Abstract
This paper addresses the stability problem of a class of switched nonlinear time-delay systems modeled by delay differential equations. Indeed, by transforming the system representation under the arrow form, using a constructed Lyapunov function, the aggregation techniques, the Borne-Gentina practical stability criterion associated with the M-matrix properties, new delay-independent conditions to test the global asymptotic stability of the considered systems are established. In addition, these stability conditions are extended to be generalized for switched nonlinear systems with multiple delays. Note that the results obtained are explicit, they are simple to use, and they allow us to avoid the problem of searching a common Lyapunov function. Finally, an example is provided, with numerical simulations, to demonstrate the effectiveness of the proposed method.
Highlights
Switched systems are a class of important hybrid systems which consist of a finite number of subsystems that are governed by differential or difference equations and a switching law which defines a specific subsystem being activated during a certain interval of time
Due to the physical properties or various environmental factors, many real-world systems can be modeled as switched systems such as computer science, autonomous transmission systems, computer disc drivers, control systems, electrical engineering and technology, automotive industry, air traffic management, chemical systems, power systems and communication networks, and other applications [ – ]
Based on the construction of a common Lyapunov function as well as the use of the Borne-Gentina practical stability criterion [, – ] associated with the M-matrix properties [, ], new delay-independent sufficient stability conditions for continuous-time switched nonlinear time-delay systems under arbitrary switching are established. These obtained results are extended to be generalized for continuous-time switched nonlinear systems with multiple delays. Note that these proposed results can guarantee stability under arbitrary switching and allow us to avoid searching of a common Lyapunov function, which is very difficult in this case
Summary
Switched systems are a class of important hybrid systems which consist of a finite number of subsystems that are governed by differential or difference equations and a switching law which defines a specific subsystem being activated during a certain interval of time. Based on the construction of a common Lyapunov function as well as the use of the Borne-Gentina practical stability criterion [ – , – ] associated with the M-matrix properties [ , ], new delay-independent sufficient stability conditions for continuous-time switched nonlinear time-delay systems under arbitrary switching are established. These obtained results are extended to be generalized for continuous-time switched nonlinear systems with multiple delays Note that these proposed results can guarantee stability under arbitrary switching and allow us to avoid searching of a common Lyapunov function, which is very difficult in this case. In the case that the nonlinear elements of Tc(·) are isolated in the last row (Assumption is satisfied) the eigenvector v(t, x(t)) relative to the eigenvalue λm is constant [ ] where λm is such that Re(λm) = max{Re(λm), λ ∈ λTc(·)} To complete this proof, we assume that Tc(·) is the opposite of an M-matrix. According to the M-matrix properties, we can find a vector ρ ∈
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