Neural network‐based adaptive control of piezoelectric actuators with unknown hysteresis

This paper proposes a neural network (NN)‐based adaptive control of piezoelectric actuators with unknown hysteresis. Based on the classical Duhem model described by a differential equation, the explicit solution to the equation is explored and a new hysteresis model is constructed as a linear model in series with a piecewise continuous nonlinear function. An NN‐based dynamic pre‐inversion compensator is designed to cancel out the effect of the hysteresis. With the incorporation of the pre‐inversion compensator, an adaptive control scheme is proposed to have the position of the piezoelectric actuator track the desired trajectory. This paper has three distinct features. First, it applies the NN to online approximate complicated piecewise continuous unknown nonlinear functions in the explicit solution to Duhem model. Second, an observer is designed to estimate the output of hysteresis of piezoelectric actuator based on the system input and output. Third, the stability of the controlled piezoelectric actuator with the observer is guaranteed. Simulation results for a practical system validate the effectiveness of the proposed method in this paper. Copyright © 2008 John Wiley & Sons, Ltd.


Introduction
Hysteresis phenomenon occurs in all smart material-based sensors and actuators, such as shape memory alloys, piezoceramics and magnetostrictive actuators (Bank & Smith, 2000;Tan & Baras, 2004). In order to study this phenomenon, different models were proposed (Brokate & Sprekels, 1996;Visintin, 1994). Normally, hysteresis models are classified into two categories, physics-based model such as Jiles-Atherton model (Jiles & Atherton, 1986) and phenomenology-based model such as Preisach operator (Brokate & Sprekels, 1996;Visintin, 1994) and Duhem model (Visintin, 1994). From control systems point of view, hysteresis is generally non-differentiable, nonlinear, and unknown. As a result, systems with hysteresis usually exhibit undesirable inaccuracies or oscillation and even instability. Mitigating the effect of hysteresis becomes necessary and important, thus it has received increasing attention in recent years (Tao & Kokotovic, 1995, Su, et al, 2000, Su, et al, 2005. Many of these studies are related to modeling of hysteresis and their control issues. With the development of artificial intelligent (AI), AI is being applied to dealing with nonlinearities in systems (Ge & Wang, 2002). Only a few studies have been carried out by using NN to tackle hysteresis modeling and compensation (Makaveev, et al, 2002;Beuschel et al, 1998;Zhao & Tan, 2006). In the paper (Makaveev, et al, 2002), a NN model is used to describe the hysteresis behavior in different frequencies with the knowledge of some properties of magnetic materials, such as loss separation property to allow the separate treatment of quasi-static and dynamic hysteretic effects. Beuschel et al used (Beuschel et al, 1998) a modified Luenberger observer and NN are used to identify a general model of hysteresis. These researches demonstrate that NN can work as an unknown function approximator to describe the characteristics of hysteresis. Recently, two papers (Zhao & Tan, 2006;Lin et al 2006) applied the approximation property of NN to coping with the identification of Preisach-type hysteresis in piezoelectric actuator, and the hysteresis estimation problem for piezo-positioning mechanism based on hysteresis friction force function, respectively. It should be noted that the aforementioned results share a common assumption that the output of hysteresis is measurable. In practical systems, smart actuators are integrated into the systems, which makes the measurement of output of hysteresis hard. Hence it is a challenge to design an observer for the unavailable output of hysteresis. Due to the unavailability of the output of hysteresis, the major obstacle of pre-inversion compensator for hysteresis is the lack of effective observer design methods for piezoelectric actuators. Especially, the traditional "Luenbergertype" nonlinear observer design (Krener & Isidori, 1983) or the "high-gain" observer (Krener & Kang, 2003) cannot be applied directly, since the hysteresis is highly nonlinear. The sliding-mode observer was developed to estimate the internal friction states of LuGre model for the servo actuators with friction (Xie, 2007). This observer needs a low-pass filter to remove the high-frequency components in the estimated state variable, which is not applicable in this paper. Yang and Lin (Yang & Lin, 2004) proposed homogeneous observers design for a class of n-dimensional inherently nonlinear systems whose Jacobian linearization is neither controllable nor observable. Inspired by NN's universal approximation property, and the aforementioned facts in observer design, we propose an observer-based adaptive control of piezoelectric actuators with unknown hysteresis in this paper. The main contribution of this paper is the following: First, it applies the NN to on-line approximate complicated piecewise continuous unknown nonlinear functions in the explicit solution to Duhem model. Second, an observer is designed to estimate the output of hysteresis of piezoelectric actuator based on the system input and output. Third, the stability of the controlled piezoelectric actuator with the observer is guaranteed by using Lyapunov extension . The organization of the paper is as follows. In Section II, a Duhem model of hysteresis and the problem statement are given. The main results on NN-based compensator for hysteresis are presented in Section III. Section IV provides an example to show the feasibility of the proposed method. Conclusions are given in Section V.

Duhem model of hysteresis
Many different mathematic models are built to describe the hysteresis behavior, such as Preisach model, Prandtl-Ishlinkii model and Duhem model (Coleman & Hodgdon, 1987;Macki et al, 1993). Considering its capability of providing a finite-dimensional differential model of hysteresis, we adopt classical Duhem model to develop the adaptive controller for the piezoelectric actuator.
The Duhem model is a rate independent operator, with input signal v , v & and output signal τ . The Duhem model describes hysteresis by the following mathematical model (Coleman & Hodgdon, 1987;Macki et al, 1993).
It can also be represented as (Coleman & Hodgdon, 1987;Macki et al, 1993): where α is the same positive number in (1), ) (v g is the slope of the model, and ) (v f is the average value of the difference between upward side and downward side. Property 1 (Coleman & Hodgdon, 1987;Macki et al, 1993): Property 2 (Coleman & Hodgdon, 1987;Macki et al, 1993): It has been shown that Duhem model can describe a large class of hysteresis in various smart materials, such as ferromagnetically soft material, or piezoelectric actuator by appropriately choosing ) (v f and ) (v g (Coleman & Hodgdon, 1987;Macki et al, 1993). One widely used pair of functions of The above equation can be solved for τ In order to describe the piezoelectric actuator, we choose the same functions (Banning, et al, 2001), which is a special case of the foregoing choice of Substituting (5) and (6) Equation (8) can be also expressed as: Following the definition of the indicator functions, we get We can also write the derivative of τ as

Augmented Multilayer Perceptron (MLP) Neural Network
The MLP NN has been explored to approximate any function with arbitrary degree of accuracy (Hornik et al, 1989). However, it needs a large number of NN nodes and training iterations to approximate non-smooth functions (i.e. piecewise continuous), such as friction, hysteresis, backlash and other hard nonlinearities. For these piecewise continuous functions, the MLP needs to be augmented to work as a function approximator. Results for approximation of piecewise continuous functions or functions with jumps are given in the www.intechopen.com paper (Selmic & Lewis, 2000). We use the augmented NN to approximate the piecewise continuous function in hysteresis model. Let S be a compact set of n R and define ) (S C n be the space such that the x ∈ , which has a jump at c x = and is continuous from the right as  (16), the piecewise continuous function F2 will be approximated by the augmented NN. In this paper, it is assumed that there exists weight matrix W such that x ∈ , and the Frobenius norm of each matrix is bounded by a known constant

NN-based compensator and controller design
Given the augmented MLP NN and hysteresis model, a NN-based pre-inversion compensator for the hysteresis is designed to cancel out the effect of hysteresis. In this section, a novel approach is developed to compensate the hysteretic nonlinearity and to guarantee the stability of integrated piezoelectric actuator control system.

Problem statement
Consider a piezoelectric actuator subject to a hysteresis nonlinearities described by Duhem model. It can be identified as a second-order linear model preceded by hysteretic nonlinearity as follows: where v(t) is the input to piezoelectric actuator, ) (t y denotes the position of piezoelectric actuator, m , b , k denote the mass, damping and stiffness coefficients, respectively, ) (• H represents the Duhem model (1).
In order to eliminate the effect of hysteresis on the piezoelectric actuator system, a NNbased hysteresis compensator is designed to make the output from hysteresis model pr τ approach the designed control signal pd τ . After the hysteresis is compensated by the NN, an www.intechopen.com adaptive control for piezoelectric actuator is to be designed to ensure the stability of the overall system and the boundedness of output tracking error of the piezoelectric actuator with unknown hysteresis. We consider the tracking problem, in which A filtered error is defined as Differentiating ) (t r p and combining it with the system dynamics Eq. (18), one may obtain: The tracking error dynamics can be written as

NN-based Compensator for Hysteresis
In presence of the unknown hysteresis nonlinearity, the desired control signal pd τ for the piezoelectric actuator is different from the real control signal pr τ . Define the error as Here we utilize a second first-layer-fixed MLP to approximate the nonlinear function 2 F .
It is assumed that the Frobenius norm of weight matrix W1 is bounded by a known constant The estimated nonlinear function 2 F is constructed by using the neural network with the weight matrix 1 W : Hence the restructure error between the nonlinear functions 2 F and 2 F is derived as: Remark 1 When the input changes its sign derivative (Beuschel et al, 1998), the augmented MLP can approximate the piecewise continuous functions. In the process, the "jump functions" leads to vertical segments in the feed-forward pre-inversion compensation, where the "functional restructure error" can be confronted by the adaptive controller in Section III.C (Selmic & Lewis, 2000). A hysteresis pre-inversion compensator is designed: Define error matrix as: Inserting (26), (28) into (25), we obtain We choose weight matrix update rule as where Γ is a positive adaptation gain diagonal matrix, and 1 p k is a positive constant.
Design the update rule of parameter μ in pre-inversion compensator v & as where η is positive constant, Proj(.) is a projection operator, which is defined as follows: The adaptive NN-based pre-inversion compensator v & is developed to drive the adaptive control signal pd τ to approach the output of hysteresis model pr τ so that the hysteretic effect is counteracted.

Controller Design Using Estimated Hysteresis Output
It is noticed that the output of hysteresis is not normally measurable for the plant subject to unknown hysteresis. However, considering the whole system as a dynamic model preceded by Duhem model, we could design an observer to estimate the output of hysteresis based on the input and output of the plant. The velocity of the actuator ) (t y & is assumed measurable. Define the error between the outputs of actuator and observer as Then the observer is designed as: The error dynamics of the observer is obtained based on the actuator model and hysteresis model.
where the parameter error is defined as a a a www.intechopen.com By using the observed hysteresis output pr τˆ, we may define the signal error between the adaptive control signal pd τ and the estimated hysteresis output as: The derivative of the signal error is: A hysteresis pre-inversion compensator is designed: By substituting the neural network output and pre-inversion compensator output into the derivative of the signal error, one obtains: The weight matrix update rule is chosen as: And the update rule of parameter μ in pre-inversion compensator v & is designed with the same projection operator as (32): The update rule of parameter a K in the observer (35) is designed with the same projection operator as (32): Hence we design the adaptive controller and update rule of control parameter as: where the projection operator is With the adaptive robust controller, pre-inversion hysteresis compensator and hysteresis observer, the overall control system of integrated piezoelectric actuator is shown in Fig. 3. The stability and convergence of the above integrated control system are summarized in Theorem 1.
Theorem 1 For a piezoelectric actuator system (18) with unknown hysteresis (1) and a desired trajectory ) (t y d , the adaptive robust controller (44), NN based compensator (39) and hysteresis observer (34) and (35) The derivative of Lyapunov function is obtained: Introducing control strategies (39), (44) and the update rules (41) From the Property 1 of Chapter 2 in the recent book (Ikhouane & Rodellar, 2007) We select the control parameters pd k , b k and observer parameters 1 L , 2 L and pr K satisfying the following inequalities: we can easily conclude that the closed-loop system is semi-globally bounded (Su & Stepanenko, 1998 Remark 2 It is worth noting that our method is different from (Zhao & Tan, 2006;Lin et al 2006) in terms of applying neural network to approximate hysteresis. The paper (Zhao & Tan, 2006) transformed multi-valued mapping of hysteresis into one-to-one mapping, whereas we sought the explicit solution to the Duhem model so that augmented MLP neural networks can be used to approximate the complicated piecewise continuous unknown nonlinear functions. Viewed from a wavelet radial structure perspective, the WNN in the paper (Lin et al 2006) can be considered as radial basis function network. In our scheme, the unknown part of the solution was approximated by an augmented MLP neural network.

Simulation studies
In this section, the effectiveness of the NN-based adaptive controller is demonstrated on a piezoelectric actuator described by (18) The control objective is to make the output signal y follow the given desired trajectory d y . From The system responses are shown in Fig.2, from which it is observed that the tracking performance is much better than that of adaptive controlled piezoelectric actuator without hysteretic compensator. The input and output maps of NN-based pre-inversion hysteresis compensator and hysteresis are given in Fig. 3, respectively. The desired control signal and real control signal map (Fig. 3c) shows that the curve is approximate to a line which means the relationship between two signals is approximately linear with some deviations. In order to show the effectiveness of the designed observer, we compare the observed hysteresis output pr τˆ and the real hysteresis output pr τ in Fig. 4. The simulation results show that the observed hysteresis output signal can track the real hysteresis output. Furthermore, the output of adaptive hysteresis pre-inversion compensator ) (t v is shown in

Conclusion
In this paper, an observer-based controller for piezoelectric actuator with unknown hysteresis is proposed. An augmented feed-forward MLP is used to approximate a complicated piecewise continuous unknown nonlinear function in the explicit solution to the differential equation of Duhem model. The adaptive compensation algorithm and the weight matrix update rules for NN are derived to cancel out the effect of hysteresis. An observer is designed to estimate the value of hysteresis output based on the input and output of the plant. With the designed pre-inversion compensator and observer, the stability of the integrated adaptive system and the boundedness of tracking error are proved. Future work includes the compensator design for the rate-dependent hysteresis.