毕业设计（论文）外文文献翻译院系：电气与自动化学院年级专业：2011级自动化2姓名：学号：附件：System compensationSystem compensation1 IntroductionIt was mentioned earlier that performance of a control system is measured by its stability, accucacy , and speed of response .in general these items are specified when a system is being designed to satisfy a specific task .Quite often the simultaneous satisfaction of all these requirements cannot be achieved by using the basic elements in the control system .Even after introducing controllers and feedback , we are limited as to the choice we may exercise in selecting a certain transient response while requiring a small steady state error. We will show how the desired transient as well as the steady state behavior of a system may be obtained by introducing compensatory elements (also called equalizer networks)into that control system loop .These compensation elements are designed so that they help achieve system performance , i. e .bandwidth, phase margin ,peak overshoot ,steady state error ,etc. without modifying the entire system in a major way .Form our experience so far we recognize that any changes in system performance can be achieved only though varying the forward loop gain .Consider the third-order unity feedback system with the following forward loop transfer function,()()()K G s s s a s b =++ From the Routh-Hurwitz criterion we know that stability requires ()K ab a b ≤+ We also know that the steady state error to a ramp input is211lim []1()ss s ab e s s G s K→=⋅=+ Obviously if it is necessary to minimize the steady state error, the gain K should be increased. Since K is constrained to a maximum value of a b （a +b ）,the minimum steady state error becomesmin 1[]ss e a b=+ A further decrease in the error requires an increase in K which in turn has a destabilizing effect on the system ,It is therefore clear that the forward “gain game ”is rather limited .2 the stabilization of unstable systemsSince the increasing of the forward loop gain K tends to destabilize a system, we must find ways it compensate it on such a way as to stabilize it again .It was established in Chapter 6 that the addition ofa pole in G(s)H(s) tends to have a destabilizing influence on system response .Can we the reverse the argument and say that the addition of a zero tends to have a stabilizing influence on system response? Let us answer this by considering an example .Consider the control system with its transfer function given in Example 6-5.This system is unstable if K>c KNow consider the same system but with the addition of a zero,312(1)()()(1)(1)K s G s H s s s s τττ+=++This is the type of function we obtain if we were to add derivative and proportional control to a third-order servomechanism .The characteristic equation becomes321212312()(1)0(1)(1)s s K s Ks s s τττττττ+++++=++And the zeros of the characteristic equation are determined by3212123()(1)0s s K s k τττττ+++++=The Routh array becomes 3s 12ττ (31K τ+)2s 12ττ+ K 1b =3121212(1)()K K τττττττ++-+1s 1b 2b 2b =0 0s 1c ω 1c K =For stability 10b ≥, and therefore13231212()()0K ττττττττ+-++>Clearly ,with a proper selection of the time constants ,this may be satisfied .The Nyquist plot for this is shown in Fig.1Fig.13 CASCADED COMPENSATIONAs indicated inFig.2, cascaded compensation consists of placing elements in series with the forward loop transfer function .Such compensation may be classified into the following categories:(a) Phase-lag compensation (b) Phase-lead compensation(a) Cascade or series compensation(c)feedback compensation ωω(d) Feedforward compensation Fig.2 Type of compensation(e) Lag-lead compensation (f) Compensation by cancellation.The details of these methods is the subject of this section. Phase-lag compensationConsider a unity feedback control system whose forward loop transfer function represents a third-order system with its Nyquist plot show in Fig.3.It is required that the gain be K 1 for satisfying the margin of stability but K 2 for satisfying the steady state performance .This seemingly contradictory requirement may be satisfied if we were to reshape the plot to the one indicated by the dotted lines .The reshaped plot may be obtained if the low-frequency part of K 1 is rotated clockwise while the high-frequency part of K 1 must lag ,the type of compensation used to achieve this is phase-lag compensation. Such compensation is obtained by a phase-lag element.C(s)Fig.3When the output of an element lags the input in phase and the magnitude decrases as a function of frequency ,the element is called a phase-lag element .Consider the lag network.The transfer function for this is211()()()1s c saT E s G s E s T +==+ Where2aT R C =; 212R a R R =+The Bode , Nyquist ,and root locus plots are show in Fig.4.We observe that⨯Fig.4the magnitude decreases with increasing frequency and lagging phase angle .The minimum phaseω=0ω=01K2KRI(a)ω(b)σj(c)m φoccurs at m ωwhich is the geometric average of the corner frequencies111log (log log )2m T aTω=+m ω=The phase angle becomesa r c t a n a r c t a nm m m a T T φωω=- ()tan 1()()mm m m T aT aT T ωφωω-+=+tan (1)/m a φ=- or sin (1)/(1)m a a φ=-+The maximum phase lag is strictly a function of a .Let us investigate how such a network alters performance of a feedback control system.The phase-lag method of compensation achieves the following:(1) Reduces high-frequency gain and improves the phase margin; (2) Increases the velocity error constant for a fixed relative stability;(3) The gain crossover frequency is decreased .This also reduces the bandwidth of the system; (4) The time response usually gets slower. Phase-lead compensationLet us return to the Nyquist plot shown in Fig .3.We could have reshaped the plot by beginning with the Nyquist plot for K 2 and rotating the high-frequency part in the counter clockwise direction but without altering the low-frequency part .Since the phase of the high-frequency part must now lead ,the type of compensation used to achieve this is phase-lead compensation .Such compensation is achieved by a phase-lead element.When the output of an element leads the input in phase and the magnitude increases as a function of frequency, the element is called a phase-lead element .Consider the lead element shown.The transfer function is 图21()11()1E s sTE s sTαα+=+Where 1212R R C T R R =+ 122R R R α+=The Bode plot ,polar plot ,and root locus plot are shown in Fig.5.We note that the magnitude increases with increasing frequency .The value of αdetermines the separation on the root locus .Themaximum phase leadm ωoccurs at m ω.Using the previous method ,itFig.5May be show thatm ω=1sin 1m αφα-=+ For systems compensated by phase-lead networks the following is concluded (1)The velocity error constant is increased and therefore the steady state error to a ramp input is decreased for a given relative stability.(2)The damping ratio is increased and the overshoot is reduced while the phase margin is increased(3) The gain crossover frequency is increased and the bandwidth is usually increased (4)The rise time is fasterPhase lag-lead compensation(a)ω(b)σj ω(c)1T1TαωPhase-lag compensation was seen to improve the steady state response although the rise time became slower .The phase-lead compensation, on the other hand, decreased the rise time and decreased the overshoot rather substantially. It is often necessary to combine these different properties for simultaneously satisfying the steady as well as transient behavior of control system 。