project2 - INTRODUCTION This project is a continuation to...

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INTRODUCTION: This project is a continuation to Project One where the transfer function of the DC Motor has been obtained and analyzed. In this project we would improve the design of the system by designing and then implementing a PID controller and employing feedback. In order to design an appropriate PID controller, the system will be needed to be approximated. We will use the pole/zero cancellation and approximation method to obtain the design. Of course there would be some requirements for the system such as a 12% overshoot and the steady state error reduced to the unit ramp function. The performance of the system is to be analyzed after implementing the PID controller. The system will be analyzed in terms of: 1. Transient and steady state performance 2. Sensitivity requirement based on Project One 3. Disturbance rejection at low frequencies 4. Noise rejection at high frequencies The following notations will be used in the report: G: the original transfer funciton of the DC Motor D: PID controller T: Transfer function N: Noise W: Disturbance X: a simplified function S: sensitivity function K: Gain K v : Velocity constant (to the unit ramp function) The reader should notice that the MATLAB codes illustrated in the project are connected to each other that is all variables defined in some code would be defined in all of the other codes. 1
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The transfer functioin for the DC Motor that was abtained in Project One is: G(s) = 5e^(8) s^4 + 139.4615s^3 + 17884.6154s^2 + 1.29385s + 32e^8 Solving for the roots of this function yields: G(s) = 5e^8 (s + 17.44 ± j105.52) (s + 52.29 ± j7.96) We notice that we can cancel two poles which are far on the real left axis to approximate the function as: G(s) = 5e^8 2797.6057( s + 17.44 ± j105.85) (this was achieved by setting s to zero for one of the poles) G(s) 1 = 178724.257 s^2 + 34.88s + 11438.624 The step response of the original function is shown on the following page: Matlab Code for Figures 1 and 2: %%%%%%%%%%%%%%%%%%%%% %Computing the poles of the system num=[5e8]; den=[1 139.4615 17884.6154 1.29385e6 32e6]; y=tf(num,den); [z,p,k]=tf2zp(num,den); %%%%%%%%%%%%%%%%%%%%%% %Step Response of the Original System t=0:0.0001:0.3; step(y, 'r' ,t); title( 'Figure 1 of Step Response to Original Function' ); %%%%%%%%%%%%%%%%%%%%%% %The Simplified System u=[(17.44 + 105.52i)(17.44 - 105.52i)]; poly(u); num1=[178724.257]; den1=[1 34.88 11438.624]; y1=tf(num1,den1); %%%%%%%%%%%%%%%%%%%%%% %Step Response of Both Functions step(y, 'r' ,y1, 'b' ,t); title(‘ Figure 2 – Step Response of Original and Simplified System ’); 2
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