1、Chapter 8 The Design of Feedback Control System8-1 Review on the Performance Indices and their Regulation8-2 Concept of Compensation8-3 Compensation Networks A.dynamic performance indices(for transient response of underdamped systems)rise time peak time settling time P.O.rdt21pdnt2/1%100%e3 4snt rts
2、tpt%8-1 The Performance Indices and their Regulations Note:P.O.depends only on !1.dynamic indices in the time domain and their regulationsTime trRisePeak time tpABP.O.%=AB100%Settling time tsDynamic performance indices time constant attenuation factor damped frequency phase angle damping ratio natur
3、e frequencyB.more indices and parametersnj021dnn arccos1n1nd21dnn (poles moving left,more stable)Note:and depend only on !(more steady)(faster)，(faster,not synchronously with and )then:，(responds slower)if fixed,and (more damped)C.The regulation of performance indicesrtptptstrtd%n (more steady)%Prin
4、ciples:determining according to the specification of first,and then determining according to other requirements!(faster)，(faster,synchronously with and )then:，(faster)if fixed,and (less steady)C.The regulation of performance indicesrtptptstrtdn (less steady)%n peak magnitude (attained at resonant fr
5、equency)(less steady)resonant frequency (when )(faster)phase margin ：，(proportional to )bandwidth (faster)D.The regulation of frequency domain specificationsptpMptb0.707prtprtbpM%100 (1)Attaching extra CL zeros that can not be neglected,results the effect of decreasing of the damping ratio and other
6、 corresponding effects.The nearer the zeros approach the origin,the more remarkable the effects are.E.other measures regulating performance indicesY(s)R(s)Gu(s)+-H(s)KGc(s)()1cG sTs (2)Attaching extra CL poles that can not be neglected,results the effect of increasing of the damping ratio and other
7、corresponding effects.The nearer the zeros approach the origin,the more remarkable the effects are.Y(s)R(s)Gu(s)+-H(s)KGc(s)1()1cG sTs (3)Attaching extra OL zeros will move the CL root locus to the left.This is favorable to the increasing of phase margin,the system stability,etc.Y(s)R(s)Gu(s)+-H(s)K
8、Gc(s)()1cG sTs (4)Attaching extra OL poles will move the CL root locus to the right.This is unfavorable to the system stability.It is not used in general.These measures will be studied in detail in the coming lectures.Y(s)R(s)Gu(s)+-H(s)KGc(s)1()1cG sTs (5)The moving modes of higher order systems ar
9、e determined by the poles.The farther one pole is from the imaginary axis,the less its impact on the transient dynamic behavior.(6)The transient behavior of a higher order system is mainly determined by the dominant poles.The nearer they are approaching the imaginary axis,the more dominant they are,
10、and therefore,the better the lower-order approximate model is.Lets go to the review on the steady-state performance.2.Steady-state performance indices and their regulation A.Steady-state performance indices Error constants.They are defined for different test input signals,namely:position error const
11、ant velocity error constant acceleration error constant The bigger the error constants are,the better the steady-state performance is.0 A1(t)A1+kA kA kAt000At2/2A1(t)AtAt2/2kkk000?123Kp=?Kv=?Ka=?Error ConstantsSteady-state Error B.Regulating the steady-state performance 1.Type number V(attaching OL
12、poles at origin)V tracking ability stability 2.OL gain K (most widely used)K Steady-state error affect stability and transient performance 3.disturbance compensation 4.choosing high quality devices C、in the frequency domain:directly represented by the phase margin and magnitude margin ,stability 3.R
13、egulating on the stability(relative)A.in the S plane:directly represented by the attenuation factor stability gkgk B.OL gain K the impact of K on the stability can be judged by using the root locus.D.Attaching extra OL zeros is favorable to the system stability,etc.4.From OL to CL Some indices and p
14、arameters are defined,and some methods are developed from OL transfer function,but used for the CL analysis and design.They are:Phase margin Magnitude margin Break frequency Type number V OL gain K Root locus method Nyquist criterion Nicolis plotcgK8-2 Concept of compensation1.Why do we need compens
15、ation?All we have learnt about the system modeling,system analysis,and etc.are preparations for the final object:Design and implement satisfactory control systems!Lets restudy an example first.Example 4.4(Proportional regulation):Given the OL transfer function of an unit negative feedback system as:
16、)5.34(5)(ssKsGA let the input to be unit step signal,calculate the transient performance indices of the system when the gain is KA=200.Investigate the change of the performance when KA is increased to 1500,or decreased to 13.5?We can not decease the settling time and P.O.at the same time only by adj
17、usting K.They are changing contradictorily,or the design requirements are conflict with each other.First choice in control system design:try our best to optimize the process itself by selecting good devices,carefully adjusting its parameters,etc.But,there are limitations on the regulation ability an
18、d possibility of the adjustment of parameters.We have to consider the change of the structure of the system and redesign it in order to obtain desired performance.That is why we need compensations.8-2 Concept of Compensation 1.Definition&Types Definition Redesigning or adjusting the system structure
19、 to alter the system response,in order to make up for deficiencies or inadequacies,by inserting additional components or devices within the structure of the feedback control system.The additional components are called compensators.Compensation types（1）Cascade （2）Feedback（3）Prefilter （4）Noise Process
20、Feedback CompensationRYPrefilter Compensation ProcessRYControllerProcess Noise CompensationRYN 2.Choose the Type of Compensation Often used:Cascade Compensation&Feedback Compensation.The choice depends on many factors,it is easier to design Cascade Compensation,so it is mostly preferred,especially i
21、n class.In the case of high standard system,Cascade Compensation and Feedback Compensation will be used at the same time.3.Approaches to System Design The performance requirements are of great importance to control system design.It can be described in terms of the time-domain or frequency-domain ind
22、ices,thus,approaches to control system design are usually determined by the form of the given or required specifications.If the performance requirements are given in time-domain,we adopt the root locus method to design the compensator to be added.If the performance requirements are given in frequenc
23、y-domain,we design the compensator to be added using Bode diagram.Often,we need transfer the set of time-domain requirements to frequency-domain requirements,or reverse.8-3 Compensation Networks1.Phase-lead network（differentiator-type）Transfer function is1()1caTsaGsTsR2R1V2V121()1 1()1VsaT sVsaT sTh
24、us:122RRaRWhere:CCRRRRT2121 The phase-lead compensation in cascade will results:1.The open-loop gain of the system will be attenuated by 1/a,thus,for an acceptable final design,raise the gain of the amplifier in order to account for the attenuation.2.The closed-loop root locus will be reshaped leftw
25、ard,this will be in favor of the system stability.j021()1 1()1V saTsV saTsPole-zero diagram of the phase-lead network20db0.1/aT1/aT10/aT 45 90 w 222221lg201lg20TTaL TtgaTtg111/T20lga10lgammm1mTa()0dd 112matga 0 L2.Phase-lag Network（integrator-type）Transfer function is:where：Phase-lag network has a f
26、requency response like an integrator over a finite range of frequencies,the smaller b is,the heavier the attenuation in high frequency band is.R2R1V2V1 2111VsbTsVsTs12()TRR C212RRRb-20db0.1/T1/T 10/T-45-90 222221lg201lg20TTbL TtgbTtg111/bT20lgb10lgbmmm()0dd 01mTb112mbtgb L We can easily tell from th
27、e Bode diagram,the maximum phase lead or lag angel occurs at .Motivation phase-lead network results in adding a phase-lead angel and amplifying the magnitude to the OL frequency response,thus,it will raise the phase margin and the dumping ratio of the compensated CL system,altering and improving the
28、 transient response.Similarly but with evident difference,phase-lag network results in attenuation in magnitude and a phase-lag angel instead within a range of frequency,so,it is mainly used to improve the steady-state accuracy of the compensated CL system.m3.Phase-lead&Phase-lag Network(1)(1)()1,1(1)(1)abcabaT sbT sG sabT sT s
