Observer-based robust control for N-coupled wave equations

In this article, we consider robust output regulation for an MIMO wave system with disturbance and reference produced from an exosystem. The approach we adopt is the observer- based approach from partial differential equation (PDE) perspective. The main differences from our previous works are two points: (a) the eigenvalues of the exosystem are not necessarily required to be algebraically simple; (b) tracking error feedback dynamic controls are designed directly from the uncertain PDE itself, not a nominal system where disturbance and reference are specially chosen so that the nominal waves and exosystem are detectable. To this purpose, we introduce transformations to transfer the original system into a disturbance free system, and at the same time, each reference is transformed into a linear combination of the exosystem. An observer is designed to estimate the states of the transformed disturbance free system and the unknown linear combinations of the exosystem. The feedforward controls are designed where the input disturbances are then written based on the linear combinations of the exosystem. A set of observer-based feedback controls is then designed by replacing the state and disturbance with their estimates from the observer. The robustness of the closed-loop system is discussed by analyzing the stability of the transformed tracking error system and the observer error system. The numerical simulations are presented to validate visually the effectiveness of the proposed tracking error feedback control.

Automated summary

In this article, we address the problem of robust output regulation for an MIMO wave system with disturbance and reference from an exosystem. We propose an observer-based approach from the perspective of partial differential equations (PDEs). The main differences from previous works lie in our consideration of exosystems with non-algebraically simple eigenvalues and the direct design of tracking error feedback controls from the uncertain PDEs themselves. We introduce transformations to convert the original system into a disturbance-free system and each reference into a linear combination of the exosystem, allowing for the design of an observer and feedforward controls. The robustness of the closed-loop system is discussed through stability analysis.