Direct Adaptive Neural Control Design for a Class of Nonlinear Multi Input Multi Output Systems

This paper proposes a direct adaptive neural control law for a class of affine nonlinear multi-input-multi-output (MIMO) systems of the form <inline-formula> <tex-math notation="LaTeX">$\dot {x}=f(x)+G(x)u$ </tex-math></inline-formula> using feedback linearization when both <inline-formula> <tex-math notation="LaTeX">$f(x)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$G(x)$ </tex-math></inline-formula> are to be estimated. It is challenging to estimate <inline-formula> <tex-math notation="LaTeX">$f(x)$ </tex-math></inline-formula>, a vector, and <inline-formula> <tex-math notation="LaTeX">$G(x)$ </tex-math></inline-formula>, a matrix, to be used in synthesizing the control law, simultaneously, because of the dimensional inconsistency arising with the available neural structures, which do not have multiple layers of outputs. This problem is addressed in this paper by exploiting the power of matrix vectorization and reshaping techniques using the Kronecker product. The strategy may be visualized as equivalent to the neural structure consisting of multiple layers of outputs that result from the appropriate manipulation of matrices corresponding to the proposed estimations. The weight update laws, for both the radial basis function neural networks that estimate both <inline-formula> <tex-math notation="LaTeX">$f(x)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$G(x)$ </tex-math></inline-formula>, are derived such that the proposed control law achieves the twin objective of the derived tracking performance as well as closed-loop system stability in the sense of Lyapunov. The ratios <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> are proposed in line with the widely used concept of Rayleigh’s quotient adopted in structural dynamics to evaluate the natural frequency of a system. The simulation results obtained from the use of a twin rotor MIMO system are presented here to demonstrate the feasibility and effectiveness of the proposed control law. The superiority of this approach lies in the development of suitable control law for a MIMO system in the absence of knowledge about the nonlinearities.