Methods for finding solutions of the state equations of descriptor fractional discrete-time and continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform and the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples.
The positivity and linearization of a class of nonlinear continuous-time system by nonlinear state feedbacks are addressed. Necessary and sufficient conditions for the positivity of the class of nonlinear systems are established. A method for linearization of nonlinear systems by nonlinear state feedbacks is presented. It is shown that by a suitable choice of the state feedback it is possible to obtain an asymptotically stable and controllable linear system, and if the closed-loop system is positive then it is unstable.
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Two related problems, namely the problem of the infinite eigenvalue assignment and that of the solvability of polynomial matrix equations are considered. Necessary and sufficient conditions for the existence of solutions to both the problems are established. The relationships between the problems are discussed and some applications from the field of the perfect observer design for singular linear systems are presented.
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Notions of externally and internally positive singular discrete-time linear systems are introduced. It is shown that a singular discrete-time linear system is externally positive if and only if its impulse response matrix is non-negative. Sufficient conditions are established under which a single-output singular discrete-time system with matrices in canonical forms is internally positive. It is shown that if a singular system is weakly positive (all matrices E, A, B, C are non-negative), then it is not internally positive.
CONTENTS Introduction.......................................................................................................... 5 1. Multifunctions and selections............................................................................... 7 1. Multifunctions and selections.................................................................. 7 2. Continuous multifunctions and selections........................................... 9 3. Measurable multifunctions and selections............................................ 16 2. Multifunctions of two variables............................................................................... 19 4. Carathéodory multifunctions and selections......................................... 19 5. The Scorza Dragoni property..................................................................... 25 6. Implicit function theorems......................................................................... 32 3. The superposition operator................................................................................... 33 7. The superposition operator in the space S........................................... 34 8. The superposition operator in ideal spaces......................................... 39 9. The superposition operator in the space C........................................... 47 4. Closures and convexifications.............................................................................. 49 10. Strong closures........................................................................................ 49 11. Convexifications....................................................................................... 52 12. Weak closures.......................................................................................... 56 5. Fixed points and integral inclusions..................................................................... 59 13. Fixed point theorems for multi-valued operators................................ 60 14. Hammerstein integral inclusions........................................................ 63 15. A reduction method................................................................................... 68 6. Applications............................................................................................................... 72 16. Applications to elliptic systems.............................................................. 72 17. Applications to nonlinear oscillations................................................. 75 18. Applications to relay problems.............................................................. 78 References.................................................................................................................... 81 Index of symbols........................................................................................................... 93 Index of terms................................................................................................................ 95
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