related Orthonormal Functions - Part I: Linear And Uncertain Models

This paper provides an overview of system identification using orthonormal basis function models, such as those based on Laguerre, Kautz, and generalised orthonormal basis functions. The paper is separated in two parts. In this first part, the mathematical foundations of these models as well as their advantages and limitations are discussed within the context of linear and robust system identification. The second part approaches the issues related with non-linear models. The discussions comprise a broad bibliographical survey of the subjects involving linear models within the orthonormal basis functions framework. Theoretical and practical issues regarding the identification of these models are presented and illustrated by means of a case study involving a polymerisation process. Copyright © 2011 Inderscience Enterprises Ltd.1401/02/15121132Aguirre, L.A., Correa, M.V., Cassini, C., Nonlinearities in NARX polynomial models: Representation and estimation (2002) IEE Proc. Control Theory and Applications, 149, pp. 343-348Akcay, H., At, N., Membership set identification with periodic inputs and orthonormal regressors (2006) Signal Processing, 86 (12), pp. 3778-3786. , DOI 10.1016/j.sigpro.2006.03.025, PII S016516840600123X, Multimodal Human-Computer InterfacesAkcay, H., Ninness, B., Rational basis functions for robust identification from frequency and time-domain measurements (1998) Automatica, 34 (9), pp. 1101-1117. , PII S0005109898000521Allgower, F., Zheng, A., (2000) Nonlinear Model Predictive Control, , Birkhauser Verlag, Basel, SwitzerlandAraújo, H.X., Oliveira, G.H.C., An LMI approach for output feedback robust predictive control using orthonormal basis functions (2009) Proc. 48th Conference on Decision and Control, , Shanghai, ChinaBalbis, L., Ordys, A.W., Grimble, M.J., Pang, Y., Tutorial introduction to the modelling and control of hybrid systems (2006) International Journal of Modelling, Identification and Control, 2 (4), pp. 259-272Belt, H.J.W., Den Brinker, A.C., Optimality condition for truncated generalized Laguerre networks (1995) International Journal of Circuit Theory and Applications, 23 (3), pp. 227-235Biagiola, S.I., Figueroa, J.L., Wiener and Hammerstein uncertain models identification (2009) Mathematics and Computers in Simulation, 79 (11), pp. 3296-3313Bodin, P., Villemoes, L.F., Wahlberg, B., Selection of best orthonormal rational basis (2000) SIAM Journal of Control and Optimization, 38 (4), pp. 995-1032Broome, P.W., Discrete orthonormal sequences (1965) Journal of the Association for Computing Machinery, 12 (2), pp. 151-168Camacho, E.F., Bordons, C., (1999) Model Predictive Control, , Springer- Verlag, London, UKCampello, R.J.G.B., Amaral, W.C., Favier, G., Optimal Laguerre series expansion of discrete Volterra models (2001) Proc. European Control Conference, pp. 372-377. , Porto, PortugalCampello, R.J.G.B., Do Amaral, W.C., Favier, G., A note on the optimal expansion of Volterra models using Laguerre functions (2006) Automatica, 42 (4), pp. 689-693. , DOI 10.1016/j.automatica.2005.12.003, PII S0005109806000069Campello, R.J.G.B., Favier, G., Amaral, W.C., Optimal expansions of discrete-time Volterra models using Laguerre functions (2004) Automatica, 40 (5), pp. 815-822Clarke, D.W., (1994) Advances in Model Based Predictive Control, , Oxford University Press, New York, USAClowes, G.J., Choice of time-scaling factor for linear system approximation using orthonormal Laguerre functions (1965) IEEE Transactions on Automatic Control, 10 (4), pp. 487-489Da Rosa, A., (2009) Identification of Nonlinear Systems Using Volterra Models Based on Kautz Functions and Generalized Orthonormal Functions, , PhD thesis, School of Electrical and Computer Engineering of the State University of Campinas (FEEC/UNICAMP), Campinas-SP, Brazil, in PortugueseDa Rosa, A., Campello, R.J.G.B., Amaral, W.C., Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions (2007) Automatica, 43 (6), pp. 1084-1091. , DOI 10.1016/j.automatica.2006.12.007, PII S0005109807000738Da Rosa, A., Campello, R.J.G.B., Amaral, W.C., An optimal expansion of Volterra models using independent Kautz bases for each kernel dimension (2008) International Journal of Control, 81 (6), pp. 962-975Da Rosa, A., Campello, R.J.G.B., Amaral, W.C., Exact search directions for optimization of linear and nonlinear models based on generalized orthonormal functions (2009) IEEE Transactions on Automatic Control, 54 (12), pp. 2757-2772Da Silva, I.N., (1995) Robust Parametric Estimation Using Artificial Neural Networks, , MSc thesis, School of Electrical and Computer Engineering of the State University of Campinas (FEEC/UNICAMP), Campinas-SP, Brazil, in PortugueseDen Brinker, A.C., Sarroukh, B.E., Pole optimisation in adaptive laguerre filtering (2004) Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing, pp. 649-652. , Montreal, CanadaDen Brinker, A.C., Optimality conditions for truncated kautz series (1996) IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 43 (2), pp. 117-122. , PII S105771309601511XDoyle III, F.J., Ogunnaike, B.A., Pearson, R.K., Nonlinear model-based control using second-order Volterra models (1995) Automatica, 31 (5), pp. 697-714Doyle III, F.J., Pearson, R.K., Ogunnaike, B.A., (2002) Identification and Control using Volterra Models, , Springer- Verlag, London, UKDumont, G.A., Fu, Y., Non-linear adaptive control via Laguerre expansion of Volterra kernels (1993) Int. J. Adaptive Control and Signal Processing, 7 (5), pp. 367-382Favier, G., Arruda, L.V.R., Review and comparison of ellipsoidal bounding algorithms (1996) Bounding Approaches to System Identification, pp. 43-68. , Milanese, M. (Ed.), Plenum Press, New YorkFavier, G., Kibangou, A.Y., Campello, R.J.G.B., Nonlinear system modelling by means of orthonormal basis functions (2003) Proc. 2nd IEEE Int. Conference on Signal, Systems, Decision and Information Technology, , Sousse, TunisiaFigueroa, J.L., Biagiola, S.I., Agamennoni, O.E., An approach for identification of uncertain Wiener systems (2008) Mathematical and Computer Modelling, 48 (1-2), pp. 305-315. , DOI 10.1016/j.mcm.2007.09.012, PII S0895717707003214Fu, Y., Dumont, G.A., Optimum time scale for discrete Laguerre network (1993) IEEE Transactions on Automatic Control, 38 (6), pp. 934-938. , DOI 10.1109/9.222305Garcia Carlos, E., Prett David, M., Morari Manfred, Model predictive control: Theory and practice - A survey (1989) Automatica, 25 (3), pp. 335-348. , DOI 10.1016/0005-1098(89)90002-2Hacioglu, R., Williamson, G.A., Reduced complexity Volterra models for nonlinear system identification (2001) Eurasip Journal on Applied Signal Processing, 2001 (4), pp. 257-265. , DOI 10.1155/S1110865701000324Henson, M.A., Nonlinear model predictive control: Current status and future directions (1998) Computers and Chemical Engineering, 23 (2), pp. 187-202. , DOI 10.1016/S0098-1354(98)00260-9, PII S0098135498002609Heuberger, P.S.C., Van Den Hof, P.M.J., Bosgra, O.H., A generalized orthonormal basis for linear dynamical systems (1995) IEEE Trans. on Automatic Control, 40 (3), pp. 451-465Heuberger, P.S.C., Van Den Hof, P.M.J., Wahlberg, B., (2005) Modelling and Identification with Rational Orthogonal Basis Functions, , Springer- Verlag, London, UKKautz, W.H., Transient synthesis in the time domain (1954) IRE Transactions on Circuit Theory, 1 (3), pp. 29-39Kibangou, A.Y., Favier, G., Hassani, M.M., Generalized orthonormal basis selection for expanding quadratic Volterra filters (2003) Proc. 13th IFAC Symposium on System Identification, pp. 1119-1124. , Rotterdam, The NetherlandsKibangou, A.Y., Favier, G., Hassani, M.M., Selection of generalized orthonormal bases for second-order Volterra filters (2005) Signal Processing, 85 (12), pp. 2371-2385. , DOI 10.1016/j.sigpro.2005.02.020, PII S0165168405001258Ljung, L., (1999) System Identification: Theory for the User, , 2nd ed. Prentice HallMäkilä, P.M., Approximation of stable systems by Laguerre filters (1990) Automatica, 26 (2), pp. 333-345Maner, B.R., Doyle III, F.J., Ogunnaike, B.A., Pearson, R.K., Nonlinear model predictive control of a simulated multivariable polymerization reactor using second-order Volterra models (1996) Automatica, 32 (9), pp. 1285-1301. , DOI 10.1016/0005-1098(96)00086-6, PII S0005109896000866Masnadi-Shirazi Mohammad, A., Ahmed, N., Optimum Laguerre networks for a class of discrete-time systems (1991) IEEE Transactions on Signal Processing, 39 (9), pp. 2104-2108. , DOI 10.1109/78.134447Milanese Mario, Belforte Gustavo, Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: Linear families of models and estimators (1982) IEEE Transactions on Automatic Control, AC-27 (2), pp. 408-414Mo, S.H., Norton, J.P., Fast and robust algorithm to compute exact polytope parameter bounds (1990) Mathematics and Computers in Simulation, 32 (5-6), pp. 481-493. , DOI 10.1016/0378-4754(90)90004-3Moreira, V.D., (2006) Robust Model Based Predictive Control of Uncertain Hybrid Systems Integrating Constraints, Logic, and Dynamics Based on Orthonormal Series Function, , PhD thesis, School of Electrical and Computer Engineering of the State University of Campinas (FEEC/UNICAMP), Campinas-SP, Brazil, in PortugueseNelles, O., (2001) Nonlinear System Identification, , Springer- Verlag, London, UKNgia, L.S.H., Separable nonlinear least-squares methods for efficient off-line and on-line modeling of systems using Kautz and Laguerre filters (2001) IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 48 (6), pp. 562-579. , DOI 10.1109/82.943327, PII S1057713001075000Ninness, B., Gustafsson, F., Orthonormal bases for system identification (1995) Proc. 3rd European Control Conference, 1, pp. 13-18. , Rome, ItalyNinness, B., Gustafsson, F., A unifying construction of orthonormal bases for system identification (1997) IEEE Transactions on Automatic Control, 42 (4), pp. 515-521. , PII S0018928697028080Ninness, B., Hjalmarsson, H., Gustafsson, F., The fundamental role of general orthonormal bases in system identification (1999) IEEE Transactions on Automatic Control, 44 (7), pp. 1384-1406Silva Oliveira, E.T.A.M., Optimality conditions for truncated Laguerre networks (1994) IEEE Trans. Signal Processing, 42 (9), pp. 2528-2530Silva Oliveira, E.T.A.M., On the determination of the optimal pole position of Laguerre filters (1995) IEEE Trans. Signal Processing, 43 (9), pp. 2079-2087Silva Oliveira, E.T.A.M., Optimality conditions for truncated Kautz networks with two periodically repeating complex conjugate poles (1995) IEEE Trans. Automatic Control, 40 (2), pp. 342-346Oliveira, G.H.C., (1997) Predictive Control for Processes with Structured Uncertainty Based on Orthonormal Series Function, , PhD thesis, School of Electrical and Computer Engineering of the State University of Campinas (FEEC/UNICAMP), Campinas-SP, Brazil, in PortugueseOliveira, G.H.C., Amaral, W.C., Latawiec, K., CRHPC using Volterra models and orthonormal basis functions: an application to CSTR plants (2003) Proc. IEEE Conference on Control Applications, pp. 718-723. , Istanbul, TurkeyOliveira, G.H.C., Amaral, W.C., Favier, G., Dumont, G., Constrained robust predictive controller for uncertain processes modeled by orthonormal series functions (2000) Automatica, 36 (4), pp. 563-572Oliveira, G.H.C., Da Rosa, A., Campello, R.J.G.B., Machado, J.M., Amaral, W.C., An introduction to models based on Laguerre, Kautz and other related orthonormal functions - part II: Nonlinear models International Journal of Modelling, Identification and Control, , forthcomingOliveira, G.H.C., Favier, G., Dumont, G., Amaral, W.C., Uncertainties identification using Laguerre models with application to paper machine headbox (1998) Proc. of IEEE/IMACS Multiconference on Computational Engineering in Systems Applications - CESA, Symposium on Control Optimization and Supervision, 1, pp. 329-334. , TunisiaPatwardhan, S.C., Shah, S.L., From data to diagnosis and control using generalized orthonormal basis filters. Part I: Development of state observers (2005) Journal of Process Control, 15 (7), pp. 819-835. , DOI 10.1016/j.jprocont.2004.08.006, PII S0959152405000211Schetzen, M., (1980) The Volterra and Wiener Theories of Nonlinear Systems, , Krieger Publishing Company, Malabar, Florida, USASjöberg, J., Zhang, Q., Ljung, L., Benveniste, A., Delyon, B., Glorennec, P.-Y., Hjalmarsson, H., Juditsky, A., Nonlinear black-box modeling in system identification: A unified overview (1995) Automatica, 31 (12), pp. 1691-1724Soeterboek, R., (1992) Predictive Control: A Unified Approach, , Prentice Hall, Upper Saddle River, NJ, USATakenaka, S., On the orthogonal functions and a new formula of interpolation (1925) Japanese Journal of Mathematics, 2, pp. 129-145Tanguy, N., Morvan, R., Vilbe, P., Calvez, L.C., Online optimization of the time scale in adaptive Laguerre-based filters (2000) IEEE Transactions on Signal Processing, 48 (4), pp. 1184-1187. , DOI 10.1109/78.827551Tanguy, N., Morvan, R., Vilbe, P., Calvez, L.-C., Pertinent choice of parameters for discrete Kautz approximation (2002) IEEE Transactions on Automatic Control, 47 (5), pp. 783-787. , DOI 10.1109/TAC.2002.1000273, PII S0018928602047633Tanguy, N., Vilbé, P., Calvez, L.C., Optimum choice of free parameter in orthonormal approximations (1995) IEEE Trans. on Automatic Control, 40 (10), pp. 1811-1813Van Den Bosch, P.P.J., Van Der Klauw, A.C., (1994) Modeling Identification and Simulation of Dynamical System, , CRC Press Inc., Boca Raton, Florida, USAVan Den Hof, P.M.J., Heuberger, P.S.C., Bokor, J., System identification with generalized orthonormal basis functions (1995) Automatica, 31 (12), pp. 1821-1834Wahlberg, B., System identification using Laguerre models (1991) IEEE Trans. on Automatic Control, 36 (5), pp. 551-562Wahlberg, B., System identification using Kautz models (1994) IEEE Trans. on Automatic Control, 39 (6), pp. 1276-1282Wahlberg, B., Ljung, L., Hard frequency-domain model error bounds from least-squares like identification techniques (1992) IEEE Trans. on Automatic Control, 37 (7), pp. 900-912Wahlberg, B., Makila, P.M., On approximation of stable linear dynamical systems using Laguerre and Kautz functions (1996) Automatica, 32 (5), pp. 693-708. , DOI 10.1016/0005-1098(95)00198-0Walter, E., Piet-Lahanier, H., Estimation of parameter bounds from bounded-error data: A survey (1990) Mathematics and Computers in Simulation, 32 (5-6), pp. 449-468Wiener, N., (1958) Nonlinear Problems in Random, , Theory, MIT Press, Cambridge, MA, USAZervos, C.C., Dumont, G.A., Deterministic adaptive control based on Laguerre series representation (1988) Int. J. Control, 48 (6), pp. 2333-2359Zhu, Y., System identification for process control: Recent experience and outlook (2006) International Journal of Modelling, Identification and Control, 6 (2), pp. 89-103Ziaei, K., Wang, D.W.L., Application of orthonormal basis functions for identification of flexible-link manipulators (2006) Control Engineering Practice, 14 (2), pp. 99-106. , DOI 10.1016/j.conengprac.2004.11.020, PII S096706610500036