1. Automatic control systems I
Automatic control systems I. Basic knowledge of automation Short overview

2. Structure of control system
actuator
controller
actuating
drives
actuating
units
Process
or plant
Safety
system
transmitter
transducers
sensors
Standard signals
Operator
desk
Variables
Material flow

3. Divide into simpler part
Which variables of the process need control. All controlled variable is an independent simpler part of the process.
(Sometimes it is not possible, but further we assume this.)
All simpler part of the process needs a control strategy.
(Open loop or closed loop philosophy)
The points of view to choose control philosophy. Which measured variables are required to define a model.
Is the controlled variable performance on/off or continuous? Is the continuous model linear or not, time invariant or not?
Can one describe the plant between the input variables and the output controlled variable by a precise model.
Open loop the accuracy of the model determines how accurate the control.
Economic efficiency points of view.
* Sometimes it’s not possible, because not only one controlled variable are influenced significantly by one manipulated variable.
Fail safe: üzembiztos

4. Dimensionless technique
manipulated
variable
controlled
variable
action
signal
feedback
signal
max
max
20 mA
20 mA
A/D conversion
control task
D/A conversion
Domain of
variability
4 mA
4 mA
min
min
Frequently used continuous signals: 4 – 20 mA 0 – 20 mA, 0 – 10 V, 2 – 10 V, (0.4 – 2 Bar)
Frequently used on /off signals: 0 – 24 VDC 0 – 110 / 230 VAC, (0 – 4 Bar)

5. Control strategies
Open loop control One can describe the plant between the manipulated and disturbance variables and the controlled variables by a precise model and so using the required measured variables one can develop a control algorithm. If all signals and variables are On/Off, than a precise model always exists. Advantages: There isn’t stability problem. This method is so punctual such as the model, the control action doesn’t require an error. Disadvantages: Sometimes this solution isn’t economical.
Closed loop control ( Feedback ) The reference signal represents the required value of the controlled variable. The controlled variable and the reference signal continuously compare and if the detecting and the reference signal are not equal, than an adequate action signal attempt to eliminate the error. Advantages: Sometimes this solution is economical. Disadvantages: The key is to appear an error and needs a transient time to eliminate this error. The controlled variable isn't always punctual. There is stability problem.
One can always choose the open loop strategy if the performance of controlled variable is on/off.
The open loop strategy is always better than closed loop, but this solution often isn’t economical because it needs more transmitter and a lot of measuring on the process for developing an appropriate model.
If the parameters of the mathematical model a little bit change in time, than it causes an error. The closed loop control can eliminate this error, but the open loop control not.

6. Block representation
We assume the input variables are within a range and the output variable remain in this range.
Actuating path of signals and variables
One input and one output block represents the context between the
the output and input signals or variables in time or frequency domain
Summing junction
Take-off point (The same signal actuate both path)
First we assume that the input and output signal set is continuous, continuous in time, and time invariant.
The nature of the block can be self-aligning (in steady-state the output is a constant), integrating (the output changes, if the input doesn't equal zero), and differentiating (the output is zero, if the input doesn't change).
w(t)
y(t)

7. Self-aligning block
In frequency domain the dynamic behaviour is describe by transfer function.
In time domain the dynamic behaviour is describe by differential equation. Y
y(t)
W1+w(t)
Y1+y(t)
WP2
WP1 Y1 t W1 W
Assuming: The area between working point 1 and 2 can be seemed as linear, than the Laplace transform and inverse Laplace transform convert each other the time and frequency domain.
w(t) t 7

8. There is no differentiating independently of the other blocks.
Integrating block
There is no differentiating independently of the other blocks.
w1(t)
If the input equal zero, than the output is steady-state
w(t)
y(t) Y
w2(t)
y(t) Y2
WP2
WP1
One of them or both of two components of input can be changed. Y1 t W1
Assuming: The area between working point 1 and 2 can be seemed as linear, other words the superposition is available.
w1(t)-w2(t) t

9. Only the frequency domain is true the next:
Block manipulation
Only the frequency domain is true the next: G1 G2
G1G2 G1 G1 G1 G1
G1+G2 G2 G1 G2 G1 G1

10. Correlation between frequency and time domain of linear systems
Fourier and inverse Fourier transform
The above is true if:
Laplace and inverse Laplace transform

11. Laplace transform Rules of Laplace transform
Laplace transform of standard signals t t t
The sinusoidal signal is also a standard signals.
If pole of sY(s) are in the left half of the s-plane
the final value theorem:
is available.

12. Using Laplace transform
Using the rules of Laplace transform to convert a differential equitation to operator
frequency form.
x(t)
y(t)
x(j)
y(j)

13. Terms of feedback control
controller
plant
GW(s)
GR(s)
GC(s)
GA(s)
GP1(s)
GP2(s)
GT(s)
compensator or control task
transmitter
controlled variable y
manipulated variable uM
error signal e
comparing element
or error detector
actuator
disturbance variable w
reference signal r
feedback signal yM
reference input element
action signal u
block model of the plant

14. Transfer functions of closed loop
GW(s)
GR(s)
GC(s)
GA(s)
GP1(s)
GP2(s)
GT(s)

15. Open loop transfer function
Bode form of opened-loop transfer function
The “i” is the type of the closed-loop system. It shows the
number of the integrating blocks inside the loop.
The type of system is used to determine the set-point tracking
and disturbance suppression. 15

16. Stability of a closed loop system
Definition of stability of linear system: Energised the closed-loop system input a short impulse which move the system from it’s steady-state position and wait. When the transient response has died and the system remove the original position or remain inside a predicted defined small area of the original position the system is stable and linear.
Definition of stability of non-linear system: Energised the closed-loop system input a short impulse which move the system from it’s steady-state position and wait. If the system remove and remain inside a predicted defined small area of the original position the system is stable.
If the transient response hasn’t died the system is unstable. 16

17. Stability examination from closed loop transfer function
Unit feedback model
r(s)
y(s)
A dynamic system is stable if the transient part of the differential equitation has died. It means the real part of the roots of the characteristics equitation have negative value.
If  negative.
The denominator of the operator transfer function is very similar to the characteristic equitation and the roots are the same. 17

18. Continuation of stability examination from closed loop
The transient solution is given by the roots of the differential equation’s characteristic equation.
The roots of the differential equation and the denominator of closed loop
transfer function (named poles) are equal and so it gives transient solution too.
A dynamic system is stable if the transient part of the differential equitation has died. It means the real part of the roots of the characteristics equitation have negative value.
If  negative.
The denominator of the operator transfer function is very similar to the characteristic equitation and the roots are the same.
Definition:
If the roots of the denominator of closed loop transfer function (named poles) have negative real part, than the closed-loop system is stable. 18

19. Stability examination from transfer function of opened-loop
Opened-loop transfer function
From definition of stability:
The frequency at which the open-loop system gain is unity is termed the gain-crossover frequency. The frequency at which the open-loop system phase-shift is -180° is termed the phase-crossover frequency.
The system is stable if plots of the opened-loop transfer function at the gain-crossover frequency the phase-shift less than -180° and at the phase-crossover frequency the amplitude gain less than unity.
Gain-crossover frequency: vágási frekvencia 19

20. Gain margin and phase margin
Opened-loop transfer function
The gain margin is the reciprocal of open-loop gain at the phase-crossover frequency. Expressed in decibel the gain margins is: -20log(open-loop’s gain at the phase-crossover frequency) dB
The phase margin is the difference of the phase-shift of the system and -180° at the gain-crossover frequency.
The system is stable if plots of the opened-loop transfer function at the gain-crossover frequency has a positive the phase margin and at the phase-crossover frequency has a positive the gain margin.
Gain-crossover frequency: vágási frekvencia 20

21. Stability examination from transfer function of opened-loop
Plots on j (s) plane is named Nyquist diagram.
This is a point on the s plane.
If the area of the s plane which is bordered by the positive real axis and the Nyquist plots doesn’t cover this point than the system is stable.
Independent plots the amplitude gain and phase-shift on lg axis is named Bode diagram.
It is the x axis line on the amplitude plots and the -180 phase-shift line on the phase-shift plots.
If the gain and phase margin criterion is
true the system is stable. 21

22. Time domain performance specification
y(t)
(rv-fv) = Steady-state error
90%
Tolerance band
Second peak value (spv)
Settling time
Rise time
10% t
Final value (fv) of response It’s 100%
Peak value (pv)
Required value of response (rv)

23. Quality performance and pole-zero arrangement
Poles and zeros arrangement
Minimum damping ratio of the second order parts of transfer function
Acceptable region for poles
Using the results derived for first and second-order systems enables a bounded design region to be drawn on the s plane within which the poles produce an acceptable performance.
Maximum settling time 23

24. Setpoint tracking
The setpoint tracking means that the controlled variable (y(s)) is able to follow the prescribed value by the changes of the reference signal (r(s)) after the transient has died.
We assume that the disturbance variable (W0(s)) equals the designed value (w(s)=0).
We can examine this with the error transfer function:
The setpoint tracking depends on:
Type of the feedback system.
The loop‘s gain (K0).
The form of the reference signal

25. Setpoint tracking 1 2 Order of the feedback system Reference signal
Standard signal form 1 2 t t t

26. Disturbance suppression
The disturbance suppression means that the controlled variable (y(s)) is able to hold the prescribed value by the changes of the disturbance variable (w(s)) after the transient has died. Assuming that the reference signal (R0(s)) equals the working point value (r(s)=0). We can examine this with the disturbance-error transfer function:
The disturbance suppression depends on:
Type of the feedback system.
The loop‘s gain (K0).
The form of the disturbance signal.
The attack point of the disturbance variable.

27. Disturbance suppression
Order of the feedback system 1 2
Entering point of disturbance
Standard signal 1 1 2 t t t