Teknik kendali

priyatmadi

FEEDBACK CONTROL SYSTEM

• Process control is methods to force process parameters to have specific values.

• Objective to maintain the value of some quantity at some desired level regardless of influences

Process Types

• Self-regulating Processes– These are uncontrolled processes. The process

variables are not regulated. Example!• Manual Controlled Processes

– These proceses are controlled by human Example!• Automatic Controlled Processes

– These process are controlled by automatic controller. There are 2 types:

• feed forward control system. Example!• feedback control system (closed-loop control system).

Example!– We concern with the analysis and design of closed-

loop control system.

Self-regulating Processes• The process outputs are not regulated, its value will easily change. • following is an example of self regulating process

steam out

cold water in

hot water outsteam in

heat exchanger

Manual controlled process

Heat exchange process when under Manual control. The dot line represent the closed loop of the controller and the process

Automatic controlled process

Heat exchange process when under Automatic control.

set pointController

3-15 psi

air supply

4-20 mA

4-20 mA

temperature transmitter

hot water out

steam out

cold water insteam in

heat exchanger

I/P

current to pneumatic pressure converter

MODELS OF PHYSICAL SYSTEM

• Mathematical model of a system is defined as a set of equation used to represent physical system.

• It should be understood that no mathematical model of physical system is exact, although we may increase the accuracy by increasing the complexity of the equations

• In this module we only concern with LTI system, whose equation can be solved using Laplace transform and can be represented by a transfer function.

models of electrical elements

)()( tRitv

dt

diLtv )(

t

vdttiC

tv0

)0()(1

)(

v(t)

i(t)

v(t)

i(t)

v(t)

i(t)

)()( sRIsV

)(1

)( sIsc

sV

)()( sLsIsV

Component Diff. equ. Laplace transform

Electrical circuit example modeling

• In this circuit we consider v1 to be the input and v2 to be the output.

• here we have to write a set equations whose solution will yield v2(t) as function of v1(t) or V2(s) as a function of V1(s)

t

tvdiC

tiRtiR0

121 )()(1

)()(

i(t)

v1(t)

R1

C

R2

v2(t)

)()(1

)(

)()(1

)()(

22

121

sVsIsC

sIR

sVsIsC

sIRsIR

t

tvdiC

tiR0

22 )()(1

)(

From the first Laplace transformed equation

)()(1

)()( 121 sVsIsC

sIRsIR

)()(1

)( 22 sVsIsC

sIR

we solve for I(s)

Substituting I(s) to the second equation

)/1(

)()(

21

1

sCRR

sVsI

we get

)()/1(

)/1()( 1

21

22 sV

sCRR

sCRsV

The transfer function of this system is

)1)(

1

)(

)()(

21

2

1

2

sCRR

sCR

sV

sVsG

• thus the circuit can be modeled by:– two differential equation– two equation in the LAPLACE transform

variable, or– a transfer function

• another model using state space will be discussed next time

Electrical circuit example modeling

Op-amp example modeling

-

+

Zi (s)

Zf (s)

Vi (s)Vo (s)

)(

)(

)(

)()(

sZ

sZ

sV

sVsG

i

f

i

o

Block diagram and signal flow graph

E(s) C(s)G(s)

E(s) C(s)G(s)

G(s)

H(s)

E(s) C(s)R(s) E(s) C(s)G(s)R(s) 1

H(s)

C(s)= G(s)E(s)

)()()(1

)()( sR

sHsG

sGsC

+

Mechanical Translational System modeling

KBsMssF

sXsC

2

1

)(

)()(

Kxdt

dxB

dt

xdMtf

2

2

)(

)()()()()()( 22 sXKBsMssKXsBsXsXMssF M

K B

J

K

B

,

KBsJssT

s

2

1

)(

)(

Mechanical Rotational System modeling

STATE VARIABLE MODELING

• Purpose: to develop presentation which preserves the input output relationship, but which is expressed in n first order equation

• Advantage: in addition to the input-output characteristic, the internal characteristic of the system is represented

• Computer aided analysis and design of state models are performed more easily

• We feedback more information (internal/state variable) about the plant

• Design procedure that result in the best control system are almost all based on state variable models.

STATE VARIABLE MODELING

)()(

)(1

)()()(

1

212

21

txty

tfM

txM

Btx

M

Ktx

xx

2

1

2

1

2

1

01)(

1010

x

xty

fMx

x

M

B

M

Kx

x

)(1

)()(

)()()(

12

22

2

2

tfM

txM

Ktx

M

B

txdt

tdx

dt

tyd

KBsMssF

sYsG

2

1

)(

)()(

)()(

)()(

2

2

tKydt

tdyBtf

dt

tydM

Let us start with an example

For second order system we define

two state variables x1(t) and x2(t) as

x1(t) = y(t)

)()(

)( 11

2 txdt

tdxtx

Then we may write

Written in specific format, we have

In matrix notation, we have

A second order D.E. has been modified into two first order D.E’s. we used two state variables x1 and x2. For one n order D.E there will be n first order D.E’s having n state variables.

STATE VARIABLE MODELING

Definition: The state of a system of any time t0 is the amount of information at t0 that together with all inputs for t t0, uniquely determines the behavior of the system for all t t0

• The standard form of the state equation is dx(t)/dt = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) where

– x(t) = state vector– A = (nn) system matrix– B = (nr) input matrix– u(t) = input vektor = (r1) vector composed of the system input function– y(t) = output vektor = (p1) vector composed of the defined output– C = (pn) output matrix– D = (pr) matrix to represent direct coupling between input and output

STATE VARIABLE MODELING

Recall that standard form of the state equation is dx(t)/dt = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) The first equation, called the state equation, is a 1st order D.E and x(t)

is the solution of the equation. The second one is the output equation. Given x(t) and u(t) y(t) can be

found. Usually matrix D is zero. Nonzero D indicates that there are some path

coupled input and output. On the first equ only the first derivatives of the state var may appear

on the left side of equation and no derivatives on the right side No derivatives may appear on the output equation. The standard format of the state equation valid for multiple input and

output system.

STATE VARIABLE MODELING

1015242

23112111

ukykyky

ukuykyky

1034253

23121122

21

ukxkxkx

ukuxkxkx

xx

32

11

xy

xy

Example 3.1 Consider a D.E as follows:

xy

uxx

100

001

0

1

00

0

0

010

6

3

45

12

k

k

kk

kk

where u1 and u2 are inputs and y1 and y2 are outputs. Let us define the states:

23

11211

yx

xyxyx

from (1) and (2) we write :

(1)

(2)

with the outputs equation

this equation equations may be written in matrix form

SIMULATION DIAGRAM We have presented the way of finding state model from differential Equation. We will present a method of finding state model form transfer function. The method is based on simulation diagram. It is a block diagram or flow graph consisted of gain, summing junction and integrator only.

Y(s)X(s) s-1

s

1Y(s)X(s)x(t) y(t)

Symbols of integrator

We will construct simulation diagram of )(1

)()()( tfM

tyM

Kty

M

Bty

s

1 y(t)

s

1 )(ty)(tyM

1f(t) +

M

K

-

M

B

-

SIMULATION DIAGRAM

s

1x2s

1u(t) +

1a

-2a

-

0a

s

1x3

-

b0

b1

b2

++

+

y(t)

x1

•There are two common form of simulation diagram

012

23

012

2)(

asasas

bsbsbsG

The transfer function of third order system is:

•The first one is the control canonical form

SIMULATION DIAGRAM

The second one is the observer canonical form :

012

22

012

2)(

asasas

bsbsbsG

u(t)

s

1x3

b1

a1

s

1x2

b2

a2

b0

a0

+

+

+

s

1+ + y(t)

x1

The diagrams can be easily expanded to higher order system

SIMULATION DIAGRAM

Once simulation diagram of transfer function is constructed, a state model of the system is easily obtained. The procedure has two step

1. Assign a state variable to the output of integrator2. Write an equation for the input of each integrator

and an equation for each system output . These equation are written as function of integrator outputs and the system inputs

This procedure yields the following state equation

STATE EQUATION FROM SIMULATION DIAGRAM (control canonical form)

s

1x2s

1u(t) +

1a

- 2a

-

0a

s

1x3

-

b0

b1

b2

++

+y(t)

x1

xy

210 bbb

uxx

1

0

0

100

010

210 aaa

u(t)

s

1x3

b1

a1

s

1x2

b2

a2

b0

a0

+

+

+

s

1+ + y(t)

x1

STATE EQUATION FROM SIMULATION DIAGRAM (observer canonical form)

xy 001

uxx

0

1

2

0

1

2

00

10

01

b

b

b

a

a

a

Example

)(01)( tty x

)(10

)(0

1)( tf

Mt

M

BM

K

t

xx

MK

sMB

s

MKBsMssF

sYsG

22

11

)(

)()(

)(1

0)(

10)( tft

M

B

M

Kt

xx

Consider the mechanical system with the following transfer function

)(01

)( tM

ty x

The state model of control canonical form is

The state model of observer canonical form is