37 NYQUIST PLOT 140 37 Nyquist Plot Consider the attached images; here are a few notes. In the plant impulse response, the initial condition before the impulse is zero. The frequency scale in the transfer function magnitude plotsis10−3 − 104 radians per second. In the plot of P (s) loci, the paths taken approach the origin from ±90◦, and do not come close to the critical point at −1+0j, which is shown with an x. In the plot of P (s)C(s) loci, the unit circle and some thirty-degree lines are shown with dots. Also, the two paths in this plot connect off the page in the right-half plane. Answer the following questions by circling the correct answer. 1. The overshoot evident in the open-loop plant is about (a) 120% (b) there is no overshoot since this is not a step response (c) 70% (d) 40% 2. The natural frequency in the open-loop plant is about (a) one Hertz (b) one radian per second - To compute this, you need a whole cycle. (c) 1.2 radians per second (d) six radians per second 3. Basedonthe plantbehavior, P (s) probably has (a) no zeros and one pole (b) one zero and one pole (c) no zeros and two poles (d) one zero and two poles - This plant has a zero at +1 (yes, a right-half plane zero, also known as an unstable zero) and two poles at −0.1 ± j. You can tell it has two complex,stable poles because of the ringing in this impulse response. You can tell it has a zero because the output instantaneously moves to a nonzero value during the impulse - this could only be caused by a differentiator. 4. Compare the abilities of the plant hooked up in a unity feedback loop (i.e., with C(s) = 1), and of the designed closed-loop system, to follow low-frequency commands: (a) The P (s)C(s) case has a lot more magnitude above one radian per second, and so it has a better command-following (b) P (s) is nice and flat at low frequencies, so it is better at command-following37 NYQUIST PLOT 141(c) P(s)C(s) has increasing values at lower frequencies and this makes itbetter - Setting C(s) = 1 will achieve about 10% tracking accuracy at low fre-quencies. The designed P(s)C(s) has a pole at or near the origin and hence is anintegrator; this gives us no tracking error in the steady state.The plant is stable, but lightly damped and it has an unstable zero. As you canguess from a quick check with a ro ot locus, this is a difficult control problem, intu-itively because the plant always moves in the wrong direction first. A PID cannotstabilize this system! I ended up using the loopshaping method in MATLAB’sLTI design tool; this gave a third-order controller with two zeros.(d) The peak in P(s) is not shared by the other plot and this makes P (s) better atcommand-following.5. Is the unity feedback loop stable, based on the loci of P (s)?(a) No: The path encircles the critical point once in the clockwise directionand that is all it takes, because the poles of P (s)C(s) are in the left-half plane - Note that the unstable zero in the plant is immaterial by itself.Nyquist’s rule is that stability is achieved if and only if p = ccw, where p is thenumber of unstable poles in P (s)C(s), and ccw is the number of counter-clockwiseencirclements of the critical point.(b) Yes: The path encircles the critical point once in the counter-clockwise direction(c) Yes: The path encircles the critical point once clockwise and this is matched bya plant zero in the right-half plane(d) No: The path encircles the critical point twice whereas it should only circle itonce.6. The designed compensator creates a stable closed-loop system, as is seen in the stepresponse plot. The gain and phase margins achieved are approximately:(a) 0.2 upward gain margin, 1.2 downward gain margin, and ±40◦phase margin(b) 1.2 upward gain margin, infinite downward gain margin, and ±30◦phasemargin(c) infinite upward gain margin, 1.2 downward gain margin, and ±30◦phase margin(d) 1.2 upward gain margin, infinite downward gain margin, and ±60◦phase margin37 NYQUIST PLOT 142 0 1 2 3 4 5 6 7 8 9 10 −15 −10 −5 0 5 10 y Plant Impulse Response time, seconds Magnitude |P(s)| Magnitude |P(s)C(s)| 30 50 dB 20 40 10 30 200 10 dB−10 0 −20 −10 −30 −20 −40 −30 −50 −40 001010rad/s rad/s37 NYQUIST PLOT 143 Loci of Plant P(s) −60 −40 −20 0 20 40 60 imag(P(s)) ω > 0 ω < 0 −50 0 50 100 real(P(s))37 NYQUIST PLOT 144 Loci of Loop Transfer Function P(s)C(s) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 imag(P(s)*C(s)) ω > 0 ω < 0 −1.5 −1 −0.5 0 0.5 1 1.5 real(P(s)*C(s)) Closed−loop Step Response 0 1 2 3 4 5 6 7 8 9 10 time, seconds −4 −3 −2 −1 0 1 2 yMIT OpenCourseWarehttp://ocw.mit.edu 2.017J Design of Electromechanical Robotic Systems Fall 2009 For information about citing these materials or our Terms of Use, visit:
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