Suboptimal closed‐form feedback control of input‐affine non‐linear systems

Abstract

A closed-form feedback solution to finite-time optimal control problems for input-affine non-linear systems is presented here. Based on the state-dependent Riccati equation (SDRE) framework, this suboptimal solution, called balanced-cost SDRE (BCSDRE), is proposed to minimise the overall control effort required for driving the system from an arbitrary initial condition to an acceptable neighbourhood around origin, within a pre-specified final time. The state error is dynamically penalised to balance the overall state error and the control cost, resulting in a modified SDRE. Unlike the existing finite-horizon SDRE methods, the proposed technique does not require to solve algebraic Riccati and Lyapunov equations at every time step, which makes this approach computationally efficient. Based on the Lyapunov theory, the origin of the BCSDRE controlled system is shown to be locally exponentially stable. Furthermore, the state profile is also shown to be locally predefined time ultimately bounded, under appropriate tuning of the proposed controller. The proposed technique is implemented to design a controller for the Van der Pol oscillator and impact angle guidance design problem, and its performance is compared with the existing finite-horizon SDRE-based techniques.