Abstract
In this paper, we develop a linear programming approach to the synthesis of stabilizing fixed structure controllers for a class of linear time invariant discrete-time systems. The stabilization of this class of systems requires the determination of a real controller parameter vector (or simply, a controller), \(K\), so that a family of real polynomials, affine in the parameters of the controllers, is Schur. An attractive feature of the paper is the systematic approximation of the set of all such stabilizing controllers, \(K\). This approximation is accomplished through the exploitation of the interlacing property of Schur polynomials and a systematic construction of sets of linear inequalities in \(K\). The union of the feasible sets of linear inequalities provides an approximation of the set of all controllers, \(K\), which render \(P(z,K)\) Schur. Illustrative examples are provided to show the applicability of the proposed methodology. We also show a related result, namely, that the set of rational proper stabilizing controllers for single-input single-output linear time invariant discrete-time plants will form a bounded set in the controller parameter space if and only if the order of the stabilizing cannot be reduced any further. Moreover, if the order of the controller is increased, the set of higher order controllers will necessarily be unbounded.
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Ackermann J (1993) Robust control systems with uncertain physical parameters. Springer, Berlin
Bengtsson G, Lindahl S (1974) A design scheme for incomplete state or output feedback with applications to boiler and power system control. Automatica 10:15–30
Bernstein D (1992) Some open problems in matrix theory arising in linear systems and control. Linear Algebra Appl 162–164:409–432
Bhattacharyya SP, Keel LH, Howze J (1988) Stabilizability conditions using linear programming. IEEE Trans Automat Contr 33:460–463
Blondel V, Gevers M, Lindquist A (1995) Survey on the state of systems and control. European J Contr 1:5–23
Brasch FM, Pearson JB (1970) Pole placement using dynamic compensator. IEEE Trans Automat Contr AC-15:34–43
Buckley A (1995) Hubble telescope pointing control system design improvement study. J Guid Control Dyn 18:194–199
Datta A, Ho MT, Bhattacharyya SP (2000) Structure and synthesis of PID controllers. Springer, London
El Ghaoui L, Oustry F, AitRami M (1997) A cone complementarity linearization algorithm for static output feedback and related problems. IEEE Trans Automat Contr 42–48:1171–1176
Goodwin GC, Graebe SF, Salgado ME (2001) Control system design. Prentice-Hall, Upper Saddle River
Henrion D, Sebek M, Kucera V (2003) Positive polynomials and robust stabilization with fixed-order controllers. IEEE Trans Automat Contr 48(7):1178–1186
Iwasaki T, Skelton RE (1995) The XY-centering algorithm for the dual LMI problem: a new approach to fixed order control design. Int J Contr 62(6):1257–1272
Keel LH, Rego JI, Bhattacharyya SP (2003) A new approach to digital PID controller design. IEEE Trans Automat Contr 48(4):687–692
Malik WA, Darbha S, Bhattacharya SP (2004) On the synthesis of fixed structure controllers satisfying given performance criteria. In: 2nd IFAC Symposium on System, Structure and Control
Malik WA, Darbha S, Bhattacharyya SP (2007a) On the boundedness of the set of stabilizing controllers. Int J Robust Nonlinear Contr, page in print
Malik WA, Darbha S, Bhattacharyya SP (2007b) A linear programming approach to the synthesis of fixed structure controllers. IEEE Trans Automat Contr, page in print
Polya G, Szego G (1998) Problems and theorems in analysis II—theory of functions, zeros, polynomials, determinants, number theory, geometry. Springer, Berlin
Syrmos VL, Abdullah CT, Dorato P, Grigoriadis K (1997) Static output feedback—a survey. Automatica 33(2):125–137
Zhu G, Grigoriadis K, Skelton R (1995) Covariance control design for the Hubble space telescope. J Guid Control Dyn 18(2):230–236
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Malik, W.A., Darbha, S., Bhattacharyya, S.P. (2011). Synthesis of Fixed Structure Controllers for Discrete Time Systems. In: Van Dooren, P., Bhattacharyya, S., Chan, R., Olshevsky, V., Routray, A. (eds) Numerical Linear Algebra in Signals, Systems and Control. Lecture Notes in Electrical Engineering, vol 80. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0602-6_17
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DOI: https://doi.org/10.1007/978-94-007-0602-6_17
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