— The Lyapunov’s second method for the stability analysis of nonlinear dynamic systems requires f... more — The Lyapunov’s second method for the stability analysis of nonlinear dynamic systems requires finding Lyapunov functions. Unfortunately, finding a suitable Lyapunov function is a tedious process for a given complex nonlinear system if it is not impossible. On the other hand there are several algebraic approaches like the eigenvalue method for analyzing or designing linear time invariant systems. In this paper, we develop the eigenvalue method for the stability analysis of extended homogeneous nonlinear systems. In the case of polynomial homogeneous systems of zero degree it is shown that the zero equilibrium state of system is globally asymptotically stable if and only if all the homogeneous eigenvalues have negative real parts. Ultimately, an example is presented to describe the approach. Index Terms-- homogeneous nonlinear system, asymptotical stability, nonlinear eigenvalue.
The Lyapunov stability analysis method for nonlinear dynamic systems requires a non positive firs... more The Lyapunov stability analysis method for nonlinear dynamic systems requires a non positive first derivative of the Lyapunov functions along the system's trajectories. In this paper, a new method is developed to relax this requirement. A new sufficient condition is developed for the stability analysis of nonlinear systems, introducing some inequalities for higher order derivatives of the Lyapunov function. The differential inequalities can be considered as a new controllable canonical form linear time invariant system with negative inputs. The stability analysis of a given nonlinear system is then reduced to check if the characteristic equation for the new linear time invariant system is Hurwitz. Some examples are presented to establish the approach.
— The Lyapunov’s second method for the stability analysis of nonlinear dynamic systems requires f... more — The Lyapunov’s second method for the stability analysis of nonlinear dynamic systems requires finding Lyapunov functions. Unfortunately, finding a suitable Lyapunov function is a tedious process for a given complex nonlinear system if it is not impossible. On the other hand there are several algebraic approaches like the eigenvalue method for analyzing or designing linear time invariant systems. In this paper, we develop the eigenvalue method for the stability analysis of extended homogeneous nonlinear systems. In the case of polynomial homogeneous systems of zero degree it is shown that the zero equilibrium state of system is globally asymptotically stable if and only if all the homogeneous eigenvalues have negative real parts. Ultimately, an example is presented to describe the approach. Index Terms-- homogeneous nonlinear system, asymptotical stability, nonlinear eigenvalue.
The Lyapunov stability analysis method for nonlinear dynamic systems requires a non positive firs... more The Lyapunov stability analysis method for nonlinear dynamic systems requires a non positive first derivative of the Lyapunov functions along the system's trajectories. In this paper, a new method is developed to relax this requirement. A new sufficient condition is developed for the stability analysis of nonlinear systems, introducing some inequalities for higher order derivatives of the Lyapunov function. The differential inequalities can be considered as a new controllable canonical form linear time invariant system with negative inputs. The stability analysis of a given nonlinear system is then reduced to check if the characteristic equation for the new linear time invariant system is Hurwitz. Some examples are presented to establish the approach.
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Papers by Vahid Meigoli