Lecture 3a. Review Chapter 10 Nise. Skill Assessment Problems. G. Hovland 2004

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1 METR4200 Advanced Control Lecture 3a Review Chapter 10 Nise Skill Assessment Problems G. Hovland 2004 Skill-Assessment Exercise 10.1 a) Find analytical expressions for the magnitude and phase responses of s) ( s 1 + 2)( s + 4) b) Make plots of the log-magnitude and the phase, using log-frequency in rad/s as the ordinate c) Make a polar plot of the frequency response

2 Exercise 10.1a jw) ( jw + 1 2)( jw + 4) w j6w + 8 (8 1 2 w ) + j6w 2 (8 w ) j6w 2 2 (8 w ) + (6w) 2 For Bode plots it is easier to stop here!! For Nyquist plots: Get rid of j in denominator Exercise 10.1a Inverse!

3 Exercise 10.1b Exercise 10.1c

4 Class Quiz Statement: Closed-loop systems with open-loop phase between -90 o and +90 o are always stable, regardless of gain. Explain why? Hint: Think Nyquist. Skill-Assessment Exercise 10.2 Draw the Bode log-magnitude and phase plots for the system shown below, where s) ( s ( s + 20) + 1)( s + 7)( s + 50)

5 Exercise 10.2 ( s + 20) s) ( s + 1)( s + 7)( s + 50) Skill-Assessment Exercise 10.3 Sketch the Nyquist diagram for the system shown below, where s) ( s + 1 2)( s + 4)

6 Exercise 10.3 Class Quiz 1: Can this system ever become unstable? Class Quiz 2: Compare with polar-plot from 10.1c and explain difference! Skill-Assessment Exercise 10.4 For the system below, where do the following s) ( s + K 2)( s + 4)( s + 6) a) Plot the Nyquist diagram b) Use your Nyquist diagram to find the range of gain K, for stability

7 Exercise 10.4a Exercise 10.4b This demonstrates a quick way of finding the stability range without having to draw the Nyquist plot

8 Skill-Assessment Exercise 10.5 Find the gain margin and the 180 o frequency for the problem in Exercise 10.4 if K 100 Skill-Assessment Exercise 10.6 For the system below, where s) ( s + 5)( s K + 20)( s + 50) do the following Draw the Bode log-magnitude and phase plots Find the range of K for stability from your Bode plots Evaluate gain margin, phase margin, zero db frequency, and 180 o frequency from your Bode plots for K 10,000.

9 Exercise 10.6a Why does the magnitude start here -74 db? Exercise 10.6b

10 Exercise 10.6c Skill-Assessment Exercise 10.7 Find the closed-loop bandwidth required for 20% overshoot and 2-seconds settling time Formulas: zeta: Equation 4.39 wbw: Equation Such formulas will be given! But you need to know how to use them! Note: These are approximations based on a 2 nd order pole!

11 Skill-Assessment Exercise 10.8 Given the system shown below, where s) ( s )( s + 20)( s + 50) plot the closed-loop log-magnitude and phase frequency response plots using the following methods: a) M and N circles b) Nichols chart Exercise 10.8a

12 Exercise 10.8b Exercise 10.8 a&b Closed-loop response from M&N or Nichols

13 Skill-Assessment Exercise 10.9 Using the open-loop frequency response for the system below, where s) 100 s( s + 5) estimate the percent overshoot, settling time, and peak time for the closed-loop step response. Exercise dB w bw Again these formulas will be given. In general, you need to find ζ and w bw and then use the formulas

14 Skill-Assessment Exercise Find the static error constants for a stable unity feedback system whose open-loop transfer function has the Bode magnitude plot shown below Exercise This table will normally be given

15 Skill-Assessment Exercise For the system below, where 10 s) s( s + 1) find the phase margin if there is a delay in the forward path of a) 0 sec b) 3 sec c) 7 sec Exercise 10.11a Direct computation as an alternative to Bode or Nyquist plots

16 Exercise b&c Skill-Assessment Exercise Estimate s), whose Bode log-magnitude and phase plots are shown below

17 Exercise Exercise 10.12

18 Exercise Summary You can expect similar style questions on class test and final exam. Calculations will be reasonably easy to avoid w 4, w 6, etc Questions will tend towards understanding rather than number crunching skills (for which we have Matlab).

Example. Given a unity feedback system that has a forward TF. , do the following

Example. Given a unity feedback system that has a forward TF. , do the following Example Given a unity feedback system that has a forward TF K ( s + ) G( s) =, do the following ( s s + ) (a) Obtain a RL using Matlab Roots: +i (b) Find the complex poles that crosses on the imaginary