Experimental model inverse-based hysteresis compensation on a piezoelectric actuator

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1 Experimental model inverse-based hysteresis compensation on a piezoelectric actuator R. OUBELLIL, 1 L. Ryba, 1 A. Voda, 1 M. Rakotondrabe, 2 1 GIPSA-lab, Grenoble Images Parole Signal Automatique 2 FEMTO-ST, Franche-Comté Electronique Mécanique Thermique et Optique Sciences et Technologies 9 octobre 2015 () Journées d Automatique GDR MACS 9 octobre / 27

2 Outline 1 Problem statement and aims of work. 2 System description and experimental setup. 3 Hysteresis modeling. Preisach model. Prandtl-Ishlinskii model. Modified Prandtl-Ishlinskii model. 4 Open-loop hysteresis compensation. Multiplicative inverse structure of Preisach model. Inverse Prandtl-Ishlinskii model. Inverse Modified Prandtl-Ishlinskii model. 5 Experimental results. 6 Conclusion. 2 / 27

3 Introduction Piezoelectric actuators are used at micro-/nanoscale (see Binning & Rohrer 1986). Hysteresis nonlinearity degrades piezoactuators performance (see Abramovitch 2007). Preisach model of hysteresis (see Preisach 1935) is the most used approach. Several alternatives to Preisach model (see Bobbio 1993). Comparison of Preisach model with its alternative models : classical Prandtl-Ishlinskii model. Modified Prandtl-Ishlinskii model (see Kuhnen 2001). 3 / 27

4 Introduction Piezoelectric actuators are used at micro-/nanoscale (see Binning & Rohrer 1986). Hysteresis nonlinearity degrades piezoactuators performance (see Abramovitch 2007). Preisach model of hysteresis (see Preisach 1935) is the most used approach. Several alternatives to Preisach model (see Bobbio 1993). Comparison of Preisach model with its alternative models : classical Prandtl-Ishlinskii model. Modified Prandtl-Ishlinskii model (see Kuhnen 2001). 3 / 27

5 Introduction Piezoelectric actuators are used at micro-/nanoscale (see Binning & Rohrer 1986). Hysteresis nonlinearity degrades piezoactuators performance (see Abramovitch 2007). Preisach model of hysteresis (see Preisach 1935) is the most used approach. Several alternatives to Preisach model (see Bobbio 1993). Comparison of Preisach model with its alternative models : classical Prandtl-Ishlinskii model. Modified Prandtl-Ishlinskii model (see Kuhnen 2001). 3 / 27

6 Introduction Piezoelectric actuators are used at micro-/nanoscale (see Binning & Rohrer 1986). Hysteresis nonlinearity degrades piezoactuators performance (see Abramovitch 2007). Preisach model of hysteresis (see Preisach 1935) is the most used approach. Several alternatives to Preisach model (see Bobbio 1993). Comparison of Preisach model with its alternative models : classical Prandtl-Ishlinskii model. Modified Prandtl-Ishlinskii model (see Kuhnen 2001). 3 / 27

7 Introduction Piezoelectric actuators are used at micro-/nanoscale (see Binning & Rohrer 1986). Hysteresis nonlinearity degrades piezoactuators performance (see Abramovitch 2007). Preisach model of hysteresis (see Preisach 1935) is the most used approach. Several alternatives to Preisach model (see Bobbio 1993). Comparison of Preisach model with its alternative models : classical Prandtl-Ishlinskii model. Modified Prandtl-Ishlinskii model (see Kuhnen 2001). 3 / 27

8 Introduction Piezoelectric actuators are used at micro-/nanoscale (see Binning & Rohrer 1986). Hysteresis nonlinearity degrades piezoactuators performance (see Abramovitch 2007). Preisach model of hysteresis (see Preisach 1935) is the most used approach. Several alternatives to Preisach model (see Bobbio 1993). Comparison of Preisach model with its alternative models : classical Prandtl-Ishlinskii model. Modified Prandtl-Ishlinskii model (see Kuhnen 2001). 3 / 27

9 Introduction Hysteresis in piezoelectric (PE) materials y x (a) U=0 Piezo Element at rest (b) U=0 Piezo Element expansion x retraction (c) U=0 Piezo Element x Operating principle of piezoelectric actuators. Hysteresis nonlinearity in PE actuators v p x Piezoelectric actuator capacitive sensor xp Piezoelectric actuator in open-loop. x p is the piezoactuator displacement. v px is the control input voltage. 4 / 27

10 Introduction Hysteresis in piezoelectric materials 1 60 V Hz Displacement (µm) V 50 V Displacement (µm) Hz 595 Hz Input voltage (V) Static hysteresis Input voltage (V) Dynamic or rate-dependent hysteresis. Displacement (µm) Displacement (µm) Dead-zone Saturation Input voltage (V) Symmetric hysteresis Input voltage (V) Asymmetric hysteresis. 5 / 27

11 Introduction Context & and problem statement Context : Accurate micropositioning with piezoelectric actuators. Problem statement : Hysteresis generates positioning errors. Displacement (µm) reference piezoactuator with hysteresis Tracking error (µm) Time (s) Triangle waveform tracking Time (s) Tracking error. Choose a precise and an easily implementable compensation approach. 6 / 27

12 Introduction Aims of work x -1 p r xp Open-loop compensation of hysteresis. H is the hysteresis. ˆ H 1 is the inverse hysteresis model Mathematical inverse. Multiplicative inverse structure. Modeling hysteresis for compensation. Classical Preisach model. Classical Prandtl-Ishlinskii (PI) model Modified Prandtl-Ishlinskii (MPI) model Experimental comparison : Multiplicative inverse structure of Preisach model, Inverse PI model, and Inverse MPI model. 7 / 27

13 System description and experimental setup Experimental setup of micro-/nanopositioning platform, GIPSA-lab The horizontal direction (X) Piezoelectric actuator Tritor T Capacitive sensor. Development PC. 8 / 27

14 Hysteresis modeling Classical Preisach model Preisach operator. Block diagram of the Preisach model. x p (t) = H(v px (t)) = µ(α,β)γ[v px (t)]dαdβ (1) α β µ(α, β) is the Preisach weighting function, γ[v px (t)] is the Preisach operator having an output of +1 or 0, α, β are the upper and lower switching values of the input v px (t). 9 / 27

15 Hysteresis modeling Classical Preisach model v px (t) > 0 x p (t) = N k=1 [X p(α k, β k 1 ) X p (α k, β k )] + X p (v px (t),β N ) (2) v px (t) < 0 x p (t) = N 1 k=1 [X p(α k, β k 1 ) X p (α k, β k )] + X p (α N,β N 1 ) X p (α N,v px (t)) (3) X p (α,β) = x pα x pαβ (4) x pα is the displacement when the input voltage increases from 0 to α. x pαβ is the displacement when the input voltage decreases from α to β. 10 / 27

16 Hysteresis modeling Classical Preisach model Approximation of X p (α,β) X p (α,β) = θ 0 β + θ 1 α + θ 2 β 2 + θ 3 βα + θ 4 α 2 +θ 5 β 3 + θ 6 β 2 α + θ 7 βα 2 + θ 8 α 3 (5) 40 Experimental Simulated data Displacement (µm) Input voltage (V) Experimental and identified Preisach model. + Capture asymmetric hysteresis. Its inverse is numerical. 11 / 27

17 Hysteresis model Classical Prandtl Ishlinskii Model Weighted backlash operator. x p (t) = H xrh [v px,x p0 ](t) = max{v px (t) r xh,min{v px (t) + r xh,x p (t T s )}}, (6) H r xh is the backlash operator, x p0 the initial state of the backlash, r xh is the threshold, T s is the sampling time. 12 / 27

18 Hysteresis model Classical Prandtl Ishlinskii Model backlash -r xh1 gain w x Hi r xh1 vpx(t) -r xh2 r xh2 xp(t) -r xhn rx HN Superposition of backlash operators w xh is the vector of weights. x p (t) = H x [v px ](t) = w xh T H r xh [v px,x p0 ](t), (7) Thresholds and initial states initialization : r xh i = i n + 1 max{v px}, i = 0,...,n, x p0i = 0, i = 0,...,n. (8) 13 / 27

19 Hysteresis modeling Classical Prandtl Ishlinskii Models (PI) Weights identification E x [v px, x p ](t) = H x [v px ](t) x p (t) (9) x p (t) is the measured output displacement. 40 Experimental Simulated data 40 Experimental Simulated data Displacement (µm) Displacement (µm) Input voltage (V) Before identification (PI). + Its inverse is analytic. Can not Capture asymmetric hysteresis Input voltage (V) After identification (PI). 14 / 27

20 Hysteresis modeling Modified Prandtl Ishlinskii Model Weighted backlash operator. Weighted dead-zone operator. x ps (t) = S rxs [x p ](t) = { max{x p (t) r xs,0}, r xs > 0 x p (t), r xs = 0 (10) S r xs r xs is the dead-zone, is threshold of dead-zone. 15 / 27

21 Hysteresis modeling Modified Prandtl Ishlinskii Model backlash -r xh1 r xh1 gain w x Hi backlash -rx S1 r X S1 gain w x S i vpx(t) -r xh2 r xh2 xp(t) xp(t) -r X S2 r X S2 xps(t) -r xhn rx HN Superposition of backlash operators. -r X SM r X SM Superposition of dead-zone operators. x ps (t) = S x [H x [v px ]](t) = w xs T S rxs [w xh T H rxh [v px,x p0 ]](t) (11) w xs is the vector of dead-zones weights. 16 / 27

22 Hysteresis modeling Modified Prandtl Ishlinskii Model (MPI) backlashes (r xh i,w xhi ) inverse dead-zones (r x S i,w x Si ) vpx(t) backlash -r xh1 r xh1 -r xh2 r xh2 gain w x Hi xp(t) Error xp(t) backlash -rx -r S1 X S2 r X S1 r X S2 gain w x S i xps(t) -r xhn rx HN Superposition of backlash operators. -r X SM r X SM Superposition of dead-zone operators. Weights identification E x [v px, x p ](t) = H x [v px ](t) Sx 1 [ x p ](t) = w T x H H rxh [v px,x p0 ](t) w T x S S r xs [ x p ](t) (12) w T x S are the weights of inverse dead-zones, S r x are the inverse dead-zones. 17 / 27

23 Hysteresis modeling Modified Prandtl Ishlinskii Model (MPI) Inverse dead-zones thresholds initialization : r x S i = i m + 1 max{x p}, i = 0,...,m, (13) r x S i is the inverse dead-zones thresholds. 40 Experimental Simulated data 40 Simulated data Experimental Displacement (µm) Displacement (µm) Input voltage (V) Before identification (MPI). + Can Capture asymmetric hysteresis. + Its inverse is analytic Input voltage (V) After identification (MPI). 18 / 27

Open-loop hysteresis compensation Multiplicative inverse structure for Preisach model Multiplicative inverse structure of Preisach model.

24 Open-loop hysteresis compensation Multiplicative inverse structure for Preisach model Multiplicative inverse structure of Preisach model. x p (t) = H(v px (t)) = ρ(v px (t)) + Γ(v px (t)) (14) v px (t) = ρ 1 (v px (t))[x pr (t) Γ(v px (t))] (15) Approximated multiplicative inverse structure v px (t) = ρ 1 (v px (t T s ))[x pr (t) γ(v px (t T s ))] (16) T s is the sampling time. 19 / 27

25 Open-loop hysteresis compensation Inverse Classical and Modified Prandtl-Ishlinskii models inverse backlashes (r x H i?,w x Hi?) inverse dead-zones (r x S i,w x Si ) backlash -r xh1 r xh1 gain w x Hi backlash -rx S1 r X S1 gain w x S i vpx(t) -r xh2 r xh2 xp(t) xp(t) -r X S2 r X S2 xps(t) -r xhn rx HN -r X SM r X SM r x H i = i j=0 w xh j (r xh i r xh j ), i = 0,...,n, (17) w xh i w x H i = ( )( ),w w + i xh 0 w xh j w xh 0 + i 1 x H 0 = w xh j j=1 j=1 1 w xh 0,i = 1,...,n (18) 20 / 27

26 Experimental results Displacement (µm) uncompensated PI Preisach MPI Desired displacement (µm) Experimental hysteresis compensation for 1 Hz triangle input signal. In the saturated zone Uncompensated system : 10 %. Compensated system with inverse PI : 3.3 %. Compensated system with multiplicative inverse of Preisach : 1.2 %. Compensated system with inverse MPI : 0.5 %. 21 / 27

27 Experimental results Displacement ref PI MPI Preisach uncomp Tracking error (µm) PI MPI Preisach uncomp Time (s) Triangular waveform tracking (1 Hz) Time (s) Tracking error (1 Hz). Displacement (µm) ref PI GPI Preisach uncomp Tracking error (µm) PI MPI Preisach uncomp Time (s) Tracking waveform tracking (10 Hz) Time (s) Tracking error (10 Hz). 22 / 27

28 Experimental results 40 ref uncomp PI MPI Preisach 2 PI MPI Preisach uncomp Displacement (µm) Tracking error (µm) Time (s) Triangular waveform tracking (50 Hz) Time (s) Tracking error (50 Hz). Table: RMS and maximal tracking error - numerical values 1 Hz 10 Hz 50 Hz uncomp Preisach PI MPI rms (µm) max (µm) rms (µm) max (µm) rms (µm) max (µm) / 27

29 Conclusion Experimental validation on the horizontal (X) axis. Inverse multiplicative structure of Preisach model, + analytic, + compensates for asymmetric hysteresis. computationally intensive. Inverse Prandtl-Ishlinskii (PI) model, + analytic, + less time consuming. does not Compensate for asymmetric hysteresis. Modified inverse Prandtl-Ishlinskii (MPI) model, + analytic, + less time consuming, + compensate for asymmetric hysteresis. 24 / 27

30 References G. Binning and H. Rohrer, Scanning Tunneling Microscopy, IBM Journal of Research and Development, vol. 30, no. 4, pp , D. Y. Abramovitch, S. B. Andersson, L. Y. Pao, and G. Schitter, A Tutorial on the Mechanisms, Dynamics, and Control of Atomic Force Microscopes, ACC 2007, pp , Jul S. Bobbio and G. Marrucci, A possible alternative to Preisach model of static hysteresis, Il Nuovo Cimento D, vol. 15, no. 5, pp , F. Preisach, Über die magnetische Nachwirkung, Zeitschrift für physik, vol. 94, no. 5-6, p , K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities : A Modified Prandtl-Ishlinskii approach, European Journal of Control, vol. 9, no. 4, pp , / 27

31 THANK YOU FOR YOUR ATTENTION Raouia OUBELLIL PhD student GIPSA-lab, Grenoble Insititut of Technology 11 rue des Mathématiques, Saint Martin d Hères cedex Phone : Fax : gipsa-lab.grenoble-inp.fr 26 / 27

32 Questions? 27 / 27

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