Engineering: Lyapunov Functions - Research Paper Example

For a number of ODEs classes, the presence of Lyapunov functions is considered as being a sufficient and necessary stability condition. Now that there are no general formulae of coming up with ODE’s Lyapunov functions, in most cases, the construction of these functions are normally known (Burton, 2002). Lyapunov functions can also be looked at in an informal perspective in that it takes values that are positive in every situation except for the values at the equilibrium after which it either increases or decreases following an ODE trajectory.

The Lyapunov functions principle merit based on the analysis of stability of ODEs states that the definite solution of the ODE is not necessary -be it numerical or analytical (Aydin and Bulgak, 2004). History Lyapunov function of control was named after Mikhailovich Lyapunoy. Lyapunoy was the first mathematician to take interest in possible modifications of nonlinear systems in relation to the linear stability theory with the equilibrium point being his point of reference. For many years, his work got minimal attention.

It was after the Cold War that most people suddenly started getting interested in his work as they realized it was applicable in aerospace stability guidance system. This system initially had several nonlinearities which could not be solved by other methods. This discovery led to the initial Lyapunov functions to be referred to as the “Second Method of Lyapunov”. The period after this witnessed the publication of a large number of Lyapunov works. The latest development on the initial Lyapunov function is the Lyapunov exponent that is closely associated with the chaos theory.

Lyapunov stability has since then had several applications including calculating equilibrium positions (Campbell, 1993). Applications Lyapunov applications are not limited to the process of proving stability but can also be applied in the synthesis through Lyapunov functions control, in the analysis of robustness, while working with systems that are uncertain, in the estimation of attraction basins of points in equilibrium, used in proving periodic orbits that do not exist or are not stable, can be applied in analyzing performances such as convergence rates, used to prove combinatorial algorithms convergence like consensus algorithms and estimating the joint spectral radius bounds of matrices (Sivasundarum and Vassilyey, 1998).

The above applications indicate that the Lyapunov theory of direct method has attracted many interests. Even with the increased developments in progress to come up with systematic and efficient search methods for Lyapunov control functions, it is much easier to locate systems that are stable by using Lyapunov function as compared to the other methods. Even in situations where such methods manage to find Lyapunov functions, the structures of such methods are often found to be so complicated. Such setbacks make most people to rethink if the Lyapunov theorem conditions are exceedingly conservative (Elliott, 2002).

The Lyapunov functions can also be applied as a primary nonlinear control tool in solving controller synthesis and stability of nonlinear polynomial dynamics systems. This can be easily reached at by having a Lyapunov function candidate picked at random and combined with any given system; if the combination satisfies the requirements of the Lyapunov theorem, then the dynamic system is stable. The synthesis approach