This paper proposes a direct adaptive neural control law for a class of affine nonlinear multi-input-multi-output (MIMO) systems of the formẋ = f (x) + G(x)u using feedback linearization when both f (x) and G(x) are to be estimated. It is challenging to estimate f (x), a vector, and G(x), 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 f (x) and G(x), 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 α and β 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.
This paper proposes a systematic formulation of inverse optimal control (IOC) law based on a rather straightforward reduction of control Lyapunov function (CLF), applicable to a class of second-order nonlinear systems affine in the input. This method exploits the additional design degrees of freedom resulting from the non-uniqueness of the state dependent coefficient (SDC) formulation, which is widely used in pseudo-linear control techniques. The applicability of the proposed approach necessitates an apparently effortless SDC formulation satisfying an SDC matrix criterion in terms of the structure and characteristics of the state matrix, [Formula: see text]. Subsequently, a sufficient condition for the global asymptotic stability (g.a.s) of the closed-loop system is established. The SDC formulations conforming to the sufficient condition ensure the existence and determination of a smooth radially unbounded polynomial CLF of the form [Formula: see text], while offering a benevolent choice for the gain matrix [Formula: see text], in the CLF. The direct relationship between the gain matrix [Formula: see text] and state weighing matrix [Formula: see text] ensures optimization of an equivalent [Formula: see text]. This feature enables one to rightfully choose the gain matrix [Formula: see text] as per the performance requisites of the system. Finally, the application of the proposed methodology for the speed control of a permanent magnet synchronous motor validates the efficacy and design flexibility of the methodology.
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