Abstract
In this chapter, we consider the problem of practical output regulation and output tracking for a linear \(2\times 2\) hyperbolic Partial Differential Equation (PDE) system with actuation and load dynamics. Indeed, it is actuated via an Ordinary Differential Equation (ODE) at one boundary and the output to be controlled is the output of an ODE at the other boundary. The main focus is on load tracking. Here, we propose to extend existing results on approximate output regulation to a class of systems similar to that considered in [8] and to extend filtering techniques to a dynamically augmented system with finite-dimensional exosystems considering possible trajectory and disturbance inputs. Issues with respect to small delays in the state reconstruction and feedback loop are considered. Due to the nature of the disturbances, the state estimation and disturbance reconstruction problems are also considered. This scenario finds applications in many systems of engineering interest, such as drilling systems [3], pneumatic systems [17], or electric transmission lines [22].
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Notes
- 1.
Note that an explicit expression of \(g_w, g_\upsilon \) could have been obtained using the transform \(\mathcal {L}_{\text{ obs }}\) with direct formulation. However, it would have led to more complex terms in the boundary conditions (57).
- 2.
Due to the use of an adequate low-pass filter, this will however not prevent the output feedback control law to be robust with regard to small delays.
- 3.
In this section, note that since the first component of X(t) is a flat output, a scalar control should be sufficient.
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Redaud, J., Bribiesca-Argomedo, F., Auriol, J. (2022). Practical Output Regulation and Tracking for Linear ODE-hyperbolic PDE-ODE 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_7
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