Robust Compensation of Delay and Diffusive Actuator Dynamics Without Distributed Feedback

Sanz, Ricardo; Garcia, Pedro Castillo; Krstić, Miroslav

doi:10.1109/tac.2018.2887148

Cited by 10 publications

(6 citation statements)

References 27 publications

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“…We remark that in the case of the wave equation, we must first stabilize it using the control law in [16] and then apply Theorem 3.4. Using Theorem 3.7, we can solve the robust output feedback PDE-ODE stabilization problems considered in [15] for a transport equation and a diffusion equation. Finally applying Theorem 4.4, we can solve the ODE-PDE estimation problem considered in [11] for a transport equation, in [8] for a diffusion equation (see also Remark 4.6) and in [9] and [1] for a wave equation (see Example 5.2).…”

Section: Discussionmentioning

confidence: 99%

“…where In [8] and [15], the actuator is modeled as a 1D diffusion equation with Dirichlet boundary control. This model can be written as an abstract linear system with state space L 2 (0, 1), input space R and output space R. Its state, control, observation and feedthrough operators are defined as follows:…”

Section: Ode Plant With Pde Actuatormentioning

confidence: 99%

“…These operators satisfy the hypothesis in Remark 3.8. Hence for the PDE-ODE cascade systems in [8] and [15], stabilizing controllers can be designed using the approach described in the remark.…”

Section: Ode Plant With Pde Actuatormentioning

confidence: 99%

“…The controllers that solve the stabilization problem in the above works are of the state feedback form. Recently, a dynamic output feedback controller was proposed in [15] for solving the stabilization problem when the PDE (actuator) is either a first-order hyperbolic equation or a 1D diffusion equation. In [25], combining the backstepping approach with the active disturbance rejection control method, an output feedback controller has been developed for stabilizing a wave PDE and ODE cascade system subject to boundary disturbance.…”

Section: Introductionmentioning

confidence: 99%

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Compensating PDE actuator and sensor dynamics using Sylvester equation

Natarajan¹

2020

Preprint

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We consider the problem of stabilizing PDE-ODE cascade systems in which the input is applied to the PDE system whose output drives the ODE system. We also consider the dual problem of constructing an observer for ODE-PDE cascade systems in which the output of the ODE system drives the PDE system, whose output is measured. The PDE in these problems is stable and the ODE is unstable. While the ODE system models the plant in both the problems, the PDE system models the actuator in the stabilization problem and the sensor in the dual problem. In the literature, these problems have been solved for specific PDE models using the backstepping approach. In contrast, in the present work we consider these problems in an abstract framework by letting the PDE system be any regular linear system. Using a state transformation obtained by solving a Sylvester equation with unbounded operators, we first diagonalize the state operator corresponding to the cascade systems. We then solve the stabilization problem and the dual estimation problem, provided they are solvable, by solving certain finite-dimensional counterparts. We also derive necessary and sufficient conditions for verifying the solvability of these problems. We show that the controller which solves the stabilization problem is robust to certain unbounded perturbations. We illustrate our theory by designing a stabilizing controller for a PDE-ODE cascade in which the PDE is a 1D diffusion equation and an observer for a ODE-PDE cascade in which the PDE is a 1D wave equation.

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Section: Discussionmentioning

confidence: 99%

Section: Ode Plant With Pde Actuatormentioning

confidence: 99%

Section: Ode Plant With Pde Actuatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Compensating PDE actuator and sensor dynamics using Sylvester equation

Natarajan¹

2020

Preprint

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show abstract

“…The idea is to design an observer that provides an estimate of

Π (t)

, which is subsequently used by the controller. This is motivated by the recent work [28], which pursues that goal for input delay systems and systems with actuator dynamics described by diffusion PDEs. We make the following assumption regarding plant (1).…”

Section: Output‐feedback Designmentioning

confidence: 99%

Backstepping boundary control design for a cascaded viscous Hamilton–Jacobi PDE–ODE system

Nikdel

Sheikholeslam

Zekri

et al. 2021

IET Control Theory & Appl

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This paper develops state-feedback and output-feedback control design methodologies for boundary stabilization of a class of systems that involves cascade connection of a nonlinear viscous Hamilton-Jacobi partial differential equation (PDE) and a possibly unstable linear ordinary differential equation (ODE). First, an explicit state-feedback controller is designed based on the infinite-dimensional backstepping method and by utilizing a locally invertible feedback linearizing transformation whose role is to convert the non-linear viscous Hamilton-Jacobi PDE to a linear heat equation. Next, an output-feedback controller is proposed which makes the closed-loop system exponentially stable through the measured output of the system. The main feature of the proposed output-feedback scheme is that the distributed integral terms are avoided in the feedback law. This is achieved by exploiting the derived state-feedback controller and introducing a virtual ODE-PDE system, whose ODE state determines the stabilizing feedback law. Then, an observer is proposed that generates state estimates of the virtual system. In both schemes, local exponential stability is shown via Lyapunov analysis and an estimate of the region of attraction is provided. Finally, simulation examples for an unstable ODE are presented to validate the effectiveness of the proposed results.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Reduced-order controllers for multi-agent consensus using output regulation viewpoint

Singh

Natarajan

2020

Automatica

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Robust Compensation of Delay and Diffusive Actuator Dynamics Without Distributed Feedback

Cited by 10 publications

References 27 publications

Compensating PDE actuator and sensor dynamics using Sylvester equation

Compensating PDE actuator and sensor dynamics using Sylvester equation

Backstepping boundary control design for a cascaded viscous Hamilton–Jacobi PDE–ODE system

Reduced-order controllers for multi-agent consensus using output regulation viewpoint

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