Stability Analysis and Stabilization of Systems With Input Backlash

Tarbouriech, Sophie; Queinnec, Isabelle; Prieur, Christophe

doi:10.1109/tac.2013.2273279

Cited by 46 publications

(48 citation statements)

References 27 publications

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“…with M 1 defined in (14) (see at the top of the next page), then, for any initial admissible conditions (x(0), Ψ 1 (0)), with x(0) ∈ S 1 , the resulting trajectories of the closed-loop system (12) converge to the set S 0 , where the sets S 1 and S 0 are defined as follows:…”

Section: B Regional Stabilitymentioning

confidence: 99%

See 1 more Smart Citation

Stability analysis of systems with nested saturation and backlash in the loop via nonstandard anti-windup compensation

Tarbouriech

Queinnec

Prieur

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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This paper deals with control systems subject to backlash and saturation in the loop. One of the goals is to characterize, or at least to approximate, the attractor of such nonlinear dynamical systems. Then, the regional or global asymptotic stability of the closed loop with respect to this attractor is handled. An anti-windup inspired loops is added aiming at improving the quality of the attractor approximation in which converge the closed-loop trajectories and of the basin of attraction of such an attractor. Numerically tractable algorithms with feasibility guarantee are provided, as soon as the linear closed-loop system, obtained by neglecting the backlash and saturation effects, is asymptotically stable. The interest of the results is drawn through an illustrative example.

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Section: B Regional Stabilitymentioning

confidence: 99%

“…We want to verify that there exists a class K function α such thatV (x) ≤ −α(V (x)), for all x such that x P x ≥ 1 and x P x ≤ η −1 (i.e. for any x ∈ S 1 \S 0 ), and for all nonlinearities Ψ 1 satisfying Lemma 1 in [14].…”

Section: B Regional Stabilitymentioning

confidence: 99%

Stability analysis of systems with nested saturation and backlash in the loop via nonstandard anti-windup compensation

Tarbouriech

Queinnec

Prieur

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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“…Another constraint than the saturation can be considered. For instance the backlash studied in [40] or the quantization [14].…”

Section: Simulationsmentioning

confidence: 99%

Global Stabilization of a Korteweg--De Vries Equation With Saturating Distributed Control

Marx¹,

Cerpa²,

Prieur³

et al. 2017

SIAM J. Control Optim.

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Abstract. This article deals with the design of saturated controls in the context of partial differential equations. It focuses on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. Two different types of saturated controls are considered. The well-posedness is proven applying a Banach fixed point theorem, using some estimates of this equation and some properties of the saturation function. The proof of the asymptotic stability of the closed-loop system is separated in two cases: i) when the control acts on all the domain, a Lyapunov function together with a sector condition describing the saturating input is used to conclude on the stability; ii) when the control is localized, we argue by contradiction. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation.Key words. Korteweg-de Vries equation, stabilization, distributed control, saturating control, nonlinear system AMS subject classifications. 93C20, 93D15, 35Q531. Introduction. In recent decades, a great effort has been made to take into account input saturations in control designs (see e.g [39], [15] or more recently [17]). In most applications, actuators are limited due to some physical constraints and the control input has to be bounded. Neglecting the amplitude actuator limitation can be source of undesirable and catastrophic behaviors for the closed-loop system. The standard method to analyze the stability with such nonlinear controls follows a two steps design. First the design is carried out without taking into account the saturation. In a second step, a nonlinear analysis of the closed-loop system is made when adding the saturation. In this way, we often get local stabilization results. Tackling this particular nonlinearity in the case of finite dimensional systems is already a difficult problem. However, nowadays, numerous techniques are available (see e.g. [39, 41,37]) and such systems can be analyzed with an appropriate Lyapunov function and a sector condition of the saturation map, as introduced in [39].In the literature, there are few papers studying this topic in the infinite dimensional case. Among them, we can cite [18], [29], where a wave equation equipped with a saturated distributed actuator is studied, and [12], where a coupled PDE/ODE system modeling a switched power converter with a transmission line is considered. Due to some restrictions on the system, a saturated feedback has to be designed in the latter paper. There exist also some papers using the nonlinear semigroup theory and focusing on abstract systems ([20], [34], [36]).Let us note that in [36], [34] and [20], the study of a priori bounded controller is tackled using abstract nonlinear theory. To be more specific, for bounded ([36], [34]) and unbounded ([34]) control operators, some conditions are derived to deduce, from the asymptotic stability of an infinite-dimensional linear system in abstract form, the asymptotic stability when closing the loop with saturating co...

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“…This kind of additional constraints can be easily expressed in a linear matrix inequality form; see Chilali and Gahinet (1996). A typical choice is to consider as region the closed circle centered in (−ω, 0) with radius r > 0, where ω is a positive real scalar, i.e., {z ∈ C : |z + ω| ≤ r}; such a condition is guaranteed by adding the following constraint: see Tarbouriech, Queinnec, and Prieur (2014):…”

Section: Controller Designmentioning

confidence: 99%

“…Therefore, as a second step, by combining Proposition 1 along with a griding procedure for the scalar τ , by solving a finite number of linear matrix inequalities, one may attempt to tighten the set S u . This is a typical approach pursued in the literature; see, e.g., Tarbouriech et al (2014). In particular, throughout this further stage one may also account for different measures for the set S u than the trace criterion considered during the design stage.…”

Section: Controller Designmentioning

confidence: 99%