You’re working as an electric machine control engineer and have been tasked with designing a proper controller for a DC motor for a specific operating point. The general equation of DC motors is shown below:
The output is torque, which is calculated using the following equation:
Where T is the electric machine torque, ωe speed of the electric machine in rad/s which is ωe = RPM, and ia and va are the armature currents and voltage respectively. The constant parameters are presented in Table 1:
Table 1 Parameters of the DC motor
For this project:
1 State space representation
1.1 Input, output and states [2 marks]
Determine inputs, outputs and states of the systems.
1.2 State space representation [2 marks]
Rearrange the equation into the standard form. of state-space representation of the system. Specify the state and output equations.
2 Linearisation
The operating point at which you’re asked to design a controller is T* = 50 Nm and RPM = 1500 rpm. For this operating point:
2.1 Operating point condition [3 marks]
Identify and calculate the operating point conditions, encompassing inputs, outputs, disturbances and states.
2.2 Linearisation [4 marks]
Linearise the system around the operating point in the form. of:
δX = AδX + BδU
δY = CδX + DδU
Determine the A, B, C and D matrices. Where all variables with δ show incremental variables.
2.3 Implementation and Validation [4 marks]
Simulate both the nonlinear system and the linear system in Simulink, and contrast their responses to a sinusoidal input that varies around the operating point input , with an amplitude of 5% of the operating point input and a frequency of 1 rad/s, i.e. Show the block diagram of the system you used to compare linear and nonlinear systems and discuss the result. Assume that the speed signal remains constant at 1500 rpm.
3 Transfer function
Convert the state space to a transfer function and analyse the system.
3.1 Convert to transfer function [2 marks]
Convert the state space to a transfer function using ss2tf in MATLAB. When deriving the transfer function, there's no need to incorporate disturbances and Ba matrix.
3.2 Pole location [2 marks]
Find the poles and zeros of G(S) and report them. Use pole and zero command in MATLAB.
3.3 Unit step response [2 marks]
Illustrate the unit step response of G(S) and report it, using the step command in MATLAB. Report the settling time.
4 Controller Design
You have been tasked to design a linear controller to control the torque of the DC electric machine at a specific operating point. It is assumed that we have access to a sensor that can measure the generated torque.
4.1 Design of controller for linearized system
4.1.1 Desing of the controller [2 marks]
For the control design task, you are required to use the “Control System Designer” application in MATLAB. The objective is to eliminate any steady-state error and achieve a settling time of approximately 1 second for the output around the operating point T* = Nm and RPM* = 1500 rpm. The linear controller shouldn’t show any oscillation or overshoot.
Please display the reference tracking time response and the controller effort time response.
4.1.2 Bandwidth [2 marks]
Illustrate the frequency response of the reference tracking and controller effort. Based on these graphs, determine the bandwidth of the controller effort and the reference tracking, also please demonstrate the frequency of the bandwidth on the graphs. Discuss these values and explain what they reveal about the performance and requirements of the controller.
4.1.3 Controller transfer function [1 mark]
Report the proportional and integral coefficients (if only one exists, indicate that the coefficient of the other is zero). Furthermore, present the transfer function of the controller, C(s).
Hint: It is recommended to use a PI controller.
4.2 Validation of the control system
In this section, you are required to implement the linear controller you have designed on the nonlinear system and demonstrate its performance. First, present the architecture of the controller, then implement it in Simulink, and finally discuss the results.
4.2.1 Architecture of the controller system [2 marks]
Describe the control architecture you used for the control design and provide a block diagram of the linear controller applied to the nonlinear system. Additionally, explain how you implemented the linear controller to manage the nonlinear system.
4.2.2 Implement the controller without a load [3 marks]
Implement the controller in Simulink and test the torque control performance. Use the nonlinear plant and assume the speed is constant at 1500 rpm. Demonstrate the tracking performance of a step torque command increasing from 50 Nm to 60 Nm at 2 sec and run the simulation for 10 sec.
4.2.3 Implement the controller with a load [4 marks]
Implement the controller in Simulink and test the torque control performance. For this test, use the nonlinear plant model for the electric machine and connect it to the provided load model (load.slx). This setup will provide a variable load, as the electric machine is connected to a load, causing the speed to no longer remain constant. Demonstrate the tracking performance of a step torque command increasing from 50 Nm to 60 Nm at 2 sec and run the simulation for 10 sec. Discuss any potential degradation in tracking performance.
5 marks of this PMA are dedicated to standard scientific report formatting, referencing, and structured reporting.
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Item
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Mark
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Quality of figures (including the type of figure, axis titles, and units)
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1
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Captions of figures and tables and referring to the captions in the main text
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1
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Structured format for the entire report
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1
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Using references and the correct format for referencing
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1
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Academic format for the report (including table of contents, list of figures, page numbers, and appropriate font size)
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1
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