Fredholm Backstepping Control of Coupled Linear Parabolic PDEs With Input and Output Delays

Deutscher, Joachim; Gabriel, Jakob

doi:10.1109/tac.2019.2945035

Cited by 19 publications

(13 citation statements)

References 19 publications

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“…for an Mc ≥ 1, all ICs w(•, 0) ∈ (L 2 (0, 1)) n − , piecewise differentiable ICs x(z, 0) ∈ R n and any c > 0 so that αc + c < 0. Proof: Observe that (10a-d) is similar to the target system already investigated in the proof of Theorem 1 in [7]. While only parallel transport equations are considered in [7], (10a-b) is a set of cascaded transport equations, which, however, does not change the result.…”

Section: B Controller Decoupling Transformationmentioning

confidence: 99%

“…Proof: Observe that (10a-d) is similar to the target system already investigated in the proof of Theorem 1 in [7]. While only parallel transport equations are considered in [7], (10a-b) is a set of cascaded transport equations, which, however, does not change the result. Hence, the states ε− (t) = {ε − (z, t), z ∈ [0, 1]} and w(t) = { w(z, t), z ∈ [0, 1]} are piecewise continuous and thus bounded on [0, D − ].…”

Section: B Controller Decoupling Transformationmentioning

confidence: 99%

“…11.2]). First backstepping results considered cascaded hyperbolic-parabolic PDE-PDE systems in [7], [19], which arise naturally from the modelling of input and output delays by transport equations. Recently, the much more challenging problem of the backstepping state feedback design for bidirectionally coupled scalar diffusion-reaction and transport equations has been dealt with in [3], [4] using the one-step approach.…”

Section: Introductionmentioning

confidence: 99%

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Output Feedback Control of Coupled Linear Parabolic ODE–PDE–ODE Systems

Deutscher

Gehring

2021

IEEE Trans. Automat. Contr.

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This paper considers the backstepping state feedback and observer design for hyperbolic and parabolic PDEs, which are bidirectionally interconnected in a general coupling structure. Both PDE subsystems consist of coupled scalar PDEs with the heterodirectional hyperbolic PDE subsystem subject to actuation and sensing. By making use of a multi-step approach to construct the transformation into a stable target system, it is shown that a backstepping state feedback and observer design only requires to solve the well-known kernel equations for the hyperbolic and parabolic subsystems as well as additional decoupling equations. The latter are standard initial boundary value problems for parabolic PDEs. This significantly facilitates the well-posedness analysis and the numerical computation of the backstepping controller. Exponential stability is verified for the state feedback loop, the observer error dynamics, and the closedloop system using an observer-based compensator. The proposed backstepping design procedures are demonstrated for numerical examples.

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Section: B Controller Decoupling Transformationmentioning

confidence: 99%

Section: B Controller Decoupling Transformationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Output Feedback Control of Coupled Linear Parabolic ODE–PDE–ODE Systems

Deutscher

Gehring

2021

IEEE Trans. Automat. Contr.

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“…Various transport and flow phenomena are modeled by hyperbolic PDE systems in practice 1‐5 . The typical application examples include traffic flow in road, 6 oil flow in drill wells, 7 time‐delays in systems, 8 current propagation in transmission lines 9 and so forth.…”

Section: Introductionmentioning

confidence: 99%

Output‐feedback fault‐tolerant tracking control for coupled 2 × 2 hyperbolic PDE systems with multiplicative sensor faults

Guo

Qin

Qiu

et al. 2023

Intl J Robust & Nonlinear

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This paper proposes an output-feedback fault-tolerant tracking control scheme for coupled 2 × 2 hyperbolic PDE systems with multiplicative sensor faults. In order to estimate the system states, an observer is designed by constructing four filters, which impose time delays to their input signals. For sensor fault identification problem, the gradient descent method is utilized based on a parametric model to estimate the fault parameter. With the filter-based observer and the estimated fault parameters, a novel control strategy is proposed via backstepping design technique to achieve the tracking objective. Finally, the simulation studies are given to verify the tracking performance of the proposed fault-tolerant tracking control scheme.

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“…Actually, the proposed finite-dimension feedback linearization method was infinite dimensional extension of the backstepping approach. It is well known that backstepping transformation is mainstream method to deal with boundary stabilization problems of parabolic PDEs, such as, linear parabolic PDEs [15][16][17][18][19], nonlinear parabolic PDEs [20][21][22], quasi-linear parabolic PDEs [23], coupled parabolic PDEs and ODE [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Boundary Stabilization of Heat Equation with Multi-Point Heat Source

Jin

Yan

2021

Mathematics

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In this paper, we consider boundary stabilization problem of heat equation with multi-point heat source. Firstly, a state feedback controller is designed mainly by backstepping approach. Under the designed state controller, the exponential stability of closed-loop system is guaranteed. Then, an observer-based output feedback controller is proposed. We prove the exponential stability of resulting closed-loop system using operator semigroup theory. Finally, the designed state and output feedback controllers are effective via some numerical simulations.

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Fredholm Backstepping Control of Coupled Linear Parabolic PDEs With Input and Output Delays

Cited by 19 publications

References 19 publications

Output Feedback Control of Coupled Linear Parabolic ODE–PDE–ODE Systems

Output Feedback Control of Coupled Linear Parabolic ODE–PDE–ODE Systems

Output‐feedback fault‐tolerant tracking control for coupled 2 × 2 hyperbolic PDE systems with multiplicative sensor faults

Boundary Stabilization of Heat Equation with Multi-Point Heat Source

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