Control System Notes by HPK Kumar

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CONTROL SYSTEM NOTES (For Bachelor of Engineering) of Engineering) Notes by:

PROF. SHESHADRI G. S

Soft Copy material designed by:

KARTHIK KUMAR H P Your feedbacks can be mailed to:

[email protected]

INDEX

(1) Introduction to Control system (2) Mathematical model of linear of linear systems (3) Transfer functions (4) Block diagram (5) Signal Flow Graphs (6) System Stability (7) Root Locus Plots (8) Bode Plots















Control Systems

Introduction to Control Systems By: CIT, Gubbi.

Control Syste System m means any quanti quantity ty of inte i nterest in in a machine achine or mechanism chanism is maintai aintaine ned or alte alterred iin n accordance accordance with desir sired manner. OR A syste systemwhich hich controls control s the output quanti quantity ty is call cal led a control control syste system.

Definitions: 1. Cont Co ntro roll lled ed Variab Var iab le: le : I t is is the the quantity quantity or condi condition tion that that is is measured asured & controll controlled.

2. Cont Co ntro roll ller er:: means measuring asuri ng the value value of the control controllled variabl variable e of the system & applyi applying ng the Controller m manipul anipulate ated d variabl vari able e to the syste system m to corr correct or to li limit the deviati viation on of the measured val value ue to the desired value.

3. Plant: A plant is is a piec piece e of equi equipm pment, which is i s a set of machine achine parts functi functioni oning ng together. The The purpose purpose of which hich is to perform forma particular particular operation. ation. Example ple: Furnace, Space Space craft craft etc.,

4. System: A system is is a combinati bination on of components that works orks toget togethe her & perform forms certain tain obje objective ctive.

5. Dist Di stur urba banc nce: e: A disturbance is a signal signal that tends tends to to affe affect the value of the outp output ut of a syste system m. If a disturbance disturbance is created inside the system, it is called internal. While an external disturbance disturbance is is gener nerated ated outside outside the the system.

6. Feedback Control: I t is is an operation operati on that, in i n the pres presence of distur disturbance bance tends tends to reduce the diff difference erence betwe between the output of a system& some reference reference input. input.

7. Servo Mechanism: A servo mechanism is is a feedback control controllled system in whi which ch the output is is some mechanical chanical position, velocity or acceleration. ration.

8. Open loop System: I n an an Open loop System, the control action action is independent of the desi desirred output. output. OR When When the output quantity quantity of the control control sys system temis not fed back to the input input quantity, quantity, the control control system sys temis call called an Open loop System.

9. Clos Cl osed ed loop lo op Syst Sy stem em:: I n the the Closed loop Control System the control action action is is depe dependent on the desir sired output, where the output quantity quantity is is considerably considerably controll controlled by sending nding a command signal to inpu inputt quanti quantity. ty.

1















2

Introduction to Control System

10. 10 . Feed Back: Normally, the feed back signal has opposite polarity to the input signal. This is called negative feed back. The The advantage advantage is the resultant signal si gnal obtaine obtained fr from the comparator being diff diffe erence of the two two signal signals s is is of small aller magnitude agnitude. It I t can be be handled easil asily b by y the control control syste system m. The The resul esulti ting ng si signal is called Actuating Signal This Thi s signal signal has zero value value when the desir sired output output is obtaine obtained. d. I n that condition, control system will not operate. Effects of Feed Back:

Let the syste system has open loop gain gain I nput nput signal signal . The Then the feed back sign ignal is, is,

E(S)

R(S) B(S)

feed back loop gain gain

Output signal signal

&

G(S)

-

C(S)

&

–

Hence,

H(S) = With this this eqn.

(a) (a )

(1) , we can write the effe ffects of fee feed back as foll ollows. ows.

Overall Gain:

Eqn. shows shows that the gain gain of the open loop sys system temis reduced educed by a factor in a fe feed back syste system. Here the the feed back signal signal is negati negative ve. I f the feed feed back gain has positi positive ve val value ue,, the overall all gain gain will be reduced. If I f the feed back gain gain has ne negative gativevalue value,, the overall all gain gain may increase.

(b) (b )

Stability:

I f a systemis able able to foll fol low the input comm command signal signal,, the system stemis said said to be Stable. A system is said said to be Unstable , if its output output is out of contr control. ol. I n eqn. qn. , if the output output of the syste system is infi infini nite te for any fini finite te input input.. This his shows that a stable stable system may become unstab unstablle for cer certain tain value value of a fe feed back gain. gain. The Therefor fore if the feed feed back is is not prope properly used, the syste ystem mcan be harmful. ul.

(c) (c )

Sensitivity: This This depends on the systemparam rameters. rs. For For a good control rol system, it is desirab irable that the system should be inse insensitive nsiti ve to its its param parameter ter changes. Sensitivity, SG =

This This fun function ion of the system can be reduced by inc increasing ing the valu of . This This can be done by selecti ecting ng proper feed feed back.















Control Systems

(d) (d )

Noise:

Examples are brush & commutation noise in electrical machines, Vibrations in in moving oving system etc.,. tc.,. The The effe ffect of feed feed back on the these noise noise signal signals s will wil l be greatly greatly infl influe uence nced by the the poi point nt at whi which ch these these signal signals s are are introduce introduced in in the syste system. I t is is possi possible ble to reduce reduce the the effe ffect of noise noi se by pr proper design sign of fe feed back system.

Classification of Control Control Systems The The Con Control rol Sy Systemcan be clas lassified mainly inly depending ing upon, (a) Method of of anal analysi ysis & design, design, as Linear & Non- Linear Systems. (b) The The type of the sign ignal, as Time Varying, Time Invariant, Continuous data, Discrete data systems etc., (c) The The type of systemcomponents, as Electro Mechanical, Hydraulic, Thermal, Pneumatic Control systems etc., (d) The The main purpo rpose, as Position control & Velocity control Systems.

1. Linear & Non -Linear Systems: I n a li l i near near syste system m, the principl pri nciple e of supe superposition rpositi on can be be app appll i ed. I n non- l i near near syste system m, this principle can’t be applied. Therefore a linear system is that which obeys superposition principle & homogeneity.

2. Time Varying & Time Invariant Systems: Whi Whi l e operating operati ng a control control system system, i f the param parameters are are unaffe unaff ected by by the tim ti me, then then the system system i s call call ed Time Invariant Control System. Most physical physical systems have param parameter eters changing changing wi wi th tim time. If I f this thi s var varii ation ati on is is measur easurable abl e during duri ng the system systemoperation operati on then then the system system is called Time Varying System. I f ther there i s no non-l non-l i near nearii ty in in the tim ti me var varyi ying ng syste system m, then then the system may be call cal l ed as Linear Time varying System.

3. Disc Di scre rete te Data Da ta Syst Sy stem em s: I f the signal signal is i s not continuou continuousl sly y varying varying wi wi th time time but it it is i s in the form for m of pulse pulses. s. Then Discrete Data Control System. the control syste system m i s call cal l ed Discrete I f the signal signal is i s in in the form for m of pulse pul se data, then then the syste system m i s call call ed Sampled Data Control System. Here the information supplied intermittently at specific instants of time. This has the advantage of Time sharing system. On the other hand, if the signal is in the form of digital code, the system is called Digital Coded System. Here use of Digital computers, µp, µc is made use use of such system systems are are analyze anal yzed d by the Z- transf transfor orm m theory. theory.

4. Cont Co ntin inuo uous us Data Da ta Syst Sy stem ems: s: I f the signal signal obtained at vari various ous parts of the syste system m are are var varying ying continuous continuousll y wi wi th tim time, then the system system i s call cal l ed Continuous Data Control Systems.

5. Adap Ad apti tive ve Cont Co ntro roll syst sy stem ems: s: I n some control control system systems, certai n param parameters are ar e ei ther not constant or vary vary in an unknow unknown n manner anner. I f the parame parameter ter variati vari ations ons are lar large ge or rapid, apid, it i t may be be de desir sirable able to to desi design gn for the capability of continuously measuring them & changing the compensation, so that the ste for criteri cri teria a can al satisfi satisfi d. Thi This s is call call d Adaptive Control Systems.

3















4

Introduction to Control System Identification & Parameter adjustment

R(s)

E(s)

Compensator

System

+

C(s)

B(s) H(S) 6. Optimal Control System: Optimal Control System i s obtained obtained by mi nim ni mi zing zing and/or and/or maxim aximi zing zing the per perform ormance

i ndex. ndex. This Thi s inde index x depe depends nds upon upon the the physi physical cal system system & skil ski l l .

7. Single Variable Control System: I n si si mple ple control control system system there wi wi l l be One input i nput & One One output such such syste system ms are call call ed Single I nput nput & Sing Singll e Output). Output). Single variable System (SISO – Single

8. Mult Mu ltii Va Vari riab able le Cont Co ntro roll Syst Sy stem em:: I n Multivariable control system ther there wi l l be more ore than one i nput nput & corr correspondi spondingly ngly more output’s output’s (MI (MIMO MO - Multipl Mul tiple e I nput nputs s & Multipl Mul tiple e Outpu Outputs ts). ).

Comparison between between Open loop & Closed loop Gain Open Loop System

Closed Loop System

1. An open loop loop system has the abil abi l i ty to perform accurately, if its calibration is good. I f the cali cali bration bration is is not not perf perfe ect its i ts pe perform rf ormance ance will go down.

1. A closed loop system has got the ability to per perform orm accuratel accurately y because of the feed back.

2. I t is eas easii er to bui bui l d. 3. I n gene generral i t is i s more ore stabl stable e as the the fee feed back is absent.

2. It is difficult to build. 3. L ess Stabl Stabl e Comparative parati vel y.

4. If non- linearity’s are pre present; sent; the system operation is not good.

4. Even under the presence of nonl i near nearii ty’s ty’s the syste system m operates better better than ope open n l oop system system.

5. F eed back i s absen absent. t. Example ple: (i) Traffic Traffic Con Control rol Sy System. (ii) Control of furnace for coal heating. (iii) An Electric Washing Machine.

5. F eed back is is present. Example: (i) Pressure Control System. (ii) Speed Control System. (iii) Robot Control System. (iv) Tem Temperat rature Con Control trol Sy System.

Note:

Any control system which operates on time basis is an Open Loop System.

















Control Systems

Block Diagram of Closed Loop System: Controller

E(S) Ref. i/p

Control Elements

Plant

Controlled o/p

Feed Back elements

Actuator

Therm Thermo ometer Bloc Block Diag Diagram ram of Temperat rature Con Control rol Sy System: Thermo meter

A/D Converter

Interface

Electric Furnace Heater

Relay

Amplifier

Interface

Programmed i/p

Temperature Control of Passenger Compartment Car: Sun

Sensor Radiation Heat Sensor

Controller

Air Conditioner

Ambient Temperature

Passenger Car

5

















Control Systems

Mathematical Models of Linear Systems By: CIT, Gubbi.

A physical system is a collection of physical objects connected together to serve an objective. An idealized physical system is called a Physical model. Once a physical model is obtained, the next step is to obtain Mathematical model. When a mathematical model is solved for various i/p conditions, the result represents the dynamic behavior of the system. An al og ou s System:

The concept of analogous system is very useful in practice. Since one type of system may be easier to handle experimentally than another. A given electrical system consisting of resistance, inductance & capacitances may be analogous to the mechanical system consisting of suitable combination of Dash pot, Mass Ma ss & Spri Sp ring ng.. The Th e adva ad vant ntag ages es of elec el ectr tric ical al syst sy stem emss are, ar e, 1. 2.

Many Ma ny circ ci rcui uitt theo th eore rems ms,, impe im peda danc ncee conc co nc epts ep ts can ca n be appl ap plic icab able le.. An Elec El ectr tric ical al engi en gine neer er fami fa mili liar ar with wi th elec el ectr tric ical al syst sy stem emss can ca n easi ea sily ly anal an alyz yzee the th e syst sy stem em unde un derr study & can predict the behavior of the system. 3. The electrical analog system is easy to handle experimentally. Translational System: Translational System:

It has ha s 3 type ty pess of forc fo rces es due du e to elem el emen ents ts.. mass, 1. Inertial Force: Due to inertial mass,

 M

F(t)



2   Fmt  M.a M. a     2 Where,    .

   .   .

damping, it is proportional proportional to velocity & is given by, Damping]: Due to viscous damping, 2. Damping Force [Viscous Damping]:

     .. 

Damping force is denoted denoted by either D or B or F

D

1

















2

Mathematical Models of Linear Systems Where, Rotational system:

                

            1. Iner In erti tial al Torq To rque ue:: 2. Dampi Da mpi ng Torq To rque ue :      ..  3. Spring Torque   :         

Analogo Ana logous us quantities in translational & Rotational system: The electrical analog of the mechanical system can be obtained by, (i) (ii)

Sl. No.

Force Voltage analogy: (F.V) Force Current analogy: (F.I)

Mechanical Translational System

Mechanical Rotational System

F.V Analogy

F.I Analogy

1.

Force (F)

Torque (T)

Voltage (V)

Current (I)

2.

Mass (M)

Moment of Inertia (M)

Inductance (L)

Capacitance (C)

3.

Viscous friction (D or B or F)

Viscous friction (D or B or F)

Resistance (R)

Conductance (G)

4.

Spring stiffness (k)

5.

Linear displacement displacement ( )

Torsional spring stiffness Reciprocal of Capacitance (1/C) ( ) Charge (q) Angular displacement ( )

Reciprocal of Inductance (1/L) Flux ( )

6.

Linear velocity ( )

Angular Velocity (w)

Voltage (v)









Current (i)



D’Alemberts Principle: The static equilibrium of a dynamic system subjected to an external driving force obeys the following principle, “For any body, the algebraic sum of externally applied forces resisting motion in any given direction is zero”.

Example Problems: (1) Obtain the electrical analog (FV & FI analog circuits) for the Machine system shown & also write the equations.

Free Body diagram D2

2 2

 

M1

    

















Control Systems

1

Transfer Functions By: CIT, Gubbi.

The input- output relationship in a linear time invariant system is defined by the transfer function.

The features of the transfer functions are, (1) It is applicable to Linear Linear Time Invariant system. system. (2) It is the ratio between the the Laplace Transform of the the o/p variable to the the Laplace Transform of the the i/p variable. (3) It is assumed that initial initial conditions conditions are zero. (4) It is independent of i/p i/p excitation. (5) It is used to obtain systems systems o/p response. An equation describing the physical system has integrals & differentials, the step involved in obtaining the transfer function are; (1) Write the differential equation of the system.

 

by ‘S’ &

 

(2)

Replace the terms

by 1/S.

(3)

Eliminate all the the variables except the desired desired variables. Impulse Response of the of the Linear System: Linear System:

G(S)

R(S)

 G(S) =       1

C(S)

In a control system, when there is is a single i/p of unit impulse impulse function, then there will be some response of the Linear System. The Laplace Transform of the i/p will be R(S) = 1

    .    . 1      i.e., the Laplace Transform of the system o/p will be simply the ‘Transfer function’ of the system. Taking L-1

    Here G(t) will be impulse response response of the Linear System. System. This is called Weighing Weighing Function. Hence LT of the the impulse response is the Transfer function of the system itself.

R

ROBLEMS: P ROBLEMS

















2

Transfer Functions

..,    1   &        1    .       .     1           1 1  Where,  = RC  (2) Obtain the TF of TF of the the mechanical system mechanical system shown in circuit 2. circuit 2.



















Control Systems

3

(3) Transfer Function Transfer Function of an of an Armature Armature Controlled DC Controlled DC Motor Motor in in circuit 3.0: circuit 3.0:

La

Ra

I f = Constant

Let,

ia

V i

‘Ra’  Resistance Resistance of armature in

ia

E b

F

V f

Inductance of armature in H’s. ‘La’  Inductance Field current. ‘ia’  Armature current. & ‘i f ’ Field

       

Ω’s.

Applied armature voltage. ‘V i’  Applied

circuit 3.0

T m

J,

Back e.m.f in volts. volts. ‘E b’  Back

F



 Angular

‘T m’  Torque Torque developed by the motor in N-m. displacement displacement of motor shaft in radians

J  Equivalent Equivalent moment of inertia of motor & load referred to the motor shaft. F  Equivalent Equivalent Viscous friction co-efficient of motor & load referred to the motor shaft.

 i.e.,       . 

The air gap flux ‘ ’ is proportional to the field current.

Where, ‘K f ’ is a constant.

The torque developed by the motor ‘T m’ is proportional to the product of the arm current & the air gap flux.

         Since the field current is constant,

Where, K a & K f are the constants.

     

Where, K T Motor – torque constant. T is



















4

Transfer Functions

    

   2           

------------------------------ (2)

-------------------------- (B)

Taking LT for back e.m.f equation, we get

      

------------------------------- (C)

Substituting the values of I a (S) & E b (S) from equation equation (C) & (2) in equation equation (1), we get

                                  .                                The block diagram block diagram representation of armature of armature controlled DC controlled DC Motor Motor can can be obtained as obtained as follows, follows,

From equation (A),

V i(S)

-

1     

E b(S) From equation (B),

I a(S)

     

(S)

I a(S)



















Control Systems

5

(4) Transfer function function of Field of Field Controlled Controlled DC DC Motor Motor in in circuit 4.0: circuit 4.0:

Let,

L f

R f

I a = Constant

R f  Field Field winding resistance. L f  Field Field winding inductance.

V f

i f

V f  Field Field control voltage.

T m



I f  Field Field current.

T m  Torque Torque developed by motor. J  Equivalent Equivalent moment of inertia of motor & load referred to the motor shaft.

circuit 4.0 J, F

F  Equivalent Equivalent Viscous friction co-efficient of motor & load referred to the motor shaft.



Angular displacement of motor  Angular

shaft. In the field controlled DC motor, the armature armature current is fed from a constant current source. source.

        

Where, K a & K f are the constants.

The KVL equation for the field circuit is,

          On Laplace Transform,

                  .       

---------------------------------- (1)



















6

Transfer Functions

 2    .        

       2                 The block diagram block diagram representation of field controlled field controlled DC DC Motor Motor can can be obtained as obtained as follows, follows,

From equation (A),

V f (S)

1     

I f (S)

From equation (B),

I f (S)

     

(S)

The complete block diagram is as shown below,

V f (S)

1     

(5) Obtain the TF

Solution:

I f (S)

      

(S)

 for the network shown in circuit 5.0: circuit 5.0:  for the network shown

Laplace Transformed Transformed network

R

R

V i

1 

C

R

circuit 5.0

C

V o



















Ω Ω

100 k (6) Find the Find the TF

Solution:

 for the network shown in circuit 6.0: circuit 6.0:  for the network shown V i

Laplace Transformed Transformed network

10

10

1M



10 F

Circuit 6.0

V i (S)

 Loop 1

10 V (S) 

10  

0

Loop 2

Writing KVL for loop (1), we get

5 5 10 10 5             1  10     2    .   2  1  105 1 ------------------------------------------- (1) 1 Writing KVL for loop (2), we get

         10 

  

0

   2  10111 -------------------------------------------- (2)    2. 106   2  106.  --------------------------------- (3) Substituting for I 1 (S) from equation equation (2) in (1), we get get

 2 10   11  . 



2

Control Systems

Ω



1 F

7





















8

Transfer Functions

                      









































2

Block Diagrams P ROBLEMS :

Reduce the Block Diagrams shown below:

(1)

+ -

-

Soluti Solution: on: By el eliminating nating thefeedfeed-back back paths, paths, we we get + -

-





















Control Systems (2)

R(S)

C(S)

-

Solution: Solution:

Shi Shifting fti ng thetake-off

-

‘

’

beyond the bl block

‘

’

, we get

R(S)

-

C(S)

-

3





















4

Block Diagrams

R(S)

-

-

(3)

Soluti Solution: on:

-

C(S)

Re-arr Re-arrangi anging the block block diagram diagram, we get R(S)

-

Eliminati nating

-

loop & combining, bining, we get

-

C(S)























Control Systems

Signal Flow Graphs By: CIT, Gubbi.

F or compli plicated syste systems, Bl Block diagr diagram am reduction uction method becomes tedi tedious ous & tim time consum consuming. An An alte alterrnate method is is that si signal flow fl ow gr graphs developed by S.J . Mason. ason. I n these graphs, each node node represents a syste system variabl vari able e & each branch branch conne connected cted between two node nodes acts as Signal Signal Multi ultipl pliier. The di direction of signal flow is indicated by an arrow.

Definitions:

1

























































































Control Systems

R(S)

-

R(S)

C(S)

C(S)

5

























6

Signal Flow Flow Graphs Eliminating feed feed back loop R(S)

, we get C(S)













































8

Signal Flow Flow Graphs Eliminati nating

loop, oop, we get



































































































































































































































































































































































Control System Notes by HPK Kumar

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