Abstract
The standard controllable and observable canonical realizations have A-matrices whose eigenvalue locations are highly sensitive to small perturbations to the nonzero entries of the A-matrix that correspond to the coefficients of the characteristic polynomial. On the other hand, the eigenvalue locations of tridiagonal A-matrices are relatively insensitive to similar perturbations. Given a set of known poles and zeros from which the transfer function of a SISO is computed, this paper details an approach to state space realizations with tridiagonal A-matrices using a genetic algorithm and the equivalence of the realization with the controllable canonical form. In addition to the insensitivity of the tridiagonal realization to perturbations, an additional genetic algorithm is used to reconfigure input and output matrices when nonzero gains in the B- and C-matrices are perturbed to zero, i.e., disconnections (shorts/opens) which account for over 70% of field failures. The reconfigured matrices achieve an equivalent realization, thereby maintaining the same input–output behavior. Reconfigurable systems are critical to maintaining exact input–output behavior (automatically) without being able to repair the disconnection.
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References
H. Bowdler, R.S. Martin, C. Rinsch, J.H. Wilkinson, The QR and QL algorithms for symmetric matrices. Numer. Math. 11, 293–306 (1968)
J.W. Brewer, Kronecker products and matrix calculus in system theory. IEEE Trans. Circuits Syst. CAS-25, 772–781 (1978)
R.A. DeCarlo, Linear Systems (Prentice Hall, Englewood Cliffs, 1989)
D.W. Faris, T.E. Paré, A. Packard, J.P. How, Controller fragility: what’s all the fuss? Proc. Allerton Conf. Commun. Control Comput. 36, 600–609 (1998)
Golub, Van Loan, Matrix Computations (Johns Hopkins University Press, USA, 1996)
C.S. Indulkar, K. Ramalingam, Estimation of transmission line parameters from measurements. Int. J. Electr. Power Energy Syst. 30, 337–342 (2008)
Y. Lee, S.D. Sudhoff, Energy Systems Analysis Consortium (ESAC) Genetic Optimization and Systems Engineering Tool, v. 2.0 Manual, School of Electrical and Computer Engr., Purdue Univ., West Lafayette, IN, 2004
D.G. Luenberger, Canonical forms for linear multivariate systems. IEEE Trans. Autom. Control 12, 290–293 (1967)
T. McKelvey, A. Helmersson, State-Space Parameterizations of Multivariable Linear Systems Using Tridiagonal Matrix Forms, in Proc. of 35th CDC, Kobe, Japan, 1996
B.N. Parlett, Reduction to tridiagonal form and minimal realizations. SIAM J. Matrix Anal. Appl. 13, 567–593 (1992)
W.J. Rugh, Linear System Theory (Prentice Hall, Upper Saddle River, 1996)
G. Strang, Linear Algebra and Its Application (Thompson, USA, 1988)
D. Turer, Robust and retunable state realizations of transfer functions with known poles and zeros. Thesis (MSECE) Purdue University (2005)
M.E. Van Valkenburg, Analog Filter Design (Oxford University Press, New York, 1982)
J.H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford University Press, London, 1965)
H. Zhang, Numerical condition of polynomials in different forms. Electr. Trans. Numer. Anal. 12, 66–87 (2001)
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Turer, D., DeCarlo, R. Robust and Retunable State Realizations of Transfer Functions with known Poles and Zeros. Circuits Syst Signal Process 29, 795–813 (2010). https://doi.org/10.1007/s00034-010-9185-5
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DOI: https://doi.org/10.1007/s00034-010-9185-5