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The most powerful unfalsified model for data with missing values. (English) Zbl 1342.93037

Summary: The notion of the most powerful unfalsified model plays a key role in system identification. Since its introduction in the mid 80s, many methods have been developed for its numerical computation. All currently existing methods, however, assume that the given data is a complete trajectory of the system. Motivated by the practical issues of data corruption due to failing sensors, transmission lines, or storage devices, we study the problem of computing the most powerful unfalsified model from data with missing values. We do not make assumptions about the nature or pattern of the missing values apart from the basic one that they are a part of a trajectory of a linear time-invariant system. The identification problem with missing data is equivalent to a Hankel structured low-rank matrix completion problem. The method proposed selects rank deficient complete submatrices of the incomplete Hankel matrix. Under specified conditions the kernels of the submatrices form a nonminimal kernel representation of the data generating system. The final step of the algorithm is reduction of the nonminimal kernel representation to a minimal one. Apart from its practical relevance in identification, missing data is a useful concept in systems and control. Classic problems, such as simulation, filtering, and tracking control can be viewed as missing data estimation problems for a given system. The corresponding identification problems with missing data are “data-driven” equivalents of the classical simulation, filtering, and tracking control problems.

MSC:

93B30 System identification
93C05 Linear systems in control theory
93B51 Design techniques (robust design, computer-aided design, etc.)
93C41 Control/observation systems with incomplete information

Software:

IDENT; CVX
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References:

[1] Willems, J. C., From time series to linear system—Part I. Finite dimensional linear time invariant systems, Automatica, 22, 5, 561-580 (1986) · Zbl 0604.62090
[2] Willems, J. C., From time series to linear system—Part II. Exact modelling, Automatica, 22, 6, 675-694 (1986) · Zbl 0628.62088
[3] Willems, J. C., From time series to linear system—Part III. Approximate modelling, Automatica, 23, 1, 87-115 (1987) · Zbl 0628.62089
[4] Van Overschee, P.; De Moor, B., Subspace Identification for Linear Systems: Theory, Implementation, Applications (1996), Kluwer: Kluwer Boston · Zbl 0888.93001
[5] Markovsky, I.; Willems, J. C.; Van Huffel, S.; De Moor, B., Exact and Approximate Modeling of Linear Systems: A Behavioral Approach (2006), SIAM, March · Zbl 1116.93002
[6] Verhaegen, M.; Dewilde, P., Subspace model identification, Part 1: The output-error state-space model identification class of algorithms, Internat. J. Control, 56, 1187-1210 (1992) · Zbl 0772.93020
[7] Chou, C.; Verhaegen, M., Subspace algorithms for the identification of multivariate errors-in-variables models, Automatica, 33, 10, 1857-1869 (1997) · Zbl 0889.93011
[8] Chiuso, A., Asymptotic variance of closed-loop subspace identification methods, IEEE Trans. Automat. Control, 51, 8, 1299-1314 (2006) · Zbl 1366.93672
[9] Goethals, I.; Van Gestel, T.; Suykens, J.; Van Dooren, P.; De Moor, B., Identification of positive real models in subspace identification by using regularization, IEEE Trans. Automat. Control, 48, 1843-1847 (2003) · Zbl 1364.93828
[10] Rapisarda, P.; Trentelman, H., Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data, Automatica, 47, 8, 1721-1728 (2011) · Zbl 1226.93048
[12] Kalman, R. E., On partial realizations, transfer functions, and canonical forms, Acta Polytech. Scand., 31, 9-32 (1979) · Zbl 0424.93020
[13] Kalman, R. E.; Falb, P. L.; Arbib, M. A., Topics in Mathematical System Theory (1969), McGraw-Hill · Zbl 0231.49001
[14] Schoukens, J.; Vandersteen, G.; Rolain, Y.; Pintelon, R., Frequency response function measurements using concatenated subrecords with arbitrary length, IEEE Trans. Instrum. Meas., 61, 10, 2682-2688 (2012)
[15] Söderström, T.; Stoica, P., System Identification (1989), Prentice Hall · Zbl 0714.93056
[16] Ljung, L., System Identification: Theory for the User (1999), Prentice-Hall: Prentice-Hall Upper Saddle River, NJ
[17] Markovsky, I., Structured low-rank approximation and its applications, Automatica, 44, 4, 891-909 (2008) · Zbl 1283.93061
[18] Markovsky, I., Low Rank Approximation: Algorithms, Implementation, Applications (2012), Springer · Zbl 1245.93005
[19] Markovsky, I., Recent progress on variable projection methods for structured low-rank approximation, Signal Process., 96PB, 406-419 (2014)
[20] Wallin, R.; Hansson, A., Maximum likelihood estimation of linear SISO models subject to missing output data and missing input data, Internat. J. Control, 87, 11, 2354-2364 (2014) · Zbl 1308.93234
[21] Usevich, K.; Markovsky, I., Variable projection for affinely structured low-rank approximation in weighted 2-norms, J. Comput. Appl. Math., 272, 430-448 (2014) · Zbl 1294.65063
[22] Markovsky, I.; Usevich, K., Structured low-rank approximation with missing data, SIAM J. Matrix Anal. Appl., 814-830 (2013) · Zbl 1271.93017
[23] Markovsky, I., A software package for system identification in the behavioral setting, Control Eng. Pract., 21, 1422-1436 (2013)
[24] Fazel, M., Matrix rank minimization with applications (2002), Elec. Eng. Dept., Stanford University, (Ph.D. thesis)
[25] Liu, Z.; Vandenberghe, L., Interior-point method for nuclear norm approximation with application to system identification, SIAM J. Matrix Anal. Appl., 31, 3, 1235-1256 (2009) · Zbl 1201.90151
[26] Willems, J. C., Fitting data sequences to linear systems, (Byrnes, C.; Datta, B.; Gilliam, D.; Martin, C., Progress in Systems and Control (1996), Birkhäuser), 405-416 · Zbl 0864.93061
[27] Sain, M.; Massey, J., Invertibility of linear time-invariant dynamical systems, IEEE Trans. Automat. Control, 14, 141-149 (1969)
[28] Willems, J. C., Paradigms and puzzles in the theory of dynamical systems, IEEE Trans. Automat. Control, 36, 3, 259-294 (1991) · Zbl 0737.93004
[29] Usevich, K.; Comon, P., Quasi-Hankel Low-rank Matrix Completion: A Convex Relaxation (2015)
[30] Boyd, S.; Vandenberghe, L., Convex Optimization (2001)
[31] Fazel, M.; Pong, T. K.; Sun, D.; Tseng, P., Hankel matrix rank minimization with applications in system identification and realization, SIAM J. Matrix Anal. Appl., 34, 3, 946-977 (2013) · Zbl 1302.90127
[33] Willems, J. C.; Rapisarda, P.; Markovsky, I.; De Moor, B., A note on persistency of excitation, Systems Control Lett., 54, 4, 325-329 (2005) · Zbl 1129.93362
[34] Polderman, J.; Willems, J. C., Introduction to Mathematical Systems Theory (1998), Springer-Verlag: Springer-Verlag New York
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