Modeling and inverse adaptive control of asymmetric hysteresis systems with applications to magnetostrictive actuator

Control Engineering Practice 33 (2014) 148–160

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Modeling and inverse adaptive control of asymmetric hysteresis systems with applications to magnetostrictive actuator Zhi Li a, Chun-Yi Su a,n, Xinkai Chen b a b

Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada Department of Electronic and Information Systems, Shibaura Institute of Technology, Minuma-ku, Saitama-shi, Saitama 337-8570, Japan

art ic l e i nf o

a b s t r a c t

Article history: Received 4 February 2014 Accepted 9 September 2014

When uncertain systems are actuated by smart material based actuators, the systems exhibit hysteresis nonlinearities and corresponding control is becoming a challenging task, especially with magnetostrictive actuators which are dominated by asymmetric hystereses. The common approach for overcoming the hysteresis effect is inverse compensation combining with robust adaptive control. Focusing on the asymmetric hysteresis phenomenon, an asymmetric shifted Prandtl–Ishlinskii (ASPI) model and its inverse are developed and a corresponding analytical expression for the inverse compensation error is derived. Then, a prescribed adaptive control method is applied to mitigate the compensation error and simultaneously guaranteeing global stability of the closed loop system with a prescribed transient and steady-state performance of the tracking error without knowledge of system parameters. The effectiveness of the proposed control scheme is validated on a magnetostrictive actuated platform. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Asymmetric hysteresis Asymmetric shifted Prandtl–Ishlinskii (ASPI) model Prescribed adaptive control Magnetostrictive actuator

1. Introduction Hysteresis is a nonlinear phenomenon that appears in many different areas. Ferromagnetic hysteresis (Brokate & Sprekels, 1996) and plastic hysteresis (Jiles & Atherton, 1986) are two typical examples. Hysteresis exhibited in the smart material-based actuators, such as magnetostrictive and piezoelectric actuators (Smith, 2005), reveals a looped and branched nonlinear relation between the input excitation and the output displacement. Hysteretic behaviors also arise in aerodynamics, where the aerodynamic forces and moments show hysteresis when the attack angle of the airplane varies. Others are encountered in mechanical systems, economics, neuroscience and electronics engineering (Esbrook, Xiaobo, & Khalil, 2013; Xiao & Li, 2013), etc. When a control system involves the hysteresis nonlinearity such as actuated by smart material-based actuators, the hysteresis nonlinearity generates an undesired and detrimental effect, which will deteriorate the system performance and cause inaccuracy or oscillations. The common approach for remedying its effect is to construct a hysteresis inverse in putting in cascade as a compensator to cancel the hysteresis effect (Krejci & Kuhnen, 2001; Kuhnen, 2003; Li, Hu, Liu, Chen, & Yuan, 2012). For constructions of the hysteresis inverse, two approaches are generally used: direct construction of complete inverse function of the hysteresis function (model) (Krejci & Kuhnen, 2001;

n

Corresponding author. E-mail address: [email protected] (C.-Y. Su).

http://dx.doi.org/10.1016/j.conengprac.2014.09.004 0967-0661/& 2014 Elsevier Ltd. All rights reserved.

Kuhnen, 2003), and use of an inverse multiplicative structure (Rakotondrabe, 2011; Zhou, Wen, & Li, 2012) of the models to compensate the complicated component in the model, in which the development of the complete inverse function of the hysteresis model is not required. Direct construction of inverse function of the hysteresis model is mainly for operator based models such as Preisach (1935) model and Prandtl–Ishlinskii (PI) model (Krejci & Kuhnen, 2001). In both, the hysteresis is modeled by a superposition of elementary relay or play operators. However, only the PI model possesses the analytic form of the inverse (Krejci & Kuhnen, 2001), which explains why the PI model is becoming dominant in the direct inverse compensation approaches. In this paper, we will focus on the direct inverse approaches owing to unknown function in hysteresis models. It is recognized that the operator based models except the Preisach model generally describe the symmetric hysteresis effects. However, there are many cases that the hysteresis exhibits asymmetric behaviors such as magnetostrictive actuators, shape memory alloys (SMA) actuators. To keep the feature of PI model with the analytic inverse, the extension of the PI model to describe the asymmetric hysteresis behavior has been exploited in the literature, including (1) cascading a nonlinear operator with the PI model. In Kuhnen (2003), a modified PI model, superposition of one-sided dead-zone operators proceeded by the PI model, was proposed. This model can describe the asymmetric hysteresis behavior and has analytical solutions of its inversion, but it cannot describe the saturated hysteresis behaviors; (2) modifying the elementary play operator. In Jiang, Ji, Qiu, and Chen (2010), the elementary play operator was redefined as right-side play operator

Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

and left-side play operator. In Jiang, Deng, and Inoue (2008), a nonsymmetric play operator was considered as the elementary operator. In Janaideh, Mao, Rakheja, Xie, and Su (2008), a generalized play operator with envelope functions was proposed, where the analytical inversion was provided with requirement of first-order derivative of the input signals. To avoid the drawbacks of the those PI extensions and preserve the advantage of the existing analytic inverse of the PI model while still being able to describe the asymmetric hysteresis behavior, an asymmetric shifted Prandtl–Ishlinskii (ASPI) model is proposed in this paper, which is constructed by three components: a PI model, a shift model and an auxiliary function. The advantages of the proposed model are (1) it is able to represent the asymmetric hysteresis behavior; (2) it facilitates the construction of the analytic inverse by directly utilizing the available PI inverse result in Krejci and Kuhnen (2001) without the requirement of first-order derivative of the input signals; (3) the analytical expression of the error of the inverse compensation can be derived for the asymmetric case, which will be explained as follows. As reported in the literature, using the inverse for hysteresis compensation generally exhibits notable compensation errors, which are attributed to hysteresis characterization errors. The use of an estimated hysteresis model in deriving the model inverse would be expected to yield some degree of hysteresis compensation error. This error yields tracking error in the closed-loop control system. To accommodate such a compensation error, the analytical expression of the error of the inverse compensation is urged in the controller design. Along this line, in Tao and Kokotovic (1993, 1995) the analytical inverse compensation error for backlash hysteresis was derived and a corresponding adaptive control scheme was then developed. For the PI model, an analytical error expression was obtained and an adaptive backstepping control scheme was developed in Janaideh, Su, and Rakheja (2012). However, the analytical error expression was obtained only for the PI model, it has not yet being exploited for its extensions. Therefore, as listed in the third advantage, the inverse compensation error for the proposed ASPI model is analytically derived for the purpose of the controller design when the nonlinear system is preceded by the asymmetric hysteresis. In order to ensure the transient and steady-state performance of the tracking error a prescribed adaptive control scheme is employed. The developed prescribed adaptive control approach guarantees the global stability of the nonlinear system and achieves the prescribed transient and steady-state performance of the tracking error without knowledge of system parameters. To validate the developed ASPI model and the adaptive inverse hysteresis control scheme, experimental results on a magnetostrictive actuated platform are presented.

149

P1: The system state x(t) tracks a desired signal xd(t) and all signals in the closed-loop are bounded; P2: Both transient and steady-state performance of tracking error e1 ðtÞ ¼ xðtÞ  xd ðtÞ should be within the prescribed area. Comparing with general nonlinear control for the system (1) only, the control signal u(t) becomes the output of the hysteresis operator uðtÞ ¼ Π ½vðtÞ, where the actual control signal is v(t). As it is well known, the hysteresis nonlinearity will deteriorate the system performance and cause inaccuracy or oscillations. Therefore, it imposes a challenge to handle this cascaded term with a basic requirement that u(t) is not available/measurable. The common approach for remedying the effect is to construct a hysteresis inverse as a feedforward compensator. Then the control law can be designed with available control methods. The complete control scheme is shown in Fig. 1. Throughout the paper the following standard assumptions are required: Assumption 1. The sign of uncertain parameter b is known. Without losing generality, it is selected as b 4 0 in this paper. Assumption 2. The desired trajectory xd(t) and its (n  1)th-order derivatives are continuous. Furthermore, ½xd ; x_ d ; …; xnd T A Ωd  Rn þ 1 with Ωd being a compact set. 3. Modeling and inverse construction of asymmetric hysteresis 3.1. The Prandtl–Ishlinskii (PI) model The Prandtl–Ishlinskii (PI) model was first used for describing the hysteresis behavior of elasto-plasticity by Prandtl in 1928, which is defined as Brokate and Sprekels (1996) P½vðtÞ ¼ p0 vðtÞ þ

Z Λ

pðrÞF r ½vðtÞ dr

ð3Þ

0

where p0 is a positive constant; p(r) is a given density function, R1 satisfying pðrÞ Z 0 with 0 rpðrÞ dr o 1. Since the density function p(r) vanishes for large values of r, the choice of Λ as the upper limit of integration in the literature is just a matter of convenience (Su, Wang, Chen, & Rakheja, 2005). F r ½v is the play operator: F r ½vð0Þ ¼ f r ðvð0Þ; 0Þ

ð4Þ

F r ½vðtÞ ¼ f r ðvðtÞ; F r ½vðt i ÞÞ

ð5Þ

for t i o t r t i þ 1 ; 0 r i r N  1, with f r ðv; wÞ ¼ maxðv  r; minðv þ r; wÞÞ

2. Problem statement Consider a dynamic nonlinear system consisting of an actuator with hysteresis nonlinearity Π ½vðtÞ and a nonlinear plant as k

_ xðnÞ ðtÞ þ ∑ ai Y i ðxðtÞ; xðtÞ; …; xðn  1Þ ðtÞÞ ¼ buðtÞ

ð1Þ

uðtÞ ¼ Π ½vðtÞ

ð2Þ

i¼1

where v(t) denotes the input and u(t) denotes the output, Yi are known continuous, linear or nonlinear functions. Parameters ai and control gain b are unknown constants, Π ½vðtÞ denotes the output of the hysteresis operator, which will be described in the following development. The control objective is to design a control signal v(t) for system (1), such that:

ð6Þ

where 0 ¼ t 0 o t 1 o⋯ o t N is a partition of ½0; t N , such that the function v(t) is monotone on each of the subintervals ½t i ; t i þ 1 . The main feature of the PI model is its unique property of being analytically invertible (Krejci & Kuhnen, 2001), which makes it possible to utilize its inverse as a feedforward compensator to cancel the hysteresis effect. However, it should be noted that owing to the definition of the play operator, the PI model can only describe symmetric hysteresis (Brokate & Sprekels, 1996).

Fig. 1. The control scheme.

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Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

shape of the shift operator. When c 4 1, Ψ c ½vðtÞ is called left shift operator; when 0 o c o 1, Ψ c ½vðtÞ is called right shift operator. The last term gðvÞðtÞ in (7) is an auxiliary function, which assists to represent the saturation behavior of hysteresis nonlinearity. As an illustration, with a selected density function pðrÞ ¼ 0:4e  0:01ðr þ ð1=6ÞÞ , χ ðcÞ and g(v) can be correspondingly constructed, which are listed in Table 1. Fig. 2 shows responses of the PI model and the ASPI model with this selected density functions and the bound for H½v being selected as R1 ¼ 10. From Fig. 2, it shows that the proposed ASPI model can indeed describe the asymmetric hysteresis effect.

3.2. The asymmetric shifted Prandtl–Ishlinskii (ASPI) model In order to extend the PI model to describe the asymmetric hysteresis effect, still possessing its unique property of being analytically invertible, an asymmetric shifted Prandtl–Ishlinskii (ASPI) model is defined in this section, which is composed of three components: a Prandtl–Ishlinskii (PI) model, a shift model and an auxiliary function. The purpose for introducing the shift model is to change the symmetric characteristics of the ASPI model and the auxiliary function is used for representing the saturated phenomenon. The ASPI model is thus defined as uðtÞ ¼ Π ½vðtÞ ¼ P½vðtÞ þ H½vðtÞ   H 1 ½v H½v ¼ R1 sat R1 H 1 ½v ¼ Ψ ½vðtÞ þ gðvÞðtÞ

3.2.1. Discussion on the selection of shift model Ψ ½v and function g(v) The purpose for introducing the shift model is to describe the asymmetric hysteresis. In order to reflect characteristics of the asymmetric hysteresis, we firstly give the definition of the hysteresis gap eg as

ð7Þ

where the first term P½vðtÞ is the PI model defined in (3), satðϖ Þ is the saturation function defined as 8 > < þ 1 if ϖ 4 1 satðϖ Þ ¼

> :

ϖ

1

eg ¼ ydownscale  yupscale

if  1 r ϖ r 1

where ydownscale, yupscale denote the downscale and the upscale of the shift operator in Fig. 3, respectively. vp in Fig. 3 denotes the input value when the hysteresis gap reaches its peak value. For an asymmetric hysteresis, if vp is at the left of the center line v¼ 0, v A ½  v0 ; v0 , it is called left asymmetric hysteresis. Otherwise, it is called right asymmetric hysteresis. Fig. 3 shows the left shift operator ðc 41Þ, right shift operator ð0 o c o 1Þ and their hysteresis gap curves. Therefore, we can superpose left shift operators with c 4 1 to describe the left asymmetric hysteresis. Otherwise, we select right shift operators with 0 o c o 1.

if ϖ o 1

R1 4 0 is a design parameter to define the bound of H½v. Ψ ½vðtÞ is defined as the superposition of the weighted shift operators: Z C1 Ψ ½vðtÞ ¼ χ ðcÞΨ c ½vðtÞ dc ð8Þ C0

where χ ðcÞ Z 0 is the density function with Ψ c ½vðtÞ is the shift operator defined as

R C1 C0

cχ ðcÞ dc ¼ Lχ o 1.

Ψ c ½vð0Þ ¼ ψ c ðvð0Þ; 0Þ

ð9Þ

Ψ c ½vðtÞ ¼ ψ c ðvðtÞ; ψ c ½vðt i ÞÞ

Remark. As a matter of fact, the right asymmetric hysteresis Γ ½v can be transformed into the left asymmetric hysteresis by Π ½v ¼  Γ ½  v. Therefore, we can always select the left shift operators ðc 4 1Þ to represent the asymmetric hysteresis. As an illustration, to get the left asymmetric hysteresis, Fig. 4 demonstrates the superposition of the left shift operators with the PI model to describe the left asymmetric hysteresis.

ð10Þ

for t i ot r t i þ 1 ; 0 r i r N  1, with

ψ c ðv; wÞ ¼ maxðcv; minðv; wÞÞ

ð11Þ

where 0 ¼ t 0 o t 1 o ⋯ o t N is the same partition of ½0; t N  as defined in (6). c A IR þ , IR þ ≔fx A IRjx Z0g is a parameter to determine the

Since the PI model and the shift model are the monotone models, the derivatives of the upscale of the hysteresis loops described by these models are monotone under piecewise monotone input signal v(t). However, for some saturated asymmetric hysteresis, such as the asymmetric hysteresis in SMA actuators, the derivative of the upscale of the hysteresis loop is non-monotone, see Fig. 5. Thus, we introduce the auxiliary function g(v) into the ASPI model to decrease the derivative of the upscale of the ASPI model so as to capture the characteristics of the saturated

Table 1 Coefficients of the PI model and the ASPI model. The selected function

The PI model

The ASPI model

p(r) r A ½0; 4 χðcÞ c A ½1; 4 g(v)

0:4e  0:01ðr þ ð1=6ÞÞ

0:4e  0:01ðr þ ð1=6ÞÞ 0:02e  0:1ðc  1Þ 0:8 arctanð3v  2Þ  0:05v2 þ 0:05v

4

3

ASPI Model

PI Model 2

1

Output

Output

2

0

0

−2

−1 −2 −4

ð12Þ

−2

0 Input

2

4

−4 −4

−2

0 Input

Fig. 2. Input–output responses of the PI model and the ASPI model.

2

4

Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

151

2.5

10 c=2 Hysteresis gap

2

Output

5 downscale 0

1.5 1 0.5 0

−5 −4

−2

upscale 0 2 Input

4

−2

0 Input

2

4

2

4

c=0.5

2

Hysteresis gap

1

1 Output

v <0 p

−0.5 −4

0 downscale upscale

0.5

−1 0 −2 −4

−2

0 Input

2

4

v >0 p

−4

−2

0 Input

Fig. 3. The shift operator and its hysteresis gap function. (a) The left shift operator and its hysteresis gap. (b) The right shift operator and its hysteresis gap.

3

The left shift model

The PI model

The asymmetric hysteresis

2

4

downscale

Output

2

downscale

1 2

1

0

downscale

0 0

−1 upscale

−1 −4

−2

0

2

upscale 4

−2 −4

−2

0 Input

2

−2 4

−4

upscale −2

0

2

4

−2

0

2

4

Hysteresis gap

0.4 1.5

1.5

1

1

0.5

0.5

0.3 0.2 0.1 0 −4

−2

0

2

4

0 −4

−2

0 Input

2

4

0 −4

Fig. 4. The modeling illustration for the asymmetric hysteresis phenomenon.

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Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

Derivative of upscale

Displacement(mm)

0.2

10

5

0.15

0.1

0.05

upscale 0

0

100 Temperature ° C

0

200

0

100 Temperature ° C

200

Fig. 5. The derivative of the upscale of the hysteresis loop in SMA actuators.

phenomenon. For such a purpose, g(v) needs to satisfy the following condition: Condition 1. 0

(1) For the monotonic increase input v(t), g ðvÞ 4  ðP ½v þ Ψ ½vÞ; (2) gððv1 þ v2 Þ=2Þ 4 ðgðv1 Þ þ gðv2 ÞÞ=2, where v1 ; v2 A ½vmin ; vmax . 0

0

Remarks. (1) The selection of the g(v) is not unique, any function can be qualified as far as it is Lipschitz continuous, derivative and satisfying the above condition. (2) Similar to the PI model, the exact shape of the hysteresis depends on the selection of the density functions, c, g(v), which can be determined once the hysteresis loops are available.

The challenge addressed in this section is to develop an inverse model Π  1 of the ASPI model for the purpose of mitigating the hysteresis effect. In fact, the inverse of Π can be constructed by utilization of the inverse result developed in Krejci and Kuhnen (2001) for the PI model itself. In the following development, we will show the procedures for this construction. From (7), it is obvious that if we can find an Π  1 so that Π ½Π  1 ½uðtÞ ¼ uðtÞ, then such a Π  1 can be qualified as an inverse of the ASPI model. Since u in (7) is expressed as uðtÞ ¼ Π ½vðtÞ ¼ P½vðtÞ þ H½vðtÞ. Then, P½vðtÞ can be re-expressed as ð13Þ

ð14Þ

where P  1 ½ denotes the inverse model of the PI model, which is defined as P½u  1 ðtÞ ¼ p 0 uðtÞ þ

Z Λ

pðrÞF r ½uðtÞ dr

Π  1 ½uðtÞ ¼ P  1 ½u  H½vðtÞ ¼ vðtÞ

The merit of the above construction is the utilization of the analytic inverse result for the PI model in Krejci and Kuhnen (2001). Only an extra signal H½vðtÞ is included to the input of P  1 for the inverse construction of Π ½vðtÞ. The existence and uniqueness of Π  1 are established in the following theorem. Theorem 1. The ASPI model Π ½vðtÞ is a bijection of C½0; T onto C½0; T, it satisfies for every u A C½0; T, C½0; T denotes the space of continuous functions on [0,T], there exists a unique v A C½0; T, such that u ¼ Π ½v. Before the proof, some lemmas and definitions are given as follows.

where ‖  ‖1 denotes the infinite norm, LH 40. Lemma 2. For every x; y A C½0; T; t A ½0; T ‖P  1 ½xðtÞ P  1 ½yðtÞ‖1 r LP  1 ‖xðtÞ  yðtÞ‖1

ð15Þ

denotes the inverse of the PI model, which has been where P proposed in Krejci and Kuhnen (2001). Definition. For a given u A C½0; T, we define an operator as

Θ½xðtÞ ¼ uðtÞ  H½P  1 ½xðtÞ

pðrÞ ¼ ðφ

ð23Þ

Then, we have jΘ½xðtÞ  Θ½yðtÞj ¼ jH½P  1 ½xðtÞ  H½P  1 ½yðtÞj Z t r LH ‖P  1 ½xðϱÞ  P  1 ½yðϱÞ‖1 dϱ 0

φðrÞ ¼ p 0 r þ

Z 0

r

ð17Þ pðξÞðr  ξÞ dξ

ð18Þ

Z

t 0

‖xðϱÞ  yðϱÞ‖1 dϱ

ð24Þ

Thus, for every t A ½0; T according to Krejci and Kuhnen (2001), ‖Θ ½xðtÞ  Θ ½yðtÞ‖1 r n

Þ″ðrÞ

ð22Þ

x ¼ P½vðtÞ

r LH LP  1 ð16Þ

1

ð21Þ

1

0

1 p0

ð20Þ

0

where p0 ¼

ð19Þ

where

Taking the inverse of P in (13) on both sides, one has vðtÞ ¼ P  1 ½P½vðtÞ ¼ P  1 ½u H½vðtÞ

Π  1 is obtained:

Lemma 1. For every v1 ; v2 A C½0; T; t A ½0; T Z t ‖v1 ðϱÞ  v2 ðϱÞ‖1 dϱ jH½v1 ðtÞ  H½v2 ðtÞj r LH

3.3. The inverse construction for the ASPI model

P½vðtÞ ¼ uðtÞ  H½vðtÞ

Thus, the following inverse

n

ðLH LP  1 tÞn ‖xðtÞ  yðtÞ‖1 n!

ð25Þ

For n sufficiently large, it satisfies ðLH LP  1 tÞn =n! o 1. Then Θn is a contraction in C½0; T. According to the Banach contraction principle (Granas & Dugundji, 2003), the following equation has the

Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

153

^  1 Þ″ðrÞ p^ ðrÞ ¼ ðφ

unique solution:

Θ½xðtÞ ¼ xðtÞ

ð26Þ

φ^ ðrÞ ¼ p^ 0 r þ

Substituting (22) and (23) into (26), we have uðtÞ ¼ P½vðtÞ þH½vðtÞ ¼ Π ½vðtÞ Then, (27) also has the unique solution. Therefore, is unique.

ð27Þ

Π  1 exists and

Z

r 0

ð35Þ p^ ðξÞðr  ξÞ dξ

ð36Þ

Thus, by applying the composition theorem on the P½ðtÞ and 1 P^ ½ðtÞ yields uðtÞ ¼ P○P^

1

½ud ðtÞ Z Λ 0 ¼ ϕ ð0Þud ðtÞ þ ϕ″ðrÞF r ½ud ðtÞ dr

4. Analytical error of the inverse compensation for the asymmetric shifted Prandtl–Ishlinskii (ASPI) model

ð37Þ

0

As a matter of fact, the inverse construction should be conducted ^ ½v (there always exists a modeling using estimated hysteresis model Π error between the estimated model and the realistic hysteresis). Therefore, the inverse compensation error is unavoidable and generally cannot be ignored in controller designs. Until recently, the analytic inverse compensation error of the PI model appeared in Janaideh et al. (2012). However, such an approach is only limited to the symmetric hysteresis case. Focusing on the asymmetric hysteresis, in the following development we will derive the analytic inverse compensation error for ASPI model. 4.1. Overview of composition theorem applied to the PI model In order to use the composition theorem, we need to rewrite the PI model as (Brokate & Sprekels, 1996) Z Λ P½uðtÞ ¼ φ0 ð0ÞuðtÞ þ φ″ðrÞF r ½uðtÞ dr ð28Þ 0

where φðrÞ denotes the initial loading curve which uniquely determines the shape of hysteresis loop described by the PI model and is defined as Z r φðrÞ ¼ p0 r þ pðκ Þðr  κ Þ dκ ð29Þ 0

where ud is the desired input signal. The compensation error epi(t) can be analytically written as epi ðtÞ ¼ ud ðtÞ  uðtÞ 0

¼ ð1  ϕ ð0ÞÞud ðtÞ 

Z Λ

ϕ″ðrÞF r ½ud ðtÞ dr:

0

ð38Þ

4.2. Analytical error of the inverse compensation for the asymmetric shifted Prandtl–Ishlinskii (ASPI) model ^ ½uðtÞ to Due to the presence of the estimation error, we use Π estimate the true hysteresis phenomenon Π ½uðtÞ, which is expressed as ^ Π^ ½vðtÞ ¼ P^ ½vðtÞ þ H½vðtÞ ! H^1 ½v ^ H½v ¼ R1 sat R1

^ H^1 ½v ¼ Ψ^ ½vðtÞ þ gðvÞðtÞ

ð39Þ

The output of the composition between the inverse compensation 1 Π^ ½uðtÞ and true hysteretic behavior Π ½uðtÞ is expressed as ^  1 ½u ðtÞ ¼ P○P^  1 ½u  H½vðtÞ ^ uðtÞ ¼ Π ○Π þ H½vðtÞ d d

ð40Þ

According to the combination results in (30), (40) becomes Z Λ 0 ^ ^ þ ϕ″ðrÞF r ½ud  H½vðtÞ dr þ H½vðtÞ uðtÞ ¼ ϕ ð0Þðud  H½vðtÞÞ

ð41Þ

Thus, φ ð0Þ ¼ p0 is a positive constant, φ″ðrÞ ¼ pðrÞ denotes the density function. According to the composition theorem presented in Krejci (1986), the composition between two PI models P γ ½ðtÞ and P δ ½ðtÞ is expressed as

Because of Er ½vðtÞ þ F r ½vðtÞ ¼ vðtÞ, where Er ½vðtÞ denotes the stop operator as

P ϕ ½uðtÞ ¼ P γ ○P δ ½uðtÞ

Er ½vð0Þ ¼ er ðvð0Þ  w  1 Þ

ð42Þ

Er ½vðtÞ ¼ er ðvðtÞ  vðt i Þ þEr ½vðt i ÞÞ

ð43Þ

0

0

¼ ϕ ð0ÞuðtÞ þ

Z Λ 0

ϕ″ðrÞF r ½uðtÞ dr

ð30Þ

where ϕðrÞ ¼ γ ○δðrÞ, γ ðrÞ and δðrÞ denote the initial loading curves of the P γ ½ðtÞ and P δ ½ðtÞ, separately. Since in practice, the exact density function p(r) in the PI model may not be available. It needs to be estimated based on the measured data. In this case, the inverse model should be constructed ^ based on the estimated density function, which is denoted as pðrÞ. Let P^ ½ðtÞ denotes the estimation of the actual hysteretic behavior P½ðtÞ as Z Λ ^ 0 ð0ÞuðtÞ þ P^ ½uðtÞ ¼ φ φ^ ″ðrÞF r ½uðtÞ dr ð31Þ 0

^ ðrÞ is defined as where φ Z r ^ κ Þðr  κ Þ dκ φ^ ðrÞ ¼ p^ 0 r þ pð

ð32Þ

1 P^ ½ðtÞ denotes the inverse of P^ ½ðtÞ as Z Λ p^ ðrÞF r ½uðtÞ dr P^ ½u  1 ðtÞ ¼ p^ 0 uðtÞ þ

ð33Þ

for t i o t r t i þ 1 and 0 r i r N  1, with er ðvÞ ¼ minðr; maxð  r; vÞÞ

ð44Þ

w  1 is the initial value. Then, we have Z Λ 0 ^ ^ þ ϕ″ðrÞðud  H½vðtÞÞ dr uðtÞ ¼ ϕ ð0Þðud  H½vðtÞÞ 

Z Λ 0 0

0

^ ϕ″ðrÞEr ½ud  H½vðtÞ dr þ H½vðtÞ

^ ^ ¼ ϕ ð0Þðud  H½vðtÞÞ þ ðϕ ðΛÞ  ϕ ð0ÞÞðud  H½vðtÞÞ Z Λ ^  ϕ″ðrÞEr ½ud  H½vðtÞ dr þ H½vðtÞ 0

0

0 0

0

0

where 1 p^ 0 ¼ p^ 0

0

ð34Þ

¼ ϕ ðΛÞud  db ðtÞ

ð45Þ RΛ

^  H½vðtÞ þ 0 ϕ″ðrÞEr ½ud  H^ ½vðtÞ dr. The where db ðtÞ ¼ ϕ ðΛÞH½vðtÞ estimation (inverse compensation) error e(t) of the ASPI model is therefore expressed as 0

0

eðtÞ ¼ ud ðtÞ  uðtÞ ¼ ð1  ϕ ðΛÞÞud ðtÞ þ db ðtÞ

ð46Þ

^ ½ðtÞ It should be noted that if the estimated hysteresis operator Π 0 is equal to the true hysteresis Π ½ðtÞ, it yields ϕðrÞ ¼ r, ϕ ðrÞ ¼ 1, ϕ″ðrÞ ¼ 0, then in (46) ϕ0 ðΛÞ ¼ 1, db ðtÞ ¼ 0, leading eðtÞ ¼ 0. Before showing the way to utilize the estimation error in the next section,

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Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

the following lemma is exploited to facilitate the robust controller design.

unconstraint one. Define SðÞ a smooth and strictly increasing function and z1 a transformed error as

Lemma 3. The term db(t) in (46) is bounded, i.e. jdb ðtÞj r D where D is a bounded constant, for any time t Z0.

e1 ðtÞ ¼ ρðtÞSðz1 Þ

Proof. Based on the definition of the stop operator (Brokate & Sprekels, 1996), one has jEr ½ðtÞ drj r r r Λ

ð47Þ

From (47), we have Z Λ Z Λ ^ ϕ″ðrÞEr ½ud  H½vðtÞ dr r Λ ϕ″ðrÞ dr 0

0

0

0

r Λðϕ ðΛÞ  ϕ ð0ÞÞ

ð48Þ

Thus, Z   Λ    0 0 ^ ϕ″ðrÞEr ½ud  H½vðtÞ dr  r jΛðϕ ðΛÞ  ϕ ð0ÞÞj   0 

ð49Þ

R1 rH½v r R1

ð50Þ

^ r ϕ ðΛÞR1  ϕ ðΛÞR1 r ϕ ðΛÞH½v 0

0

0

ð51Þ

From (50) and (51), we have 0 ^  H½vj r jðϕ0 ðΛÞ þ 1ÞR1 j jϕ ðΛÞH½v

ð52Þ

Therefore, based on (49) and (52), 0

0

0

jdb ðtÞj rjðϕ ðΛÞ þ 1ÞR1 jþ jΛðϕ ðΛÞ  ϕ ð0ÞÞj ¼ D:

□

ð53Þ

5. Prescribed adaptive control

(1) M o Sðz1 Þ o M (2) limz1 - þ 1 Sðz1 Þ ¼ M, limz1 -  1 Sðz1 Þ ¼ M Since SðÞ is strictly increasing as well as ρðtÞ 4 0, the inverse transformation can be written as   e1 ðtÞ ð56Þ z1 ¼ S  1 ρðtÞ Assume z1 ðtÞ remains bounded z1 A L1 , 8 t Z 0, then M oSðz1 Þ o M holds, and hence the (54) can be guaranteed. A candidate function SðÞ is selected as Mez1 þMe  z1 ez1 þe  z1

ð57Þ

Conduct inverse transformation on (57), yielding   e1 ðtÞ 1 e1 ðtÞ=ρðtÞ  M ¼ ln z1 ¼ S  1 2 M  e1 ðtÞ=ρðtÞ ρðtÞ

ð58Þ

Then the derivative of z1 with respect to time can be written as   ∂S  1 e1 ðtÞ z_ 1 ¼ e1 ðtÞ ρðtÞ ∂ ρðtÞ " #  1 1 1 e_ 1 ðtÞ e1 ðtÞρ_ ðtÞ   ¼ 2 e1 ðtÞ=ρðtÞ  M e1 ðtÞ=ρðtÞ  M ρðtÞ ρ2 ðtÞ ¼ r 1 ðx_ 1  x_ d  e1 ðtÞρ_ ðtÞ=ρðtÞÞ

Different from the standard procedure of backstepping control presented in the literature (Su et al., 2005; Zhou et al., 2004; Zhang & Lin, 2011), the transient and steady-state performance of tracking error are incorporated in the design procedure of prescribed adaptive control. This control approach is originally developed in Bechlioulis and Rovithakis (2009), which is the first time that provides a systematic procedure to accurately compute the required bounds, thus making tracking error converge to a predefined arbitrarily small residual set, with convergence rate no less than a pre-specified value, exhibiting a maximum overshoot less than a sufficiently small preassigned constant (Bechlioulis & Rovithakis, 2009, 2008). 5.1. Prescribed performance function and error transformation The performance function is introduced in Bechlioulis and Rovithakis (2009) for the purpose of depicting a convergent zone in which the trajectory of tracking error which starts from a point in the zone remains for all future time. The performance function is a decreasing smooth function, which is defined as ρ : R þ -R þ with limt-1 ρðtÞ ¼ ρ1 4 0. It is noted that the control objective P2 can be guaranteed by satisfying M ρðtÞ o e1 ðtÞ o M ρðtÞ

SðÞ conforms the following conditions:

Sðz1 Þ ¼

According to the definition in (7), it yields

ð55Þ

ð54Þ

for all t Z0, where M o 0; M 4 0 are selected parameters. M ρð0Þ and M ρð0Þ represent the upper bound of the maximum overshoot and the lower bound of the undershoot, respectively. The constant ρ1 denotes the maximum tracking error at the steady state. Thus, the performance function and the parameters M, M prescribe the convergent zone for the transient and steady state performance of the tracking error. In order to meet the requirements P1 and P2 together with (54), an error transformation is developed (Bechlioulis & Rovithakis, 2009) by transforming the original nonlinear system (1) into an equivalent

ð59Þ

where r1 ¼

" # 1 1 1  : 2ρðtÞ e1 ðtÞ=ρðtÞ  M e1 ðtÞ=ρðtÞ  M

It is noted that both e1 ðtÞ and ρðtÞ in (59) are available and they can be involved in controller design. 5.2. Prescribed adaptive controller design The system (1) can be re-written as x_ 1 ¼ x2 x_ 2 ¼ x3 ⋮ x_ n  1 ¼ xn x_ n ¼ aT Y þbuðtÞ

ð60Þ T

T

where a ¼ ½  a1 ;  a2 ; …;  ak  and Y ¼ ½Y 1 ; Y 2 ; …; Y k  . The parameters a, b are unknown. u(t) denotes the system input with the inverse compensation as 0

uðtÞ ¼ ϕ ðΛÞud  db ðtÞ

ð61Þ

Considering the time derivative of transformed error (59) and nonlinear system (60), the transformed nonlinear system dynamics are given by z_ 1 ¼ r 1 ðx2  x_ d  e1 ðtÞρ_ ðtÞ=ρðtÞÞ x_ 2 ¼ x3 ⋮ x_ n  1 ¼ xn x_ n ¼ aT Y þbp ud ðtÞ  dðtÞ 0

ð62Þ

where bp ¼ bϕ ðΛÞ, dðtÞ ¼ bdb ðtÞ. The controller design is achieved by using the recursive back-stepping technique and is summarized

Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

as follows. The control law is developed as ud ðtÞ ¼ ζ^ ud1 ðtÞ

ð63Þ

with ^ ud1 ðtÞ ¼  kn zn  zn  1 þ α_ n  1 þ xdðnÞ  a^ Y þ sgnðzn ÞD T

ð64Þ

where z1 ¼

1 e1 ðtÞ=ρðtÞ  M ln 2 M  e1 ðtÞ=ρðtÞ

zi ¼ xi  xðid  1Þ  αði  1Þ ;

ð65Þ i ¼ 2; 3; …; n

α1 ¼  k1 z1 =r 1 þ e1 ðtÞρ_ ðtÞ=ρðtÞ

ð66Þ ð67Þ

α2 ¼  k2 z2 þ α_ 1  r 1 z1

ð68Þ

Eqs. (77) and (78) show that V(t) is nonincreasing. Therefore, ^ μ^ p and D^ are bounded. By utilizing the Lasalle– zi ði ¼ 1; …; nÞ, ζ^ , a, Yoshizawa theorem in Kristic and Kokotovic (1995) to (78), it further follows that zi -0ði ¼ 1 ¼ 1; …; nÞ as t-1, which concludes the tracking error is bounded within the prescribed zone. □ Remark on numerical implementation of the inverse compensator: It should be noted that the proposed control scheme is generally numerically implemented by microprocessors. From Fig. 1, the proposed control scheme can be described by two components: a feedforward inverse compensator and an adaptive control law. To implement numerically the inverse compensator (19), a discrete expression of (7) can be obtained as follows:

Π ½vðtÞ ¼ P½vðtÞ þ H½vðtÞ n

¼ p0 vðtÞ þ ∑ pi F ri ½vðtÞ þ R1 sat i¼1

αi ¼  ki zi þ α_ i  1  zi  1

ð69Þ

^ and where ki are positive design parameters. The parameters ζ^ , D the vector a^ are updated by the following adaptation laws:

ζ^ ¼  ηζ ud1 ðtÞzn

_

ð70Þ

a^_ ¼ Γa Yzn

ð71Þ

^_ ¼  η jzn j D D

ð72Þ

The stability of the closed-loop system is established in the following theorem. Theorem 2. For the transformed nonlinear system (1) preceded by ASPI model in (61), the prescribed adaptive controller presented by (63)–(72) guarantees that

!

þ gðvÞðtÞ=R1

m

!

∑ qi Ψ cj ½vðtÞ =R1

j¼1

ð79Þ

where pi denotes the weights of the play operator, F ri ½vðtÞ is the play operator at the threshold of ri, n is the number of the play operator used for identification, qi denotes the weight of the shift operator, Ψ ci ½vðtÞ is the shift operator at the slope of ci, m is the number of the elementary shift operator used for identification. gðvÞðtÞ is a local Lipschitz-continuity function. The numerical calculation of (19) can be listed as follows. Step 1: Identifying the weights pi, qi and parameters of the selected function g(v) in (7) by using the input and output experimental data of the magnetostrictive actuator as described in Section 6.2. Step 2: Construct the inverse model Π  1 in (19) by utilizing the analytic inverse P  1 in Krejci and Kuhnen (2001) as n

(i) All signals in the closed-loop system remain bounded. (ii) The tracking control with prescribed performance condition (54) is preserved.

155

P  1 ½uðtÞ ¼ p 0 vðtÞ þ ∑ p i F r i ½uðtÞ

ð80Þ

i¼1

where i

l1

Proof. From (59), and (65)–(69), and with bp ud ðtÞ ¼ bp ζ^ ud1 ¼ ud1  bp ζ~ ud1 , we have

r i ¼ p0 r i þ ∑ ∑ bj ðr l  r l  1 Þ

ð81Þ

z1 z_ 1 ¼ rz1 z2  k1 z21

ð73Þ

p 0 ¼ 1=p0

ð82Þ

z2 z_ 2 ¼ z2 z3  k2 z22  rz1 z2

ð74Þ

pi ¼  

zi z_ i ¼ zi zi þ 1 ki z2i  zi  1 zi

ð75Þ

T ^  dðtÞ  bp ζ~ u Þ zn z_ n ¼ zn ð  kn zn  zn  1 þ a~ Y þsgnðzn ÞD d1

ð76Þ

^ To establish the global ^ Let D~ ¼ D  D. where ζ~ ¼ ζ  ζ^ , a~ ¼ a  a. boundedness, the following Lyapunov function candidate is adopted: n

bp ~ 2 1 2 1 T 1 1 ~2 D zi þ a~ Γa a~ þ ζ þ 2 2 2 2 η ηD i¼1 ζ

VðtÞ ¼ ∑

ð77Þ

l¼1j¼1

p0 þ∑ij ¼ 1 pj

p i  p0 þ ∑ij ¼11 pj

ð83Þ

It is obvious that the computation of the inverse compensator only involves a small number of elementary operators and a selfdesigned function. For the control laws (63) and adaptive laws (70)–(72), their structures only include computations of simple scalar functions. Therefore, the proposed controller is computationally efficient and implementable.

6. Experimental results

The derivative of V(t) with regard to the time is n

T ^ n V_ ðtÞ ¼  ∑ ki z2i þ a~ Y zn  bp ζ~ ud1 zn þ sgnðzn ÞDz i¼1

T 1 dðtÞzn þ a~ Γa a~_ þ

bp ~_ ~ 1 ~_ ~ ζ ζ þ DD

ηζ

ηD

n

1 T r  ∑ ki z2i þ a~ ðY zn þ Γa a~_ Þ  bp ζ~ ðud1 zn i¼1

 1 _ 1 ~_  ζ~ Þ þ μ~ ðvðtÞzn  D~ jzn j  D Þ

ηζ

n

¼  ∑ ki z2i i¼1

ηD

ð78Þ

In this section, the prescribed adaptive controller designed above will be experimentally applied to a magnetostrictive actuator platform. Magnetostrictive materials are a class of materials that change their shape in presence of an external magnetic field. The characteristics of the magnetostrictive materials are called magnetostriction (MS) (Olabi & Grunwald, 2008) which was first discovered by James Joule in the 1842. In terms of this property, the magnetostrictive materials can be utilized as magnetostrictive actuators to drive various manipulation devices, especially in applications where large force and small displacements are required over a wide range of frequency with high precision (Li, Su, & Chai,

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Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

2014). Therefore, magnetostrictive actuators appear poised to play an increasingly important role in applications of vibration control (Zhang, 2004), high dynamic servo valve (Karunanidhi & Singaperumal, 2010), high-frequency micro-pump (Wang, Cheng, & An, 2010), and micro/nano-positioning (Yang, Yang, & Xu, 2012), etc. 6.1. Experimental setup In order to verify the proposed ASPI model and the developed robust controller, an experimental platform is established based on the system description shown in Fig. 6, where the actuator, MFR OTY77 manufactured by Etrema Products, consists of a wound wire solenoid surrounding two Terfenol-D rods, which are preloaded by a compression bolt and a spring washer. Three permanent magnets are installed along the Terfenol-D rods to provide a magnetic field with a bias. Fig. 7 shows main components of the actuator. A capacitive sensor (Lion Precision, model C23-C250) is used for measurement of the actuator displacement response with a sensitivity of 80 mV/μm, bandwidth of 15 kHz and a resolution of 35.53 nm. The excitation current to the actuator is applied through the power amplifier LVC2016 produced by AE Techron Inc. The displacement response of the actuator, measured by an integrated capacitive sensor, is obtained via the dSPACE control board equipped with 16-bit analog-to-digital converters (ADC) and 16-bit digital-to-analog converters (DAC). 6.2. Asymmetric hysteresis modeling and its inverse compensation Fig. 7. Magnetostrictive actuator (Aljanaideh, Al Janaideh, Rakheja, & Su, 2013).

Π ½vðtÞ ¼ P½vðtÞ þ H½vðtÞ n

¼ p0 vðtÞ þ ∑ pi F ri ½vðtÞ þR1 sat i¼1

þ gðvÞðtÞ=R1



m

!

15

10

Displacement (µ m)

Fig. 8 shows the hysteresis nonlinearity with minor loops in the magnetostrictive actuator which demonstrates that the hysteresis effects are asymmetric. In order to clearly illustrate this asymmetric phenomenon, the gap values of the hysteresis loops are calculated and plotted in Fig. 9 under inputs uðtÞ ¼ B0 sin ð2π tÞ; B0 ¼ 3; 5; 8. As is shown in the figure, the gap values of the hysteresis loops are not symmetric with x¼ 0 (the dotted line in Fig. 9), and all the peak values (the circle marker in Fig. 9) fall in left side of x¼0, which indicates that the hysteresis loops are left asymmetric. To capture the asymmetric hysteresis effects, the proposed ASPI model is therefore employed. In order to facilitate parameters identification of the ASPI model, the discrete expression in (84) is expressed as

5

0

−5

−10

∑ qi Ψ cj ½vðtÞ =R1

j¼1

ð84Þ

According to Condition 1, gðvÞðtÞ is selected as gðvÞðtÞ ¼  g 1 v2 ðtÞ þ g 2 , where g 1 Z 0; g 2 Z 0, R1 is selected as R1 ¼ 100. Ψ ci ½vðtÞ is

−15 −8

−6

−4

−2

0

2

4

6

8

Current (A) Fig. 8. Hysteresis effects with minor loop phenomenon exhibited in magnetostrictive actuator.

Fig. 6. The experimental platform.

Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

selected as the left shift operator, where ci 4 1. For simplicity, eight play operators and four left shift operators are chosen (n¼8, m¼4). The weights pi, qi and coefficients g1, g2 can be found by the following constrained quadratic optimization:

7

8A

Hysteresis gap ( µ m)

6 5

minf½C Λ  dT ½C Λ  dg

4 3

ΛðiÞ Z 0;

5A

iA f1; 2; 3; …; 14g

ð86Þ

Λ ¼ ½p0 ; …p8 ; q1 ; q2 ; q3 ; g1 ; g 2  , C ¼ ½F r0 ; …; F r8 ; Ψ c1 ; Ψ c2 ; where Ψ c3 ;  vðtÞ2 ; 1, with ri ¼ ½0; 0:1000; 1:2726; 1:5657; 1:8589; 2:1520; 2:7383; 3:0314; 3:6177, ci ¼ ½1:1; 1:2; 1:8, d is the output of the magnetostrictive actuator under a designed amplitude decreasing sinusoidal input signal. Then, the nonlinear least-square optimization toolbox in MATLAB is utilized to identify those parameters. Since the modeling error is inevitable, we can only obtain the estimated parameters as p^ i ¼ ½0:0284; 0:8593; 0:1166; 0:1348; 0:2873; 0:1529; 0:1704; 0:0348; 0:3617, q^ 1 ¼ 0:0567, q^ 2 ¼ 0:0391, q^ 3 ¼ 0:2777, g^ 1 ¼ 0:0497, g^ 2 ¼ 0:6205. Fig. 10 shows the comparison of input– output responses between the magnetostrictive actuator and ASPI model. Fig. 11 shows the modeling error which is defined as T

2 1 0 −8

−6

−4

−2

0

2

4

6

8

Current (A) Fig. 9. The gap values of the hysteresis in the magnetostrictive actuator.

Experimental data Model simulation data

15 10

Displacement ( µ m)

ð85Þ

with the constraints

3A

em ðtÞ ¼

5 0

4.5 4

−5

3.5 −10

3

−15 −6

−4

−2

1.5 2

0

2

2.5 4

6

Current (A) Fig. 10. Comparisons of input–output responses between experimental data and ASPI model.

1.5 1 0.5

Error(%)

157

0 −0.5 −1 −1.5 −2 −2.5

0

2

4

6

8

Time(sec) Fig. 11. Modeling error.

10

100ðxðtÞ  uðtÞÞ maxðxðtÞÞ

ð87Þ

where x(t) and u(t) denote the output of the magnetostrictive actuator and the ASPI model. The comparative results and the modeling error (less than 2.5%) suggest that the proposed ASPI model corresponds well with the experimental data. By using the above identified parameters of the PI model and the 1 analytical inverse P^ ½ðtÞ results in (81), the thresholds and weights 1 ^ of P ½ðtÞ are calculated as r i ¼ ½0:0028; 1:0438; 1:3381; 1:6721 ; 2:0902; 3:0161; 3:5290; 4:5752, p 0 ¼ 35:2113, p i ¼ ½  34:0848;  0:1308;  0:1178;  0:1768;  0:0679;  0:0617;  0:0111;  0:0944. Then, according to (19), the inverse ASPI model is constructed as ^ shown in Fig. 12, where H½ðtÞ ¼ 100 satðð0:0567Ψ c1 ½vðtÞ þ 0:0391 Ψ c2 ½vðtÞ þ 0:2777Ψ c3 ½vðtÞ  0:0497v2 ðtÞ þ 0:6205Þ=100Þ. The inverse compensator was implemented in the Matlab/ simulink and the codes were transformed into real-time control codes and downloaded to the dSPACE board. A desired tracking signal ud ðtÞ ¼ B1 sin ð2π tÞ; B1 ¼ 2; 5; 10 was applied to the compensator with an amplitude of 2 μm, 5 μm, 10 μm and a frequency of 1 Hz, and the output was applied to the magnetostrictive actuator through the power amplifier (LVC2016). The measured actuator displacement responses were subsequently obtained using capacitive sensor (Lion Precision, model C23-C) and uploaded into the dSPACE board. In order to compare with the experimental results, the simulation of inverse compensation is also conducted in the simulink. Fig. 13 shows the comparative results between the simulation and experimental test of the inverse compensation. The simulation results in Fig. 13 shows a perfect linear input– output relation between the desired displacement and actual displacement, which indicates that the hysteresis effects are completely canceled. However, in the experimental results, due to the existence of the modeling error, the inverse compensation error is unavoidable. For example, in Fig. 13(a) (the experimental result) the input and output relationship still shows a hysteretic loop, which suggests that the hysteresis is not completely canceled. Therefore, the prescribed adaptive control scheme was applied to accommodate this compensation error. 6.3. Prescribed adaptive control implementation

Fig. 12. The inverse compensator.

The entire control scheme is illustrated in Fig. 1. As a matter of fact, the magnetostrictive actuated platform can be modeled as the series connection of a hysteresis operator and a first-order dynamic component (Riccardi, Naso, Turchiano, & Janocha, 2013),

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Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

1.5

2

2µm Simulation Result

Actual displacement ( µ m)

Actual displacement ( µ m)

2

1 0.5 0 −0.5 −1 −1.5

1.5

2µm Experimental Result

1 0.5 0 −0.5 −1 −1.5

−2 −2

−1

0

1

−2 −2

2

Desired displacement (µ m)

Actual displacement ( µ m)

4 3

1 0 −1 −2 −3 −4

2

5µm Experimental Result 3 2 1 0 −1 −2 −3 −4

−5 −5

−2.5

0

2.5

−5 −5

5

Desired displacement (µ m) 10

10 µ m Simulation Result

5

0

−5

−10 −10

−5

0

5

−2.5

0

2.5

5

Desired displacement (µ m)

Actual displacement ( µ m)

Actual displacement ( µ m)

1

4

2

10

0

5

5µm Simulation Result

Actual displacement ( µ m)

5

−1

Desired displacement (µ m)

10

Desired displacement (µ m)

10 µ m Experimental Result

5

0

−5

−10 −10

−5

0

5

10

Desired displacement (µ m)

Fig. 13. The simulation and experimental inverse compensation results. (a) Simulation and experimental results (2 μm). (b) Simulation and experimental results (5 μm). (c) Simulation and experimental results (10 μm).

namely we select n¼ 1, Y ¼ xðtÞ in (60). The control objective is to force the output of the magnetostrictive-actuated system to follow the desired signal xd ¼ 5 sin ðtÞ and ensure the transient and steady-state performance of the tracking error within the prescribed function area. The prescribed performance function is selected as ρ ¼ ð1  0:07Þe  t þ 0:07 with M ¼ 10, M ¼  10. The parameters in the control and adaptive laws are selected as c1 ¼ 45, ηζ ¼ 1, Γ a ¼ 1, ηD ¼ 40. The initial state is chosen as xð0Þ ¼ 1:8. In addition, in the implementation the function sgnðzn Þ is replaced by satðzn Þ to avoid the chattering effect. The

experimental results are shown in Figs. 14–16. Fig. 14 shows the tracking error. It can be seen that a fairly satisfactory tracking performance is achieved after a small transient response and the tracking error converges to a small neighborhood of zero. To further illustrate the effectiveness of the adopted controller and compare with the open loop inverse compensation Fig. 13(b), the input–output relationship of the magnetostrictive actuated platform is demonstrated in Fig. 15. Although there are some transient response at beginning, the input and output responses of the actuator gradually converge to a nearly linear relationship instead

Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

10

0.45 µ m

0.5 0

6

Tracking Error ( µ m)

Several distinct features of this paper are summarized as follows:

1

8

−0.5

4

−1 40

2

42

44

0 −2 −4 −6 −8 −10

0

5

10

15

20

25

30

35

40

45

Time (sec) Fig. 14. The tracking error. 5

5µm

Actual displacement ( µ m)

4

(1) An asymmetric shifted Prandtl–Ishlinskii (ASPI) model was first proposed, being composed of three components: a Prandtl– Ishlinskii (PI) operator, a shift operator and a Lipschitz-continuous function. The benefits for such a model are that it can represent the asymmetric hysteresis behavior and facilities construction of the analytic inverse model for the purpose of mitigating the hysteresis effects of the magnetostrictive actuator. (2) Due to the presence of estimated error, the inverse compensation error is unavoidable. By means of the composition theorem, we firstly obtained the analytical inverse compensation error based on the ASPI model, which constitutes the main contribution of this paper. (3) To achieve global stability of the nonlinear system and guarantee the transient and steady-state performance of the positioning system, an adaptive prescribed control approach was utilized. Experimental results attained on the magnetostrictive actuated platform demonstrate the feasibility and effectiveness of the proposed approach.

3 2

References

1 0 −1 −2 −3 −4 −5 −5

−4

−3

−2

−1

0

1

2

3

4

5

Desired displacement ( µ m) Fig. 15. The input–output relation of the magnetostrictive actuator with prescribed adaptive controller. 3

2

Controller Output (A)

159

1

0

−1

−2

−3

0

5

10

15

20

25

30

35

40

45

Time (sec) Fig. 16. The control input signal to the system.

of nonlinear compensation error showing in Fig. 13(b). Fig. 16 shows control signal from the power amplifier to the magnetostrictive actuated platform. From these experimental results, the proposed prescribed adaptive controller clearly demonstrates excellent tracking performance.

7. Conclusion This paper deals with the robust adaptive control of a class of uncertain nonlinear system with asymmetric hysteresis input.

Aljanaideh, O., Al Janaideh, M., Rakheja, S., & Su, C.-Y. (2013). Compensation of ratedependent hysteresis nonlinearities in a magnetostrictive actuator using an inverse Prandtl–Ishlinskii model. Smart Materials and Structures, 22(2), 025027. Bechlioulis, C. P., & Rovithakis, G. A. (2008). Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed performance. IEEE Transactions on Automatic Control, 53(9), 2090–2099. Bechlioulis, C. P., & Rovithakis, G. A. (2009). Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems. Automatica, 45(2), 532–538. Brokate, M., & Sprekels, J. (1996). Hysteresis and phase transitions. New York: Springer-Verlag. Esbrook, A., Xiaobo, T., & Khalil, H. K. (2013). Control of systems with hysteresis via servo compensation and its application to nanopositioning. IEEE Transactions on Control Systems Technology, 21(3), 725–738. Granas, A., & Dugundji, J. (2003). Fixed point theory. New York: Springer-Verlag. Janaideh, A., Mao, J., Rakheja, S., Xie, W., & Su, C. (2008). Generalized Prandtl– Ishlinskii hysteresis model: Hysteresis modeling and its inverse for compensation in smart actuators. In IEEE conference on decision and control (pp. 5182–5187). Mexico: Cancun. Janaideh, A. M., Su, C. -Y., & Rakheja, S. (2012). Inverse compensation error of the Prandtl–Ishlinskii model, IEEE conference on decision and control (pp. 1597–1602). Maui, Hawaii, USA. Jiang, C., Deng, M., & Inoue, A. (2008). A novel modeling of nonlinear plants with hysteresis described by non-symmetric play operator. In The 7th world congress on intelligent control and automation (pp. 2221–2224). China: Chongqing. Jiang, H., Ji, H., Qiu, J., & Chen, Y. (2010). A modified Prandtl–Ishlinskii model for modeling asymmetric hysteresis of piezoelectric actuators. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 57(5), 1200–1210. Jiles, D. C., & Atherton, D. L. (1986). Theory of ferromagnetic hysteresis. Journal of Magnetism and Magnetic Materials, 61(1–2), 48–60. Karunanidhi, S., & Singaperumal, M. (2010). Design, analysis and simulation of magnetostrictive actuator and its application to high dynamic servo valve. Sensors and Actuators A: Physical, 157(2), 185–197. Krejci, P. (1986). Hysteresis, convexity and dissipation in hyperbolic equation. In International series of math science and applications. Krejci, P., & Kuhnen, K. (2001). Inverse control of systems with hysteresis and creep. IEE Proceedings of Control Theory and Applications, 148(3), 185–192. Kristic, I. K. M., & Kokotovic, P. V. (1995). Nonlinear and adaptive control design. New York: Wiley. Kuhnen, K. (2003). Modeling, identification and compensation of complex hysteretic nonlinearities. European Journal of Control, 9(4), 407–418. Li, Z., Hu, Y., Liu, Y., Chen, T., & Yuan, P. (2012). Adaptive inverse control of nonlinear systems with unknown complex hysteretic non-linearities. IET, Control Theory and Applications, 6(1), 1–7. Li, Z., Su, C.-Y., & Chai, T. (2014). Compensation of hysteresis nonlinearity in magnetostrictive actuators with inverse multiplicative structure for Preisach model. IEEE Transactions on Automation Science and Engineering, 11(2), 613–619. Olabi, A. G., & Grunwald, A. (2008). Design and application of magnetostrictive materials. Materials and Design, 29(2), 469–483. Preisach, F. (1935). Über die magnetische nachwirkung. Zeitschrift für Physik, 94, 277–302. Rakotondrabe, M. (2011). Bouc–Wen modeling and inverse multiplicative structure to compensate hysteresis nonlinearity in piezoelectric actuators. IEEE Transactions on Automation Science and Engineering, 8(2), 428–431.

160

Z. Li et al. / Control Engineering Practice 33 (2014) 148–160

Riccardi, L., Naso, D., Turchiano, B., & Janocha, H. (2013). Adaptive control of positioning systems with hysteresis based on magnetic shape memory alloys. IEEE Transactions on Control Systems Technology, 21(6), 2011–2023. Smith, R. (2005). Smart material systems: Model development. Society for Industrial and Applied Mathematics. Su, C.-Y., Wang, Q., Chen, X., & Rakheja, S. (2005). Adaptive variable structure control of a class of nonlinear systems with unknown Prandtl–Ishlinskii hysteresis. IEEE Transactions on Automatic Control, 50(12), 2069–2074. Tao, G., & Kokotovic, P. V. (1993). Adaptive control of systems with backlash. Automatica, 29(2), 323–335. Tao, G., & Kokotovic, P. V. (1995). Adaptive control of plants with unknown hystereses. IEEE Transactions on Automatic Control, 40(2), 200–212. Wang, C.-L., Cheng, X., & An, P. (2010). Modeling and simulation of a high-frequency micro-pump based on giant magnetostrictive material (gmm). Journal of Coal Science and Engineering (China), 16(2), 206–209. Xiao, S., & Li, Y. (2013). Modeling and high dynamic compensating the ratedependent hysteresis of piezoelectric actuators via a novel modified inverse

Preisach model. IEEE Transactions on Control Systems Technology, 21(5), 1549–1557. Yang, B.-T., Yang, D.-H., & Xu, P.-Y. (2012). Large stroke and nanometer-resolution giant magnetostrictive assembled actuator for driving segmented mirrors in very large astronomical telescopes. Sensors and Actuators A: Physical, 179(0), 193–203. Zhang, T. L. (2004). Giant magnetostrictive actuators for active vibration control. Smart Materials and Structures, 13(3), 473. Zhang, X. Y., & Lin, Y. (2011). Adaptive tracking control for a class of pure-feedback non-linear systems including actuator hysteresis and dynamic uncertainties. IET, Control Theory and Applications, 5(16), 1868–1880. Zhou, J., Wen, C., & Zhang, Y. (2004). Adaptive backstepping control of a class of uncertain nonlinear systems with unknown backlash-like hysteresis. IEEE Transactions on Automatic Control, 49(10), 1751–1759. Zhou, J., Wen, C., & Li, T. (2012). Adaptive output feedback control of uncertain nonlinear systems with hysteresis nonlinearity. IEEE Transactions on Automatic Control, 57(10), 2627–2633.