Abstract
In the paper [CL1] the notion of a convex invertible cone,cic, of matrices was introduced and its geometry was studied. In that paper close connections were drawn between thiscic structure and the algebraic Lyapunov equation. In the present paper the same geometry is extended to triples of matrices andcics of minimal state space models are defined and explored. This structure is then used to study balancing, Hankel singular values, and simultaneous model order reduction for a set of systems. State spacecics are also examined in the context of the so-called matrix sign function algorithm commonly used to solve the algebraic Lyapunov and Riccati equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. A. Baltzer, Accelerated Convergence of the Matrix Sign Function Method of Solving Lyapunov, Riccati and Other Matrix Equations,Internat. J. Control,32 (1980), 1057–1078.
G. P. Barker, Common Solution to the Lyapunov Equations,Linear Algebra Appl.,16 (1977), 233–235.
C. Beck, J. Doyle, and K. Glover, Model Reduction of Multi-Dimensional and Uncertain Systems,IEEE Tram. Automat. Control,41 (1996), 1466–1477.
C.-T. Chen, A Generalization of the Inertia Theorem,SIAM J. Appl. Math.,25 (1973), 158–161.
N. Cohen and I. Lewkowicz, Convex Invertible Cones and the Lyapunov Equation,Linear Algebra Appl.,250 (1997), 105–131.
N. Cohen and I. Lewkowicz, Convex Invertible Cones of Rational Matrix Functions, preprint.
E. D. Denman and A. N. Beavers, Jr., The Matrix Sign Function and Computation in Systems,Appl. Math. Comput.,2 (1976), 63–94.
K. V. Fernando and H. Nicholson, Reciprocal Transformation in Balanced Model-Order Reduction,Proc. IEE-D,130 (1983), 359–362.
M. Fu and B. R. Barmish, Stability of Convex and Linear Combinations of Polynomials and Matrices Arising in Robustness Problems,Proceedings of the Conference on Information Sciences and Systems, 1987, pp. 16–21.
K. Glover, All Optimal Hankel-Norm Approximations of Linear Multivariable Systems and TheirL ∞ Error Bounds,Internat. J. Control,39 (1984), 115–1193.
M. Green and D. J. N. Limebeer,Linear Robust Control, Prentice-Hall, Englewood Cliffs, NJ, 1995.
U. Helmke, Balanced Realization for Linear Systems: A Variational Approach,SIAM J. Control Optim,31 (1993), 1–15.
R. A. Horn and C. R. Johnson,Matrix Analysis, Cambridge University Press, Cambridge, 1985.
R. A. Horn and C. R. Johnson,Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
C. R. Johnson and L. Rodman, Convex Sets of Hermitian Matrices with Constant Inertia,SIAM J. Algebraic Discrete Methods,6 (1985), 351–359.
C. Kenney and G. Hewer, Necessary and Sufficient Conditions for Balancing Unstable Systems,IEEE Trans. Automat. Control,32 (1987), 157–160.
P. Lancaster and L. Rodman,Algebraic Riccati Equations, Oxford University Press, Oxford, 1995.
M. G. Safonov and R. Y. Chiang, A Schur Method for Balanced-Truncation Model Reduction,IEEE Trans. Automat. Control,34 (1989), 729–733.
C. P. Therapos, Balancing Transformation for Unstable Non-Minimal Linear Systems,IEEE Trans. Automat. Control,34 (1989), 455–457.
H. K. Wimmer, Inertia Theorems for Matrices, Controllability and Linear Vibrations,Linear Algebra Appl,8 (1974), 337–343.
F. Wu, InducedL 2 Norm Model Reduction of Polytopic Linear Uncertain Systems,Automatica,32 (1996), 1417–1426.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cohen, N., Lewkowicz, I. Convex invertible cones of state space systems. Math. Control Signal Systems 10, 265–286 (1997). https://doi.org/10.1007/BF01211507
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01211507