Methods of Controller Tunin1.docx

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Methods of Controller Tuning

i) Ziegler-Nichols Method

This method is used as a rule where the rule of heuristic PID produce good values for three PIDgain parameters. Those three PID gain parameters are the controller path gain, Kp, the

controller’s integrator time constant, Ti and the controller’s derivative time constant, Td.

From the measurements it is derived the two measure feedbac loop parameters which are the

period of the oscillation fre!uenc", Tu at the stabilit" limit and the ultimate gain margin Ku for

loop stabilit". #sing the characteristic e!uation of $% GOL & $ %'v'm'p'c & (

From the value of Ku and Tu that obtained from the derivation, the value of Kc,τ I and

τ D

can be calculated b" using the iegler*+ichols Tuning hart-

K cu τ I τ D

P control K u/2 * *

PI control K u/2.2 T u /1.2 *

PID control K u/1.7 T

u /2

T u /8

The other wa" of using this method is b" using the formula

τ = Pu

2 π tan (

π ( Pu−2θ ) Pu

)

τ = Pu

2 π √ ( K K cu)

2−1

ii) Direct Synthesis

Direct s"nthesis method are based upon predescribing a desired form for the s"stem’s response

which then a controller strateg" and parameters is found to give the response.

For first order process of direct s"nthesis controller strateg", the formula is given b"


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~G=

K p

τ p s+1

/nd setting as PI control it gives

K c= τ p

K p τ c

0here τ i=τ p

/s for first order plus time dela" 1F2PTD) 3odel, the standard model is given b"

~G (s)=

K p Ke−θs

τs+1

/nd the controller settings is

K c= 1

K

τ ❑

θ+τ c

0here τ i=τ ❑

iii) Internal Model Control (IMC) Method

I3 method is similar lie D4 where the controller is based on the assumed process model and a

desired closed*loop transfer function which leads to anal"tical e5pressions for the controller

settings.

In order to get the values for the controller settings, $6$ Pade appro5imation and the First 2rder

Ta"lor 4eries appro5imation can be used.

$) $6$ Pade /ppro5imation


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e−θs=

1−θ

2s

1+θ

2s

K c= 1

K

2( τ θ )+12( τ cθ )+1

τ I =θ

2+τ

τ D= τ

2( τ θ )+1

7) First 2rder Ta"lor series appro5imation

e−θs=1−θs

K c= 1

K

τ

τ c+θ

τ I =τ

Selection of design parameter,τ c

In choosing the design parameter, τ c , it is a e" decision for both D4 and I3 design

method. 'enerall", when τ c increases, it produces more conservative controller. That is

because K c decreased when τ I increases.

The guidelines for choosing the τ c is as follows

$. 8c69 : (.; and 8c : (.$81)7. 8 : 8c : 9 1hien and Fruehauf, $==()

?. 8c & 9 14ogestad, 7((?)

!S"#T

i) Ziegler-Nichols Method

@" using the formula,

$ % '2A & $ % 'v'm'p'c & ( *************** 1$)

/ssuming 'm&'v& $,

'iven 'p &1

2 s+3e−0.5 s

, which then divided b" ? to mae in general form of , 'p & K

τs+1

.


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10

17

PV vs Time Graph (Ziegler-Nichols)

Z-N PID Controller

This gives, 'p &0.333

0.6667 s+1

@" using Pade appro5imation, e−0.5s

&1−0.25 s1+0.25 s

'p &0.333

0.6667 s+1(1−0.25 s1+0.25 s

)

&0.333−0.0833 s

0.6667 s2+0.917 s+1

From 1$), $%(0.333−0.0833 s) Kc

0.6667 s2+0.917 s+1 & (

4ubstituting s & i.B, and replace i7 & *$,

*(.$>C B7 % (.=$C i.B %$ % (.???Kc (.(;??1i.B)Kc&(

4eparating the imaginar" and real terms,

1$%(.???Kc*(.$>CB7) % 1(.=$CB*(.(;??BKc)i&(

(.=$CB*(.(;??BKc&( , $%(.???Kc*(.$>CB7

Kc&$$.($ B&E.7= Tu &

$.$;;

hoosing PID mode, the + table

3ode K cu 8I 8Drror 1I4) rror 1I/) Displa"

PID K c /1.7 T u /2 T u /8

>.GC> (.E=G (.$G;E C.G>>?G ;.CC$$C *$.>GG$C

?.7?; (.7=C (.(CG7E ?.$>? $(.$ $

The first value calculated results in oscillated graph 1as shown in Figure $a) with large value of

error. The graph oscillated with large amplitude therefore it is not stable. The value is then tuned

b" dividing each value with 7 which gives I4, I/ error responses of ?.$>? and $(.$

meanwhile the PID controller response is $. This value is stable as shown in Figure $b as it

reached the set point value.

B &2 π

Tu &: Tu

&2 π

ω


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0 50 100 150 200 250

Time (seconds)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

P

V

0 50 100 150 200 250

Time (seconds)

0

0.2

0.4

0.6

0.

1

1.2

1.4

P

V

PV vs Time (Ziegler-Nichols)

Z-N PID Controller

Figure $a- The response of + PID controller before tuning

Figure $b- The response of + PID controller after tuning

ii) Direct Synthesis Method

In order to get the values of Kc and the PI controller, the First 2rder Plus Time Dela" 1F2PTD)

formula is used.


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K c= 1

K

τ

θ+τ c andτ I =τ

From the transfer function, K & (.???, 8 & (.>>C, 9 & (.E

Halue of τ c based on $.;$; 7.(=; G.E $

0ith 8c of (.E, it gives the value Kc of 7 and it gives a smaller I4 error compared to when 8c is(.>.

Figure 7- The

response of D4

PI controller with Kc e!uals to 7.


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iii) Internal Model Control (IMC) Method

In order to get the value Kc,τ I and

τ D , Pade appro5imation is used. The formula is

0ith K & (.???, 8 & (.>>C, 9 & (.E, Kc= 1

K

2

( τ

θ

)+1

2( τ cθ )+1τ 1=θ

2+τ τ D= τ

2( τ θ )+1

τ 1=0.917 and τ D=0.182

8c Kc rror 1I4) rror 1I/) Displa"

(.E ?.>>C $.>7C ?.?=G $

$ 7.7 $.>?> ?.7CC $

$.E $.EC$ $.CC= ?.7C; $

/t 8c of $,the Kc

gives the

value of

7.7 has

the least

error

shown when response has approached the set point.


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Figure ?- The graph of I3 PD controller at 8c of $ and Kc of 7.7

DISC"SSI$N

The methods of controller tuning are iegler*+ichols 1+), Direct 4"nthesis 1D) and

Internal 3odel ontrol 1I3). 0ith the given transfer function and simulation stop time of 7E(

second, the stud" is to observe and determine which method gives the best optimum value

according to the response and error value obtained.

The first method of controller tuning is iegler*+ichols. For igler +ichols method we

used PID controller to obtain the response. 0e used $%'2A & ( to find the value of K c and Tu and

using both value to find K cu,τ I , and τ D respectivel". From calculation, value of K c & $$.($

and Tu & $.$;;. @ased on the Figure $a, the graph that we obtained was constant at ero before

oscillated start at 7((s. The oscillation increases with time. Thus, the process did not achieve

stable response as it was not at its optimum value, $ and resulting the error to be huge in number

with C.G>>?G and ;.CC$$C using I4 and I/ respectivel". Therefore, tuning is needed to be

done to achieve a stable response. >C and 9 & (.E were taen from the transfer function. #sing a given e!uation with different

values of 8c,, the value of K c can be calculated. Halues for 8c were assumed above (.G as we used


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τ D , Pade appro5imation was used and values of 8c were assumed with values of (.E, $ and

$.E. Thus, values of K c calculated were ?.>>C, 7.7 and $.EC$ respectivel" and the lowest error

obtained for I4 was $.>7C for K c & ?.>>C and error for I4/ was ?.7CC for K c & $.


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time dela" which adds the stabilit" to the loop. /s the conclusion, all of the methods can be used

with the given transfer function and I3 has the best response.

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