Abstract
In this chapter we give an overview on energy shaping control for Distributed Parameter Systems defined on a 1D spatial domain using the port Hamiltonian framework. We consider two different cases: when actuators and sensors are located within the spatial domain and when the actuator is situated at the boundary of the spatial domain, leading to a boundary control system (BCS). In the first case we show how dynamic extensions and structural invariants can be used to change the internal properties of the system when the system is fully actuated, and how it can be done in an approximate way when the system is actuated using piecewise continuous actuators stemming from the use of patches. Asymptotic stability is achieved using damping injection. In the boundary controlled case we show how the closed loop energy function can be partially shaped, modifying the minimum and a part of the shape of this function and how damping injection can be used to guarantee asymptotic convergence.
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Notes
- 1.
Note that we have used the lower indexes \(\zeta \) and t to refer to the partial derivative with respect to that index.
- 2.
\(\otimes \) is the Kronecker product and \(\mathbf {1}_{k\times 1}\) the vector of dimension k containing only ones.
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Le Gorrec, Y., Ramirez, H., Wu, Y., Liu, N., Macchelli, A. (2022). Energy Shaping Control of 1D Distributed Parameter Systems. In: Auriol, J., Deutscher, J., Mazanti, G., Valmorbida, G. (eds) Advances in Distributed Parameter Systems. Advances in Delays and Dynamics, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-94766-8_1
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