Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations

Klein, Olaf; Krejčı́, Pavel

doi:10.1016/s1468-1218(03)00013-0

Cited by 36 publications

(21 citation statements)

References 7 publications

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“…T ∈ R n ; f j (·) and f n (·) are unknown smooth functions; d(t) is a bounded disturbance; y ∈ R is the output of the system; and u ∈ R is the input of the system and the output of the hysteresis nonlinearity, which is represented by the generalized Prandtl-Ishlinskii model in [7] as follows…”

Section: A Problem Formulationmentioning

confidence: 99%

“…the backlash-like hysteresis and the conventional PrandtlIshlinskii hysteresis model discussed in the above works [3]- [6], the generalized Prandtl-Ishlinskii hysteresis model proposed in [7], can capture the hysteresis phenomenon more accurately and accommodate more general classes of hysteresis shapes, by adjusting not only the density function, but also the input function. However, the difficulty in dealing with the generalized Prandtl-Ishlinskii hysteresis model lies in that the input function in the generalized Prandtl-Ishlinskii hysteresis model is unknown and non-affine.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive neural control for uncertain nonlinear systems in pure-feedback form with hysteresis input

Ren

Lee

et al. 2008

2008 47th IEEE Conference on Decision and Control

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Abstract-In this paper, adaptive neural control is investigated for a class of unknown nonlinear systems in purefeedback form with the generalized Prandtl-Ishlinskii hysteresis input. The non-affine problem both in the pure-feedback form and in the generalized Prandtl-Ishlinskii hysteresis input function is solved by adopting the Mean Value Theorem. By utilizing Lyapunov synthesis, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded (SGUUB), and the tracking error converges to a small neighborhood of zero. Simulation results are provided to illustrate the performance of the proposed approach.

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Section: A Problem Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Adaptive neural control for uncertain nonlinear systems in pure-feedback form with hysteresis input

Ren

Lee

et al. 2008

2008 47th IEEE Conference on Decision and Control

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“…3. An evolution equation like (3.5) that involves an hysteresis operator with input perturbation arises, for example, if one is dealing with a model for visco-elasto-plasticity with hysteresis operators as in [10]. Using the Andrews trick (see [1]) to estimate the time and space dependent displacement, one introduces a function p, being the antiderivative with respect to space of the velocity, i.e., of the derivative of the displacement with respect to time.…”

Section: Uniform Estimates For Ode Involving Outward Pointing Scalar mentioning

confidence: 99%

“…(iii) Conditions to check the outward pointing properties for the Stop operator, the Play operator, the Prandtl-Ishlinskii operator, and generalized Prandtl-Ishlinskii operators can be found in [10,11]. In [2,3] such conditions have been formulated for Preisach operators and for the inverse of Preisach operators one can find corresponding conditions in [7,8].…”

Section: Causal Operator That Is Pointing Outwards With Boundmentioning

confidence: 99%

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Outward pointing properties for vectorial hysteresis operators and some applications

Klein¹

2008

J. Phys.: Conf. Ser.

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Outward pointing properties for vectorial hysteresis operators and some applications Abstract. In a number of papers, the concept of outward pointing hysteresis allowed the application of the method of invariant regions to derive uniform bounds for solutions to evolution equations involving scalar hysteresis operators and input perturbations.The aim of this note is to generalize the "pointing outward" properties to vectorial hysteresis operators, and to show how these properties can be used to derive uniform estimates for differential equations involving vectorial hysteresis operators and input perturbations. Introduction and statement of the problemThe method of invariant regions is quite often used to derive uniform estimates for solutions to evolution equations involving nonlinear scalar superposition operators and some input perturbations, such that asymptotic stability results can be shown, see, e.g., [16,17].In a number of papers, the concept of outward pointing hysteresis allowed the application of the method of invariant regions also to evolution equations involving scalar hysteresis operators and input perturbations, see [2,3,7,8,9,10,11].The aim of this note is to generalize the "pointing outward" properties to vectorial hysteresis operators, and to show how these properties can be used to derive uniform estimates for differential equations involving vectorial hysteresis operators and input perturbations.After recalling the definition of hysteresis operators in Section 2, in Section 3 the outward pointing definitions for scalar hysteresis operators will be recalled and some theorems will show that one can derive uniform bounds for evolution equations with hysteresis operators in a similar way as is done with the method of invariant regions for evolution equations with superposition operators.In Section 4, the definition of the outward pointing condition for vectorial hysteresis operators will be formulated, some examples for vectorial operators satisfying this condition will be presented, and it will be shown how one can derive uniform bounds for ODEs involving this kind of operator. Moreover, there will be a discussion of which problems arise if one also takes input perturbations into account. To overcome these problems, the more restrictive componentwise outward pointing condition for vectorial hysteresis operators is introduced in Section 5, and uniform estimates for ODEs with componentwise outward pointing vectorial hysteresis operators and input perturbations are derived.

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Adaptive dynamic surface control of a class of pure feedback systems preceded by generalized P‐I hysteresis

Liu

et al. 2020

Asian Journal of Control

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Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations

Cited by 36 publications

References 7 publications

Adaptive neural control for uncertain nonlinear systems in pure-feedback form with hysteresis input

Adaptive neural control for uncertain nonlinear systems in pure-feedback form with hysteresis input

Outward pointing properties for vectorial hysteresis operators and some applications

Adaptive dynamic surface control of a class of pure feedback systems preceded by generalized P‐I hysteresis

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