Output feedback stabilization of coupled reaction-diffusion processes with constant parameters

The problem of output feedback boundary stabilization is considered for n coupled plants, distributed over the one-dimensional spatial domain [0; 1] where they are governed by linear reaction-diffusion partial differential equations (PDEs). All plants have constant parameters and each is equipped with its own scalar boundary control input, acting at one end of the domain. First, a state feedback law is designed to exponentially stabilize the closed-loop system with an arbitrarily fast convergence rate. Then, collocated and anticollocated observers are designed, using a single boundary measurement for each plant. The exponential convergence of the observed state towards the actual one is demonstrated for both observers, with a convergence rate that can be made as fast as desired. Finally, the state feedback controller and the preselected, either collocated or anticollocated, observer are coupled together to yield an output feedback stabilizing controller. The distinct treatments are proposed separately for the case in which all processes have the same diffusivity and for the more challenging scenario where each process has its own diffusivity. The backstepping method is used for both controller and observer designs. Two main classes of coupled PDEs are studied in the paper. In the first one, all processes possess a Dirichlet-type boundary condition (BC) at the uncontrolled side. With reference to this class, the state feedback and observer-based output feedback designs are successfully solved in both the equi-diffusivity and distinct-diffusivity scenarios, and, particularly, the kernel matrices of the underlying transformations are derived in analytical form by using the method of successive approximations to solve the corresponding kernel PDEs. Thus, the resulting control laws and observers become available in explicit form. The second and more general class of coupled PDEs considered in the paper entails a subset of the processes having Dirichlet-type BCs at the uncontrolled side, whereas all remaining processes possess Neumann-type BCs. With reference to this wider class of systems with heterogenous BCs, it turns out that the state feedback design can only be solved in the equi-diffusivity case, although the resulting kernel matrix is no longer available in explicit form, whereas the same approach yields an overdetermined kernel PDE admitting no solution in the distinct-diffusivity case. Anticollocated observer design and output feedback designs are additionally developed in the equi-diffusivity scenario. Interestingly, the observer gains are still available in explicit form also in this case. Capabilities of the proposed synthesis and its effectiveness are supported by a numerical study made for two coupled systems with heterogeneous BCs.