Contents I. Introduction
Because of their function approximation ability, fuzzy systems or neural networks are usually used to model unknown nonlinear functions for adaptive fuzzy/neural control of nonlinear systems. Many remarkable results on this field have been reported during the past two decades. For instance, see [1]– [12] and the references therein. In some practical systems, state variables may be unmeasurable or just partly measurable. In this case, those control strategies via state feedback cannot be well implemented in practice. To overcome the difficulty caused by the lack of precise information about system state, an observer is set up to estimate those unmeasurable state variables. Further, these estimated state variables can be used for feedback control design. On the other hand, there are also cases where the state variables are measurable, but their estimation, instead of their direct measurement, may be interesting to reduce costs with sensors. So, observer-based output feedback control strategy has been paid considerable attention. In the early research on observer-based neural/fuzzy adaptive control, some pioneering results have been reported, respectively. In [13], a high-gain observer was constructed for nonlinear systems and a neural adaptive output feedback control scheme was proposed via estimated state feedback. For nonlinear systems in output feedback form, in which the unknown functions only contain the output variable, the neural/fuzzy adaptive output feedback control was investigated in [14]– [16]. In [14], a linear state observer was designed. The unknown nonlinear functions were approximated by fuzzy systems and the control laws were constructed by the classical adaptive control design method. In [15] and [16], with the unknown functions approximated by neural networks, the neural adaptive control laws were proposed. In [17], a neural adaptive output feedback control strategy was designed for multi-input and multi-output (MIMO) nonlinear discrete-time systems in the strict-feedback form. A system representation transformation was introduced, by which the controlled systems were first transformed to the state-output model and then neural adaptive output feedback control laws were proposed. The design method was further extended to more general MIMO nonlinear discrete-time systems in [18] and [19]. Since then, observer-based fuzzy/neural adaptive control has been further extended from nonlinear systems in output feedback structure to those in strict feedback structure. In [20], a fuzzy adaptive observer was designed and a fuzzy adaptive output-feedback control method was proposed for nonlinear strict-feedback systems by using small gain theorem and backstepping. By following the same line, the proposed fuzzy adaptive output feedback control schemes were further applied to nonlinear strict feedback systems with unknown dead-zone input [21]–[23]. In [24], the problem of neural adaptive output feedback control was addressed for nonlinear time-delay systems in strict feedback form. A memoryless neural adaptive observer was constructed and a neural adaptive controller was presented to guarantee a good control performance. By further applying this method, a neural adaptive output feedback control design scheme was developed in [25] for a class of large-scale nonlinear time-delay systems and neural adaptive decentralized output feedback controllers were constructed. In [26], for nonlinear systems in the strict feedback form, a linear observer was first designed and then a fuzzy adaptive output feedback control strategy was developed to guarantee the stability of the closed-loop systems. In [27], the authors considered neural adaptive output feedback control for the large-scale nonlinear systems with unknown interconnections. An observer-based decentralized neural adaptive control law was designed. These observer-based fuzzy/neural adaptive control design approaches have been extended to nonlinear stochastic systems in the strict feedback structure in [28]–[30] . Recently, a linear observer was constructed in [31], and then small-gain theorem was used to develop a fuzzy adaptive output feedback controller. By adopting Nussbaum function approach, a neural adaptive output-feedback control scheme was presented for nonlinear systems with output constrained in [32]. More recently, observer-based adaptive fuzzy/neural control strategies have been further extended to nonstrict-feedback systems [33], to MIMO nonlinear systems [34], to switched nonlinear pure-feedback systems [35], and to second-order multiagent systems [36].
