Lecture introduction to control systems chapter 2 mathematical models of continuous control systems (dr huynh thai hoang)

Block diagram algebraTransfer function of systems in series... Block diagram algebra cont’Transfer function of systems in parallel... Block diagram algebra cont’Transfer function of feed

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Lecture Notes

Introduction to Control Systems

Instructor: Dr Huynh Thai Hoang Department of Automatic Control Faculty of Electrical & Electronics Engineering

Ho Chi Minh City University of Technology

Email: hthoang@hcmut.edu.vn

huynhthaihoang@yahoo.com Homepage: www4.hcmut.edu.vn/~hthoang/

20 September 2011 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 1

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Chapter 2

MATHEMATICAL MODELS OF CONTINUOUS CONTROL SYSTEMS

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æ The concept of mathematical model

Content

æ The concept of mathematical model

æ Transfer function

æ Block diagram algebra

æ Signal flow graph

æ State space equation

æ Linearized models of nonlinear systems

Ø Nonlinear state equation

Ø Linearized state equation

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The concept of mathematical models

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æ If you design a control system what do you need to know about the

Question

æ If you design a control system, what do you need to know about the plant or the process to be control?

æ What are the advantages of mathematical models?

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æ Practical control systems are diverse and different in nature

Why mathematical model?

æ Practical control systems are diverse and different in nature

æ It is necessary to have a common method for analysis and design of different type of control systems ⇒ Mathematics

+ + + − ( ) ( ) ( ) )

(

1 1

0 d y t a d y t a dy t a y t

a

n n

1 1

0 d u t b d u t b du t b u t b

m m

+ +

+

Invariant System

+ +

+ + 1 −1 −1 ( )

dt

a dt

dt

b dt

b m + m− + + m− + m

n: system order, for proper systems: n ≥m

a i , b i: parameter of the system

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Example: Car dynamics

)()

(

)

(

t f t

M: mass of the car B friction coefficient: system parameters

M: mass of the car, B friction coefficient: system parameters

f(t): engine driving force: input

v(t): car speed: output

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Example: Car suspension

)()

(

)()

(

2

t f t

Ky dt

t

dy B dt

t y

d

dt dt

M: equivalent mass

B friction constant, K spring stiffness

f(t): external force: input

y(t): travel of the car body: output

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Example: Elevator

M t

K M

t

dy B

t y

d

2

g M t

K g

M dt

y B dt

Cabin &

load

M L: mass of cabin and load, M B: counterbalance

B friction constant, K gear box constant

τ(t): driving moment of the motor: input

y(t): position of the cabin: output

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æ Difficult to solve differential equation order n (n>2)

Disadvantages of differential equation model

æ Difficult to solve differential equation order n (n>2)

= +

+ +

)

(

1 1

dt

t dy a

dt

t y d

a dt

t y d

n n

n

L

dt dt

dt

) (

) ( )

( )

(

1 1

dt

t du b

dt

t u d

b dt

t u d

m m

m

+ +

+

æ System analysis based on differential equation model is difficult

æ System design based on differential equation model is almostimpossible in general cases

æ It is necessary to have another mathematical model that makes the

æ It is necessary to have another mathematical model that makes theanalysis and design of control systems easier:

Ø transfer function

Ø state space equation

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The Laplace transform of a function f(t) defined for all real numbers

Definition of Laplace transform

The Laplace transform of a function f(t), defined for all real numbers

t ≥ 0, is the function F(s), defined by:

()

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Gi th f ti f(t) d (t) d th i ti L l

Properties of Laplace transform

Given the functions f(t) and g(t), and their respective Laplace

sF dt

t

df

L

s F

t

)(

f

t

)

()

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Laplace transform of basic functions

æ Unit step function:

f 0

0 t

f

1)

(

i

t u

t

01

f

0 t

f

0)

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Laplace transform of basic functions (cont’)

f 0

0t

f

)

()

(

i

t t

tu t

s

t u

at e

f(t)

{ at } 1

)(

0

0f

)

(.)

(

ti

t

i

t u e

t u

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Laplace transform of basic functions (cont’)

f 0

0t

f

sin)

()

(sin)

(

i

t t

u t t

+

=

s

t u t

L

æ Table of Laplace transform: f p f Appendix A, Feedback control ofpp f dynamic systems, Franklin et al.

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æ Consider a system described by the differential equation:

Definition of transfer function

æ Consider a system described by the differential equation:

Linear time

=+

++

)

(

1 1

a

n n

L

invariant system

++

a dt

)(

)()

()

(

1 1

d

t

du b

d

t u

d b d

t u

1 1

0

dt dt

are zeros, we have:

=+

++

)()

()

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æ Transfer function:

Definition of transfer function (cont’)

æ Transfer function:

m m

b s

Y s

1

1 1

0)

()

æ Definition: Transfer function of a system is the ratio between the

n n

a s

1

1 1 0

)(

)

(

L

yLaplace transform of the output signal and the Laplace transform

of the input signal assuming that initial conditions are zeros

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Transfer function of components

Procedure to find the transfer function of a component

æ Step 1: Establish the differential equation describing the output relationship of the components by:p p p y

input-Ü Applying Kirchhoff's law, current-voltage relationship ofresistors, capacitors, inductors, for the electrical components

Ü Applying Newton's law the relationship between friction and

Ü Applying Newton s law, the relationship between friction andvelocity, the relationship between force and deformation ofsprings for the mechanical components

function of the component

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Transfer function of some type of controllers

+

=

RCs

s G

G

1

)(

+

RCs

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Transfer function of some type of controllers (cont’)

s

2 1

2

R R

1 2

R R

C R

R T

<

+

=

R R

α

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I

1

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Active controllers

æ Proportional Derivative controller (PD)

s K K

K K

s

G( ) = P + I + D

2 1

2 2 1

1

C R

C R C

R

K P = − +

2 1

1

C R

K I = −

1

2C R

K D = −

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Transfer function of DC motors

Equivalent diagram of a DC motor

− L a : armature induction − ω : motor speed

− R a : armature resistance − M t : load inertia

− U a : armature voltage − B : friction constant

− E a : back electromotive force − J : moment of inertia of the rotor

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æ Applying Kirchhoff's law for the armature circuit:

)(

)

()

dt

t

di L R

t i t

dt

)()

Φ : excitation magnetic flux

æ Applying Newton’s law for the rotating part of the motor:

t

d J t

B t

M t

)()

()

dt

J t

B t

M t

)()

()

where: M (t) = KΦi a(t)

(3)(4)

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æ Taking the Laplace transform of (1), (2), (3), (4) leads to:

(5))

()

U a = a a + a a + a

(6)(7)

)()

E a = Φω

)()

()

(8))

()

(s K i s

æ Denote:

a a

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æ From (5) and (7) we have:

æ From (5) and (7), we have:

)1

(

)()

()

(

s T R

s E s

U s

( T s

R a + a

)1

(

)()

()

(

s T B

s M s

/ 1 )

/

1 a

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Transfer function of a thermal process

Temperature of th

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Transfer function of a thermal process (cont’)

æ The approximate transfer function of the thermal Y (s)

æ The approximate transfer function of the thermal

process can be calculated by using the equation: ( )

)

()

(

s U

s

Y s

æ The approximate output is:y(t) = f (t −T1)

æ The approximate output is: y(t) f (t T1)

where: ( ) (1 t/T2 )

e K

t

The Laplace transform of f (t) is:F( ) K

The Laplace transform of f (t) is:

)1

(

)(

2s T s

s F

+

=

)(

1

Ke s

Y

s T

Applying the time delay theorem:

)1

(

)(

2s T s

s Y

+

=Applying the time delay theorem:

)

()

(

1

−

Ke s

Y G

s T

1)

(

)

()

U

s G

⇒

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Transfer function of a car

(

)

(

t f t

Bv dt

F

s

V s

)(

)

()

G B

Ms s

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Transfer function of an suspension system

M: equivalent car massq

Ky dt

t

dy B dt

t y

d

dt dt

Ms s

F

s G

++

=

)(

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Transfer functions of sensors

H ( ) =

æ Ex: Suppose that temperature of a furnace changing in the range y(t)

= 0÷5000C, if a sensor converts the temperature to a voltage in the

range y fb (t) 0÷5V, then the transfer function of the sensor is:

)/

(01.0)

(500/

)(5)

s H

fb

+

=1)

(

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Transfer functions of control systems

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Block diagram

æ Block diagram is a diagram of a system in which the principal parts

æ Block diagram is a diagram of a system, in which the principal parts

or functions are represented by blocks connected by lines, that showthe relationships of the blocks

æ A block diagram composes of 3 components:

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Block diagram algebra

Transfer function of systems in series

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Block diagram algebra (cont’)

Transfer function of systems in parallel

(

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Transfer function of feedback systems

)()

(1

)

(

s H s G

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Transfer function of feedback systems

(1

)

()

(

s H s G

s

G s

G cl

−

=

)(1

)

()

(

s G

s

G s

G cl

−

=)

()

(

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Transfer function of multi loop systems

æ For a complex system consisting of multi feedback loops, we perform equivalent block diagram transformation so that simple connecting q g p g

blocks appears, and then we simplify the block diagram from the inner loops to the outer loops

æ Two block diagrams are equivalent if their input-output relationship are the same

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Moving a pickoff point behind a block

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Moving a pickoff point ahead a block

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Moving a summing point behind a block

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Moving a summing point ahead a block

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Interchanging the positions of the two consecutive summing points

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Splitting a summing point

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Note

point :

p

two summing points if there exists

a pickoff point between them:

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Block diagram algebra – – Example 1 Example 1

æ Find the equivalent transfer function of the following system:

Y(s)

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Block diagram algebra – – Example 1 (cont’) Example 1 (cont’)

()

(s G3 s G4 s

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)(s G1 s

)]

()

() [(

1

)()

()

(1

)

()

s G s

G s

G

s G s

G s G

s

G s

G C

−+

=+

=

)]

()

().[

(1

)()

(

1+ G2 s G A s + G2 s G3 s G4 s

æ Equivalent transfer function of the system:

)()

()

)()]

(1

[)

().[

(1

)(

4 3

=

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Y(s)

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æ Interchanging the positions of the summing points and ¡

Moving the pickoff point ¢ behind the block G2(s)

Y(s)

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æ GB(s) = feedback loop [G2(s), H2(s)]

GC(s) = [GA(s)// unity block]

GC(s) [GA(s)// unity block]

Y(s)

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H G

G B

+

=

H G

2

1 2

2

11

1

*

G

H G

G

H G

=+

=

2 2

1 3 3

2 3

2

1 2

2 2

2 3

11

*

H G

H G G

G G

G

H G

G G

=

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H G G

3 1 3 3

3 2 2

2

1 3 3

2 1

3 3

2

2 2

1 3 3

2

11

*

H H G H

G G H

G

H G G

G H

H G G

G

H G

H G G

G

H G

G G

D

D E

++

+

=+

+

=+

=

3 1 3 3

3 2 2

2 3

2 2

3

1

H G

D

++

æ E i l t t f f ti f th t

1 3 3

2 1

3 1 3 3

3 2 2

2 1

1 1

1

.1

1.1

*

H H G H

G G H

G

H G G

G G

H H G H

G G H

G G

G

G G

E

E eq

++

+

++

+

=+

=

3 1 3 3

3 2 2

2

1+ G H + G G H + G H H

1 3 1 3

2 1

1

H G G G

G G H

H G H

G G H

G

H G G G

2 1 3

1 3 3

3 2 2

2

1 G H G G H G H H G G G G G H

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Y(s)

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Hint to solve example 3

æ Move the summing point ¡ ahead the block G1(s),

then interchange the position of the summing points and¡

Hint to solve example 3

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S l ti t E l 3

S l ti t E l 3 Solution to Example 3

æ Students calculate the equivalent transfer function themselves usingthe hints in the previous slide

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Remarks on block diagram algebra

æ When calculating the equivalent transfer function, it is necessary tomanipulate many calculations on algebraic fractions This could be apotential source of error if the system is complex enough

potential source of error if the system is complex enough

⇒ Block diagram algebra is only appropriate for finding equivalenttransfer function of simple systems

To find equivalent transfer function of complex systems, signal flowgraph method (to be discussed later) is more effective

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Definition of signal flow graph

Y(s)

æ Signal flow graph : a networks consisting of nodes and branches.

æ Node: a point representing a signal or a variable in the system

æ Branch: a line directly connecting two nodes, each branch has anarrow showing the signal direction and a transfer functionrepresenting the relationship between the signal at the two nodes ofthe branch

æ Source node: a node from which there are only out-going branches.

æ Sink node: a node to which there are only in-going branches.

æ Hybrid node: a node which both has in going branches and out

æ Hybrid node: a node which both has in-going branches and

out-going branches

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Definition of signal flow graph (cont’)

æ Forward path: is a path consisting of continuous sequence of

branches that goes in the same direction from a source node to a sinknode without passing any single node more than once

æ Path gain is the product of all transfer functions of the branches

æ Path gain is the product of all transfer functions of the branchesbelonged to the path

æ Loop: is a closed path consisting of continuous sequence of

branches that goes in the same direction without passing any singlenode more than once

Loop gain is the product of all transfer functions of the branches

belonged to the loop

LoopForward path

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Mason’s formula

æ The equivalent transfer function from a source 1

æ The equivalent transfer function from a source

node to a sink node of a system can be found

by using the Mason’s formula:

∑ΔΔ

̇ P k: is the gain of k th forward path from the considered source node

to the considered drain node

̇ Δ: is the determinant of the signal flow graph

K+

−+

−

=

g nontouchin

m j i

m j i g

nontouchin

j i

j i i

L

, , ,

1

̇ Δ: is the determinant of the signal flow graph

g nontouchin g

nontouchin

i

L : is the gain of the i th loop

̇ Δk: is the cofactor of the k th path

Δk is inferred from Δ by removing all the gain(s) of the loop(s)touching the forward path P k

æ Note: Nontouching loops do not have any common nodes A loop and

a path touch together if they have at least one common node

Tiêu đề	Mathematical Models Of Continuous Control Systems
Người hướng dẫn	Dr. Huynh Thai Hoang
Trường học	Ho Chi Minh City University of Technology
Chuyên ngành	Automatic Control
Thể loại	Lecture Notes
Năm xuất bản	2011
Thành phố	Ho Chi Minh City

Định dạng
Số trang	120
Dung lượng	1,73 MB