6.302 Feedback Systems Recitation 0: Nyquist Stability Prof. Joel L. Dawson

Transcription

1 6.302 Feedbac Systems Today s theme is going to be examples of using the Nyquist Stability Criterion. We re going to be counting encirclements of the -/ point. But first, we must be precise about how we count encirclements. EXAMPLE : Problem: In each region, how many positive (clocwise) encirclements are there? The numbers in the regions are the answers. Notice that when counting encirclements, we are free to split the contour into any convenient number of continuous circles or loops. CLASS EXERCISE Label the number of clocwise encirclements in the Nyquist plots below: Page Cite as: Joel Dawson, course materials for Feedbac Systems, Spring 2007.

2 6.302 Feedbac Systems Now there is only one more diffculty to resolve: what to do with singularities on the axis? 1 = s We actually have two choices: B A σ radius goes to zero radius goes to zero Choice # Choice #2 Im( ) Im( ) -/ B A Re( ) Re( ) -/ Page 2 Cite as: Joel Dawson, course materials for Feedbac Systems, Spring 2007.

3 6.302 Feedbac Systems Let s do the accounting for the two cases. Remember that we re calculating Z = N + P, and we want Z = 0 if the system is to be stable. Choice # Range of P N Z Stable? - < < 0 0 NO 0 < < YES Choice #2 - < < 0 0 NO 0 < < - 0 YES Either choice gives you the right answer! And both choices give agreement with root locus: σ 0 <0 We have all of the basic tools now...we can proceed now to even more complex examples. There s one more tric that can help you, which is that bode plots can be a tremendous help in drawing complicated Nyquist contours. EXAMPLE: 1 = (s+1)(10-2 s+1) Page 3 Cite as: Joel Dawson, course materials for Feedbac Systems, Spring 2007.

4 6.302 Feedbac Systems Consider the Bode Plot: 1 (s+1)(10-2 s+1) log ω L 0 (s) Hits ω = 0 = (10j+1)(0.1j+1) 0. 1 Im( ) σ Re( ) σ=- -/ (,0) σ=-00-0.j Page Cite as: Joel Dawson, course materials for Feedbac Systems, Spring 2007.

5 6.302 Feedbac Systems We can now do the normal accounting: Range of P N Z Stable? - < < - 0 NO - < < YES 0 < < YES And chec our results with root locus: σ σ 0 <0 And now an impressive example, just in case you thought EXAMPLE was some ind of joe...! (0-2 s + ) = 2 (s+) 3 (0-3 s + ) 2 Begin by doing a Bode Plot Page 5 Cite as: Joel Dawson, course materials for Feedbac Systems, Spring 2007.

6 6.302 Feedbac Systems log ω L 0 (s) ω Page 6 Cite as: Joel Dawson, course materials for Feedbac Systems, Spring 2007.

7 6.302 Feedbac Systems Im( ) N=2 N=0 N=2 Re( ) Root Locus σ Page 7 Cite as: Joel Dawson, course materials for Feedbac Systems, Spring 2007.

8 6.302 Feedbac Systems This is a PRIME EXAMPLE of a conditionally stable system. Now loo bac at the Bode Plot, and observe where the phase crossings of 80 are taing place. The bubble on the Nyquist plot where N = 0 is located at around / = Let =0 6.5 then, and return to the Bode Plot. You ll discover that the first time the phase hits 80, the magnitude of the loop transmission is greater than unity!! But for this value of, the Nyquist Criterion says the system is stable. Nyquist is right...we can t trust our intuition in this case. Page 8 Cite as: Joel Dawson, course materials for Feedbac Systems, Spring 2007.