Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system

Transcription

1 Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng functon response cause effect 1

2 Defnton of the transfer functon: Let G(s) denote the transfer functon between an nput, x, and an output, y. Then, by defnton where: G s Y s X s L y( t) Y s L x( t) X s 2

3 Development of Transfer Functons Example: Strred Tank Heatng System Fgure 2.3 Strred-tank heatng process wth constant holdup, V. 3

4 Recall the prevous dynamc model, assumng constant lqud holdup and flow rates: dt Vρ C = wc T T + Q dt (1) Suppose the process s ntally at steady state: ( 0 ) =, ( 0 ) =, ( 0) = ( 2) T T T T Q Q where T steady-state value of T, etc. For steady-state condtons: 0 = wc T T + Q (3) Subtract (3) from (1): dt V ρ C = wc T T T T Q Q dt + (4) 4

5 But, dt dt = ( T) because s a constant (5) d T dt T Thus we can substtute nto (4-2) to get, dt VρC = wc T T + Q dt ( ) (6) where we have ntroduced the followng devaton varables, also called perturbaton varables : T T T, T T T, Q Q Q (7) Take L of (6): ( = ) = Vρ C st s T t 0 wc T s T s Q s (8) 5

6 Evaluate T ( t = ) 0. By defnton, T T T. Thus at tme, t = 0, T 0 = T 0 T (9) But snce our assumed ntal condton was that the process was ntally at steady state,.e., T( 0) = Tt follows from (9) that T 0 = 0. Note: The advantage of usng devaton varables s that the ntal condton term becomes zero. Ths smplfes the later analyss. Rearrange (8) to solve for T ( s) : K 1 T s Q s T s τs+ 1 τs+ 1 = + (10) 6

7 where two new symbols are defned: 1 V ρ K and τ 11 wc w Transfer Functon Between Q and T Suppose T s constant at the steady-state value. Then, T() t = T T ( t) = 0 T ( s) = 0. Then we can substtute nto (10) and rearrange to get the desred TF: T s K = Q s τ s+ 1 (12) 7

8 Transfer Functon Between and T T : Suppose that Q s constant at ts steady-state value: Qt = Q Q t = 0 Q s = 0 Thus, rearrangng Comments: ( s) T 1 = T s τ s+ 1 (13) 1. The TFs n (12) and (13) show the ndvdual effects of Q and on T. What about smultaneous changes n both Q and? T T 8

9 Answer: See (10). The same TFs are vald for smultaneous changes. Note that (10) shows that the effects of changes n both Q and T are addtve. Ths always occurs for lnear, dynamc models (lke TFs) because the Prncple of Superposton s vald. 2. The TF model enables us to determne the output response to any change n an nput. 3. Use devaton varables to elmnate ntal condtons for TF models. 9

10 Propertes of Transfer Functon Models 1. Steady-State Gan The steady-state of a TF can be used to calculate the steadystate change n an output due to a steady-state change n the nput. For example, suppose we know two steady states for an nput, u, and an output, y. Then we can calculate the steadystate gan, K, from: y y K = u u (4-38) For a lnear system, K s a constant. But for a nonlnear u, y. system, K wll depend on the operatng condton 10

11 Calculaton of K from the TF Model: If a TF model has a steady-state gan, then: K = s 0 lm G s (14) Ths mportant result s a consequence of the Fnal Value Theorem Note: Some TF models do not have a steady-state gan (e.g., ntegratng process n Ch. 5) 11

12 2. Order of a TF Model Consder a general n-th order, lnear ODE: n n 1 m d y dy dy d u an + a n n 1 + a n a0y = bm + m dt dt dt dt m 1 d u du bm b b0u (4-39) m dt dt Take L, assumng the ntal condtons are all zero. Rearrangng gves the TF: G s Y s = = U s m = 0 n = 0 bs as (4-40) 12

13 Defnton: The order of the TF s defned to be the order of the denomnator polynomal. Note: The order of the TF s equal to the order of the ODE. Physcal Realzablty: For any physcal system, n mn (4-38). Otherwse, the system response to a step nput wll be an mpulse. Ths can t happen. Example: ay 0 = b du 1 + bu 0 and step change n u (4-41) dt 13

14 3. Addtve Property Suppose that an output s nfluenced by two nputs and that the transfer functons are known: Y( s) 1 and 2 Y s = G s = G s U s U s 1 2 Then the response to changes n both U1 and U2can be wrtten as: = + Y s G s U s G s U s The graphcal representaton (or block dagram) s: U 1 (s) U 2 (s) G 1 (s) G 2 (s) Y(s) 14

15 4. Multplcatve Property Suppose that, Y s U2 s = G2( s) and U2 s U3 s = G3( s) Then, Y s = G s U s and U s = G s U s Substtute, Or, Y s = Y s G s G s U s U3 s = G s G s U s G s G s Y s 15

16 Lnearzaton of Nonlnear Models So far, we have emphaszed lnear models whch can be transformed nto TF models. But most physcal processes and physcal models are nonlnear. - But over a small range of operatng condtons, the behavor may be approxmately lnear. - Conclude: Lnear approxmatons can be useful, especally for purpose of analyss. Approxmate lnear models can be obtaned analytcally by a method called lnearzaton. It s based on a Taylor Seres Expanson of a nonlnear functon about a specfed operatng pont. 16

17 Lnearzaton (contnued) Consder a nonlnear, dynamc model relatng two process varables, u and y: dy dt = f ( y u), (4-60) Perform a Taylor Seres Expanson about u = u and y = y and truncate after the frst order terms, f f f ( u, y) = f ( u, y) + u + y (4-61) u y s s where u = u u, y = y y, and subscpt s denotes the steady state, ( u, y). Note that the partal dervatve terms are actually constants because they have been evaluated at the nomnal operatng pont, s. 17

18 Lnearzaton (contnued) Substtute (4-61) nto (4-60) gves: dy f f = f ( u, y) + u + y (A) dt u y s Because u, y s a steady state, t follows from (4-60) that s f ( u, y) = 0 (B) Also, because y y- y, t follows that, dy dy = (C) dt dt Substtute (B) and (C) nto (A) gves the lnearzed model: dy f f = u + y dt u y s s (4-62) 18

19 q Example: Lqud Storage System dh Mass balance: A q q (1) dt = Valve relaton: q = Cv h (2) h A = area, C v = constant Combne (1) and (2) and rearrange: dh 1 Cv = q h = f( h, q) (3) dt A A Eq. (3) s n the form of (4-60) wth y = h and u = q. Thus, we can utlze (4-62) to lnearze around h= h and q = q, q dh f f = h + dt h s q s q (4) 19

20 where: f = h s C v 2A h (5) f q s 1 = A Substtute nto (4) gves the lnearzed dh 1 Cv = q h dt A 2A h model: (6) (7) Ths model can be expressed n terms of a valve resstance, R dh h = q h wth R (8) dt A AR C v 20

21 State-Space Models Dynamc models derved from physcal prncples typcally consst of one or more ordnary dfferental equatons (ODEs). In ths secton, we consder a general class of ODE models referred to as state-space models. Consder standard form for a lnear state-space model, x = Ax + Bu + Ed (4-90) y=cx (4-91) 21

22 where: x = u = the state vector the control vector of manpulated varables (also called control varables) d = y = the dsturbance vector the output vector of measured varables. (We use boldface symbols to denote vector and matrces, and plan text to represent scalars.) The elements of x are referred to as state varables. The elements of y are typcally a subset of x, namely, the state varables that are measured. In general, x, u, d, and y are functons of tme. The tme dervatve of x s denoted by ( = x t) Matrces A, B, C, and E are constant matrces. x d /d. 22

23 Example: CSTR Model Consder the prevous CSTR model. Assume that T c can vary wth tme whle c A, T, q and w are constant. Nonlnear Model: Lnearzed Model: where: 23

24 Example 4.9 Show that the lnearzed CSTR model of Example 4.8 can be wrtten n the state-space form of Eqs and Derve state-space models for two cases: (a) Both c A and T are measured. (b) Only T s measured. Soluton The lnearzed CSTR model n Eqs and 4-85 can be wrtten n vector-matrx form: dc A a11 a12 c A 0 dt = + Tc dt a21 a22 T b2 dt (4-92) 24

25 Let x1 c A and x2 T, and denote ther tme dervatves by x 1 and x 2. Suppose that the coolant temperature T c can be manpulated. For ths stuaton, there s a scalar control varable, u T c, and no modeled dsturbance. Substtutng these defntons nto (4-92) gves, x 1 a11 a12 x1 0 u x = 2 a21 a + 22 x 2 b 2 A B (4-93) whch s n the form of Eq wth x = col [x 1, x 2 ]. (The symbol col denotes a column vector.) 25

26 a) If both T and c A are measured, then y = x, and C = I n Eq. 4-91, where I denotes the 2x2 dentty matrx. A and B are defned n (4-93). b) When only T s measured, output vector y s a scalar, y = T and C s a row vector, C = [0,1]. Note that the state-space model for Example 4.9 has d = 0 because dsturbance varables were not ncluded n (4-92). By contrast, suppose that the feed composton and feed temperature are consdered to be dsturbance varables n the orgnal nonlnear CSTR model n Eqs and Then the lnearzed model would nclude two addtonal devaton varables, c A and T. 26