Tableau methods for analysis and design of linear systems

-Certain classes of elementary operations on the system matrix of a linear system are defined, and then their use IS illustrated for the calculation of canonical forms, the largest (A,B)-mvarmnt and controllablhty subspaces in ker(C), transmission zeros, and minimal inverses Questions of numerical accuracy are discussed The results are directly codable as efficient numerical algorithms, and may be extended to the use of orthogonal t ransformatmns 1 I N T R O D U C T I O N A NUMBER of algebrmc design methods and theoretical results for linear systems can be slmphfied by studylrrg the operations of hnear algebra reqmred for their algorithmic implementation In this paper a tableau method for the analysis and design of tlme-mvarlant hnear systems will be described The paper might well be summarized as an appllcatton of the 'systemmatrix' methods of Rosenbrock to the abstract algebrmc results of Wonham and others Tableau methods have the following desirable properties (a) The concepts and operations reqmred in a number of algebraic design algorithms are unified (b) Because the internal structure of hnear systems is very easily put m evidence, certain abstract proofs become s~mple or even transparent when concretized using a tableau *Recetved 1 December 1977, rewsed 28 June 1978, revised 24 January 1979 The original version of this paper was presented at the 4th IFAC Symposium on Mult ivanable Technological Systems which was held m Fredencton, New Brunswick, Canada during July 1977 The pubhshed Proceedings of this IFAC Meeting may be ordered from Pergamon Press L td , Headlngton Hill Hall, Oxford OX3 0BW, England This paper was recommended for pubhcatlon in revised form by associate edttor E J Davison tPor t lons of this work were performed while the author was on leave at the Laboratolre d 'Automat lque et d 'Analyse des Systemes du CNRS, Toulouse, France This research was supported by the Natural Science and Engineering Research Council of Canada under grant A7380 :~Department of Electrical Engineering, Umvers~ty of Waterloo, Waterloo, Canada N2L 3G1 (c) The method results m efficient, easily programmable numerical algorithms using the elementary operations of linear algebra (d) Certain questions of genermty and numerical stability are quickly answered for given system &mensmns (e) The method is &rectly extendable to timevarying index-variant systems The usual n-state, m-mput, p-output hnear system ~ x = A x + B u (S1)