Decoupling the Equations of Isospectral Flow

2006 ◽  
Vol 5-6 ◽  
pp. 481-490
Author(s):  
Seamus D. Garvey ◽  
Peter Van Eetvelt ◽  
U. Prells
Keyword(s):  
Differential Equations ◽  
Continuous Functions ◽  
Order System ◽  
Scalar Parameter ◽  
Flow Equations ◽  
Rates Of Change ◽  
Isospectral Flow ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Vibrating Systems

Three (m × n) matrices {K, D, M} represent a second-order system in the form (K + Dλ+ Mλ2). If m = n, system eigenvalues are defined as the values of λ for which det(K + Dλ+ Mλ2) = 0. If {K, D, M} are continuous functions of a real scalar parameter, σ, eigenvalues and dimensions of the associated eigenspaces remain constant if and only if the rates of change of {K, D, M} obey certain ODEs called the isospectral flow equations. The integration of these matrix differential equations is of interest here. This paper explains the motivation behind this work in terms of vibrating systems and it reports two related hypotheses concerning how the solutions to these equations may be decoupled. Work underway towards proving and using these hypotheses is presented. No existing known solutions allow this decoupling in general.

scholarly journals Design of Terminal Sliding Mode Controllers for Disturbed Non-Linear Systems Described by Matrix Differential Equations of the Second and First Orders

2019 ◽  
Vol 9 (11) ◽  
pp. 2325 ◽  
Author(s):  
Paweł Skruch ◽  
Marek Długosz
Keyword(s):  
Differential Equations ◽  
Sliding Mode ◽  
Sliding Surface ◽  
Terminal Sliding Mode ◽  
Sliding Mode Controllers ◽  
System Trajectory ◽  
Non Linear ◽  
Matrix Differential Equations ◽  
Non Linear Systems ◽  
Matrix Differential

This paper describes a design scheme for terminal sliding mode controllers of certain types of non-linear dynamical systems. Two classes of such systems are considered: the dynamic behavior of the first class of systems is described by non-linear second-order matrix differential equations, and the other class is described by non-linear first-order matrix differential equations. These two classes of non-linear systems are not completely disjointed, and are, therefore, investigated together; however, they are certainly not equivalent. In both cases, the systems experience unknown disturbances which are considered bounded. Sliding surfaces are defined by equations combining the state of the system and the expected trajectory. The control laws are drawn to force the system trajectory from an initial condition to the defined sliding surface in finite time. After reaching the sliding surface, the system trajectory remains on it. The effectiveness of the approaches proposed is verified by a few computer simulation examples.

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