Funnel Control for the Boundary Controlled Heat Equation

Reis, Timo; Selig, Tilman

doi:10.1137/140971567

Cited by 19 publications

(21 citation statements)

References 24 publications

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“…Therefore, (48) holds which, in turn, implies that α k is bounded (by α(ε * k )) and that γ k = α k e k is bounded (by ε * k α(ε * k )). By boundedness of e k+1 , γ k and essential boundedness ofγ k−1 , it follows from (40), together with (42) and (43), thatė k is essentially bounded and so e k ∈ W m .…”

Section: Resultsmentioning

confidence: 95%

“…For such systems, the feasibility of funnel control has to be investigated directly for the (nonlinear) closed-loop system, see e.g. [48] for a boundary controlled heat equation, [47] for a general class of boundary control systems, [6] for the monodomain equations (which represents defibrillation processes of the human heart) and [4] for the Fokker-Planck equation corresponding to the Ornstein-Uhlenbeck process.…”

Section: Resultsmentioning

confidence: 99%

“…We first show that ∀ t ∈ [0, ω) : e k (t) 2 ≤ ε * k (48) by the contradiction argument of Step 3 (mutatis mutandis). Suppose that (48) is false.…”

Section: Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Funnel control of nonlinear systems

Berger

Ilchmann

Ryan

2021

Math. Control Signals Syst.

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Tracking of reference signals is addressed in the context of a class of nonlinear controlled systems modelled by r-th-order functional differential equations, encompassing inter alia systems with unknown “control direction” and dead-zone input effects. A control structure is developed which ensures that, for every member of the underlying system class and every admissible reference signal, the tracking error evolves in a prescribed funnel chosen to reflect transient and asymptotic accuracy objectives. Two fundamental properties underpin the system class: bounded-input bounded-output stable internal dynamics, and a high-gain property (an antecedent of which is the concept of sign-definite high-frequency gain in the context of linear systems).

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Section: Resultsmentioning

confidence: 95%

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Funnel control of nonlinear systems

Berger

Ilchmann

Ryan

2021

Math. Control Signals Syst.

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“…Funnel control is not restricted to systems of ordinary differential equations, but can also be used for infinite-dimensional systems (see e.g. [18,21]) and systems of differentialalgebraic equations (see e.g. [1,2]).…”

Section: Introductionmentioning

confidence: 99%

Funnel control for nonlinear systems with higher relative degree

Berger

Lê

Reis

2018

Proc Appl Math and Mech

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We consider tracking control for nonlinear multi-input, multi-output systems which have arbitrary strict relative degree and input-to-state stable internal dynamics. For a given reference signal, our aim is to design a controller which achieves that the tracking error evolves within a prespecified performance funnel around the reference signal. To this end, we introduce a new controller which involves the first r − 1 derivatives of the tracking error, where r is the strict relative degree of the system.

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“…The heat process, also known as reaction-diffusion process, is used widely in science and engineering and a great deal of contributions have been given to them [1][2][3][4][5][6]. It is well known that the boundary stabilisation problem of integer-order unstable heat system is solved in [7][8][9][10][11][12], where the boundary control law, known as backstepping control law, is in the form of an integral operator with a known, continuous kernel function.…”

Section: Introductionmentioning

confidence: 99%

Boundary feedback stabilisation for the time fractional‐order anomalous diffusion system

Chen

Kou

2016

IET Control Theory & Applications

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In this study, the authors attempt to explore the boundary feedback stabilisation for an unstable heat process described by fractional-order partial differential equation (PDE), where the first-order time derivative of normal reaction-diffusion equation is replaced by a Caputo time fractional derivative of order α ∈ (0, 1]. By designing an invertible coordinate transformation, the system under consideration is converted into a Mittag-Leffler stability linear system and the boundary stabilisation problem is transformed into a problem of solving a linear hyperbolic PDE. It is worth mentioning that with the help of this invertible coordinate transformation, they can explicitly obtain the closed-loop solutions of the original problem. The output feedback problem with both anti-collocated and collocated actuator/sensor pairs in one-dimensional domain is also presented. A numerical example is given to test the effectiveness of the authors' results.

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Funnel Control for the Boundary Controlled Heat Equation

Cited by 19 publications

References 24 publications

Funnel control of nonlinear systems

Funnel control of nonlinear systems

Funnel control for nonlinear systems with higher relative degree

Boundary feedback stabilisation for the time fractional‐order anomalous diffusion system

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