Solving of a Minimal Realization Problem in Maple

In the computer algebra system Maple, we have created a package MinimalRealization to solve the minimal realization problem for a discrete-time linear time-invariant system. The package enables to construct the minimal realization of a system starting with either a finite sequence of Markov parameters of a system, or a transfer function, or any non-minimal realization. It is designed as a user library and consists of 11 procedures: ApproxEssPoly, ApproxNullSpace, Approxrank, ExactEssPoly, FractionalFactorizationG, FractionalFactorizationMP, MarkovParameters, MinimalityTest, MinimalRealizationG, MinimalRealizationMP, Realization2MinimalRealization. The realization algorithm is based on solving of sequential problems: (1) determination of indices and essential polynimials (procedures ExactEssPoly, ApproxEssPoly), (2) construction of a right fractional factorization of the transfer function (FractionalFactorizationG, FractionalFactorizationMP), (3) construction of the minimal realization by the given fractional factorization (MinimalRealizationG, MinimalRealizationMP, Realization2MinimalRealization). We can solve the problem both in the case of exact calculations (in rational arithmetic) and in the presence of rounding errors, or for input data which are disturbed by noise. In the latter case the problem is ill-posed because it requires finding the rank and the null space of a matrix. We use the singular value decomposition as the most accurate method for calculation of the numerical rank (Approxrank) and the numerical null space (ApproxNullSpace). Numerical experiments with the package MinimalRealization demonstrate good agreement between the exact and approximate solutions of the problem.

Keywords

discrete-time linear time-invariant systems; fractional factorization; minimal realization; algorithms for solving of realization problem.

References

1. Adukov V.M. On a Problem of Minimal Realization. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming & Computer Software, 2013, vol. 6, no. 3, pp. 5-17. (in Russian)
2. Adukov V.M. Generalized Inversion of Block Toeplitz Matrices. Linear Algebra and Its Applications, 1998, vol. 274, pp. 85-124. DOI: 10.1016/S0024-3795(97)00304-2
3. De Schutter B. Minimal State-Space Realization in Linear System Theory: An Overview. Journal of Computational and Applied Mathematics, Special Issue on Numerical Analysis in the 20th Century. Vol. I: Approximation Theory, 2000, vol. 121, no. 1-2, pp. 331-354.
4. Pushkov S.G., Krivoshapko S.Y. On the Problem of Realization in the State-Space Model for Interval Dynamic Systems. Computing Technologies, 2004, vol. 9, no. 1, pp. 75-85. (in Russian)
5. Sinha N.K. Minimal Realization of Transfer Function Matrices. A Comparative Study of Different Methods. International Journal of Control, 1975, vol. 22, no. 5, pp. 627-639. DOI: 10.1080/00207177508922109
6. Kailath T. Linear Systems. N.J., Prentice-Hall, Inc., Englewood Cliffs, 1980.
7. Foster L.V., Davis T.A. Algorithm 933: Reliable Calculation of Numerical Rank, Null Space Bases, Pseudoinverse Solutions, and Basic Solutions Using SuitesparseQR. ACM Transactions on Mathematical Software (TOMS), 2013, vol. 40, issue 1, Article no. 7 (23 pages).
8. Adukov V.M. Fractional and Wiener-Hopf Factorizations. Linear Algebra and Its Applications, 2002, vol. 340, no. 1-3, pp. 199-213.
9. Adukov V.M. Generalized Inversion of Finite Rank Toeplitz and Hankel Operators with Rational Matrix Symbols. Linear Algebra and Its Applications, 1999, vol. 290, pp. 119-134. DOI: 10.1016/S0024-3795(98)10216-1
10. Tether A.J. Construction of Minimal Linear State-Variable Models from Finite Input-Output Data. IEEE Transactions on Automatic Control, 1970, vol. AC-15, no. 4, pp. 427-436. DOI: 10.1109/TAC.1970.1099514