# Design a state feedback controller by placing the poles for a second order system ap- proximation with damping ratio

An electric rail car is powered through its roof using a contact assembly to high power voltage lines. The assembly dynamics are shown below, where the input to the system is the Force Up,fup(t), and the output,y, is the distance between the Head Mass and the Contact Wire,y(t) =yh(t) ycat(t), yielding a transfer functionGp(t) =Y(s). The objective is to maintain a constant distance from the Fup(s) wire assembly. The system parameters that are known are summarized in the table below (Kaveis the average of the tension wire spring constants). (a) (10 points) Model the system in Matlab. You were able to characterize the frequency response of the system using a vector signal analyzer as shown in the bode plot below, is your model accurate? (Explain with plots and analysis. The values for the plot are available on the Course Website) (b) (15 points) After characterizing the model, use root locus techniques to design a PID controller that yields: •0.5 second settling time•No more than 50% overshoot •Zero Steady State Error In order to remove high frequency oscillations use a notch filter with the following transfer function: GN(s) =(s+ 8 + 95.5824j)(s+ 8 95.5824j) (s+ 60)2 For the following figure, the values are: Name: Parameter Mh Mf fvf fvh Kh Ks Kave Value Units 9.1 kg 17.2 kg 30 130 Ns m Ns m 7x103N m 82.3x103N m 1.535x106N m MEM 355 – Winter 2018 Project Page 2 of 4 Name: Which would yield the following state space system: 24Mfs2+(fvf+fvh)s+Kh (fvhs+Kh) 0 Define the transfer function as: 03524yf3524f35Ksyh= 0 (1) (2) (3) Gp(s) =Yh Ycat= F Kave+Ksycat0 1.5802×108(4.98875x107s+ 2.68625×109) (fvhs+Kh) Mhs2+fvhs+Kh+Ks Ks (s2+ 158.469s+ 9282.942)(s2+ 8.1191s+ 376.3225) 2. (20 points) Given the following system in a unity feedback configuration: G(s) =(s+8)(s+10) (s+4)(s+6)(s+7) Using any method design a Compensator for the following requirements. You must show all work and plots for full credit. •• 3. (20 points) An aircraft roll dynamics have the following transfer function:G(s) =(s)=5(4) V(s) (s+2)(s+5) Where is the roll of the aircraft andVis the applied voltage to the actuation system. Using any method design a Compensator for the following requirements. You must show all work and plots for full credit. ••Steady state error of 4. For the following system: G(s) =K(s+ 4)(s2+5s+100)(s2+2s+10) (5) MEM 355 – Winter 2018 Project Page 3 of 4 Name: (a) (5 points) Plot the Bode plot and Nyquist plot. (b) (5 points) Determine the stability regions using both the plots in (a). (c) (5 points) Design a proportional compensator in a closed loop configuration that is suciently stable and improves the transient response using Loop Shaping Techniques (Bode Plot). 5. For the following system: G(s) =s+ 2(6) (s+5)(s+9) (a) (5 points) Determine the controllability of the system. (b) (10 points) Design a state feedback controller by placing the poles for a second order system ap- proximation with damping ratio,?=.7, and natural frequency of!n= 96. [Do not use the Matlab functions place or acker] (c) (5 points) Simulate your system in (b) using both matlab and Simulink and show the step response.

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