Journal of Applied Research and Technology

Journal of Applied Research and Technology

www.jart.icat.unam.mx

Journal of Applied Research and Technology 17 (2019) 313-325

Venkatachalam Veeraragavan^a*, Prabhakaran Duraisamy^b, Ramakrishnan Krishnan^c

𝑥̇(𝑡) = 𝐴𝑥(𝑡) + 𝐴𝑑𝑥(𝑡 − 𝜏(𝑡)), 𝑡 > 0

𝑥(𝑡) = ∅(𝑡), ∀𝑡 ∈ [−𝜏̅, 0], 𝜏̅ > 0

𝜏(𝑡)

𝜏̅ 𝜇

;

0 1 0 0

1 0 0

0 1 0

0 0 1 0





























−

−

−

=

F F

F H H

H V

T T K T T K T

A



























 − −

=

0 0 0 0

0 0 0 0

0 0 0 0

0 0

V V I V

V P

d

T K K T

K K

A

𝑥(𝑡) 𝑥(𝑡 − 𝜏(𝑡))

0 ≤ 𝜏(𝑡) ≤ 𝜏̅, 𝜏̇(𝑡) ≤ 𝜇, ∀𝑡 ≥ 0

𝑥(𝑡) ∈ ℝ^4×1 𝐴 ∈ ℝ^4×4

𝐴𝑑∈ ℝ^4×4 ∅(𝑡)

𝜏̅ 𝑎𝑛𝑑 𝜇

[𝑅 𝑆

𝑆^𝑇 𝑅] ≥ 0; 𝑃 > 0; 𝑄𝑖> 0, 𝑖 = 1,2; 𝑅 > 0 𝛱 < 0.

𝛱 = 𝑒1𝑃𝑒4𝑇+ 𝑒4𝑃𝑒1𝑇+ 𝑒1(𝑄₁+ 𝑄2)𝑒₁^𝑇+ 𝑒2(−(1𝜇)𝑄₁)𝑒₂^𝑇

− 𝑒3𝑄2𝑒3𝑇+ 𝑒4(𝜏²𝑅)𝑒4𝑇

− [𝑒₂^𝑇− 𝑒₃^𝑇

𝑒1𝑇− 𝑒2𝑇]

𝑇

[𝑅 𝑆

𝑆^𝑇 𝑅] [𝑒₂^𝑇− 𝑒₃^𝑇

𝑒1𝑇− 𝑒2𝑇]

𝑒₁= [𝐼 0 0]^𝑇, 𝑒₂= [0 𝐼 0]^𝑇, 𝑒₃= [0 0 𝐼]^𝑇, 𝑒₄= (𝐴𝑒₁^𝑇+ 𝐴_𝑑𝑒₂^𝑇)^𝑇.

𝑉₁(𝑡) = 𝑥^𝑇(𝑡)𝑃𝑥(𝑡)

𝑉2(𝑡) = ∫_{𝑡−𝜏(𝑡)}^𝑡 𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+ ∫_𝑡−𝜏^𝑡 𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠

𝑉₃(𝑡) = 𝜏 ∫ ∫_−𝜏⁰ _𝑡+𝜃^𝑡 𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠𝑑𝜃

𝑃, 𝑄𝑖, 𝑖 = 1,2, 𝑅

η(t)= [x^T(t) x^T(t-τ(t)) x^T(t-τ)]^T

𝑉_𝑖(𝑡), 𝑖 = 1,2,3

𝑉₁(𝑡)

𝑉̇₁(𝑡) = 𝑥̇^𝑇(𝑡)𝑃𝑥(𝑡) + 𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡),

𝑉̇1(𝑡) = 𝜂^𝑇(𝑡)(𝑒₄𝑃𝑒₁^𝑇+ 𝑒1𝑃𝑒4𝑇)𝜂(𝑡)

𝑉₂(𝑡)

𝑉̇₂(𝑡) = 𝑥^𝑇(𝑡)(𝑄₁+ 𝑄₂)𝑥(𝑡) − (1 − 𝜏̇(𝑡))𝑥^𝑇

(𝑡 − 𝜏(𝑡))𝑄₁𝑥(𝑡 − 𝜏(𝑡)) − 𝑥^𝑇(𝑡 − 𝜏)𝑄₂𝑥(𝑡 − 𝜏)

𝜏̇(𝑡) ≤ 𝜇 < 1, 𝑉̇₂(𝑡)

𝑉̇2(𝑡) ≤ 𝑥^𝑇(𝑡)(𝑄₁+ 𝑄2)𝑥(𝑡) − (1 − 𝜇)𝑥^𝑇(𝑡 − 𝜏(𝑡))𝑄1𝑥

(𝑡 − 𝜏(𝑡)) − 𝑥^𝑇(𝑡 − 𝜏)𝑄₂𝑥(𝑡 − 𝜏) (12)

𝑉̇₂(𝑡) ≤ 𝜂^𝑇(𝑡)(𝑒₁(𝑄₁+ 𝑄₂)𝑒₁^𝑇)𝜂(𝑡) − 𝜂^𝑇(𝑡)

(𝑒₂(1 − 𝜇)𝑄₁𝑒2𝑇)𝜂(𝑡) − 𝜂^𝑇(𝑡)(𝑒₃𝑄2𝑒3𝑇)𝜂(𝑡)

𝑉₃(𝑡)

𝑉̇₃(𝑡) = 𝑥̇^𝑇(𝑡)(𝜏²𝑅)𝑥̇(𝑡) − 𝜏 ∫_𝑡−𝜏^𝑡 𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠.

𝑉̇₃(𝑡) ≤ 𝜂^𝑇(𝑡) ((𝑒₄(𝜏²𝑅)𝑒₄^𝑇) −

[𝑒₂^𝑇− 𝑒₃^𝑇 𝑒1𝑇− 𝑒2𝑇]

𝑇

[𝑅 𝑆

𝑆^𝑇 𝑅] [𝑒₂^𝑇− 𝑒₃^𝑇

𝑒1𝑇− 𝑒2𝑇]) 𝜂(𝑡)

[𝑅 𝑆

𝑆^𝑇 𝑅] ≥ 0.

𝑉𝑖(𝑡)

𝑉̇(𝑡) ≤ ∑³_𝑖=1𝑉_𝑖(𝑡)

𝑉̇(𝑡) ≤ 𝜂^𝑇(𝑡)𝛱𝜂(𝑡) 𝛱 < 0,

𝑥̇(𝑡) = 𝐴𝑥(𝑡) + 𝐴_𝑑𝑥(𝑡 − 𝜏(𝑡)) + 𝑓(𝑥(𝑡), 𝑡)

+𝑔(𝑥(𝑡 − 𝜏(𝑡)), 𝑡), 𝑥(𝑡) = ∅(𝑡), ∀𝑡 ∈ [−𝜏̅, 0], 𝜏̅ > 0

𝐴, 𝐴𝑑, 𝜏(𝑡) ∅(𝑡)

𝑓(𝑥(𝑡), 𝑡) 𝑔(𝑥(𝑡 − 𝜏(𝑡)), 𝑡)

𝑓(0, 𝑡) = 𝑔(0, 𝑡) = 0

𝑓^𝑇(. )𝑓(. ) ≤ 𝛼²𝑥^𝑇(𝑡)𝐹^𝑇𝐹𝑥(𝑡)

𝑔^𝑇(. )𝑔(. ) ≤ 𝛽²𝑥^𝑇(𝑡 − 𝜏(𝑡))𝐺^𝑇𝐺𝑥(𝑡 − 𝜏(𝑡)),

‖𝑓(. )‖ ≤ 𝛼‖𝑥(𝑡)‖ + 𝛽‖𝑥(𝑡 − 𝜏(𝑡))‖

𝛼 ≥ 0 𝛽 ≥ 0

𝑓^𝑇(. )𝑓(. ) ≤ 𝛼²𝑥^𝑇(𝑡)𝐹^𝑇𝐹𝑥(𝑡)

+𝛽²𝑥^𝑇(𝑡 − 𝜏(𝑡))𝐺^𝑇𝐺𝑥(𝑡 − 𝜏(𝑡))

𝐹 𝐺

𝛼 𝛽

𝑃, 𝑄_𝑖, 𝑖 = 1, 2, 𝑅 𝜀 ≥

0; 𝑆

[𝑅 𝑆

𝑆^𝑇 𝑅] ≥ 0;

𝑃 > 0; 𝑄_𝑖> 0, 𝑖 = 1,2; 𝑅 > 0 𝛱̅ < 0.

𝛱̅ = 𝑒̅₁𝑃𝑒̅₅^𝑇+ 𝑒̅₅𝑃𝑒̅₁^𝑇+ 𝑒̅₁(𝑄₁+ 𝑄₂+ 𝜖𝛼²𝐹^𝑇𝐹)𝑒̅₁^𝑇

+𝑒̅₂(−(1 − 𝜇)𝑄₁+ 𝜖𝛽²𝐺^𝑇𝐺)𝑒̅₂^𝑇− 𝑒̅₃𝑄₂𝑒̅₃^𝑇− 𝑒̅₄(𝜖𝐼)𝑒̅₄^𝑇

+𝑒̅₅(𝜏²𝑅)𝑒̅₅^𝑇− [𝑒̅₂^𝑇− 𝑒̅₃^𝑇 𝑒̅1𝑇− 𝑒̅2𝑇]

𝑇

[𝑅 𝑆

𝑆^𝑇 𝑅] [𝑒̅₂^𝑇− 𝑒̅₃^𝑇 𝑒̅1𝑇− 𝑒̅2𝑇]

 

 

 



0 0 0



,

, 0 0 0

, 0 0 0

, 0 0 0

4 3 2 1

T T T T

I e

I e

I e

I e

=

=

=

=

𝑒̅₅= (𝐴𝑒̅₁^𝑇+ 𝐴_𝑑𝑒̅₂^𝑇+ 𝑒̅₄^𝑇)^𝑇.

𝑥̇(𝑡) = [𝐴 + ∆𝐴(𝑡)]𝑥(𝑡) + [𝐴𝑑+ ∆𝐴𝑑(𝑡)]𝑥(𝑡 − 𝜏(𝑡))

𝑥(𝑡) = ∅(𝑡), ∀𝑡 ∈ [−𝜏̅, 0], 𝜏̅ > 0.

∆𝐴(𝑡), ∆𝐴_𝑑(𝑡)

𝐴(𝑡) 𝐴_𝑑(𝑡)

[∆𝐴(𝑡) ∆𝐴𝑑(𝑡)] = 𝐷 𝐹(𝑡)[𝐸𝑎 𝐸_𝑏]

𝐹^𝑇(𝑡)𝐹(𝑡) ≤ 𝐼, ∀𝑡

𝐹(𝑡)

𝐷, 𝐸𝑎, 𝐸𝑏

𝐹(𝑡)

𝜏 > 0, 𝜇 > 0,

𝜏(𝑡)

𝜏̅ > 0 𝜇 > 0

[𝑅 𝑆

𝑆^𝑇 𝑅] ≥ 0; 𝑃 > 0; 𝑄_𝑖> 0, 𝑖 = 1,2; 𝑅 > 0

0

*

*

*

*

*

0

*

*

*

*

0

*

*

*

0 0

0

*

*

0 2

) 1

* (

~

2 2 2

1 2

2 2

1











































−

−

−

−

− + − +

−

−

−

−

− + +

+ +

=



I I

RD R

R Q

E R

A S R S S R

Q

E PD R A S S R PA R Q

Q PA P A

T b T

d T T

T a T

T T d T











 





𝛱 <

0 𝐴 𝐴𝑑

∆𝐴(𝑡) = 𝐴 + 𝐷𝐹(𝑡)𝐸_𝑎 ∆𝐴_𝑑 = 𝐴_𝑑 + 𝐷𝐹(𝑡)𝐸_𝑏 𝛱 < 0

𝛱 + 𝛱_𝑢+ 𝛱_𝑢^𝑇 < 0

,

0

*

*

*

0 0

*

*

) ) ( 0 (

0

*

) ) ( 0 (

) ) ( ( ) ) ( (

) ) ( (

2 2













































 + 

=

 +



R E t DF

R E t E DF

t DF P E t DF P

P E t DF

T b

T a b

a T a u T u





𝛱_𝑢 𝛱_𝑢= 𝐻𝐹(𝑡)𝐸.

𝛱 + 𝐻𝐹(𝑡)𝐸 + (𝐻𝐹(𝑡)𝐸)^𝑇< 0

𝑃,𝑄max₁,𝑄₂,𝑅𝜏

[𝑅 𝑆

𝑆^𝑇 𝑅] ≥ 0; 𝑃 > 0; 𝑄_𝑖> 0, 𝑖 = 1,2; 𝑅 > 0 𝛱 < 0

𝐾𝐻= 34 𝜏𝐻= 30

𝐾𝑉= 1.25 𝜏𝑉= 3

𝐾𝐹= 0.08 𝜏𝐹= 2

) 1

( = +

s s K G

V V

V  () 1

= + s s K G

H H

H 

) 1

( = +

s s K G

F F

F 

) (t

e⁻s^

− +

PI Controller Law

Communication

delay Valve

Heat Exchanger

Sensor ) 

ref(s



)

s(s



)

h(s

 )

e(s

 _v(s)

Controller s KP+K^I

𝜏̅

𝑥̇(𝑡) = (𝐴 + 𝐴𝑑)𝑥(𝑡)

(𝐴 + 𝐴_𝑑) 𝑃𝐼 −

𝑗𝜔

𝐾𝐼

𝐾_𝐼 𝐾_𝑃

∆𝜃𝐻(𝑡)

𝐾_𝐼

𝑲_𝑷 𝑲_𝑰

(𝑨 + 𝑨_𝒅)

±

±

±

±

±

±

±

±

±

𝐾𝑃 & 𝐾𝐼) 𝜏̅

𝐾_𝑃= 0.75 𝐾_𝐼= 0.05, 𝜇 = 0.05

𝜏̅

0 5 10 15 20 25 30 35 40 45 50

-30 -20 -10 0 10 20 30 40 50

t sec

output temperature

Responses of thermal control system for three different values of time-delays (K_p= 0.75: K_I=0.05: =0.5)

=8.6

=9

=8.6801

1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8

Delay bounds for thermal system with nonlinealy laod perturbation

Kp

max

UnStable Region

stable region

==0, K I= 0.05

=0.025, =0, K_I= 0.05

==0.025, K_I= 0.05 𝐾𝑃

𝐾_𝑃, 𝛼 𝑎𝑛𝑑 𝛽

𝝁 = 𝟎. 𝟎𝟓 𝝁 = 𝟎. 𝟏𝟎 𝝁 = 𝟎. 𝟐𝟎 𝝁 = 𝟎. 𝟑𝟎 𝝁 = 𝟎. 𝟓𝟎 𝑲𝑷= 𝟎. 𝟕𝟓

𝑲𝑰= 𝟎. 𝟎𝟓

𝑲_𝑷= 𝟏 . 𝟎 𝑲_𝑰= 𝟎. 𝟎𝟓

𝑲_𝑷= 𝟏 . 𝟐𝟎 𝑲_𝑰= 𝟎. 𝟎𝟓

𝑲_𝑷= 𝟎. 𝟕𝟓 𝑲_𝑰= 𝟎. 𝟎𝟕𝟓

𝑲𝑷= 𝟐. 𝟎 𝑲𝑰= 𝟎. 𝟎𝟏

𝑲𝑷= 𝟏 𝑲𝑰= 𝟎. 𝟏𝟓

𝑲𝑷= 𝟑. 𝟎 𝑲𝑰= 𝟎. 𝟏

𝑲𝑷= 𝟓. 𝟎 𝑲𝑰= 𝟎. 𝟎𝟓

μ

0.1𝐼𝑛

𝑛 𝑥(𝑡)

𝜏

𝛼 𝛽

𝐾_𝑃 𝐾_𝐼

𝛼 𝛽

𝐾_𝑃 (1 − 7) 𝐾_𝐼= 0.05, 𝜇 = 0.8

𝜏̅

𝛼 = 𝛽 = 0; 𝛼 = 0.025, 𝛽 = 0; 𝛼 = 𝛽 = 0.025

𝜏

𝜏_{𝑉𝑚𝑎𝑥}

𝐾𝑃 𝐾𝐼

𝜏_{𝑉𝑚𝑎𝑥}

𝜏̅

𝑲𝑷 𝑲𝑰 𝜶 = 𝜷 = 𝟎

𝜇 = 0.05 𝜇 = 0.10 𝜇 = 0.20 𝜇 = 0.30 𝜇 = 0.50

𝜏̅

𝑲_𝑷 𝑲_𝑰 𝜶 = 𝟎. 𝟎𝟐𝟓; 𝜷 = 𝟎

𝜇 = 0.05 𝜇 = 0.10 𝜇 = 0.20 𝜇 = 0.30 𝜇 = 0.50

𝜏̅

𝑲_𝑷 𝑲_𝑰 𝜶 = 𝟎. 𝟎𝟐𝟓; 𝜷 = 𝟎. 𝟎𝟐𝟓

𝜇 = 0.05 𝜇 = 0.10 𝜇 = 0.20 𝜇 = 0.30 𝜇 = 0.50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

5.5 6 6.5 7 7.5 8 8.5 9

Delay bounds for Temperature control system with parametric uncertainty

_v

max

-maximum delay margin

K_p=0.75, K_I=0.05, =0.10 K_p=1.0, K_I=0.05, =0.10 Kp=1.20, K

I=0.05, =0.10

system under different parametric uncertainties.

The deduced result of this work is more realistic in operating conditions in real time temperature control system.

acknowledgement Dr. K. Ramakrishnan (Associate Professor of Electrical and Electronics Engineering, Pondicherry Engineering College, Pondicherry, India) and Dr.

𝝉𝑽_𝒎𝒂𝒙= 𝟎. 𝟐𝟎 𝝉𝑽_𝒎𝒂𝒙= 𝟎. 𝟒𝟎 𝝉𝑽_𝒎𝒂𝒙= 𝟎. 𝟔𝟎 𝝉𝑽_𝒎𝒂𝒙= 𝟎. 𝟖𝟎 𝜇

= 0.10 𝜇

= 0.30 𝜇

= 0.50 𝜇

= 0.10 𝜇

= 0.30 𝜇

= 0.50 𝜇

= 0.10 𝜇

= 0.30 𝜇

= 0.50 𝜇

= 0.10 𝜇

= 0.30 𝜇

= 0.50 𝑲_𝑷= 𝟎. 𝟕𝟓

𝑲_𝑰= 𝟎. 𝟎𝟓 𝑲𝑷= 𝟏. 𝟎 𝑲𝑰= 𝟎. 𝟎𝟓 𝑲_𝑷= 𝟏. 𝟐𝟎 𝑲_𝑰= 𝟎. 𝟎𝟓 𝑲𝑷= 𝟎. 𝟕𝟓

𝑲𝑰

= 𝟎. 𝟎𝟕𝟓 𝑲_𝑷= 𝟐. 𝟎 𝑲_𝑰= 𝟎. 𝟎𝟏 𝑲𝑷= 𝟏. 𝟎 𝑲𝑰= 𝟎. 𝟏𝟓 𝑲_𝑷= 𝟑. 𝟎 𝑲_𝑰= 𝟎. 𝟏𝟎 𝑲𝑷= 𝟓. 𝟎 𝑲𝑰= 𝟎. 𝟎𝟓

15.5𝑛²+ 3.5𝑛 4𝑛²+ 4𝑛 2.5𝑛²+ 2.5𝑛

3𝑛²+ 2𝑛 3𝑛²+ 2𝑛 + 1

ℝ^𝑛→ 𝑛

∈→

𝜏̅ →

𝑡 ∈ [−𝜏̅, 0] → −𝜏̅ ≤ 𝑡 ≤ 0, 𝑡& − 𝜏̅ > 0:

𝑋^𝑇→ 𝛼 & 𝛽 → 𝜇 →