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The problem of sensor placement for second order infinite dimensional systems is examined within the context of a disturbance-decoupling observer. Such an observer takes advantage of the knowledge of the spatial distribution of disturbances to ensure that the resulting estimation error dynamics are not affected by the temporal component of the disturbances. When such an observer is formulated in a second order setting, it results in a natural observer. Further, when the natural observer is combined with a disturbance decoupling observer, the necessary operator identities needed to ensure the well-posedness of the observer, are expressed in terms of the stiffness, damping, input and output operators. A further extension addresses the question of where to place sensors so that the resulting natural disturbance decoupling observer is optimal with respect to an appropriately selected performance measure. This paper proposes this performance measure which is linked to the mechanical energy of second order infinite dimensional systems. The proposed sensor optimization is demonstrated by a representative PDE in a second order setting. 

Abstract We study a temperature and velocity output tracking problem for a two-dimensional room model with the fluid dynamics governed by the linearized translated Boussinesq equations. Additionally, the room model includes finite-dimensional models for actuation and sensing dynamics; thus, the complete model dynamics are governed by an ODE–PDE–ODE cascade. As the main contribution, we design a low-dimensional internal model-based controller for robust output tracking of the room model. The controller’s performance is demonstrated through a numerical example. 

Abstract The problem of parameter identification appears in many physical applications. A parameter of particular interest in cancer treatment is permeability, which modulates the fluidic streamlines in the tumor microenvironment. Most of the existing permeability identification techniques are invasive and not feasible to identify the permeability with minimal interference with the porous structure in their working conditions. In this paper, a theoretical framework utilizing partial differential equation (PDE)-constrained optimization strategies is established to identify a spatially distributed permeability of a porous structure from its modulated external velocity field measured around the structure. In particular, the flow around and through the porous media are governed by the steady-state Navier–Stokes–Darcy model. The performance of our approach is validated via numerical and experimental tests for the permeability of a 3D printed porous surrogate in a micro-fluidic chip based on the sampled optical velocity measurement. Both numerical and experimental results show a high precision of the permeability estimation. 

This work is concerned with the use of mobile sensors to approximate and replace the full state feedback controller by static output feedback controllers for a class of PDEs. Assuming the feedback operator associated with the full-state feedback controller admits a kernel representation, the proposed optimization aims to approximate the inner product of the kernel and the full state by a finite sum of weighted scalar outputs provided by the mobile sensors. When the full state feedback operator is time-dependent thus rendering its associated kernel time-varying, the approximation results in moving sensors with time-varying static gains. To calculate the velocity of the mobile sensors within the spatial domain the time-varying kernel is set equal to the sensor density and thus the solution to an associated advection PDE reveals the velocity field of the sensor network. To obtain the speed of the finite number of sensors, a domain decomposition based on a modification of the Centroidal Voronoi Tessellations (µ-CVT) is used to decompose the kernel into a finite number of cells, each of which contains a single sensor. A subsequent application of the µ-CVT on the velocity field provides the individual sensor speeds. The nature of this µ-CVT ensures collision avoidance by the very structure of the kernel decomposition into non-intersecting cells. Numerical simulations are provided to highlight the proposed sensor guidance.