Currently, in many industrialized sectors, DC (Direct Current) motors are being used in many different ways from robotics to automobiles, small-sized and medium-sized dynamic applications frequently features a DC motors for broad ranges of functionalities. The DC motor is an electric motor running on direct current. The most common actuator in many control system is DC motors. This types of motors provides direct rotational motion and, combined with cables and wheels or drums, which provides translational motion. Figure 1 below illustrates the free-body diagram and the electric circuit of the rotor (Carnegie, May 2017):
In this assignment, it deals with Continuous-time and discrete-time PI and PID controls of Direct Current motor angular velocity. The main control system which is considered in this assignment is the PID (Proportional plus Integral plus Derivative) controller. The control system is designed in computer software (*MATLAB software) which analyses the stability and causality of the designed controller using continuous time domain. The system designed in continuous time domain is then discretized and a comparison is made between the continuous time and discrete time systems.
PI (Proporsonal plus Integral) controllers are largely used in eliminating the steady state errors which result from P-controllers. Nevertheless, the overall stability and response speed of these systems has negative impacts. These kind of controllers are frequently applied in places where speed of systems does not matter. Because PI controllers are unable to predict the future errors of systems, it is unable to eliminate the oscillations and also reduce the rise time.
PID (Proportional plus Integral plus derivative) controllers has been used for some period of time in industries for processes control application. PID controllers have broadly applied in practical industries due to its attractive attributes like facile design, simple architectures, and parameters tuning with no complicated computations. Nevertheless, the PID controllers generally needs certain priori manuals retuning for making effective industrial applications (Kaur, 2017). Bypassing the mentioned problem, an adaptive PID controllers are currently proposed, which are comprises PID controllers and a fuzzy compensators (Narvekar & Upadhye, 2016). The PID controllers can be automatically tuned online the controls gain that is located on the incline descent methods and the fuzzy compensators are implemented for elimination of effects of the approximations errors that may be introduced by PID controllers in systems stability in the Lyapunov senses. PID controllers are extensively applied because their simple design and reliable operations. A PID design consist of three (3) main controls which
includes: Kd, Ki and Kp respectively referring to derivative, integration, and Proportional gains (Kaur, 2017) (Bishop & Dorf, 2009) (Owen, 2013). Table 1 below illustrate how these three (3) terms affect the PID percentage overshoot, steady state error and rise time.
|
Elements |
Overshoot |
Rise time |
|
|
Kp |
increases |
Reduces |
Reduces |
|
Ki |
increases |
Eliminates |
Reduces |
|
Kd |
Reduces |
No effect |
No/small change |
Table 1: Factors affecting PID controllers (Turevskiy, et al., 2011)
J.G. Ziegler (2006) discussed the concepts and design procedures of genetic algorithms as an optimization tools. Additional, it proposes the procedures to realize the workability and applicability of GA for processes controls applications. The simulated result presented a better optimizations of hybrid genetic algorithms controller than and conventional controllers and fuzzy standalone (Ziegler, 2006).
Algreer (2008), presents the optimization of the experimental ways adopted for traditional PID controller parameters, an optimization method based on improved ant colony optimizations for PID parameters of BP neural network (Algreer, 2008).
Te-Jen Su, et al (2008) described an efficient and fast tuning techniques based on modified genetic algorithms (MGA) structure for finding the optimal parameters of the PID controllers as a result the desired system specifications are met. The step response of closed loop system wes compared with that of GA where the results showed that the performances of PID controlled systems was improved by the MGA- based methods (Te-Jen Su, et al., 2008).
Maryand and Marimuthu (2009); presented the developments of a fuzzy-neural-network (FNN) PI-/PD like controllers with online learning for speed trajectories tracking of a brushless drive systems. The design used extended Kalman filter (EKF) to train FNN structure as part of the PI-/PD-like fuzzy design. The conventional PID controllers was exchanged with the proposed FNN PI-/PD-like controller with EKF learning mechanisms which showed that performances provided has an improvement of the FNN PI-/PD-like controller over PID control (Maryand & Marimuthu, 2009).
Rubaai (2011) designed genetic neural fuzzy PID (GNFPID) controller by merging the Sugeno fuzzy logic, radial basis function neural network (RBF- NN) and genetic algorithm (GA) for regulating the optimal parameters of a PID controllers. The method used a rule based of the Sugeno fuzzy systems and fuzzy PID controllers of the automatic voltage regulators (AVR) for progressing the system sensitive responses. It was affirmed that GNFPID has higher efficiency and more robust in refining the sensitive responses of an AVR systems (Rubaai, 2011).
Li and Liu (2011) presented two (2) different speed controllers which are fuzzy PID oversees online ANFIS controllers and fuzzy online gain tuned anti wind up PID controllers for the speed control of brushless dc motor. MATLAB simulated results showed a better performance with fuzzy PID managed online ANFIS controllers under all running conditions of the drives (Li & Liu, 2011).
Saikia, et.al (2015) offered the controller gains were optimized using an optimization method termed as “Cuckoo Search”. Research shown that dynamic response of the systems with BES were better than without BES. The sensitivity analysis shown that the optimum controllers gain in with BES was robust (Saikia, et al., 2015).
. Kaur et al.(2016) presented an adaptive PID controller which showed much improvement on any electromechanical system (kaur, et al., 2016).
The PID controllers has many merits which includes: First the controller implementation and design is easy The PID controller can be constructed using the analogue circuits or a logic gates circuits or MCU or resistor-inductor circuits. On the other hand, PID controllers needs suitable and acceptable sampling time to design which needs to be very accurate.
Second, the PID controllers are much stable if the three (3) constants (i.e. Kp, Ki, and Kd) are chosen wisely. If chosen wisely the PID controllers has ability to withstand external disturbances such as vibrations, noise, etc. In case these constant are not chosen accurately, the system becomes unstable. Thirdly, the robustness of the system where it can be attained when the performance and the stability of PID Controllers are not affected by a small difference in plant or operating conditions. The benefit of the PID controllers are that they are robust.
The forth is the performance of the PID controllers which is evaluated by its ability to overcome the internal and external disturbances effect referred to as the disturbance rejections of the control systems. A smaller value for derivative are needed since it is very sensitive to disturbance which may result into unstable systems. High value of derivative values results to oscillations of the systems, hence unstable systems. The response and rise time of this controller needs to be less than 2% of the output of the system and have a stable state. Furthermore, speed of peak time requires to be considerably fast to reach the peak value in a given system.
The fifth is the power consumption of the system, less power can be consumed by PID controller for a stable designed controller. On the other hand, unstable system consumes surplus power consequently PID has to be introduced for the system to gain stability resulting to dissipation of less energy hence less power is consumed.
Lastly, the PID reduces the Steady state error of the system. A steadtdy state error is the difference value between the exact output produced by the system as compared to the desired output of the similar system. PID controllers minimizes the steady state error in the control system over time and the error rates (Novoteknik, 2010). The controller minimizes the sse and the error rate of a given system, The desired output is met when sse is zero (Owen, 2013).
There are two (2) modern alternative to PID controlllers for slow processes and systems with indeterminate parameters which are the good gain control technique and the Ziegler-Nicholas technique. The two (2) techniques are lab techniques used in tuning PID controllers (William-Son, 2015).
The Ziegler–Nichols technique is exploratory technique where the PID controller is tuned through setting the Integral, I and derivative, D gains to zero. The proportional gains, K p is raised until it attains the final gain. At this point the outputs of the control loop has stable and consistent oscillations. The optimum gain reached and the oscillation periods are used setting the proportional, P, integral, I and derivative, D gains depending on controllers type used. The technique is used in simulation and it is also applicable the most common electromechanical system.
The Good Gain techniques are used to achieve a better stability to the control loops which is better stability as compared to Ziegler-Nichols’ methods (Ogatan, 2014). The Good Gain technique, is simple to design and it is used on both simulated systems and on real process (that is without the knowledge about the processes to be controlled). The technique provides better stability and doesn’t require the control loops to get into oscillations while tuning (Ogatan, 2014).
Conclusion
On this section, the design parameters of PID control is discussed, the merits and demerits of PID and where applicable are discussed. Literature review of PI and PID has been discussed fully and the progress of the PID controllers. Lastly, the section discusses the modern alternatives to PID control and where applicable.
The shaft position of any DC motor is majorly controlled by variation of the input/applied voltage U or the output voltage V, as illustrated in figure 4 below. The Motor illustrated below attached an inertial load. The input voltage to the armature is represented by U while that of field side of the motor is denoted by V. The resistance and inductance of the armature are given by R and L, on the other hand, resistance and inductance of the field side the DC motor is given by:and respectively (Bishop & Dorf, 2009).
The motor torque produced by the Motor is directly proportional to the product of field current and armature current, which is.
Where k is constant of proportionality.
For an Field-controlled current motors, armature currents are kept constant, varying the field voltage, do control the field current, therefore torque of the motor varies with the field current (Bishop & Dorf, 2009). i.e.
Where is torque of the motor
Now, if the Laplace transform of equation (2) is taken, then the following equation (3) is given as:
On the field side of the DC motor the voltages/current association is specified by the voltage across the field inductor and the voltage across the field resistor, giving the field voltage (Kuhnert & sailan, 2016): That is:
Where is the voltage across field resistor and is the voltage across field inductor.
On the other hand, back emf is produced which is direct proportional to the angular speed of the motor, as described in equation (5) below
Here is the back emf constant.
The Electromechanical Equation can be described by taking the Laplace transform of equation (5) which gives equation (6) below:
Taking the Laplace transform of equation (4) gives and making the subject of the formula gives;
The transfer function from the input voltage to the resulting current is given by:
The input voltage to the resulting motor torque transfer function can be determined by merging equations (3) and (8) to give:
The Dynamics of Mechanical System is given by equations (8) below:
However, T (t) is given by equation (11) below:
Where J is the motor’s moment of inertia and b is the viscous friction coefficient of the motor. Taking the Laplace transform of equation (9) and rewriting the equation as input.
Inserting the given parameter values in equation (16) gives the transfer function in equation (14) as obtained in MATLAB: The following MATLAB code define the parameters and finds the transfer function of the system
Conclusions.
In whole of section 2, it deals with the control systems of field-controlled DC motor. First, the understanding of how DC motor works is learned by deriving the transfer function of the system where later the parameters are provided for analysis in MATLAB. The system implemented here is found to be stable since the poles were found to be on the left hand side near the origin of the s-plane. Also using the Ziegler-Nichols tuning method and Chien-Hrones-Reswick tuning method concludes that the system is stable.
For different types PID turning, the results demonstrated that each method had its specific merits when comparison is made. When DC motor speed control system given, the performance of each tuning is different. For example, the Ziegler-Nichols tuning technique produced faster step response of the system with acceptable overshoot. On the other hand. Chien-Hrones-Reswick tuning technique the PID has a larger rise time and a smaller overshoot having suitable system transient response.
a): The open loop and closed loop transfer functions of digitized system.
The DC motor system implemented in section 2 in s-domain can be digitized to z-domain. The discretized system is given in open loop and closed loop Z-domain transfer function. At this point, the transfer function in equation (14) is being converted to the discrete z-domain from the continuous S-domain by help of MATLAB. The command c2d command is used in the conversion from S-domain to Z-domain. The MATLAB command requires three parameters which include: the model of the system, the sampling time and the type of hold circuit. In this section the system model is given by equation (13) while the type of hold circuit I will use Zero Order Hold (ZOH) since it is not specified on which to use.
The MATLAB commands below is used to find the open-loop and closed-loop transfer function of the DC field controlled DC motor..
The PID controllers can either be used to discretize continuous-time PID controllers or by straight creating discrete-time PID controllers in MATLAB. With PID command, the approaches used to discretize the derivative terms and integral terms can be specified independently.
The command, pidstd() in MATLAB is being used in creating a discrete-time controller. Equation (17) is the discrete time controller (MathWork, 1994-2018):
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