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1、Phase Lead Compensation (3 hours),MODULE 19,Module19-20 采用bode图的方法，设计控制器（校正器），使得系统性能满足要求,Generally, a single performance requirement may be met by selecting a particular value of K.,Example19.1 The bode diagram of a minimum-phase system is shown in following figure. Design K to make the system to be
2、 stable.,-60db/dec pass through 0db line closed-loop system is unstable.,Increase K -20db/dec pass through 0db line closed-loop system is stable.,Multiple-design constrain：It is not possible to satisfy both system performance requirements with a single value of gain K,Example19.2 (P396),ess 0.32% fo
3、r the unit ramp input. PO% 20%.,Solution:,Letting K=316,from fig.4.17, 0.45, PM45,When K=316PM0,K=316,It is not possible to satisfy both system performance requirements with a single value of gain K !,K=316, bode diagrams:,When PM=45K=6.3,19.1 Compensation (校正) and Compensator （校正器）,(P398) The shape
4、 of the Bode diagram has to be altered to allow it to achieve both performance requirements. This alteration of the Bode diagram is known as system compensation.,为了满足闭环系统多个性能的要求，需要修改开环bode图的形状，这种bode图的改变称之为校正。,(1) Compensation （校正）,(2) Compensator （校正器）,A compensator is an additional component or ci
5、rcuit that is inserted into a control system.,Cascade compensation,Feedback compensation:,加入compensator 开环传函为Gc(s)G(s)H(s) Bode diagram形状改变 performance 改变,20lg|GcGH(j)|=20lg|Gc(j)|+20lg|GH(j)|,Connatural characteristic determined by plant and sensor,Compensator characteristic determined by designer,
6、Desired characteristic determined by requirement,GcGH(j) = Gc(j) + GH(j),Change the shape of the bode diagram,Keep K=316, modify the phase angle plot,Method 1:,Method 2:,Keep K=316, modify the magnitude plot,Example19.2,Proportion compensator; Phase Lead Compensator;Phase Lag Compensator;Phase Lead-
7、Lag Compensator;Proportional plus Integral Compensator (PI); Proportional plus Differential Compensator (PD); Proportional plus Integral plus Differential Compensator (PID);,(3) Basic compensator Gc(s),(6) Design compensator Gc(s),了解GC(s)的作用根据闭环系统性能要求,选择恰当的校正器确定校正器的具体参数,19.2 Phase Lead Compensation
8、(相位超前校正),CircuitTransfer function The characteristics Determine the parameter,(1) Phase lead element (circuit),(2) Transfer function,通常,希望校正环节的增益为1 再串联的放大器,Phase Lead：,(3) Bode diagrams,20db/dec,超前相角,在频率1与2的几何中心线出现最大的超前相角max,过大 高频噪声大； 限制15max60,(4) The effect of the Phase lead compensator （超前校正器的作用）
9、,Increase PM by adding phase lead,Example19.2 (P396),ess0.32% for the unit ramp input. PO20.,Solution:,ess=1/K312.5,PO0.45, PM=45,Letting K=316,Letting =0.45,K=316,adding Phase lead Compensator m=45,but,PM45,Example19.2 (P396),ess0.32% for the unit ramp input. PO20%.,K=316,幅值为-10lg处定为新的穿越频率c,超前校正器的设
10、计思路： 将m放在新的穿越频率c处；,由于m对应的幅值为10lg；,PM=45,Gc(s)provide a positive phase increasing PM,Gc(s)provide a positive slope to ensure -20de/dec pass through 0db line,校正后 adding Gc(s)： 低频: 不变稳态性能不变； 中频：PMPO； cTs; 稳定性、瞬态性能变； 高频：高频噪声衰减能力下降；,Phase lead compensation process (校正步骤 确定，T参数）,Step 1: Calculate the max
11、of the compensator；,max=(PMd PM)110% 考虑10的裕量,Step 2: Calculate the parameter ;,Step 3 Calculate another parameter T,验算：,Uncompensated System:,Compensated system:,PMPO c n Ts,Step response:,19.3 Comment on the Applicability and Results of Phase Lead Compensation (对超前校正应用和效果的评价 ）,(1) Low frequency is
12、unchanged稳态性能不变 (2) Middle-frequency slope is changed (变为-20db/dec) The overshoot is decreased. ( PMPO）Gain crossover frequency c increase, or the open-loop and closed-loop bandwidths are increased. (带宽增大） Faster response, but High-frequency noise is increased. (3) If we need PM , we need double pha
13、se lead compensator or other approach.,Example19.3 (P417),The steady state error due to a unit ramp input of less than 25% . A closed-loop damping ration of about 0.7.,Solution:,ess=1/Kv25% Kv=K 4, =0.7 PM70,Letting K=4,max=57,1=1.62,校验：Computer-generated Bode diagram,The unit step response,%=10% 0.
14、7,还需将m再提高些,Transfer function of phase lead compensator.characteristics of phase lead compensator.how to use the phase lead compensator.,conclusion,At the end of this Module students should be able to know,Question:,Where does the phase lead element add? Does it affect on low frequency? Does it affect on high frequency? Does the lead phase can increase infinitely? If the necessary additional phase is too large, what can we do?,
