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Piezoelectric-driven stick slip actuators have been drawn more and more attention in the nano- positioning application due to the high accuracy and theoretical unlimited displacement. However, the hysteresis of piezoelectric actuator (PEA) and the nonlinear friction force between the end- effector and the stage make control of piezoelectric-driven stick slip actuator challenge. This paper presents the development of an autoregressive exogenous (ARX)-based proportional-integral-derive (PID)-sliding mode control (SMC) for the velocity tracking control of the piezoelectric-driven stick slip actuator. Stability is guaranteed by rigorously choosing the appropriate PID parameters and the zero steady state error is achieved. To verify the effectiveness of the proposed method, experiments were carried out on a commercially-available piezoelectric-driven stick slip actuator. The tracking errors were compared with the traditional PID controller, illustrating that in spite of existing of modeling error, the ARX-based PID-SMC is able to better improve the velocity tracking performance of piezoelectric-driven stick slip actuator, compared with the traditional PID controller.

Piezoelectric-driven stick-slip actuators play important roles in nano-positioning applications due to their simple configuration, high accuracy and theoretically unlimited displacement [

The hysteresis of PEA and the friction dynamics of the end-effector make the control of stick-slip actuator challenge. An analog electronic circuit was developed in [

All the research mentioned above related to the displacement tracking control of the piezoelectric-driven stick-slip actuators under step or ramp reference signal. The velocity tracking control has not been reported yet. This work focuses on the development of control method for the velocity tracking control of the piezoelectric- driven stick-slip actuators.

PID/PI/PD controller has shown great potential in the control application for nano-positioners. The challenge of PID/PI/PD control in the velocity tracking control of stick-slip piezoelectric-driven stage is maintaining the system stability in the presence of uncertainty and disturbance. It also has an issue with low gain margin in high frequency applications [

As a state tracking control scheme, the PID-SMC developed in [

It is noted that if the dynamics of the plant can be represented by an auto-regressive (ARX) model without

zeros, for example,

time instant k respectively, such a model will be readily transferred to a state space model with its state being the system output and the state tracking based PID-SMC is able to be directly applied for the dynamics compensation. Inspired by this, an ARX-based PID-SMC is developed in this paper and applied in the velocity control of a one-DOF stick-slip PEA. The effectiveness of the proposed method is experimentally verified and compared with the traditional PID controller introduced in [

The discrete nth order plant might be generally described by the following transfer function

where

Equation (2) is identical to a discrete transfer function without zeros. The neglect of zeros definitely leads to model errors, which is considered to be disturbance in Equation (2).

The ARX model can be rewritten in a state space form

where

discrete state space model. The state vector is represented in terms of the outputs in the past history, which suggest the state tracking is essentially the output tracking. Therefore, the state tracking SMC design method can be applied.

Denote the desired output vector to be

tive of SMC is to force the error state

Similar to the general SMC design approach introduced in [

It is noted that Equation (5) is non-casual if the future desired output is unknown. In such a case,

Substituting Equation (5) into Equation (4) yields

For the system described by Equation (3), one has

For the sliding function that takes the following form of

The control action can be considered consisting two parts, i.e.,

where

Substituting Equation (10) into Equations (8) and (9) yields

For the convenience of following discussion, Equations (9) and (11) are rewritten as

where

where

where P, I and D are parameters of the discrete PID-based SMC; T is the sampling period.

Theorem 1: If the closed-loop system (13) is stable, the zero steady state error can be achieved [

Theorem 2: There exist some P, I and D such that the closed-loop control system (13) is stable [

To verify the effectiveness of the proposed method, experiments were carried out to control the motion velocity of a stick-slip piezoelectric-driven actuator (

M_{e} is the mass of the end-effector, F_{f} is the friction force between the PEA stage and the end-effector. The hysteresis model H is cascaded with the vibration dynamics model. The output―PEA displacement is feed- forward to the friction model, which generates the friction force and the motion parameters of the end-effector.

To apply the ARX-based PID-SMC, the PEA and the end-effector are considered to be an integrated system with its input being the driven voltage u and its output being the motion velocity of the end-effector, as shown in

To drive the end-effector, a saw-tooth generator is required, as seen in

tooth signal is demanded to move the end-effector to the backward direction, as shown in

where T is the tooth width, mod represents the modulus after division.

The displacement of the end-effector is measured by the inductive sensor. The average velocity v is estimated by

where

The dynamics of the integral system can be regarded approximately as a second order system according to our previous research [

The ARX-based PID-SMC developed in this study was implemented in experiments to control the velocity of the piezoelectric-driven slick-slip actuator. The sliding surface was defined according to Equation (9), where

It can be seen that the tracking errors approach to zero through the use of I component in the PID regulator. The velocity step response of the piezoelectric-driven stick-slip actuator controlled by the proposed method is faster than that controlled by the traditional PID controller. For example, when a 10 μm/s step reference input was provided, the rising time of the velocity step response controlled by the ARX-based PID-SMC is 0.13 s, 0.2 ms less than that controlled by the PID controller. Faster velocity response is also observed for the 20 μm/s step reference input. However, since the dynamics performance of the piezoelectric-driven stick-slip actuator varies with the amplitude of the input voltage, the model error increased when the 20 μm/s step reference input was provided to the actuator. As a result, the overshoot increased to 10% in this case.

Reference inputs | Controller | Rising time (s) | Overshoot (%) |
---|---|---|---|

10 µm/s step input | SMC | 0.13 | 0 |

PID | 0.33 | 0 | |

20 µm/s step input | SMC | 0.11 | 10 |

PID | 0.21 | 0 |

To test the control performance of the proposed method, the same velocity step tracking experiments were also carried out with an increased mass of the end-effector. The same parameters is applied and compared with the traditional PID controller.

It is noted that the rising time of the velocity step response controlled by the ARX-based PID-SMC is 0.09 s. This is different from the result with an decreased mass of the end-effector, since the dynamics of the piezoelectric-driven stick-slip actuator also changes with the weight of the end-effector.

To further show the effectiveness of the proposed control method, sinusoidal tracking experiments with different

Reference inputs | Controller | Rising time (s) | Overshoot (%) |
---|---|---|---|

10 µm/s step input | SMC | 0.09 | 0 |

PID | 0.21 | 0 |

frequencies were carried out on the piezoelectric-driven stick-slip actuator. The same parameters were applied and the control performance was compared with the same PID controller.

It is noted that the velocity tracking performance deteriorates when the motion direction of the end-effector changes. For example, for 10 μm/s 1 Hz sinusoidal reference velocity, the maximum tracking error in time interval 0.5 - 0.75 s and 1.5 - 1.75 s is 50% of the amplitude if controlled by the ARX-based PID-SMC. The maximum tracking error even reaches 100% of the reference amplitude of the traditional PID controller is applied. This might be due to the nonlinear friction force between the end-effector and moving stage when the velocity changes from positive to negative. Obviously, the ARX-based PID-SMC partially compensates the nonlinearity. Consideration of the nonlinear friction model and its integration in the controller design are required for further improvement on the velocity tracking performance of the piezoelectric-driven stick-slip actuator.

This paper presents the development of an ARX-based PID-SMC for the velocity control of the piezoelectric- driven stick-slip actuator. Specifically, by applying ARX model, the output-tracking problem is defined as one of state tracking, while the “bang-bang” switching control in SMC is replaced with PID-based one. With the developed control scheme, chattering or the state oscillation at a high frequency can be eliminated and also the zero steady state error can be achieved. To verify the effectiveness of the developed control scheme, experiments were carried out on a piezoelectric-driven stick-slip actuator, whose dynamics was identified by experiments. The results of velocity tracking performance with the proposed control scheme were compared to that with the traditional PID controller. It was shown that both the step response and the sinusoidal tracking performance can be further improved by applying the proposed method, although the ARX model error exists. Howev-

Reference inputs | Frequency (Hz) | Tracking error (µm/s) | |
---|---|---|---|

ARX-based PID-SMC | PID | ||

10 µm/s sinusoidal input | 0.25 | 1.12 | 1.57 |

0.5 | 2.13 | 2.89 | |

1 | 2.82 | 4.72 |

er, the control performance of the ARX-based PID-SMC depends on the accuracy of the identified dynamic model. This might be solved by employing a dynamic model with adaptive parameters and an adaptive ARX-based PID-SMC, which will be the focus of the future work. Furthermore, nonlinear friction model should be considered in the SMC design to further improve the velocity tracking performance of the piezoelectric-driven stick- slip actuator.

The support to the present study from the China Scholarship Council (CSC) and the Natural Sciences and Engineering Research Council (NSERC) of Canada is acknowledged.