ELEC 372 LECTURE NOTES, WEEK 4 Dr. Amir G. Aghdam Concordia University

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1 ELEC 37 LECTURE NOTES, WEE 4 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 uo, Joh Wiley & So, Ic, 00 - Hydraulic actuator: A hydraulic actuator i ued for the liear poitioig of a ma ad ca provide large power amplificatio - Figure 4 how the operatio of a hydraulic actuator Low-preure oil retur High-preure oil Low-preure oil retur x( M, B y( Figure 4: A hydraulic actuator

2 - Whe x ( > 0, the high-preure oil eter the right ide of the large pito chamber, forcig the pito to the left Thi caue the low-preure oil to flow out of the valve chamber from the leftmot chael Similarly, whe x ( < 0, the high-preure oil eter the left ide of the large pito chamber, forcig the pito to the right Thi caue the low-preure oil to flow out of the valve chamber from the rightmot chael - To obtai a model for the hydraulic actuator, it i aumed that the compreibility of the oil i egligible (i practice, the compreibility of oil may caue ome reoace becaue it act like a tiff prig) It i alo aumed that the highpreure hydraulic oil i provided by a cotat preure ource - The iput x ( ad the output y ( are related through a ecod-order oliear differetial equatio ad after liearizatio aroud x ( = 0 ad implificatio, we will have the followig trafer fuctio for a hydraulic actuator: Y ( ) = X ( ) ( M + B) M i the ma of the pito ad the attached load ad B are fuctio of the pito area, frictio, ad the flowig oil - The trafer fuctio of the hydraulic actuator i imilar to that of the electric θ ( ) motor (armature-cotrolled DC motor) give by E ( ) ( + ) m a Time domai aalyi Firt-order ytem: The trafer fuctio of a firt-order ytem i a follow: x ( y ( + - We have:

3 3 - The impule repoe of the ytem i: - The tep repoe of the ytem i: Y ( ) H ( ) = = X ( ) + dy( + y( = x( dt h( = e u( Y ( ) = = ( + ) y( = ( e t t + ) u( - For > 0 we will have the followig teady tate value for the tep repoe: y = lim y( = t - Note that i geeral, if all pole of H () are i the LHP, y ca be foud uig the fial-value theorem a how below: y = lim y( = lim Y ( ) = lim H ( ) = H (0) t H (0) i called the DC gai of the ytem (for a table ytem) - Sice for a firt-order ytem y = H ( 0) =, thi mea that i order to have o teady-tate error for the tep iput, mut be equal to - The tep repoe of a firt-order ytem for = i give i the followig figure ( H ( ) = ): +

4 4 - i called the time cotat of the ytem ad a maller mea a fater ytem - The pole of the firt-order ytem i located at = ad i idicated i the followig figure: Im{} -plae Re{} - I geeral, pole cloer to the imagiary axi repreet lower time repoe - The ettlig time t i the time it take the ytem traiet to decay More preciely, it i the time required for the ytem output to ettle withi a certai percetage of it teady-tate value The mot commoly ued percetage are %, % ad 5% - For the firt-order ytem with % meaure we have t = 4 ad for 5% meaure we have t = 3 We will ue the % meaure for the ettlig time i thi coure

5 5 - Small ettlig time i deirable i the deig of cotrol ytem - The ramp repoe i the repoe of the ytem to a uit ramp igal x ( = tu( whe the iitial coditio are zero We will have: ( X ) = Y ( ) = = + ( + ) y( = ( t + e t ) u( + + / - The teady-tate error for k = ca be obtaied whe t e = lim e( = lim tu( y( = lim t t + = t t t, ad i give by: - The ramp repoe of a firt-order ytem for = ad = i give i the followig figure ( H ( ) = ): + - From the reult obtaied for uit tep repoe ad uit ramp repoe, it ca be cocluded that a table firt-order ytem with uit DC gai H ( ) = ha + zero teady-tate error for the tep iput ad cotat teady-tate error for the ramp iput

6 6 Secod-order ytem: The trafer fuctio of a ecod-order ytem i: b + b0 x ( y ( + a + a 0 - The differetial equatio relatig the output to the iput i give by: d y( dy( dx( + a + a0 y( = b + b0 x( dt dt dt - Let u aume that b = 0, which mea that the ytem ha o zero The: H ( ) Y ( ) b 0 = = X ( ) + a + a0 - Uually it i impler to ormalize the ecod-order trafer fuctio uch that the DC gai i oe ( b 0 = a0 ) ad the ue the followig tadard form to decribe the ytem: H ( ) = + ζ + - i equal to a 0 ad ζ i equal to a a 0 - Note that a ecod-order ytem with the tadard trafer fuctio ca be reulted from the followig cloed-loop ytem: + R () Y () ( + ς ) - - May of the practical ecod-order ytem (uch a a cloed-loop poitio cotrol ytem with a DC motor) have i fact the above cloed-loop tructure

7 7 - The pole of the ecod-order trafer fuctio H () are located at: - For ξ we have two real pole, = ζ ± ζ - For ξ < we have two complex pole which alway come i complex cojugate pair - The ecod-order ytem i table if ad oly if ζ > 0 (which reult i two pole i the LHP) - The behaviour of a ecod-order ytem deped highly o ζ - Stable ecod-order ytem ( ζ > 0 ): ζ > Overdamped Im{} -plae Re{} ζ = Critically damped Im{} -plae Re{}

8 8 0 < ζ < Uderdamped Im{} -plae Re{} - Utable ecod-order ytem ( ζ 0 ): ζ = 0 Udamped Im{} -plae Re{} ζ Negatively damped Im{} -plae Re{}

9 9 < ζ < 0 Negatively damped Im{} -plae Re{} - We are oly itereted i table ecod-order ytem: overdamped, critically damped ad uderdamped - Overdamped ytem: A overdamped ecod-order ytem ha two real pole i the LHP ad o it ca be coidered a the parallel itercoectio of two firtorder ytem - Uderdamped ytem: A uderdamped ecod-order ytem ha a pair of complex cojugate pole:, = ζ ± j - i called atural frequecy or atural udamped frequecy - ζ i called dampig ratio d

10 0 - i called the damped atural frequecy, or damped frequecy, or ζ coditioal frequecy ad i deoted by d Thi i, i fact, the frequecy of the decayig ocillatio i the tep repoe, a we will ee i the followig page - ζ i called the dampig factor or dampig cotat (becaue it determie the rate of rie or decay of the tep repoe, a dicued later) ad i deoted by α d α θ Im{} -plae Re{} Figure 4 - The uit tep repoe of the ecod-order ytem H ( ) = + ζ + i: For 0 < ζ < : For ζ = 0 : For ζ = : y( = e d αt i( t + θ ), θ = co π y( = i( t + ) = co( t y( = e ( + d ζ