LECTURE (2)

Mathematical Modeling of

Systems

Assist. Prof. Amr E. Mohamed

Objectives

In this lecture, we lead you through a study of mathematical models of

physical systems.

After completing the chapter, you should be able to

Describe a physical system in terms of differential equations.

Understand the way these equations are obtained.

Realize the use of physical laws governing a particular system such as

Newton’s law for mechanical systems and Kirchhoff’s laws for electrical

systems.

Realize that deriving mathematical models is the most important part of

the entire analysis of control systems.

2

Mathematical Model

Mathematical modeling of any control system is the first and foremost task that acontrol engineer has to accomplish for design and analysis of any control engineeringproblem.

A mathematical model of a dynamic system is defined as a set of differential equationsthat represents the dynamics of the system accurately, or at least fairly well.

Note that a mathematical model is not unique to a given system. A system may berepresented in many different ways and, therefore, may have many mathematicalmodels, depending on one’s perspective. For example,

In optimal control problems, it is good to use state-space representations.

On the other hand, for the transient-response or frequency-response analysis of single-input-single-output, linear, time-invariant systems, the transfer functionrepresentation may be more convenient than any other.

Once a mathematical model of a system is obtained, various analytical andcomputational techniques may be used for analysis and synthesis purposes. Becausethe systems under consideration are dynamic in nature, the equations are usuallydifferential equations. If these equations can be linearized, then the Laplacetransform may be utilized to simplify the method of solution. 3

Different Mathematical Models

Commonly used mathematical models are

Differential equation model (Time Domain).

Transfer function model (S-Domain).

State space model (Time Domain).

Use of the models depends on the application. For example, to find the

transient or steady state response of SISO (Single Input Single Output)

LTI (Linear Time Invariant) system transfer function model is useful. On

the other hand for optimal control application state space model is

useful.

4

Control systems Classifications

5

Non-Linear System OR Linear System

Time Varying System OR Time Invariant System

Single Variable Control OR Multivariable Control

Classical Representation

(Classical Control)OR

State Space Representation

(Modern Control)

Manual Control System OR Automatic Control System

Open-Loop Control system OR Closed-Loop Control system

Laplace Transform

Laplace Transforms: method for

solving differential equations,

converts differential equations

in time 𝑡 into algebraic

equations in complex variable 𝑠.

6

Laplace Transform Properties

7

The approach to dynamic system problems can be as follows:

1. Define the system and its components.

2. Formulate the mathematical model and list the necessary assumptions

3. Write the differential equations describing the model.

4. Solve the equations for the desired output variables.

5. Examine the solution and the assumptions.

6. If necessary reanalyze or redesign the system.

8

TRANSFER FUNCTION

Transfer functions are commonly used to characterize the input—output

relationships of components or systems that can be described by linear,

time-invariant, differential equations.

The transfer function of a linear, time-invariant, differential equation

system is defined as “the ratio of the Laplace transform of the output

(response function) to the Laplace transform of the input (driving

function) under the assumption that all initial conditions are zero”.

9

TRANSFER FUNCTION

The general form of the differential equation for LTI-System is given by

𝒂𝟎 𝒚(𝒏)

+ 𝒂𝟏 𝒚(𝒏−𝟏)

+ … + 𝒂𝒏−𝟏 𝒚 + 𝒂𝒏 𝒚 = 𝒃𝟎 𝒙(𝒎)

+ 𝒃𝟏 𝒙(𝒎−𝟏)

+ … + 𝒃𝒎−𝟏 𝒙 + 𝒃𝒎 𝒙

where y is the system output and x is the input of the System

The transfer function of this system is obtained by taking the Laplace

transforms of both sides of Equation (under the assumption that all initial

conditions are zero),

𝑎0𝑆𝑛𝑌 𝑠 + ⋯ + 𝑎𝑛−1𝑆

1𝑌 𝑠 + 𝑎𝑛 𝑌 𝑠 = 𝑏0𝑆𝒎𝑿(𝑠) + ⋯ + 𝑏𝑚−1𝑆

𝟏𝑿(𝑠) + 𝑏𝑚 𝑿(𝒔)

Then: 𝑎0𝑆𝑛 + ⋯+ 𝑎𝑛−1𝑆

1 + 𝑎𝑛 𝑌 𝑠 = 𝑏0𝑆𝒎 + ⋯+ 𝑏𝑚−1𝑆

𝟏 + 𝑏𝑚 𝑿(𝒔)

10

TRANSFER FUNCTION

Then the transfer function is

𝑻𝒓𝒂𝒏𝒔𝒇𝒆𝒓 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏 = 𝑮 𝒔 =𝑳𝒂𝒑𝒍𝒂𝒄𝒆 𝒐𝒇 𝑶𝒖𝒕𝒑𝒖𝒕

𝑳𝒂𝒑𝒍𝒂𝒄𝒆 𝒐𝒇 𝑰𝒏𝒑𝒖𝒕 𝑨𝒔𝒔𝒖𝒎𝒊𝒏𝒈 𝒁𝒆𝒓𝒐 𝒊𝒏𝒊𝒕𝒊𝒂𝒍 𝑪𝒐𝒏𝒅𝒊𝒕𝒊𝒐𝒏

𝐺 𝑠 =𝑌(𝑠)

𝑋(𝑠)=

𝑏0𝑆𝒎 + ⋯+ 𝑏𝑚−1𝑆

𝟏 + 𝑏𝑚

𝑎0𝑆𝑛 + 𝑎1𝑆

𝑛−1 + ⋯+ 𝑎𝑛−1𝑆1 + 𝑎𝑛

Poles: are roots of the denominator (Values of s such that transfer

function becomes infinite)

Zeros: are roots of the numerator (Values of s such that transfer

function becomes 0)

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To derive the transfer function

1. Write the differential equation for the system.

2. Take the Laplace transform of the differential equation, assuming all

initial conditions are zero.

3. Take the ratio of the output Y(s) to the input R(s). This ratio is the

transfer function.

12

Describing Differential Equations for Electrical and Electronic Elements

13

Electrical Circuits

14

Op Amps in system control Signal amplification in the sensor circuits

Filters used for compensation purposes

Modeling of the “real world” systems

Lead or lag networks

Design of controllers

15

u(s)

e(t) OPAMP+

-

OUT

R1

R2

u(t)

e(t)

OPAMP+

-

OUT

R2

R1

Inverting amplifier Non-inverting amplifier

teR

Rtu

1

2 teR

Rtu

1

21

Op Amps in system control

16

Summation circuit

teteteR

Rtu 321

1

2 OPAMP+

-

OUT

R1

R1

R1

R2

u(t)

e1(t)

e2(t)

e3(t)

U(s)

E(s) OPAMP+

-

OUT

Z1

Z2

1

2

Z

Z

sE

sU

Transfer function of op amp

Op Amps in system control

17

U(s)

E(s)

C1

OPAMP+

-

OUT

R1

Integrator

ITssCRZ

Z

sE

sU

11

111

2

Differentiator

11

2

1

2

sCR

sCR

Z

Z

sE

sU

Op Amp lead-or-lag network

18

111

11

sCR

RZ 122

22

sCR

RZ

U(s)

E(s)

C2

OPAMP+

-

OUT

R1

R2

C1

1

11

22

2

1

2 1

1 R

sCR

sCR

R

Z

Z

sE

sU

Op Amp lead-lag network

19

1

1

131

3111

sCRR

RsCRZ

1

1

242

4222

sCRR

RsCRZ

Eo(s)

Ei(s)

C2

OPAMP+

-

OUT

R3

R4

C1R1

R2

OPAMP+

-

OUT

R5

R6

Example: Write the integodifferential equations and the transfer function of the following circuit

20

v(t))ii(

dt

dLRi

2111

0

02

1

1222

t

dtic

)ii(

dt

dLRi

Take Laplace Transform of both

sides then find G(s) = I2(s)/V(s)

Describing Differential Equations for Translation Mechanical Elements

21

Example: Write the differential equations and the transfer function of the Spring Mass Damping System shown

22

s-domain

)()( sFsVs

KfMs v

K

vf

M)(tf

)(tv

)()()()(

2

2

tftKxdt

tdxf

dt

txdM v

differential equation

transfer function

KsfMs

s

sF

sV

v

2)(

)(

Example: Write the differential equations to model the system shown

23

1K1M

)(tf )(1 tx2K

vf 2M

)(2 tx

3K

For the Mass M1

K2 (x1(t) – x2(t)) + K1x1(t) + fv d/dt (x1(t) – x2(t)) + M1 d2/dt2 x1(t) = f(t)

For the Mass M2

K2 (x1(t) – x2(t)) + fv d/dt (x1(t) – x2(t)) = K3x2(t) + M2 d2/dt2 x2(t)

InputOutput

Describing Differential Equations for Rotational Mechanical Elements

24

Transfer Function for:(a) Angular (b) Torque Displacement

Describing Differential Equations for Electro Mechanical Elements

25

DC Motor

dt

tdKv m

bb

)( Vb = Back electomotive force

Kb = the constant back emf

)()( sIKsT atm Tm = Torque of the motor

Kt = motor torque constant

)()()( 2 ssDsJsT mmmm

Mechanical Relation

Electrical Relation

)(1)(

)(

a

btm

m

mat

a

m

R

KKD

Jss

JRK

sE

s

Ia(s) = (Ea(s)-Vb(s))/(Ra+sLa)

The Transfer Function is

Block Diagram Models

A block diagram of a system is a pictorial representation of the

functions performed by each component and of the flow of signals.

Such diagram depicts the interrelationships that exist among the various

components. Differing from a purely abstract mathematical

representation, a block diagram has the advantage of indicating more

realistically the signal flows of the actual system.

Transfer function can be represented as a block diagram:

26

)(sR )(sC0

1

1

0

1

1

asasa

bsbsbn

n

n

n

m

m

m

m

Components Of a block diagram for a LTI system

27

Procedures for drawing block diagram

1. Write the equations that describe the dynamic behavior for each

component.

2. Take Laplace transform of these equations, assuming zero initial

conditions.

3. Represent each Laplace-transformed equation individually in block

form.

4. Assembly the elements into a complete block diagram.

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Example:

Example derive the D.E. and the transfer function then draw the block

diagram for the following circuit:

Take Laplace transform:

29

R

Cei eo

i

R

sEsE

R

tetesI oioi )()()()()(

L

C

idte

R

eei o

oi

Cs

sI

C

idtsEo

L

Block Diagram Reduction

Rules for reduction of the block diagram:

1. Any number of cascaded blocks can be reduced by a single block

representing transfer function being a product of transfer functions of

all cascaded blocks.

30

Block Diagram Reduction

2. Moving a summing point

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(a) Behind the block

(b) Ahead of the block

Block Diagram Reduction

3. Moving a pickoff point

32

(a) Behind the block

(b) Ahead of the block

Block Diagram Reduction

4. Equivalent transfer function for parallel subsystems is the sum of

their transfer functions

33

Block Diagram Reduction

5. Feedback control system

34

Example reduce the following block diagram:

35

R

_+

_

+1G 2G 3G

1H

2H

++

C

Moving the summing point ahead of G1, we have:

36

R

_+

_

+1G 2G 3G

1H

1

2

G

H

++

C

Combing G1 and G2 in Cascade, we get:

37

R

_+

_

+21GG 3G

1H

1

2

G

H

++

C

Eliminating the feedback loop G1, G2 and H1 we get:

38

R

_+

_

+21GG 3G

1H

1

2

G

H

++

C

Combing the two blocks in Cascade, we get

39

R

_+

_

+121

21

1 HGG

GG

3G

1

2

G

H

C

Similarly eliminating the second feedback loop we get:

40

R

_+

_

+121

321

1 HGG

GGG

1

2

G

H

C

Similarly eliminating the third feedback loop we get:

41

R

_+232121

321

1 HGGHGG

GGG

C

The system is reduced to the following block diagram:

42

R

321232121

321

1 GGGHGGHGG

GGG

C

Conclusions of block diagram reduction Technique

1. Numerator of the closed-loop transfer function C(s)/R(s) is the product

of the transfer functions of the feedforward path.

2. The denominator of the closed-loop transfer function C(s)/R(s) is equal

to:

1 − Σ( 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑜𝑢𝑛𝑑 𝑒𝑎𝑐ℎ 𝑙𝑜𝑜𝑝)

3. The positive feedback loop yields a negative term in the denominator.

43

Example of system with two inputs R(s) and D(s)1. Find 𝑌1(𝑠)/𝐷(𝑠) when 𝑅(𝑠) = 0

2. Find 𝑌2(𝑠)/𝑅(𝑠) when 𝐷(𝑠) = 0

3. Deduce the total response 𝑌(𝑠) of the

control system when R(s) and D(s) ≠ 0

𝑌1 𝑠

𝐷 𝑠𝑅 𝑠 =0

=𝐺2

1 + 𝐺2(𝐻1 − 𝐻2)

𝑌2 𝑠

𝑅 𝑠𝐷 𝑠 =0

=𝐺1𝐺2

1 + 𝐺2(𝐻1 − 𝐻2)

𝑌 𝑠 = 𝑌1 𝑠 + 𝑌2 𝑠 = 𝑌1 𝑠

𝐷 𝑠𝑅 𝑠 =0

. 𝐷(𝑠) + 𝑌2 𝑠

𝑅 𝑠𝐷 𝑠 =0

. 𝑅(𝑠)

𝑌 𝑠 =𝐺2

1 + 𝐺2 𝐻1 − 𝐻2𝐷 𝑠 +

𝐺1𝐺2

1 + 𝐺2 𝐻1 − 𝐻2𝑅(𝑠)

44

Signal Flow Graph Models

Definitions:

45

input node (source)

b1x a

2xc

4x

d

1

3x

3x

output node (sink)

mixed node

input node (source)

mixed node

forward pathpath

loop

branch

node

transmittance

Signal Flow Graph Models

Node: a point representing a signal or variable.

Branch: unidirectional line segment joining two nodes.

Path: a branch or a continuous sequence of branches that can be

traversed from one node to another node.

Loop: a closed path that originates and terminates on the same node

and along the path no node is met twice.

Nontouching loops: two loops are said to be nontouching if they do not

have a common node.

46

Flow graphs of control systems

47

)(sR

)(sG

)(sC)(sG)(sR )(sC

block diagram signal flow graph

)(sR

_+

)(sH

)(sG

)(sC)(sE)(sG

)(sR

)(sC1

)(sE )(sH

Mason’s Signal Flow Graph Gain Formula

The transfer function T(s) of a closed loop control system is:

Where

∆ = 1 – Σ(All different loop gains)

+ Σ(Gain products of all combinations of two non-touching loops)

- Σ(Gain products of all combinations of three non-touching loops)

+ …

Pk : The paths connecting the input R(s) and the output Y(s)

∆k : is ∆ with the loops touching the kth path removed

48

k kp

k

sR

sYsT

)(

)()(

Example:

1. Calculate forward path transfer function Pk for each forward path k.

2. Calculate all loop TF’s.

3. Consider nontouching loops 2 at a time.

Loops L1 do not touch Loops L3 and L4

Loops L2 do not touch Loops L3 and L4

49

87652

43211

GGGGP

GGGGP

774663

332221

,

,,

HGLHGL

GHLHGL

Example:

5. Calculate Δ.

6. Calculate Δk for each forward path.

7. The TF of the system is

50

4232413143211 LLLLLLLLLLLL

212

431

1

1

LL

LL

423241314321

2187654343212211

1

11)(

)(

)(

LLLLLLLLLLLL

LLGGGGLLGGGGPPsT

sR

sY

Block Diagram to Signal Flow Graph The SFG can be constructed from the block diagram as show in the

following example:

To get the transfer function, we use the Mason’s Gain Formula. 51

52