Control system analysis and design via the second method of lyapunov: (I) continuous-time systems (II) discrete time systems

1959 ◽  
Vol 4 (3) ◽  
pp. 112-112 ◽  
Author(s):  
R. Kalman ◽  
J. Bertram
Keyword(s):  
Control System ◽  
Discrete Time ◽  
Continuous Time ◽  
System Analysis ◽  
System Analysis And Design ◽  
Analysis And Design ◽  
Discrete Time Systems ◽  
Continuous Time Systems ◽  
Time Systems ◽  
Second Method Of Lyapunov

Control System Analysis and Design Via the “Second Method” of Lyapunov: II—Discrete-Time Systems

1960 ◽  
Vol 82 (2) ◽  
pp. 394-400 ◽  
Author(s):  
R. E. Kalman ◽  
J. E. Bertram
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
System Analysis ◽  
Companion Paper ◽  
Data Systems ◽  
Sampled Data ◽  
Analysis And Design ◽  
Continuous Time Systems ◽  
Time Systems ◽  
Second Method Of Lyapunov

The second method of Lyapunov is applied to the study of discrete-time (sampled-data) systems. With minor variations, the discussion parallels that of the companion paper on continuous-time systems. Theorems are stated in full but motivation, proofs, examples, and so on, are given only when they differ materially from their counterparts in the continuous-time case.

Control System Analysis and Design Via the “Second Method” of Lyapunov: I—Continuous-Time Systems

1960 ◽  
Vol 82 (2) ◽  
pp. 371-393 ◽  
Author(s):  
R. E. Kalman ◽  
J. E. Bertram
Keyword(s):  
Control System ◽  
System Analysis ◽  
Transient Behavior ◽  
System Optimization ◽  
Companion Paper ◽  
Analysis And Design ◽  
Continuous Time Systems ◽  
The Subject ◽  
Time Systems ◽  
Second Method Of Lyapunov

The “second method” of Lyapunov is the most general approach currently in the theory of stability of dynamic systems. After a rigorous exposition of the fundamental concepts of this theory, applications are made to (a) stability of linear stationary, linear nonstationary, and nonlinear systems; (b)estimation of transient behavior; (c) control-system optimization; (d) design of relay servos. The discussion is essentially self-contained, with emphasis on the thorough development of the principal ideas and mathematical tools. Only systems governed by differential equations are treated here. Systems governed by difference equations are the subject of a companion paper.

scholarly journals Analysis and comparison of the stability of discrete-time and continuous-time linear systems

2016 ◽  
Vol 26 (4) ◽  
pp. 551-563
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Asymptotic Stability ◽  
Discrete Time ◽  
Continuous Time ◽  
Asymptotically Stable ◽  
Discrete Time Systems ◽  
Continuous Time Systems ◽  
The Matrix ◽  
Discretization Step ◽  
Time Systems ◽  
Time Linear

Abstract The asymptotic stability of discrete-time and continuous-time linear systems described by the equations xi+1 = Ākxi and x(t) = Akx(t) for k being integers and rational numbers is addressed. Necessary and sufficient conditions for the asymptotic stability of the systems are established. It is shown that: 1) the asymptotic stability of discrete-time systems depends only on the modules of the eigenvalues of matrix Āk and of the continuous-time systems depends only on phases of the eigenvalues of the matrix Ak, 2) the discrete-time systems are asymptotically stable for all admissible values of the discretization step if and only if the continuous-time systems are asymptotically stable, 3) the upper bound of the discretization step depends on the eigenvalues of the matrix A.

On positive reachability of time-variant linear systems on time scales

2013 ◽  
Vol 61 (4) ◽  
pp. 905-910 ◽  
Author(s):  
Z. Bartosiewicz
Keyword(s):  
Time Scales ◽  
Discrete Time ◽  
Continuous Time ◽  
Gram Matrix ◽  
General Criterion ◽  
Positive Systems ◽  
Discrete Time Systems ◽  
Continuous Time Systems ◽  
Time Invariant ◽  
Time Systems

Abstract Positive reachability of time-variant linear positive systems on arbitrary time scales is studied. It is shown that the system is positively reachable if and only if a modified Gram matrix corresponding to the system is monomial. The general criterion is then specified for particular cases of continuous-time systems and various classes of discrete-time systems. It is shown that in the case of continuous-time systems with analytic coefficients the conditions for positive reachability are very restrictive, similarly as for time-invariant systems

scholarly journals Local Observability of Systems on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zbigniew Bartosiewicz
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Time Scale ◽  
Analytic Functions ◽  
Analytic Geometry ◽  
Discrete Time Systems ◽  
Continuous Time Systems ◽  
Real Analytic ◽  
Time Systems ◽  
Real Analytic Geometry

Analytic systems on an arbitrary time-scale are studied. As particular cases they include continuous-time and discrete-time systems. Several local observability properties are considered. They are characterized in a unified way using the language of real analytic geometry, ideals of germs of analytic functions, and their real radicals. It is shown that some properties related to observability are preserved under various discretizations of continuous-time systems.

scholarly journals Similarity to contraction and asymptotics of a class of infinite-dimensional discrete-time systems

2005 ◽  
Vol 2005 (1) ◽  
pp. 87-99 ◽  
Author(s):  
Joseph J. Yamé
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Discrete Systems ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Discrete Time Systems ◽  
Power Bounded Operators ◽  
Continuous Time Systems ◽  
Time Systems ◽  
Power Bounded

A class of infinite-dimensional discrete-time state operators is exhibited as concrete instances of power-bounded operators that are not similar to contractions. It is shown that such discrete-time systems arise from sampled feedback control of unstable continuous-time systems. The asymptotic behavior of the state operators of these discrete systems is not intimately related to their spectral radius.

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