16.06 Principles of Automatic Control Lecture 27 Nonminimum Phase Systems Our design rules so far are based on the bode gain-phase theorem, which applies to stable, minimum phase systems. The RHP zeros or time delays of NMP systems place fundamental limitations on the achievable performance of any closed-loop systems. Example: Consider the plant Gpsq “ 1 ´ s{10 sp1 ` s{1q Our goal is to design a closed-loop controller with bandwidth as large as possible. How well can we do? Bode plot: |G| 10 2 -1 10 0 -2 10 10 -1 −2 −4 10 −2 10 −1 ω 10 1 0 10 1 10 2 The slope at high frequency is ´1, so it seems that we should be able to cross-over anywhere. However, in this case we need to look at the phase plot, sicne gain-phase theorem does not apply: G 0 −50 −100 −150 −200 −250 −300 −350 10 −1 10 0 10 1 10 2 10 3 ω Note that additional phase due to zero at s “ `10 is negative. So if we use pure gain, the crossover frequency must be below about ωc “ 3. Let’s add compensation to make slope ´1 everywhere: Kpsq “ k 1 ` s{1 Ð cancels plant pole 1 ` s{10 Ð stable pole ñ KpsqGpsq “ Bode Plot: 2 1 1 ´ s{10 s 1 ` s{10 Magnitude of GK 10 10 10 10 2 should crossover below here for good margins 0 -1 −2 ω −4 10 −1 10 0 10 1 10 2 10 3 Phase, degrees 0 −100 −200 −300 ω 10 −1 10 0 10 ω 1 10 2 10 3 So NMP zero causes signiﬁcant phase lag (relative to the phase expected from slope) at frequencies up to one decade below crossover. Suppose we could accept PM as low as PM “ 30˝ . What would control system look like? Solve for k: =GK “ ´ 90˝ ´ 2 tan ´1 ω{10 “ ´ 150˝ ñ ωc “5.77 r/s |GK| “k{ω k “ ωc “5.77 Therefore, 1`s 1 ` s{10 KpsqGpsq T psq “ 1 ` KpsqGpsq 1 ´ s{10 “ s2 ? ` 2p0.278qs `1 57.7 57.7 Kpsq “5.77 3 Im(s) 7.3j Re(s) -2.1 10 -7.3j See step response plotted below: 1.5 Step Response 1 0.5 0 0 0.5 1 1.5 Time, t (sec) 2 2.5 3 Note that Mp « 49%. In addition, there is a 20% undershoot (wrong way behavior). The bottom line is that a non-minimum phase zero places fundamental limitations on the bandwidth of the closed-loop system. As a practical matter, if the NMP zero is at s “ a, we must have ωc ď a{2 More realistically, to achieve reasonable phase margins and step response, we need ωc ď a{3 Even at ω “ a{10, the NMP zero adds 12˝ of anomalous phase lag. 4 Time Delay The eﬀect of a pure time delay is similar to that of a NMP zero. Indeed, a time delay is non-minimum phase. The transfer function os a T-second delay is e´sT “ e´j ωT So the additional phase lag is ωT . As a practical matter, must cross over at ωc ď 1{T but more reasonably should have ωc ď 0.6{T Unstable systems For an unstable system, the Bode gain-phase theorem does not apply either. In this case, however, the disagreement between slope and phase occurs at low frequency (when viewed properly). Example: Gpsq “ 10 s´1 Bode: Magnitude of G 10 10 10 2 0 should crossover above here for good margins −2 10 −1 10 0 10 1 10 2 10 3 Phase, degrees −50 −100 −150 −200 −1 10 10 0 10 ω 5 1 10 2 10 3 Using arguments similar to those made for NMP zeros, can see that we need to crossover at least at ωc ě 2p where p is the location of the unstable pole. Note that this is a fuzzy requirement - inthe example, can stabilize the system wtih any ωc ą 0, but margins will be poor unless ωc ě 2 r/s. 6

16.06 Principles of Automatic Control, Lecture 27

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