Mathematical modeling of dynamic systems: Chapter 3: Mathematical modeling of dynamic systems. Linearization of nonlinear systems:
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1 hpter : themticl modelig of dymic ytem Itructor: S. Frhdi themticl modelig of dymic ytem: Simplicity veru ccurcy: it i poible to improve the ccurcy of mthemticl model by icreig it compleity. Lier ytem: ytem i clled lier if priciple of uperpoitio pplie. Lier time ivrit ytem d lier time vrit ytem: differetil equtio i lier if the coefficiet re cott or of the idepedet vrible. Emple of Nolier Sytem: d d d d d ASi d ω t Lieriztio of olier ytem: I cotrol egieerig orml opertio of the ytem my be roud equilibrium poit, d the igl my be coidered mll igl roud equilibrium. If the ytem operte roud equilibrium poit d if igl ivolved re mll igl, the it i poible to pproimte the olier ytem by lier ytem
2 Trfer fuctio d impule- repoe fuctio: I cotrol theory, fuctio clled trfer fuctio re commoly ued to chrcterize the iput-output reltiohip of compoet or ytem tht c be decribed by lier, time ivrit, differetil equtio. The trfer fuctio of lier, time ivrit, differetil equtio i defied the rtio of the Lplce trform of the output to the Lplce trform of the iput uder the umptio tht ll iitil coditio re zero. Emple: oider the tellite ttitude cotrol ytem how i the figure, the trfer fuctio i determied follow d θ J T J Θ T Θ T J ovolutio Itegrl: Impule-repoe fuctio: For lier, time ivrit ytem trfer fuctio i Where X i the Lplce trform of the iput d Y i the Lplce trform of the output, where we ume tht ll itil coditio ivolved re zero, hece: So the ivere Lplce fuctio i give by the followig covolutio itegrl: Y X Y X y t t t τ g t τ dτ g t τ τ dτ oider the output of ytem to uite-impule iput whe the iitil coditio re zero. Sice the Lplce trform fuctio of uit impule fuctio i uity, the Lplce trform of the output of the ytem i: Y Thu the ivere Lplce trform of the output i give by: y t L [ ]
3 odellig Emple odellig Emple Thi digrm how preure cotroller. y i the equilibrium poitio of the device. At t there i udde icree i preure. The motio i govered by: m && y cy& ky P ut Fid yt. P i the force due to the pplied preure. Figure: Kreyzig, pg 6 Firt trform the ODE: [ Y y y' ] c[ Y y ] m e ky P The ytem i iitilly t ret o thi implifie to: my cy ky P P Now olve for Y: odellig Emple Y P m c k Now the hrd prt... we mut ivert thi for yt. Sice Y i i the form of oe polyomil divided by other we will epd it uig prtil frctio. c P Y c c m c k odellig Emple Determiig ukow coefficiet we hve: P k Y m c k k m c k... which c t let be iverted term by term.
4 odellig Emple From tble of Lplce trform we hve: odellig Emple Puttig every together we hve: L L b b b b t bt e be t bt e e P αt c y t e co ωt i ωt k k ωmk From firt prtil frctio From rel prt of the ecod prtil frctio. Block digrm Block digrm of cloe loop ytem. A ytem my coit of umber of compoet. To how the fuctio performed by ech compoet, we commoly ue digrm clled the block digrm.
5 Ope loop trfer fuctio d feedforwrd trfer fuctio: E fuctio FeedForwrd trfer H E B fuctio Opeloop trfer S H B Feedbck igl loed-loop trfer fuctio: H R B R E S E ] [ H R H R Elimitig E from the bove equtio yield: Thu cloe loop trfer fuctio i obtied : loed-loop ytem ubjected to diturbce: By uperpoitio the iput d the diturbce effect, the output trfer fuctio i determied : [ ] D R H D R Emple: coider the R circuit how i the figure, uig the ytem dymic equtio, the overll block digrm of the ytem c be obtied it i drow:
6 Rule of Block Digrm Algebr Emple: the block digrm how i figure c be implified tep by tep it i how i figure b through e Emple: implify the block digrm how.
7 licl or Frequecy-Domi Techique: odelig i the Time Domi - Stte-Spce: Advtge overt differetil equtio ito lgebric equtio vi trfer fuctio. Rpidly provide tbility & triet repoe ifo. Didvtge Applicble oly to Lier, Time-Ivrit LTI ytem or their cloe pproimtio. LTI LTI limittio limittio becme becme problem problem circ circ whe whe pce pce pplictio pplictio becme becme importt. importt. Stte-Spce or oder or Time-Domi techique Advtge Provide uified method for modelig, lyzig, d deigig wide rge of ytem uig mtri lgebr. Nolier, Time-Vryig, ultivrible ytem Didvtge Not ituitive clicl method. lcultio required before phyicl iterprettio i ppret Stte-Spce Repreettio A LTI ytem i repreeted i tte-pce formt by the vector-mtri differetil equtio DE : t & At But yt t Dut Dymic equtio euremet equtio with t t d iitil coditio t. The vector, y, d u re the tte, output d iput vector. The mtrice A, B,, d D re the ytem, iput, output, d feedforwrd mtrice. Defiitio Sytem vrible: Ay vrible tht repod to iput or iitil coditio. Stte vrible: The mllet et of lierly idepedet ytem vrible uch tht the iitil coditio et d pplied iput completely determie the future behvior of the et. Lier Idepedece: A et et of of vrible i i lierly idepedet if if oe of of the the vrible c c be be writte lier combitio of of the the other.
8 Defiitio Stte vector: A colum vector whoe elemet re the tte vrible. Stte pce: The -dimeiol pce whoe e re the tte vrible. The miimum umber of tte vrible i equl to: the order of the DE decribig the ytem. the order of the deomitor polyomil of it trfer fuctio model. the umber of idepedet eergy torge elemet i the ytem. Remember the the tte vrible mut be be lierly idepedet! If If ot, ot, you you my my ot ot be be ble ble to to olve for for ll ll the the other ytem vrible, or or eve write the the tte equtio. I eerl: Lieriztio
9 Emple: overtig Trfer Fuctio to Stte Spce Stte vrible re ot uique. A ytem c be ccurtely modeled by everl differet et of tte vrible. Sometime the tte vrible re elected becue they re phyiclly meigful. Sometime becue they yield mthemticlly trctble tte equtio. Sometime by covetio.. oider the DE Phe-vrible Formt d y d y d y L y bu where y i the meure vrible d u i the iput.. The miimum umber of tte vrible i ice the DE i of th order.. hooe the output d it derivtive tte vrible. y y& & Firt row of tte equtio & d y & d y L bu Lt row of tte equtio overtig from Stte Spce to Trfer Fuctio t & At But yt t Dut with t t d zero iitil coditio. Tkig the Lplce trform, X AX BU Y X DU Y U [ [ I A] B ] [ I A] D U dj B D X [ I A] [ I A] BU BU DU I the ce of SISO Sigle - Iput, Sigle - Output ytem: [ I A] B det[ I ] det[ I A] A D
10 Phe-vrible Formt 4. Arrge i vector-mtri formt [ ] y b d L L L L K K Note the trfer fuctio formt Note the trfer fuctio formt Y U b L Emple: equivlet block digrm howig phe-vrible. Note: yt ct Trfer Fuctio with Numertor Polyomil Trfer Fuctio with Numertor Polyomil cotiued. From the firt block: X R /. Therefore, d u
11 Trfer Fuctio with Numertor Polyomil cotiued Emple:. The meuremet obervtio equtio i obtied from the ecod trfer fuctio. [ ] Y b b b X b X bx b X But, X X d X X So, Y b X bx bx y b b b If forcig fuctio ivolve derivtive term:
12 echicl ytem The fudmetl lw goverig mechicl ytem i Newto ecod lw Emple Trfer Fuctio Stte-pce model omprig with tdrd form: give
13 Emple Electricl Sytem Bic lw goverig electricl circuit re Kirchhoff curret lw d voltge lw. Kirchhoff curret lw: The lgebric um of ll curret eterig d levig ode i zero Kirchhoff voltge lw: The lgebric um of the voltge roud y loop i electricl circuit i zero
14 Emple: RL circuit omple impedce: I drivig trfer fuctio for electricl circuit, we frequetly fid it coveiet to write Lplce trformed equtio directly, without writig differetil equtio. The comple Impedce Z of two termil circuit i the rtio of E, the Lplce trform of the voltge cro the elemet to I, the Lplce trform of the curret through the elemet. If the two termil elemet i reitce R, cpcitce, or iductce L, the the comple impedce i give by R, /, or L repectively. If the comple impedce re coected i erie, the totl impedce i the um of idividul comple impedce. Emple Emple
15 Liquid level ytem For turbulet flow V H For lmir flow Emple Emple
16 Therml ytem Therml ytem re thoe tht ivolve the trfer of het from oe ubtce to other. To implify the lyi we ume tht therml ytem c be repreeted by lmped prmeter model, tht ubtce tht re chrcterized by reitce to het flow hve egligible het cpcitce d tht ubtce tht re chrcterized by het cpcitce hve egligible reitce to het flow. Here we oly coider coductio d covectio.
17 Therml reitce d therml cpcitce Emple Lierized ervo hydrulic ytem Aroud orml opertig poit :
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