第三章时间响应8-12学时.ppt

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Chapter 3,Time Response Analysis,Module 3 to Module 11,Control design methodology,Controller Design,Root-Locus PI Control,Requirement Analysis,Modeling,analytical system IDs,Dynamic model,Control algorithm,Performance Specifications,Satisfy,Design Goals,Performance Specifications,Stability,Transient response,Steady-state error,Robustness,Disturbance rejection,Sensitivity,Performance Specs,Stability,BIBO stability:bounded input results in bounded output.,A LTI system is BIBO stable if all poles of its transfer function are in the LHP(,p,i,Rep,i,0).,Performance Specs,Stability,Unstable,Stable,Performance specifications,Settling time,Overshoot,Controlled,variable,Time,Reference,%,Steady State,Transient State,Steady state error,Feedback Control,Chapter 3,Sheet 31,Response versus pole location,Transfer function,STABLE:poles 0,time constant:,example:,Feedback Control,Chapter 3,Sheet 32,0,1,2,3,4,5,0,0.2,0.4,0.6,0.8,1,Time,First-order system response,Natural response=impulse response,response,Feedback Control,Chapter 3,Sheet 33,Pole location:real roots,0,0.5,1,1.5,2,2.5,3,3.5,4,4.5,5,-0.5,0,0.5,1,1.5,2,Time,h(t),example:,Feedback Control,Chapter 3,Sheet 34,Pole locations in the s-plane,Im,Re,0,1,2,3,4,5,-15,-10,-5,0,5,Time(,secs,),Amplitude,0,1,2,3,4,5,-0.4,-0.2,0,0.2,0.4,0.6,0.8,1,Time(,secs,),Amplitude,0,1,2,3,4,5,-1,-0.5,0,0.5,1,Time(,secs,),Amplitude,0,1,2,3,4,5,0,0.5,1,1.5,2,2.5,3,Time(,secs,),Amplitude,0,1,2,3,4,5,0,0.2,0.4,0.6,0.8,1,Time(,secs,),Amplitude,0,1,2,3,4,5,-1,-0.5,0,0.5,1,Time(,secs,),Amplitude,0,1,2,3,4,5,0,0.2,0.4,0.6,0.8,1,Time(,secs,),Amplitude,0,1,2,3,4,5,0,0.2,0.4,0.6,0.8,1,Time(,secs,),Amplitude,0,1,2,3,4,5,0,2,4,6,8,10,12,14,Time(,secs),Amplitude,Feedback Control,Chapter 3,Sheet 35,Complex poles,Complex pairs,s,w,d,Re,Im,q,w,n,Feedback Control,Chapter 3,Sheet 36,Response of system with complex poles,Transfer function,impulse response,Feedback Control,Chapter 3,Sheet 37,Impulse response of second order system with complex poles,02,4,6,8,10,-1,-0.5,0,0.5,1,Time,h(t),z,=0.0,0.1,0.2,0.3,0.4,0.5,0.7,1.0,Feedback Control,Chapter 3,Sheet 38,Step response of second order system with complex poles,0,2,4,6,8,10,0,0.5,1,1.5,2,Time,z=,0.0,0.1,0.2,0.3,0.4,0.5,0.7,1.0,w,n,=1,Feedback Control,Chapter 3,Sheet 39,Pole location/damping ratio,Re,Re,Re,Im,Im,Im,Feedback Control,Chapter 3,Sheet 40,Oscillatory time responses,0,5,10,15,20,-1,-0.5,0,0.5,1,Time,h(t),Feedback Control,Chapter 3,Sheet 41,Oscillatory time responses,0,1,2,3,4,5,-2,-1.5,-1,-0.5,0,0.5,1,1.5,2,time,h(t),Feedback Control,Chapter 3,Sheet 42,Time domain specifications0,0.2,0.4,0.6,0.8,1,1.2,1.4,time,y(t),t,r,t,s,M,p,t,p,t,r,rise time,t,s,settling time,t,p,peak-time,M,p,overshoot,90%,10%,Feedback Control,Chapter 3,Sheet 43,Specifications,For second order systems:,Feedback Control,Chapter 3,Sheet 44,Overshoot versus damping ratio,z,0,0.2,0.4,0.6,0.8,1,0,0.2,0.4,06,0.8,1,z,M,p,Feedback Control,Chapter 3,Sheet 45,Specifications,w,n,Re,Re,Re,Im,Im,Im,q,s,Feedback Control,Chapter 3,Sheet 46,Specifications in the s-plane,Re,Im,example:,-5 -4 -3 -2 -1 0 1,3,2,1,0,-1,1,1,Feedback Control,Chapter 3,Sheet 47,Zeros and additional poles,for a=1 and a=2 pole cancelledwith extra zero,with extra pole,Feedback Control,Chapter 3,Sheet 48,Second order system with zero,0,2,4,6,8,10,0,0.5,1,1.5,2,Time,stepresponse,1,2,4,100,zero,a,Feedback Control,Chapter 3,Sheet 49,System with zero,H,d,(s),0,2,4,6,8,10,-0.5,0,0.5,1,1.5,2,Time,stepresponse,H,d,(s),H,0,(s),H,(,s),H,0,(s),Feedback Control,Chapter 3,Sheet 50,Nonminimum,-phase,behaviour,stepresponse,zero:,RHP,0,2,4,6,8,10,12,-1.5,-1,-0.5,0,0.5,1,1.5,Time,H,d,H,0,H,Feedback Control,Chapter 3,Sheet 51,Aircraft response,Altitude control of Boeing-747,0,0.5,1,1.5,2,2.5,3,-5,0,5,10,15,20,Time,Altitude,theory,MATLAB,Feedback Control,Chapter 3,Sheet 52,Additional pole,0,2,4,6,8,10,0,0.2,0.4,0.6,0.8,1,1.2,Time,y(t),a,1,2,5,100,Feedback Control,Chapter 3,Sheet 53,Effects of poles and zeros,Second order system,rise time:,settling time:,overshoot:,zero:,overshoot,if zero 4.,s,if zero in RHP,pole:,rise time,if pole 4.,s,Feedback Control,Chapter 3,Sheet 54,Obtaining models from experiments,from transients,frequency response,stochastic response,step/impulse,sinusoidal signals,noise signals,transient response,Feedback Control,Chapter 3,Sheet 55,0,1,2,3,4,5,6,0,0.2,0.4,0.6,0.8,1,0,1,2,3,4,5,6,-5,-4,-3,-2,-1,stepresponsetime,ln,(1-y(t),time,y(t),0,Feedback Control,Chapter 3,Sheet 56,0,1,2,3,4,5,6,-25,-20,-15,-10,-5,0,0,1,2,3,4,5,6,0,0.2,0.4,0.6,0.8,1,time,time,ln,(1-y(t)-,Ae,-,a,t,),y(t),y(t),Feedback Control Systems,Sheet 16,Chapter 1,Poles&Zeros,Zeros of G(s)roots numerator,Poles of G(s)roots denominator,Characteristis,equationdenominator of G(s)=0,Im,Re,pole-zero pattern,poles,zeros,Feedback Control Systems,Sheet 17,Chapter 1,Transient Response,partial fraction expansion:,residues:,response:,0,1,2,3,4,5,0,1,Time(,secs,),c,Feedback Control Systems,Sheet 18,Chapter 1,Response to initial condition,Differential equation:,with initial conditions:,Laplace,transform:,Feedback Control Systems,Sheet 19,Chapter 1,Transient Response,inputr(t),responsec,1,(t),initial condition c(0)=0,inputr(t)=0,responsec,2,(t),initial condition c(0),total response,+,c,1,(t)+c,2,(t),Feedback Control Systems,Sheet 20,Chapter 1,Graphical Determination of Residues,Im,Re,.,.,-,a,b,A,a,Example,Im,Re,-3 -2 -1,Feedback Control Systems,Sheet 21,Chapter 1,Responses with Repeated Poles,with residues,Feedback Control Systems,Sheet 22,Chapter 1,Example,First order system:,Input:ramp function,G(s),R(s),C(s),Output:,0,2,4,6,8,10,0,2,4,6,8,10,Time(,secs,),c,T,T,r,c,Feedback Control Systems,Sheet 23,Chapter 1,Complex conjugate poles,System:,Stepresponse,:,Poles:,Im,Re,Feedback Control Systems,Sheet 26,Chapter 1,Example:Third order system,transfer function,stepresponse,Im,Re,0.8,j,-0.5,-2,28,0,58,0,0,2,4,6,8,10,0,0.2,0.4,0.6,0.8,1,12,Time(,secs,),c,Example,Utilization control in a video server,Periodic task T,i,corresponding to each video stream i,ci:processing time,pi:period,Stream is requested CPU utilization:ui=ci/pi,Total CPU utilization,:U(t)=,k,uk,k is the set of active streams,Completion rate,:,R,c,(t)=(,kc,um)/,t,where m is the set of terminated video streams during t,t+,t,Unknown,Admission rate,:R,a,(t)=(,ka,uj)/,t,where j is the set of admitted streams during t,t+,t,Problem:design an admission controller to guarantee U(t)=,U,s,regardless of,R,c,(t),Model,Differential equation,U(t),R,a,(t),C?,U,s,-,CPU,R,c,(t),Model(differential equation):,Error:E(t)=U,s,-U(t),Controller C?E(t),R,a,(t),Three ways of system modeling,A Diverge to Math,System representations,u(t),g(t),y(t),Time domain:convolution;differential equations.,U(s),G(s),Y(s),s(frequency)domain:multiplication,s-domain is a simple&powerful“language”for control analysis,Block diagram:pictorial,Laplace,transform of a signal f(t),A Diverge to Math,Laplace,transform,where s=,+i is a complex variable.,Laplace,transform is a translation from time-domain to,s-domain,Differential equation,Polynomial function,Basic translations,Impulse functionf(t)=,(t)F(s)=1,Step signal,f(t)=a1(t),F(s)=1/s,Ramp signalf(t)=a,t F(s)=a/s,2,Exp signal,f(t)=e,at,F(s)=1/(s-a),Sinusoid signalf(t)=sin(at)F(s)=a/(s,2,+a,2,),Composition rules,Linearity,L,af,(t)+,bg,(t)=,a,L,f(t)+,b,L,g(t),Differentiation,L,df,(t)/,dt,=,sF,(s)f(0,-,),Integration,L,t,f,()d,=F(s)/s,A Diverge to Math,Laplace,transform,A Diverge to Math,Transfer function,Modeling a linear time-invariant(LTI)system,G(s)=Y(s)/U(s),Y(s)=G(s)U(s),U(s),G(s),Y(s),E.g.,a second order system with,poles,p,1,and p,2,A Diverge to Math,Poles and Zeros,The response of a linear time-invariant(LTI)system,p,i,are,poles,of the function and decide the system behavior,A Diverge to Math,Time response vs.pole location,f(t,)=,e,pt,p,=,a,+,bj,Unstable,Stable,A Diverge to Math,Block diagram,A pictorial tool to represent a system based on transfer functions and signal flows,Represent a feedback control system,C(s),R(s),Y(s)G,o,(s),R(s),Y(s),G,c,(s),Back to,Our utilization control example,U(t),R,a,(t),C?,U,s,-,CPU,R,c,(t),Model(differential equation):,Error:E(t)=U,s,-U(t),Controller C?E(t),R,a,(t),Model,Transfer,func,.&block,diag,.,Inputs:reference U,s,(s)=U,s,/s;completion rate,R,c,(s),Close-loop system transfer functions,U,s,(s)as input:G,1,(s)=C(s)G,o,(s)/(1+C(s)G,o,(s),R,c,(s)as input:G,2,(s)=G,o,(s)/(1+C(s)G,o,(s),Output:U(s)=G,1,(s)U,s,/s+G,2,(s),R,c,(s),CPU is modeled as an integrator,R,c,(s),G,o,U,s,/s,R,a,(s),U(s),C(s),Control design methodology,Controller Design,Root-Locus PI Control,Requirement Analysis,Modeling,analytical system IDs,Dynamic model,Control algorithm,Performance Specifications,Satisfy,Design Goals,Performance Specifications,Stability,Transient response,Steady-state error,Robustness,Disturbance rejection,Sensitivity,Performance Specs,Stability,BIBO stability:bounded input results in bounded output.,A LTI system is BIBO stable if all poles of its transfer function are in the LHP(,p,i,Rep,i,0).,Performance Specs,Stability,Unstable,Stable,Performance specifications,Settling time,Overshoot,Controlled,variable,Time,Reference,%,Steady State,Transient State,Steady state error,Example:Control&Response in an Email Server(IBM),Control,(,MaxUsers,),Response,(queue length),Good,Slow,Bad,Useless,Performance Specs,Steady-state error,Steady state(tracking)error of a stable system,r(t)is the reference input,y(t)is the system output.,How accurately can a system achieve the desired state?,Final value theorem,:if all poles of,sF,(s)are in the open left-half of the s-plane,then,Easy to evaluate system long term behavior without solving it,Performance Specs,Steady-state error,Steady state error of a CPU-utilization control system,U(t),e,ss,=-20%,U,s,Performance SpecsRobustness,Disturbance rejection,:steady-state error caused by external disturbances,Can a system track the reference input despite of external disturbances?,Denial-of-service attacks,Sensitivity,:relative change in steady-state output divided by the relative change of a system parameter,Can a system track the reference input despite of variations in the system?,Increased task execution times,Device failures,Performance Specs,Goal of Feedback Control,Guarantee stability,Improve transient response,Short settling time,Small overshoot,Small steady state error,Improve robustness,wrt,uncertainties,Disturbance rejection,Low sensitivity,Control design methodology,Controller Design,Root-Locus PID Control,Requirement Analysis,Modeling,analytical system IDs,Dynamic model,Control algorithm,Performance Specifications,Satisfy,Controller Design,PID control,Proportional-Integral-Derivative(PID)Control,C(s),R(s),Y(s),-G,o,(s),E(s),X(s),Proportional Control,Integral control,Derivative control,Classical controllers with well-studied properties and tuning rules,Controller Design,CPU Utilization Control,Inputs:set-point U,s,(s)=U,s,/s;task completion,R,c,(s),Close-loop system transfer functions,U,s,(s)as input:G,1,(s)=C(s)G,o,(s)/(1+C(s)G,o,(s),R,c,(s)as input:G,2,(s)=G,o,(s)/(1+C(s)G,o,(s),C(s)=?to achieve zero steady-state error:U(t),U,s,CPU is modeled as an integrator,R,c,(s),G,o,U,s,/s,R,a,(s),U(s),C(s),Proportional Control,Stability,Proportional Controller,r,a,(t)=,Ke,(t);,C(s)=K,Transfer functions,U,s,/s as input:G,1,(s)=K/(s+K),R,c,(s)as input:G,2,(s)=1/(s+K),Stability,Pole p,0,=-K0,Note:System may shoot to 100%if K0!,R,c,(s),G,o,U,s,/s,R,a,(s),U(s),C(s),Proportional Control,Steady-state error,Assume completion rate,R,c,(t)keeps constant for a time period longer than the settling time:,R,c,(s)=,R,c,/s,System response is,Compute steady-state err using final value theorem,P-control cannot achieve the desired CPU utilization U,s,;instead it will end up lower by,R,c,/K,Oops!,The larger the proportional gain K is,the closer will CPU utilization approach to U,s,CPU Utilization,Proportional ControlU(t),e,ss,=-20%,U,s,Proportional-Integral Control,Stability,Proportional Controller,r,a,(t)=K(e(t)+,K,i,t,e,(,)d,),C(s)=K(1+,K,i,/s),Transfer functions,U,s,/s as input:G,1,(s)=,(Ks+,KK,i,)/(s,2,+Ks+,KK,i,),R,c,(s)as input:G,2,(s)=s/(s,2,+Ks+,KK,i,),Stability,Poles Rep,0,0,Rep,0,0&,K,i,0,R,c,(s),G,oU,s,/s,R,a,(s),U(s),C(s),Proportional Control,Steady-state error,Assume completion rate,R,c,(t)keeps constant for a time period longer than the settling time:,R,c,(s)=,R,c,/s,System response is,Compute steady-state err using final value theorem,PI control can accurately achieve the desired CPU utilization U,s,Control analysis gives design guidance,CPU Utilization,Proportional-Integral Control,U(t),e,ss,=0,t,s,t,r,t,p,M,p,=0,U,s,Controller Design,Summary&pointers,PID control:simple,works well in many systems,P control:may have non-zero steady-state error,I control:improves steady-state tracking,D control:may improve stability&transient response,Linear continuous time control,Root-locus design,Frequency-response design,State-space design,G.F.Franklin et.al.,Feedback control of dynamic systems,Discrete Control,More useful for computer systems,Time is discrete;sampled system,denoted k instead of t,Main tool is z-transform,f,(,k,),F,(,z,),where,z,is complex,Analogous to,Laplace,transform for s-domain,Discrete Modeling,Difference equation,V,(,m,)=,a,1,V,(,m,-1)+,a,2,V,(,m,-2)+,b,1,U,(,m,-1)+,b,2,U,(,m,-2),z,domain:,V,(,z,)=,a,1,z,-1,V,(,z,)+,a,2,z,-2,V,(,z,)+,b,1,z,-1,U,(,z,)+,b,2,z,-2,U,(,z,),Transfer function,G,(,z,)=(,b,1,z,+,b,2,)/(,z,2,-a,1,z,-,a,2,),V,(,m,):output in,m,th,sampling window,U,(,m,):input in,m,th,sampling window,Order,n,:#sampling-periods in history affects current performance,SP=30 sec,and n=2,Current system performance depends on previous 60 sec,Root Locus analysis of Discrete Systems,Stability boundary:|,z,|=1(Unit circle),Settling time=distance from Origin,Speed=location relative to,Im,axis,Right half=slower,Left half=faster,Effect of discrete poles,|,z|,=1,Longer settling time,Re(s),Im,(s),Unstable,Stable,Higher-frequency,response,Feedback control works in CS,U.Mass:network flow controllers(TCP/IP RED),IBM:Lotus Notes admission control,UIUC:Distributed visual tracking,UVA,Web Caching,QoS,Apache Web Server,QoS,differentiation,Active queue management in networks,Processor thermal control,Online data migration in network storage(with HP),Real-time embedded networking,Control middleware,Feedback control real-time scheduling,Advanced Control Topics,Robust Control,Can the system tolerate noise?,Adaptive Control,Controller changes over time(adapts),MIMO Control,Multiple inputs and/or outputs,Stochastic Control,Controller minimizes variance,Optimal Control,Controller minimizes a cost function of error and control energy,Nonlinear,systems,Neuro,-fuzzy control,Challenging to derive analytic results,Issues for Computer Science,Most systems are non-linear,But linear approximations may do,eg,fluid approximations,First-principles,modeling,is diffi