The fundamental problem in terms of identification, state estimation, and control of distributed systems is the sensor and actuator locations [1,2,3,6,7]. In particular, the important aspect of a successful controller synthesis in the scope of distributed parameter systems is the placement of actuators/sensors, as inadequate actuator placements may affect important system properties, like its controllability and stabilizability, while inappropriate sensor location placements may deteriorate performance characteristics of the model based controller. Having this in mind, the best controller design (robust, cheap, easily implementable) implies the best selection of sensor and actuator locations. Specifically, by using the optimal control law formulation based on the model predictive control, the idea is to bring the actuator/sensor locations problem into the control design problem. Namely, if the optimal problem is to minimize the performance index by finding the control policy, then to find a minimum over all optimal values with respect to actuator/sensor locations it indeed appears to be truly optimal way of choosing the sensor and actuator locations[4,5].

In particular, in this work the modal model predictive control (MMPC) design method- ology is introduced in the framework of optimal actuation policies which arise due to the presence of input and state constraints for a class of distributed parameter systems modeled by parabolic PDEs. The predictive control law accounts for input and state constraints and with respect to actuator/sensor position placement it generates an optimal actuation policy that switches the applied control action among available pre-specified actuator locations. The proposed constrained predictive control law utilizes a low order modal representation in the optimization functional while higher modes are included only in the PDE state constraints. Accordingly, the proposed model predictive control is then formulated by the minimization algorithm whereby optimization is performed over all available preset collocated actuator/sensor positions. In this sense, the minimizing algorithm provides the control law that chooses among best collocated actuator/sensor positions available with respect to the lowest optimal cost among these positions. An example of a diffusion-reaction process, with spatially-uniform unstable steady state, subject to flux boundary conditions is considered. Simulation results demonstrate a successful application of the proposed predictive control technique that achieves the infinite-dimensional closed-loop system stability and input and state constraints satisfaction through implementation of optimal actuation policies among preset actuator/sensor positions.

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