CHE302 LECTURE VI DYNAMIC BEHAVIORS OF REPRESENTATIVE PROCESSES. Professor Dae Ryook Yang

Transcription

1 CHE30 LECTURE VI DYNAMIC BEHAVIORS OF REPRESENTATIVE PROCESSES Professor Dae Ryook Yang Fall 00 Dep. of Chemical and Biological Engineering orea Universiy CHE30 Process Dynamics and Conrol orea Universiy 6-

2 REPRESENTATIVE TYPES OF RESPONSE For sep inpus U(s) Process G(s) Y(s)? Y() Type of Model, G(s) Nonzero iniial slope, no overshoo or nor oscillaion, s order model s order+time delay Underdamped oscillaion, nd or higher order Overdamped oscillaion, nd or higher order Inverse response, negaive (RHP) zeros Unsable, no oscillaion, real RHP poles Unsable, oscillaion, complex RHP poles Susained oscillaion, pure imaginary poles CHE30 Process Dynamics and Conrol orea Universiy 6-

3 dy() d ST ORDER SYSTEM Firs-order linear ODE (assume all deviaion variables) Transfer funcion: Sep response: Wih U() s = A/ s, L = y () + u() ( s+ ) Ys () = U() s A Y s y A e s( s+ ) Ys () = U() s ( s+ ) L / () = () = ( ) / y( ) = A( e ) 0.63A A e A Time consan CHE30 Process Dynamics and Conrol orea Universiy 6-3 y() A 0.63A Gain / ( ) (Seling ime=4 5 ) y = Ae = A (0) / / / 0 (Nonzero iniial slope) = 0

4 Impulse response Wih U() s = A, A L A Y() s = y () = e ( s + ) / y() A Ramp response Wih U( s) = a/ s, a Y s y a e a s ( s+ ) L / ( ) = () = + ( ) y() u() / Sinusoidal response [ ω ] ω ω Wih U() s = L Asin = /( s + ), Y() s Aω L = ( s+ )( s + ω ) A y e ω + / () = ( ω ωcosω + sin ω ) y() φ = an ω A ω + ω + u () CHE30 Process Dynamics and Conrol orea Universiy 6-4

5 Ulimae sinusoidal response ( ) A y e ω + A = ( ω cosω+ sin ω) ω + A = + = ω + / () = lim ( ω ω cosω + sin ω ) Ampliude 0 sin( ω φ) ( φ an ω ) Phase angle The oupu has he same period of oscillaion as he inpu. Bu he ampliude is aenuaed and he phase is shifed. Normalized Ampliude Raio (AR N ) = < ω + Phase angle = an High frequency inpu will be aenuaed more and phase is shifed more. ω CHE30 Process Dynamics and Conrol orea Universiy 6-5

6 BODE PLOT FOR ST ORDER SYSTEM AR plo asympoe AR N ( ω 0) = lim = ω 0 ω + AR N ( ω ) = lim = ω ω + ω Phase plo asympoe φω ( 0) = liman ω = 0 ω 0 o φω ( ) = liman ω = 90 ω o I is also called low-pass filer CHE30 Process Dynamics and Conrol orea Universiy 6-6

7 ST ORDER PROCESSES Coninuous Sirred Tank dca V = qcai qca d CA() s q = = C () s Vs+ q ( V / qs ) + Ai c Ai, q i V h c A, q Wih consan hea capaciy and densiy dt ( Tref ) ρvcp = ρqcp( T0 Tref ) d ρqc ( T T ) T() s q = = T () s Vs+ q ( V / qs ) + 0 p ref T 0, q i V h T, q CHE30 Process Dynamics and Conrol orea Universiy 6-7

8 INTEGRATING SYSTEM dy () d L = u() sy() s = U() s Ys () Transfer Funcion: = U() s s Sep Response Wih U() s = / s, Y() s = L y () = s y() Slope= The oupu is an inegraion of inpu. Impulse response is a sep funcion. Non self-regulaing sysem CHE30 Process Dynamics and Conrol orea Universiy 6-8

9 INTEGRATING PROCESSES Sorage ank wih consan oule flow Oule flow is pumped ou by a consan-speed, consanvolume pump Oule flow is no a funcion of head. q i dh A qi q d = Hs () = Q() s As i H() s = Qs () As V Area = A h q CHE30 Process Dynamics and Conrol orea Universiy 6-9

10 nd order linear ODE Transfer Funcion: Sep response ND ORDER SYSTEM d y () dy() y () u() ( s s ) Ys () U() s d ζ L + + = d + ζ + = Ys () = U s s s () ( + ζ + ) Varies wih he ype of roos of denominaor of he TF. Real par of roos should be negaive for sabiliy: ζ 0 Two disinc real roos ( ζ > ): overdamped (no oscillaion) Double roo ( ): criically damped (no oscillaion) Complex roos ( ζ = 0 ζ < Time consan ): underdamped (oscillaion) CHE30 Process Dynamics and Conrol orea Universiy 6-0 Gain Damping Coefficien

11 Case I ( ζ > ) wih U(s)=/s e e Ys () = = y () = s s s s s s / / L ( + ζ + ) ( + )( + ) ( ) Case II ( ζ = ) L Ys () = = y () = + / e s( s + s+ ) s( s+ ) / ( ) Case III (0 ζ < ) L ζ / ζ ζ () = () = cosα + sin α ( α = ) Ys y e s( s + s+ ) α Naural frequency CHE30 Process Dynamics and Conrol orea Universiy 6-

12 Ulimae sinusoidal response [ ω ] Wih U() s = L Asin, Y() s y Aω L = ( s + s+ )( s + ω ) A () = sin( ω+ φ) ( φ = an ) ( ω ) + ( ζω ) Oher mehod o find ulimae sinusoidal response ζω ω -( α + jω) - jω For ( s+ α + jω), y () has e and i becomes e as ( α > 0). s jω Gs () = G( jω ) = ( s + ζs+ ) ( ω ) + jζω AR= G( jω ) = = ( ω ) + jω ( ω ) + ( ζω ) φ Im( G( jω)) ζω = G( jω) = an = an Re( G( jω)) ω CHE30 Process Dynamics and Conrol orea Universiy 6-

13 BODE PLOT FOR ND ORDER SYSTEM AR plo Phase plo AR ( ω ) = lim = N ω ( ) ( ) ω + ζω ( ω ) ζω ζ φω ( ) = liman = liman = 80 ω ω ω o ω Resonance d( ARN )/ dω = 0 ζ ωmax = for 0< ζ < Resonance The ampliude of oupu oscillaion is bigger han ha of inpu when he resonance occurs. CHE30 Process Dynamics and Conrol orea Universiy 6-3

14 ST ORDER VS. ND ORDER (OVERDAMPED) Iniial slope of sep response { } As A s order: y (0) = lim sys () = lim 0 s s = s + { } As nd order: y (0) = lim sys () = lim = 0 s s s+ ζ s+ Shape of he curve (Convexiy) = < > / s order: y () e 0 (For 0) No inflecion / / A e e nd order: y () = ( ) ( + as ) Inflecion CHE30 Process Dynamics and Conrol orea Universiy 6-4

15 CHARACTERIZATION OF SECOND ORDER SYSTEM nd order Underdamped response Rise ime ( r ) r = ( nπ cos ζ)/ ζ ( n= ) Time o s peak ( p ) p = π / ζ Seling ime ( s ) s / ζ ln(0.05) + exp ζπ ζ ( ζ ) + exp / + exp 3ζπ ζ Overshoo (OS) π π 3π ( ) OS= a/ b = exp πζ / ζ ζ ζ ζ Decay raio (DR): a funcion of damping coefficien only! ( ) DR= c/ a= ( OS ) = exp πζ / ζ Period of oscillaion (P) P = π / ζ CHE30 Process Dynamics and Conrol orea Universiy 6-5

16 ND ORDER PROCESSES Two anks in series If v =v, criically damped. Or, overdamped (no oscillaion) CA() s == C () s (( V / qs ) + )(( V / qs ) + ) Ai c Ai, q i V V c A, q Spring-dashpo (shock absorber) By force balance ( mg+ f()) ky cv= ma my = ky cy + ( mg + f()) m c m y + y + y = f () k 4mk k ζ (can be <: underdamped) CHE30 Process Dynamics and Conrol orea Universiy 6-6

17 Underdamped Processes Many examples can be found in mechanical and elecrical sysem. Among chemical processes, open-loop underdamped process is quie rare. However, when he processes are conrolled, he responses are usually underdamped. Depending on he conroller uning, he shape of response will be decided. Sligh overshoo resuls shor rise ime and ofen more desirable. Excessive overshoo may resuls long-lasing oscillaion. CHE30 Process Dynamics and Conrol orea Universiy 6-7

18 POLES AND ZEROS Gs () m m N() s bs ( m + bm s + + bs + ) = = n n Ds () ( as n + an s + + as + ) Poles (D(s)=0) Where a ransfer funcion canno be defined. Roos of he denominaor of he ransfer funcion Modes of he response Decide he sabiliy Zero (N(s)=0) Where a ransfer funcion becomes zero. Roos of he numeraor of he ransfer funcion Decide weighings for each mode of response Decide he size of overshoo or inverse response They can be real or complex CHE30 Process Dynamics and Conrol orea Universiy 6-8

19 Real pole from s = / Mode: e ( s + ) y() Im / Re If he pole is a he origin, i becomes inegraing pole. If he pole is in RHP, he response increases exponenially. Complex pole from ζ ζ s = ± j = α ± jβ ζ + ζ s = (funcion of only) = ζ s =± an (funcion of ζ only) ζ ζ = cosθ CHE30 Process Dynamics and Conrol orea Universiy 6-9 ( s + ζs+ ) (-< ζ < ) ζ / Im ζ θ ζ / ζ / Re

20 Modes: α± jβ α e = e (cosβ± jsin β) Assume is posiive. ζ / ζ ζ = e (cos ± jsin ) If ζ < 0, he exponenial par will grow as increases: unsable If ζ > 0, he exponenial par will shrink as increases: sable If ζ = 0, he roos are pure imaginary: susained oscillaion Effec of zero Ns () Gs () = = w + + wn ( s+ p ) ( s+ p ) ( s+ p ) ( s+ p ) n n The effecs on weighing facors are no obvious, bu i is clear ha he numeraor (zeros) will change he weighing facors. CHE30 Process Dynamics and Conrol orea Universiy 6-0

21 Lead-lag module N() s ( as+ ) Gs () = = Ds () ( s ) EFFECTS OF ZEROS Depending on he locaion of zero M( as ) a Ys () + M = = + s( s+ ) s s+ (a) a > > 0 The lead dominaes he lag. (b) 0 a + Lag < The lag dominaes he lead. Lead a / y () = M e (c) 0 > a Inverse response CHE30 Process Dynamics and Conrol orea Universiy 6-

22 Overdamped nd order+single zero sysem Gs () Ys () N() s ( as+ ) = = Ds () ( s+ )( s+ ) (a) a The lead dominaes he lags. (b) 0 < a The lags dominae he lead. (c) Inverse response M( as+ ) ( a ) ( a ) M ( + )( + ) + + = = + + s s s s s s a y () = M + e + e / a / > > 0 (assume > ) 0 > a CHE30 Process Dynamics and Conrol orea Universiy 6-

23 Oher inerpreaion y () y () Gs () ( as+ ) = = + ( s+ )( s+ ) ( s+ ) ( s+ ) s = = ( s + ) ( ) ( a + ) ( a) s= / s = = ( s + ) ( ) ( a + ) ( a ) s= / ( s + ) Since >, is slow dynamics and is fas dynamics. a U(s) y () y() ( s + ) > y y () a () a > 0 < 0 < 0 > 0 < y() y () y() > 0 < ε Y (s) Y (s) Y(s) > 0 < 0 > CHE30 Process Dynamics and Conrol orea Universiy 6-3

24 EFFECTS OF POLE LOCATION Faser response Faser response Im(s) More oscill. Less oscill. Shorer period of oscill. Unsable Region Faser response Re(s) Shorer period of oscill. CHE30 Process Dynamics and Conrol orea Universiy 6-4

25 EFFECTS OF ZERO LOCATION Zero a origin: Im(s) Complex LHP zero: valley Inverse response region Dominan Pole Deeper valley Complex RHP zero: 0 Re(s) Real LHP zero: More overdamped More overshoo Real RHP zero: Bigger inverse response CHE30 Process Dynamics and Conrol orea Universiy 6-5