Class 07 Time domain analysis Part II 2 nd order systems

Class 07 Time domai aalysis Part II d order systems

Time domai aalysis d order systems iput S output Secod order systems of the type α G(s) as + bs + c

Time domai aalysis d order systems iput S α as + bs + c output Secod order systems of the type α G(s) as + bs + c

Time domai aalysis d order systems Secod order systems: α that is: iput Y(s) R(s) as + bs + as + α bs Y(s) R(s) + c c s + output α a b s a + c a K o ω ζω ω

Time domai aalysis d order systems iput K ω o + ζω + ω s s output K o gai of the system ζ dampig coefficiet ω atural frequecy the trasfer fuctio ca be rewritte as: Y(s) R(s) s + K o ζω ω s + ω

Time domai aalysis d order systems iput K ω o + ζω + ω s s output K o gai of the system ζ dampig coefficiet ω atural frequecy Besides these 3 above parameters, we also have ω d dampig frequecy ω d ω 1 ζ 0 < ζ 1

Time domai aalysis d order systems Example 1: Y(s) R(s) 4s 3 + 1s K o 3 ζ 3 ω 0,5 + 1 poles: s,914 s 0,086 poles are real ad distict Example : Y(s) R(s) s + 3 s + 1 poles: s 1 (duple) K o 3 ζ 1 ω 1 ω d 0 poles are real ad repeated

Time domai aalysis d order systems Example 3: Y(s) R(s) s K o 1,5 ζ 0,5 ω 1 + 3 s + ω d 0,866 poles: s 0,5 ± 0,866j complex cojugate poles Example 4: Y(s) R(s) s K o 3 ζ 0 ω ω d 1 3 + 1 poles: s ± j (pure imagiary) complex cojugate poles

Time domai aalysis d order systems Characteristic equatio: p(s) s + ζω 4 ζ ω 4 ω 4 ω (ζ 1) s + ω > 0 (ζ 1) > 0 ζ > 1 ζ > 1 0 (ζ 1) 0 ζ 1 ζ 1 < 0 (ζ 1) < 0 ζ < 1 ζ < 1 ζ > 1 poles are real ad distict ζ 1 poles are real ad repeated 0 < ζ < 1 poles are complex cojugate

Time domai aalysis d order systems iput K ω o + ζω + ω s s output uit step iput What is the output? (step respose)

Time domai aalysis d order systems iput K ω o + ζω + ω s s output Y(s) K oω + ζω + ω s s R(s) ad sice r(t) uit step: Y(s) ω o + ζω + ω K 1 s s s

Time domai aalysis d order systems iput K ω o + ζω + ω s s output y(t) L 1 [ Y(s) ] the uit step respose depeds o the value of ζ: a) 0 < ζ < 1 (uder dampig) b) ζ 1 (critical dampig) c) ζ > 1 (over dampig)

Time domai aalysis d order systems So, i the case 0 < ζ < 1 (uder dampig) the uit step respose is: y(t) where iput ω y(t) L 1 [ Y(s) ] ζω K o 1 t ζ e cos ωdt + se ωdt, t > 1 ζ d ω 1 ζ K ω o + ζω + ω s s ( dampig frequecy ) output 0

Time domai aalysis d order systems iput K ω o + ζω + ω s s output uit step respose : y(t) K o 1 e ζω t cos ω d t + ζ 1 ζ se ω d t

Time domai aalysis d order systems uit step respose: y(t) K o 1 e ζω t cos ω d t + ζ 1 ζ se ω d t

Time domai aalysis d order systems iput K ω o + ζω + ω s s output y(t) L 1 [ Y(s) ] I the case ζ 1 (critical dampig) the uit step respose is: y(t) K [ ( )] 1 e ζω t 1+ ω t, t 0 o >

Time domai aalysis d order systems iput K ω o + ζω + ω s s output uit step respose : y(t) K o [ ( )] 1 e ζω t 1+ ω t

Time domai aalysis d order systems uit step respose : y(t) K o [ ( )] 1 e ζωt 1+ ω t

Time domai aalysis d order systems iput K ω o + ζω + ω s s output I the case ζ > 1 (over dampig) the uit step respose is: where y(t) y(t) L 1 [ Y(s) ] p1t p t ω e e K o 1 +, t > 1 p1 p ζ ( ) ζ ± ζ 1 p1, ζω m ω ζ 1 ω 0 System has real poles

uit step respose: ζ ω + t p 1 t p p p 1 1 K y(t) 1 o e e o K s s ω + ζω + ω output iput Time domai aalysis d order systems

uit step respose: ζ ω + t p 1 t p p p 1 1 K y(t) 1 o e e Time domai aalysis d order systems

Time domai aalysis d order systems uit step respose ζ 1 ζ

Time domai aalysis d order systems uit step respose ζ 4 ζ 1

Time domai aalysis d order systems The uder dampig case

Time domai aalysis d order systems I the case 0 < ζ < 1, the uit step respose y(t) ζω K o 1 t ζ e cos ωdt + se ωdt, t > 1 ζ 0 ca have may differet forms, depedig o the values of ζ (dampig coefficiet), ω (atural frequecy) e K o (gai) Observe that ω d depeds o ζ ad ω ω d ω 1 ζ (dampig frequecy)

Time domai aalysis d order systems uit step respose ζ 0,13 ω 0,57 ζ 0,1 ω

Time domai aalysis d order systems uit step respose ζ 0,5 ω 1 ζ 0,5 ω 0,

Time domai aalysis d order systems uit step respose ζ 0,65 ω ζ 0,7 ω 0,8

Time domai aalysis d order systems uit step respose ζ 0,8 ω 1,4 ζ 0,85 ω 0,7

Time domai aalysis d order systems ζ 0 ω 0, uit step respose

Time domai aalysis d order systems Now let us cocetrate i the case 0 < ζ < 1 ad calculate some parameters.

Time domai aalysis d order systems calculate some parameters/variables for y(t) the uit step respose of the d order system.

steady state output y ss

Time domai aalysis d order systems y ss steady state respose ( steady state output ) K o ω Y(s) s + ζω s + ω R(s) R (s) 1 s y ss lim y(t) t lim s o s Y(s) lim s 0 K o s + K o ω ζω s s + ω 1 s y ss K o

Time domai aalysis d order systems y K ss o

risig time t r

Time domai aalysis d order systems t r risig time time it takes the step respose y(t) to reach the fial value y ss K o for the first time.

Time domai aalysis d order systems t r risig time Is the istat of time that y(t) reaches the fial value K o for the first time. y(t r ) K 1 e ζω t r ζ cosω t + se ω t 1 ζ o d r d r K o 1 ζω e t ζ r cos ωdt r + se ωdt r 1 ζ 0 tg( ω d t r ) 1 ζ ζ ω ζω d

Time domai aalysis d order systems t r risig time Depeds o the values of ζ ( dampig coefficiet ), ad of ω ( atural frequecy ) t r ω arctg ζω ω d d arctg 1 ζ ζ ω d t r arctg( ω d /ζω ) / ω d

Time domai aalysis d order systems t r arctg ω ω ζω d d

peak time t p

Time domai aalysis d order systems t p peak time is the istat of time i which the step respose y(t) reaches the first peak.

Time domai aalysis d order systems peak time y ss K o ( gai ) ζ ( dampig coefficiet ), ω ( atural frequecy ) y dy dt K o ζω e ζω t cos ω d t r + ζ 1 ζ se ω d t r + e ζω t ω d seω d t r + ζ ω d 1 ζ cos ω d t r ω

Time domai aalysis d order systems peak time dy dt K o e ζω t ζω + ω d cos ω seω d d t + ζ t ζω 1 ζ ω cos se ω d t) ω d t + K o e ζω t se ω d t ζ ω 1 ζ + ω (1 ζ 1 ζ ) t ω K o e ζω se ωdt 1 ζ 0

Time domai aalysis d order systems peak time dy dt 0 se ω d t 0 ω d t 0, π, π, 3π, L π t p ω d t p π / ω d

Time domai aalysis d order systems t p π ω d

overshoot M p

Time domai aalysis d order systems

Time domai aalysis d order systems

Time domai aalysis d order systems

Time domai aalysis d order systems M p overshoot it is the percetage above of the fial value y ss that is reached by the first peak.

Time domai aalysis d order systems The overshoot M p ca be expressed as a value betwee 0 ad 1.

Time domai aalysis d order systems Or, istead, it ca be expressed as a value betwee 0% ad 100%.

Time domai aalysis d order systems 0 overshoot M p 1 or 0% overshoot M p 100%

o o ζ 1 t o p K K ) se ( ζ ) ( cos 1 K M p ζ π π ω e overshoot ss ss max p y y y M o o p p K K ) y(t M ou ( ) o o / o o p K K K K M d ζ + ω π ω e 1 0 y ss K o ( gai ) ζ ( dampig coefficiet ), ω ( atural frequecy ) Time domai aalysis d order systems

Time domai aalysis d order systems overshoot ζ π M p K o e K 1 o ζ ζ π M p e 1 ζ M p depeds oly o ζ ( dampig coefficiet )

Time domai aalysis d order systems overshoot M p depeds oly o ζ ( dampig coefficiet ) M p 1 ζπ e or ζ ζ π M p e 1 ζ 100%

Time domai aalysis d order systems M p e 1 ζ π ζ

settlig time t s

Time domai aalysis d order systems t s settlig time is the time required for the step respose y(t) to reach ad remai withi a give error bad aroud the fial value y ss.

Time domai aalysis d order systems This error bad ca be of 5% above ad 5% below of the fial value y ss.

Time domai aalysis d order systems or, istead, from % above ad % below the fial value y ss.

Time domai aalysis d order systems The settlig time is obtaied from the equatios of y e (t), the curves that ecompass y(t). y e (t) K [ ] ζω t 1 ± e, t 0 o >

Time domai aalysis d order systems That is, the settlig time t s is obtaied by calculatig y e (t s ) 1,05 K o for the case of t s with 5% tolerace, ad y e (t s ) 1,0 K o for the case of t s with % tolerace, obtaiig the followig values: t s (5%) 3 ζω t s (%) 4 ζω

Aálise o domíio do tempo - Sistemas de ª ordem settlig time Note that the settlig time t r is iversely proportioal to ζω, which is the distace of the real part of the poles to the origi. hece: t ad t s s (5%) (%) 3 ζω 4 ζω t s (5%) 3 / ζω t s (%) 4 / ζω

Time domai aalysis d order systems t s (5%) 3 ζω

Time domai aalysis d order systems t s (%) 4 ζω

Time domai aalysis d order systems Note that we have see cases i which 0 < ζ < 1 ζ 1 ζ > 1 that is: ζ > 0 However, if ζ < 0 the: the system is ustable

Time domai aalysis d order systems ζ < 0 ustable system (a example)

Time domai aalysis d order systems ζ < 0 ustable system (aother example)

Departameto de Egeharia Eletromecâica Thak you! Felippe de Souza felippe@ubi.pt