Inverse optimal control of a class of affine nonlinear systems

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, A ( x ) . 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 V ( x ) = x T P ( x ) x , while offering a benevolent choice for the gain matrix P ( x ) , in the CLF. The direct relationship between the gain matrix P ( x ) and state weighing matrix Q ( x ) ensures optimization of an equivalent s ( x ) = x T Q ( x ) x . This feature enables one to rightfully choose the gain matrix P ( x ) 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.