Itroductio to Cotrol Sytem CLASSIFICATION OF MATHEMATICAL MODELS Icreaig Eae of Aalyi Static Icreaig Realim Dyamic Determiitic Stochatic Lumped Parameter Ditributed Parameter Liear Noliear Cotat Coefficiet Variable Coefficiet Dicrete Time Cotiuou Time Firt Order Secod Order Higher Order Peter Avitabile Mechaical Egieerig Departmet Uiverity of Maachuett Lowell 1 Dr. Peter Avitabile
Claificatio of Mathematical Model Source: Dyamic Sytem Vu & Efadiari Dr. Peter Avitabile
Itroductio to Cotrol Sytem Defiitio Cloed Loop Sytem there alway exit ome feedback of oe or more variable that ifluece the iput excitatio. Diturbace Iput igal that exit but o cotrol i poible. Negative/Poitive Feedback igal that are added to or ubtracted from a iput igal. Ope Loop Sytem o feedback igal are available to the cotrol ytem. Plat device, ytem or compoet that i decribed by ome trafer relatio. Eetial Compoet of a Cotrol Sytem Plat, Seor, Actuator, Cotroller 3 Dr. Peter Avitabile
Itroductio to Cotrol Sytem INPUT R () + (Referece) E a () G c () Cotroller U () + G a () + Actuator Diturbace D() Plat G p () Geeral Block Diagram of a Cotrol Sytem H() Seor Y() CONTROLLED OUTPUT 4 Dr. Peter Avitabile
Example Cotrol Sytem (a) (c) (b) Source: Dyamic Sytem Vu & Efadiari 5 Dr. Peter Avitabile
Itroductio to Cotrol Sytem Pole ad Zero Kowledge of the ytem Pole ad Zero i importat iformatio i the deig ad aalyi of dyamic ytem, vibratio, ad cotrol. Aume a ytem trafer fuctio uch a: T.F. N() D() umerator deomiator zero pole The umerator polyomial ca be factored to idetify the ytem trafer fuctio ZEROS. The deomiator polyomial ca alo be factored to idetify the ytem trafer fuctio POLES. T.F. 0.5 3 + 0.5 ( + 1)( ) + + + ZERO 0.5 + deoted by O POLES 1, deoted by X POLE X Im ZERO XO - -1 Re 6 Dr. Peter Avitabile
Time Cotat Firt Order Sytem Differetial Equatio τ& v + v f (t) Characteritic Equatio Pole τ + 1 0 1 τ 1 τ X Im Re 7 Dr. Peter Avitabile
Time Cotat Firt Order Sytem Coider the mechaical ytem how c m x,x& f m && x + cx& f (t) OR m v& + cv f (t) the m c v& + v f (t) c OR τ& v + v f (t) c The time cotat i τ m c 8 Dr. Peter Avitabile
Firt Order Sytem RC Circuit Coider the fuctio of a RC circuit h() 140 ( + 1)( + 7)( + 0) X F 1 7 0 ( + 1)( + 7)( + 0) Writte i POLE-ZERO form, the block diagram i F 1 7 0 + 1 + 7 + 0 Slowet Pole Quicker Pole Fatet Pole X 9 Dr. Peter Avitabile
Firt Order Sytem RC Circuit A SIMULINK Model ca be ued to quickly ee the repoe of each pole. STEP ( + 1)( + 7)( + 0) 1 + 1 7 + 7 140 X 1 0 + 0 X 7 The pole cloet to the jω axi domiate the time repoe of the ytem. X 0 X SUM 10 Dr. Peter Avitabile
SIMULINK Model 11 Dr. Peter Avitabile
Time Cotat Secod Order Sytem && x Differetial Equatio + ζω x& + ω x f (t) Pole - Overdamped 1, ζω ± ω ζ Pole Critically damped 1, ω (repeated) Pole Uderdamped 1, ζω ± jω 1 σ ± jω d Pole Udamped 1, ± jω 1 ζ Characteritic Equatio + ζω + ω 0 Im X X X Im Re X Re Im X Im X X Re Re 1 Dr. Peter Avitabile
Time Cotat Secod Order Sytem Coider the mechaical ytem m && x + cx& + kx with ω ζ c c c k m c mω f (t) OR c c k && x + x& + x m m km f (t) m which i writte i tadard form a && x + ζω x& + ω x f (t) m 13 Dr. Peter Avitabile
Traiet Repoe Specificatio Term Maximum Percet Overhoot M p % The maximum overhoot i the maximum amplitude expreed a a percetage of the teady tate value. M p e πζ 1 ζ Settlig Time - t The time required for the repoe to reach a mall percetage of teady tate value. t t % 5% t t ' ' % 5% 4τ 3τ 4 ζω 3 ζω 4 σ 3 σ 14 Dr. Peter Avitabile
Traiet Repoe Specificatio Term Rie Time t r The time for a uderdamped d order ytem to rie from 0 to 100% of the fial teady tate value. t r 1 ω ta 1 1 ζ ζ Peak Time t p The time required to reach maximum overhoot i called peak time. t p π ω d 15 Dr. Peter Avitabile
Traiet Repoe Specificatio Term 16 Dr. Peter Avitabile
Trafer Fuctio Buildig Block Methodology All trafer fuctio ca be broke dow ito piece (called buildig block) for evaluatio ad aemet. Thee ca be categorized a cotat firt order pole (or zero) at origi firt order pole (or zero) ot at origi ecod order pole (or zero) with dampig ratio le tha 1.0 For Example: H() Cotat 750( + 90) ( + + 101)( + 500) 1 t Order Zero (ot at origi) 1 t Order Pole (at origi) d Order Pole Dampig < 1.0 1 t Order Pole (ot at origi) 17 Dr. Peter Avitabile
Trafer Fuctio Buildig Block Methodology Firt Order Pole At Origi 1 G () 1 G(jω) jω Firt Order Pole 1 G() + a 1 G(jω) jω + Secod Order Pole ω G() + ζω + ω G(jω) a ω ( jω) + ζω ( jω) + ω db Mag db Mag db Mag log f log f log f 0 db/octave 18 Dr. Peter Avitabile 0 db/octave 40 db/octave
Trafer Fuctio Buildig Block Methodology Firt Order Zero At Origi G () G(jω) jω Firt Order Zero + a G() a jω + a G(jω) a Secod Order Zero + ζω + ω G() ω G(jω) ( jω) + ζω ( jω) ω + ω db Mag db Mag db Mag log f log f log f 0 db/octave 19 Dr. Peter Avitabile 0 db/octave 40 db/octave
Cotroller Type O/Off Differetial O/Off Proportioal Cotrol (P cotrol) Derivative Cotrol (D cotrol) Itegral Cotrol (I cotrol) PD Cotrol PI Cotrol PID Cotrol Other (Advaced Method) Lead Network Cotrol Lag Network Cotrol Lead-Lag Network Cotrol Feed Forward Cotrol 0 Dr. Peter Avitabile
Cotroller Type O/Off 1 0 e(t) OR M 1 M 0 Differetial Example: Home heatig ytem Differetial 1 Dr. Peter Avitabile
Cotroller Type Proportioal Cotrol P Cotrol i a imple gai ad hould be alway tried firt. U() G ()E K p () c a K E () E a () U() p a Derivative Cotrol D Cotrol add dampig to the ytem ad therefore provide tability to the ytem. U() G ()E () c a K E E a () K d U() d a () Dr. Peter Avitabile
Cotroller Type Itegral Cotrol I Cotrol reduce teady tate error but ca icreae itability. U() Gc ()Ea () K I K E E a () U () I a () 3 Dr. Peter Avitabile
Cotroller Type PID Cotrol U() Deired Value of Cotrol Variable G ()E () K K E () K E I a () K p + + K d U () + c a K p + K I + I p + + K K d d a M() Plat 4 Dr. Peter Avitabile
5 Dr. Peter Avitabile