Adaptive local linear modelling and control of nonlinear dynamical systems

Abstract

Systems theory is a well-established and mature area of engineering research, where many strong general mathematical results are available. Especially the analysis of linear identification and control systems have been pursued by many researchers leading to a complete understanding of various mechanisms that are effective in the stability, controllability and observability of these. Due to the availability of such an extensive knowledge base about linear systems, modern industrial control applications are still typically designed utilising the results from linear control systems theory. Nevertheless, academic research has been concentrating around problems involving the stability, identification and control of nonlinear dynamical systems in the last few decades. These efforts have now also matured into a broad theory of nonlinear systems, their identification and control. Initial efforts in this area pursued parametric approaches, inspired by the established linear systems theory, where the system dynamical equations are generally assumed to be known from physical principles, possibly with some uncertainty in the values of certain parameters. In this framework, the system identification and system control problems are decoupled, therefore can be solved sequentially. More recently, adaptive system identification and control methodologies have also been investigated, once again leading to a very good understanding of the adaptation in linear systems and a satisfactorily general insight to

Feedforward Nonlinear Control Using Neural Gas Network

“…After obtaining this NG model, the design of the local linear controller is simpler than that of the global nonlinear controller. Local linear mapping using another prototype-based algorithm such as SOM was successfully tested at the NASA facilities [12].…”

“…Prototypes are already obtained in (8), whereas the adaptation rules of and ∇ are calculated considering Newton's method for energy cost (10). The learning rules for and ∇ are shown in (11) and (12), respectively:…”

Técnicas de extracción del conocimiento basadas en data mining visual para la supervisión de procesos industriales. Análisis de la dinámica basado en mapas auto-organizados

“…De hecho, el grado de no linealidad del modelo conjunto puede controlarse por medio del número de submodelos que se utilizan para construirlo y, por tanto, se puede adaptar a la distribución de los datos (McNames, 1999). El diseño de un sistema de control a partir de este modelado también es más sencillo, pues se reduce a diseñar múltiples controladores locales más simples que se intercambian o cooperan (Erdogmus et al, 2005). Esta estrategia es escalable, pues su complejidad se incrementa linealmente con el número de submodelos.…”

“…La reconstrucción da lugar a la trayectoria en el espacio de estados que, si se cumplen las condiciones anteriores, preserva los invariantes dinámicos y garantiza que no haya cruces entre trayectorias en el espacio de reconstrucción. Se han propuesto varios métodos para estimar d e , como el análisis de falsos vecinos más cercanos o elíndice de Lipschitz (Erdogmus et al, 2005). alguna medida de distancia y la aplicación de su modelo asociado proporcionan la aproximación lineal por tramos a la dinámica no lineal global.…”