Reset integral control for improved settling of PID-based motion systems with friction

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

doi:10.1016/j.automatica.2019.06.017

Cited by 38 publications

(70 citation statements)

References 25 publications

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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: 81%

“…We emphasize that Assumption 1 is stronger than (implies) Assumption 2, but characterizes a simplified setting addressed in [8] and more recently addressed in our works [22,17]. Assumption 2 is weaker and therefore requires more advanced techniques, presented in [18].…”

Section: Assumption 2 (Stribeck Friction)mentioning

confidence: 99%

“…As compared to (6), here matrix A takes an observable canonical form. A first reason for introducing the new representation (17) is that the set of equilibria A in (9) simplifies to…”

Section: Stick and Slip Observed From Insightful Coordinatesmentioning

confidence: 99%

“…Under Assumption 1, the compact set A in (9) is globally KL asymptotically stable for (6). Equivalently, the compact set in (18) is globally KL asymptotically stable for (17). Namely, there exists a class KL function β such that all solutions x satisfy…”

Section: Stick and Slip Observed From Insightful Coordinatesmentioning

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: 81%

Section: Assumption 2 (Stribeck Friction)mentioning

confidence: 99%

“…As compared to (6), here matrix A takes an observable canonical form. A first reason for introducing the new representation (17) is that the set of equilibria A in (9) simplifies to…”

Section: Stick and Slip Observed From Insightful Coordinatesmentioning

confidence: 99%

Section: Stick and Slip Observed From Insightful Coordinatesmentioning

confidence: 99%

See 2 more Smart Citations

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

Self Cite

View full text Add to dashboard Cite

show abstract

“…in which is the position of each system with respect to its equilibrium, , are its velocity and mass, respectively, and ( ) is a known external input force. We suppose that the masses are connected by a spring with stiffness coefficient and subject to Coulomb frictions described as in [26,27] by the set-valued maps ⇉ Ψ ( ),…”

Section: Numerical Examplementioning

confidence: 99%

Observer design via interconnections of second‐order mixed sliding‐mode/linear differentiators

Andrieu

Astolfi

Bernard

2020

Intl J Robust & Nonlinear

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High-gain observers and sliding mode observers are two of the most common techniques to design observers (or differentiators) for lower triangular nonlinear dynamics. While sliding mode observers can handle globally bounded nonlinearities, high-gain linear techniques can deal with globally Lipschitz nonlinearities. To gain in generality and avoid the usual assumption that the plant's solutions are bounded with known bound, we propose here to mix both designs in the more general case where the nonlinearities satisfy a global incremental affine bound. We inspire from the recently-developed low-power high-gain observer technique, which relies on the interconnection of several second order high-gain observers. Adding slidingmode correction terms into this low-power structure enables to guarantee global convergence of the estimation error in finite-time with gains depending only on the parameters of the incremental affine bound of the nonlinearities. The estimation error is also proved to be uniformly stable along solutions starting from any compact sets of initial conditions.

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Stick‐slip and convergence of feedback‐controlled systems with Coulomb friction

Ruderman

2021

Asian Journal of Control

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An analysis of stick-slip behavior and convergence of trajectories in the feedback-controlled motion systems with discontinuous Coulomb friction is provided. A closed-form parameter-dependent stiction region, around an invariant equilibrium set, is proved to be always reachable and globally attractive. It is shown that only asymptotic convergence can be achieved, with at least one but mostly an infinite number of consecutive stick-slip cycles, independent of the initial conditions. Theoretical developments are supported by a number of numerical results with dedicated convergence examples.

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Reset integral control for improved settling of PID-based motion systems with friction

Cited by 38 publications

References 25 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

Observer design via interconnections of second‐order mixed sliding‐mode/linear differentiators

Stick‐slip and convergence of feedback‐controlled systems with Coulomb friction

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