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1. Controlling PDEs with mobile actuators constrained over time-varying reachability sets

   https://doi.org/10.1109/CDC42340.2020.9304472  

   Demetriou, Michael A. ( December 2020 , 2020 59th IEEE Conference on Decision and Control (CDC))

   The use of mobile actuators for the control of spatially distributed systems governed by PDEs results in both implementational and computational challenges. First it requires the backward-in-time solution to the actuator guidance and the backward-in-time solution to the control operator Riccati equation. A way to address this computational challenge is to consider a continuous-discrete alternative whereby the mobile actuator is repositioned at discrete instances and resides in a specific spatial location for a certain time interval. In order to find optimal paths for a given time interval, a set of feasible locations is derived using the reachability set. These reachability sets are further constrained to take into account the time it takes to travel to any spatial position with a prescribed maximum velocity. The proposed hybrid continuous-discrete control and actuator guidance is demonstrated for a 2D diffusion PDE that uses no constraints and angular constraints on the actuator motion. 

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2. Incorporating impact of hazardous and toxic environments on the guidance of mobile sensor networks used for the cooperative estimation of spatially distributed processes

   https://doi.org/10.1109/CDC.2018.8619588  

   Demetriou, Michael A. ( December 2018 , 2018 IEEE Conference on Decision and Control (CDC))

   This work incorporates the effects that hazardous environments have on sensing devices, in the guidance of mobile platforms with onboard sensors. Mobile sensors are utilized in the state reconstruction of spatiotemporally varying processes, often described by advection-diffusion PDEs. A typical sensor guidance policy is based on a gradient ascent scheme which repositions the sensors to spatial regions that have larger state estimation errors. If the cumulative measurements of the spatial process are used as a means to represent the effects of hazardous environments on the sensors, then the sensors are considered inoperable the instance the cumulative measurements exceed a device-specific tolerance level. A binary guidance policy considered earlier repositioned the sensors to regions of larger values of the state estimation errors thus implementing an information-sensitive policy. The policy switched to an information-averse guidance the instance the cumulative effects exceeded a certain tolerance level. Such a binary policy switches the sensor velocity abruptly from a positive to a negative value. To alleviate these discontinuity effects, a ternary guidance policy is considered and which inserts a third guidance policy, the information-neutral policy, that smooths out the transitions from information-sensitive to information-averse guidance. A novelty in this ternary guidance has to do with the level-set approach which changes from a guidance towards large values of the state estimation error towards level sets of the state estimation error and eventually towards reduced values of the state estimation error. An example on an advection-diffusion PDE in 2D employing a single interior mobile sensor using both the binary and ternary guidance policies is used to demonstrate the effects of hazardous environments on both the sensor life expectancy and the performance of the state estimator. 

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3. 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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4. Robust control of PDEs with disturbances using mobile actuators constrained over time-varying reachability sets

   https://doi.org/10.1109/CDC45484.2021.9683015  

   Demetriou, Michael A. ; Iftime, Orest V. ; Zhuk, Sergiy ( December 2021 , 2021 60th IEEE Conference on Decision and Control)

   We design a practical mobile actuator guidance policy for linear parabolic equations in 2D: the guidance is chosen so that H2-measure of uncertainty is minimized provided the system is subject to a distributed disturbance. We first present a guidance policy where the mobile actuator location to be selected will be fixed over a certain time interval of interest. Further we add extra complexity by taking into account the dynamics of the mobile actuator over the 2D domain of interest under reachability constraints. The proposed approach is illustrated through numerical studies. 

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5. "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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