ELEC 372 LECTURE NOTES, WEEK 1 Dr. Amir G. Aghdam Concordia University

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1 EEC 37 ECTURE NOTES, WEEK Dr Amir G Aghdam Cocordia Uiverity Part of thee ote are adapted from the material i the followig referece: Moder Cotrol Sytem by Richard C Dorf ad Robert H Bihop, Pretice Hall Feedback Cotrol of Dyamic Sytem by Gee F Frakli, J David Powell ad Abba Emami-Naeii, Pretice Hall Automatic Cotrol Sytem by Farid Golaraghi ad Bejami C Kuo, Joh Wiley & So, Ic, Itroductio - Sytem: A itegrated whole eve though compoed of divere, iteractig, pecialized tructure ad ub-juctio Note: Ay ytem ha a umber of objective ad the weight placed o them may differ widely from ytem to ytem () A ytem perform a fuctio ot poible with ay of the idividual part Complexity of the combiatio i implied (IEEE Stadard Dictioary of Electrical ad Electroic Term) - A ytem ca be a idutrial proce, automobile, airplae, etc I a wider ee, the cocept of ytem ca be exteded to ecoomical, biological ad ocial procee a well - A ytem (proce or pla i mathematically characterized by a operatio betwee iput() ad output() ad i deoted a follow: x( Itput Sytem y( Output y( = f ( x( ) x( y( ecture Note Prepared by Amir G Aghdam

2 - Cotrol ytem: A ytem i which a deired effect i achieved by operatig o the variou iput to the ytem util the output, which i a meaure of the deired effect, fall withi a acceptable rage of value (IEEE Stadard Dictioary of Electrical ad Electroic Term) - I oe etece, cotrol i the proce of makig a ytem variable adhere to a particular value, called the referece value (G F Frakli, J D Powell ad A E Naeii, Feedback Cotrol of Dyamic Sytem) - A cotrol ytem i a ytem that will provide a deired ytem repoe - Cotrol i foud i all ector of idutry ad ometime i referred to a a hidde techology becaue of it eetial importace to o may device ad ytem while beig maily out of ight (G F Frakli, J D Powell ad A E Naeii, Feedback Cotrol of Dyamic Sytem, 6 th Editio, Pretice Hall, ) - Example of cotrol ytem iclude uality cotrol of maufactured product, automatic aembly lie, machie-tool cotrol, pace-vehicle ytem, miile guidace ytem, robotic ytem, dik drive, traportatio ytem, power ytem, MicroElectroMechaical Sytem (MEMS), aotechology - More recet applicatio of moder cotrol theory iclude uch o-egieerig ytem a biological, biomedical, ad ecoomic ytem - A a imple example, coider the followig water-tak ytem h - I the water tak height cotrol ytem it i deired to cotrol the height of the water i the tak ( h ), which i alo called cotrolled variable, by chagig the flow rate of the iput water ( ) ad/or the flow rate of the output water ( ), which are alo called maipulated variable - Differet clae of cotrol ytem iclude: ecture Note Prepared by Amir G Aghdam

3 3 Multi-iput multi-output (MIMO) cotrol ytem veru igle-iput igleoutput (SISO) cotrol ytem Dicrete-time cotrol ytem veru cotiuou-time cotrol ytem Ope-loop cotrol ytem veru cloed-loop cotrol ytem - A example of MIMO cotrol ytem i give below h h 3 - Robot are example of dicrete-time cotrol (computer-cotrolled) ytem - A example of a cloed-loop cotrol ytem i the automobile, which i cotrolled by the driver The driver make chage i teerig ad velocity by meaurig the relative poitio of the car through hi/her eye ad comparig it with the deired poitio - Figure (a), (b) how the ope-loop ad cloed-loop cotrol of the height of the water A fixed poitio for the valve h Figure (a): Ope-loop cotrol of the height of the water ecture Note Prepared by Amir G Aghdam

4 4 A DC motor whoe output i a fuctio of the height of the water A floatig device to meaure height of the water through a potetiometer h Figure (b): Cloed-loop cotrol of the height of the water - Block diagram of a baic cloed-loop cotrol ytem i a follow Iput (deired outpu + Error Actual - Cotrol device Actuator Proce output Meaured output Seor Feedback - Regulator: A ytem deiged to maitai a output fixed agait ukow diturbace i called a regulatig cotrol or a regulator For example the cruie cotrol of a automobile i a regulator (G F Frakli, J D Powell ad A E Naeii, Feedback Cotrol of Dyamic Sytem) - Servo: A ytem deiged to follow a chagig referece igal i called trackig cotrol or a ervo (G F Frakli, J D Powell ad A E Naeii, Feedback Cotrol of Dyamic Sytem) For example a ati-aircraft miile cotrol ytem i a ervo - Typical cotrol objective: Good traiet repoe Good teady-tate repoe Diturbace rejectio Robute ecture Note Prepared by Amir G Aghdam

5 5 Mathematical Foudatio - iearity: A liear ytem i a ytem that poee the property of uperpoitio, which i for ay cotat value a ad b, the followig euatio i atified: x ( y(, x( y( ax ( + bx( ay( + by ( - It ca be eaily verified that for liear ytem, a iput which i zero at all time, reult i a output which i zero at all time - A example of a liear ytem: y ( = t x( - Time ivariace: A ytem i time ivariat if the behaviour ad characteritic of the ytem are fixed over time Time ivariat ytem have the followig property: x( y( x( t t) y( t t) - A example of a time ivariat ytem: y ( = x ( - Cauality: A ytem i caual (or oaticipative) if the output at ay time deped oly o value of the iput at the preet time ad i the pat - A capacitor i a example of a caual ytem whoe iput ad output are related through the euatio: t v( = i(τ ) dτ C - Bouded-Iput Bouded-Output (BIBO) Stability: A BIBO table ytem i oe i which a bouded iput reult i a bouded output - A reitor i a example of a BIBO table ytem whoe iput ad output are related through the followig euatio: v ( = Ri( - A cotat amplifier y ( = K x( i alo a example of a BIBO table ytem - Covolutio: For liear time ivariat (TI) ytem, the iput x (, output y ( ad the impule repoe (the output whe the iput i a uit impule fuctio) h ( are related through the followig euatio which i called covolutio + y( = x( * h( = x( τ ) h( t τ ) dτ = x( t τ ) h( τ ) dτ + ecture Note Prepared by Amir G Aghdam

6 6 - Thi mea that a TI ytem ca be completely characterized by it impule repoe - For TI ytem with impule repoe h ( : Cauality h ( =, t < BIBO tability + h ( < - aplace traform: Give a real-valued time fuctio x( R, the uilateral aplace traform i defied a follow: t X = x t = + ( ) ( ( )) x( e, C, X ( ) C - The aplace traform pair i alo deoted by x( X ( ) - Not every igal ha a aplace traform For example, igal that grow fater tha e at, a, uch a t e do ot have a aplace traform - Propertie of aplace traform: For a caual igal (a igal which i eual to zero for all t < ), we have: iearity: a x ( a x ( a X ( ) + a X ( ), a a C +, t Time hiftig: x t t ) e X ( ), t > ( Time calig: x( a X ( ), a > a a Covolutio property: x x ( X ( ) X ( ) Differetiatio i the time domai: d x( X ( ) ( d x( ) d x( ) x( ) I particular: dx( d x( dx( ) X ( ) x( ), ad X ( ) x( ) d Differetiatio i the -domai: tx( X ( ) d t Itegratio i the time domai: x( τ ) dτ X ( ) ecture Note Prepared by Amir G Aghdam

7 7 Iitial value theorem: For a igal x ( which cotai o igularity at t =, + we have x( ) = lim X ( ) Fial value theorem: A igal for which X () i aalytic for all o the imagiary axi ad the right half plae, ha the followig property: lim x( = lim X ( ) t - The aplace traform of ome importat fuctio are give below ( δ ( ad u ( repreet uit impule fuctio ad uit tep fuctio, repectively) δ ( t u( ( )! e u( + a at (coω u( + ω ω (i ω u( + ω - Example : Fid the olutio of the followig differetial euatio: dy( + y( = δ (, y( ) = Solutio: By takig the aplace traform of both ide of the above differetial euatio, we will get the followig algebraic euatio: Y ( ) y( ) + Y ( ) = Thi reult i: t Y ( ) = y( = e u( + - I thi coure, we are maily cocered with the ytem whoe iput ad output are related through a liear cotat coefficiet differetial euatio of the followig form: ecture Note Prepared by Amir G Aghdam

8 8 - If N = M - If N > M trictly proper N = a d y( M = m= b m m d x(, m, the the ytem i called proper a N =, N M, the the ytem i called trictly proper All phyical ytem are - Trafer fuctio: The trafer fuctio of a TI ytem i defied a the ratio of the aplace traform of the output variable to the aplace traform of the iput variable, with all iitial coditio aumed to be zero - The trafer fuctio of a ytem with the differetial euatio give by () i a ratioal fuctio of a follow: H ( ) : M M Y ( ) bm + bm + + b + b = : = N N X ( ) a + a + + a + a = N N b( ) a( ) - The root of b () are called the zero ad the root of a () are called the pole of the trafer fuctio - The behaviour of a ytem deped highly o the locatio of it pole ad zero - A caual ytem with a ratioal trafer fuctio ad N M i BIBO table if ad oly if all of it pole lie i the ope left-half plae (HP) For practical reao, a ytem with imple pole o the imagiary axi ad o pole i the right-half plae (RHP) i referred to a a margially table ytem becaue oly a bouded iput with exactly the ame freuecy a the imagiary pole will reult i a ubouded output For example if a ytem ha a pair of pole = ± jω ad all other pole are i the HP, oly the iput igal A co( ω t + θ ) (which i a bouded igal) for ay value of A ad θ will caue a ubouded output Sometime a itegrator (which ha a igle pole i the origi) i coidered table - For example, coider ytem with the followig pole-zero cofiguratio: () ecture Note Prepared by Amir G Aghdam

9 9 Im{} Im{} Im{} -plae -plae -plae Re{} Re{} Re{} Utable Stable Stable Im{} -plae Im{} -plae Im{} -plae Re{} Re{} Re{} Margially table Utable Utable - Step repoe of a TI ytem i the output of the ytem whe the iput i a uit tep ad all iitial coditio are zero - Step iput i oe of the commoly ued tet igal i the cotrol ytem For example chagig the et-poit i the automobile cruie cotrol mea applyig a tep iput to the ytem - Impule repoe h ( i eual to the derivative of the tep repoe ( a follow: d( h( = - Example : Fid the tep repoe of the followig ytem: x ( H ( ) = y( - Solutio: Y ( ) =, X ( ) =, X ( ) t Y ( ) = = (partial fractio expaio) y( = ( e ) u( ( ) ecture Note Prepared by Amir G Aghdam

10 Thi ytem i utable (becaue of the RHP pole, tep repoe goe to ifiity a time goe to ifiity I other word the output i ot fiite while the iput i fiite) It i to be oted that the fial value theorem caot be applied to y ( a Y ( ) = i ot aalytic i the RHP (it ha a pole i the RHP) If oe ( ) applie the fial value theorem to y ( by mitake, it will reult i y ( ) = which i ot correct ecture Note Prepared by Amir G Aghdam