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90 PERFORMANCE SPECIFICATIONS AND LIMITATIONS The Bode's gain and phase relation can be extended to stable and nonminimum phase transfer functions easily.Let 21,22,...,zk be the right-half plane zeros of L(s), then L can be factorized as o)=二s+as+丝二s+丝Lm( s+刘18+2 8十2k where Lmp is stable and minimum phase and (jw)=Lmp(jw).Hence ∠L(io）=∠Lmp(jwo)+Ⅱjo+a 0+2i =1 1 ncoth+∑∠n+ jwo+zi which gives ZL(jwo)= 1 lnoth以d+∑∠io+产 dv (o.12) jwo+zi Since∠二jwo+名≤0 for each,anon-minimumphaseeonibutesan additional w0+2i phase lag and imposes limitations upon the rolloff rate of the open-loop gain.For example,suppose L has a zero at z>0 then (w/2):=∠jiwo+2 =-90°，-53.130，-280 1w0+2 lw-z,z/2,3/4 as shown in Figure 0.10.Since the slope of L near the crossover frequency is in general no greater than-1 which means that the phase due to the minimum phase part,Lmp, of L will in general be no greater than-900,the crossover frequency (or the cosed-loop bandwidth)must satisfy We<z/2 (o.13) in order to guarantee the closed-loop stability and some reasonable closed-loop perfor- mance. Next suppose L has a pair of complex right half zeros at z=x+jy with x >0 then (wo/小el0:=∠二iwo+之二jwo+z jw0+2jw0+2 w=|z,lz/2,|zl/3,l/4 -180°，-10a.2.°，-73.70， -5,Re(z)≥S(z) -180°， -8.70,-55.9° -41.3,Re(z)≈S(z -30°， 00, 0. 0°，Re(z)≤S(z) PERFORMANCE SPECIFICATIONS AND LIMITATIONS The Bodes gain and phase relation can be extended to stable and nonminimum phase transfer functions easily Let z zzk be the righthalf plane zeros of L s then L can be factorized as L s s z s z s z s z s zk s zk Lmp s where Lmp is stable and minimum phase and jL jj jLmp jj Hence L j Lmp j Y k i j zi j zi Z d ln jLmpj d ln coth jj d X k i j zi j zi which gives L j Z d ln jLj d ln coth jj d X k i j zi j zi Since j zi j zi for each i a nonminimum phase zero contributes an additional phase lag and imposes limitations upon the rollo rate of the openloop gain For example suppose L has a zero at z then z j z j z zzz as shown in Figure Since the slope of jLj near the crossover frequency is in general no greater than which means that the phase due to the minimum phase part Lmp of L will in general be no greater than the crossover frequency or the closedloop bandwidth must satisfy c z in order to guarantee the closedloop stability and some reasonable closedloop perfor mance Next suppose L has a pair of complex right half zeros at z x jy with x then jzj j z j z j z j z jzjjzjjzjjzj Re z z Re z z Re z z