1. Automatic Control System
Modelling

2. Modelling dynamical systems
Engineers use models which are based upon mathematical relationships between two variables.
We can define the mathematical equations:
Measuring the responses of the built process (black model)
Not interested in the number and the real value of the time constants of the process, only it is enough that the response is similar sufficient accuracy
Using the basic physical principles (grey model).
In order to simplification of mathematical model the small effects are neglected and idealised relationships are assumed.
Developing a new technology or a new construction nowadays it’s very helpful applying computer aided simulation technique.
This technique is very cost effective, because one can create a model from the physical principles without building of process.

3. Grey box model

4. Modelling mechanical systems
Newton’s law one-dimensional translation and rotational systems.
F: force [N]; FD: absorber’s force; Fs: spring’s force
M: moment about centre of mass of body [Nm]
I: the body’s moment of inertia [kgm2]
s, : displacement [m; rad]
v, ω: velocity [m/sec; rad/sec]
a, : acceleration [m/sec2; rad/sec2]
C: friction constant [Nsec/m]
k: spring constant [N/m]
velocity
acceleration
Assign variables and sufficient to describe an arbitrary position of the object.
Draw a free-body diagram of each components and indicate all forces acting.
Apply Newton’s law in translation and rotational form.
Combine the equations to eliminate internal forces.

5. Modelling mechanical systems
Road surface
The car’s wheel vertical motion is assumed one-dimensional and the mass hasn’t got extension. The sock absorber is represented by a dashpot symbol with friction constant C.
The force from the spring acts on both masses in proportional their relative displacement with spring constant k.
The equilibrium positions of the two mass are offset from the spring’s unstretched positions because of the force of gravity.
Nowadays the mathematical programs allow one to create a better model.
The system can be approximated by the simplified system shown in left. t y m2 m1 k2 k1 C x t t r

6. Modeling mechanical systemsk2 k1 C x t
The positive displacements or velocity of mass, and so the positive force are signed by up arrows. t r

7. Simulating the system by MATLAB

8. Difference equations y7 y6 y5 y4 y3 y2 y1 y0
A block input is energised by x(t), and the response
is y(t). Because of the response needs any time or in
simulation the calculation of the response also needs any time, and so y(iT0) can be calculated from previous value than x(iT0). The sampled block has a delay time (T0).
Example:
T T T0 4T0 5T0 6T0 7T0

9. Using difference equation

10. Using Laplace transform to model mechanical systemsk2 k1 C

11. Block modeling mechanical system

12. Block modeling mechanical system Block reduction

13. Modeling electromechanical systems
DC motor’s voltages.
The electromotive force based on:
DC motor’s tongues.
B: magnet field [T: Tesla]; Ta armature torque
Ia: armature current Tf friction torque
UE: electromotive force [V] TL load torque
UL: voltage on inductance [V] C: friction constant [Nsec/m]
UR: voltage on resistance [V]

14. Ke electromotive force constant
Modeling DC motor
 angular displacement
Ta armature torque
Tf friction torque
TL load torque
Ia armature current
Ra armature resistance
La armature inductance M
armature inertia
Generated electromotive force (emf) against the applied armature voltage
Ke electromotive force constant
Ka motor torque constant
C rotational friction constant

15. Simulating the system by MATLAB

16. Example: Block modeling DC motor

17. In time domain using difference equations

18. In operator frequency domain
Assuming than Udc is constant and the system is steady-state when one change the value of Udc
Examination of dynamic behaviour can be used the Laplace transform. M
G(s)

19. Block modeling DC motor

20. Simpler block model of DC motor

21. Models of electronic circuit
Kirchhoff’s current law: The algebraic sum of current leaving a junction or node equals the algebraic sum of the current entering that node.
Kirchhoff’s current law: The algebraic sum of all voltages taken around a closed path in a circuit is zero.
Resistor
Capacitor
Inductor

22. Models of electronic circuit
All resistance equal R and all capacitance equal C. Point “A” is nearly ground and such as a summing junction for the currents.
“B” is a take off point for U2.
The OpAmp amplitude gain is Au(s) I2 I1 B A U1 U2 U1 U1 I1 U2 I1 I2 I2 U2 I3 U2 I3

23. Modeling heat flow
Heat energy flow.
Thermal conductivity:
A: cross-sectional area l: length of the heat-flow path
k: thermal conductivity constant
Temperature as a function of heat-energy flow:
Specific heat:
q: heat energy flow J/sec R: thermal resistance °C/J T: temperature °C
C: thermal capacity J/°C

24. Heat flow models m: the mass of the substanceq2 R0
m: the mass of the substance
cv: specific heat constant q1
room T1 R1
The net heat-energy flow into a substance:
The heat energy flows through substances (across the room’s wall):
The heat can also flow when a warmer mass flow into a cooler mass or vice versa:

25. Heat flow models The total heat-energy flow:
It’s non-linear, except T1=T2.

26. Modelling a heat exchanger
water
steam
The time delay between the measurement and the exit flow of the water:
As: area of the steam inlet valve, Aw: area of the water inlet
Ks: flow coefficient of the inlet valve, Kw: flow coefficient of the water inlet
cvs: specific heat of steam, cvw: specific heat of water
Tsi: temperature of inflow steam, Twi: temperature of inflow water
Ts: temperature of outflow steam, Ts: temperature of outflow water
Cs=mscvs thermal capacity of the steam, Cw=mwcvw thermal capacity of the water
R: thermal resistance (average over the entire exchanger)

27. Simulating heat exchanger

28. Laplace form heat exchanger
The equation is nonlinear because the state variable Ts is multiplied by the the control input As. The equation can be linearized at the working point Ts0, and so Tsi-Ts0=Ts nearly constant. To measure all temperature from Twi, it’s eliminated Twi=0.

29. Block model of heat exchanger

30. Modeling by reaction curve
Black box model
Modeling by reaction curve

31. Feedback control GW(s) GC(s) GA(s) GP1(s) GP2(s) GT(s)
plant
GW(s)
controller
GC(s)
GA(s)
GP1(s)
GP2(s)
A/M
GT(s)
Process field
When auto / manual switch is manual
position (open), then GC(s)=1 W
GW(s)
R0+r
U0+u
GC(s)
GA(s)
GP(s)
A/M
YM0+yM
GT(s)

32. Modelled the process from reaction curve by dead-time proportional first order transfer function HPT1
Apply a small step-change to the controller output and record the open-loop response.
The first step is to find the maximum slope of the reaction curve and draw a tangent.
The next step is to determine the “effective delay time” and the “effective time constant” of the plant, where the line of maximum slope crosses the initial and final value of the response.

33. The error of the model The principle of the less squares:
The better model the less sum of value of the squares.
The error of the model:

34. A better model of process from reaction curve by dead-time second order transfer function HPT2
It needs computer! The beginning parameters:

35. Modelled the process from reaction curve by “n” order transfer function PTn

36. Modelled the process from reaction curve by dead-time integral first order transfer function HIT1