Keywords: Adaptive nonlinear optimal control, non-affine in control nonlinear systems, model free recursive control; trajectory tracking. During the last decades, the control of nonlinear systems which concerns normally the stabilization or trajectory tracking control is generally considered difficult and becomes worldly active and hot research subjects . The combined consideration of twofold for a controlled systems, which means the realization not only the control trajectory tracking or stabilization , but also the optimization of a prescribed performance index such as minimization the trajectory tracking error, the fuel consumption, or particles emissions, etc are not easy.
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Currently there exists no adequate work relating the optimal control of nonlinear systems, especially for the non-affine in control nonlinear systems, since their corresponding Hamilton-Jacobi- Bellman HJB equation are difficult to resolve . Compared to linear or nonlinear affine in control systems to solve their corresponding Riccati Equation RE , Algebraic Riccati Equation ARE and HJB, the nonlinear non-affine in control systems are more complex and has no feasible solutions when their systems dynamics are not explicit.
Recently, online adaptive approximation- based optimal control which refers to an online approximator based Adaptive Critic Designs ACD is gradually proposed .
While in reference , a relative new optimal adaptive control OAC is proposed for MIMO nonlinear systems in the form of strict feedback. The optimal adaptive feedback scheme is introduced for the affine systems to estimate the solution of HJB equation online which becomes the optimal feedback control input for the closed-loop system. In adaptive control literature, various nonlinear systems and methods are appeared and discussed .
Nonlinear systems in form of strict feedback in a variety of ways and their stability are studied with the application of the normal Backstepping scheme without any optimality.
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Recently, the inverse optimal control for strict feedback systems which consist on an associated cost function based control law is introduced in . Furthermore, in many applications, the inverse function for nonlinear systems is difficult or even impossible to find. In many practical cases, systems full states are not fully accessible.
Then output feedback nonlinear controllers which use model-based observers were developed when their systems dynamics are known.
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While in many cases where their systems dynamics are unknown, the control of unknown strict feedback systems using adaptive neural-network based scheme is given to approximate their uncertain dynamics . Therefore in our previous paper, a composed adaptive controller which is based on a recursive model free stabilization sub-controller RMFSSC and a recursive uncertain dynamic compensation sub-controller RUDCSC are proposed for nonlinear systems in .
Using PCS theory, piecewise-continuous controllers were firstly developed to enable sampled trajectory tracking of linear systems . Then for improving trajectory tracking performance, a derived piecewise continuous controller and recursive model free controller were proposed in . Unfortunately, the referred proposed RMFSSC is defined without any prescribed cost function to minimize and not considered for the case of non-affine in control nonlinear systems.
This referred ANORC controller which employs only the output of the non-affine in control nonlinear systems and a L 2 norm to minimize the trajectory tracking error has a relative simple architecture and can be implemented to the case where the internal dynamics and parameters of the controlled non-affine in control nonlinear systems are unknown. It aims to highlight the encountered difficulties and the proposed solutions to achieve the tracking objective.
Based on a literature overview of linear and nonlinear reference tracking, the achievements and the limitations of the existing strategies are highlighted. This motivates the present work to propose clear control algorithms for perfect and approximate tracking controls of nonlinear systems described by Takagi-Sugeno models. First, perfect nonlinear tracking control is addressed and necessary structural conditions are stated.
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If these conditions do not hold, approximate tracking control is proposed and the choice of the reference model to be tracked as well as the choice of the criterion to be minimized are discussed with respect to the desired objectives. The case of constrained control input is also considered in order to anticipate and counteract the effect of the control saturation.
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