Linear matrix inequalities (LMIs) based observer and controller design for second order parabolic PDE

In this work, a linear matrix inequality (LMI) methodology is used to design a controller to stabilize the second order parabolic PDE in the presence of input constraints. The novel feature of the proposed synthesis is to construct the LMI formulation within the modal space by using the spectrum analysis, which transfers the original infinite dimensional PDE state into the modal state, described by the abstract state space representation. The state feedback controller and Luenberger observer are developed by taking the entire infinite number of modal states into account, thereby stabilizing the system and reconstructing the state rigorously. The manipulated variable constraints, naturally existing in most real applications, are also considered in the design framework. Finally, since the initial value of modal states are hardly known in advance, the LMI formulation is designed to maximize the region of attraction (ROA) for augmented state space, including the modal state, input, estimation error, together with the fast modal output bound, such that the controller and the observer are robust enough to the error associated with initial conditions. Using a numerical example of an unstable second order parabolic PDE, we demonstrate that if feasible, the state feedback controller and observer, generated by LMI, have enough capability to drive the process modal state into the equilibrium.