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Part I: Control of the patient lift:
We have modelled and simulated a patient lift made up of a DC motor, a gearbox, a pulley, a potentiometer and an accelerometer. We set the values of all the elements according to the datasheets of every part. The Simulink project of this lift is given along with this assignment. To obtain a desirable behaviour, your task is to control the system without changing the lift model.
Task 1: Analysis of the lift in s-domain and frequency domain. Step.1: Stability
We would like to find the root locus of the poles of the closed loop system. Cut the feedback line and linearize the system (by finding the step response in lineariser). Transfer the linearized system (probably called linsys1) to MATLAB’s workspace. Find and report the root locus plot of the open loop transfer function. (For the root locus, zoom in to show 1st and 2nd most dominant poles.
Report the Bode plot of the system with margins. Using both methods (root locus and Bode plots), suggest a ???? (proportional gain to amplify the error), by which the system becomes unstable. Set the value of ???? two times (once try a value less than the critical value, then a value more than the found value,) and report the simulation results.
Step 2.: Controlling acceleration
Set ???? = 1 and
Set the value of ???? = 60 and Run the simulation and report the value of acceleration of the load in a scope plot. What is the highest absolute acceleration value? We would like to add the measured acceleration to the error to make the system stop if the acceleration gets too high. However, the question is if this makes the system unstable. set the gain for the feedback of the potentiometer and the accelerometer to 1 (explain why we do this.). Feed the accelerometer with the acceleration of the load and cut the feedback after the acceleration and set it as output. Find the root locus of this open loop system. (Remember that the potentiometer remains closed loop). Report the value of weight for the accelerometer that makes the system unstable. Also sketch the Bode plot with the margins.
Task 2: Obtaining desirable behaviour using a PID controller.
Set the weight of the load to 100 Kg with no disturbance weight. Feed the accelerometer with the linear position of the load and set both weights of the feedback to 0.5. Set up a PID controller instead of ???? and tune it using “Response optimization”. You can choose your own signal to track and initial values for PID. Explain the signal you would like the lift to track and mention it in the report along with your choice of initial values of PID gains. Then let MATLAB tune your PID. Report the step response before and after applying PID.
Part 2 (Optional): Control of a landing rocket:
The transfer function that relates thrust angle ??(??) to the vehicle angle of landing ??(??) can be modelled via the following equation:
?(??) ????1/?? ??(??) = = T(??) ??2 - (??????1/??)

where the various constants are: ?? = thrust; ??1 = distance from the centre of mass (C) to the centre of pressure (P); ??2 = distance from the centre of mass (C) to the rocket engine; ?? = Moment of inertia of the rocket vehicle about C; ???? = linearized aerodynamic constant relating pitch moment to angle of
????1/??=1.0 attack. But in this assignment, we simplify the problem by setting and ??????1/?? = 1.0. So, the transfer function would simply be:
??(??) = ?? 2 - 1
The rocket has a proper vertical velocity and should land in 3 seconds; However, the current angle is 1 radian off the vertical axis (perpendicular to the ground.)
The task: Make it land (stabilize the closed loop system). We would the output to have a steeling time of 3 seconds with 3% overshoot. Sketch the block diagram. Justify the choice of controller by showing the behaviour of the system using roughly sketched root locus plot and Bode plots. Describe the optimization process (with snapshots of optimization) and show the plots of the final response.