Sampled-data H∞ filtering of a 2D heat equation under pointlike measurements

The existing sampled-data observers for 2D heat equations use spatially averaged measurements, i.e., the state values averaged over subdomains covering the entire space domain. In this paper, we introduce an observer for a 2D heat equation that uses pointlike measurements, which are modeled as the state values averaged over small subsets that do not cover the space domain. The key result, allowing for an efficient analysis of such an observer, is a new inequality that bounds the L 2 -norm of the difference between the state and its point value by the reciprocally convex combination of the L 2 -norms of the first and second order space derivatives of the state. The convergence conditions are formulated in terms of linear matrix inequalities feasible for large enough observer gain and number of pointlike sensors. The results are extended to solve the H ∞ filtering problem under continuous and sampled in time pointlike measurements

Sinc-Galerkin estimation of diffusivity in parabolic problems

A fully Sinc-Galerkin method for the numerical recovery of spatially varying diffusion coefficients in linear partial differential equations is presented. Because the parameter recovery problems are inherently ill-posed, an output error criterion in conjunction with Tikhonov regularization is used to formulate them as infinite-dimensional minimization problems. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which displays an exponential convergence rate and is valid on the infinite time interval. The minimization problems are then solved via a quasi-Newton/trust region algorithm. The L-curve technique for determining an approximate value of the regularization parameter is briefly discussed, and numerical examples are given which show the applicability of the method both for problems with noise-free data as well as for those whose data contains white noise

Delayed H∞ control of 2D diffusion systems under delayed pointlike measurements

Up to now, robust control of multi-dimensional diffusion systems was confined to averaged measurements. In this paper, we consider 2D diffusion systems with delayed pointlike measurements. A pointlike measurement is the state value averaged over a small subdomain that approximates its point value. The main novelty enabling the study of such measurements is a new inequality, which we call the reciprocally convex variation of Friedrich’s inequality. It bounds the difference between a function and its point values in the L2-norm using the function’s derivatives. Combining this result with a new Lyapunov–Krasovskii functional, which has a spatially-varying kernel, we solve the H∞ control and filtering problems in the presence of time-varying input and output delays. We show that any 2D semilinear diffusion system with pointlike measurements can be stabilized by static output feedback applied through characteristic functions if the controller gain and number of sensors/actuators are large enough while the input and output delays are sufficiently small. The results are demonstrated on a 2D catalytic slab model