System Modeling. Lecture-2. Emam Fathy Department of Electrical and Control Engineering

Transcription

1 System Modeling Lecture-2 Emam Fathy Department of Electrical and Control Engineering 1

2 Types of Systems Static System: If a system does not change with time, it is called a static system. Dynamic System: If a system changes with time, it is called a dynamic system. 2

3 Static Systems A system is said to be static if its output y(t) depends only on the input u(t) at the present time t. Following figure gives an example of static systems, which is a resistive circuit excited by an input voltage u(t). Let the output be the voltage across the resistance R 3, and according to the circuit theory, we have y t = R 2 R 3 u t R 1 R 1 + R 3 + R 2 R 3 3

4 Dynamic Systems A system is said to be dynamic if its current output may depend on the past history as well as the present values of the input variables. Mathematically, y( t) [ u( ), 0 t] u : Input, t : Time Example: A moving mass y u Model: Force=Mass x Acceleration M M y = u

5 Dynamic Systems examples: RC circuit, Bicycle, Car, Pendulum (in motion) 5

6 Ways to Study a System System Experiment with actual System Experiment with a model of the System Physical Model Mathematical Model Analytical Solution Simulation Frequency Domain Time Domain Hybrid Domain 6

7 Model A model is a simplified representation or abstraction of reality. Reality is generally too complex to copy exactly. Much of the complexity is actually irrelevant in problem solving. 7

8 What is Mathematical Model? A set of mathematical equations (e.g., differential eqs.) that describes the input-output behavior of a system. What is a model used for? Simulation Prediction/Forecasting Diagnostics Design/Performance Evaluation Control System Design

9 Black Box Model When only input and output are known. Internal dynamics are either too complex or unknown. Input Output 9

10 Grey Box Model When input and output and some information about the internal dynamics of the system is known. u(t) y[u(t), t] y(t) 10

11 White Box Model When input and output and internal dynamics of the system is known. u(t) dy ( t) du( t) d y( t) 3 dt dt 2 dt 2 y(t) 11

12 Transfer Function Transfer Function G(S) is the ratio of Laplace transform of the output to the Laplace transform of the input. Assuming all initial conditions are zero. u(t) Plant y(t) G( S) Y( S) U( S) 12

13 Electrical Systems

14 Example: RC Circuit Find out the transfer function of the RC network shown in figure. Assume that the capacitor is not initially charged. R u i C + y _ u is the input voltage applied at t=0 y is the capacitor voltage u = Ri + 1 C idt 14

15 Example Find the transfer function relating the capacitor voltage, Vc(s), to the input voltage, V(s)

16 Example Differential equation di( t) 1 L Ri( t) i( ) d v( t) dt C t 0

17 Example Redraw the circuit using Laplace transform. 1 V ( s) L* s R I( s) c* s 1 V C ( s) I( s) c* s I( s) VC ( s)* c* s

18 1 V C s) I( s) c* s 1 V ( s) Ls R I( s) c* S ( I( s) V ( s)* c* s C.. (1).. (2) From (1) & (2) 1 V ( s) Ls R VC ( s)* c* c* s s V C ( s) V ( s) c* s s 2 1 L* s R 1/ L* c R 1 s L L* c c 1 * s cls 2 1 Rcs 1

19 Electric Network Transfer Functions We can also present our answer in block diagram

20 Electric Network Transfer Functions Solution summary laplace Using mesh analysis

21 HW Find the transfer function, I 2 (s)/v(s) Output I 2 (s) Input V(s)

22 Mechanical Systems

23 Translational Mechanical System Transfer Function We are going to model translational mechanical system by a transfer function. In electrical we have three passive elements, resistor, capacitor and inductor. In mechanical we have spring, mass and viscous damper.

24 Example Consider the following system (friction is negligible) k F M x Free Body Diagram f k F M Where fk and fm are force applied by the spring and inertial force respectively. f M 24

25 Example f k F M f M F f k f M Then the differential equation of the system is: F = M x + kx Taking the Laplace Transform of both sides and ignoring initial conditions we get 2 F( s) Ms X ( s) kx( s) 25

26 Example 2 F( s) Ms X ( s) kx( s) The transfer function of the system is X ( s) F( s) Ms 1 2 k if M 1000kg k 2000Nm 1 X ( s) F( s) s

27 Example-2 Find the transfer function X 2 (s)/f(s) of the following system. Free Body Diagram f k1 f k2 f B f k1 f B M 2 M 1 k 2 F(t) f M 2 f M 1 F( t) f k 1 f k 2 f M 2 f B 0 f k 1 f M 1 f B 27

28 End Of Lec 2 28