Lecture 5 First Order Transfer Function Models Process Control Prof. Kannan M. Moudgalya IIT Bombay Tuesday, 30 July 2013 1/38 Process Control First Order Models
Outline 1. Thermal sensor as a first order system 2. Scilab demonstration of curve fitting 3. First order systems in series Scilab demonstration 2/38 Process Control First Order Models
1. Thermal sensor as a I order system 3/38 Process Control First Order Models
Thermocouple to Measure Temperature ◮ Shaded: sensor T a ◮ Mass m and heat capacity C p ◮ Area available for heat transfer T ◮ Initially, sensor and ambient are at T ◮ All of a sudden, the ambient temperature goes to T a ◮ Model this: get T as a fn. of T a 4/38 Process Control First Order Models
Model of temp. measurement system T a dT(t) mC p = hA(T a − T(t)) dt T ◮ Initially, sensor & ambient at T ◮ Ambient temperature goes to T a ◮ Subtract the steady state and express in deviational variables ◮ ∆T(s) = G(s)∆T a (s) 1 ◮ G(s) = τ s + 1. Gain? Why? ◮ where, τ = mC p hA ◮ What is the unit of τ ? 5/38 Process Control First Order Models
A sample problem statement ◮ A thermometer at 30 ◦ C is suddenly immersed into boiling water ◮ Thermometer be modelled as a first order system with time constant of 0.5min ◮ Calculate the temperature profile as indicated by the thermometer. 6/38 Process Control First Order Models
Solution to thermometer problem T a ◮ Y(s) = G(s)U(s) K T ◮ G(s) = 0 . 5s + 1, K = 1 ◮ U(s) = ambient temperature ◮ Y(s) = temperature indicated by the thermometer, deviational ◮ y(t) ↔ Y(s), u(t) ↔ U(s) ◮ Initial temperature = 30 ◦ C 1. y(t) = T(t), u(t) = T a (t). 2. y(t) = T(t) − 30, u(t) = T a (t). 3. y(t) = T(t), u(t) = T a (t) − 30. 4. y(t) = T(t) − 30, u(t) = T a (t) − 30. right 7/38 Process Control First Order Models
Can also use appropriate variables 1 ◮ ∆T(s) = τ s + 1∆T a (s) ◮ ∆T(t) ↔ ∆T(s), ∆T a (t) ↔ ∆T a (s) ◮ τ = 0 . 5min 70 ◦ C ◮ ∆T a (t) = ◮ ∆T a (s) = 70 s ◮ ∆T(t) = 70 � 1 − e − t / 0 . 5 � ◮ T(t) = 30 + ∆T(t) 8/38 Process Control First Order Models
2. Scilab Demonstration of Curve Fitting 9/38 Process Control First Order Models
Recall: Step response of a first order system Let the plant have a 1st order transfer function: K y(s) = τ s + 1u(s) Its step response is given by, � 1 − e − t /τ � y(t) = K 10/38 Process Control First Order Models
Scilab code to plot step response of a first order system: 02-step.sce s = %s 1 tau = 0 . 5 ; 2 3 G = 1/( tau ∗ s +1) t = 1 : 0 . 1 : 5 ; 4 5 y = csim ( ’ step ’ , t ,G) ; p l o t 2 d ( t , y ) 6 11/38 Process Control First Order Models
Step response of I order system in Scilab 12/38 Process Control First Order Models
Curve Fitting ◮ Given model parameters τ and K, we can obtain step response ◮ Can we back-calculate? ◮ Given the step response, can we calculate τ and K? 13/38 Process Control First Order Models
Identification of first order TF through step response ◮ Experimental data have noise ◮ Does it affect identification? ◮ Demonstrate with 2 . 5 G(s) = 0 . 75s + 1 with noise superimposed 14/38 Process Control First Order Models
Data generation to simulate a noisy system: step-noise.sce I s = %s 1 tau = 0 . 7 5 ; 2 3 G = 2 . 5 / ( tau ∗ s +1) t = 0 : 0 . 0 1 : 5 ; 4 5 y = csim ( ’ step ’ , t ,G) ; p l o t 2 d ( t , y ) 6 x l a b e l ( ” t ” ) 7 y l a b e l ( ”y” ) 8 x t i t l e ( ”y ( t ) vs . t ” ) 9 15/38 Process Control First Order Models
Data generation to simulate a noisy system: step-noise.sce II n o i s e = rand (1 , l e n g t h ( y ) , ’ normal 10 ’ ) ; y n o i s e = y + 0.25 ∗ n o i s e ; 11 / / p l o t 2 d ( t , y n o i s e ) 12 y 2 n o i s e = y + 0.025 y . ∗ n o i s e ∗ 13 ; p l o t 2 d ( t , y 2 n o i s e ) 14 16/38 Process Control First Order Models
We will use noise corrupted y as the plant data 17/38 Process Control First Order Models
Can we extract the model parameters by curve fitting using plant data? 18/38 Process Control First Order Models
Curve fitting by optimisation ◮ Plant data: y = noise + output of the plant with G = 2 . 5 / (0 . 75s + 1) ◮ Fit this with the output y prediction from a plant with transfer function with G = K / ( τ s + 1) ◮ Get K and τ by minimising || y − y prediction || 2 ◮ What should we get for K and τ ? 19/38 Process Control First Order Models
Scilab code to identify a first order transfer function: order 1.sce I exec ( ’ step − n o i s e . sce ’ ) ; 1 exec ( ’ c o s t f 1 . s c i ’ ) ; 2 3 y = y 2 n o i s e ; g l o b a l ( ’ y ’ , ’ t ’ ) ; 4 5 6 x0 = [3 2 ] ; [ f , xopt , gopt ] = optim ( c o s t f 1 , ’ b 7 ’ , [ 0 . 1 0 . 1 ] , [ 5 5 ] , x0 , ’ ar ’ ,500 ,500) 8 kp = xopt (1) ; 20/38 Process Control First Order Models
Scilab code to identify a first order transfer function: order 1.sce II tau = xopt (2) ; 9 y p r e d i c t i o n = kp ( 1 − exp( − t ∗ 10 / tau ) ) ; format ( ’ v ’ ,6) ; 11 p l o t 2 d ( t , y ) 12 p l o t 2 d ( t , y p r e d i c t i o n ) 13 21/38 Process Control First Order Models
Scilab code for objective function: costf 1.sci I f u n c t i o n [ f , g , i n d ] = c o s t f 1 ( x , 1 i n d ) 2 kp = x (1) ; tau = x (2) ; y p r e d i c t i o n = kp ( 1 − exp( − t ∗ 3 / tau ) ) ; f = ( norm ( y − y p r e d i c t i o n , 2 ) ) ˆ2; 4 5 g = numdiff ( func 1 , x ) ; e n d f u n c t i o n 6 7 22/38 Process Control First Order Models
Scilab code for objective function: costf 1.sci II f u n c t i o n f = f u n c 1 ( x ) 8 9 kp = x (1) ; tau = x (2) ; y p r e d i c t i o n = kp ( 1 − exp( − t ∗ 10 / tau ) ) ; f = ( norm ( y − y p r e d i c t i o n , 2 ) ) ˆ2; 11 e n d f u n c t i o n 12 23/38 Process Control First Order Models
3. First order systems in series 24/38 Process Control First Order Models
More realistic model of thermometer ◮ Modelled as a first order system ◮ A more realistic model: T a T T w ◮ The sheath is also modelled ◮ Its temperature is denoted as T w , T-wall ◮ Derive the second order relation between ∆T a (s) and ∆T(t) 25/38 Process Control First Order Models
More realistic model of thermometer T a T T w ◮ Model of the sensor: dT ◮ mC p dt = h sw A sw (T w − T) ◮ Model of the wall: dT w ◮ m w C pw = dt h wa A wa (T a − T w ) − h sw A sw (T w − T) ◮ Derive the relation between T a and T ◮ Same procedure as before: deviational ◮ Order of the transfer function: second 26/38 Process Control First Order Models
Model derivation method ◮ Laplace transform models are in deviational form ◮ For each equation, subtract the corresponding steady state ◮ Assume small deviation, if required ◮ Take Laplace transform of each equation and simplify ◮ Substitute the expression for T w into the other equation ◮ You will get T as a function of T a only ◮ The transfer function will be second order: denominator will be a second degree polynomial 27/38 Process Control First Order Models
Will do it as an assignment 28/38 Process Control First Order Models
Another second order example 29/38 Process Control First Order Models
Recall: Model of Flow Control System Q i ( t ) h ( t ) Q ( t ) = x ( t ) h ( t ) ◮ Suppose that x is constant K 1 ◮ ∆h(s) = τ s + 1∆Q i (s) ◮ τ = A / x s , K 1 = 1 / x s , K 2 = h s / x s . ◮ ∆Q(s) = x s ∆h(s) 30/38 Process Control First Order Models
Two tanks in series Q i (t) ◮ Let out flow rate of I tank = ∆Q 1 (s) h(t) ◮ ∆Q 1 (s) = G 1 (s)∆Q i (s) ◮ ∆Q(s) = G 2 (s)∆Q 1 (s) ◮ = h(t) G 2 (s)G 1 (s)∆Q i (s) Q(t) = x(t)h(t) How do you model this? 31/38 Process Control First Order Models
Comparing response of first and second order systems ◮ Response of ◮ a first order system ◮ two first order systems in series are similar 32/38 Process Control First Order Models
Scilab code to plot step response of a seond order system: step-2nd.sce s = %s 1 2 G 1 = 1/( s +1) 3 G 2 = 1/( s +2) 4 G = G 1 ∗ G 2 t = 1 : 0 . 1 : 1 0 ; 5 6 y = csim ( ’ step ’ , t ,G) ; p l o t 2 d ( t , y ) 7 x l a b e l ( ” t ” ) 8 y l a b e l ( ”y” ) 9 x t i t l e ( ”y ( t ) vs . t ” ) 10 33/38 Process Control First Order Models
Step response of second order system 34/38 Process Control First Order Models
Recall: step response of a first order system: 1 / (0 . 5s + 1) 35/38 Process Control First Order Models
Can we interchange data and model? 36/38 Process Control First Order Models
What is learnt today ◮ First order system ◮ Second order systems ◮ Examples ◮ Scilab demonstration 37/38 Process Control First Order Models
Thanks 38/38 Process Control First Order Models
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