Model Reference Adaptive Control Based on KP Model for Magnetically Controlled Shape Memory Alloy Actuators

Abstract

Introduction

Magnetically controlled shape memory alloy (MSMA) actuators take advantages of their large deformation and high controllability. However, the intricate hysteresis nonlinearity often results in low positioning accuracy and slow actuator response.

Methods

In this paper, a modified Krasnosel'skii-Pokrovskii model was adopted to describe the complicated hysteresis phenomenon in the MSMA actuators. Adaptive recursive algorithm was employed to identify the density parameters of the adopted model. Subsequently, to further eliminate the hysteresis nonlinearity and improve the positioning accuracy, the model reference adaptive control method was proposed to optimize the model and inverse model compensation.

Results

The simulation experiments show that the model reference adaptive control adopted in the paper significantly improves the control precision of the actuators, with a maximum tracking error of 0.0072 mm.

Conclusions

The results prove that the model reference adaptive control method is efficient to eliminate hysteresis nonlinearity and achieves a higher positioning accuracy of the MSMA actuators.

Keywords

Adaptive recursive algorithm Hysteresis nonlinearity Krasnosel'skii-Pokrovskii model Magnetically controlled shape memory alloy Model reference adaptive control

Introduction

Along with the improvements of technology, magnetically controlled shape memory alloy (MSMA) actuators have emerged as a novel type of actuator in recent years. Compared to other smart materials, MSMA actuators have advantages of higher response frequency, higher control accuracy, and larger deformation rate (1–2–3–4). However, the hysteresis nonlinearity, which often exists in MSMA actuators, greatly affects the positioning accuracy and actuator performance (5). Therefore, it is essential to eliminate the hysteresis nonlinearity of MSMA actuators.

Several effective control methods have been adopted to eliminate negative effects of the nonlinearity in smart materials. Sayyaadi and Zakerzadeh (6) established an inverse PI model to compensate complex hysteresis of the MSMA actuators. The valid proportional-integral controller was adopted in this paper. Experimental results prove that tracking error of the hybrid control was reduced by 60% compared to the feed-forward control. Ping and Jouaneh (7, 8) proposed a classical Preisach model as a controller for the hysteresis nonlinearity compensation of the piezoelectric actuators. A hybrid control consisting of proportional-integral-derivative (PID) feed-back control and feed-forward control was adopted to further increase the positioning precision. Experimental results demonstrate that the maximum error of the hybrid control was reduced by 50% compared to the general PID control. Wang et al (9) proposed an inverse MPI model as a controller to increase the positioning accuracy of the piezoelectric actuators. Simulation results show that the positioning accuracy was doubled when the controller was introduced to compensate the intricate nonlinearity. Song et al (10) employed the inverse classical Preisach model as a controller to eliminate complex hysteresis of piezoelectric ceramic actuators for real-time control. Simulation results show that the error decreased by 50% to 70% when the hysteresis compensation was added to the feed-forward control. Feng et al (11) designed a robust adaptive controller based on the Duhem model to describe the severe hysteresis nonlinearity. Experimental results prove that the controller could effectively eliminate the nonlinearity. Ru et al (12) proposed an effective mathematical model to compensate severe nonlinearity of the piezoelectric actuators. Besides, the adopted control based on the hysteresis model was established to eliminate the intricate nonlinearity. The maximum tracking error was less than 3%, which proves that the adopted method can significantly increase the positioning precision. Zhou et al (13) established the control method based on an inverse PI model to compensate the complex nonlinearity. A hybrid control was proposed, which was composed of the feed-forward compensation and the neural network controller. The maximum error rate of the hybrid control was reduced from 1.72% to 1.37% compared to the feed-forward control. Experimental results demonstrate that the hybrid control could increase the positioning precision effectively. Kuhnen (14) proposed a new compensator method based on the modified Prandtl-Ishlinskii (PI) operator. An inverse controller was used to compensate the severe hysteresis of magnetostrictive actuators. Experimental results prove that the tracking error was decreased from 50% to 3% compared to conventional magnetostrictive actuators.

In this paper, a modified Kransnosel'skii-Pokrovskii (KP) model is established to compensate complex hysteresis of the MSMA actuators. Adaptive recursive algorithm is employed to obtain unknown parameters of the KP model. To increase the positioning precision of the MSMA actuators, a model reference adaptive control is proposed which can adjust the parameters of the KP model and compensation signals of the inverse model at real time. Simulation result verifies the validity of the adopted control.

Methods

Modeling of the MSMA actuators based on the KP model

MSMA is a novel smart material, which can produce great strain and has ability of shape memory under magnetic field (15, 16). The strain is often due to the transformation between the twin-variants of MSMA. Figure 1 illustrates the strain mechanism of MSMA. MSMA has deformation in direction of the axis when excited by magnetic field H in the direction of the y axis (17). As shown in Figure 1A variant 1 can transform to variant 2 when the magnetic field H passes through the MSMA perpendicular to the direction of the induced strain (18). However, stronger magnetic field can make the MSMA reach maximum strain. As variant 2 returns to variant 1, the actuator has to rely on the elastic force produced by the spring, as shown in Figure 1B. Compared to other new functional materials, MSMA has special performance and wide development space, but the hysteresis nonlinearity caused by the twin-variants greatly influences the application of the MSMA actuators. Thus, developing a valid hysteresis model is a promising way to solve the hysteresis nonlinearity of the MSMA actuators.

Fig. 1

The deformation mechanism of magnetically controlled shape memory alloy (MSMA).

The KP model is regarded as a modified Preisach model. It can comprehensively compensate the intricate hysteresis nonlinearity. Hysteresis nonlinearity can be considered as a superposition of several hysteresis operators. The mathematical expression of the KP hysteresis model is shown as:

u (t) = H [v] (t) = \int_{p} k_{p} [v, ξ_{p}] μ (p) d p

Eq. [1]

where u(t) is hysteresis output, v is input of the system, H()is transformation factor, kp[v,ξp](t) is hysteresis operator, ξp is the extreme value of the KP operator, p is the integral domain of the Preisach plane, and µ(p) is density function of the Preisach plane (19).

The traditional KP hysteresis operator is shown in Figure 2A. According to the actual output displacement of the MSMA actuators, the range interval changes from (-1.1) to (0.1), which places all the hysteresis operators in the first quadrant. The modified KP hysteresis operator is shown in Figure 2B.

Fig. 2

(A) Traditional KP hysteresis operator; (B) modified KP hysteresis operator.

The expression of the modified KP operator is:

k_{p} [v, ξ_{p}] (t) = {\begin{matrix} \max {ξ_{p} (t), r [v (t) - p_{2}]} & \dot{v} > 0 \\ \max {ξ_{p} (t), r [v (t) - p_{1}]} & \dot{v} \leq 0 \end{matrix}

Eq. [2]

ξp (t) could be expressed as:

ξ_{p} (t) = {\begin{matrix} 0 & t = t_{0} \\ k_{p} [v, ξ_{p}] (t) & t = t_{q} \geq t_{q - 1} ν sgn \dot{v} (t^{+}) = - sgn \dot{v} (t^{-}) \\ ξ_{p} (t_{q - 1}) & t_{q} \geq t \geq t_{q - 1} ν sgn \dot{v} (t^{+}) = sgn \dot{v} (t^{-}) \end{matrix}

Eq. [3]

where q is the number of sign changes of v.(t), q = 1, 2, 3…..n.

In Eq. (2), r(t) is the ridge function of the boundaries of the adopted KP operator. It can be shown as:

r (t) = {\begin{matrix} 0 & v (t) < 0 \\ v (t) / a & 0 \leq v (t) \leq a \\ 1 & v (t) > a \end{matrix}

Eq. [4]

where a = 1/L-1.

To distinguish the integral form of the KP model more easily, it should be transformed into a linearly parameterized form by dividing the Preisach plane T into a mesh grid of l*l, and l is the number of bisectrix lines in T. The Preisach model after discretization can be modified as:

u (t) = H [v] (t) = \sum_{i = 1}^{l + 1} \sum_{j = 1}^{i} k_{i j} [v (t), ξ_{p}] μ_{i j}

Eq. [5]

where kij[v(t),ξ_p], is the related hysteresis operator of each mesh grid, and µij is the related average density of each mesh grid.

Each hysteresis operator kij[v(t),ξ_p] is obtained according to the input signal v(t) in Eq. (2), and the discrete parameter l is determined on the basis of the actual control accuracy requirement.

When kij[v(t),ξ_p] is replaced by kij. Parameters Γ and θ are set to

Γ = {[k_{11}, k_{21}, k_{22}, \dots k_{i j} \dots k_{(l + 1) (l + 1)}]}^{T}

Eq. [6]

θ = {[μ_{11}, μ_{21}, μ_{22} … μ_{i j} … μ_{(l + 1)}_{(l + 1)}]}^{T}

Eq. [7]

then Eq. (5) can be written as:

u (t) = Γ^{T} θ

Eq. [8]

In this section, an accurate inverse model is established to decrease the severe hysteresis of the MSMA actuators. And the input signal v(t) of the inverse model is obtained, based on the desired output ud(t). Here, the specific method can be expressed through the following steps (20–21–22):

The input v2 is set to v2 = v_{d present}, and the corresponding output u2 is set to u2 = u_{d present}.

If ud>ud present, the desired output is larger than the current output. As a result, the process has an increasing tendency, which is shown in Figure 3A.

Fig. 3

Diagrams of establishing the inverse KP model. (A) The variation of the desired output when the process has an increasing tendency; (B) The desired output when u₂ is greater than u_d for the first time; (C) The variation of the desired output when the process has a decreasing tendency; (D) The desired output when u₂ is lesser than u_d for the first time.

Set v₁ = v₂ and u₁ = u₂.

Update the value of v₂ through v₂ = v₁ + dv, and update the calculated u₂ = H(v₂).

If u₂<u_d, then go back up to step 1.

When u₂ is greater than u_d for the first time, as expressed in Figure 3B the desired input v can be expressed by Eq. (9).

\overset{\land}{v} = v_{1} + d v \frac{u_{d} - u_{1}}{u_{2} - u_{1}}

Eq. [9]

If u_d >u_{d present}, the desired output is smaller than the current output, and the process has a decreasing tendency, which is shown in Figure 3C.

Set v₁ = v₂ and u₁ = u₂.

Update the value of v₂through v₂ = v₁ - dv, and update u₂ = H(v₂) through calculating.

If u₂ <ud, then go back up to step 1.

When u₂ is lesser than u_d for the first time, as expressed in Figure 3D the desired input v can be expressed by Eq. (10).

\hat{v} = v_{1} - d v \frac{u_{d} - u_{1}}{u_{2} - u_{1}}

Eq. [10]

Model reference adaptive control based on KP model

In the paper, the hysteresis model is established to be a reference model. An adaptive KP model is obtained when the adaptive recursive algorithm is employed to identify unknown parameters of this adopted model. The structure of this adaptive compensation based on an inverse KP model is expressed in Figure 4.

Fig. 4

Structure of the adaptive compensation based on the inverse KP model.

At any time t, r(t), is the input (desired output displacement), i(t) is the control signal obtained by the accurate inverse KP hysteresis model. The MSMA actuator Γ is driven by the control signal i(t), and y(t) is the deformation. Y_m(t) is the estimated output of KP model $\hat{Γ} (t)$ of the MSMA actuators. Error e(t) between the model output and actual output is adjusted by an adaptive recursive algorithm. The model $\hat{Γ} (t)$ is regarded as a mathematic model of the model reference adaptive control. The related inverse model ${\hat{Γ}}^{- 1}$ is established to generate a new control signal i(t + 1). The adaptive KP model is adjusted and updated depending on the changes of the error at real time, which further improves the positioning accuracy.

It is assumed that e is the error of MSMA actuators. θ is the unknown density parameter. The control purpose is to adjust the parameters of the controller which makes e(∞) = 0. The performance index function is:

J = J (θ) = \frac{1}{2} e^{2}

Eq. [11]

In order to obtain the minimum value of J, parameters are adjusted along the negative gradient direction of J, which is:

\dot{θ} = \frac{d θ}{d t} = - γ \frac{\partial J}{\partial θ} = - γ \frac{\partial J}{\partial e} \frac{\partial e}{\partial θ} = - γ e \frac{\partial e}{\partial θ}

Eq. [12]

where ∂e / ∂θ is the sensitivity coefficient, γ is the adjustment rate. This parameter adjustment control law is called adaptive control (23–24–25).

In any time t, it is assumed that θ is the actual density parameter of the KP model, $\hat{θ}$ is the related estimated value, and Γ is the KP operator matrix related to each density parameter. Then the actual output and the model output are shown as:

u = H (v) = {\hat{θ}}^{T} Γ

Eq. [13]

\hat{u} = H (v) = {\hat{θ}}^{T} Γ

Eq. [14]

Error e is:

e = u - \hat{u} = (θ^{T} - {\hat{θ}}^{T}) Γ

Eq. [15]

The standard error function is

ε = \frac{u - \hat{u}}{m^{2}} = \frac{θ^{* T} Γ}{m^{2}}

Eq. [16]

where $\overset{*}{θ} = θ - \hat{θ}$ , and m2 is the normalized function, which can be expressed as:

m^{2} = 1 + n s, n s = Γ^{T} Γ

Eq. [17]

The performance index function is set to:

J (\hat{θ}) = \frac{ε^{2} m^{2}}{2} = \frac{{(u - {\hat{θ}}^{T} Γ)}^{2}}{2 m^{2}}

Eq. [18]

Parameter $\hat{θ}$ is changed by the negative gradient direction of J, that is:

\dot{\hat{θ}} = - γ \nabla J (\hat{θ})

Eq. [19]

where γ = γ T >0 is the adaptive gain, and the function of $\nabla J (\hat{θ})$ is expressed as:

\nabla J (\hat{θ}) = - \frac{(u - {\hat{θ}}^{T} Γ) Γ}{m^{2}} = - ε Γ

Eq. [20]

Replacing Eq. (20) into Eq. (19), which can be expressed as:

\dot{\hat{θ}} = γ ε Γ

Eq. [21]

Eq. (21) is the adaptive control rule of the KP model.

Results

In this section, a series of experiments and simulations are used to prove the effectiveness of the adopted method. Figure 5B shows the density parameter. If the discretization line is greater, the modeling accuracy is better. Meanwhile, the calculated amount will be larger. In consideration of calculated amount and modeling accuracy, the number of the discretization line is chosen as l = 10 after several experiments.

Fig. 5

Results of the adaptive identification based on KP model. (A) Density parameter (the number of the discretization line l is chosen as l = 10); (B) Displacement tracking curves of the adaptive KP model output and actual output; (C) Displacement tracking error curves of the adaptive KP model output and actual output.

Simulation experiments could be shown in Figures 5A and 5C. Figure 5A is the displacement tracking curves of the adaptive KP model output and actual output. The blue solid line is the actual output displacement curve, and the red dashed line represents the model output displacement curve. It shows that the model output can track the actual output accurately. Figure 5C is the displacement tracking error curve of the adaptive KP model output and actual output. It could be shown that at the beginning of the identification, the model tracking error is a little large, and the maximum error is 0.0118 mm. The tracking error decreases as the adaptive recursive algorithm adjusts the density parameters. The maximum error is reduced to 0.008 mm, which proves that the adaptive recursive algorithm is effective in modeling and accurately controlling the MSMA actuators at real time.

Figure 6A shows the control signal curve, the system input curve and the output displacement tracking curve of the model reference adaptive control system. The purple solid line represents the control signals of the inverse model, and the blue dashed line represents the actual input displacement curve. The red dotted line represents the output displacement curve. Figure 6B demonstrates the input-output curve of this model reference adaptive control system. It can be shown that the obvious relationship between the input and output is linear approximately, which proves that the proposed control could decrease the effect of the intricate nonlinearity. Figure 6C is the displacement tracking error curve of the employed control system.

Fig. 6

Results of the model reference adaptive control. (A) The control signals and input-output displacement tracking curves of the adopted system. (B) Input-output relationship curves of the adopted system. (C) Displacement tracking error curves of the adopted system.

It can be seen from Figures 6A-C that the proposed model reference adaptive control can achieve a high positioning accuracy, and the displacement tracking error is decreased to 0.0072 mm. Simulation results demonstrate that the model reference adaptive control method can significantly increase the positioning precision of the MSMA actuators.

Discussion

In this paper, a modified KP model is employed to compensate the hysteresis of the MSMA actuators and the adaptive recursive algorithm is used to identify the unknown density parameters of the KP model. Simulation experiment results demonstrate that the identification error is 0.0118 mm, which occurs at the beginning of the identification. The identification error decreases as the adaptive recursive algorithm adjusts the density parameters. The maximum error is reduced to 0.008 mm, which proves that the algorithm can greatly improve the identification accuracy.

For the sake of eliminating the intricate hysteresis of the MSMA actuators, an inverse KP model is built as a compensator for the MSMA actuators. To further improve the positioning accuracy, an effective model reference adaptive control is proposed based on an inverse KP model. Simulation experiments prove that the controller accomplishes a better control of the system actual output, with a maximum error of 0.0072 mm. Compared to the control accuracy in classical methods of MSMA actuators (26, 27), the results prove that the model reference adaptive control method is efficient to eliminate hysteresis nonlinearity and achieves a higher positioning accuracy of the MSMA actuators.

Footnotes

Financial support: This study is supported by the National Natural Science Foundation of China (No: 51675228), Program of Science and Technology Development Plan of Jilin province of China (No: 20140101062JC), and Program of the Twelfth Five-Year Science and Technology Research Plan of Education Department of Jilin province of China (No: 2014B023).

Conflict of interest: The authors declare that they have no conflicts of interest.

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