Decentralized controller design with asymmetric input constraints for unknown unmatched interconnected systems

In this paper, we explore the decentralized control problem through the lens of adaptive dynamic programming for continuous-time nonlinear systems, particularly those with unknown mismatched interconnections and asymmetric input constraints. First, the unknown interconnection term is addressed by approximating it with a radial basis function neural network. This approximation relies on the local states of isolated subsystems and the reference states of coupled subsystems, thus sidestepping the common assumption that interconnections are matched and upper bounded. Following this, the challenge of designing a decentralized optimal controller design is reframed as a series of local optimal controller design problems. This reframing is facilitated by adaptive critic networks and considers the asymmetric constraints of subsystems. The application of the Lyapunov stability theorem demonstrates that controllers, even with asymmetric input constraints, can rapidly stabilize the large-scale system. More importantly, we conclude that the control laws designed here serve as the decentralized control strategies for large-scale nonlinear systems. Our methodology employs the radial basis function neural network and the critic neural network. The former approximates interconnection terms, while the latter deals with cost functions, enabling to derive optimal decentralized control strategies under asymmetric constraints. The uniform ultimate boundedness of observation error and weight approximation error are assured by using the Lyapunov theorem. This is further supported by the introduction of a state observer to estimate the state of the interconnected subsystems and the use of the critic neural network to approximate an improved cost function. This approach allows for an approximate solution to the Hamilton–Jacobi–Bellman equation, resulting in optimal decentralized control strategies satisfying the asymmetric input constraints. At the same time, based on the weight updating rules of the critic neural network, we can guarantee that weight approximation errors are uniformly ultimately bounded by selecting the suitable Lyapunov function. The selection of neural networks in this study is driven by considerations of convergence speed and computational burden, leading to the choice of two specific types of networks. The effectiveness of the developed control method is then rigorously tested through simulation and comparative experiments implemented in a MATLAB environment. Comparative experiments underscore the advancements made by the algorithm developed in this paper, especially under asymmetric control constraints. Contrasting our approach with unimproved cost functions and strategies lacking control constraints, we showcase significant improvements. The simulation results are shown in Figs. 1–9, which fully verify the effectiveness of our established scheme. We can derive that the developed control method significantly enhances stabilization speed and performance.