Keywords

1 Introduction

Separately excited DC motors transform electrical energy into mechanical energy and consist of electric-mechanical components. This type of DC motors maintains an intrinsic relation between torque and speed which has benefited its use and application in industrial processes such as trains, position system and robot arms [1].

Researchers have developed various control system methodologies to obtain an appropriated output for DC motor speed control. In this regard, PID controllers offer a feedback loop mechanism to tune their coefficients and provide an appropriate response for linear systems. Although this controller is well-known in industrial applications and applied in other systems that require module control adaptation, its application is not suitable when torque load increases [2].

On the other hand, Fuzzy Logic Controller (FLC) imitates the human understanding to provide an optimal response evading mathematical calculation for linguistic ideology. This controller expresses the inputs in terms of logical variables which could be easily translated in control design. The interpretation of continuous values provides an appropriate output signal regarding to system requirements. In particular, FLC offers advantages for higher order systems and facilitates continuous parameters modification [2, 3].

Although previous works apply PID and FLC controllers for separately excited DC motors in an individual manner [2, 3], this research proposes a comparative study between these controllers. PID and FLC are tested for parameters such as torque load rejection and percentage overshoot to determine the benefits and drawbacks of each controller to obtain an appropriated output for DC motors.

The remainder of this paper is organized as follows. Section 2 provide background about DC motor, PID controllers and different methods to determine the coefficients for calculating DC motor speed control. The simulations and results of the performance of the controllers are described in Sect. 3. Section 4 summarizes the conclusions of this paper.

2 Background

2.1 DC Motor

DC motor separately excited presents two distinctive sections known as field and armature as illustrated in Fig. 1. The magnetic field is generated in the field section, while, in the armature field, an armature current (Ia) is produced by a voltage (Va) in order to generate an electro-mechanical force (E). The electro-mechanical force is related to the angular velocity (ω), and this physical phenomenon is the purpose of motors.

Fig. 1.
figure 1

Structure of DC motor.

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The DC motor equations are:

$$ V_{a} = R_{a} .i_{a} + L_{a} .\frac{{di_{a} }}{dt} + E $$
(1)
$$ E = K_{e} .\omega .\phi $$
(2)
$$ T_{e} = K_{T} .i_{a} .\phi $$
(3)
$$ T_{e} = T_{L} + J\dot{\omega } + B\omega $$
(4)

Where, Ra is armature resistance in [Ω], La is armature inductance in [H], TE is electro-mechanic torque in [N.m], TL is torque load in [N.m], J is momentum of inertia in [Kg/m2], B is coefficient of friction in [N.m.s], KT is torque motor gain in [N.m/A], KE is back electro-mechanical force coefficient in [V.s/rad], ω is angular speed in [rad/s] and φ is the flux in [Wb].

By applying Laplace transforms to the Eqs. (1), (2), (3) and (4), it is possible to elaborate a Simulink model to obtain a DC motor representation.

$$ V_{A} \left( S \right) = R_{A} I_{A} + L_{A} I_{A} S + E\left( S \right) $$
(5)
$$ T_{E} = T_{L} + J\dot{\omega }\left( S \right)S + B\omega \left( S \right) $$
(6)

From Eqs. (5) and (6), the angular speed transfer function is calculated as follows:

$$ \frac{\omega \left( S \right)}{E\left( S \right)} = \frac{K}{{JL_{A} S^{2} + \left( {JR_{A} + BL_{A} } \right)S + \left( {BR_{A} + K} \right)}} $$
(7)

2.2 PID Controllers

PID controllers have been implemented in more than 90% of the industrial applications and processes because of its robust performance and simplicity of parameters [4, 12]. Specifically, these controllers consist of three components: proportional (KP), integrative (KI) and derivative (KD) variables. Proportional gain defines the response to the present error (e(t)), which corresponds to the difference between the reference and the response. By accumulating the recent errors, integrative coefficient designs the optimal responses. Derivative coefficients assign the action based on the rate at which the error has been changed [1, 4]. According to some specific parameters, PID controllers provides a control variable (u(t)) by adjusting their gains, in order to obtain an output signal. The structure of a PID controller is described in Fig. 2.

Fig. 2.
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Structure of PID controller.

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The structure of this controller is represented by the following equation:

$$ u\left( t \right) = \left[ {K_{P} .e\left( t \right) + K_{I} \mathop \smallint \nolimits_{0}^{t} e\left( t \right)d\tau + K_{D} \frac{de\left( t \right)}{dt}} \right] $$
(8)

Different methods could be employed to find the coefficients, such as: Ziegler-Nichols open loop response technique, closed loop or resonance technique and manual tuning.

2.3 Ziegler-Nichols Open Loop Method

Ziegler-Nichols open loop method has been implemented to acquire the PI and PID coefficients. These two types of controllers are widely applied in different industrial processes [5, 6]. By sketching a tangent line in the open loop step response, it is possible to obtain the following parameters: gain in steady-state (K), delay time (L) and time constant (T); Fig. 3 provides an overview of this process.

Fig. 3.
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Response for open loop methodology.

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By applying the equation, A = (K.L)/T, it could obtain the PID gains through the Table 1.

Table 1. Open loop table parameters.
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2.4 Ziegler-Nichols Closed Loop Resonance Method

This method initiates by setting integral and derivative coefficients to zero. The next step corresponds to increase the proportional coefficient from zero to a gain (Ku), where the step response shows a continuous oscillation [5, 6]. Finally, the response period (Tu) could be determined by measuring the time length of one cycle. Figure 4 shows this method and its parameters.

Fig. 4.
figure 4

Closed loop response methodology.

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The Table 2 provides the values to obtain PID gains as follow:

Table 2. Close loop resonance table parameters.
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Manual tuning is another alternative method to find PID coefficients, providing an output signal more affordable related to system requirements [2]. For industrial locations, a significant amount of processes or applications are determined for high order or nonlinearity systems. PID controllers operates effectively for first and second order linear systems. However, these controllers could be affected by load torque increasing or components physical deterioration. To provide a different methodology, scientist have established a technique to attend to complex systems.

2.5 Fuzzy Logic Controller

In 1965, Dr. Lotfi A. Zadeh established the concept of Fuzzy logic controller to deal with known drawbacks of PID controllers. FLC is able to control complex systems by executing a linguistic ideology achieved from the experience and knowledge of the operator [7, 8]. This controller offers a procedure for symbolizing and applying an interpretation of programmer experience, in order to establish a signal that works over the plant. Fuzzifier, knowledge base, inference mechanism and defuzzifier are the FLC components, where e(t) is the error, ce(t) is the change of error and u(t) is the control variable.

Fuzzifier.

This component transforms a standardized input gain into an etymological term. For example, it is able to obtain an exemplification as negative small (NS) or big positive (BP) by estimating the membership function. The most used shape is triangular waveform. Figure 5 shows a general idea of this component.

Fig. 5.
figure 5

Example of the membership function of e(t) and ce(t).

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Knowledge Base.

This module offers the database and rules for the FLC. Knowledge base is the responsible component for providing information to all FLC elements. Rule base consists of a set of linguistic interpretation related to the input and the desired output.

Inference Mechanism.

Inference process defines the FLC output by calculating the knowledge base commands. The programmer skill or knowledge establish the commands that offer the output values, for example:

$$ {\text{If}}\,e(t)\,{\text{is}}\,{\text{Em}}\,{\text{and}}\,ce(t)\,{\text{is}}\,{\text{dEn}}\,{\text{then}}\,f(t)\,{\text{is}}\,{\text{Cmn}}. $$

Where the functions incomes are defined by Em and dEn, f(t) corresponds to the controller output and Cmn is the output function.

Defuzzifier.

In this section, FLC output is transformed into numerical gain. There are different types of defuzzifiers such as mean of maximum, center of gravity, center of average.

PD, PI and PD + I are the most applied FLC schematic. The well-known drawbacks of the FLC correspond to elaborate the scheme of the rules, select the fitting defuzzi-fier and find an appropriate response [8, 9, 11].

3 Simulation and Results

Matlab/Simulink offers a valuable software tool to study the performance of the controllers. This section analyzes the behavior for speed control of a separately excited DC motor tuned by (1) open, (2) closed loop methodology, (3) manual tuning and (4) Fuzzy PD + I controllers. These methods are evaluated under different factors such as disturbance rejecting and multiple steps input.

For the first section of this examination, 1000 rad/s step input is applied to each controller. Table 3 shows the DC motor parameters for the simulation.

Table 3. DC motor parameters.
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Applying the Eq. 7, angular speed transfer function is shown as follows:

$$ \frac{\omega \left( S \right)}{E\left( S \right)} = \frac{1558.4}{{S^{2} + 70.02S + 1872}} $$
(9)

Method 1: Ziegler Nichols Open Loop Response

Ziegler Nichols open loop response provides a quick methodology to find the PID parameters – A and L. By applying a step response to the DC motor, the value of these variables is 0.0286 and 0.0375 respectively. For this study, PID configuration is used due to its robustness. Table 4 shows the PID gains and Fig. 6 displays the step response for PID controller. The percentage overshoot is 12.4% and rise time is 0.0315 s.

Table 4. PID coefficients obtained by open loop response methodology.
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Fig. 6.
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Step response for PID controller tuned by open loop methodology.

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Method 2: Ziegler Nichols Closed Loop Response

This methodology is essential to calculate the parameters Ku and Tu whose values are 75 and 0.5 respectively. The coefficients of these controllers are shown in Table 5. For this examination, PID controller is the most suitable to be applied due to its effectiveness and efficient. Figure 7 shows the step response, obtaining a percentage overshoot of 12% and rise time of 0.105 s.

Table 5. PID coefficients obtained by closed loop response methodology.
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Fig. 7.
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Step response for PID controller tuned by closed loop.

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Method 3: Manual Tuning

For PID controller tuned by manual adjusting, Table 6 shows its gains and Fig. 8 displays the step response. The overshoot percentage and rise time are 0% and 0.104 s, respectively. As a previous methodology, PID controller provides an optimal response for the requirements specification.

Table 6. PID coefficients obtained by manual tuning.
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Fig. 8.
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Step response for PID controller tuned by manual adjusting.

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Method 4: Fuzzy PD + I

According to the aforementioned explanation, FLC offers a feasible solution for non-linear systems by executing an etymological philosophy, avoiding any calculation, in order to define the action of system control variable [10, 11]. For this examination, the two inputs of the fuzzy PD + I controller are error and change of error, and the output works over the system.

By a trial and error methodology, it is possible to obtain the membership functions of the FLC, related to the system requirements [2, 10]. The membership functions of the Fuzzy PD + I controller are shown in the Fig. 9.

Fig. 9.
figure 9

Error, change of error and output membership functions for Fuzzy PD + I controller.

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Table 7 shows 25 fuzzy rules which can be established by the observation and testing of the system performance. Table 8 displays the fuzzy PD + I gains of the system.

Table 7. FLC rules.
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Table 8. Fuzzy PD + I Coefficients.
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Figure 10 displays the step response for Fuzzy PD + I controller, the percentage overshoot is 0% and rise time is 0.291 s. This response is produced because the integral term acts as an accumulator of the previous error.

Fig. 10.
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Step response for Fuzzy PD + I controller.

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This study contains an examination of variable steps input. The multiple input initiates at 1000 rad/s; at 2 s, the reference growths to 1200 rad/s. At the end, the input decreases to 800 rad/s at 4 s. The Fig. 11 displays the responses for the three controllers. For multiple steps inputs, the behavior of each controller maintains its performance. Ziegler Nichols open and closed loop response offer an output signal with a significant overshoot (over 12%). This effect is produced due the proportional and integral action. In contrast, manual tuning provides a response without overshoot because its integral coefficient is considerable small. Figure 12 shows the responses for Fuzzy PD + I controller.

Fig. 11.
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Multiple steps response for PID controllers.

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Fig. 12.
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Multiple steps response for Fuzzy PD + I.

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At 3 s, a 5% torque load disturbance is executed to the DC motor, for open loop response methodology PID controller response drops to 998.1 rad/s and the time that takes to return to the reference point is 0.21 s. For closed loop methodology, the response decays to 998.3 rad/s and the time to recover the reference is 0.62 s. For manual tuning, the output signal drops to 973.5 rad/s and the time to recover the reference is 0.63 s. Figure 13 shows the torque load rejection for PID controllers. Basically, this time can be eliminated by growing the integral term, however, this effects produce a rise of the overshoot percentage. According to Fuzzy PD + I, the response drops to 925 rad/s. As demonstrated in the preceding examination, Fuzzy PD + I presents a slow performance because the integral gain works as sum of the output before to it. FLC’s linguistic representation offers a methodology to eliminate complex calculations. Figure 14 displays the disturbance rejection for Fuzzy PD + I controllers.

Fig. 13.
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Torque load rejection for PID controllers.

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Fig. 14.
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Torque load rejection for Fuzzy PD + I controller.

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4 Conclusion

This paper aims to obtain an appropriated output for separately excited DC motor by comparing different control methodologies including PID controllers and FLC. These controllers have been tested under different conditions such as overshoot percentage, rise time, torque load disturbance and multiple steps input. According to the results, Ziegler Nichols approach provides an output with a considerable overshoot percentage and a slow torque load rejection. On the other hand, PID controller tuned by manual adjusting shows a reduction in proportional and integral terms which in turn produces an output response without overshoot and an appropriated torque load rejection. In the case of FLC, the output response does not present an overshoot and acceptable torque load rejection, but the response is slow since the integral term act as an accumulator of the previous error. To summarize, PID controller tuned by manual adjusting seems to be the most appropriated scheme for DC motor speed control based on the above described parameters. As a future study, the efficient of the proposed controller can be analyzed and compared with an optimizer.