An approximate stability analysis of nonlinear systems described by Universal Learning Networks
Stability is one of the most important subjects in control systems. As for the stability of nonlinear dynamical systems, Lyapunov's direct method and linearized stability analysis method have been widely used. But, it is generally recognized that finding an appropriate Lyapunov function is fairly difficult especially for the nonlinear dynamical systems, and also it is not so easy for the linearized stability analysis to find the locally asymptotically stable region. Therefore, it is crucial and highly motivated to develop a new stability analysis method, which is easy to use and can easily study the locally asymptotically stable region at least approximately, if not exactly. On the other hand, as for the calculation of the higher order derivative, Universal Learning Networks (ULNs) are equipped with a systematic mechanism that calculates their first and second order derivatives exactly. So, in this paper, an approximate stability analysis method based on ? approximation is proposed in order to overcome the above problems and its application to a nonlinear dynamical control system is discussed. The proposed method studies the stability of the original trajectory by investigating whether the perturbed trajectory can approach the original trajectory or not. The above investigation is carried out approximately by using the higher order derivatives of ULNs. In summarizing the proposed method, firstly, the absolute values of the first order derivatives of any nodes of the trajectory with respect to any initial disturbances are calculated by using ULNs. If they approach zero at time infinity, then the trajectory is locally asymptotically stable. This is an alternative linearized stability analysis method for nonlinear trajectories without calculating Jacobians directly. In the method, the stability analysis of time-varying systems with multi-branches having any sample delays is possible, because the systems are modeled by ULNs. Secondly, the locally asymptotically stable region, where asymptotical stability is secured approximately, is obtained by finding the area where the first order terms of Taylor expansion are dominant compared to the second order terms with ? approximation assuming that the higher order terms more than the third order are negligibly small in the area. Simulations of an inverted pendulum balancing system are carried out. From the results of the simulations, it is clarified that the stability of the inverted pendulum control system is easily analyzed by the proposed method in terms of studying the locally asymptotically stable region.
stability analysis nonlinear systems universal learning networks higher order derivatives calculation neural networks