×

zbMATH — the first resource for mathematics

Set-membership errors-in-variables identification of MIMO linear systems. (English) Zbl 1388.93031
Summary: In this paper, we consider the problem of set-membership identification of Multiple-Input Multiple-Output (MIMO) linear models when both input and output measurements are affected by bounded additive noise. Firstly, we propose a general formulation that allows the user to take into account possible a-priori information on the structure of the MIMO model to be identified. Then, we formulate the problem in terms of a suitable polynomial optimization problem that is solved by means of a convex relaxation approach. To show the effectiveness of the proposed approach, we test the original MIMO identification algorithm on a simulation example, as well as on a set of input-output experimental data, collected on a multiple-input multiple-output electronic process simulator.

MSC:
93B30 System identification
93C35 Multivariable systems, multidimensional control systems
93C05 Linear systems in control theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Åström, K.; Hagander, P.; Sternby, J., Zeros of sampled systems, Automatica, 20, 1, 31-38, (1984) · Zbl 0542.93047
[2] Casini, M.; Garulli, A.; Vicino, A., Input design in worst-case system identification with quantized measurements, Automatica, 48, 12, 2997-3007, (2012) · Zbl 1255.93045
[3] Castaldi, P., Diversi, R., Guidorzi, R., & Soverini, U. (1999). Identification of multivariable errors-in-variables models. In 1999 European control conference.
[4] Cerone, V., Feasible parameter set for linear models with bounded errors in all variables, Automatica, 29, 6, 1551-1555, (1993) · Zbl 0800.93237
[5] Cerone, V., Parameter bounds for ARMAX models from records with bounded errors in variables, International Journal of Control, 57, 1, 225-235, (1993) · Zbl 0774.93021
[6] Cerone, V.; Lasserre, J. B.; Piga, D.; Regruto, D., A unified framework for solving a general class of conditional and robust set-membership estimation problems, IEEE Transactions on Automatic Control, 59, 11, 2897-2909, (2014) · Zbl 1360.93330
[7] Cerone, V.; Piga, D.; Regruto, D., Enforcing stability constraints in set-membership identification of linear dynamic systems, Automatica, 47, 11, 2488-2494, (2011) · Zbl 1228.93038
[8] Cerone, V.; Piga, D.; Regruto, D., Improved parameter bounds for set-membership EIV problems, International Journal of Adaptive Control and Signal Processing, 25, 3, 208-227, (2011) · Zbl 1222.93054
[9] Cerone, V.; Piga, D.; Regruto, D., Bounded error identification of Hammerstein systems through sparse polynomial optimization, Automatica, 48, 10, 2693-2698, (2012) · Zbl 1271.93161
[10] Cerone, V.; Piga, D.; Regruto, D., Set-membership error-in-variables identification through convex relaxation techniques, IEEE Transactions on Automatic Control, 57, 2, 517-522, (2012) · Zbl 1369.93670
[11] Cerone, V.; Piga, D.; Regruto, D., Bounding the parameters of block-structured nonlinear feedback systems, International Journal of Robust and Nonlinear Control, 23, 1, 33-47, (2013) · Zbl 1263.93059
[12] Cerone, V.; Piga, D.; Regruto, D., A convex relaxation approach to set-membership identification of LPV systems, Automatica, 49, 9, 2853-2859, (2013) · Zbl 1364.93824
[13] Cerone, V.; Piga, D.; Regruto, D., Fixed-order FIR approximation of linear systems from quantized input and output data, Systems & Control Letters, 62, 12, 1136-1142, (2013) · Zbl 1282.93272
[14] Cerone, V.; Regruto, D., Parameter bounds evaluation of Wiener models with noninvertible polynomial nonlinearities, Automatica, 42, 10, 1775-1781, (2006) · Zbl 1114.93096
[15] Cerone, V., & Regruto, D. (2008). Set-membership identification of LPV models with uncertain measurements of the time-varying parameter. In 2008 47th IEEE conference on decision and control (pp. 4491-4496).
[16] Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., Solving quadratic distance problems: an LMI-based approach, IEEE Transactions on Automatic Control, 48, 2, 200-212, (2003) · Zbl 1364.90240
[17] Chisci, L.; Garulli, A.; Vicino, A.; Zappa, G., Block recursive parallelotopic bounding in set membership identification, Automatica, 34, 1, 15-22, (1998) · Zbl 0908.93021
[18] Chiuso, A., The role of vector autoregressive modeling in predictor-based subspace identification, Automatica, 43, 6, 1034-1048, (2007) · Zbl 1282.93264
[19] Diversi, R., & Guidorzi, R. (2012). A covariance-matching criterion in the Frisch scheme identification of MIMO EIV models. In Proc. of 16th IFAC symposium on system identification (SYSID) (pp. 1647-1652). Brussels, Belgium.
[20] Fogel, E.; Huang, Y., On the value of information in system identification-bounded noise case, Automatica, 18, 2, 229-238, (1982) · Zbl 0433.93062
[21] Garulli, A., Tight error bounds for projection algorithms in conditional set membership estimation, Systems & Control Letters, 37, 5, 293-300, (1999) · Zbl 0948.93018
[22] Garulli, A.; Vicino, A.; Zappa, G., Conditional central algorithms for worst case set-membership identification and filtering, IEEE Transactions on Automatic Control, 45, 1, 14-23, (2000) · Zbl 0971.93072
[23] Gibson, S.; Ninness, B., Robust maximum-likelihood estimation of multivariable dynamic systems, Automatica, 41, 10, 1667-1682, (2005) · Zbl 1087.93054
[24] Green, M.; Anderson, B., Identificationi of multivariable errors in variable models with dynamics, IEEE Transactions on Automatic Control, 31, 5, 467-471, (1986) · Zbl 0607.93061
[25] Kojima, M.; Kim, S.; Waki, H., Sparsity in sums of squares of polynomials, Mathematical Programming, 103, 1, 45-62, (2005) · Zbl 1079.90092
[26] Larimore, W. (1983). System identification, reduced-order filtering and modeling via canonical variate analysis. In Proceedings of the 1983 American control conference (pp. 445-451).
[27] Lasserre, J. B., Global optimization with polynomials and the problem of moments, SIAM Journal on Optimization, 11, 3, 796-817, (2001) · Zbl 1010.90061
[28] Lasserre, J. B., Convergent SDP-relaxations in polynomial optimization with sparsity, SIAM Journal on Optimization, 17, 3, 822-843, (2006) · Zbl 1119.90036
[29] Lasserre, J., (Moments, positive polynomials and their applications, Optimization series, vol. 1, (2010), Imperial College Press)
[30] Marshall, M., Representation of non-negative polynomials, degree bounds and applications to optimization, Canadian Journal of Mathematics, 61, 205-221, (2009) · Zbl 1163.13019
[31] Milanese, M.; Belforte, G., Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: linear families of models and estimators, IEEE Transactions on Automatic Control, 27, 2, 408-414, (1982) · Zbl 0479.93060
[32] (Milanese, M.; Norton, J.; Piet-Lahanier, H.; Walter, E., Bounding approaches to system identification, (1996), Springer US) · Zbl 0845.00024
[33] Milanese, M.; Novara, C., Set membership identification of nonlinear systems, Automatica, 40, 6, 957-975, (2004) · Zbl 1109.93020
[34] Milanese, M.; Taragna, M., H-infinity set membership identification: A survey, Automatica, 41, 12, 2019-2032, (2005) · Zbl 1090.93010
[35] Milanese, M.; Vicino, A., Optimal estimation theory for dynamic systems with set membership uncertainty, Automatica, 27, 6, 997-1009, (1991) · Zbl 0737.62088
[36] Mu, B.-Q.; Chen, H.-F., Recursive identification of multi-input multi output errors-in-variables Hammerstein systems, IEEE Transactions on Automatic Control, 60, 3, 843-849, (2015) · Zbl 1360.93727
[37] Nie, J., Optimality conditions and finite convergence of lasserre’s hierarchy, Mathematical Programming, 146, 1, 97-121, (2014) · Zbl 1300.65041
[38] Norton, J. P., Special issue on bounded-error estimation, International Journal of Adaptive Control and Signal Processing, 8, 1, (1994) · Zbl 0825.93057
[39] Norton, J. P., Special issue on bounded-error estimation, International Journal of Adaptive Control and Signal Processing, 9, 1, (1995)
[40] Parrilo, P. A., Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming, 96, 2, 293-320, (2003) · Zbl 1043.14018
[41] Pouliquen, M., Pigeon, E., & Gehan, O. (2011). Output error identification for multi-input multi-output systems with bounded disturbances. In 2011 50th IEEE conference on decision and control and European control conference (pp. 7200-7205).
[42] Schoukens, J., Suykens, J., & Ljung, L. (2009). Wiener-Hammerstein benchmark. In Proc. of 15th IFAC symposium on system identification (SYSID). Saint-Malo, France.
[43] Schweppe, F., Recursive state estimation: unknown but bounded errors and system inputs, IEEE Transactions on Automatic Control, 13, 1, 22-28, (1968)
[44] Söderström, T., Errors-in-variables methods in system identification, Automatica, 43, 6, 939-958, (2007) · Zbl 1193.93090
[45] Söderström, T., A generalized instrumental variable estimator for multivariable errors-in-variable, International Journal of Control, 85, 3, 287-303, (2012) · Zbl 1282.93248
[46] Stoica, P.; Jansson, M., MIMO system identification: state-space and subspace approximations versus transfer function and instrumental variables, IEEE Transactions on Signal Processing, 48, 11, 3087-3099, (2000) · Zbl 1006.93592
[47] Sturm, J. F., Using sedumi 1.02, a Matlab toolbox for optimization over symmetric cones, Optimization Methods & Software, 11, 1-4, 625-653, (1999) · Zbl 0973.90526
[48] van den Bosch, P. P.J.; van der Klauw, A. C., Modeling, identification and simulation of dynamical systems, (1994), CRC Press · Zbl 0897.93002
[49] Van Overschee, P.; De Moor, B., Subspace identification for linear systems: Theory implementation applications, (1996), Springer Science & Business Media · Zbl 0888.93001
[50] Verhaegen, M., Identification of the deterministic part of MIMO state space models given in innovations form from input-output data, Automatica, 30, 1, 61-74, (1994) · Zbl 0791.93054
[51] Viberg, M., Subspace-based methods for the identification of linear time-invariant systems, Automatica, 31, 12, 1835-1851, (1995) · Zbl 0846.93023
[52] Waki, H.; Kim, S.; Kojima, M.; Muramatsu, M.; Sugimoto, H., Algorithm 883: sparsepop—a sparse semidefinite programming relaxation of polynomial optimization problems, ACM Transactions on Mathematical Software, 35, 2, 15:1-15:13, (2008)
[53] Walter, E.; Piet-Lahanier, H., Estimation of parameter bounds from bounded-error data: a survey, Mathematics and Computers in Simulation, 32, 5, 449-468, (1990)
[54] Wang, X.; Yang, C.; Zhang, B.; Wang, L., Interval algorithm for set-membership identification of MIMO LTI system, 126-136, (2013), Springer Berlin Heidelberg Berlin, Heidelberg
[55] Wills, A.; Ninness, B., On gradient-based search for multivariable system estimates, IEEE Transactions on Automatic Control, 53, 1, 298-306, (2008) · Zbl 1367.93683
[56] Zaiser, S., Buchholz, M., & Dietmayer, K. (2014a). Interval system identification for MIMO ARX models of minimal order. In 53rd IEEE conference on decision and control (pp. 1774-1779).
[57] Zaiser, S., Buchholz, M., & Dietmayer, K. (2014b). MIMO order and state-space model identification from interval data. In Prof. of 2014 IEEE conference on control applications (CCA) (pp. 134-139).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.