Canonical form for the matrices of linear discrete-time systems

Some applications of canonical matrices to linear continuous-time systems are reviewed. A canonical form is proposed for real nonderogatory convergent matrices, such as the A matrices which occur in the description of linear discrete-time dynamical systems by vector-matrix difference equations of the form xk+1 = Axk + Buk. The new canonical form is applied to the generation of particular and general solutions of the matrix equation ATLA − L = −K, which occurs in the application of Lyapunov theory to the analysis and design of such systems.

[1]  Comment: Solution of the Lyapunov matrix equation with a diagonal input matrix, obtained without matrix inversion , 1967 .

[2]  Second method of Liapunov and Routh's canonical form , 1963 .

[3]  H. Power.  Similarity transformations involving the schwarz and companion matrices , 1969 .

[4]  D. K. Cheng,et al.  Analysis of linear systems , 1960 .

[5]  P. Parks,et al.  Liapunov and the Schur-Cohn stability criterion , 1964 .

[6]  C. Storey,et al.  Some Applications of the Lyapunov Matrix Equation , 1968 .

[7]  General Performance Indices for the time and Frequency Response for the Free Motion of Linear Discrete Control Systems , 1965 .

[8]  W. Wonham,et al.  Optimal bang-bang control with quadratic performance index , 1964 .

[9]  R. Butchart.  An Explicit Solution to the Fokker-Planck Equation for an Ordinary Differential Equation† , 1965 .

[10]  P. Parks.  A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  H. Power,et al.  A note on the matrix equation A' LA-L = -K , 1969 .

[12]  R. E. Kalman,et al.  On the Hermite-Fujiwara theorem in stability theory , 1965 .

[13]  O. Taussky.  Matrices C with Cn → 0 , 1964 .

[14]  H. W Turnbull.  An introduction to the theory of canonical matrices, by H.W. Turnbull and A.C. Aitken , 1945 .

[15]  Henry M. Power,et al.  The Mechanics of the Bilinear Transformation , 1967 .

[16]  Henry M. Power,et al.  The companion matrix and Liapunov functions for linear multivariable time-invariant systems , 1967 .

[17]  J. Diamessis.  A new method for calculating system performance measures , 1964 .

[18]  H. L. Mason,et al.  The dynamics of automatic controls , 1948 .

[19]  Equivalence of Lyapunov matrix equations for continuous and discrete systems , 1967 .

[20]  N. N. Puri,et al.  Calculation of Quadratic Moments of High-Order Linear Systems via Routh Canonical Transformation , 1964, IEEE Transactions on Applications and Industry.