NL q Theory

In this Chapter we develop a model based neural control framework which consists of neural state space models and neural state space controllers. Like in modern (robust) control theory standard plant forms are considered. In order to analyse and synthesize neural controllers within this framework, the so-called NL _q system form is introduced. NL _q s represent a large class of nonlinear dynamical systems in state space form and contain a number of q layers of an alternating sequence of linear and static nonlinear operators that satisfy a sector condition. All system descriptions are transformed into this NL _q form and sufficient conditions for global asymptotic stability, input/output stability with finite L₂-gain and robust performance are derived. It turns out that NL _q s have a unifying nature, in the sense that many problems arising in neural networks, systems and control can be considered as special cases. Moreover, certain results in H_∞ and μ control theory can be interpreted as special cases of NL _q theory. Examples show that following principles from NL _q theory, stabilization and control of several types of nonlinear systems are possible, including mastering chaos.