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1. "Employing mobile sensor density to approximate state feedback kernels in static output feedback control of PDEs,"

   https://doi.org/10.23919/ACC50511.2021.9483372  

   M. A. Demetriou (May 2021, 2021 American Control Conference (ACC),)

   This work considers the replacement of a full-state feedback controller by a static output feedback controller employing a finite number of point sensors. This is achieved by the approximation of the feedback kernel associated with the full state feedback operator. The feedback kernel is partitioned into equiareal cells and an appropriately selected centroid within each cell serves as the sensor location. This allows one to approximate the inner product of the feedback kernel and the full state by the finite weighted sum of static output feedback measurements. By equating the feedback kernel with the density of a hypothetical sensor network, the problem of approximating the sensor density becomes that of partitioning the sensor density using the proposed computational-geometry based decomposition that is based on a modification of Centroidal Voronoi Tessellations. When the control is considered over a finite horizon and/or the actuator itself is repositioned within the spatial domain, the resulting feedback kernel is rendered time-varying. This requires its partitioning at each time leading to mobile sensors within the spatial domain. Two guidance policies are proposed: one uses the partitioning of the kernel method at each time to find the optimal sensors thus resulting in moving sensors. The other method uses the kernel partitioning only at the initial time and subsequently uses the sensor density as the initial condition for an advection PDE that represents the evolution of the sensor density. This advection PDE is solved for the velocity thereby providing the velocity of the density of the sensor network. Projecting the sensor density velocity onto the same partitioning used for the kernel provides the sensor velocities. A numerical example of an advection diffusion PDE is presented to provide an understanding of this computational geometry based partitioning of feedback kernels. 

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2. Using local reachability sets in guiding mobile actuator and its orbiting sensors to approximate state feedback kernels in output control of PDEs

   https://doi.org/10.1109/CDC51059.2022.9993034  

   Demetriou, Michael A. (December 2022, 2022 IEEE 61st Conference on Decision and Control (CDC))

   This paper considers a class of distributed parameter systems that can be controlled by an actuator onboard a mobile platform. In order to avoid computational costs and control architecture complexity associated with a joint optimization of actuator guidance and control law, a suboptimal policy is proposed that significantly reduces the computational costs. By utilizing a continuous-discrete optimal control design, a mobile actuator moves to a new position at the beginning of a new time interval and resides for a prescribed time. Using the cost to go with variable lower limit, the optimization simplifies to solving algebraic Riccati equations instead of differential Riccati equations. Adding a hardware feature whereby the mobile sensors are constrained to stay within the proximity of the mobile actuator, a feedback kernel decomposition scheme is proposed to approximate a full state feedback controller by the weighted sum of sensor measurements. 

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3. Finite Dimensional Functional Observer Design for Parabolic Systems

   https://doi.org/10.1109/CDC45484.2021.9682848  

   Demetriou, Michael A.; Hu, Weiwei (December 2021, 2021 60th IEEE Conference on Decision and Control)

   This paper combines two control design aspects for a class of infinite dimensional systems, and each of the designs aims at significantly reducing the implementation complexity and computational load. A functional observer, and its extension of an unknown input functional observer, aims to reconstruct a functional of the infinite dimensional state. The resulting compensator only requires the solution to an operator Sylvester equation plus one differential equation for each dimension of the control signal, as opposed to an infinite dimensional filter evolution equation and an associated operator Riccati equation for the filter operator covariance. When the functional to be estimated coincides with the expression of a full state feedback control signal, then the functional observer becomes the minimum order compensator. When the parabolic system admits a decomposition whereby the system is decomposed into a lower finite dimensional subspace comprising the unstable eigenspectrum and an infinite stable subspace, then the functional observer-based compensator design becomes the minimum order compensator for the finite dimensional subsystem. This approach dramatically reduces the computation for solving the ARE needed for the full state controller and the associated Sylvester equation needed for the functional observer. Numerical results for a parabolic PDE in one and two spatial dimensions are included. 

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4. Design and optimization of minimum-order compensators of distributed parameter systems via functional observers and unknown input functional observers

   https://doi.org/10.23919/ECC54610.2021.9654879  

   Demetriou, Michael A.; Hu, Weiwei (June 2021, 2021 European Control Conference (ECC))

   This paper revisits the design of compensator-based controller for a class of infinite dimensional systems. In order to save computational time, a functional observer is employed to reconstruct a functional of the state which coincides with the full state feedback control signal. Such a full-state feedback corresponds to an idealized case wherein the state is available. Instead of reconstructing the entire state via a state-observer and then use this state estimate in a controller expression, a functional observer is used to estimate the product of the state and the feedback operator, thus resulting in a significant reduction in computational load. This observer design is subsequently integrated with a sensor selection in order to improve controller performance. An appropriate metric is used to optimize the sensor location resulting in improved performance of the functional observer-based compensator. The integrated design is further extended to include a controller with an unknown input functional observer. The results are applied to 2D partial differential equations and detailed numerical studies are included to provide an appreciation in the significant savings in both operational and computational costs. 

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5. Optimal Communication Topology and Static Output Feedback of Networked Collocated Actuator/Sensor Pairs in Distributed Parameter Systems

   https://doi.org/10.23919/ACC45564.2020.9147660  

   Demetriou, Michael A.; Hadjicostis, Christoforos N. (July 2020, 2020 American Control Conference (ACC))

   This paper is motivated by economic aspects of fixed initial and operating costs for control of spatially distributed systems. In particular, the paper investigates the possibility of a large number of inexpensive actuating and sensing devices, as an alternative to (a reduced number of) expensive high capacity devices. While such an alternative reduces the fixed initial costs associated with actuators and sensors, it may also lead to increased operating costs resulting from communication requirements between the now-networked actuator-sensor-control units. To simplify the controller architecture, a proportional controller is assumed that amounts to a static output feedback controller. In a network of n actuator-sensor pairs, an all-to-all communication topology results in a fully populated static output feedback matrix with as much as n(n-1) communication links. In addition to a traditional performance index used to obtain the static output feedback gain matrix, this paper proposes a mixed index wherein both the traditional performance index and the number of communication links (representing operating costs associated with information exchange links), are taken into account. As an example, the proposed scheme is applied to a parabolic partial differential equation having four actuator-sensor pairs. The resulting optimization produces a sparse static gain matrix with a communication topology that has half the graph edges of the fully connected case and with essentially the same performance. 

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