A Jordan canonical form for reachable linear systems

Abstract We construct a new canonical form for reachable matrix pairs (A, B) under the similarity action. The canonical pairs are explicitly characterized by the property that A is in Jordan canonical form and B is in a novel block echelon form whose block sizes are determined by the Jordan structure of A. The reachable pairs are classified by lists of indices, and it is shown that the canonical form is continuous on the Jordan strata consisting of all reachable pairs with the same index list. The Jordan strata are characterized topologically, and explicit dimension formulas are derived. Finally we indicate some open problems related to the Jordan decomposition of the space of reachable systems.

[1]  G. Birkhoff,et al.  A survey of modern algebra , 1942 .

[2]  Okko H. Bosgra,et al.  Input-output invariants for linear multivariable systems , 1980 .

[3]  D. Hinrichsen,et al.  Generalized Hermite Matrices and Complete Invariants of Strict System Equivalence , 1983 .

[4]  M. Denham.  Canonical forms for the identification of multivariable linear systems , 1974 .

[5]  V. Popov.  Invariant Description of Linear, Time-Invariant Controllable Systems , 1972 .

[6]  Tosio Kato.  A Short Introduction to Perturbation Theory for Linear Operators , 1982 .

[7]  Diederich Hinrichsen,et al.  A wild quiver in linear systems theory , 1987 .

[8]  D. Prätzel-Wolters,et al.  Canonical forms for linear systems , 1983 .

[9]  Jordan echelon forms for linear state and input transformations , 1986, 1986 25th IEEE Conference on Decision and Control.

[10]  A. Tannenbaum.  Invariance and System Theory: Algebraic and Geometric Aspects , 1981 .

[11]  U. Helmke,et al.  Canonical Forms and Orbit Spaces of Linear Systems , 1986 .

[12]  A Jordan control canonical form for singular systems , 1988 .

[13]  C. Desoer,et al.  A proof of controllability of Jordan form state equations , 1968 .

[14]  P. Lancaster,et al.  Invariant subspaces of matrices with applications , 1986 .

[15]  J. Ackermann.  Parameter space design of robust control systems , 1980 .

[16]  Petre Stoica,et al.  Decentralized Control , 2018, The Control Systems Handbook.

[17]  M. Heymann.  A unique canonical form for muItivariable linear systems , 1970 .

[18]  U. Helmke.  The topology of a moduli space for linear dynamical systems , 1985 .

[19]  C. Byrnes,et al.  Decidability criteria for the similarity problem, with applications to the moduli of linear dynamical systems , 1977 .

[20]  Thomas Kailath,et al.  Linear Systems , 1980 .

[21]  R. Guidorzi.  Canonical structures in the identification of multivariable systems , 1975, Autom..