Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference r r K u G y Figure Open loop conrol yem Figure how a andard feedbac conrol yem wih he ame purpoe ha he oupu y follow he reference r r e K u G y Figure Cloed loop feedbac conrol The performance of a conrol yem are precribed in erm of - he abiliy of he yem - range of eady-ae error e lim e lim r y - deired maximum allowed percen overhoo POV < POV - deired maximum allowed eling ime < - deired maximum rie ime r < r n hi lecure we are only concerned wih he abiliy and he eady ae error performance Deign of conroller which aify he re of he performance will be dicued in fuure lecure Open loop conrol reduce o finding he deired conrol inpu rajecory which will produce a deired oupu of he conrolled yem n effec i require accurae nowledge on he model of he yem a lea of he aic characeriic of he yem Uing a inpu of he model he deired oupu of he yem, he model will oupu a ignal which, when applied a he yem inpu, will produce ha pecified deired oupu Tae for example he eady ae error for an open loop conrol yem when he reference i a uni ep hen e lim e lim r y lim Y 0 0 lim [ KG ] lim[ KG ] K0 G0 0
Thu he eady ae error in hi cae i zero only if K0 / G0 ie exac nowledge on he DC gain of he yem i available Le for a momen conider ha we now exacly he value of G 0 even if hi i no generally rue Wha if an exernal diurbance ignal affec i added o he conrol inpu lie hown in Figure 3? d r K u G y Figure 3 Open loop conrol yem wih diurbance Calculae he value of he eady ae error if he diurbance i a uni ep See ha in hi cae he eady ae error will never be zero Feedbac conrol algorihm or feedbac conroller calculae he conrol inpu baed on he error difference beween he deired yem oupu ie he reference and he meaured preen yem oupu in he ene of cancelling he error difference Can uch conroller cancel he eady ae error in he preence of uncerainie on he yem dynamic and/or in he preence of exogenou diurbance? f o, how can we deign uch conroller? Seady-ae error The eady ae error i defined a he value of he error ignal e r y a The eady ae error i hown in he figure for a uni ep and a uni ramp reference y e r 0 Figure 4 Nonzero eady-ae error o a ep reference
e y r 0 Figure 5 Nonzero eady-ae error o a ramp reference Feedbac can improve he racing capabiliie of a plan by maing he eady-ae error maller, preferably zero The cloed-loop ranfer funcion i given by remember Maon formula Y KG T KG However, o find he eady-ae error e lim e, one deermine E Y T KG To find he eady-ae error in repone o a uni ep reference, elec Final Value Theorem o obain e e E KG lim lim lim 0 0 and ue he Example Given he yem in he nex figure d r e K G y Figure 6 where he plan i G and he conroller i he inegral compenaor K wih he inegral gain, calculae he eady ae error in repone o a uni ep reference and a uni ep diurbance 3
Noe ha if he feedbac loop ha a gain differen ha hen he error ignal will no how up in he diagram Thu even if in hi cae he error can be calculaed uing Maon formula in general i i beer o ue he definiion on page a Find he Seady-Sae Error in epone o a Uni Sep Diurbance d To find he requeed eady-ae error, e 0 and D / The ranfer funcion from d o e i given by G E D KG E D D Uing now he Final Value Theorem yield e lim E lim 0 0 0 Noe ha hi mean he oupu y goe o zero, ince he reference i r0 b Find he Seady-Sae Error in epone o a Uni Sep eference r To find he requeed eady-ae error, e / and D 0 The ranfer funcion from r o e i given by E KG E Uing now he Final Value Theorem yield e lim E lim 0 0 0 Noe ha hi mean he oupu y goe o one, ince he reference i ru - c Find he POV For a Uni Sep eference The cloed-loop ranfer funcion from he command r o he oupu y i given by GK Y GK 4
5 Y The cloed-loop characeriic polynomial i Comparing hi o he andard form n n ω ζω one ee ha n n ω ζ ω One can herefore elec he inegral gain o obain any deired value of damping raio, and hence of POV which i given by 00% / ζ πζ e POV d Find he Oupu y if, u e d u e r The ranfer relaion beween he wo inpu and he oupu i given by GK G Y D GK GK or D D Y Seing now, D one obain Y For he given value of one may now ue he invere Laplace ranform o deermine he oupu y
Feedbac conrol yem deign for zero eady ae error n general, o follow a reference wih zero eady-ae error, he pah beween he reference and he yem oupu hould conain a erm lie For inance, o follow a ramp velociy reference / one require a lea wo inegraor in he pah from he reference o he oupu Tha i, for zero eady-ae error in repone o a given reference, he conrol yem hould conain a model of he deired reference rajecory Alo, o rejec he eady ae componen of a diurbance of nonzero mean he pah beween he reference and he diurbance inpu hould conain a erm lie D Thee wo rule expre he inernal model principle Problem Conider he yem in he figure d r e K G y Figure 7 Le he plan be given in he form m b0 b bm n G n a a n d N n N The compenaor i given a K, GK, and aume ha he plan doe no have N d N any pole in zero A Le D 0 and r, r 0 Wha i he minimum value of N uch ha he eady ae error i zero? B Le 0 and D d, d 0 Wha i he minimum value of N uch ha he eady ae error i zero? n C epea poin A and B conidering ha G, g 0 d g The advanage of feedbac conrol - a well deigned conroller can cancel he eady ae error uch ha he oupu of he yem will follow, afer he ranien repone i finihed, he deired pecified oupu ie he reference, even in he preence of uncerainie relaive o he model of he yem or he appearance of exogenou diurbance Some co of feedbac conrol - increaed complexiy of he conrol yem: enor are required and enor have heir own dynamic; moreover one ha o deal wih meauremen noie 6
- in ome cae feedbac can mae he yem unable for pecific conroller rucure and cerain value of he conroller parameer Sable feedbac conrol Bad deign of feedbac conroller can lead o an unable cloed loop yem Thu while chooing a rucure for he conroller you mu alway chec o ee if he cloed loop yem will remain able you can ue ouh e on he characeriic equaion or calculae he pole of he yem and pecify he condiion uch ha all he pole are in he lef half -plane and for which value of conroller parameer The nex example illurae hi idea Problem Tae he feedbac conrol yem in he figure d r e K G y Figure 8 where D 0, K a proporional conroller, and he yem ha he ranfer funcion G a Calculae he pole and he zero of he yem b he yem minimum phae? Y c Calculae he ranfer funcion of he cloed loop yem H d For which value of will he cloed loop yem remain able? Noe ha here he conroller i a proporional conroller e Calculae he eady-ae error in repone o a uni ep reference f n order o obain zero eady-ae error we ae he conroller o be K Thi i an inegral conroller According o he inernal model principle we now ha we need an inegraor in he conroller Bu will i be able o alo abilize he yem? f For which value of will he yem be able? f Calculae he eady ae-error in repone o a uni ep reference g Le he conroller be K p Thi conroller i a P proporional and inegraor g For which value of will he yem be able? g Calculae he eady ae error in repone o a uni ep reference 7