Control Systems, PID Controller & Discretization

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Please be advised that this is an INDIVIDUAL coursework. It is acceptable to exchange ideas, but the report, including the descriptive Section 1 and all the design, analytical and simulation analysis must be your own individual work, written with your own words and very clearly referenced when citations are made.

The DC motor system of Section 2 is now discretised. Use Matlab to determine the open loop and closed loop Z-domain transfer functions of the discretised system.

1. b) Perform a root locus analysis of the discretised system in the Z domain. Select three different values of the sampling period to evaluate the effect of discretisation on stability.
2. c) Apply PID control in discrete time using Matlab simulations with the same sampling time periods as in 3b.
3. d) Compare the performance of PID control of 3c with an equivalent continuous time PID implementation using performance metrics of your own choice.
4. e) Provide your conclusions from this simulation analysis.

Backup Policy for Coursework

Control is crucial in industrial applications since it would be quite challenging to perform the tasks as assigned to a machine accurately. The controls systems are meant to direct, instruct, and regulate the tools and devices in the industries. The control systems are applied in mechanical, automotive, electrical, and pneumatic systems. The mechanical systems, for instance, need a control where the input and intended output are controlled. For all the aforesaid different types of systems, there are two classifications in control engineering. The PID controller finds application in chemical industries based on its rate of success in practical applications of this nature. The systems can either be open-loop systems or closed-loop systems.

The systems are represented using block diagrams to show the processes and their interaction with inputs and outputs. For an open-loop system, the information is fed into the plant or system which performs processing, and the result or manufacturing is yielded. One typical example is the washing machine. In the open loop system, the input and output are not related such that in a washing machine, the efficiency of cleaning does not depend on the information instead of the present time.

Another typical example of the open loop system is the burglar alarm system. The system operates by detecting motion or movement within the vicinity of a home using a sensor. Any change recognized triggers an alarm. The alarm can only be stopped manually.

The Burglar alarm system in an open loop system

One critical caveat experienced with the open loop control system is the inability to make automatic adjustments especially when the input changes. The free loop system is, therefore, not suitable for use as a sophisticated control system in industrial applications. Some of the outputs obtained may be of very high magnitude or even negligible magnitudes which could affect the operation of systems. This call for a feedback loop to enable system control. The introduction of a feedback loop in order makes the system a closed loop control system. The output of the control system is used to adjust the input signal. The system compares the output with the intended result, and it takes action to improve the input signal to ensure that the expected output is obtained. The use of the feedback loop increases the accuracy of the production.

The feedback can either be positive or negative depending on the magnitude of the output. The positive feedback seeks to make the yielded output deviates from the present command status while the negative feedback causes the new production to move closer towards the current command status (Gu, et al., 2013). For instance, when an amplifier is put close to the microphone, the input volume kept increasing hence the interference increased causing a lot of noise pollution. The feedback loop may require the input reduced by moving the mic away from the amplifier. The modern industrial applications have embraced the closed-loop control systems as they need automation of the arrangements. Some of the conventional closed-loop system applications are air conditioners, refrigerators, automatic rice cookers, electronic ticketing machines. Due to the complexity of the systems, they are more expensive and complicated to design. One typical example is the air conditioner,

Importance of Control Systems in Industrial Applications

In the industrial applications, the PID controller is commonly implemented to provide control in both small-scale and large-scale devices.

The proportional-integral-derivative controller is a control loop feedback mechanism. The PID algorithm has three major components namely, the proportional component, the integral part, and the derivative. Each element affects the output signal to get the optimal performance of the system.

From the control system, the PID algorithm revolves around correcting the error. The error is usually the difference between the process variable and the setpoint. Some drawbacks are seen by the very high value of proportional control leads to the oscillation of the Process variables. The controller tends to generate an offset value. These controllers increase the maximum overshoot of the system. The yielded signal tends to have the offset minimized by the essential part. The preceding section and the recent trajectory of the controller error have no impact on the proportional term as it is.

The equivalent corrects instances of failure, the correct integral accumulation of failure, and the derivative corrects the present error in respect to the error in the previous check. The effect of the derivative is to counter the overshoot by the proportional and integral parts. When the error is significant, the two coefficients affect the controller yield. The controller response will imply a change in the error as fast as possible hence the need for a secondary function.

The algorithm of a PID controller is given as,

Unfortunately, further increase in the constant proportional causes a tendency to oscillation. The effect of adding the integral component is that the strength of the integral action increases with a decrease in integral time. The integral component deals with the steady state error. The PI controller encounters oscillations, and the introduction of the second component causes damping to increase. The damping of oscillations increases with increase in the derivative time but decreases again when the derivative time becomes too large. The proportional response is given as,

The proportional constant used need not be very high as it may result in the system instability. Similarly, when the proportional gain is too low, the control action may tend to be negligible hence fails to respond actively to the system disturbances. The proportional controller has the effect of minimizing the rise time, and it may reduce the steady-state error but not eliminate it. The commonly used value is such that the proportional band is expressed as a percentage,

Difference Between Open-Loop and Closed-Loop Systems

The integral controller has an impact on the magnitude of the error as well as the duration of the failure. The fundamental is the sum of the instantaneous mistakes over a period. The essential component eliminates the steady-state error, but it introduces the transient response and worsens it. The comprehensive control is given as,

The secondary component is obtained by determining the slope of the error over a given period and multiplying the value with the gain factor. The controller slows the rate of change of the controller output. The controller has the effect of increasing the stability of the system, it reduces the overshoot of the yielded output, and improves the transient response. It is given as,

The exact nonlinear dynamics of a humanoid with the constraint is derived as shown in the equations below,

There is a relationship observed between the input power, the external power, and the power consumed through the resistor when the law of conservation of power is implemented such that,

The linear system is linear if the output of the scaled sum of two input signals is the equivalent scaled sum of outputs. The system is time-invariant if the system output is the same and given the same input signal regardless of time. The system can be tuned using the different tuning methods available. A new approach to the control of the electric drives with permanent magnet synchronous motors, based on feedback linearization has been developed and experimentally proven. Three various prescribed dynamics to speed demands were achieved.

The sampled data signal is a discrete time signal resulting from sampling a continuous time signal. The digital signal is obtained by two processes of sampling and quantizing. The digital controls are used for achieving optimal performance in the form of maximum productivity, maximum profit, minimum cost, and the use of minimum energy in production activities. The advantages of the digital controls are that it has reduced price as there is the bulk manufacturing of the control systems, they are flexible in response to design changes, noise immunity, and they are more useful in the modern control systems. Some analog control systems exhibit good performances as compared to the digital control systems. Digital control systems tend to introduce a delay in the loop. The loop contains both analog and digital signals to provide a means for conversion from one form to another to be used by each sub-system. The systems are designed in the Z-domain. Hence, they must be converted from the S-domain to the Z-domain for discretization of the control systems (Dougherty & Cooper, 2003).

PID Controller: Functionality and Components

The use of online optimization and advanced control. The new system allows for non-linear systems to be controlled in the complex multi-variable interactions; operating constraints are put in place such as safety limits, input saturation constraints, and product quality constraints. The control over wide operating range are the process non-linearities and the changing process parameters or conditions. The conventional approach is the multi-loop PI which is difficult to tune. The ad-hoc constraint handling using logic programming and the lack of coordination. The gain scheduling handles the non-linearity of systems. The use of multi-loop control in industrial processes for multivariable and exhibit strong interaction among the variables. The stability of the system is determined when the system has its characteristic equation to zero (Arbogast & Dougherty, 2003).

In a nutshell, many industries are interested in controlling the temperature level of their process tanks. The feedback loop of the PID measures the actual level of the parameter under study and compares it to the input or reference value to determine the error. The PID controller finds application in many industries and fields where process and plants have been installed. There are new and alternative models for the implementation of the system. The digital PID controllers are quite easy to implement and operate as they do not require a specialist or very highly trained person to run. Other alternatives such as the pneumatic controller are applied for the pneumatic system control to ensure that systems are controlled cost-effectively. According to many researchers and practical tests carried out in the industry, it is recommended that the PID controller is utilized for processes with low to medium order plant transfer functions with small time delays.

The conventional control scheme has multiple single inputs and single output PID controllers used for the control of plants. The consequence of loop interaction is the lack of coordination between different PID loops. Neighboring PID loops can cooperate with each other or end up opposing or disturbing each other. Such advanced control systems can be used to solve the Tennessee Eastman problem. The model predictive control does multivariable control based on the on-line use of the dynamic model. The modified form of classical optimal control problem can systematically and optimally handle multivariable interactions, operating input and output constraints, and process non-linearities. Another approach is the use of the data-driven models. The development of linear state space and transfer models starting from the first principles or gray box models is quite an abstract proposition.

Advantages of Digital Control Systems

The Research and Development control sections have focused on the application of the new approach to enhance the control systems for the outer loop based on MRAC or SMC to improve the precision of control. The discrete time signals are defined only at specific time instances. The discrete time signals have amplitudes between two consecutive time instants which is not determined but only on the particular time instance.

• Electrical and mechanical equations for the system

The DC motor electrical-mechanical equivalent circuit is as illustrated below,

The system introduces a back electromotive force at the armature section. The back electromotive force is directly proportional to the angular velocity of the rotor within the motor section. The motor generates a torque which is linearly related to the armature current. It is given that at constant input voltage, the armature current and the angular velocity and torque are constant.

The mechanical behavior of the systems is further analyzed using the equation,

The equations demonstrate the dynamic nature of the DC motor. When the electrical term is neglected, the magnitude of the system reduces as the mechanical time constant is left in the equation as the governing factor,

When the equation for armature current is substituted to the mechanical equation, we obtain,

Having obtained the mechanical and electrical system functions for the DC motor, the transfer function of the system can be obtained as,

Two external forces come to affect the system namely the input voltage and the generated torque at the armature. These two forces tend to exhibit linear proportionality such that,

The back emf, e, is proportional to the angular velocity of the shaft by a constant factor.

The Laplace transform of the system equation is given as,

The open loop transfer function is obtained by eliminating the electrical component from the two equations. The rotational speed is considered as the output, and the armature voltage is regarded as the input,

Continuous-time state-space model.

The system has two poles located on the left as

When poles are on the right-hand side, the system is said to be unstable. In this case, the poles are on the left of the imaginary axis hence the system is stable. The analysis of the root locus is determined mainly by the location of the poles and zeros.

• Discussion (PID Control)

Ease of implementation – it is feasible and easy to implement. The PID gains can be designed by the system parameters to achieve the yield and estimations precisely. The gain of PID controller can be determined simply by the system tracking error, and the system is analyzed under the requirement set up.

Discretization of Control Systems

Stabilization requirements -the open loop system is an unstable system as the output obtained is not the desired output. Further, the output yielded may tend to be of huge magnitude hence destroying the cascaded sub-sections attached to the output. The control system corrects the magnitude by ensuring the output provided is corrected until it is achieved as the desired yield.

Performance – the closed-loop control system guarantees that the efficiency of the system is attained to close to an ideal value. The performance of a system is tested based on the output obtained. The system is tested on may scale, and it should perform well and ensure that the system parameters are stable all through.

Robustness – a system is considered robust when it can accommodate the uncertainties and disturbances in the surrounding. The system may not be in a position to resist the noises and hence depict low robustness as demonstrated in the open-loop systems. The PID may have a low robust ability compare to some of the modern alternatives of the PID used in the various industrial applications. The robust controllers enable the system to encounter the challenges of operating in a very dynamic uncertain environment in the industrial sector. Some of the environmental disturbances are weather changes, temperature variations, and especially power fluctuations such as the power surge (Arbogast & Dougherty, 2003).

Energy consumption – as much as an industry wishes to automate their processes and plants, there is need to consider the consumption of energy by the equipment installed. The open loop does not consider the energy consumed or even regulate it; rather it follows the initial settings. For instance, when using an air conditioner, the system should detect temperature changes and change modes when suitable to reduce the energy consumption levels.

Steady state error – the actuators can saturate and hence the error continues to accumulate. When the controller regains control, the response may end up being exaggerated. The proportional controller minimizes the steady-state error in the system, but the introduction of the integral component eliminates the effects of the steady state error in the plant.

• Ziegler Nichols PID tuning method (MATLAB implementation)

In practice, when the Ziegler-Nichols tuning is implemented in an industrial plant the system transfer function of the plant is unknown to the control engineers. The control engineers are required to fine tune the device a little bit to attain the appropriate control for the system. The system plant, G(s), is a second order system that has a lag and integration for its transfer function. There are some tuning methods used in controlling system plants such as:

• Ziegler-Nichols open loop method (yields low results for percent overshoot)
• CHR method for 0% overshoot
• Ziegler-Nichols closed-loop method (Has a shorter settling time)

Non-Linear Control of Multi-Variable Systems

The measuring output for an impulse or step input and it uses system models may be determined using system identification techniques. These PID controllers are tuned on-site due to machine and process variations. Every plant has its settings and algorithms used to set up the controllers. As highlighted, a few tuning rules can be by-passed using trial and error methods to get the right fit for controlling a unique model and process. About 95 percent of the closed-loop industrial processes use the Ziegler-Nichols method to control the system plant,

The Ziegler-Nichols method uses the Monte Carlo method to develop tuning rules. The system enables the tuning of a system model using a computer simulation system or a digital input point when the specific transfer function of the system and controller is not known to the engineer. A number of the industrial processes are non-linear and dynamic as a result of exposure to external noises. The introduction of external disturbances makes the system quite difficult to work with. An alternative method implemented is the good gain method. The method is obtained from a series of laboratory experiments. When compared to the Ziegler-Nichols closed loop method, it does not require the control system to be brought into sustained oscillations in the tuning phase. Unlike the Nichols tuning method which is implemented on the PID controllers, the good gain method is used in PI controllers.

• Analysis of controller performance when exposed to external disturbances

When using the PID, the proportional controller, integral component, and the derivative component were involved as indicated in the setup. The different values used in the control show that the steady state error is minimized when the proportional gain factor is greater than one, it is further eliminated when the integral component is introduced. When the integral component was raised from 0 to 5 the oscillatory portion of the output signal was eliminated and the steady state error as well. The derivative controller stabilizes the system completely to ensure that the system is robust, stable, and performs as intended. The components are able to curb the noise disturbances in the environment while correcting the error at the PID input from the summer. The system is tuned using the forward Euler tuning to ensure that the component values set are in line with the required values to ensure stability of a system. It was observed that a very large value on the proportional gain factor may cause more damage than good as the signal tends to increase in magnitude to very high levels. The integral value seeks to stabilize this by eliminating the steady state error. The Ziegler-Nichols method is used in the plants with neither the integrators nor the dominant complex-conjugates poles. The unit step response models an S-shaped curve or the reaction curve. The tuning method use the gain estimator chart to determine the reaction of the different controller modes,

All contrllers seek to solve two problems: faster responses imply poor system stability while the better system stability states are only attained when the response is slow. The control systems aim at reaching a compromise where there is acceptable stability and medium fastness of response. The Ziegler-Nichols method is implemented in the following transfer function models for different levels of systems,

Another alternative method is the Tyreus-Luyben method. It is implemented like the Ziegler-Nichols method but has different final controller settings. The controllers are implemented for the PI and the PID controllers unlike the Ziegler-Nichols method which is suitable for PID controllers solely. The controller setting for the Tyreus-Luyben method are,

Another method was proposed by Chien, Hrones, and Reswich is the CHR method which modifies the Ziegler-Nichols open-loop method. It achieves a very quick response with or without 20 percent overshoot in the output signal. To use this model, it is required that the engineer or designer determines the first order and the dead time model of the system. To achieve the faster response with very low overshoot, the proportional gain constant and the derivative time is less while the integral component time is larger such that the integral action is amplified over the proportional and the derivative. The later components are responsible for reducing steady-state error and rise time respectively.

• Discretized system. Matlab for open and closed loop z-domain transfer functions.

The first step is to convert the system from continuous to discrete mode. The analysis of the system can, after that, be done in the discrete mode. There are analog and digital controllers implemented in series with the industrial plants or processes. The digital controllers are being used more than the analog ones in the modern age. There are several advantages of the digital controls such as the reduced costs in the bulk manufacturing of the controller equipment. These controllers tend to be very flexible in response to the design changes, noise immunity, and they are more useful in the modern control systems. Some of the analog controllers tend to exhibit very good performance as compared to the digital controllers. The systems that utilize the digital controllers are designed in the z-domain. Hence, they must be converted from the continuous time domain or the s-domain to the discrete time domain or z domain.

• PID control in discrete conditions

Signals in real-life applications are set up in continuous time while for easier analysis they need to be converted to digital mode. The discrete time mode occurs when the amplitude between two consecutive time instants are not defined. The sampled data signal is obtained by sampling the continuous time signals about a given period. These systems are used to attain the optimal performance. They are obtained in the form of the maximum productivity; they provide maximum profit at minimal manufacturing costs or minimal energy use. The design of the digital controllers is carried out in the z-domain. The discrete time signals have amplitudes between two consecutive time instants which is not defined but only on the particular time instance. The sampled data signal is a discrete time signal resulting from sampling a continuous time signal.

Signals carry energy from one position to another. Linear systems and non-linear systems are used to find out on the energy carried. Signals are patterns that form and the signals carry certain information. The continuous time signals are the world’s most common signals are the scale used is infinitesimal. Some of the parameter measured are the velocity of a mobile device as well as the voltage of the electrical circuits. The discrete time signals are sampled continuous time signals. Signals are considered periodic in that they tend to repeat themselves. The even and odd signals are described as those whose negative sign means differently. The pulse signals are those that are nearly completely zero and closely related to the short spike. A step signal, as the one used in the simulation, is zero to a certain time and then tends to be constant, thereafter. These attributes describe a list of useful signals which are input into systems and they produce output signals. The continuous time systems represent how continuous signals are transformed via differential equations. The discrete time systems represent how discrete signals are transformed via difference equations.

The digital signal is obtained by two processes of sampling and quantizing. The digital controls are used for achieving optimal performance in the form of maximum productivity, maximum profit, minimum cost, and the use of minimum energy in production activities. The advantages of the digital controls are that it has reduced cost as there is the bulk manufacturing of the control systems, they are flexible in response to design changes, noise immunity, and they are more useful in the modern control systems. Some analog control systems exhibit good performances as compared to the digital control systems. Digital control systems tend to introduce a delay in the loop. The loop contains both analog and digital signals to provide a means for conversion from one form to another to be used by each sub-system. The systems are designed in the Z-domain. Hence, they must be converted from the S-domain to the Z-domain for discretization of the control systems (Dougherty & Cooper, 2003).

• Discussion on the performance of PID control

Non-linear differential equations formulated in the magnetic field-fixed d, q coordinate system describe the permanent magnet SM for the system development. The discrete time two-phase oscillator produces the transformation matrix elements needed for the transformation blocks. A high gain proportional control law with voltage saturation limits was used for simulation. The bang-bang control law operating in the sliding mode is the switching strategy for the two-phase version and is satisfactorily determined. Electric drive with synchronous motor consists of a synchronous machine with nominal parameters (Dorf , n.d.).

• Simulation analysis conclusions

The PID controller implements a unity feedback loop. In some cases, it uses the varied feedback depending on whether a sensor has been used to read the actual output and sends it to the summer to check for errors. The continuous time signals are the world’s most common signals are the scale used is infinitesimal. Some of the parameter measured is the velocity of a mobile device as well as the voltage of the electrical circuits. The controller tends to generate an offset value. These controllers increase the maximum overshoot of the system. The yielded signal tends to have the offset minimized by an integral part when the integral element is added to the proportional element. The past section and the recent trajectory of the controller error have no impact on the proportional term as it is. The proportional corrects instances of error, the correct integral accumulation of error, and the derivative corrects the present error in respect to the error in the previous check. For many cases, we cannot obtain the same desired results in terms of theoretical and practical cases. For that project, we have to make theoretical power calculations for DC motors to obtain the desired DC motor speed. However, in practice, we could not obtain the same results as it is calculated theoretically. For that purpose, we have to use controllers to minimize the error between actual and theoretical results

References

“A General Approach to Control a Positive Buck Boost Converter to Achieve Robustness against Input Voltage Fluctuations and Load Changes,” IEEE Trans. Power Electronics, vol. 18, pp. 438-446, Jan 2003

Arbogast, J. & Dougherty, D., 2003. A multiple model adaptive control strategy for DMC. [Online]  Available at: https://ieeexplore.ieee.org/document/1243451/

Chien-Ming Lee et. al, “LabVIEW Implementaton of an Auto-Tuning PID Regulator via Grey-predictor,”
in Proc. IEEE-CISC, 2006, pp. 1-5

Dougherty, D. & Cooper, D., 2003. A practical multiple model adaptive strategy for single loop MPC. [Online]
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Gu, D.-W., Petkov, H. P. & Konstantinov, M. M., 2013. Robust Control Design with Matlab (R). 2nd Edition ed. London: Springer Nature.

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