16.06 Principles of Automatic Control, Lecture 23

16.06 Principles of Automatic Control Lecture 23 Stability Margins Stability margins measure how close a closed-loop system is to instability, that is, how large or small a change in the system is required to make it become unstable. The two commonly used measures of stability are the gain margin and the phase margin. • The gain margin (GM) is the factor by which the gain can be increased before the system becomes unstable. • The phase margin (PM) is the amount of additional phase lag that would make the phase be ´180˝ where |KGpjωq| “ 1. The GM and PM are important not only because they measure how close the closed-loop system is to instability, but also because they (but especially the PM) can be used to predict the transient behavior of the closed-loop system. Gain and phase margin on Nyquist diagram: Im(s) Re(s) -1 PM 1 GM GM and PM on Bode diagram: 1 Magnitude 10 10 10 10 2 0 GM −2 −4 10 −1 10 0 10 1 10 2 Phase (deg) 0 −100 PM -180 −200 −300 −1 10 0 10 10 Frequency, ω (rad/sec) 1 10 2 Relationship between PM and damping When the phase margin is small, the closed-loop system is close to instability, so that there will be closed-loop poles near the jω´axis. That is, low PM ñ low damping ratio. This result can be made explicit by considering the closed-loop system r a s(s+b) - The closed-loop transfer function is T psq “ s2 a ` bs ` a So, ωn “a b ζ“ ? 2 a Can show that, for this system, 2 ¨ ˛ PM “ tan ´1 ˝ ba 2ζ ‚ 1 ` 4ζ 2 ´ 2ζ 2 The functional form isn’t really important - the important point is that ζ is nearly a linear function of PM: 1 actual ζ Damping Ration, ζ 0.8 0.6 ζ= 0.4 PM 100 0.2 0 0 20 40 Phase Margin (deg) 60 80 So can often predict (effective) damping ratio using approximation ζ« PM 100 (PM in degrees) Even when system is not second order, PM is a good predictor of peak overshoot (Mp ), and resonant peak magnitude (Mr ). PM is often specified as a design requirement. Bode’s Gain-Phase Relationship We saw that for poles and zeros in the left-half-plane, the phase of Gpjωq is proportional to the slope of the magnitude curve (on a log-log scale), but smeared-out. That is, =Gpjωq « 90˝ ˆ slope of |G| This idea can be made precise via Bode’s gain-phase theorem: For any stable, minimum phase system, the phase of Gpjωq can be determined uniquely from the magnitude of Gpjωq. 3 The phase is in fact given by 1 =Gpjωq “ π ż8 ´8 dM W puqdu du where M “ log |Gpjωq| (natural log) u “ logpω{ω0 q dM “ slope of Bode plot magnitude du W puq “ weighting function 101 “ logpcothp qq 2 Note that this is a funny sort of convolution - we are convolving a weighting function with the slope of another function, but working on logarythmic axes! The weighting function looks like: W 5 4 3 2 1 u -4 -2 0 2 4 Note that 92% of area of W puq is within ˘1 decade of the center. So the phase is nearly completely determined by the slope of M within ~1 decade. Why is this result important? It implies that in almost every case, a well-designed control loop will have a magnitude plot with slope -1 at the crossovr frequency!1 1 Actually, in some cases, the slope might be +1, but this is rare. 4 |GK| 0 Typical loop gain for well-designed control system -2 -1 ωC ω -2 In this case, the phase at cross-over will be a weighted average of ´90˝ (weighted a lot), ´180˝ (weighted some), and 0˝ (weighted hardly at all). So the phase will be between ´90˝ and ´180˝ , with probably reasonable PM. 5