Compensation of play operator-based Prandtl-Ishlinskii hysteresis model using a stop operator with application to piezoelectric actuators
Zhi Li, Omar Aljanaideh, Subhash Rakheja and Chun-Yi Su* Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd. W., Montreal, Quebec, H3G 1M8, Canada E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] *Corresponding author
Mohammad Al Janaideh Department of Mechatronics Engineering, The University of Jordan, Amman, 11942, Jordan E-mail: [email protected]
Abstract: Piezoelectric actuators exhibit limited tracking performance in precision control due to their inherent hysteresis non-linearity. In this paper, the hysteresis behaviour is described by a play operator-based Prandtl-Ishlinskii (PPI) model. A corresponding stop operator-based Prandtl-Ishlinskii (SPI) model is utilised to compensate for the hysteresis non-linearity. For this purpose, the two parameters describing the SPI model, the thresholds and the weights, are analytically derived based on the PPI model, which constitutes a main contribution of the paper. As an illustration, the effectiveness of the compensator is demonstrated through experimental results attained with a piezoelectric micro-positioning stage. The experimental results show that the SPI model can serve as an effective feedforward compensator and can thus enhance the tracking/positioning precision of the piezoelectric actuators.
Reference to this paper should be made as follows: Li, Z., Aljanaideh, O., Rakheja, S., Su, C-Y. and Al Janaideh, M. (2012) ‘Compensation of play operator-based Prandtl-Ishlinskii hysteresis model using a stop operator with application to piezoelectric actuators’, Int. J. Advanced Mechatronic Systems, Vol. 4, No. 1, pp.25–31.
Biographical notes: Zhi Li is currently a PhD student at the Department of Mechanical and Industrial Engineering at Concordia University, Montreal, QC, Canada. His research interests include the application of automatic control theory to smart materials.
Omar Aljanaideh is currently a PhD student at the Department of Mechanical and Industrial Engineering at Concordia University, Montreal, QC, Canada. His research interests include modelling and control of hysteresis non-linearity.
Subhash Rakheja received his PhD from Concordia University, Montreal, QC, Canada. He is currently a Professor of Mechanical Engineering and the Concordia Vehicular Ergodynamics Research Chair at the CONCAVE Research Centre, Concordia University, Montreal, Canada.
Chun-Yi Su received his PhD from South China University of Technology, Guangzhou, China in 1990. He is currently a Professor and Concordia Research Chair at Concordia University, Montreal, QC, Canada. His research interests include the application of automatic control theory to mechanical systems and control of systems involving hysteresis non-linearities.
Mohammad Al Janaideh received his MS and PhD in Mechanical Engineering and Mechatronics from Concordia University, Montreal, QC, Canada in 2004 and 2009, respectively. He is currently with the Department of Mechatronics Engineering, University of Jordan, Amman, Jordan. His research interests include control smart actuators, modelling rate-dependent and rate-independent hysteresis non-linearities, and compensation of hysteresis.
26 Z. Li et al.
This paper is a revised and expanded version of a paper entitled ‘Compensation of a piezoceramic actuator hysteresis nonlinearities using the stop operator-based Prandtl-Ishlinskii model’ presented at the 2010 International Conference on Modelling, Identification and Control, Okayama, Japan, 17–19 July 2010.
1 Introduction
Piezoelectric actuators have the advantages of producing large force, fast response and high precision (Smith, 2005; Ge and Jouaneh, 1996; Shen et al., 2008) and have been widely used in nano-positioning systems (Devasia et al., 2007), such as AFM (Leang et al., 2009), SPM (Esbrook et al., 2010), dual-stage servo system for HDDs (Li and Horowitz, 2001) and active aperture antenna (Granger et al., 2000). However, piezoelectric actuators exhibit limited tracking performance in precision control due to their inherent hysteresis non-linearity, which can severely limit the system positioning performance by giving rise to undesirable inaccuracy or oscillations, even leading to an instability of the closed-loop system (Hu and Mrad, 2002; Tao and Kokotovic, 1995). Therefore, considerable efforts have been made towards design of controllers for compensating the hysteresis effect. The majority of the reported approaches are inversion based compensation approaches (e.g., Tao and Kokotovic, 1995; Krejci and Kuhnen, 2001).
There are several models available for describing hysteresis behaviour, such as the Preisach model, the Prandtl-Ishlinskii (PI) model, the Bouc-Wen model and the Duhem model. Among them, the PI model is attractive due to its simplicity and suitability to construct feedforward compensator for the purpose of mitigating the hysteresis effect. A PI model could be constructed through superposition of two different operators. One is the stop operator-based Prandtl-Ishlinskii (SPI) model, and another is the play operator-based Prandtl-Ishlinskii (PPI) model. The SPI model, which was formulated for describing elastoplasticity hysteresis behaviour of materials (Brokate and Sprekels, 1996), yields concave clockwise hysteresis loops. However, the PPI model yields convex counter-clockwise hysteresis loops. Thus, it makes the SPI model a potential feedforward compensator if proper weights and thresholds can be analytically derived. Kuhnen and Janocha (1999) attempted a SPI model to compensate for hysteresis described by a PPI model, where the thresholds and weights of the SPI model were calculated in an adaptive manner based on an error function. In this paper, the PPI model is utilised to describe the hysteresis behaviour of the piezoelectric actuator. The thresholds and the weights of the SPI model are subsequently analytically derived for the first time in the literature, which constitutes the main contribution of the paper. The effectiveness of the SPI model in compensating the hysteresis non-linearity is investigated in the laboratory by implementing the SPI model with a piezo micro-positioning stage in a feedforward manner. It should be noted that because the inverse of the PPI model and the derived SPI model serve the same role as
a compensator of the PPI model, the derived SPI model can be thought as an alternative to the inverse compensation.
2 The Prandtl-Ishlinskii model
2.1 The PPI model
The PPI model integrates the play operator together with a weight function to characterise the hysteresis non-linearity. The properties of the PPI model have been described in details in Brokate and Sprekels (1996). The PPI model is defined as:
0[ ]( ) ( ) [ ]( )
Rrv t p r v t dr= Γ∏ ∫ (1)
where p(r) is the weight function, satisfying p(r) ≥ 0 with
0( ) ,rp r dr
∞< ∞∫ R is a constant so that the weight function
p(r) vanishes for large value of R. Γr[v](t) is the play operator defined by Brokate and Sprekels (1996):
[ ](0) ( (0),0)[ ]( ) ( ( ), [ ]( ))
r r
r r r i
v f vv t f v t v t
Γ =Γ = Γ
(2)
for ti < t ≤ ti+1; 0 ≤ i ≤ N − 1, with
( ) ( )( ), [ ] max ,min , [ ]r r rf v v v r v r vΓ = − + Γ (3)
where 0 = t0 < t1 < … < tN = tE is a partition of [0, tE] such that the function v(t) ∈ C[0, T], C[0, T] denotes the space of continuous function on [0, T], is monotone on each of the subintervals [ti, ti+1]. The input-output relationship of a play operator is shown in Figure 1. The play operator is rate-independent, Lipchitz-continuous and monotone. The PPI model can also be formulated in the discrete form as:
( )0
[ ]( ) [ ]( )j
n
j rj
v k w r v k=
= Γ∑∏ (4)
where n is the number of the play operators, rj are thresholds, 0 = r0 < r1 < … < rn = R, and w(rj) are the weights defined as:
( ) ( )( )1j j j jw r p r r r+= − (5)
For r0 = 0, the discrete expression of the PPI model can be expressed as:
( )1
[ ]( ) ( ) [ ]( )j
n
j rj
v k qv t w r v k=
= + Γ∑∏ (6)
where q = w(0) = w0.
Compensation of play operator-based Prandtl-Ishlinskii hysteresis model 27
Figure 1 Input-output relationship of the play operator
2.2 The SPI model
The Prandtl-Ishlinskii model can also be constructed using stop operators for describing hysteresis non-linearity. Unlike the PPI model, the SPI model exhibits clockwise hysteresis loops, attribute to the properties of the stop operator. The input-output relationship of a stop operator is illustrated in Figure 2. The output of the stop operator for the input v(t) ∈ C[0, T] can be expressed as:
( ) ( )( )[ ](0) ( (0))
[ ]( ) ( ) [ ]s s
s s i s i
v e v
v t e v t v t v t
Ω =
Ω = − +Ω (7)
for ti < t ≤ ti+1; 0 ≤ i ≤ N − 1,, with
min( ,max( , ))se s s v= − (8)
where 0 = t0 < t1 < … < tN = tE is a partition of [0, tE] such that the function v(t) ∈ C[0, T] is monotone on each of the subintervals [ti, ti+1].
Figure 2 Input-output relationship of the stop operator (see online version for colours)
Some of the essential properties of the stop operator can be described as follows:
• Clockwise operator: the stop operator yields clockwise input-output curves, while the play operator results in counter-clockwise input-output curves.
• Monotonicity: the stop operator Ωs is a monotone operator. For a given input v(t) ∈ C[0, T], the following property holds (Brokate and Sprekels, 1996):
( )[ ]( ) [ ](0))( ( ) (0) 0s sv t v v t vΩ −Ω − ≥ (9)
• Lipschitz-continuity: for a given input v(t) ∈ C[0, T], the stop operator is Lipschitz continuous (Brokate and Sprekels, 1996).
The output of the SPI model is formulated upon integration of the stop operator corresponding to different thresholds s in conjunction with the positive and integrable weight function g(s), such that:
0[ ]( ) ( ) [ ]( )
Ssv t g s v t dsΦ = Ω∫ (10)
The discrete expression of the SPI model can also be described by n stop operators, 0 < s1 < … < sn < s0 = S, such that:
( )0
[ ]( ) [ ]( )n
j rjj
v k a s v k=
Φ = Γ∑ (11)
where
( ) ( )( )1j j j ja s g s s s+= − (12)
Owing to the Lipschitz continuity of the stop operator and integrable weight function, it can be concluded that the SPI model is Lipschitz-continuous for a given input v(t) ∈ C[0, T]. It can be further concluded that the SPI model is a monotone operator, since the stop operator is monotone and the weight function is integrable and positive.
3 Compensation of hysteresis via SPI model
3.1 Feedforward compensation
The feedforward compensation, shown in Figure 3, was used in this paper to obtain identity mapping between the desired input vd(t) and the output v(t) as:
[ ]( ) ( )dv t v t⎡ ⎤= Φ⎣ ⎦∏ (13)
As shown in Section 2, the SPI model yields clockwise hysteresis loops, which is essential to introduce the SPI model as a compensator to mitigate the hysteresis effects described by the PPI model.
Figure 3 The feedforward compensation
Lemma 3.1: The SPI model Φ[v](t) is a compensator of the PPI model Π[v](t) if, for any initial output Π(0), there exists a Φ(0), such that the series connection of the compensator and the model yields identity transformation starting from the initial states Π(0) and Φ(0), and v(t) = Π [Φ[vd]](t) = Id[vd](t) = vd(t).
Figure 4 shows the inner structure for the SPI model and the PPI model. The two parameters, thresholds sj and weights aj of SPI model can be analytically derived based on the initial loading curve and the given thresholds rj and weights wj of the PPI model.
28 Z. Li et al.
Figure 4 Inner structure of the SPI model and the PPI model
3.2 Analytical implementation of the feedforward compensator by using SPI model
In this section, the initial loading curve of the PPI model is presented as a tool to analytically calculate the thresholds and weights of the SPI model. This curve describes the possible hysteresis loops generated by the Prandtl-Ishlinskii model. The initial loading curve of the PPI model can be expressed as:
( )0
( ) max 0,n
j jj
r r r w=
Θ = −∑ (14)
where r ∈ [r0, rn], and r0 = 0, n is the number of the play operator.
The function Θ : R+ → R+ is convex and increasing function. In order to obtain the parameters of the compensator, the initial loading curve of the SPI model is defined as:
( )0
( ) min ,n
j jj
s s s a=
=∑Ψ (15)
where the function Ψ : R+ → R+ is concave and increasing, and n is the number of the stop operator. s ∈ [0, s0], s0 is set to be a large positive real number, satisfying s0 > max(v(t)), to ensure strict monotonicity of the SPI model.
As shown in Figures 5(a) and 5(b), the initial loading curve of the SPI model is concave, while the initial loading curve of the PPI model is convex. Owing to the convex initial loading curve, the hysteresis loops resulting from the PPI model would be counter clockwise loops. On the other hand, the hysteresis loops resulting from the SPI model would be clockwise loops. This denotes that a composition between the SPI and PPI models could yield identity in input-output curves with proper thresholds and weights. Furthermore, both the initial loading curves should be symmetric about the line y = x, as shown in Figure 5(c).
Figure 5 (a) Initial loading curve of the SPI model (b) initial loading curves of the PPI model (c) composition of the initial loading curves of the SPI and the PPI models
(a) (b)
(c)
Figure 6 The relationship between the initial loading curves associated with the SPI and the PPI models
Figure 6 shows the relationship of the initial loading curves between the SPI model and the PPI model. In order to obtain the proper weights and thresholds of the SPI model which can effectively mitigate the hysteresis non-linearity described by the PPI model, the thresholds and the initial loading curve of the SPI model must satisfy:
( )j js r= Θ (16)
( )j js r=Ψ (17)
According to equations (16) and (17), for any point B(rk, Θ(rk)) on the initial loading curve of the PPI model, it can always find a point C(sk, Ψ(sk)) on the initial loading curve of the SPI model. Furthermore, points C and B are symmetric about the line y = x, and OA = CD and AB = OD. The thresholds of the SPI model may be related to those of the PPI model in the following manner:
Compensation of play operator-based Prandtl-Ishlinskii hysteresis model 29
( )( ) ( )
( ) ( )
1 1 0
2 2 1 1 2 0
3 3 1 1 3 2 2 3 0
1 1 2 2 0n n n n
s r ws r r w r w
s r r w r r w r w
s r r w r r w r w
=
= − +
= − + − +
= − + − + +…
(18)
Then the weights of the SPI model aj can be calculated according to (17) as
1 0 1 1 1 2 1 1 1
2 0 1 1 2 2 2 2 2
0 1 1 2 2
k n
k n
k k k k n k
s a s a s a s a s a rs a s a s a s a s a r
s a s a s a s a s a r
+ + + + + + =
+ + + + + + =
+ + + + + + =
… …… …
… …
0 1 1 2 2
n k k n n ns a s a s a s a s a r+ + + + + + =… …
(19)
Equation (19) includes n + 1 unknown variables, while the number of equations is n. In order to solve equation (19) and to obtain the weights aj, the weight a0 should be solved first. For this purpose, an additional point is taken on the initial loading curve of the SPI model as (sn+1, Ψ(sn+1)). By letting sn+1 = ρ + sn with ρ being a positive real number, it can be shown that ξ = Ψ(sn+1).
According to equations (16) and (17):
( ) ( )( )
0 1 1 k k
n n n
w r w r w
r w s
ξ ξ ξ
ξ ρ
+ − + + −
+ − = +
…
… (20)
( ) 0 1 1n k k n ns a s a s a s aρ ξ+ + + + + + =… … (21)
Equations (19) and (21) yield:
0 na rρ ξ= − (22)
Equation (20) can be rewritten as
( ) ( )( ) ( )( ) ( )
0 0 1
1 1
n n n
n n k
n k k n n n
r w r w r w
r r w r w
r r w r w s
ξ ξ
ξ
ξ ρ
− + + −
+ − + + −
+ − + + − = +
…
…
(23)
Simultaneous solutions of equations (22) and (23) yield a0 as:
00 1
1
k na
w w w w=
+ + + + +… … (24)
Then weights aj of the SPI model can be easily obtained by solving equation (19).
Remark: It should be noted that an inverse Prandtl-Ishlinskii model formulation was proposed in Kuhnen (2003) on the basis of the initial loading curve. The proposed stop-based model is comparable with the inverse model. Considering that the inverse and SPI models serve as compensators to cancel the effect of hysteresis, the initial loading curve of the SPI model must be same as that of the inverse of the PPI model. Because of the uniqueness of the inverse initial
loading curve, it explains why Figure 6 is similar to Figure 4 in Kuhnen (2003). To make the initial loading curve of the SPI model as shown in Figure 6, two parameters, the thresholds and the weights, have to be formulated. From the above results, the thresholds of the SPI model (18) are the same as those of the inverse PPI model [see the expression (12) in Kuhnen (2003)] because of the uniqueness of the initial loading curve. However, the weights calculated from (19) and (24) are completely different. As for the question whether to use the inverse or the SPI as a compensator, there is no clear answer and it needs to be further investigated.
4 Experimental verification
4.1 Experimental setup
In order to validate the effectiveness of the SPI model in compensating the hysteresis behaviour of the piezoelectric actuator, an experimental platform is established as shown in Figure 7. It consists of the following elements:
1 Piezoelectric actuator: P-753.31C piezoelectric actuator, manufactured by Physik Instrument Company, was considered for measurement of the hysteresis non-linearity. The actuator provided a maximum displacement of 38 μm from its static equilibrium position. The excitation voltage to the actuator ranged from 0 to 100 V.
2 Capacitive sensor: an integrated capacitive sensor was used for measurement of the actuator displacement response with a sensitivity of 2.632 V/μm.
3 Voltage amplifier: the excitation voltage to the actuator was applied through a voltage amplifier (LVPZT, E-505), with a fixed gain of 10.
4 Data acquisition system: the actuator displacement response signal was acquired in the dSpace Control Desk together with the input signal.
Figure 7 The experimental platform
The measurements of the actuator displacement response were performed under two different excitations. These included:
30 Z. Li et al.
1 a harmonic excitation at a low frequency of 1 Hz, v(t) = 30sin(2πt) + 40
The measured signals were analysed to characterise the hysteresis effects of the actuator, where the first excitation was selected to identify the major loop input-output property of the actuator, while the complex harmonic excitation was chosen to measure the major as well as minor hysteresis loops.
4.2 Hysteresis modelling
The hysteresis non-linearity of the piezoelectric actuator can be described by the PPI model. The model is formulated using the threshold and density function of the following forms:
j jr σ= (25)
( ) jjp r e βα −= (26)
where α, β and σ are positive constants. Ten play operators (n = 10) were chosen to formulate the PPI model. The model parameters, X = β, α, σ, w0, were identified through minimisation of an error sum-squared function by using MATLAB optimisation toolbox, subjected to the following constrains:
0, , , 0wσ α β > (27)
The identified parameters of the PPI model by using the input-output data of the piezoelectric actuator were found to be: σ = 3.548, α = 0.0383, β = 0.1206 and w0 = 0.2567. The validity of the PPI model is investigated by comparing the model responses and the laboratory-measured data of the piezoelectric actuator. These comparisons under the chosen input voltages are shown in Figures 8(a) and 8(b). The results clearly show that the PPI model can effectively characterise the minor as well as major hysteresis loops of the piezoelectric actuator.
4.3 Compensation of hysteresis non-linearity using SPI model
In this section, the SPI model is utilised as a feedforward compensator to compensate for the hysteresis non-linearity. The parameters of the SPI model are derived based on the obtained thresholds (25) and weights (26) of the PPI model that were presented in the previous section. The weights aj and the thresholds sj of the SPI model are obtained using (18), (19), and (24) as: 2.146, 0.4544, 0.3230, 0.2397, 0.1831, 0.1444, 0.1156, 0.0941, 0.0774, 0.0642, 0.0543 and 100, 0.9108, 1.9418, 3.0797, 4.3122, 5.6285, 7.0194, 8.4762, 9.9915, 11.5587, 13.1716. Figures 9(a) and 9(b) show the outputs of the SPI model applied to the input amplifier of the piezoelectric actuator through the output board and D/A converter. It should be noted that this output is further amplified by the voltage amplifier
(gain = 10). The measured input-output characteristics of the piezoelectric actuator with the SPI model are illustrated in Figure 10 for the two selected inputs. The figures show inputs in terms of the desired input displacement, vd, as seen in Figure 3. The experimental results illustrate that the SPI model can serve as an effective feedforward hysteresis compensator for the piezoelectric actuator.
Figure 8 Comparisons between the measured displacement responses with the results derived from the PPI model under two selected inputs, (a) sinusoidal excitation (b) complex harmonic excitation
0 20 40 60 800
5
10
15
20
25
30
Voltage (v) D
ispl
acem
ent (
μm)
Experimental data
The PPI model
(a)
0 20 40 60 800
5
10
15
20
25
30
Voltage (v)
Dis
plac
emen
t (μm
)
Experimental data
The PPI model
(b)
Figure 9 The outputs of the SPI model under two selected inputs, (a) sinusoidal excitation (b) complex harmonic excitation
0 1 2 3 4 50
1
2
3
4
5
6
7
8
Time (sec)
Vol
tage
(v)
(a)
Compensation of play operator-based Prandtl-Ishlinskii hysteresis model 31
Figure 9 The outputs of the SPI model under two selected inputs, (a) sinusoidal excitation (b) complex harmonic excitation (continued)
0 1 2 3 4 51
2
3
4
5
6
7
Time (sec)
Vol
tage
(v)
(b)
Figure 10 Input-output characteristics of the piezoelectric stage with the SPI model compensator under two selected inputs, (a) sinusoidal excitation (b) complex harmonic excitation
5 10 15 20 25 305
10
15
20
25
30
Input vd (μm)
Dis
plac
emen
t (μm
)
(a)
5 10 15 20 25 305
10
15
20
25
30
Input vd(μm)
Dis
plac
emen
t (μm
)
(b)
5 Conclusions
This paper formulates an SPI as a feedforward compensator to compensate for the hysteresis non-linearity in a piezoelectric actuator. The thresholds and the weights of the SPI model are analytically derived for the first time in the literature, which constitutes the main contribution of the paper. The results attained from laboratory experiments performed with a piezoelectric micro-positioning stage showed that the SPI model can serve as an effective
compensator for the hysteresis nonlinearities. The effectiveness of the SPI model as the feedforward compensator is demonstrated under harmonic and complex harmonic inputs involving both major as well as minor loop hysteresis non-linearities.
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