Abstract
We develop convergence criteria of an iterative learning control on the whole desired trajectory to obtain the hysteresis-compensating feedforward input in hysteretic systems. In the analysis, the Prandtl-Ishlinskii model is utilized to capture the nonlinear behavior in piezoelectric actuators. Finally, we apply the control algorithm to an experimental piezoelectric actuator and conclude that the tracking error is reduced to 0.15% of the total displacement, which is approximately the noise level of the sensor measurement.
1. Introduction
Piezoelectric actuators (PEAs) have been widely used in nanopositioning systems due to their fast response and nanometer scale resolution [1–3]. However, the hysteresis existing in PEAs can greatly limit system performance [4, 5]. Control of hysteretic system is an important area of control system research and a challenging problem [6–9]. Research on feedback and model-based feedforward control has been studied to achieve relatively high-precision positioning [10–13]. Iterative methods can be used to improve the positioning performance if the positioning application is repetitive. Therefore, many researchers study the iterative and adaptive control methods to minimize the adverse effect of hysteresis [14–18].
The main challenge in iterative approaches for hysteretic systems is to assure convergence of the iterative algorithm. Leang and Devasia divide a general desired trajectory into some monotonicity partition [15, 16]. Afterwards, they prove the convergence of iterative learning control (ILC) algorithm on each single branch. In this paper, we study the design of (ILC) algorithm to compensate for hysteresis-caused error in PEAs. The main contribution of our work is proving convergence of ILC algorithm on whole tracking trajectory.
The remainder of this paper is organized as follows. First, we state the problem in the next section. Afterwards, we briefly review the Prandtl-Ishlinskii model in the context of this work and prove convergence of the ILC algorithm we designed. Finally, we implement the ILC algorithm on experimental stage and show our experimental results and conclusions.
2. Problem Statement
Consider a hysteretic system of the following form: where is the input, is the output, and denotes the hysteresis function . For a given desired trajectory defined on the finite time interval , the objective is to find an input by way of the following iterative learning control (ILC) algorithm: where , is a constant (to be determined), and and are the input and output at the th iteration, respectively. Figure 1 depicts the block diagram of the ILC algorithm. The goal of the ILC algorithm is to generate a feedforward control input in the norm sense, where
In this paper, is used to denote the space of continuous functions on , and denotes the space of continuous monotone functions on .
Definition 1 (incremental strictly increasing operators). An operator is called incremental and strictly increasing, if, for , considering the ordering for all , there exist constants such that for any .
Lemma 2. Let the operator be incremental and strictly increasing. If and in (2) for all , then for any .
Proof. We use the mathematical induction to prove this assertion. First, we prove for any . Consider where . We can obtain . Then, we assume , for all , and we also get ; therefore, the assertion holds.
Lemma 3. Let the operator be incremental and strictly increasing with constants as defined by Definition 1. Then, for all .
Proof. See [19].
Theorem 4. Consider a system of the form . Let the operator be incremental and strictly increasing. If the constant and , for all , then the iterative control algorithm converges; that is, where and uniformly in as .
Proof. To prove the convergence of the ILC algorithm, we show contraction of the input (2). Subtracting from both sides of (2) and substituting in place of , we get for all . Taking the function norm of (9), we get By Lemma 2, we obtain for all . Since the operator is incrementally strictly increasing with constants and , then by Lemma 3 we obtain where . Because , (11) is a contraction. By induction, we obtain that therefore, the sequence as ; that is, converges to uniformly in . Proof is completed.
3. The Prandtl-Ishlinskii Hysteresis Model
The Prandtl-Ishlinskii (PI) model can be used to capture the rate-independent hysteresis nonlinearity in piezoelectric actuators. In this section, the PI model is presented.
The PI model utilizes the play or stop operators and a density function to characterize the hysteresis behavior. The hysteresis play operator is illustrated in Figure 2, while its detailed formulations have been presented in [20]. For a given input , the play operator with threshold is defined by with where is initial value of the operator , and is a partition of , such that the input function is monotone on each subinterval .
3.1. Property of Play Operator
Lemma 5. For any , the operator can be extended uniquely to a Lipschitz continuous operator . And it holds for all , for all initial values , and for all . Consider
Proof. See [20] Section 2.3.
Lemma 6. If , for all , and initial value , a Lipschitz continuous operator is incremental and strictly increasing; that is, there exist constants such that .
Proof. Consider Since , for all , and initial value , then . By induction, . There must exist a constant such that Because and , (15) can be simplified as Let and . We get Let . From (18) and (20), we obtain therefore, the assertion holds.
3.2. Property of the Prandtl-Ishlinskii Model
The PI model assumes that the output of a piezoelectric actuator is a weighted superposition of basic . The output is written as where is a given density function, satisfying with .
Lemma 7. If , for all , and all initial values in all operators , then PI operator is incremental and strictly increasing; that is, there exist constants such that for any .
Proof. Consider for any , where . Similarly, we can get for any , where . Proof is completed.
Theorem 8. Consider a hysteretic system of the form . Let the PI hysteresis operator satisfy the condition of Lemmas 6 and 7. If the constant and in (2), for all , then the iterative control algorithm converges; that is, where and uniformly in as .
Proof. The proof is identical to the proof of Theorem 4.
4. Experimental Results
4.1. Experimental Setup
The ILC algorithm is applied on an experimental piezoelectric actuator PST150/7/40VS12, which is a preloaded PZT from Piezomechanik in Germany. The natural frequency of the actuator is 20 kHz. The actuator provides a maximum displacement of 40 μm and includes an integrated high-resolution strain gauge position sensor (SGS). The excitation module comprises a voltage amplifier (HVPZT XE-501.B) with a fixed gain of 15, which provides the excitation voltage for the actuator. The AD7011-EVA controller board equipped with 12-bit ADC and12-bit DAC is utilized to generate and acquire input voltage and output displacement signals. The experimental data are acquired at a sampling frequency of 1 kHz. Figure 3 illustrates the structure of the experimental system.
4.2. Experimental Result
We apply the ILC algorithm to track a sinusoidal trajectory . The constant in (2) is chosen to be and the initial input . The experimental results are shown in Figure 4.
(a) Displacement versus time
(b) Input versus time
| (c) versus |
| (d) versus time |
Figure 4(a) shows the results of the ILC algorithm to track the desired trajectory, and the feedforward input of the ILC algorithm is depicted in Figure 4(b). The maximum error at the th iteration is shown in Figure 4(c), and it demonstrated that convergence of ILC algorithm is achieved. Figure 4(d) shows the tracking error at the 30th iteration. The maximum error is approximately 0.015 μm, which is of the total displacement range.
5. Conclusions
In this paper, we designed an ILC algorithm to compensate for hysteresis-caused tracking error in piezoelectric actuators and proved convergence of this algorithm on the whole tracking trajectory. Experiments were carried out to verify the effectiveness of the ILC algorithm. The experimental results show that the tracking error can be reduced to the noise level of the sensor measurements.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the National Natural Science Foundation of China (no. 61174044).