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In this paper, an adaptive law with an integral action is designed and implemented on a DC motor by employing a rotary encoder and tachometer sensors. The stability is proved by using the Lyapunov function. The tracking errors asymptotically converge to zero according to the Barbalat lemma. The tracking performance is specified by a reference model, the convergence rate of Lyapunov function is specified by the matrix

Various electromechanical motors have been used for industrial applications, e.g., electric motors, piezomotors and hydraulic actuators. Generally, it is necessary to enhance the motor performance with the feedback control based on state measurements. Khorrami

As digital angle-measuring sensors, rotary encoders consist of optics, mechanics and electronics. Compared to analog angle-measuring sensors, digital rotary encoders have simple structures while preserving high accuracy. In order to achieve accurate motion control, the velocity can be measured by using tachometer sensors which usually consist of tachogenerators and circuits. The tachogenerator can also give the voltage output which is proportional to the speed of the rotational motor.

To achieve precise motion, it is necessary to measure and input both the angle and the speed of motor shafts by employing a rotary encoder and tachometer sensors. Based on the angle and speed sensors, various controllers have been designed for motors [

This paper uses Model Reference Adaptive Control (MRAC), which is a convenient approach to satisfy the requirements of designers [

Though the adaptive control has shown its effectiveness in achieving robust performance without the knowledge of parameter values, a comprehensive design approach is still necessary for engineering applications. In order to obtain high-performance adaptive control in the presence of disturbances, this paper presents a comprehensive design approach in which both the bandwidth and damping ratio can be included in the proposed controller.

In this paper, two kinds of MRAC are designed and then used on a motor by employing a rotary encoder and tachometer sensors. The tracking error can converge to zero with the integral action in the presence of input disturbances. The experimental results are presented to investigate the effectiveness of the proposed control approach.

After calibration, the generated voltage is a factor of the motor speed. Then, the motor speed can be fed to the adaptive controller. Next,

This section presents a design method of the adaptive control without the integral action. The stability proof is also presented. To achieve good tracking performance, a MRAC is designed to drive the tracking error to zero. Considering simplification, the transfer function in

To specify the desired performance, this paper employs a stable reference model as shown in _{n}

The following non-adaptive control law is used:

Then, the closed loop system can be given by:

The feedback control system achieves the performance as the matching condition below:

The exact control gains
_{r}^{*} is a constant or the slow time-variant parameter (i.e., ^{*}=

The adaptive control law is nonlinear, as shown in

To give the tracking performance, the parameter errors _{x}_{r}

Then, the closed loop system can be rewritten to:

Next, let _{p}_{m}

By comparing the closed loop system in

In the reference model in _{m}

To prove the stability, a Lyapunov function candidate is used as follows:

The time derivative of the Lyapunov function candidate can be given:

Then, ^{T}x

Thus, the system is stable according to the Lyapunov theorem [_{n},θ_{x}_{r}

According to the Barbalat lemma [

Here

According to the Barbalat lemma, the tracking error _{m}_{n}‖

To improve the adaptation rate, the proportional action can be added to the adaptive law [_{1} = 1 and _{2} = 0.5, and they are used in experiments.

Experimental studies of the adaptive control are now presented. Firstly,

After several trials, the reference model is given:

The selection of

The matrix

The tracking performance of the adaptive control is shown in

To guarantee the stability of the adaptive control, it is necessary to demonstrate the boundness of _{x1}_{x2}_{I}

_{2} is larger than _{1}.

The adaptation rates can be respectively estimated:
_{1} is faster than _{2}.

During experiments, the adaptive control using _{1} and _{2} have similar tracking errors, but the control signals are different. _{2}, Γ_{2}_{1}, Γ_{1}

To further investigate the adaptive control in the presence of disturbances, this section presents the adaptive control with integral action. The integral control action can be added to the adaptive law through _{I}

The augmented plant can be rewritten to:

Similar to Section 2.2, the control law can be given by:

Then the closed loop system can be given as:

According to the matching condition in

The reference model can be given:

Actually, the exact value of
_{x}_{I}_{x}_{I}_{x}

Substituting

The error equation can be given

Choosing the Lyapunov function as

Then, the time derivative of

Denote that the adaptive law is:

Then,

It can be concluded that the closed loop is Lyapunov stable, and

The adaptive control law is shown as follows:

This section presents how to specify the requirements through the reference model _{m}_{m}_{m}_{1}:
_{m}_{1}‖ (i.e., _{1} and _{1} are dominant poles which mainly contribute to the dynamic response of the reference model.)

To satisfy the matching conditions, the reference model is:

Both

Compared with the transfer function in

The dynamics of the reference model should match the dynamics (_{n}_{m}_{21} = −57.6, _{22} = −14.4 and _{23} = −81. The reference model can be rewritten to:

The step responses of _{m1} and _{m}_{m1} contributes most of the response for the third order reference model _{m}_{m}_{m})

In Section 2, it has been found that a large

After trials,

Then,

In addition,

The experimental results of the position and velocity tracking are shown in

As shown in

In order to investigate the performance of the adaptive control, the input disturbance is considered in this section. The square wave disturbance with magnitude 1 is adopted. The reference is also square wave. During the experiment, the disturbance is added by using a LabView block as an input disturbance. The disturbance enters the close loop systems as the reference signal enters. The tracking errors of position and velocity are shown in

Finally, a comparison between experiment and simulation results is presented in this paper. According to the motor manual, the time constant is 0.25. The DC gain is 5.2.

The simulation and experiment have similar responses. The tracking errors of the position and velocity from experiment study are larger than that from simulation study. In the three cases, all the tracking errors asymptotically converge (

In order to accurately control motors, this paper employs both a rotary encoder and tachometer sensors to measure the angle-position and speed, respectively. Based on the measurements, two adaptive controllers are developed for the motor system. The stability and convergence are validated by the Lyapnov theorem and Barbalat lemma. Then, the control system is implemented by using Labview. Experimental results indicate that the tracking errors of the motor position and velocity asymptotically converge to their error ranges. In the presence of disturbances, the adaptive controller with integral action presents better performance than the case without integral action.

State estimation approaches are encouraged to investigate for possible output feedback control. In this paper, the proposed adaptive controllers needs full state information, but actually some states (e.g., motor velocity) are not provided. Thus, the state estimation must be designed in the possible output feedback control.

This work was supported by the National High Technology Research and Development Program of China (No. 2012AA120601), National Natural Science Foundation of China (No. 11202044, No. 11072044) and the Fundamental Research Funds for the Central Universities.

MRAC control sketch of the motor.

Sketch of the motor experiment.

Working principle of the rotary encoder.

Adaptive control block.

The reference position, the motor position and the tracking error (

The reference velocity, the motor velocity and the tracking error (

Control signals under (_{1}, Γ_{1}_{2}, Γ_{2}

Step responses of _{m}_{m}_{1} (star line).

The reference position, the motor position and the tracking error (

The reference velocity, the motor velocity and the tracking error (

Control signal.

Tracking errors of position (

Comparison of experiment and simulation results (case I: experimental results of Adaptive control; case II: experimental results of adaptive control with integral action; and case III: Simulation results of simulation of adaptive control with integral action).