(631i) Control and Estimation of the Heat Equation: Real-Time Experimental Application on a Metal Slab

The first is related to the infinite-dimensional nature of distributed parameter systems, which adds a degree of complexity to the design of the controller and state estimator. A simple way to solve this problem is the application of early lumping techniques, which take the set of PDEs and change them into ordinary differential equations [2]. Unfortunately, this approach might lead not only to inaccuracy of the predicted dynamics but also can be computationally intensive. Thus, using late-lumping approaches, which take into account the infinite-dimensional nature of the system and defer the model approximation until implementation [3,4].

Another challenge faced with distributed parameter systems is local actuation and measurement, which is typically the only option in these systems, as distributed actuation and measurements are generally not economically (or physically) viable [5]. Thus, generally, actuation and measurement are done and taken at points of the system, leading to an increase in the complexity of an output feedback controller design.

One of the simplest and most studied distributed parameter system phenomena is the heat transfer across a metal slab. The PDE that describes this system is a parabolic equation, representing the spatiotemporal temperature distribution in the slab.

In this contribution, we consider the real-time state estimation and controller design for a distributed system, specifically a metal slab. The controller takes into account the infinite-dimensional nature of the system, with boundary actuation and point measurement. Temperature sensors are deployed taking into account the optimal and minimal sensor placement in the system to collect real-time temperature data. This data is then used as input for the estimation algorithm. In this case, a Kalman filter based on the PDE model is employed to process the sensor data and update the state estimation of the system.

Then, the controller uses the reconstructed state to regulate the temperature profile in the metal slab using boundary control and an optimal control design. For the optimal controller, a Model Predictive Controller (MPC) design is deployed, taking into account a discrete-time representation of the system and a Linear Quadratic control problem. The MPC is used to optimally reach a desired reference function, where both measured points and reconstructed states are used as desired outputs of the system.

Finally, the implementation of the controller and state estimator is then displayed in real time using a mobile and portable experimental setup. The setup can heat and cool an isolated metal slab while measuring the temperature in specific points, which are then used for the state reconstruction on the output feedback controller developed.

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[2] Alizadeh Moghadam, A.; Aksikas, I.; Dubljevic, S.; Forbes, J.F. Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica 2013, 49, 526–533.

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[4] Curtain, R.; Zwart, H. An Introduction to Infinite Dimensional Linear Systems Theory; Springer: New York, NY, USA, 2020;

[5] Zhang L, Xie J, Dubljevic S. Sensor location selection for continuous pulp digesters with delayed measurements. AIChE J. 2022;