Stability of direct heuristic dynamic programming for nonlinear tracking control using PID neural network

The issue of designing a high performance controller to track a desired system trajectory is one of most important problems in control theory and practice. More recently, there has been a growing interest in the study of tracking control problem. In this paper, we discuss the design and stability properties of a special approximate/adaptive dynamic programming (ADP) method for a general multiple-input-multiple-output (MIMO) discrete-time nonlinear optimal tracking control problem. The direct heuristic dynamic programming (HDP) design algorithm is firstly derived by incorporating the PID control rule into neural networks (NNs). This design approach considers using not only the typical state variables but also their derivatives and cumulative sums as inputs to the controller output. It is therefore expected to retain PID controller properties with additional learning capability. Moreover, our nonlinear control problem is formulated under a general condition that system nonlinearity is unknown and therefore it introduces modelling errors for the controller design. By using a Lyapunov stability construct, we provide new results of uniformly ultimately boundedness (UUB) for the proposed PIDNN-based direct HDP controller in discrete-time nonlinear tracking setting with desired tracking performance.