Many industrial sectors today, direct current motors (also called DC motors) are used in different ways from automobiles to robotics small and medium-sized driving applications regularly features DC motors for the wide range of functionalities. A DC motor can be defined as an electric motor which runs on direct current. Common actuators in control systems are Direct Current motors. The DC motors provide direct rotary motion and, coupled with cables and drums or wheels, providing translational motion. The electric circuit of the armature and the free-body diagram of the rotor are shown in figure 1.1 below (Melkin, 2017):
Figure 1. 1: The electric circuit of the armature and the free-body diagram of the rotor
In this assignment, it deals with Continuous and discrete time PID control of DC motor angular velocity. The major control system considered in this assignment is the PID controller, which is simulated using MATLAB software. The system is then discretized and compared with the continuous time control.
During simulation and equation derivation of a DC motor, the following are assumed: The input of the system is the voltage source ((u) as shown in figure 1.1 above) applied to the armature of the motor and the output is the rotational speed of the shaft. The rotor and shaft are assumed to be rigid. Furthermore, the viscous friction model, that is, the friction torque is proportional to shaft angular velocity.
PID (Proportional+ Integral+ Derivative) controller provide a range of amendments because it contains three (3) key controls which includes P-control, I-control and D-control which may be altered. PID Controller control and handles system characteristics like settling time, percentage overshoot, stability, steady-state error, rise time, etc. Even if there are three control elements in the controller, it still has some disadvantage, because the implementation complexity increases in the system (Abu-Khalaf, et al., 2009). Though, each control element has different functions, the elements are exclusively dependent to each other; since single element can be varied by changing another element. Consequently, PID design is complex as compared to the designing P- controller, PD- controller or PI- controller (Anon., 2016). In this part, the strength and disadvantages of PID controller in terms of implementation of the controller, stabilization requirements, performances, robustness, energy consumptions and steady state errors
During implementation of PID controller, one strength on implementing the PID controller is that it is easier to construct and design. The PID controllers can be an analogue circuit or a logic gate circuit or MCU or inductors and resistor circuit. Conversely, PID controller needs acceptable and a better sampling time for implementing which requires to be very accurate
The PID controllers to be stable it needs several factor such as Kp, Ki, and Kd. When the needs are all met the PID controller is more stable. The following table 1.1 illustrate how these element affect the PID percentage overshoot, rise time and steady state error.
|
Elements |
Effect on Rise time |
Effect on overshoot |
Effect on steady state error |
|
Kp |
Reduces |
increases |
Reduces |
|
Ki |
Reduces |
increases |
Eliminates |
|
Kd |
No/small chage |
Reduces |
No effect |
Table 1. 1Factors affecting PID controllers
In obtaining a very accurate PID controlled system, these requirements indicated in table 1.1 above must be met to be able to withstand external disturbances like noise, vibrations, etc. Failure to meet the requirement, the system becomes unstable.
The performances of PID control systems is evaluated by its ability to overcome the disturbances effects referred to as the disturbance rejection of the control systems.
A small value for derivative value is required since it might result into unstable system due to the high sensitivity to disturbance such as noise and vibration. High value of derivative will result to oscillation of the system, thus unstable system.
The rise time or responses time of PID controller is required to be less than 2 percent of the system output and having a stable state. Moreover, speed of the peak time is required to be considerably faster in reaching the peak values for the given system.
The robustness of a system can be attained when the stability and the performance of PID Controllers are not affected by a smaller differences in plant or the operating condition. The advantage of the PID controller is that, the robustness is achieved for system with less robust.
The PID controller are designed to consume less power, for a system which is unstable it may consume a lot of power. But when PID contlol are introduce the system gains its stability, thus less energy is dissipated resulting to less power consumption
Steady state errors can be well-defined as the difference values between the exact output produced by the system and the desired output of the same system. PID controllers are used to minimize the steady state error in the control systems over time and the error rate (Novotecknik, 2009). For a PID, it reduces the error rate and sse of the system, When sse is zero, that means the desired output of the system is met. The integral components (Ki) sums the error term over time. The integral components increase continuously if there exist a small error. The phenomenon in which the integral component continue to increase is referred to as integral windup, which occurs when the integral actions reach to saturations and does not reduce the error to zero. The importance for the integrator are the ant windup operations for saturating the actuator (Owen, 2012).
Part b: Examples of PI-controlled system and PID controlled system.
PI controllers are mostly used in eliminating the steady state error which may result from P-controllers. Though, in relations to the speed of the responses and general stabilities of systems, it have some negative impacts. These controllers are typically used in places where system speed does not matter. Subsequently PI controllers does not have ability in predicting the future error for systems, thus the controller cannot decrease/increase the rise time and elimination of oscillations, if applied an amount of I guarantee set points overshoot.
An example is the cruise control systems control which is used to control the speed/velocity of the vehicle, similarly by regulating the throttle positions. In fact cruise controls actuate the throttle valves by a cable connection to actuators in place of pressing the car pedals. Figure 1.2 below shows the pi cruise control (Deka & Haloi, June, 2014).
Figure 1. 2: Car cruise control with PI controller
The aim of cruise control systems are maintaining constant car speed in spite of having external disturbances such as change of road grade or wind. The control is accomplished by computing the car speed, therafter the speed is compared with the reference/desired speed and automatically, the throttle is adjusted in accordance with control law (Deka & Haloi, June, 2014).
PID Controller:
PID controllers have the best control dynamic counting zero steady state errors, faster responses (shorter rise-time), higher stability and no oscillations as compared to other controllers such as PI-controller. The requirement to use a derivative gain component in addition to the PI controllers is for eliminating overshoots and any oscillation occurring in the output responses of controlled systems. One of the core advantage of PID controllers is that the controller can be used with higher order processes even more than single energy storages
An example used in PID control in many industries is a DC servo motor. The elementary components of normal servo motion systems are illustrated in figure 1.2 below By use of standard Laplace notations. In the figure 1.2, servo drive close a current loops and are made simply as linear transfer functions Obviously, the servo drive contains a peak current limit, thus the linear models are not completely accurate; nevertheless, it provides a sensible representations for the analysis. Basically, servo drives receives voltage commands that represent the preferred motor currents. The shaft torque, of the motor, is directly associated to motor current, by torque constant, Equation (1.1) below shows the mentioned above relationships.
The transfer function of the current/torque regulators can be estimated as unity for relative lower motion frequency which is needed.
Table 1. 2. The block diagram of PID Servo Control
The servo-motors are made as torque constant, a viscous damping term, , and lump inertia. The lump inertia terms contains the servo-motor and inertia of the load. There is an assumption that the loads are firmly coupled in such a way that the torsional rigidity passes the natural mechanical resonance points further than the servo controller bandwidths. In this case, it become easier to model the total systems inertia as the sum of load inertia and the motor for a frequency that can be controlled. Slightly added complicated models are required if coupler dynamic is integrated.
The real motor positions, typically estimated by by a resolve or an encoder couples directly to the motor shaft. Once more, the assumption made above also assumes that the feedback devices are firmly mounted in a way that mechanical resonant frequency can be ignored without any effect. Disturbances from external shaft torque, is added to the generated torque by the current of the motor to give the torque available for accelerating the total inertia, J (Ziegler & Nichols, 2000).
Neighboring of the servo drives and motor blocks are the servo controllers that close the position loops. A simple servo controllers normally contain both a trajectory generators and PID controllers. The trajectory generators usually provide position setpoints commands illustrated in figure 1.1 as The PID controllers operate on the position errors and output torque commands which is scaled by an approximation of the torque constantof the moto. In case that the torque constant of the motor(s) is unknown, the gains of the PID is re-scaled to get the desired gain. Since the exact values of the torque constant of the motor is commonly unknown, a symbol ^ can be used to designate an estimated values in the controllers. Generally, equation (1.2) below holds with adequate accuracy in that outputs of the servo-controllers commands the accurate amount of current for the wanted torque:
There exist three (3) gains for adjusting in PID controllers, which acts on the position errors given in equation 1.3 below. the superscript * denotes a commanded values (Ziegler & Nichols, 2000):
The outputs of PID controllers are torque signals. The mathematical expressions in time domain is illustrated in equation 1.4 below (Ziegler & Nichols, 2000):
The two modern alternative to PID controls for slow process and system with uncertain parameters are Ziegler-Nicholas method and good gain control method. These two are lab methods used in tuning PID controller (WILLIAMSON, 2015).
The Ziegler–Nichols methods are exploratory methods whereby PID controllers are tuned through setting the D (derivative) and I (integral) gains to zero (WILLIAMSON, 2015). The “P” (proportional) gain, K p is raised till it attains the final gain. This is the point at which the outputs of the control loops has consistent and stable oscillations. The maximum gain attained and the oscillation period are used to set the derivative, D, promotional, P and integral, I gains which depends on controller type used. This method can be used for simulations and it is also probably the most common to use in real life.
Figure 1. 3: Ziegler-Nichols method
The Good Gain method is used to give better stability to the control loop better stability than that of Ziegler-Nichols’ methods (OGATA, 2013). The Good Gain method, as simple as it is, can be used both on real processes (without any knowledge about the processes to be controlled), and in simulated systems. This method gives better stability and does not need the control loop to get into oscillations when tuning (OGATA, 2013). These are two benefits of this method as compared with the Ziegler-Nichols’ methods.
Figure 1. 4. The Good Gain method.
The system which is digitized it has several advantages over time continuous system. The flexibility and capability of decision making in the control program is the chief benefits of digital control systems (Anon., n.d.)among others:
The modern control applications are using process control where the power of digital processing techniques are used to perform the desired control tasks (KUO & HASELMAN, 2014). Although the majority of systems that need to be controlled are often analog nature, the modern digital control applications are using A/D- and D/A-conversions as the principal operations to achieve appropriate control of processes. This has brought the increasing need for discretization in modern control applications where sampling is carried out by point measurements (OGATA, 2013).
Motor drives requires a rotor position sensors to correctly perform phase commutations and current controls. A constant supplies of position data is essential; therefore position sensors having high resolution, such as a resolver or a shaft encoders, is characteristically used (Gamazo-Real, et al., July, 2010). For complex systems, therefore, low-cost Hall-effects sensor are typically used. Moreover, accelerometers or electromagnetic variables reluctance sensors has widely applied in measuring motor position and speed (Deka & Haloi, June, 2014). The angular motion sensors based on magnetic fields sensing principle stand out due to several inherent advantages and sensing benefits (Deka & Haloi, June, 2014).
Figure 2.0 below shows DC motor with an inertial load attached on it. The applied voltage to the armature and the field and sides of the motor can be signified respectively as and .The inductances and resistances of the armature and the field side the DC motor are given by:and where R is the armature resistance, L is armature inductance, is the field resistance and is the field inductance (Dorf & Bishop, 2001).
Figure 2. 1: The electrical circuit of a DC motor.
The motor torque produced by is proportional to armature current and field current, i.e.
Where k is constant of proportionality. For a armature-current controlled motors, is kept constant, and the armature voltage, controls the field current., thus the motor torque decreases or increases with the armature current (Dorf & Bishop, 2001). That is
Where is the motor torque. Taking the laplace transform of equation (2.2) gives the following equation:
On the side of the armature of the DC motor the current/ voltage association is given by Voltage across the resistor plus voltage across the inductor plus the back e.m.f () induced by the rotation of the armature windings gives the armature voltage (sailan & Kuhnert, 2015): That is:
The back emf is directly proportional to the motor speed:
Where is the back e.m.f. constant. Taking the Laplace transform of equation (2.5) gives: The Electromechanical Equations thus can be given as:
Taking the Laplace transform of equation (2.4) above and inserting equation (2.6) gives equation (2.7) below
The Mechanical System Dynamics is given by:
But T (t) is given by:
Where b is the viscous friction coefficient of the motor, and J is the moment of inertia of the motor. Rewriting the mechanical equation (2.9) as input output equation gives:
Consequently, the motor torque input to rotational speed transfer function changes. Giving equation 2.9 below:
Equation (2.9) is a first order system. The block diagram for equations (2.7), (2.8) and (2.9) is given in figure 2.2 below:
Figure 2. 2: The system block diagram.
From figure 2.2, the dc motor has an inherent feedback from the CEMF. This improves system stability by adding an electromechanical damping (Anon., 2017). To get the transfer function from the input voltage of armature to the output speed of the motor, the feedback formula is used to reduce the block diagram as:
Where:
Thus, the transfer function is given as:
Equation (2.10) is the transfer function of a DC motor.
Equation (2.10) can also be written as:
The physical parameters for the assignment are:
V/rad/sec
Nm/Amp
Ohm
Inputting the given values in equation (2.11) gives the transfer function. From the MATLAB the transfer function is given by:
The following MATLAB code finds the root locus of the system described by equation (2.11)
Figure 2.3 below shows the root locus of the DC motor control given above:
Figure 2. 3: The root locus of DC motor
The root locus designs are used in prediction of closed-loop responses from the root locus plots which describes possible closed-loop poles location and are drawn from the open-loop transfer functions (Thomas & Poongod, 2009). Thus, modification of root locus is impossible by adding poles and zeros through the controller for achieving the desired closed-loop responses of the system (Melkin, 2017). From the root locus of the system are
The poles are on negative, thus the system is stable. Also, both open-loop poles are closely located to the left, thus the poles affects the closed-loop dynamics.
PID controllers are generic control loop feedbacks mechanisms used widely in industrial control systems which are majorly usually used feedback controllers. PID controllers calculates the error values that exist between measured process variables and the desired responses. The PID controller tries to minimize the errors by correcting the process control inputs. The controller algorithms has three (3) parameters which includes: (I) the proportional constant which depends on the present error. In DC motor is used for increasing the response speed system and reducing the steady-state error. (II). the integral depends on the accumulation of past errors, it is used for elimination of steady-state error at all integral time constants (Owen, 2012). (III) the derivative value that predicts the future error, depending on the current rate of changes. It is used for reducing the system response overshoot (Phillips & Harbor, 2000) (Abdulameer, et al., 2016). The time constant formulae for PID controller is given in equation (2.12) below (Abdulameer, et al., 2016):
Figure 2.4 below illustrates the block diagram of DC motor control system (Abdulameer, et al., 2016)
Figure 2. 4: DC motor PID control block diagram
Using MATLAB, the following code is used for PID tunning methods for a step response to the input.
The output response of the DC motor is given in figure 2.5 below for the Ziegler-Nichols tuning method
Figure 2. 5: The Ziegler-Nichols tuning method
The control parameter of the Ziegler-Nichols tuning method are given in table 2.1 below.
|
Tuned |
Baseline |
|
|
Kp |
9.7632 |
5.7046 |
|
Ti |
2.7178 |
1.8629 |
|
Td |
0.24888 |
1.28 |
Table 2. 1: Control parameters for the Ziegler-Nichols tuning method.
The output response of the DC motor is given in figure 2.6 below for the Chien-Hrones-Reswick tuning method
Figure 2. 6: The Chien-Hrones-Reswick tuning method step response
The control parameter of the Chien-Hrones-Reswick tuning method are given in table 2.1 below.
|
Tuned |
Baseline |
|
|
s |
0.93546 |
5.7046 |
|
Ti |
2.6683 |
1.8629 |
|
Td |
0.66707 |
1.28 |
Table 2. 2: Control parameters for the Chien-Hrones-Reswick tuning method.
Table 2.4 below illustrate the robustness and performance of the PID controller as depicted from MATLAB for Ziegler-Nichols tuning method
|
Tuned |
Baseline |
|
|
Rise time |
2.73 seconds |
4.44 seconds |
|
Settling time |
1.11 seconds |
15.5 seconds |
|
Overshoot |
8.07 % |
5.87 % |
|
Peak |
1.08 |
1.06 |
|
Gain margin |
Inf dB @ NaN rad/s |
Inf dB @ NaN rad/s |
|
Phase Margin |
74.1 deg. @ 0.58 rad/s |
74.1 deg. @ 0.366 rad/s |
|
Closed-loop stability |
Stable |
Stable |
Table 2. 3: The Ziegler-Nichols tuning method robustness and performance of the DC motor
Table 2.4 below illustrate the robustness and performance of the PID controller of the DC motor as depicted from MATLAB for Chien-Hrones-Reswick tuning method
|
Tuned |
Baseline |
|
|
Rise time |
19.1 seconds |
4.44 seconds |
|
Settling time |
28.6 seconds |
15.5 seconds |
|
Overshoot |
0.0405 % |
5.87 % |
|
Peak |
1.0 |
1.06 |
|
Gain margin |
Inf dB @ NaN rad/s |
Inf dB @ NaN rad/s |
|
Phase Margin |
78.0 deg. @ 0.0919 rad/s |
74.1 deg. @ 0.366 rad/s |
|
Closed-loop stability |
Stable |
Stable |
Table 2. 4: The Chien-Hrones-Reswick tuning method robustness and performance of the DC motor
Conclusions.
In this part of the assignment, it deals with the control system of armature-controlled DC motor. The transfer function is given with the parameter, the system is found to be stable since the poles were found to be on the left hand side near the origin. For the PID turning, the results illustrates that each technique has its specific merits as compared to the other method. Taking the specified DC motor speed control transfer function in equation (2.11), it shows that the Ziegler-Nichols method produces a faster response of the system having acceptable overshoot whereas Chien-Hrones-Reswick method of tuning the PID has a smaller overshoot having suitable system transient responses.
Part a: Determining the open loop and closed loop Z-domain transfer functions of the discretized system.
The system discussed in section 2, can be discretized into open loop and closed loop Z-domain transfer functions. Here, the transfer function in equation (2.11) is changed from the continuous Laplace S-domain to the discrete z-domain by use of MATLAB. The software is used to attain the mentioned conversion via use of the c2d command. The MATLAB command needs three (3) parameters: a system model (given by equation 2.11), the type of hold circuit and the sampling time. In this part of assignment, since the type of hold circuit is not given, then ZOH (Zero Order Hold) will be assumed.
The Matlab code for is given below.
From the MATLAB, and assuming that the sampling time is 0.001 seconds, the open loop discrete time transfer fiction is given in equation 3.1 below as:
And for the closed loop discrete time transfer fiction is given in equation 3.2 below:
It can be observed that, the z-domain of a closed loop and open loop is the same, since equation (3.1) is equal to equation (3.2).
Part b: The root locus analysis.
Using three (3) different sapling time e.g. Ts1=0.0001, Ts2=0.001 and ts3=0.01 seconds, the following code performs the root locus of each time and finds the pole locations.
Figure 3.1 illustrates the root locus of discretized system with sampling time of 0.0001 seconds
.
Figure 3. 1: Root locus of discretized system with Ts=0.0001 s
The pole location is: p1=1.000, p2=0.9996
Figure 3.2 llustrates the root locus of discretized system with sampling time of 0.001 seconds.
Figure 3. 2: Root locus of discretized system with Ts=0.001 s
The pole location is: p1=0.9998, p2=0.9963.
Figure 3.3 illustrates the root locus of discretized system with sampling time of 0.01 seconds.
Figure 3. 3: Root locus of discretized system with Ts=0.01 s
The pole location is: p1=1.000, p2=0.9996
Using MATLAB, the PID controller can be used to discretize a continuous-time PID controllers or by creating a discrete-time PID controllers straight. With the pid commands, the methodologies used for discretizing the integral terms and the derivative terms can be independently specified. In MATLAB command, pidstd() is used to create a discrete-time controllers. The discrete time controller is given by the following equation 3.3 below (MathWorks, 1994-2018):
Where IF(z) and DF(z) are respectively integrator formula for the integrator and integrator formula for derivative filter given by:
In this section different discrete integrators formula are chosen by use of the DFormula and IFormula input (MathWorks, 1994-2018). In MATLAB software design and simulation sampled-data control systems can be done as shown in figure 3.4 below. The frequency domain consideration is used to design. This design leads to a pole-cancellation PID controls technique.
Figure 3. 4: Block representation of discrete PID controller (Fadali & Visioli, 2013).
The following MATLAB code is used in application of PID controller in discrete time with different sampling time as used in part b above.
From MATLAB, and
When Ts=0.0001 seconds, the followings can be deducted:
Sample time: 0.0001 seconds.
Discrete-time zero/pole/gain model
Zeros, z=-0.9999
Poles: p1=1.0000; p2= 0.9996; p3=0
kd = 1.0525e-09
When Ts=0.001 seconds, the followings can be deducted:
Sample time: 0.001 seconds.
Discrete-time zero/pole/gain model
Zeros, z=-0.9987
Poles: p1=.9998; p2= 0.9963; p3=0
kd = 1.0513e-07
When Ts=0.01 seconds, the PID controller does not work together with the system, since it becomes unstable. PI term in this part is essential for achieving the steady state zero error necessity while the PD term is used for accelerating the systems response (phillips, 2012).
Part d: Comparison between the s-domain PID and z-Domain PID.
In both controllers, they offer pneumatic or electronic functionality. PID controllers such as auto-tuning functions, MATLAB software structure selections whether to use interactive, parallel, series, cascaded, etc, and remotely networks configuration. In both cases, the controllers’ implementations has to cope with performance, robustness and stability of the system. In discrete Transform (or z-domain), algorithms has to compact with sufficient sampling in the digital forms, process wind-ups, avoidance of proportional and derivative kicks, velocity and positional implementations, bumpless parameter tuning, quantization effect in integral actions, etc. when these factors are met in discete form, the performance and robustness is higher than that of s-domain (ResearchGate, 2015). Here I esed adaptive PID control algorithm for comparison
By use of tic and toc command in MATLAb, it is obvous that the disct=rete form tkes less thie as compared to continuous time simulation of the given armature-controlled DC motor.
Part e: Conclusions.
Any modification required for the PID-controllers may be implemented with discrete PID-controllers. From part d above, it can be seen that the significant modifications for the integrator is the ant-windup-operations for saturating the actuator, a soft mode changes when switching from automatic to manual operations and/or manual to automatic, bumpless parameters value varies in adaptive PID algorithm (Also for self-tuning algorithms will behave the same) (Aalto, 2010). For example, when actuators saturate integrator grows continuously to high values. Therefore, when designing discrete PID controller some parameters are considered, although this type of PID is faster and more stable. For the armature-controlled Dc motor system provided, the performance is high and is stable as depicted in part c above.
Conclusions
The assignment consisted three (3) parts. In the first part, I was able to understand the major benefits and drawback of PID controllers. Some applications of PI and PID controller in day-todays life is discussed. Furthermore in this part, the advantages of using digital systems over continuous time systems. It is found that, discrete system are more proffered than continuous time system since discrete system has low error rate, low cost, high speed, e.t.c. Also, we found how motor angular velocity control can be relevant to complex systems in day today life.
In part two (2), the control of angular velocity of a DC motor shaft by varying an input voltage u is illustrated. Simulation of continuous time PID control system of a DC motor is done using MATLAB software. The system stability, robustness, response time and other characteristics of system are simulated. A tuning PID controller is then simulated using different metric to determine which metrics is better and preferred to control the DC motor. It is found that, the Ziegler-Nichols method produces a faster response of the system having acceptable overshoot while Chien-Hrones-Reswick method of tuning the PID has a smaller overshoot having suitable system transient responses.
In the last part of the system, the DC motor is discretized and simulated in MATLAB to compare with the continuous time system. Different metrics of discrete system is used to determine the system response, robustness, stability and easiness of implementation and the error rate of the system. It is found that, using discrete system, the DC motor is more stable, more robust, less response time and less error rate as compared to continuous time system. Although, in both continuous and discrete system, choosing the PID controller should be chosen carefully to provide a stable and robust system
References
Aalto, 2010. Discrete-time approximations of continuous controllers; discrete PID controller. s.l.:Lecture notes.
Abdulameer, A., Sulaiman, M., Aras, M. & Saleem, D., 2016. uning Methods of PID Controller for DC Motor Speed Control. Indonesian Journal of Electrical Engineering and Computer Science, 2 August, pp. 343-349.
Abu-Khalaf, M., Chen, R. & Turevskiy, A., 2009. PID Control Design Made Easy. Matlab digest.
Anon., 2016. PID control strengths and limitations. [Online]
Available at: https://forum.allaboutcircuits.com/threads/what-are-the-pid-control-strengths-and-limitations.133881/
[Accessed 18 February 2018].
Anon., 2017. DC Motor Speed: Simulink Modeling. [Online]
Available at: https://ctms.engin.umich.edu/CTMS/index.php?example=MotorSpeed§ion=ControlPID
[Accessed 15 February 2018].
Anon., n.d. Introduction to Discrete-Time Control System. [Online]
Available at: https://wikieducator.org/images/8/82/Chp_1.pdf
[Accessed 20 February 2018].
Deka, J. & Haloi, R., June, 2014. Study of Effect of P, PI Controller on Car Cruise Control System and Security. International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, III(6), pp. 980-987.
Dorf & Bishop, 2001. Modern Control Systems. 9th ed. Inc: Prentice-Hall.
Fadali, M. S. & Visioli, A., 2013. Digital control engineering : analysis and design. 2nd ed. USA: Academic Press.
Gamazo-Real, J. C., Vázquez-Sánchez, E. & Gómez-Gil, J., July, 2010. Position and Speed Control of Brushless DC Motors Using Sensorless Techniques and Application Trends. Articles from Sensors (Basel, Switzerland).
KUO, B. C. & HASELMAN, P., 2014. Matlab tools for control systems. Analysis and design. s.l.:Prentice Hall, .
MathWorks, 1994-2018. PID controller in standard form. [Online]
Available at: https://www.mathworks.com/help/control/ref/pidstd.html
[Accessed 18 February 2018].
Melkin, C., 2017. DC Motor Speed: Root Locus Controller Design. [Online]
Available at: https://ctms.engin.umich.edu/CTMS/index.php?example=MotorSpeed§ion=ControlRootLocus
[Accessed 15 February 2018].
Novotecknik, 2009. P-I-D Controller:. Introduction to PID control, Volume III.
OGATA, K., 2013. Discrete time control systems. s.l.:Prentice Hall.
Owen, F., 2012. Designing and tuning PID controllers.. In Control Systems Engineering A Practical Approach, pp. 1-41.
Phillips, C. & Harbor, R., 2000. FEEDBACK CONTROL SYSTEMS. 4th ed. New Jeresy: Prentice-Hall.
phillips, H., 2012. Discrete PID Control. UK: s.n.
ResearchGate, 2015. Discover scientific knowledge at ResearchGate. [Online]
Available at: https://www.researchgate.net/post/What_is_the_difference_between_continuous_and_discrete_PID_controllers
[Accessed 18 February 2018].
sailan, K. & Kuhnert, K.-D., 2015. DC Motor Angular Position Control using PID Controller for the porpuse of controlling the Hydraulic Pump. Germany, s.n.
Thomas, N. & Poongod, P., 2009. Position Control of DC Motor Using Genetic Algorithm Based PID Controlle. London, U.K, s.n.
WILLIAMSON, D., 2015. Digital control and implementation.. s.l.: Prentice Hall, .
Ziegler, J. & Nichols, N., 2000. Optimum Settings for Automatic Controllers. Transactions of the American Society of Mechanical Engineers (ASME), Volume 64, pp. 759-768.
Essay Writing Service Features
Our Experience
No matter how complex your assignment is, we can find the right professional for your specific task. Contact Essay is an essay writing company that hires only the smartest minds to help you with your projects. Our expertise allows us to provide students with high-quality academic writing, editing & proofreading services.
Free Features
Free revision policy
$10Free bibliography & reference
$8Free title page
$8Free formatting
$8How Our Essay Writing Service Works
First, you will need to complete an order form. It's not difficult but, in case there is anything you find not to be clear, you may always call us so that we can guide you through it. On the order form, you will need to include some basic information concerning your order: subject, topic, number of pages, etc. We also encourage our clients to upload any relevant information or sources that will help.
Complete the order form
Once we have all the information and instructions that we need, we select the most suitable writer for your assignment. While everything seems to be clear, the writer, who has complete knowledge of the subject, may need clarification from you. It is at that point that you would receive a call or email from us.
Writer’s assignment
As soon as the writer has finished, it will be delivered both to the website and to your email address so that you will not miss it. If your deadline is close at hand, we will place a call to you to make sure that you receive the paper on time.
Completing the order and download