Decoupling in the design and synthesis of singular systems. (English) Zbl 0587.93017

This paper presents necessary and sufficient conditions for decoupling of linear, time-invariant, singular systems. The control law applied is of constant-ratio proportional and derivative state feedback type. It is shown that such a control law presents an invertible transformation which transforms the singular closed loop system to a regular one. In the frequency domain this gives a duality property between regular and generalized transfer functions. Given a system that satisfies the necessary and sufficient conditions, the class of all feedback matrices which decouple the system is characterized. Finally, a synthesis technique is developed for the realization of desired closed loop pole- zero configurations.


93B50 Synthesis problems
34A99 General theory for ordinary differential equations
93C05 Linear systems in control theory
93B55 Pole and zero placement problems
93C35 Multivariable systems, multidimensional control systems
93C99 Model systems in control theory
Full Text: DOI


[1] Campbell, S.L., Singular systems of differential equations, (1980), Pitman London · Zbl 0419.34007
[2] Campbell, S.L.; Meyer, C.D.; Rose, N.J., Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM J. appl. math, 31, 411, (1976) · Zbl 0341.34001
[3] Christodoulou, M.A., Analysis and synthesis of singular systems, () · Zbl 0611.93010
[4] Cobb, D., Feedback and pole placement in descriptor variable systems, Int. J. control, 33, 1135, (1981) · Zbl 0464.93039
[5] Cobb, D., Descriptor variable systems and optimal state regulation, IEEE trans. aut. control, AC-28, 601, (1983) · Zbl 0522.93036
[6] Falb, P.L.; Wolovich, W.A., Decoupling in the design and synthesis of multivariable control systems, IEEE trans. aut. control, 12, 651, (1967)
[7] Hautus, M.L.J.; Heymann, M., Linear feedback decoupling-transfer function analysis, IEEE trans. aut. control, AC-28, 823, (1983) · Zbl 0523.93035
[8] Leontieff, W., Essays in economics, (1977), M. E. Sharpe New York
[9] Mufti, I.H., A note on the decoupling of multivariable systems, IEEE trans. aut. control, 14, 415, (1969)
[10] Mukundan, R.; Dayawansa, W., Feedback control of singular systems—proportional and derivative feedback of the state, Int. J. syst. sci., 14, 615, (1983) · Zbl 0509.34004
[11] Newcomb, R.W., The semistate description of nonlinear time-variable circuits, IEEE trans. circ. syst., 28, 62, (1981)
[12] Pandolfi, L., Controllability and stabilization for linear systems of algebraic and differential equations, J. opt. theor. appl., 33, 241, (1980) · Zbl 0397.93006
[13] Pugh, A.C., The mcmillan degree of a polynomial system matrix, Int. J. control, 24, 129, (1976) · Zbl 0329.93023
[14] Verghese, G.; Lévy, B.C.; Kailath, T., A generalized state-space for singular systems, IEEE trans. aut. control, 26, 811, (1981) · Zbl 0541.34040
[15] Wilde, R.R.; Kokotovic, P.C., Optimal open and closed loop control of singularity perturbed linear systems, IEEE trans. aut. control, 18, 616, (1973) · Zbl 0273.49053
[16] Wonham, W.M.; Morse, A.S., Decoupling and pole assignment in linear systems: a geometric approach, SIAM J. control, 8, 1, (1970) · Zbl 0206.16404
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.