Sytem Analyi Prof. Cear de Prada ISAUVA rada@autom.uva.e
Aim Learn how to infer the dynamic behaviour of a cloed loo ytem from it model. Learn how to infer the change in the dynamic of a cloed loo ytem a a function of the controller arameter. Be aware of the contraint imoed by roce (and the controller) on the achievable erformance of the cloed loo ytem
A control loo E() R U() G() LT LC
Block in erie U() G () X() G () G ()X() G ()G ()U() G()U() U() G () G() G ()G ()
Cloed Loo Tranfer Function (CLTF) E() R() U() G() G()U() [ ] E() [ ]
Cloed loo ytem E() R() U() G() H() G()U() [ H() ] E() H() [ H() ]
Diturbance E() R U() G() H() G()U() [ H() ] [ H() ] E() H() H()
TranmitterController E() R() U() % G() ºC ma ºC ma ºC ma If the controller ue the tranmitter calibration and the tranmitter dynamic i fat comared with the one of the roce, then the feedback dynamic can be omitted
Cloed loo E() R() U() G() ey relation for feedback ytem analyi and deign
Cloed loo Control ignal E() R() U() G() U() R()E() R()[ ] R()[ G()U() ] U()[ R()G()] R()[ ] R() R() U()
Time reone in cloed loo E() R() U() G() The time reone of the cloed loo ytem under change in w(t) or v(t) can be comuted from the cloed loo ole and zero uing the reviou analyi
U() Examle E() ) )( ( ) ( G() G() G() d d d d τ τ τ τ τ τ τ τ τ d d τ
Examle E() U() d τ d τ τ d( τ ) ( τ )( τ d ) For oitive, table overdamed reone with no change in concavity againt SP te change and with change in concavity and an advanced reone if the diturbance v exerience a te change
Characteritic equation The tye of reone and the tability in cloed loo are given by the ole of the cloed loo TF, which correond to the root of the characteritic equation: 0 Changing the controller R(), the cloed loo time reone can be modified. Notice that the cloed loo dynamic can be comletely different from the oen loo one
Cloed loo zero Num() Den() Num() Den() Num() Den() The oen loo zero aear alo a zero of the cloed loo TF Num() Den() Num() Den() Num() Den() Den() Num()
Right half lane zero (untable zero) Num() Den() Num() Den() Num() Den() Num() Den() Num() Den() Num() Den() Den() Num() y(t) y c (t) If the oen loo time reone i of minimum hae tye, the cloed loo time reone will be imilar, indeendently of the controller R()
Chemical reactor u F r T ri TT Reactor T, x T r At the oerating oint: T 9 ºC x 0.90 T r 75.6 ºC F r 47.8 l/m T ri 50 ºC u 4 % U() 0.9 T ri () F r () 3 3 0.784 4 5.7.45 5.566 0.499.4 5.7.45 5.566 T()
w Chemical reactor TC TT w u F r T ri Reactor T, x U() 3 T r 0.378.98 5.7.45 5.566 3 T ri () 0.784 4 5.7.45 5.566 T()
Chemical reactor 0 ( 0.378.98) 0 3 5.7.45 5.566 3 5.7.45 5.566 ( 0.378.98) 0 For 4 the cloed loo ole are:.580.08i.580.08i.0079 Alo a zero at: 5. Ste reone to a change of degree in the SP
Cloed loo E() R() U() G()
Change of the cloed loo dynamic a function of change in the controller Proortional controller E() arameter U() G() G() G() G() Ecuación caracterítica : G() 0
Root locu G() G() Characteritic equation : G() G() 0 lane The root locu i a rereentation in the lane of the cloed loo ole for different value of the controller gain (and eventually any other aramenter) It allow to know the cloed loo tability and the tye of dynamic reone that correond to different value of the controller gain. The root locu mut be ymmetric reect to the real axi
Firt order ytem E() U() τ Characteritic equation : G() 0 0 τ 0 τ τ Overdamed reone with decreaing ettling time for increaing Fater reone in cloed than in oen loo /τ lane oen loo ole The root locu tar in the oen loo ole.
U() Second order ytem E() 0 G() tic equation : Characteri n n n ω δω ω ) 4( 4 0 0 n n n n n n n n n n n n δ ± ω δω ω ω ω δ ± δω ω ω δω ω δω ω
Second order ytem If the oen loo roce i overdamed, then, when i increaed from zero, the cloed loo reone i initially alo overdamed and increaingly fater, but, above a certain gain, the reone become underdamed with contant ettling time and increaing overhoot and ocillation frequency δω n ± ω n δ Oen loo ole Plano The root locu tar in the oen loo ole.
Second order ytem δω n ± ω n δ If the oen loo roce i underdamed, then, when i increaed from zero, the cloed loo reone i alo underdamed with contant ettling time and increaing overhoot and ocillation frequency Oen loo ole lane The root locu tar in the oen loo ole.
Root locu Den() for the root locu tart in the oen loo ole for G() 0 Num() Num() Den() 0 Den() Num() the root locu end at the oen loo zero ( )( τ ) ( ) 3 τ3 ( τ )( τ ) ( τ )( τ ) Extra zero located at infinite can be conidered to exit (u to equating the number of ole and zero) 0 0 0
Root locu G() G() 0 lane For any oint on the root locu, G() ha argument π Oen loo ole Siotool
Third order ytem Imag Axi.5.5 0.5 0 0.5.5 lane.5 4 3 0 Real Axi 3 4 4 With increaing, the ytem reone i more ocillatory and can become untable
Real cloed loo ytem U() Actuator Proce A real cloed loo ytem i alway of third or higher order due to the dynamic of actuator and tranmitter. Accordingly, a high value of will tend to detabilize the cloed loo ytem. Imag Axi.5.5 0.5 0 0.5.5 Tranmitter lane.5 4 3 0 Real Axi
Root locu lane τ ωn δω ω n n lane /τ.5.5 lane Imag Axi 0.5 0 0.5.5 ( τ )( τ )( τ3 ).5 4 3 0 Real Axi
Zero in the right hand ide Imaginary Axi 8 6 4 0 4 6 Root Locu lane 8 3.5.5 0.5 0 0.5 Real Axi 3 4 4 A the root locu end at the oen loo zero, if there are untable zero in oen loo, then the cloed loo ytem will became untable for increaing value of
PIG() E() ( T i ) U() G() Characteritic equation : R()G() 0 T i G() T i 0 For a given T i one can draw the root locu of the extended ytem (T i )G()/
PI Firt order E() ( T i ) U() τ Characteritic equation : R()G() 0 Tτ i T i 0 T τ T ( T i( ( i i ) ± ) ± ) T i Tτ ( τ ( T( τ ) 0 i i ) ) 4τ 4Tτ T (T ) i i i 0 The root locu can be drawn for any given T i
PI Firt order τ 0.5 0.8 0.6 0.4 ( ) T ( i T i, τ 0.5 ) Imag Axi 0. 0 0. 0.4 0.6 0.8 3.5.5 0.5 0 0.5.5 Real Axi
PI Firt order τ 0.5 3 ( ) T ( i ) 0.5 Imag Axi 0 T i 0.5, τ 0.5 3 9 8 7 6 5 4 3 0 Real Axi SyQuake
PI G() G() (0.5 )( ).5 0.5 ( ) T ( i ) 4 Imag Axi 0 0.5 T i 4.5 3.5.5 0.5 0 0.5.5 Real Axi
PI G() G() (0.5 )( ) 5 4 3 ( ) T ( i T i ) Imag Axi 0 3 4 5 3.5.5 0.5 0 0.5.5 Real Axi
PI G() 0 G() (0.5 )( ) 8 6 4 ( ) T ( i ) 0.5 Imag Axi 0 4 T i 0.5 6 8 0 5 4 3 0 Real Axi The cloed loo dynamic can vary a lot according to the relative zero oition SyQuake
Steady tate error E() R() U() G() If the value of the et oint change or a diturbance aear, which will be the value of the error e(t) at teady tate? e lim e(t) t lim E() 0
Steady tate error, e w TC TT u F r T ri Reactor T r e T, x
Steady tate error E() R() U() G() E() [G()U() ] [E() ] E()[ ] E()
Steady tate error, te on W E() R() U() G() e lim e(t) t E() lim E() 0 lim 0 w w G(0)R(0)
Steady tate error, te on W E() R() U() G() w e G(0)R(0) If G() or R() have an (a )(...) integrator: (b )(c ) w G(0)R(0) e 0 G(0)R(0) CStation If not, the teady tate error will have a finite value, decreaing with
Steady tate error, te onv E() R() U() G() E() e lim e(t) t lim E() 0 lim 0 v D(0)v G(0)R(0) If have a zero at 0 e 0
Steady tate error, te on V E() U() R() G() D(0)v e G(0)R(0) G(0)R(0) e If ha no integrator: if (a )(...) G() or R() have one (b )(c ) ntegrator: If not, the teady D(0)v 0 tate error will G(0)R(0) have a finite value, decreaing with
Steady tate error, te on V D(0)v e G(0)R(0) If ha one integrator: If G() or R() have one integrator: GR() v v GR() D(0)v e The error will be finite GR(0) If neither G() nor R() have one integrator: D(0)v 0 0G(0)R(0) v v Increaing error
Delay E() U() R() G() w e G(0)R(0) D(0)v e G(0)R(0) The exitance of delay in G() or doe not influence the analyi of the error in teady tate e d (a )(...) (b )(c ) SyQuake
Steady tate error, ram on W E() R() U() G() E() w e lim e(t) lim E() lim t 0 0 w G(0)R(0)
Steady tate error, ram on W E() R() U() G() GR() e w G(0)R(0) e w GR(0) If G() or R() do not have an integrator: Infinite error. If they have one: finite error. Two integrator are required in in order to make the error zero ay teady tate.
4 baic TF E() R U() G() It i imortant to ay attention alo to the control effort R() R() U()
One degree of freedom controller E() R U() G() If R() i choen in order to get a good dynamic reone againt et oint change, then, the reone againt diturbance i given, and vicevera. There i no enough degree of freedom to deign the controller for the two aim imultaneouly.
w U() T Controller Two degree of freedom R() controller DOF [ T() S() ] / R S B()T() R()A() B()S() u B() Y () U() A() B / A Proce B() R()A() B()S() It i oible to elect R and S in order to get a good reone againt diturbance and elect T in order to tune the reone againt et oint change v y