Observer-based boundary control of distributed parameter systems: a port-Hamiltonian approach

Thesis project

Title: Observer-based boundary control of distributed parameter systems: a port-Hamiltonian approach

Abstract: The port-Hamiltonian approach has shown to be well suited for the modelling and control of distributed parameter systems (DPSs). Some examples of this class of systems are the waves, beams, open channels, fluid motion, piezoelectric structures, and chemical reactors. In this thesis, we have provided some tools for the design of observer-based state feedback (OBSF) controllers for a class of DPSs. The class of DPSs studied in this thesis is the class of boundary controlled port-Hamiltonian systems (BC-PHSs). These are systems described by partial differential equations with actuators and sensors located at the spatial boundaries. For the synthesis, we have used two approaches: the early-lumping and the late-lumping approach. For the early lumping approach, we have proposed two methods to design the OBSF gains. In both, we combine classical control tools as the linear quadratic regulator or pole placement with the port-Hamiltonian approach to guarantee the closed-loop stability when applying the OBSF controller to the BC-PHS. For the late-lumping approach, we have proposed infinite-dimensional observers for BC-PHSs subject to different types of measurements. Depending on the case, the observer convergence is guaranteed to be asymptotic or exponential. Finally, some first steps toward the observer-based damping injection and energy shaping have been studied using the vibrating string as an example.

Model-based control requires the model of the system to be controlled. In general, a model is an approximation of a system and therefore contains errors. For systems described by Partial Differential Equations (PDEs) with actuators and sensors located at the spatial boundaries of the domain, one can design controllers based on a discretized model of the system. The discretized model is described by Ordinary Differential Equations (ODEs) and the controller can be designed more easily than for PDEs. However, since the discretized model is an approximation of the system described by PDEs, the closed-loop stability is not always guaranteed when using classic control tools of Linear Time-Invariant (LTI) systems. We exemplify this in the following example. We consider the vibrating string with force actuators and velocity sensors at both sides of the string. The design is based on a discretized motel that contains 59 state variables (open loop eigenvalues in yellow). We design the state feedback (eigenvalues in blue) and the Luenberger observer (eigenvalues in orange). Then, we analyze the closed-loop eigenvalues (in violet) when applying the designed observer-based state feedback (OBSF) controller to a discretized model that is more precise than the one used for the design (the idea is to approach the original system described by PDEs). Some eigenvalues that were not considered during the design are destabilized in the closed-loop. In a numerical simulation, for a short time interval, the system seems stable. However, when time goes to infinity, the system blows up.

Open-loop and closed-loop eigenvalues

Simulation of the closed-loop system

In this thesis, we propose two methodologies for the design of the state feedback and the Luenberger observer such that we can guarantee closed-loop stability between the OBSF controller (described by ODEs) and the system described by PDEs. In the following figures, we show that the closed-loop eigenvalues (in violet) remain at the left side of the imaginary axis. In the simulation, we show that the observed-displacement approaches the real displacement while both converge to zero.

Open-loop and closed-loop eigenvalues

Simulation of the closed-loop system

References

Toledo-Zucco, J.P., Ramirez, H., Wu, Y., & Le Gorrec, Y. (2022). Linear Matrix Inequality Design of Exponentially Stabilizing Observer-Based State Feedback Port-Hamiltonian Controllers. IEEE Transactions on Automatic Control. Link. arXiv. Animation.
Toledo-Zucco, J.P., Wu, Y., Ramirez, H., & Le Gorrec, Y. (2020). Observer-based boundary control of distributed port-Hamiltonian systems. Automatica. Link. arXiv. Animation.
Malzer, T, Toledo-Zucco, J-P., Gorrec, Y. L., & Schöberl, M. (2020). Energy-based in-domain control and observer design for infinite-dimensional port-Hamiltonian systems. International Symposium on Mathematical Theory of Networks and Systems (MTNS), Cambridge, UK, August 2020. Link, Arxiv.
Toledo-Zucco, J-P., Wu, Y., Ramirez, H., & Le Gorrec, Y. (2019). Observer-Based State Feedback Controller for a class of Distributed Parameter Systems. Workshop on Control of Systems Governed by Partial Differential Equations (CPDE) 2019, Oaxaca, Mexico, May 2019. Link.
Mattioni, A., Toledo-Zucco, J-P., & Le Gorrec, Y. (2019). Observer Based Nonlinear Control of a Rotating Flexible Beam. IFAC World Congress, Berlin, Germany, July 2020. Link.

Infinite-dimensional observers for high order boundary-controlled port-Hamiltonian systems

In this paper, we have proposed an infinite-dimensional observer for a class of Boundary Control Systems. We have provided sufficient conditions to guarantee asymptotic or exponential convergence of the observer depending on the boundary measurements.

Toledo-Zucco, J. P., Wu, Y., Ramirez, H., & Gorrec, Y. L. (2023). Infinite-dimensional observers for high order boundary-controlled port-Hamiltonian systems. IEEE Control Systems Letter. DOI. arXiv.

References

Toledo-Zucco, J.P., Wu, Y., Ramirez, H., & Gorrec, Y. L. (2023). Infinite-dimensional observers for high order boundary-controlled port-Hamiltonian systems. IEEE Control Systems Letter. Link, arXiv. Animation.
Toledo-Zucco, J-P., Wu, Y., Ramirez, H., & Le Gorrec, Y. (2022). Observer design for 1-D boundary controlled port-Hamiltonian systems with different boundary measurements. Workshop on Control of Systems Governed by Partial Differential Equations (CPDE), Kiel, Germany, September 2022. Link.
Toledo-Zucco, J-P., Ramirez, H., Wu, Y., & Le Gorrec, Y. (2019). Passive observers for distributed port-Hamiltonian systems. IFAC World Congress, Berlin, Germany, July 2020. Link. Draft.

In [1], the authors extend the energy shaping control technique to 1D linear boundary controlled port-Hamiltonian Systems. For example, this control law can speed up the end tip position of a vibrating string. This control law is, in general, a state feedback control law. In [2], we propose an alternative to implement these type of control laws using finite-dimensional dynamic observers.

[1] Macchelli, A., Le Gorrec, Y., Ramírez, H., & Zwart, H. (2016). On the synthesis of boundary control laws for distributed port-Hamiltonian systems. IEEE transactions on automatic control, 62(4), 1700-1713.

[2] Toledo, J., Wu, Y., Ramírez, H., & Le Gorrec, Y. (2020). Observer-based boundary control of distributed port-Hamiltonian systems. Automatica, 120, 109130.

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