51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Inverse Compensation Error of the Prandtl-Ishlinskii Model Mohammad Al Janaideh, Chun-Yi Su, and Subhash Rakheja
Abstract— Compensation of hysteresis nonlinearity of smart actuators through applications of an inverse hysteresis model generally yields errors when coupled with a plant. In this study, the resulting compensation error is analytically derived considering the Prandtl-Ishlinskii model and its inverse, which has not yet been reported. The proposed analytical error formulation could be conveniently applied in controller design for effective hysteresis compensation and thereby enhanced tracking performance of the smart material actuators. The initial loading curve and its inverse are presented for formulation of the estimated Prandtl-Ishlinskii model for describing hysteresis nonlinearity of a smart actuator and its inverse. The compensation error is subsequently derived analytically from composition of the Prandtl-Ishlinskii model with its inverse based on the inverse of initial loading curve. The properties of the compensation error are further illustrated, which also exhibits hysteresis nonlinearity that can be further characterized by a Prandtl-Ishlinskii model. The integration of the proposed analytical error model with the composition resulted in only negligible hysteresis compensation error. The simulation results suggest that a significantly improved tracking performance can be achieved in the closed-loop control system using the proposed analytical error model.
I. I NTRODUCTION Smart actuators such as piezoceramic and magnetostrictive actuators, invariably, exhibit hysteresis nonlinearity, which can cause oscillations in the actuator responses in the openloop, and poor tracking performance and potential instabilities of the closed-loop control system, see for example [1] -[3]. Considerable efforts have been made to seek methods for effective compensation of hysteresis effects in order to enhance the tracking performance of smart actuators, particularly for the closed-loop control systems. A number of model-based and inverse model-based methods have been proposed for compensation of hysteresis effects in smart actuators, see for example [2]-[5]. Model-based methods employ a hysteresis model, often an operator-based model, to construct a controller design for hysteresis compensation, see for example [6]. Control algorithms based on inverse hysteresis models have been suggested to be more effective in compensating for the hysteresis effects, see for example [7]-[17]. A number of studies have reported inverse hysteresis models derived from the hysteresis models using either analytical or numerical methods, see for example [2][10]. Considering that Mohammad Al Janaideh is with Department of Mechatronics Engineering, The University of Jordan, Amman, Jordan, E-mail:
[email protected]. Chun-Yi Su and Subhash Rakheja are with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, Canada, E-mail:
[email protected],
[email protected]
978-1-4673-2066-5/12/$31.00 978-1-4673-2064-1/12/$31.00 ©2012 ©2012 IEEE IEEE
the Preisach model is not analytically invertible, numerical methods are employed to obtain approximate inversions of the model. The effectiveness of the reported approximate inversions in conjunction with different controller syntheses in hysteresis compensation has been demonstrated in a few studies, see for example [10][11]. The Prandtl-Ishlinskii model offers an attractive and unique property of being analytically invertible [18]. The studies reporting hysteresis compensation using analytical Prandtl-Ishlinskii model inverse, however, also exhibit notable compensation errors, which are partly attributed to hysteresis characterization errors, and in-part to the estimated Prandtl-Ishlinskii model [7][15]. The applications of such estimated hysteresis models in deriving the model inverse would be expected to yield some degree of hysteresis compensation error. This error yields tracking error in the closedloop control system. The magnitude of the tracking error could be greatly minimized using robust or adaptive control methods with the closed-loop control system. This would be particularly more feasible for the analytically invertible Prandtl-Ishlinskii model, where the error associated with inverse compensation could be derived analytically, although it has not yet been attempted. In this paper, the analytical error of the inverse compensation of the Prandtl-Ishlinskii model is analytically derived. The properties of the analytical error of the inverse compensation are illustrated. The results demonstrate that the error of the inverse compensation can also be described by Prandtl- Ishlinskii model. The analytical expression of the error of the inverse compensation can be used with available model based control methods. Consequently, combination between the inverse-based and model-based control methods can be carried out to achieve high tracking performance of the closed-loop control system. II. T HE P RANDTL -I SHLINSKII M ODEL AND THE I NITIAL L OADING C URVE The Prandtl-Ishlinskii model and the initial loading curve, described in [15][18], are only briefly presented in this section. A. The Prandtl-Ishlinskii model For a given input v(t) ∈ C[0, T ], the output of the PrandtlIshlinskii model Π[v](t) = qv +
∫ R 0
p(r)Γr [v](t)dr ,
(1)
where q is a positive constant, p(r) is a density function, r is positive threshold, and Γr [v](t) represents the output of
1597
the play operator. In discrete form the output of the PrandtlIshlinskii model can be expressed with the thresholds 0 = r0 ≤ r1 · · · ≤ rN ≤ rN+1 = ∞ as
7 6
(2)
Πφ[v](t)
φ(r)
N
Π[v](t) = qv + ∑ p(r j )Γr j [v](t)∆r j ,
5
5 4 3 2
j=1
−5
1
where N is the number of the play operators Γr j and p(r j ) are positive weights. The output of the play operator Γr j is defined for functions v that are monotone (non-decreasing or non-increasing) in each interval [ti−1 ,ti ] of a partition 0 = t0 < · · · < tm = T by the formula Γr j [v](t) = max(v(t) − r j , min(v(t) + r j , Γr j [v](ti−1 ))
0
0
1
2
3
4
5
6
7
−5
r
ϕ (r) = qr +
p(ζ )(r − ζ )d ζ .
0
(4)
The Prandtl-Ishlinskii model (1) can be expressed with the initial loading curve (4) as Π[v](t) = ϕ ′ (0)v +
ϕ ′ (0)
∫ R 0
ϕ ′′ (r)Γr [v](t)dr,
The output of the inverse Prandtl-Ishlinskii model is expressed analytically as Π−1 [v](t) = ψ ′ (0)v +
where = q and = p(r). In discrete form, the initial loading curve (4) can be expressed for r ∈ [r j , r j+1 ) as
ψ (s) = q−1 s +
(10)
∫ s 0
g(ζ )(s − ζ )d ζ .
(11)
The output of the inverse model can be expressed as Π−1 [v](t) = q−1 v +
∫ S 0
g(s)Γs [v](t)ds.
(12)
For 0 = s0 ≤ s1 · · · ≤ sN ≤ sN+1 = ∞ N
Π−1 [v](t) = q−1 v + ∑ g(si )Γsi [v](t)∆s j .
(6)
(13)
i=1
j
ϕ ′ (r) = q + ∑ p(ri )∆ri .
(9)
where ψ is the inverse of the initial loading curve. Analytically
i=1
and
0
′′
ψ −1 (s)Γs [v](t)ds
ψ = ϕ −1 ,
j
ϕ (r) = qr + ∑ p(ri )(r − ri )∆ri
∫ S
where s is positive threshold and ψ is concave initial loading curve [15]. Analytically
(5)
ϕ ′′ (r)
5
III. I NVERSE P RANDTL -I SHLINSKII M ODEL
B. The initial loading curve The initial loading curve, also referred to as the shape function, forms the essential basis for describing the curvature of the output-input loops of the Prandtl-Ishlinskii model and deriving the analytical inverse of the PrandtlIshlinskii model [15][18][19]. Initial loading curve of the Prandtl-Ishlinskii model could be described as the stressstrain curve under increasing load, from zero to a final value, and is defined for the thresholds r > 0 as [15][18]
0 v(t)
(a) (b) Fig. 1. (a) The initial loading curve ϕ (r), and (b) show the output of the Prandtl-Ishlinskii model constructed with the initial loading curve ϕ (r) (a).
(3)
with initial condition Γr j [v](0) = max(v(0) − r j , min(v(0) + r j , 0)).
∫ r
0
(7)
The initial loading curve can be expressed for s ∈ [s j , s j+1 ) as j
i=1
We denote the Prandtl-Ishlinskii model Π with initial loading curve ϕ as Πϕ . C. Example 1
ψ (s) = q−1 s + ∑ g(si )(s − si )∆si
(14)
i=1
and j
Let’s construct the Prandtl-Ishlinskii model using the density function p(r) = 0.1r and q = 0.17. With these parameters the Prandtl-Ishlinskii model is constructed with the initial loading curve 17 1 ϕ (r) = r + r3 . (8) 100 60 Consider an input of the form v(t) = 7 sin(π t)/(1 + 0.06t), where t ∈ [0, 13]. The chosen simulation parameters are ∆t = 0.01, r ∈ [0, 7], ∆r = 0.07. The initial loading curve (8) and output of the Prandtl-Ishlinskii model is presented in Figure 1.
ψ ′ (s) = q−1 + ∑ g(si )∆si .
(15)
i=1
We denote the inverse Prandtl-Ishlinskii model Π−1 constructed with inverse initial loading curve ψ = ϕ −1 as Πψ or Πϕ −1 . A. Example 2 Inverse Prandtl-Ishlinskii model Πϕ −1 presented in Example 1 is shown in Figure 2(a). The output of the inverse compensation Πϕ ◦ Πϕ −1 [v](t) is shown in Figure 2(b).
1598
the error of the inverse compensation for the input v(t) can be expressed as
5 Πφ o Πφ−1[v](t)
Πφ−1[v](t)
5
0
−5
e(t) = (1 − η ′ (0))v(t) −
0
0 v(t)
5
−5
(a)
0 v(t)
e(t) = v(t) − Πψ ◦ Πϕ [v](t).
Fig. 2. (a) The output of the inverse model Πϕ −1 [v](t), and (b) show the output of the inverse compensation Πϕ ◦ Πϕ −1 [v](t).
A. Analytical formulation of composition of the PrandtlIshlinskii model The resulting compensation error could be analytically derived from composition of the Prandtl-Ishlinskii model, introduced by Pavel Krejˇc´ı in [19]. The composition states that when the output of the Prandtl-Ishlinskii model Πσ [v](t), constructed with initial loading curve σ , is applied as an input to another Prandtl-Ishlinskii model Πρ , constructed with initial loading curve ρ , the composition Πρ ◦ Πσ can also be characterized by a Prandtl-Ishlinskii model Πρ ◦ Πσ , constructed with an initial loading curve θ . Analytically Πθ [v](t) = Πρ ◦ Πσ [v](t),
(16)
θ (r) = ρ ◦ σ (r).
(17)
where The output of the composition expressed as 0
θ ′′ (r)Γr [v](t)dr ,
Πψ ◦ Πϕ [v](t) = Πη [v](t) Πψ ◦ Πϕ [v](t) = η ′ (0)v(t) +
The inverse Prandtl-Ishlinskii model applied as a feedforward compensator in an open-loop manner generally yields some compensation error [15] [2]. For the first time in the literature, the error of the inverse compensation error can be is formulated analytically for the Prandtl-Ishlinskii model, which has not yet been reported in the literature and could be subsequently applied to seek better hysteresis compensation. The analytical error of the inverse compensation of the Prandtl-Ishlinskii model is presented in this section. In order to formulate this error analytically, analytical formulation of composition of the Prandtl-Ishlinskii model is presented.
(18)
where θ ′′ (r) is a density function and θ ′ (0) is a positive constant. B. Analytical error of the inverse compensation
(20)
Then the output of the inverse compensation is expressed as
and
IV. A NALYTICAL E RROR OF THE I NVERSE C OMPENSATION OF THE P RANDTL -I SHLINSKII M ODEL
∫ R
η ′′ (r)Γr [v](t)dr ,
η (r) = ψ ◦ ϕ (r). (19) Proof: The error of the inverse compensation expressed as
5
(b)
Πθ [v](t) = θ ′ (0)v(t) +
0
where
−5
−5
∫ R
∫ R 0
(21)
η ′′ (r)Γr [v](t)dr.
(22)
Then, we conclude that the error of the inverse compensation is expressed as e(t) = v(t)(1 − η ′ (0)) −
∫ R 0
η ′′ (r)Γr [v](t)dr.
(23)
Remark 1: It can be concluded that the error of the inverse compensation shows hysteresis nonlinearities presented by Prandtl-Ishlinskii model. Consequently, for high compensation error cause undesirable inaccuracies or oscillations and even instability. Remark 2: For the given Prandtl-Ishlinskii model based on initial loading curve Πϕ , if an exact inverse of the PrandtlIshlinskii model could be constructed using Πϕ −1 , then the initial loading curve in (23) will reduce to η (r) = (r). Then, η ′′ (r) = r and η ′ (0) = 1. The inverse compensation error would thus diminish, e(t) = 0 leading to Πη (r) [v](t) = v(t). Remark 3: Without proposing the inverse model as a feedforward compensator, where Πψ [v](t) = v(t) and ψ (r) = r, the error (23) can be expressed as e(t) = v(t)(1 − ϕ ′ (0)) − In this case η (r) = ϕ (r).
∫ R 0
ϕ ′′ (r)Γr [v](t)dr.
(24)
C. The weights and the thresholds of the inverse model In this section the parameters identifications for the error of the inverse compensation are presented. We follow the procedure presented in [15]. Let ψ = ϕ −1 , then η (r) can be expressed as ψ ◦ ϕ (r) = r. (25) To obtain the parameters of the inverse, for s ∈ [s j , s j+1 ), d d where j = 0, · · · , N, we use dr (ψ ◦ ϕ (r)) = dr r. Then, ′ ′ ϕ (r)ψ (s) = 1, where s j = ϕ (r j ). Then, thresholds s j can be written with r0 = s0 = 0 as j−1
We denote the inverse estimated Prandtl-Ishlinskii model as Πϕˆ −1 . Theorem 1: When the output of the inverse of the estimated Prandtl-Ishlinskii model Πψ [v](t), where ψ = ϕˆ −1 , is applied as a feedforward compensator to mitigate the hysteresis nonlinearities of the Prandtl-Ishlinskii model Πϕ ,
s j − s j−1 = (q + ∑ p(ri )∆ri )(r j − r j−1 )
(26)
i=1
and the weights g j are
1599
j
q−1 + ∑ g(si )∆si = i=1
1 j q + ∑i=1
p(ri )∆ri
.
(27)
q
1 + g(s1 )∆s1 + g(s2 )∆s2 = , q + p(r1 )∆r1 + p(r2 )∆r2 .
Πψ o Πφ[v](t)
−1
1 q+p(r1 )∆r1 ,
6
6
4
4
2
2
Πψ o φ [v](t)
Then q−1 + g(s1 )∆s1 =
0 −2 −4
.
−6
q−1 +g(s1 )∆s1 +.+g(sN )∆sN =
1 . q + p(r1 )∆r1 + . + p(rN )∆rN
−4
−5
−6
5
−5
0 v(t)
(a)
5
(b)
1
j
j−1
i=1
i=1
e(t)
0.5
p j ∆r j
(28)
0 −0.5
(q + ∑ pi ∆ri )(q + ∑ pi ∆ri ).
−1 −1.5
For p j = p(r j )∆r j and g j = g(s j )∆s j , we conclude [15] 1 g0 = , g j = − q
0 v(t)
1.5
Using Equation (27) we conclude g(s j )∆s j = −
0 −2
0
2
4
6 Time (S)
8
10
12
(c)
pj j
j−1
i=1
i=1
.
(29)
(q + ∑ pi )(q + ∑ pi ) j−1
s j − s j−1 = (q + ∑ pi )(r j − r j−1 ).
(30)
Fig. 3. (a) The output of the inverse compensation Πψ ◦ Πϕ [v](t) and Πη ∫ with ϕˆ (r) = 0.18r + 0r 0.11ζ (r − ζ )d ζ , (b) show the output of the compensation Πψ ◦Πϕ [v](t) when (a) applied as a feedforward compensator, (c) The time history of the error of the inverse compensation v(t) − Πψ ◦ Πϕ [v](t) (dashed line) and e(t) of Theorem 1 (dotted line) for the compensation (b).
i=1
V. A N A DAPTIVE C ONTROL WITH I NVERSE P RANDTL -I SHLINSKII M ODEL A controller synthesis is used to integrating the analytical error expression (23) and to investigate effectiveness of the inverse model-based hysteresis compensation when coupled
4
2
2 Πψ o φ [v](t)
4
0
−2
−4
0
−2
−5
0 v(t)
−4
5
−5
0 v(t)
(a)
5
(b)
3 2 1 e(t)
The error associated with application of the estimated inverse model and as a feedfoward compensator in an openloop manner, and the effectiveness of the analytical error model are illustrated through an example. In this example we apply the inverse estimated Prandtl-Ishlinskii model Πψ , where ψ = ϕˆ −1 , to compensate for the hysteresis nonlinearities of the Prandtl-Ishlinskii model Πϕ considered in Example 1. We use the simulation parameters of Example 1. The inverse of the Prandtl-Ishlinskii model is presented based on estimated initial loading curve. In this example, we compare the output of the inverse compensation Πϕ ◦ Πψ [v](t) when the output of the inverse Prandtl-Ishlinskii model Πψ [v](t) applied as a feedforward compensator to compensate for the hysteresis nonlinearity of Πϕ and output of the compensation Πη [v](t) obtained by Theorem 1. In order to construct the inverse estimated PrandtlIshlinskii model Πψ∫, we consider the initial loading φˆ (r) = 0.18r + 0r 0.11ζ (r − ζ )d ζ and φˆ (r) = 0.2r + curve ∫r 0 0.2ζ (r − ζ )d ζ for the simulation results in Figure 3 and in Figure 4, respectively. The example shows the capability of the Theorem 1 to compute the output of the compensation and the error of the inverse compensation. It is clear that the output of the inverse compensation is hysteresis nonlinearities. The example shows that better estimation for the parameters of the Prandtl-Ishlinskii model yields less hysteresis nonlinearities in the output of the compensation and this yields less error in the compensation.
Πψ o Π φ [v](t)
D. Example 3
0 −1 −2 −3
0
2
4
6 Time (S)
8
10
12
(c) Fig. 4. (a) The output of the compensation Πψ ◦ Πϕ [v](t) and Πη with ∫ ϕˆ (r) = 0.2r + 0r 0.2ζ (r − ζ )d ζ , (b) show the output of the compensation Πψ ◦ Πϕ [v](t) when (a) applied as a feedforward compensator, (c) The time history of the error of the inverse compensation v(t)−Πψ ◦Πϕ [v](t) (dashed line) and e(t) of Theorem 1 (dotted line) for the compensation (b).
with a controlled plant in a closed-loop manner. Su et al. [6] proposed an adaptive robust controller to control a system preceded with hysteresis nonlinearities of the PrandtlIshlinskii model without using the inverse compensation. From (23) it is shown that the inverse compensation error is still expressed by Prandtl Ishlinskii model. Therefore, the approach presented in [6] can sill be used for the controller design. It should be mentioned that the purpose of this section is: (i) to couple the output of the inverse compensation into a controlled plant in a closed-loop control system, and (ii) to study the effects of the inverse compensation and its error on the tracking error performance of the closed-loop
1600
control system. The closed-loop control system comprising the plant together with the inverse model compensator is used. The nonlinear hysteresis of the actuator is characterized by the Prandtl-Ishlinskii model Πϕ and its estimated inverse Πψ is applied as a feedforward compensator. The output of the inverse compensation Πψ ◦ϕ [u](t) is applied as an input signal to a controlled plant, generally characterized as a dynamic system of the form (t)) = bΠψ ◦ Πϕ [u](t)
i=1
(31)
Πψ ◦ Πϕ [u](t) = bη (0)u(t) + b
0
x˙ = a ′′
η (r)Γ[u](t)dr
where Yi are known continuous linear or nonlinear functions, parameters ai and control gain b are unknown constants, v(t) is the input of hysteresis, u(t) is the input to the system and y(t) is the system output. It is common assumption that the sign of sign of b is known. Without losing generality, we assume b > 0. The control objective is to design an output feedback control law for u(t) to force the plant state x(t) to follow a specified desired trajectory xd (t), i.e., x(t) → xd (t) as t → ∞. We assume that the reference signal xd (t) is a (i) smooth bounded signal and its time derivatives xd (t)(1 ≤ i ≤ n) are bounded. The dynamic system (31) can be expressed as x˙1 = x2 , ..., x˙n−1 = xn , and x˙n = − ∑ki=1 aiYi (x1 (t), x2 (t), ..., xn−1 (t)) + bΠψ ◦ Πϕ [u](t), where x1 (t) = x(t), x2 (t) = x(t), ˙ ..., xn (t) = x(n−1) (t), a = [a1 , a2 , · · · , ak ]T , and Y = [−Y1 , −Y2 , · · · , −Yk ]T To design a robust adaptive control law, the following definitions of parameters are given a˜ = aˆ − a and η˜ ′′ (t, r) = ηˆ ′′ (t, r)− η ′′ (r), where aˆ is an estimate of a, βˆ is an estimate 1/bη ′ ηˆ ′′ (t, r) is an ∫estimate of η ′′ (r). Let of β = ∫ R ′′ (0), and R ′′ ˆ ˆ (t, r) |Γr [u](t)| dr. B(t)∆ 0 η (r) |Γr [u](t)| dr and B(t)∆ 0 η ∫ R ′′ ′′ ˜ ˆ |Γr [u](t)| dr. The Then we have B(t)∆ ( η (t, r) − η (r)) 0 controller is proposed as u(t) = βˆ (t)u1 (t)
1 − e−x(t) + bΠϕ ◦ Πψ [u](t) 1 + e−x(t)
(36)
with the Prandtl-Ishlinskii model, shown in Figure 5(a). The inverse estimated model Πψ shown in Figure 5(b). In this figure we consider the exact inverse Πψ = Πϕ −1 and the estimated inverse of Πψ = 1.1Πϕ −1 and Πψ = 0.9Πϕ −1 . In Figure 5(c) we show that output of the inverse compensation Πψ ◦ Πϕ of the open-loop system when the inverse model Πψ presented in Figure 5(b) applied to compensate for the hysteresis nonlinearities considered in 5(a). Figure 5(d) shows the error of the inverse compensation for the openloop system. 100
50
50
25 Πψ[rd](t)
′
∫ R
In this section we consider a nonlinear system in the form of (31) given by [6]
0 −50
−100 −50
0 −25
−25
0 rd(t)
25
−50 −50
50
−25
(a)
0 rd(t)
25
50
(b) 30
50
20 10
0
e(t)
where
A. Simulation Example
Πφ[rd](t)
x (t) + ∑ aiYi (x(t), x(t), ˙ ...,x
(n−1)
ΠψoΠφ[u](t)
k
(n)
Πψ , and the compensation error described as (23), the adaptive controller described by (32) can guarantee the global boundedness of the closed-loop control system and the output tracking with desired precision. Proof: Theorem 1 shows that the error the inverse compensation e(t) is a Prandtl-Ishlinskii hysteresis model. Consequently, similar procedures presented in [6] can be used to prove the stability.
0 −10
(32)
where u1 = −cn zn − zn−1 − aˆ Y + uN + xdn + α˙ n−1 , where (i−1) z1 (t) = x1 (t) − xd (t), zi = xi (t) − xd − α˙ n−1 , for i = 2, 3, ..., n, α1 (t) = −c1 z1 (t), αi (t) = −ci zi (t) − zi−1 (t) + α˙ i−1 (x1 , ...., xi−1 , xd , ...., xdi−1 ), for i = 2, 3, ..., n − 1, uN = ˆ and ci > 0 = 1, 2, ..., n − 1 are design parameters. sign(zn )B, The corresponding adaptive laws are ˙ βˆ = −λ u1 zn ,
(33)
a˙ˆ = γ Y(zn ),
(34)
∂ ′′ ηˆ (t, r) = ξ |zn ||η ′′ (r)[u](t)|, (35) ∂t where λ , γ , and ξ are positive constants determining the rules of the adaptations. Theorem 2: For the plant described by (31) with the hysteresis Πϕ , the inverse of the estimated hysteresis model
−50
−50
−20
−25
0 rd(t)
(c)
25
50
−30
0
2
4 6 Time (S)
8
10
(d)
Fig. 5. Open-loop system for compensation hysteresis nonlinearities using the inverse Prandtl-Ishlinskii model. (a) Hysteresis loops of the PrandtlIshlinskii model Πϕ , (b) the output of the inverse Prandtl-Ishlinskii model Πψ , (c) the output of the compensation Πψ ◦ Πϕ [rd ](t), and (d) the time history of the output of the inverse compensation. Solid line: Πψ = Πϕ −1 , dashed line: Πψ = 1.1Πϕ −1 , and dotted line: Πψ = 0.9Πϕ −1 .
We use the adaptive controller (32) and (33)-(35) with a = b = 1, xd (t) = 30 sin(3t), t ∈ [0, 10], λ = 0.07, γ = 0.16, ξ = 0.34, βˆ (0) = 0.43, a(0) ˆ = 0.13, x(0) = 1.05. To illustrate the effusiveness of the inverse model in the closed-loop control system, simulation is carried out with the exact inverse model Πψ = Πϕ −1 and the estimated inverse model of Πψ = 1.1Πϕ −1 and Πψ = 0.9Πϕ −1 . The relationship between
1601
the control signal u(t) and the output of the compensation in the closed-loop control system with Πψ = 1.1Πϕ −1 and Πψ = 0.9Πϕ −1 are shown in Figure 6. It is obvious that closedloop control system with Πψ = 0.9Πϕ −1 exhibits higher compensation nonlinearities. Figure 7 shows the tracking error er (t) = xd (t) − y(t) of the closed-loop control system. The controller clearly demonstrates excellent tracking performance as evident from the results. Owing to the hysteresis nonlinearity in the output of the compensation, Figure 7 shows higher tracking error with Πψ = 0.9Πϕ −1 between t = 0 and t = 5 Seconds. However, the results demonstrate the robustness of the adaptive controller with inverse estimated model. Due to high hysteresis nonlinearities, Figure 8 shows high tracking error with the inverse estimated model Πψ [u] = 0.5Πϕ −1 [u] and without the inverse model Πψ [u] = u. 150 100
50
φ
Π oΠ [u](t)
ΠψoΠφ[u](t)
50
0
ψ
0 −50
−50
−100 −150 −150 −100 −50
0
50 u(t)
100 150 200
−50
0 u(t)
(a)
50
(b)
Fig. 6. The relationship between the control u(t) and the output of the compensation: (a) Πψ [u] = 0.9Πϕ −1 [u], and (b) Πψ [u] = 1.1Πϕ −1 [u].
6 4 e (t)
2 r
0 −2 −4 −6 0
1
2
3
4
5
6
7
8
9
10
Time (S)
Fig. 7. Tracking error considering the inverse estimated Prandtl-Ishlinskii model : Πψ [u] = Πϕ −1 [u] (Solid blue line), Πψ [u] = 1.1Πϕ −1 [u] (dashed black line), and Πψ [u] = 0.9Πϕ −1 [u] (dotted red line).
200
er(t)
100 0 −100 −200 0
2
4
6
8
10
Time (S)
Fig. 8. Tracking error of the closed-loop control system without the inverse estimated model Πψ [u] = u (dashed line), and with inverse estimated model Πψ [u] = 0.5Πϕ −1 [u] (solid line).
VI. C ONCLUSIONS The error of the inverse compensation of the PrandtlIshlinskii model that derived analytically shows hysteresis effects that can be expressed by Prandtl-Ishlinskii model. Due to hysteresis effects in the output of the compensation, high compensation errors may cause undesirable inaccuracies or oscillations and even instability. Since an analytical expression for the error of the inverse compensation is derived, the inverse estimated Prandtl-Ishlinskii model can be applied with model-based control methods to achieve high tracking performance in the closed-loop control system. R EFERENCES [1] R. C. Smith, Smart Material Systems. Philadelphia, PA: SpringerVerlag, 2005. [2] M. Al Janaideh, S. Rakheja, and C-Y. Su, “An analytical generalized Prandtl-Ishlinskii model inversion for hysteresis compensation in micro positioning control,” IEEE/ASME Transactions on Mechatronics, vol. 16, no. 4, pp. 734-744, 2011. [3] G. Tao and P. V. Kokotovi´c, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. New York, NY: Wiley, 1996. [4] R. Gorbet, K. Morris, and D. Wang, “Passivity-based stability and control of hysteresis in smart actuators,” IEEE Transactions on Control Systems Technology, vol. 9, no. 1, pp. 5-16, 2001, 2001. [5] X. Tan and J. Baras, “Modeling and control of hysteresis in magnetostrictive actuators”, Automatica, vol. 40, no. 9, pp. 1469–1480, 2004. [6] C. Y. Su, Q. Wang, X. Chen, and S. Rakheja, “Adaptive variable structure control for a class of nonlinear systems with unknown PrandtlIshlinskii hysteresis”, IEEE Transactions on Automatic Control, vol. 50, no. 12, pp. 2069-2074, 2005. [7] M. Al Janaideh, Generalized Prandtl-Ishlinskii hysteresis model and its analytical inverse for compensation of hysteresis in smart actuators, Phd thesis, Concordia University, Montreal, 2009. [8] A. Cavallo, C. Natale, S. Pirozzi, C. Visone, “Feedback control systems for micropositioning tasks with hysteresis compensation,” IEEE Transactions on Magnetics, vol. 40, no. 2, pp. 876879, 2004. [9] D. Davino, C. Natale, S. Pirozzi, C. Visone,“Phenomenological dynamic model of a magnetostrictive actuator,” Physica B, vol. 343, pp. 112-116, 2004. [10] R. V. Iyer, X. Tan, and P. S. Krishnaprasad, “Approximate inversion of the Preisach hysteresis operator with applications to control of smart actuators,” IEEE Transactions on Control Systems Technology, vol. 50, no. 6, pp. 798-810, 2005. [11] J. Nealis and R. C. Smith, “Model-based robust control design for magnetostrictive transducers operating in hysteretic and nonlinear regimes,” IEEE Transactions on Control Systems Technology, vol. 15, no. 1, pp. 22-39, 2007. [12] A. G. Hatch, R. C. Smith, T. De and M. V. Salapaka, “Construction and Experimental Implementation of a Model-Based Inverse Filter to Attenuate Hysteresis in Ferroelectric Transducers,” IEEE Transactions on Control Systems Technology, vol. 14, no. 6, pp. 1058-1069, 2006. [13] X. Tan and H. Khalil, “Control unknown dynamic hysteretic systems using slow adaption: preliminary results,” Proc. of the American Control Conference, NewYork, NY, USA, pp. 3294- 3299, 2007. [14] C. Visone, “Hysteresis modelling and compensation for smart sensors and actuators,” Journal of Physics: Conference Series, vol. 138, pp. 1-25, 2008 [15] P. Krejci and K. Kuhnen, “Inverse control of systems with hysteresis and creep”, IEE Proc Control Theory Application, vol. 8, pp. 185-192, 2001. [16] C. Visone and M. Sj¨ostr¨om, “Exact invertible hysteresis models based on play operators,” Physica B, vol. 343, pp. 148-152, 2004. [17] K. Kuhnen, “Modeling, identification and compensation of complex hysteretic nonlinearities- a modified Plandtl-Ishlinskii approach,” European Journal of Control, vol. 9, pp. 407-418, 2003. [18] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, New York, Springer, 1996. [19] P. Krejci, “Hysteresis, convexity and dissipation in hyperbolic equation”, Gakuto , International series of Math Science and applications, 1986.
1602
