ee 322 control theory

8/13/2019 EE 322 Control Theory

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Control Systems

Sisil Kumarawadu, PhDProfessor in Electrical Engineering

University of Moratuwa

--System (controlled) modeling--Controller design

--System analysis-stability-transient-steady state-frequency response


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Feedback (closed-loop) controlSystems

Feedbackelement

+-

Reference I/P Measured O/PController Plant

Is a signal or informationProcessing device

Controlled system

Sensors

Block diagram of a standard error feedback control system


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Plant (controlled system): is the process tobe controlled

FB Element: Sensor(s) that feeds the plantO/P back to the I/P side

Controller: Takes the error (differencebetween I/P and the FB O/P) into accountto create a control signal in a way the erroris minimized

Input (reference): is the desired O/P


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Open-loop Vs Closed-loop

ControlI/P O/P

Controller Plant

Block diagram of a open-loop control system

** O/P quantity has no influence in the I/P quantity (and hence in the controller)and hence there is no feedback in the system.

Eg : Bread Toaster: The setting of the darkness knob or timer represents the I/P,and the degree of darkness or crispness of the toasted bread is the O/P.If the degree of darkness is not satisfactory (may be coz type of breads different),there is no way to automatically alter the length of time the heat is supplied


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Analog Control (traditional)

Controllers are implemented with analogcomponents (eg: resisters, capacitors, operationalamplifiers)

Explosive growth and expanding efficiencyOf digital technology

Today, controllers are typically implementedas programmable digital hardware (digital

computers) Digital Control


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A Digital Control System(computer-based/sampled-data control systems)

Sensor (s)

+-

O/PMicrocontroller/DSP/PLD(digital computer)

Plant ADC DAC

ADC

To be usable by digital computers, analog signals need to besampled and converted to digital form by an analog-to-digitalconverter (ADC). Digital signals from the digital computer needs

to be converted back to analog by a digital-to-analog converter(DAC)


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The control strategy can be changed easily withoutnew hardware.

The system is smaller, lighter, required less power,and costs less

The system is more reliable, maintainable, andtestable

The system is more immune to noise Complex control algorithms for higher control

performance can be implemented

DSP/PLC Solutions (Digital) ascompared with traditional analog

controllers


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ADC, DAC Sensors (analog) Plant (analog)

Controller (digital)

Sensors (analog) Plant (analog)

Controller (analog)

Sampled-datacontrol systems(Digital Ctrl Sys)

Analog controlsystems


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Essence of Transforms inEngineering

Laplace transform methodis extensively used ordinaryconstant coefficient

(Linear-time-invariant or LTI)differential equations.

Fourier transform techniques


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Laplace-transform Vs Z-transform

Analog (Continuous) Digital (Discrete)

Linear differential equations Linear difference equations(represents linear systems)

Algebraic equations Algebraic equations

Signals Signals(Solutions of the differential (Solutions of the differenceEquations) Equations)

Laplace-transform Z-transform

Inverse Laplace-transform Inverse Z-transform


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Analog Control Vs Digital Control

)( sG P

)()()(1)()(

)()(

)(

)()()()(

s H sG sG sG sG

s R sC

s M

sC s E sG sG

P C

P C

P C

+-

I/P O/P

)( s H

)( s R)( sG P

)( sC )( sGC

Closed-loop

Transfer function (T/F)

)( sC

)( sGC )( sG P )( s H

)( s R

Controller T/F

Plant T/FFB element T/F

Laplace transformof the O/P

Laplace transformof the I/P


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Analog Control Vs Digital Control

)( sG P

)()()(1)()(

)()(

)(

)()()()(

z H z G z G z G z G

z R z C

z M

z C z E z G z G

P C

P C

P C

+-

I/P O/P

)( z H

)( z R)( z G P

)( z C )( z GC

Closed-loop

Digital transfer function (T/F)

)( z C

)( z GC )( z G P )( z H

)( z R

Controller T/F

Plant T/FFB element T/F

Z-transformof the I/P

Z-transformof the O/P


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Remarks:

is usually a programmed in a micro-controlleror a single-chip DSP. DACs and ADCs areassumed part of the blocks H(z) and .

**There still are several similarities betweendigital and analog control design and analysistechniques. Particularly, when T (sampling time)is small, design and analysis techniques in theanalog domain can be used in sampled-datacontrol systems as well.

)( z GC

)( z GC


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Remarks (cont..): **But T can be significantly large in certain practical

situations, for instance-- Remote control (data communication delay)-- Delays in sensor information processing ( eg :image processing when cameras are used as asensor)

In such situations, control is inherently digital

and digital control theories (eg: z-transform techniquesinstead of Laplace-transform) may be necessary toimprove control performance.


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Sampled-data Systems (controller is adigital computer)

Analog control theories areapplicable if the effects offinite sampling is negligible

(T is small: How small is casedependent and hence can not

be generalized)

Digital control theories may benecessary if the effects of finitesampling is not negligible

(T is significantly large)

Note: Control theories refer to: System (plant) modeling, control theories(control laws/Control algorithms), performance and stability analysis etc..


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Example 1: A temperature control system

Hot water

User valve

ThermocoupleSignal(Voltage signal)

Feed-water valve

Feed-water

High-levelSensor (5.5ft)(either High OR Low)

Low-levelSensor (0.5ft)

Gas

Burner valve

The PlantI/P (reference): desired water tempMeasurable O/P: temperature and water levelControl I/P: Burner valve position


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The control engineers task is to design a controller (either analog or digital) that Processes the temperature and water level to produce control I/P (valve position) toMaintain the water temperature at the desired level

Example 2: Robot arm

Plant: Robot manipulatorsI/P: Desired trajectory

Desired positionO/P: Joint angle position and velocity

Relative position between the end-effecter and the target pointSensors: Optical encoders, tacho-generators

Stereo camerasController: Numerous different methods are being used Eg : PID

Actuators: Motors

This type of control system can be modeled and a controller can be designedusing computer assisted engineering (CAE) software packages.

Eg : MATRIX, SIMULINK-- Modeling, design, and analysis tools


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ON-OFF Mode of FB Control This is the simplest mode of FB control. An On-Offcontroller operates on the controlled variable only

when it crosses the set-point (reference) The controller output has only two states, namely, fully

On and fully OFF. Take a temperature control problem, for instance.

Actual temperature can be kept around the set-point(reference point or the desired value) by turning off theheating element when the measured temp isanywhere above the set point. Likewise, the heater isturned on when the actual temperature tries to dropbelow the set-point


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ON-OFF Temperature Control Action

Temp

Time

Set-point

Heater

ON ON ON ON

OFF OFF OFF


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ON-OFF Mode of FB Control The process temperature is continually cycling.

Peak-to-peak and cycling period will depend on theprocess characteristics and the controller has nocontrol over them.

The ideal ON-OFF controller is not practical assensor noise and other electrical interferences maycause the output to cycle rapidly when themeasured value is around the set-point (referencevalue). This could be detrimental to the devicessuch as contactors and valves. To prevent this, ahysteresis can be added to the controller asfollows. This will prevent the output from chatteringif the peak-to-peak noise is less than the

hysteresis.


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ON-OFF Temperature Control Actionwith Hysteresis

Temp

Time

Set-point

Heater

ON ON ON ON

OFF OFF OFF

Hysteresis


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Transfer Curves(a) Without hysteresis

(b) With hysteresis

Temp

Temp

ON

OFF

ON

OFF

A B

C DE

FG

Set-point


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Tolerance Band Control ( Hysteresis Ctrl )

T A+

T A-

i a

i a*s A

1

2

3

1: Hysteresis comparator2, 3: Drive circuits (of the

switches)

(sinusoidal, for instance)


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Tolerance Band Control Tolerance band (hysteresis) is a parameter

that can be preset in the comparator. If the actual current, i A, tries to go beyond the

upper limit of the tolerance band, T A+ is turned

on ( T A- is turned off ). T A+ is turned off if i A triesto go below the lower limit. The load back-emf and the load resistance

determine how fast the current changesbetween the upper and lower limits.Switching frequency depends on this andalso varies along the current waveform.


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Mathematical Models of Systems

Transfer function, stability, roots ofCharacteristics Equation, Rouths Criterion

State variable methods

State space design and analysis (MIMOSystems)


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Airplane Attitude Control


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Transfer Function Block Diagram


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Consider the 3 rd order aircraft attitude controlsystem (unity FB) with OLTF

For this system, it can be proved that whenK=181.17, the max. overshoot is 78.88%

Now change the control law as follows

)000,204,13.3408(105.1

)( 27

s s s K

sG

1,17.181)( P D P K K where s K K K K


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With the new controller, Cha.Eq. of the closed-loop systems is

To apply Root Locus Method, rewrite the aboveequation as (to draw root loci, open-loop TF should take form G(s)H(s) = K N(s)/D(s) )

where

010718.2)10718.2000,204,1(3.3408 9923 s K s s D

010718.2000,204,13.3408

10718.21)(1 923

9

s s s s K

sG Deq

)6.90649.57)(6.90649.57)(3.3293(10718.2

)(9

j s j s s s K

sG Deq


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The Root-Loci


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When K D increases, one root moves fromt-3293.3 towards the origin while the twocomplex roots move left and approach theasymptotes.

Too large values of K D results in two

complex roots having reduced dampingand increased natural frequency of thesystem.

To that end, it appears that the ideallocation for the two complex roots is nearthe bend of the root locus


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f ff f


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Unit step response for different values of K D

Time (sec)

)(t y


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Conclusions

Minimum of maximum overshoot occurs at

Rise time is improved with the increase of Too high a value of K D increases the maximum

overshoot and the settling time. This is due tolowering of damping as K D is increased

indefinitely.

.002.0 D K .

D K

A der ivat ive con t ro l ac t ion can improvethe transient response of the system!!


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| G ( j w

) | ( d B

)

P h a s e

( D e g . )

ee 322 control theory

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