1. Chapter 3 mathematical Modeling of Dynamic Systems

2. Modeling Mathematical models are developed from: physical laws
chemical laws
biological laws
economic laws
etc
Mathematical models take the form of:
differential equations
transfer functions
state equations
block diagrams
signal flow graphs
Mathematical models are used to analyze dynamic characteristics and to design control systems

3. Models Simplicity vs Accuracy lumped vs distributed parameter models
linear vs nonlinear models
time-invariant vs time varying models
As simple as possible with the required accuracy

4. Transfer Function and Impulse-Response Function
Convolution Integral
Impulse-Response Function

5. Convolution Integral ILT convolution
Multiplication in the frequency domain
corresponds to
convolution in the time domain

6. Convolution

7. Impulse Response Function
The response of a differential equation to an impulse (delta function) input is the Impulse Response Function of that differential equation.
For an impulse input
The Laplace transform of the impulse response is
= transfer function
The impulse response is the inverse Laplace transform of the transfer function

8. Impulse Response, Convolution, and solution to DEs
The output of a system due to the input r(t) is the convolution of the input function and the impulse response of the system.
y(t) = conv(g(t),r(t))
There is an intimate relationship between the impulse response of a system and the response of the system to any other input.

9. Automatic Control Systems
Block Diagrams
Signals
Blocks and Transfer functions
Summing point
Branch point
Block diagram of a closed loop system
Open-loop transfer function and feedforward transfer function
Closed-loop transfer function
Automatic Controllers
Industrial Controllers
On-off, P, I, PI, PD, PID
Disturbances
Block diagram reduction

10. Start by doing what’s necessary,
then what’s possible, and
suddenly you are doing the impossible.
-St. Francis of Assisi

11. Modeling review Mechanical systems Electrical systems
I hope to come back to this topic later
Skip 3-5, 3-9, 3-10
Mechanical examples?
Electrical examples?
Thermal examples?
Transfer function?
Time constant
Mechanical examples?
Electrical examples?
Thermal examples?
Transfer function?
Undamped natural frequency
damping ratio

12. Transfer function Input signal Output signal Transfer function
Block diagram
Equation

13. Block Diagram reduction
Open loop TF?
Feedforward TF?
Closed loop TF?

14. Steady-State Error Steady-state means the output looks like the input.
Step inputs produce step (constant) outputs
Sinusoidal inputs produce sinusoidal outputs
Ramp inputs produce ramp outputs
X inputs produce X outputs
Steady-state means that the transients have died out.
The output has (at least) two terms,
steady-state (mathematically looks like the input)
transient (decays to zero)
sometimes other terms (normally a bad situation)

15. Computing SS error To compute E(s):
inverse Laplace transform to get e(t)
take limit as t goes to infinity.
Steady state error only exists if the limit exists.

16. Chapter 3 Problems A B Comments 1-5 1-8 Block diagram reduction
6-10, 12,13 9
State space 11
10-12
State space to transfer function
14-18
13-18
Mechanical models
19-22
19-24
Electrical models 23 25
Electro mechanical model
24-26
26-27
Signal flow graph 27
28-29
linearization

17. Assignments Ungraded homework: A1-5 Graded homework: B1-7
Test 1: Solving DEs via Laplace transforms, Block diagrams.
Laplace transforms
definition
2 formulas for functions
derivatives
Euler’s identities
ALGEBRA
CALCULUS
Complex numbers/algebra