16.06 Principles of Automatic Control, Lecture 6

16.06 Principles of Automatic Control Lecture 6 Time Domain Specifications: Many control systems are dominated by a second order pair of poles. So look at time response (to step input) of Hpsq “ ωn2 s2 ` 2�ωn s+ωn2 Typical response: 1.4 1.2 Mp 1 0.9 ±1% 0.5 0.1 0 0 tr 2 t4p t 6 8 s 10 Time, t (sec) 1 12 14 t Mp tr ts tp “ “ “ “ peak overshoot rise time (10% to 90%) settling time (1%) time of peak Each of the above parameters may be important in the design of the control system. For example, the designer of a hard disk drive may specify a maximum settling time of the response of the read/write head to a commanded change in position. Peak overshoot is important, both because it is a measure (to a degree) of stability, and for practical reasons, overshoot should be minimized (think of an elevator!). Rise time tr (and to a lesser extent peak time tp ) is a measure of the speed of response of the system. Often, a maximum tr will be specified. We can connect ζand ωn to Mp , tp , tr , with two important caveats: first, some of the rela­ tionships are approximate. Second, additional poles and zeros will change the results, so all of the results should be viewed as guidelines. The step response of Hpsq is ˜ hs ptq “ 1 ´ e´ζωn t 1 cospωd tq ` a sinpωd tq 1 ´ ζ2 Using elementary calculus, we can find tp and Mp (see text): tp “ π ωd Mp “e ?´πζ 1´ζ 2 “e´π tan Θ where Θ “ sin´1 ζ. Typical values: ζ 0.5 0.7 Mp 0.16 0.05 2 ¸ The rise time is approximately tr « 1.8 ωn The rise time is a bit faster for systems with less damping, a bit longer for systems with more damping, and sensitive to additional poles and zeros. The settling time can be approximated via: e´ζωn ts «0.01 4.6 Ñ ts « ζωn Note that, in reality, settling time varies discontinuously with ζ, since as damping increases, a peak may decrease from just over 1.01 to just under 1.01, so ts is drastically reduced. Desired pole locations Given specifications on tr , Mp , and ts , where should poles be? tr ďa 1.8 Ñ ďa ωn 1.8 Ñ ωn ě “ ωnmin a Im(s) x ωn min Re(s) x Likewise, to keep Mp less than a fixed value, must have ζ ě ζpMp q: 3 Im(s) x Re(s) x Finally, must have: ζωn ě 4.6 ts Im(s) x Re(s) x Putting these constraints together will yield an allowable region for the poles (see better drawing in text): Im(s) Re(s) N.B.: The allowable region is a guide. After a system is designed, the performance will have to be evaluated. 4