# zbMATH — the first resource for mathematics

Controller reduction via stable factorization and balancing. (English) Zbl 0604.93020
A procedure is given to simplify the controller obtained by the LQG method. It is based on a certain right coprime factorization $$K(s)=N(s)D^{-1}(s)$$ of the optimal controller transfer function. Using a balancing technique by Moore the authors approximate the pair D(s), N(s) by a low-order pair $$D_ 1(s)$$, $$N_ 1(s)$$ so that $$\| N(j\omega)-N_ 1(j\omega)\|_{L^{\infty}}$$, $$\| D(j\omega)-D_ 1(j\omega)\|_{L^{\infty}}$$ are sufficiently small. It is shown that the approximation $$K_ 1(s)=N_ 1(s)D_ 1^{-1}(s)$$ gives a good accuracy of the approximation of the closed-loop behavior. A numerical example for a SISO system of 8th order is described in details.
Reviewer: A.Pervozvanskij

##### MSC:
 93B50 Synthesis problems 15A23 Factorization of matrices 65F35 Numerical computation of matrix norms, conditioning, scaling 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 93C05 Linear systems in control theory 93C35 Multivariable systems, multidimensional control systems
Full Text:
##### References:
 [1] AIMMJAN V. M., Math. US.S.R. Sb 15 (1971) [2] ANOI-RSON B. D. O., Proc. IFAC Workshop on Model Error Concepts and Compensation (1985) [3] ANOIRSON B. D. O, Linear Optimal Control (1971) [4] DOI: 10.1016/S0167-6911(84)80055-9 · Zbl 0536.93070 [5] ENNS , D. , 1984 , Ph.D. thesis , Dept of Electrical Engineering, Stanford University . [6] DOI: 10.1080/00207178408933239 · Zbl 0543.93036 [7] GLOVER , K. , and LIMEBEER , D. J. N. , 1983 ,Proc. Amer. Contr. Conf. San Francisco , CA , 644 . [8] JONCKHEERE E. A., Internal. Symp. Theory Networks and Systems (1981) [9] DOI: 10.1109/TAC.1981.1102736 · Zbl 0553.93038 [10] KWAKERNAAK H., Linear Optimal Control Systems (1972) · Zbl 0276.93001 [11] DOI: 10.1016/0167-6911(85)90014-3 · Zbl 0559.93036 [12] DOI: 10.1109/TAC.1981.1102568 · Zbl 0464.93022 [13] DOI: 10.1109/TAC.1984.1103674 · Zbl 0542.93014 [14] DOI: 10.1109/TAC.1982.1102945 · Zbl 0482.93024 [15] VERRIEST E. I., Proc. 24th Symp. on Circuits and Systems 365 (1981) [16] DOI: 10.1016/0167-6911(85)90066-0 · Zbl 0564.93059 [17] DOI: 10.1109/TAC.1984.1103497 · Zbl 0535.93014
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.