Lecture 8 Transfer Function Definition Block Diagram Manipulation - PowerPoint PPT Presentation

Lecture 8 Transfer Function Definition Block Diagram Manipulation Process Control Prof. Kannan M. Moudgalya IIT Bombay Tuesday, 6 August 2013 1/30 Process Control Transfer functions, block diagram manipulation

Outline 1. Formal definition of transfer functions 2. Block diagram manipulation 3. Time delay 2/30 Process Control Transfer functions, block diagram manipulation

1. Formal definition of transfer functions 3/30 Process Control Transfer functions, block diagram manipulation

One Definition of Impulse Function, δ ◮ It is a function with a nonzero value at only one point, with area unity ◮ Rectangle of infinite height with zero width ◮ Triangle of infinite height with zero base length 4/30 Process Control Transfer functions, block diagram manipulation

Second Definition of Impulse Function ◮ Let the impulse function be δ (t) ◮ The integral � b � f(0) , a < 0 < b f(t) δ (t)dt = 0 , 0 / ∈ (a , b) a ◮ The Laplace Transform of δ (t) is � ∞ e − st δ (t)dt = 1 0 5/30 Process Control Transfer functions, block diagram manipulation

Transfer function Recall the procedure we used to derive transfer function: ◮ Considered only time invariant systems ◮ Linearised them ◮ Made initial conditions to be zero 6/30 Process Control Transfer functions, block diagram manipulation

Impulse response ◮ We studied step response, ramp response, etc. ◮ The impulse response of a system is 1. step response to a system which behaves like an impulse 2. inverse Laplace transform of its transfer function 3. don’t know what it is 7/30 Process Control Transfer functions, block diagram manipulation

Definition of transfer functions ◮ Recall the input-output property u ( t ) ↔ u ( s ) y ( s ) ↔ y ( t ) G ( s ) ◮ What will be the output, if the input is impulse? ◮ y(s) = G(s)u(s) = G(s), in case of impulse input ◮ i.e. y(s) = G(s), in case of impulse input ◮ So, g(t) (where, g(t) ↔ G(s)) is known as the impulse response ◮ Transfer function is the Laplace Transform of impulse response 8/30 Process Control Transfer functions, block diagram manipulation

Impulse response ◮ We studied step response, ramp response, etc. ◮ The impulse response of a system is 1. step response to a system which behaves like an impulse 2. inverse Laplace transform of its transfer function 3. don’t know what it is Answer: 2 9/30 Process Control Transfer functions, block diagram manipulation

2. Block diagram manipulation 10/30 Process Control Transfer functions, block diagram manipulation

Transfer functions in series u ( s ) G 1 ( s ) G 2 ( s ) y ( s ) u ( s ) y 1 ( s ) y ( s ) G 1 ( s ) G 2 ( s ) ◮ y 1 (s) = G 1 (s)u(s) ◮ Under what conditions? Initial condition is zero ◮ Or when deviational variables are used ◮ Also when the model is linear ◮ y(s) = G 2 (s)y 1 (s) = G 2 (s)G 1 (s)u(s) ◮ Overall transfer function = G 2 (s)G 1 (s) ◮ = G 1 (s)G 2 (s), in case of scalars 11/30 Process Control Transfer functions, block diagram manipulation

A good counter example ◮ When nonlinear or time varying, systems do not commute ◮ J. C. Proakis and D. G. Manolakis, Digital Signal Processing Principles, Algorithms, and Applications. Prentice Hall, Inc., Upper Saddle River, NJ and also New Delhi ◮ Repeated in K. M. Moudgalya, Digital Control. John Wiley & Sons, Chichester and also New Delhi 12/30 Process Control Transfer functions, block diagram manipulation

Transfer functions in parallel y 1 ( s ) G 1 ( s ) u ( s ) y ( s ) u ( s ) y ( s ) G 2 ( s ) G 1 ( s ) + G 2 ( s ) y 2 ( s ) ◮ y 1 (s) = G 1 (s)u(s) ◮ Once again, under zero initial conditions ◮ Or when deviational variables are used ◮ y(s) = y 1 (s) + y 2 (s) = G 1 (s)u(s) + G 2 (s)u(s) ◮ = (G 1 (s) + G 2 (s))u(s) ◮ Overall transfer function = G 1 (s) + G 2 (s) 13/30 Process Control Transfer functions, block diagram manipulation

Closed loop transfer function 14/30 Process Control Transfer functions, block diagram manipulation

Recall: Model of Flow Control System Q i ( t ) h ( t ) Q ( t ) = x ( t ) h ( t ) K 1 K 2 ∆h(s) = τ s + 1∆Q i (s) − τ s + 1∆x(s) 15/30 Process Control Transfer functions, block diagram manipulation

Recall the block diagram representation Disturbance ∆ Q i Variable K 1 τs + 1 ∆ x ∆ h − K 2 τs + 1 Manipulated Controlled Variable Variable ◮ + sign is not explicitly shown ◮ − sign has to be shown at the summing junction 16/30 Process Control Transfer functions, block diagram manipulation

How do we use this model in feedback control design? Disturbance ∆ Q i Variable K 1 τs + 1 ∆ x ∆ h − K 2 τs + 1 Manipulated Controlled Variable Variable 17/30 Process Control Transfer functions, block diagram manipulation

Feedback control system block diagram ∆ Q i K 1 τs + 1 e Setpoint ∆ x ∆ h − K 2 G c τs + 1 − ◮ G c is controller, to design in this course ◮ Setpoint is the desired value of height ◮ e is the error ◮ The output is subtracted from the setpoint ◮ What is the relation between setpoint and ∆h? 18/30 Process Control Transfer functions, block diagram manipulation

A simplified closed loop transfer function y sp e u y G c G − K ◮ What is the relationship between y and y sp ? ◮ K is the measurement transfer function ◮ G c is the transfer function of the controller ◮ The closed loop relation is GG c y(s) = 1 + KGG c 19/30 Process Control Transfer functions, block diagram manipulation

Problem from Final Exam, 2009 This problem is concerned with a scheme, known as cascade control, shown below: Y sp Y 1 1 K c 1 K c 2 s + 1 (2 s + 1)(4 s + 1) − − G m Determine the closed loop transfer function 20/30 Process Control Transfer functions, block diagram manipulation

3. Time delay 21/30 Process Control Transfer functions, block diagram manipulation

Time Shift Laplace Transform ◮ Let the Laplace Transform of f(t) be F(s) ◮ The Laplace Transform of f(t − L) is 1. F(s − L) 2. e L F(s) 3. e − L F(s) 4. e − sL F(s) ◮ Answer: e − sL F(s) ◮ So, the Laplace Transform of u(t) = [1(t) − 1(t − t 1 )] × M is � 1 s − 1 � se − t 1 s ◮ U(s) = M 22/30 Process Control Transfer functions, block diagram manipulation

Time Delay Modelling From a previous lecture 23/30 Process Control Transfer functions, block diagram manipulation

One Way to Determine Time Delay ◮ Change the input by a step ◮ Find out when the output starts changing ◮ Determine the time delay ◮ Check whether the SBHS has any time delay! 28/30 Process Control Transfer functions, block diagram manipulation

What we learnt today ◮ Formal definition of transfer function ◮ Introduction to block diagram manipulation ◮ Time delay processes 29/30 Process Control Transfer functions, block diagram manipulation

Thank you 30/30 Process Control Transfer functions, block diagram manipulation