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# ENG321 Digital Control System Design SUSS Assignment Sample Singapore

The ENG321 Digital Control System Design course introduces the design of digital control systems. The course covers the principles of digital control including z-transform analysis, transfer function representation, feedback control, and PID control. The course also covers the design of digital controllers using the LabVIEW Control Design and Simulation Module.

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### Assignment Brief 1: Discuss the discrete-time control system theory, transfer function and state-space equation of control systems.

Discrete-time control system theory refers to the study of control systems that operate in a discrete-time domain. Unlike continuous-time control systems, where signals and processes are continuous in time, discrete-time systems operate on signals and processes that are sampled at discrete time intervals. This is often necessary when dealing with digital control systems, where the signals are processed using digital circuits.

Transfer Function:

The transfer function of a discrete-time control system is a mathematical representation of the system’s input-output relationship. It is defined as the ratio of the system’s output to its input in the Z-domain, where Z represents the complex variable used in the system’s analysis. The transfer function is represented using the Z-transform of the system’s impulse response. The Z-transform is a mathematical function that converts a discrete-time signal into a complex function of Z. The transfer function is useful in analyzing the system’s stability, performance, and design.

State-Space Equation:

The state-space equation is another representation of a discrete-time control system. It represents the system’s dynamics as a set of first-order differential equations that relate the system’s state variables to its input and output signals. The state-space equation is usually written in matrix form, where the state variables, input signals, and output signals are represented as column vectors, and the system’s dynamics are represented using matrices. The state-space equation is useful in analyzing the system’s controllability, observability, and stability.

### Assignment Brief 2: Analyze the discrete-time control systems in the z-plane.

Discrete-time control systems are systems that operate on signals that are sampled at discrete time intervals. These systems are represented in the z-plane, where the z-transform is used to convert the discrete-time signal from the time domain to the z-domain.

In the z-plane, the poles and zeros of the transfer function are represented as points. The location of these points in the z-plane determines the stability and behavior of the system. Specifically, the stability of the system is determined by the location of the poles, which should be inside the unit circle for the system to be stable.

Furthermore, the behavior of the system can be analyzed by studying the frequency response of the system, which is represented by the magnitude and phase of the transfer function in the z-plane. The frequency response gives information about the gain and phase shift of the system at different frequencies, which can be used to design control strategies that optimize system performance.

In addition to analyzing the poles and zeros and frequency response, discrete-time control systems can be designed using various techniques such as pole placement, PID control, and state-space methods. These techniques allow for the design of controllers that can achieve desired system performance specifications such as stability, tracking, and disturbance rejection.

### Assignment brief 3: Determine the transfer function, stability and the transient/ steady-state response of a digital control system.

A digital control system can be represented by its transfer function, which relates the Laplace transform of the output to the Laplace transform of the input. The transfer function for a digital control system can be obtained by taking the z-transform of the difference equation that describes the system.

The stability of a digital control system can be analyzed by examining the location of the poles of the transfer function in the z-plane. If all the poles are within the unit circle in the z-plane, then the system is stable. If any pole lies outside the unit circle, then the system is unstable.

The transient response of a digital control system describes the behavior of the system as it approaches its steady-state response after a change in the input. The transient response is characterized by the time constant, damping ratio, and natural frequency of the system. The steady-state response of a digital control system is the output that results after the transient response has died out.

### Assignment Brief 4: Calculate settling time, feedback gain and other parameters associated with digital control systems.

The settling time and feedback gain are important parameters associated with digital control systems. Here are the steps to calculate them:

1. Determine the transfer function of the digital control system, which relates the output to the input.
2. Convert the transfer function to the Z-domain using the Z-transform.
3. Find the poles of the transfer function in the Z-domain. The poles are the roots of the denominator of the transfer function.
4. Calculate the settling time using the following formula:
settling time = -ln(0.05) / (real part of dominant pole)
The dominant pole is the pole with the largest magnitude, and its real part should be negative for the system to be stable.
5. Calculate the feedback gain using the following formula:
feedback gain = 1 / (1 + loop gain)
The loop gain is the product of the plant transfer function and the feedback transfer function.
6. Other parameters associated with digital control systems include rise time, peak time, and overshoot. These can be calculated using the step response of the system, which is the response of the system to a step input. The rise time is the time it takes for the response to rise from 10% to 90% of its final value. The peak time is the time it takes for the response to reach its maximum value. The overshoot is the percentage by which the response exceeds its final value before settling. These parameters can be obtained by analyzing the step response graphically or using formulas that relate them to the system’s poles and zeros.

### Assignment Brief 5: Construct the controllability matrix, observability matrix and other matrices associated with control system design.

In control system design, there are several matrices that are commonly used to analyze and design control systems.

1. Controllability matrix: The controllability matrix is a matrix that describes the controllability of a system. It is denoted by C and is defined as:

C = [B AB A^2B … A^(n-1)B]

where A is the system matrix and B is the input matrix. The controllability matrix is used to determine whether all the states of a system can be reached by applying suitable inputs. If the rank of the controllability matrix is equal to the number of states of the system, then the system is said to be completely controllable.

1. Observability matrix: The observability matrix is a matrix that describes the observability of a system. It is denoted by O and is defined as:

O = [C; CA; CA^2; … CA^(n-1)]

where C is the output matrix. The observability matrix is used to determine whether all the states of a system can be observed from the output. If the rank of the observability matrix is equal to the number of states of the system, then the system is said to be completely observable.

1. State feedback matrix: The state feedback matrix is a matrix that describes the feedback gain in a state feedback control system. It is denoted by K and is defined as:

K = R^-1B^TP

where R is the control weighting matrix, P is the solution of the algebraic Riccati equation, and ^T denotes the transpose. The state feedback matrix is used to determine the feedback gain in a state feedback control system.

1. Kalman filter gain matrix: The Kalman filter gain matrix is a matrix that describes the gain of the Kalman filter in an observer-based control system. It is denoted by L and is defined as:

L = PH^TR^-1

where P is the solution of the algebraic Riccati equation, H is the output matrix, and R is the measurement noise covariance matrix. The Kalman filter gain matrix is used to determine the gain of the Kalman filter in an observer-based control system.

1. System matrix: The system matrix is a matrix that describes the dynamics of a system. It is denoted by A and is defined as:

A = [0 1 0 0 … 0;

0 0 1 0 … 0;

0 0 0 1 … 0;

…………

0 0 0 0 … 1;

a0 a1 a2 a3 … an-1]

where a0, a1, a2, …, an-1 are the coefficients of the characteristic polynomial of the system. The system matrix is used to describe the dynamics of a system.

These matrices are important tools for analyzing and designing control systems, and they are used extensively in control system theory and practice.

### Assignment Brief 6: Design a digital control system meeting the required specifications.

A digital control system is a system that uses digital techniques to control an analog or digital process. The most common type of digital control system is a computer-based system. A digital control system may use a microprocessor, a digital signal processor, an application-specific integrated circuit, a field-programmable gate array, or other digital devices.

The advantages of digital control systems over analog control systems include increased accuracy, improved stability, and the ability to make changes to the system without physically altering the system. In addition, digital control systems can be more easily monitored and updated than analog control systems.

The disadvantages of digital control systems include the need for more complex hardware and the potential for increased system latency. In addition, digital control systems can be more expensive to implement than analog control systems.

### Assignment Brief 7: Use software tools to study digital control system design.

There are several software tools available for studying digital control system design. Here are some popular ones:

1. MATLAB: MATLAB is a widely used tool for studying digital control system design. It has a comprehensive set of functions and toolboxes for system analysis, design, and simulation. MATLAB also provides a graphical user interface (GUI) for designing control systems.
2. Simulink: Simulink is a graphical modeling tool that is used for simulating and analyzing control systems. It has a block diagram environment that makes it easy to design and test control systems.
3. LabVIEW: LabVIEW is a data acquisition and control system design software that has a graphical programming language. It is widely used in industrial automation and research applications.
4. Control System Toolbox: Control System Toolbox is a MATLAB toolbox that provides functions for analyzing and designing control systems. It includes functions for linear and nonlinear systems, frequency domain analysis, and system identification.
5. Scilab: Scilab is an open-source numerical computation software that has a toolbox for control systems design. It provides functions for modeling, simulation, and analysis of linear and nonlinear control systems.
6. Python Control Systems Library: Python Control Systems Library is a Python library that provides functions for modeling, simulation, and analysis of control systems. It includes functions for linear and nonlinear systems, state-space analysis, and optimal control.

By using any of the above-mentioned software tools, you can easily study digital control system design and simulate different control strategies to see how they perform under different conditions.

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