第六章Design-via-Frequency-Response1.ppt

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,*,Chapter 6Design via Frequency Response,2026/3/2 周一,1,Design Objectives,Produce desired transient response.,Reduce steady-state error.,Achieve closed-loop stability.,Total Response=Natural Response+,Forced Response,The closed-loop control systems natural response must not dominate!The output must follow the input.,2026/3/2 周一,2,Case Study:Antenna Position Control,The search forextraterrestrial life isbeing carried out withradio antennas like the one pictured here.,A radio antenna is anexample of a systemwith position controls.,2026/3/2 周一,3,Case Study:Antenna Azimuth Position Control System,A.System Concept,B.Detailed Layout,2026/3/2 周一,4,Case Study:Antenna Azimuth Position Control System(continued),C.Schematic diagram,D.Functional block diagram,2026/3/2 周一,5,2026/3/2 周一6,Case Study:Antenna Azimuth Position Control System Response,System normally operates to drive pointing error to zero.,Motor is driven only when there is a pointing error.,The larger the error the faster the motor turns.,Too large a signal amplifier gain could cause overshoot/instability.,Satisfactory design revolves around a balance between transient performance,steady-state performance,and stability.Adjusting gain&adding compensators are the tools a control engineer has to achieve this balance.,2026/3/2 周一,7,The Design Process,Step 1,:Determine a physical system and specifications from the requirements,Step 2,:Draw a functional block diagram,Step 3,:Transform the physical system into a schematic,Step 4,:Use the schematic to obtain a block diagram,signal-flow diagram,or state-space representation,Step 5,:If multiple blocks,reduce the block diagram to a single block or closed-loop system,Step 6,:Analyze,design,and test to see that requirements and specifications are met,2026/3/2 周一,8,Basic Approach,Compensation in the frequency domain,can be viewed as:,Adding gain,at low frequencies to improve,steady-state,performance.,Adding phase angle,at the desired phase margin frequency,to improve,transient performance,.,Phase margin frequency approximates the closed-loop,bandwidth.,Adding phase angle,can be used to design,for a,desired bandwidth and/or phase margin,.,2026/3/2 周一,9,Lag&Lead Compensator Format,2026/3/2 周一,10,Lead Compensator Frequency Response,2026/3/2 周一,11,Maximum Phase Increase forLead Compensation,MATLAB Code:,beta=0:0.01:1;,phi_max=(180/pi)*,asin,(1-beta)./(1+beta);,plot(beta,phi_max),grid,title(Maximum Phase Increase,vs,beta),ylabel,(Maximum Phase Lead-degrees),xlabel,(beta),2026/3/2 周一,12,Key Elements of Design Approach,Translate specifications,into closed-loop bandwidth,and/or phase margin specifications.,Control bandwidth,by selecting frequency at 0 dB.,gain crossover,the“crossover frequency”.,Control phase margin,by selecting correct phase angle,at crossover.,2026/3/2 周一,13,Asymptotic Approximation of Closed-Loop Frequency Response,Closed-loop bandwidth,is approximately at,open-loop,gain crossover frequency,|GH(,jwgc,)|=1(0 dB),which is also the,Phase Margin frequency,.,2026/3/2 周一,14,Damping Ratio,vs,Phase Margin,Key Approximation:,Phase Margin,100*damping ratio,2026/3/2 周一,15,Peak Overshoot,vs,Phase Margin,Note:,Phase Margins,from 40 to 60 deg,correspond to,Peak Overshoot,from 30 to 10%.,2026/3/2 周一,16,Normalized Bandwidth,vs,Damping Ratio,Note:,1)Well-damped System:,Bandwidth Natural Frequency,2),Overdamped,System:,Bandwidth,0.5*Natural,Frequency,2026/3/2 周一,17,Lead Compensator Design,1),Adjust,compensator gain,Kc,to obtain desired crossover,frequency.,2)Select,beta,parameter to obtain enough phase lead to meet,phase margin specification.,3)Select,wmax,approximately equal to crossover frequency so,that additional lead directly contributes to phase margin.,4),Complications,arise since,gain at,wmax,is not unity,but,Kc,/,sqrt,(beta);thereby,changing crossover frequency and,phase margin.,2026/3/2 周一,18,Visualizing Lead Compensation,Lead compensation gets is name from the fact that its phase angle response is positive,I.E.,its output“leads”its input in phase.,1),Kc,is selected so that A is the desired closed-loop bandwidth.,2)wmax,is selected near A so maximum additional phase is provided near gain crossover.,3)Since compensator gain at A is not unity(0 dB.)the compensator changes crossover frequency to C.,2026/3/2 周一,19,Design Procedure:Step#1,Adjust gain,to obtain crossover frequency that approximates,desired closed-loop bandwidth.,Set crossover frequency,equal to bandwidth if,overdamped,system is desired and equal to 1/2 bandwidth if,underdamped,.,If there is no bandwidth specification:,A),Low Bandwidth Alternative,:Adjust gain to meet,phase margin spec as best you can.,B),High Bandwidth Alternative,:Adjust gain to meet,steady-state error spec as best you can.,2026/3/2 周一,20,Design Procedure:Step#2,If,phase margin spec is not met,go to Steps#3 through 7,(Lead Compensation).,If,phase margin spec is met,check steady-state error spec,and go to Steps#8 through 10(Lag Compensation).,If,both specs are met design is complete,using proportional,compensation.,2026/3/2 周一,21,Transient(Lead Compensator)Design Procedure:Steps#3-7,Step#3,:,Allow for an,extra 5 to 12 deg,of phase margin.,Calculate the additional phase lead to meet phase margin spec:,beta_lead=phase margin spec,-Step#1 phase margin result,+(5 to 12).,Step#4,:,Set Lead Compensator gain(,Klead,)equal to gain,obtained in Step#1.,2026/3/2 周一,22,Design Procedure:Step#5,Find the frequency(,wgcc,),at which the uncompensated,open-loop frequency response equals:,-20*log10(,Klead,/,sqrt,(beta_lead)=,-,Klead,(dB)+10*log10(beta_lead),wgcc,is the“corrected”gain crossover frequency.,It accounts,for the fact that the gain of the lead compensator at its,maximum phase angle is not unity;thereby changing the,gain crossover frequency.,2026/3/2 周一,23,Design Procedure:Steps#6&7,Step#6:,Set,wmax,=,wgcc,=1/(,Tlead,*,sqrt,(beta_lead).,Solve for,Tlead,:,Tlead,1/(,wgcc,*,sqrt,(beta_lead),Step#7:,Obtain Bode plot of compensated,open-loop system.,Confirm crossover frequency and gain&phase margins.,Obtain Bode plot of compensated,closed-loop system.,Confirm bandwidth&resonant-peak performance.,Tune lead compensator as necessary.,2026/3/2 周一,24,Steady-State(Lag Compensator)Design Procedure:Steps#8-10,Step#8:,Compute the open-loop gain,Kol,obtained after,completion of Steps 1-7.,Step#9:,Compute the open-loop gain,Kd,needed,to meet,the steady-state error performance specifications.,Step#10:,Compute the additional gain required,Kd,/,Kol,.,If greater than one,set gain of lag compensator:,Klag,=beta_lag=,Kd,/,Kol,.,Then,set,Tlag,=(2 to 10)/,wgcc,with the larger value preferred.This increases gain without,changing bandwidth or phase margin.,2026/3/2 周一,25,Example of Lead&LagCompensation,Problem:,Consider a,unity feedback control system,with,G(s)=1/s(s+1),which represents a simplified model of a DC motor as well,as a satellite antenna.Design a series compensator to obtain,a,steady-state error of less than 0.1 for a ramp input,with,a,peak overshoot less than 25%.,Specifications:,The velocity error coefficient,Kv,must be,=1/0.1=10,&Peak overshoot=45 deg,2026/3/2 周一,26,Example:Low-bandwidth Design,Step#1a:,Plot uncompensated,open-loop frequency response.,Gp,=,tf,(1,1 1 0);margin(,Gp,),PM=51 deg,Thus,gain can be lowered to obtain PM=45 deg by finding the frequency at which the open-loop phase angle is-180+45=-135 deg.,2026/3/2 周一,27,Example:Low-bandwidth Design,Step#1a(cont.):,magnitude,angle,frequency=bode(,Gp,1),magnitude=,0.7071,angle=,-135,frequency=,1,Kp,=1/magnitude,Kp,=,1.4142,Check Resulting Phase Margin:,Gm,Pm,Wcg,Wcp,=margin(,Kp,*,Gp,),Gm=,Inf,Pm=,45.0000,Wcg,=,NaN,Wcp,=,1.0000,2026/3/2 周一,28,Example:Low-bandwidth Design,Step#2:,Since the Phase Margin spec can be met with,proportional compensation,there is no need for lead,compensation.Checking the steady-state performance:,Kv,=,Kp,*,dcgain,(,conv,(1 0,ng,),dg),Kv,=,1.4142,This does not meeting the specified value of 10.So,we must consider a,lag compensator,and move on to,Step#8,.,2026/3/2 周一,29,Example:Low-bandwidth Design,Step#8:,The uncompensated open-loop gain,Ko,=1.,Step#9:,The desired open-loop gain,Kd,=10.,Step#10:,Kclag,=,Kd,/,Ko,Kclag,=,10,beta_lag=,Kclag,beta_lag=,10,Using the crossover frequency obtained in Step 1,Wcp,Tlag,=10/,Wcp,Tlag,=,10.0000,Lag Compensator:,ngc,=,Kclag,*,Tlag,1;,dgc,=beta_lag*,Tlag,1;,Gc,=,tf,(,ngc,dgc,),Transfer function:,100 s+10,-,100 s+1,2026/3/2 周一,30,Example:Low-bandwidth DesignClosed-loop Performance,G=,Gc,*,Gp,;,Gclosed,_loop=,minreal,(G/(1+G),2026/3/2 周一,31,Example:Low-bandwidth DesignClosed-loop Step Performance,Peak_overshoot=(max(step(,Gclosed,_loop)-1)*100,Peak_overshoot=,27.8128,Overshoot 25%due to lag,zero s=-0.1.Redesign would,increase Phase Margin slightly.,Settling Time 15 sec,2026/3/2 周一,32,Example:Low-bandwidth DesignClosed-loop Frequency Response,bode(,Gclosed,_loop),Comment:,Bandwidth slightly greater than 1,rad,/sec,2026/3/2 周一,33,Example:High-bandwidth Design,2026/3/2 周一,34,Example:High-bandwidth Design,Phase Margin of 18 deg does not meet 45 deg spec.So,lead compensation is appropriate.,2026/3/2 周一,35,Example:High-bandwidth Design,Step#3:,The additional phase lead needed to meet specifications,including a 5 to 12 degree margin,is:,gm,pm=margin(G);,phimax,=45-pm,phimax,=,27.0358,With a margin of 8 degrees,phimax,=35,phimax,=,35,So,phimaxr,=,phimax,*pi/180;,beta_lead=(1-sin(,phimaxr,)/(1+sin(,phimaxr,),beta_lead=,0.2710,2026/3/2 周一,36,Example:High-bandwidth Design,Step#4:,Kclead,=,Kp,=10,Step#5:,We need to find the frequency at which the magnitude of the uncompensated,open-loop transfer function is:,(,Kclead,/,sqrt,(beta_lead)(-1),ans,=,0.0521(-26 dB),From the Bode diagram this will be in the neighborhood of 4,rad,/sec.,Now,magnitude_,Gp,=bode(,Gp,4),magnitude_,Gp,=,0.0606,Searching in the frequency domain,wgcc,=4.3;magnitude_,Gp,=bode(,Gp,wgcc,),magnitude_,Gp,=,0.0527,This accuracy is quite adequate for design purposes,2026/3/2 周一,37,Example:High-bandwidth Design,2026/3/2 周一,38,Example:High-bandwidthStep Response,Peak_overshoot=(max(step(,Gclosed,_loop2)-1)*100,Peak_overshoot24.2226,Settling Time,1.2 sec,2026/3/2 周一,39,Example:High-bandwidthFrequency Response,Bandwidth 6,rad,/sec,2026/3/2 周一,40,Exercise 1.1,2026/3/2 周一,41,Exercise 1.2,2026/3/2 周一,42,Exercise 1.3,2026/3/2 周一,43,Exercise 1.4,2026/3/2 周一,44,Exercise 1.5,2026/3/2 周一,45,Exercise 2.1,2026/3/2 周一,46,Exercise 2.2,2026/3/2 周一,47,Exercise 3.1,2026/3/2 周一,48,Exercise 3.2,2026/3/2 周一,49,Exercise 4.1,2026/3/2 周一,50,Exercise 4.2,2026/3/2 周一,51,Exercise 4.3,2026/3/2 周一,52,Exercise 5.1,五、论述题,1,论述：系统校正的主要方法分类及其各自的特点。（5分）,2,论述：最小相位系统的基本概念，以及在控制系统性能分析中的意义。（5分）,3,论述：系统稳定性的主要判别方法及其应用特点。（5分）,2026/3/2 周一,53,Ex.1-2 p.72026/3/2 周一,54,Ex.1-3 p8,例1-3 为了保持希望的温度，又温控开关接通或断开电加热器电源。在使用热水时，水箱中流出热水并补充冷水。试说明系统工作原理并画出系统原理框图。,2026/3/2 周一,55,Ex.2-1 p20,1.列写网络微分方程,2.,已知：,L=1H,C=0.2F,R=2.5,.,设初始条件为零，,Vi(t)=10*1(t)V.,试用拉氏变换法求输出信号,Vo(t).,2026/3/2 周一,56,Ex.2-3 p24,例2-3 已知参数如图所示，求折算到电动机轴上的等效转动惯量和等效粘性摩擦系数，并导出以,M,为输入，以,2为输出的齿轮系传递函数。,2026/3/2 周一,57,Ex.4-3 p104,例4-3 已知系统的传递函数为,试绘制系统的概略幅相频特性曲线。,2026/3/2 周一,58,Ex.5-10 p146,例5-10 已知单位反馈系统的开环传递函数,试用奈氏判据判断系统的闭环稳定性。,2026/3/2 周一,59,Ex.5-13 p150,2026/3/2 周一,60,Ex.5-14 p153,例,5-14 系统开环频率特性如图,a、b,所示，试判断闭环系统的稳定性。,P=0.,2026/3/2 周一,61,Ex.5-15 p154,例5-15 已知单位反馈系统的开环传递函数,试判断系统的闭环稳定性。,2026/3/2 周一,62,Ex.5-17 p156,2026/3/2 周一,63,