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# ENG319 Analogue Control System Design SUSS Assignment Sample Singapore

ENG319 Analogue Control System Design course is a third-year undergraduate course offered by the Department of Electrical and Electronic Engineering at SUSS. The course covers the design of analogue control systems, with an emphasis on feedback control theory. The course is divided into two parts: Part A and Part B. Part A covers the design of linear control systems, while Part B covers the design of nonlinear control systems.

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### Assignment Task 1: Discuss the basic control theory, Laplace transform, transfer function and state-space equation of control systems.

Control theory is the branch of engineering and mathematics that deals with the design and analysis of systems that are capable of regulating the behavior of dynamic systems. A control system is a set of interconnected components that are designed to maintain a desired output by controlling the input to the system.

The Laplace Transform is an important mathematical tool used in control theory to transform a time-domain signal into a frequency-domain representation. The Laplace Transform of a function f(t) is given by:

F(s) = ∫[0, ∞] f(t) e^(-st) dt

where s is a complex variable and e^(-st) is the exponential function.

The Laplace Transform is useful in control theory because it allows us to analyze the behavior of a system in the frequency domain. This is important because many physical systems are more easily described in terms of their frequency response rather than their time response.

The transfer function of a system is a mathematical representation of the relationship between the input and output of a system. It is defined as the Laplace Transform of the system’s output divided by the Laplace Transform of its input:

H(s) = Y(s)/X(s)

where H(s) is the transfer function, Y(s) is the Laplace Transform of the system’s output, and X(s) is the Laplace Transform of the system’s input.

The transfer function is important in control theory because it allows us to design controllers that can modify the behavior of a system by modifying its input.

The state-space representation of a system is another important mathematical tool used in control theory. It is a set of first-order differential equations that describe the behavior of a system in terms of its state variables. The state-space representation is given by:

x'(t) = Ax(t) + Bu(t)

y(t) = Cx(t) + Du(t)

where x(t) is a vector of state variables, u(t) is the system’s input, y(t) is the system’s output, A is the system’s state matrix, B is the input matrix, C is the output matrix, and D is the direct transmission matrix.

The state-space representation is useful in control theory because it provides a more complete and flexible description of a system’s behavior than the transfer function. It allows us to design controllers that can modify the behavior of a system by modifying its state variables directly.

### Assignment Task 2: Examine the feedback analogue control systems using transient and steady-state responses.

An analog control system is a feedback control system that uses continuous signals, such as voltage or current, to control a physical process. The system typically consists of a controller, which receives input signals from sensors, and actuator, which performs actions on the process.

Transient Response:

The transient response of a control system is the behavior of the system in the time domain after a sudden change in the input. It describes how the system responds to changes in the input signal over time. The transient response can be analyzed using several parameters, including rise time, settling time, and overshoot.

• Rise time: The rise time is the time taken by the system output to rise from 10% to 90% of its steady-state value. A smaller rise time indicates faster system response.
• Settling time: The settling time is the time taken by the system output to reach and remain within a specified range, typically within 2% of the steady-state value. A shorter settling time indicates a faster system response.
• Overshoot: Overshoot is the maximum percentage by which the system output exceeds its steady-state value. High overshoot can lead to instability in the system.

Steady-State Response:

The steady-state response of a control system is the behavior of the system after it has stabilized, where there are no changes in the input. The steady-state response can be analyzed using several parameters, including steady-state error, gain margin, and phase margin.

• Steady-state error: The steady-state error is the difference between the desired output and the actual output of the system in the steady state. The goal is to minimize steady-state error to achieve accurate control of the process.
• Gain margin: The gain margin is the amount by which the gain of the system can be increased before the system becomes unstable. A higher gain margin indicates greater stability of the system.
• Phase margin: The phase margin is the amount by which the phase shift of the system can be increased before the system becomes unstable. A higher phase margin indicates greater stability of the system.

### Assignment Task 3: Determine the transfer function, stability, gain and other parameters of open and closed-loop analogue control systems.

Determining the transfer function, stability, gain, and other parameters of open and closed-loop analogue control systems involves analyzing the system’s mathematical model and characteristics. Here are the steps involved:

1. Develop the mathematical model of the system: This involves representing the system’s components and their interconnections using mathematical equations. The model could be represented in various forms such as differential equations, transfer functions, state-space equations, or block diagrams.
2. Find the transfer function of the system: The transfer function of a system is the ratio of the Laplace transform of the output to the Laplace transform of the input when all initial conditions are zero. It can be obtained from the mathematical model by taking the Laplace transform of the output and input signals and then solving for the transfer function.
3. Determine the stability of the system: The stability of a system refers to its ability to return to a steady state after being subjected to disturbances. It is determined by analyzing the poles of the transfer function, which correspond to the roots of the characteristic equation. If all the poles are in the left half of the complex plane, the system is stable; otherwise, it is unstable.
4. Find the gain of the system: The gain of a system is the ratio of the output signal amplitude to the input signal amplitude at steady-state. It can be calculated by evaluating the transfer function at s=0.
5. Analyze other parameters of the system: Other parameters that can be analyzed include the frequency response, time response, damping ratio, natural frequency, and overshoot.
6. For a closed-loop system, determine the closed-loop transfer function, which is obtained by connecting the feedback path from the output to the input of the system.
7. Analyze the closed-loop system using the same methods as in the open-loop case. The stability of the closed-loop system is determined by analyzing the poles of the closed-loop transfer function.

Overall, determining the transfer function, stability, gain, and other parameters of open and closed-loop analogue control systems requires a good understanding of the system’s mathematical model and the tools used to analyze it, such as Laplace transforms, poles and zeros, and frequency response analysis.

### Assignment Task 4: Calculate phase-margin, settling time, overshoot and other parameters associated with analogue control systems.

Calculating parameters associated with analog control systems requires specific information about the system, such as the transfer function, input signal, and other characteristics. In general, the following are some of the most common parameters that can be calculated:

1. Phase Margin: The phase margin is a measure of the stability of a control system. It is defined as the amount of additional phase shift that can be added to the system before it becomes unstable. The phase margin is typically measured in degrees, and a larger phase margin indicates a more stable system.
2. Settling Time: The settling time is the time required for the system output to reach and remain within a specified percentage of the final value. Typically, the settling time is defined as the time required for the output to settle within 5% or 2% of the final value.
3. Overshoot: The overshoot is the amount by which the system output exceeds its steady-state value before settling. Overshoot is typically expressed as a percentage of the final value, and a larger overshoot indicates a more oscillatory response.
4. Gain Margin: The gain margin is a measure of how much additional gain can be added to the system before it becomes unstable. It is typically measured in decibels, and a larger gain margin indicates a more stable system.
5. Bandwidth: The bandwidth is the range of frequencies over which the system can respond to input signals. A larger bandwidth indicates a faster and more responsive system.

To calculate these parameters, it is necessary to know the transfer function of the system and the type of input signal being used. The specific method for calculating each parameter may vary depending on the particular system and input signal being used.

### Assignment Task 5: Draw the signal flow graph/ Nyquist plot/ Bode plot for analogue control systems.

Signal flow graph:

1. A signal flow graph is a graphical representation of the system’s transfer function. It shows the flow of signals through the system and the mathematical relationships between the input and output signals. The steps involved in drawing a signal flow graph are as follows:
• Identify the input and output signals of the system.
• Identify the transfer functions or blocks that describe the system’s behavior.
• Draw the blocks in a sequence that represents the signal flow from input to output.
• Label the inputs and outputs of each block.
• Connect the blocks with arrows that represent the signal flow.

Nyquist plot:

1. A Nyquist plot is a graphical representation of the frequency response of a system. It shows the magnitude and phase of the system’s transfer function as a function of frequency. The steps involved in drawing a Nyquist plot are as follows:
• Determine the transfer function of the system.
• Rewrite the transfer function in terms of complex variables.
• Evaluate the transfer function at different frequencies and plot the resulting complex values on a polar plot.
• Connect the plotted points to form a curve.
• Determine the stability of the system by analyzing the curve’s behavior as frequency approaches infinity.

Bode plot:

1. A Bode plot is a graphical representation of the frequency response of a system. It shows the magnitude and phase of the system’s transfer function as a function of frequency in a logarithmic scale. The steps involved in drawing a Bode plot are as follows:
• Determine the transfer function of the system.
• Rewrite the transfer function in terms of complex variables.
• Evaluate the transfer function at different frequencies and plot the resulting magnitude and phase values on a logarithmic plot.
• Label the plot’s frequency axis in decades or decades per second.
• Analyze the plot to determine the system’s gain and phase margins, crossover frequency, and bandwidth.

### Assignment Task 6: Design an analogue control system meeting the required specifications.

In order to design an analogue control system, we first need to determine the required specifications for the system. Here are some possible specifications for a control system:

1. Control parameter: The system must be able to control a particular parameter, such as temperature, pressure, or flow rate.
2. Setpoint: The system must be able to set a desired value for the control parameter.
3. Accuracy: The system must be able to maintain the control parameter within a certain range of the setpoint, with a specified degree of accuracy.
4. Response time: The system must be able to respond quickly to changes in the control parameter, with a specified response time.
5. Stability: The system must be stable, meaning that it must not oscillate or overshoot excessively.

Based on these specifications, here is an example of an analogue control system design:

1. Control parameter: Temperature
2. Setpoint: 100 degrees Celsius
3. Accuracy: Maintain temperature within +/- 1 degree Celsius of setpoint
4. Response time: Respond to changes in temperature within 1 second
5. Stability: No oscillation or overshoot exceeding +/- 2 degrees Celsius

To achieve these specifications, we could use a proportional-integral-derivative (PID) controller, which is a common type of analogue control system. A PID controller uses feedback to adjust the control parameter based on the difference between the setpoint and the actual value of the parameter.

Here is a possible design for the PID controller:

1. Proportional (P) controller: This component responds to the current error between the setpoint and the actual temperature. The output of the P controller is proportional to the error, and it helps the system to quickly respond to changes in temperature. The gain of the P controller is set to 0.5.
2. Integral (I) controller: This component responds to the accumulated error over time. The output of the I controller is proportional to the integral of the error, and it helps the system to eliminate any steady-state error. The gain of the I controller is set to 0.1.
3. Derivative (D) controller: This component responds to the rate of change of the error. The output of the D controller is proportional to the derivative of the error, and it helps the system to anticipate changes in temperature. The gain of the D controller is set to 0.01.

The outputs of the P, I, and D controllers are summed together to generate the final control signal. This signal is then sent to a control valve, which adjusts the flow of a heating or cooling medium to maintain the temperature within the desired range.

### Assignment Task 7: Use software tools to study analogue control system design.

There are various software tools available that can be used to study and design analogue control systems. Some of the commonly used ones are:

1. MATLAB: MATLAB is a popular software tool for control system analysis and design. It offers various functions and tools for modelling, simulation, and analysis of control systems. It has a user-friendly interface and allows for easy implementation of various control techniques.
2. Simulink: Simulink is a graphical simulation and modelling environment that is integrated with MATLAB. It allows for the creation of block diagrams for control system design, simulation and testing. It is widely used for developing and simulating control systems in industries such as aerospace, automotive, and manufacturing.
3. LTSpice: LTSpice is a free, open-source software tool that is used for simulating analogue circuits. It is widely used for the design and analysis of power electronics circuits and control systems. It offers a user-friendly interface and allows for easy simulation and testing of circuits.
4. PSpice: PSpice is a software tool that is used for the simulation and analysis of analogue circuits. It offers a wide range of modelling features and allows for the implementation of various control techniques. It also offers a user-friendly interface and allows for easy simulation and testing of circuits.
5. Proteus: Proteus is a software tool that is used for the simulation and design of electronic circuits. It offers various simulation modes, including microcontroller simulation and virtual instrumentation simulation. It also offers a user-friendly interface and allows for easy implementation of various control techniques.

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