Robust Control Simulation and Implementation for DC Motor

A direct current (DC) motor is a very old machine, is widely used in automatic systems that require precise speed changes, the applications have become more important and developed in recent years [1, 2]. Because of their high reliability, flexibility and low cost, where speed control of motor is required [3, 4].

The DC motor will have changes in electrical parameters as temperature increases, current and voltage fluctuations, time-varying load conditions, driving and operating conditions. These changes cause the DC motor to have non-linear characteristics. Therefore, the nonlinear control method is necessary. In this research, SMC method will be applied to control the speed of DC motor [5, 6].

PI is one of the classic control methods still widely used. Based on 90% of industries [7], PI to be used due to the advantage of being simple and applicable. However, a disadvantage of this method is that performance decreases if the installation is non-linear [8], A PI corrector has an integration, which makes it possible to solve the problem of the static error. On the other hand, it slows down the system, so care must be taken to respect a sufficient phase margin, but not too large either (the greater the phase margin, the slower the response and the risk of saturation increases)

For a pure integral regulator, the dynamic regime is relatively long. On the other hand, the proportional regulator will react immediately to deviations from the setting, but cannot completely eliminate static errors. The combination of proportional and integral action makes it possible to combine the advantages of proportional action (i.e. rapid response to control deviations) with the advantage of integral action, which is precise compensation for the pilot.

Sliding Mode Control (SMC) is a controller that can handle non-linear conditions in the installation. The SMG is robustness, ability to handle nonlinear systems, time varying systems [9, 10], it can be designed for fast dynamic responses and good capabilities over a wide range [11, 12].

The sliding mode controller has been suggested to achieve robust performance against parameter variations and load disturbances. It also offers a fast dynamic response, stable control system and easy hardware-software implementation, is low-cost [13-16], and small [17].

On the other hand, this control method offers some drawbacks associated with the large torque chattering that appears in steady state. Chattering involves high-frequency control switching and may lead to excitation of unmodelled high frequency system dynamics. Chattering also causes high heat losses in electronic systems and undue wear in mechanical. In order to reduce the chattering phenomenon, a sign function is used.

Howbeit, the major problem encountered in the control DC motor is the determination of scaling factors of regulators, conventional PI. In fact, it has noted that there's no precise system dimensioning, which enables us to obtain controllers gains that give the asked performance, and the determination of gains is generally through the trial and error system grounded on manual testing a number of implicit results until an acceptable result. In order to grease the tuning of regulators earnings icing optimal performance and reduced time consumption comparatively to the" trial- error", we Address by genetic algorithms" Genetic Algorithms (GAs).

The strength of genetic algorithms is their ability to search out the realm of area solutions containing the global optimum of the objective function. However, they're ineffective once it involves realize the precise value of the optimum during this space. Or this can be exactly what the native optimization algorithms perform best. It’s so natural to supposed to associate an area algorithmic program with genetic algorithmic program so as to search out the precise worth of the global optimum. We are able to simply do therefore by applying at the top of the genetic algorithmic program an area algorithmic program on the simplest item found, like the gradient or simplex algorithmic program.

The special merit of the suggested optimized controllers is a): To search the optimum operating point for DC motor in speed control mode. b) To enhance the performance of DC motor

The outline of this paper is as follows: in Section II, the modeling of the DC motor is presented. The design of a SMC for speed regulation of a DC motor is presented in Section 3. In Section 4 Hybrid genetic algorithms based mostly PI controller is proposed In Section 5 the performances of the proposed control are illustrated by some simulation and implementation results. Finally, some concluding remarks are given in Section 6.

The basic principle of the sliding mode control consists in moving the state trajectory of the system toward a surface S(X) =0 and maintaining it around this surface with the switching logic function U_n. The basic sliding mode control law is expressed as [20] (shows Figure 2):

$U=U_{e q}+U_n$         (6)

This expression uses two terms, U_eq and U_n. Where U_eq: is determined off line with a model that represents the plant as accurately as possible.

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Figure 2. Diagram block of sliding mode control (SMC)

The trajectory tracking problem consists in determining a control law u(x) which makes it possible to ensure the convergence of the state x of the system towards the desired state x_d.

The sliding variable is:

$S=\dot{e}+\lambda e$        (7)

with, λ>0.

e: The error.

$e=x_d-x_1=\Omega-\Omega_{r e f}$           (8)

$\dot{e}$: The error derivative.

$\dot{e}=\dot{x}_d-\dot{x}_1$           (9)

The transfer function:

$F(p)=\frac{\Omega(p)}{U_a(p)}=\frac{K_c}{j L_a p^2+\left(R_a j+L_a f_r\right) p+R_a f_r+K_c K_e}$

We pose:

$j L_a=A$

$R_a j+L_a f_r=B$

$R_a f_r+K_e K_c=C$

$F(p)=\frac{\Omega(p)}{U(p)}=\frac{K_c}{A p^2+B p+C}$

We divide the equation on A:

$F(p)=\frac{\Omega(p)}{U(p)}=\frac{K_c / A}{p^2+B / A p+C / A}$

In the time domain, the equation F(p) can be written:

$p^2 \Omega+\frac{B}{A} p \Omega+\frac{C}{A} \Omega=\frac{K_c}{A} U$

$\ddot{\omega}+\frac{B}{A} \dot{\omega}+\frac{C}{A} \omega=\frac{K_c}{A} U$

$\ddot{\omega}=\frac{K_c}{A} U-\frac{B}{A} \dot{\omega}-\frac{C}{A} \omega$

with, x₁=ω(t).

Then the system can be converted to the following canonical form:

$\dot{x}_1=x_2=\dot{\Omega}(t)$

$\ddot{x}_1=\dot{x}_2=\ddot{\Omega}(t)$

So:

$\dot{x}_2=\frac{K_c}{A} U-\frac{B}{A} \dot{\Omega}-\frac{C}{A} \Omega$

$S=\dot{e}+\lambda e$

$S=\left(\dot{x}_d-\dot{x}_1\right)+\lambda\left(x_d-x_1\right)$

The discontinuous command is given by:

$U_n=-\alpha \frac{S}{|S|+\delta}$        (10)

U_eq Determined from the relationship: $\dot{S}=0$.

$\dot{S}=\left(\ddot{x}_d-\ddot{x}_1\right)+\lambda\left(\dot{x}_d-\dot{x}_1\right)$

$\left.\dot{S}=\left(\ddot{x}_d-\left[\frac{K_c}{A} U_{e q}-\frac{B}{A} x_2-\frac{C}{A} x_1\right]\right)+\lambda \dot{x}_d-\lambda x_2\right)$            (11)

$\left.\dot{S}=0 \Rightarrow \ddot{x}_d-\left[\frac{K_c}{A} U_{e q}+\frac{B}{A} x_2-\frac{C}{A} x_1\right]+\lambda \dot{x}_d-\lambda x_2\right)=0$

$U_{e q}=\frac{A}{K_c}\left[\frac{B}{A} x_2-\frac{C}{A} x_1+\lambda \dot{x}_d-\lambda x_2+\ddot{x}_d\right]$

3.png

Figure 3. DC motor hardware setup

The result of numerical simulation using Matlab Software Figure 3 and the hardware implementation using the Arduino microcontroller were seen in Figure 5, we used a equation command (12). In this paper the reference speed is 500 rpm, for this speed, the current is 27,32 mA, and the electromagnetic torque 1.15 kg.cm.

5a.png

5b.png

Figure 5. Simulation of SMC and PI of system

5.1 Numerical simulation

From the system response, it can be seen that the motor speed follows the given reference. The role of the integral action is to cancel the difference between the measurement and the set point (the static error is zero).

The effects of the PI corrector are:

- Decreased rise time.

- Elimination of static error.

- Increased stabilization time.

- Increased overshoot.

According to the satisfactory results obtained in Figure 6, it can be seen that the motor speed follows its reference, which shows that the use of sliding control of the system has better performance compared to the PI which is a faster stabilization time and the response time by sliding mode is better compared to the PI corrector. Table 2 gives comparison between SMC and PI.

Table 2. System response performance

6.png

Figure 6. Numerical simulation of SMC and PI of system

5.2 Arduino hardware implementation

The hardware implementation would be done using Arduino, JGA25 DC motor, L298 driver motor, encoder and INA219 current sensor. The configuration of the device is shown in Figure 3.

The result of PI is shown in Figure 7. The control signal SMC is shown in Figure 8. It can be seen that the system was able to reach the reference quickly in SMC compared with PI. The detailed system response is shown in Table 3.

7.png

Figure 7. The control signal PI controller

8.png

Figure 8. The control signal SMC

Table 3. System response performance