Hybrid model formulation and stability analysis of a PID-controlled motion system with Coulomb friction

Bisoffi, Andrea; Beerens, R.; Zaccarian, Luca; Heemels, W.P.M.H.; Nijmeijer, Henk; Wouw, Nathan van de

doi:10.1016/j.ifacol.2019.11.760

Cited by 2 publications

(11 citation statements)

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“…The results presented in this paper provide a unified and comprehensive overview of the research accomplishments reported in [22,17,18,21,15] and the preliminary works [16,19]. As compared to those works we provide here a unified development, highlighting the importance of building hybrid models comprising logic variables to allow for the construction of smooth or Lipschitz Lyapunov functions, in addition to including a novel understanding of the exponential convergence properties of certain solutions in the Coulomb friction case.…”

Section: Introductionmentioning

confidence: 82%

“…The timerτ in H δ removes these Zeno solutions, and exploits the inherent dwell-time property of solutions to (17) established in Lemma 2 to make sure that the (unique, from Lemma 1) solution to (17) is semiglobally captured by H δ . Indeed, after solutions to H δ exit a stick phase and enter a slip phase jumping from D 1 or D −1 , the timer is reset to zero via g 1 or g −1 and enforces that a time δ elapses before solutions exit a stick phase again (due to the conditionτ ∈ [δ, 2δ]), which corresponds to the property of solutions to (17) in Lemma 2. The fact that model H δ correctly represents, in a semiglobal fashion, dynamics (17) is established in the next lemma, which is proven in [21,Lemma 2]. Lemma 3.…”

Section: Remarkmentioning

confidence: 92%

“…Then the following semi-global dwell-time result has been proven in [21,Lemma 1] for the Coulomb case of Assumption 1.…”

Section: Semiglobal Dwell Time and Hybrid Extended Modelmentioning

confidence: 99%

“…As suggested in [21], based on Lemma 2, we may introduce the following hybrid extended model capable of semiglobally representing dynamics (17). More precisely, the next model semiglobally reproduces the solutions of (17) in the sense rigorously characterized in Lemma 3 below.…”

Section: Remarkmentioning

confidence: 99%

“…This was a main motivation for introducing the hybrid extended model (23). Indeed, model (23) simplifies the Lyapunov characterization of the desirable behavior of solutions by way of introducing, in [21], an equivalent smooth version of function V , corresponding tō…”

Section: Lyapunov Functions For Proving Theoremmentioning

confidence: 99%

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To stick or to slip: A reset PID control perspective on positioning systems with friction

Bisoffi

Beerens

Heemels

et al. 2020

Annual Reviews in Control

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We overview a recent research activity where suitable reset actions induce stability and performance of PID-controlled positioning systems suffering from nonlinear frictional effects. With a Coulomb-only effect, PID feedback produces a set of equilibria whose asymptotic (but not exponential) stability can be certified by using a discontinuous Lyapunov-like function. With velocity weakening effects (the so-called Stribeck friction), the set of equilibria becomes unstable with PID feedback and the so-called "hunting phenomenon" (persistent oscillations) is experienced. Resetting laws can be used in both scenarios. With Coulomb friction only, the discontinuous Lyapunov-like function immediately suggests a reset action providing extreme performance improvement, preserving stability and inducing desirable exponential convergence of a relevant subset of the solutions. With Stribeck, a more sophisticated set of logic-based reset rules recovers global asymptotic stability of the set of equilibria, providing an effective solution to the hunting instability. We clarify here the main steps of the Lyapunov-based proofs associated to our reset-enhanced PID controllers. These proofs involve building semiglobal hybrid representations of the solutions in the form of hybrid automata whose logical variables enable transforming the aforementioned discontinuous function into smooth or at least Lipschitz ones. Our theoretical results are illustrated by extensive simulations and experimental validation on an industrial nano-positioning system.

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Section: Introductionmentioning

confidence: 82%

Section: Remarkmentioning

confidence: 92%

“…Then the following semi-global dwell-time result has been proven in [21,Lemma 1] for the Coulomb case of Assumption 1.…”

Section: Semiglobal Dwell Time and Hybrid Extended Modelmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Lyapunov Functions For Proving Theoremmentioning

confidence: 99%

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To stick or to slip: A reset PID control perspective on positioning systems with friction

Bisoffi

Beerens

Heemels

et al. 2020

Annual Reviews in Control

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Reset PID Design for Motion Systems With Stribeck Friction

Beerens

Bisoffi

Zaccarian

et al. 2022

IEEE Trans. Contr. Syst. Technol.

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We present a reset control approach to achieve setpoint regulation of a motion system with a proportionalintegral-derivative (PID) based controller, subject to Coulomb friction and a velocity-weakening (Stribeck) contribution. While classical PID control results in persistent oscillations (hunting), the proposed reset mechanism induces asymptotic stability of the setpoint, and significant overshoot reduction. Moreover, robustness to an unknown static friction level and an unknown Stribeck contribution is guaranteed. The closed-loop dynamics are formulated in a hybrid systems framework, using a novel hybrid description of the static friction element, and asymptotic stability of the setpoint is proven accordingly. The working principle of the controller is demonstrated experimentally on a motion stage of an electron microscope, showing superior performance over classical PID control. I. INTRODUCTIONFriction is a performance-limiting factor in many highprecision motion systems for which many control techniques exist in the literature. A branch of control solutions relies on developing as-accurate-as-possible friction models, used for online compensation in a control loop, see, e.g., [7], [22], [31], [32]. These model-based friction compensation methods are typically prone to model mismatches due to, e.g., unreliable friction measurements, or time-varying or uncertain friction characteristics. Model-based techniques, therefore, may suffer from over-or undercompensation of friction, thereby resulting in loss of stability of the setpoint [39], and thus limiting the achievable positioning accuracy. Adaptive control methods (see, e.g., [5], [17]) provide some robustness to timevarying friction characteristics, but model mismatches (and the associated performance limitations) still remain. Non-modelbased control schemes have also been proposed, examples of which are impulsive control (see, e.g., [36], [47]), ditheringbased techniques (see, e.g., [29]), sliding-mode control (see, e.g., [10]), or switched control [35]. Apart from properly smoothened and paramterized sliding-mode control solutions (see, e.g., [3]), these non-model-based controllers may employ high-frequency control signals, risking excitation of highfrequency dynamics, in addition to raising tuning challenges.

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Hybrid model formulation and stability analysis of a PID-controlled motion system with Coulomb friction

Cited by 2 publications

References 9 publications

To stick or to slip: A reset PID control perspective on positioning systems with friction

To stick or to slip: A reset PID control perspective on positioning systems with friction

Reset PID Design for Motion Systems With Stribeck Friction

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