research article temperature control of gas chromatograph...

Research ArticleTemperature Control of Gas ChromatographBased on Switched Delayed System Techniques

Xiao-Liang Wang, Ming-Xu Zhang, Kun-Zhi Liu, and Xi-Ming Sun

The School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Kun-Zhi Liu; [email protected]

Received 10 April 2014; Accepted 10 May 2014; Published 26 May 2014

Academic Editor: Yun-Bo Zhao

Copyright © 2014 Xiao-Liang Wang et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We address the temperature control problem of the gas chromatograph. We model the temperature control system of the gaschromatograph into a switched delayed system and analyze the stability by common Lyapunov functional technique. The PIcontroller parameters can be given based on the proposed linear matrix inequalities (LMIs) condition and the designed controllercan make the temperature of gas chromatograph track the reference signal asymptotically. An experiment is given to illustrate theeffectiveness of the stability criterion.

1. Introduction

Gas chromatograph can separate the mixture by using chro-matographic column and then the components of themixturecan be analyzed qualitatively. At present, gas chromatographhas been widely used in medicine, food safety, petrochemical[1], environmental science [2], and many other fields.

However, with gas chromatograph applied to processanalysis [3], quality testing [4], environment online monitor-ing [5], and sudden emergencymonitoring, the contradictionbetween non-real-time measurement and the demand of thereal-time measurement in various fields is becoming moreand more obvious. Therefore, the application of gas chro-matograph into the field of measure is restricted. Recently,the contradiction is solved partly by improving the speed oftemperature programming of chromatographic column or byimproving the column flow velocity, as well as by reducingthe chromatographic column inner diameter [6]. Amongthese approaches, the first method, that is, by improvingthe speed of temperature programming of chromatographiccolumn, seems more effective. As demonstrated in the paper[7], the analysis time can be shortened to 10 percent byimproving the speed of temperature programming. Since the

temperature of the chromatograph column affects directly thegas chromatograph column efficiency, separation selectivity,and the sensitivity and the stability of detector, therefore theaccurate temperature control for the thermostated oven isvery important and is our main attention in this paper. Ingeneral, the thermostated oven works at 0∘C∼400∘C. Sinceheating process of the thermostated oven is essentially aheat transfer process, time delay phenomena are inevitable.In the meanwhile, the parameters of the thermostated ovensystem model change with the variation of the temperature.Therefore, the controller design and stability analysis for thiskind of system are complicated extremely, and to the best ofthe authors’ knowledge, there are few works available in theexisting literature till now. In this paper, in order to trackthe reference temperature signal, a switching controller isintroduced, whose parameters can change with the variationof the temperature.Wemodel the temperature control systemas a switched delayed system [8]. Based on such a switcheddelayed system model, stability of gas chromatograph canbe analyzed by the common Lyapunov functional [9], andthe PI controller parameters can be given such that thetemperature of the gas chromatograph tracks the referencesignal asymptotically. An experiment is given to illustrate theeffectiveness of the stability.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 367629, 5 pageshttp://dx.doi.org/10.1155/2014/367629

2 Mathematical Problems in Engineering

Data systemand print

Detector

Detector

amplifier

InjectorGas inlets

Pneumaticcontrols Thermostated

oven

Column

Gas chromatograph

Figure 1: Structure diagram of gas chromatograph.

3024181260

90

70

50

30

10

t (min)

T(∘C)

𝜏 T

Figure 2: Illustration of ascending curve method.

2. Modeling Based on Switched DelayedSystem

Gas chromatograph consists of several parts as shown inFigure 1. The mixture to be detected is firstly gasified andthen goes into the chromatographic column through injector.The temperature programming of the thermostated oven isexecuted by the electrical control equipment. The model canbe described by the following transfer function:

𝐺 (𝑠) =

𝐾𝑒

−𝜏𝑠

1 + 𝑇𝑠

,(1)

where 𝑇 and 𝐾 are, respectively, the constant parametersand 𝜏 is the transmission delays. These parameters can beobtained by analyzing ascending curve as shown in Figure 2.Specifically, 𝐾 = 𝑦(∞)/Δ𝑈, where 𝑦(∞) is the steady statevalue of the step response and Δ𝑈 is the difference of a givenstep signal. PI controller is adopted to control the temperaturesystem as follows:

𝑃 (𝑡) = 𝐾

𝑝𝑒 (𝑡) + 𝐾

𝑖∫

𝑡

0

𝑒 (𝑡) 𝑑𝑡, (2)

where 𝐾𝑝is the proportion coefficient, 𝐾

𝑖is the integral

coefficient, and 𝑒(𝑡) is the error between the reference andoutput. Figure 3 is the control block diagram.

Y(s)U(s) +

−

E(s)kp +

kis

ke−𝜏s

1 + Ts

Figure 3: Signal flow graph.

The transfer function of the whole system can be given asfollows:

𝐸 (𝑠)

𝑈 (𝑠)

= 1 −

𝐾𝐾

𝑝𝑠𝑒

−𝜏𝑠

+ 𝐾𝐾

𝑖𝑒

−𝜏𝑠

𝑇𝑠

2

+ 𝑠 + 𝐾𝐾

𝑝𝑠𝑒

−𝜏𝑠

+ 𝐾𝐾

𝑖𝑒

−𝜏𝑠

. (3)

Let 𝑀(𝑠)/𝑈(𝑠) = 1/(𝑇𝑠

2

+ 𝑠 + 𝐾𝐾

𝑝𝑠𝑒

−𝜏𝑠

+ 𝐾𝐾

𝑖𝑒

−𝜏𝑠

) and𝑍(𝑠)/𝑀(𝑠) = 𝐾𝐾

𝑝𝑠𝑒

−𝜏𝑠

+ 𝐾𝐾

𝑖𝑒

−𝜏𝑠, where𝑀 is intermediatevariable; then, we have

𝑢 = 𝑇�� (𝑡) + �� (𝑡) + 𝐾𝐾

𝑝�� (𝑡 − 𝜏) + 𝐾𝐾

𝑖𝑚(𝑡 − 𝜏) ,

𝑧 = 𝐾𝐾

𝑝�� (𝑡 − 𝜏) + 𝐾𝐾

𝑖𝑚(𝑡 − 𝜏) ,

𝑒 = 𝑢 − 𝑧.

(4)

Set𝑥1= 𝑚 and𝑥

2= ��; then, the system’s state space equation

can be written as follows:

��

1= 𝑥

2,

��

2= −

1

𝑇

𝑥

2−

𝐾𝐾

𝑖

𝑇

𝑥

1(𝑡 − 𝜏) −

𝐾𝐾

𝑝

𝑇

𝑥

2(𝑡 − 𝜏) +

1

𝑇

𝑢,

𝑒 = 𝑢 − 𝐾𝐾

𝑝𝑥

2(𝑡 − 𝜏) − 𝐾𝐾

𝑖𝑥

1(𝑡 − 𝜏) ,

(5)

where 𝑥1and 𝑥

2are the system state. Denote 𝑥 = [𝑥

1𝑥

2]

𝑇

and 𝑥(𝑡 − 𝜏) = [𝑥

1(𝑡 − 𝜏) 𝑥

2(𝑡 − 𝜏)]

𝑇; then, (5) can bereformulated as follows:

�� = [

0 1

0 −

1

𝑇

]𝑥 +

[

[

0 0

−

𝐾𝐾

𝑖

𝑇

−

𝐾𝐾

𝑝

𝑇

]

]

𝑥 (𝑡 − 𝜏) + [

0

1

𝑇

]𝑢,

𝑒 = 𝑢 − [𝐾𝐾

𝑖𝐾𝐾

𝑝] 𝑥 (𝑡 − 𝜏) .

(6)

Set 𝑢(𝑡) = 𝑟. Let 𝑥 = 𝑥 − 𝑐𝑟, where 𝑐 = [ 1/𝐾𝐾𝑖0

]; then, wehave

𝑥 = [

0 1

0 −

1

𝑇

]𝑥 +

[

[

0 0

−

𝐾𝐾

𝑖

𝑇

−

𝐾𝐾

𝑝

𝑇

]

]

𝑥 (𝑡 − 𝜏) ,

𝑒 = − [𝐾𝐾

𝑖𝐾𝐾

𝑝] 𝑥 (𝑡 − 𝜏) .

(7)

When the thermostated oven temperature varies from0∘C to 120∘C, the system model is given as follows:

𝐺 (𝑠) =

𝐾

1𝑒

−𝜏𝑠

1 + 𝑇

1𝑠

. (8)

Mathematical Problems in Engineering 3

MCUBTA41-600B

heating equipment (GS-2010)

Gas chromatograph

Temperature measuring

transmittingcircuit

y(t)u(t) +

−

e(t)

Figure 4: Temperature control system.

The corresponding PI controller parameters are 𝐾𝑝𝑖

and𝐾

𝑖1. The obtained temperature control subsystem is given as

follows:

𝑥 =

[

[

0 1

0 −

1

𝑇

1

]

]

𝑥 +

[

[

0 0

−

𝐾

1𝐾

𝑖1

𝑇

1

−

𝐾

1𝐾

𝑝1

𝑇

1

]

]

𝑥 (𝑡 − 𝜏) ,

𝑒 = − [𝐾

1𝐾

𝑖1𝐾

1𝐾

𝑝1] 𝑥 (𝑡 − 𝜏) .

(9)

When the thermostated oven temperature varies from120∘C to 260∘C, the parameters of the thermostated oventemperature system are𝐾

2and𝑇2.The corresponding PI con-

troller parameters are𝐾𝑝2and𝐾

𝑖2.The obtained temperature

control subsystem is given as follows:

𝑥 =

[

[

0 1

0 −

1

𝑇

2

]

]

𝑥 +

[

[

0 0

−

𝐾

2𝐾

𝑖2

𝑇

2

−

𝐾

2𝐾

𝑝2

𝑇

2

]

]

𝑥 (𝑡 − 𝜏) ,

𝑒 = − [𝐾

2𝐾

𝑖2𝐾

2𝐾

𝑝2] 𝑥 (𝑡 − 𝜏) .

(10)

When the thermostated oven temperature varies from260∘C to 400∘C, the parameters of the temperature systemare 𝑇3and 𝐾

3and the parameters of the corresponding PI

controller are 𝐾𝑝3

and 𝐾𝑖3. The state space equation for the

temperature control subsystem can be written as follows:

𝑥 =

[

[

0 1

0 −

1

𝑇

3

]

]

𝑥 +

[

[

0 0

−

𝐾

3𝐾

𝑖3

𝑇

3

−

𝐾

3𝐾

𝑝3

𝑇

3

]

]

𝑥 (𝑡 − 𝜏) ,

𝑒 = − [𝐾

3𝐾

𝑖3𝐾

3𝐾

𝑝3] 𝑥 (𝑡 − 𝜏) .

(11)

We model the whole temperature control system to be aswitched delayed system as follows:

�� (𝑡) = 𝐴

𝑞(𝑡)𝑥 (𝑡) + 𝐹

𝑞(𝑡)𝑥 (𝑡 − 𝜏) , (12)

𝑒 (𝑡) = 𝐶

𝑞(𝑡)𝑥 (𝑡 − 𝜏) , (13)

where 𝑥 ∈ R2 is the state, 𝜏 ≥ 0 is the delay, 𝑞(𝑡) : R≥0

→

{1, 2, 3}, 𝐴𝑞= [

0 1

0 −1/𝑇𝑞], 𝐹𝑞= [

0 0

−𝐾𝑞𝐾𝑖𝑞/𝑇𝑞 −𝐾𝑞𝐾𝑝𝑞/𝑇𝑞], and 𝐶

𝑞=

− [𝐾

𝑞𝐾

𝑖𝑞𝐾

𝑞𝐾

𝑝𝑞]. The switched delayed system consists of

three subsystems.Denote the continuous function space from

[−𝜏, 0] to R2 by 𝐶𝜏. Let Ω

1= {(𝜙

1, 𝜙

2)

𝑇

∈ 𝐶

𝜏| 0 ≤

𝐾𝐾

𝑖𝜙

1(−𝜏) + 𝐾𝐾

𝑝𝜙

2(−𝜏) ≤ 120}, Ω

2= {(𝜙

1, 𝜙

2)

𝑇

∈ 𝐶

𝜏|

120 < 𝐾𝐾

𝑖𝜙

1(−𝜏)+𝐾𝐾

𝑝𝜙

2(−𝜏) ≤ 260}, andΩ

3= {(𝜙

1, 𝜙

2)

𝑇

∈

𝐶

𝜏| 260 < 𝐾𝐾

𝑖𝜙

1(−𝜏)+𝐾𝐾

𝑝𝜙

2(−𝜏) ≤ 400}.The system (12)

is switched to subsystem 𝑞 ∈ {1, 2, 3} when 𝑥𝑡∈ Ω

𝑞, where 𝑥

𝑡

is defined as 𝑥𝑡(𝑠) = 𝑥(𝑡 + 𝑠), 𝑠 ∈ [−𝜏, 0].

3. Stability Analysis

Next, we will give a theorem guaranteeing the uniformlyglobally asymptotical stability of system (12).

Theorem 1. The switched delayed system (12) is uniformlyglobally asymptotically stable for any large delay 𝜏 and for anyswitching signal 𝑞(𝑡) if there exist positive definite matrices 𝑃and 𝑄 such that for all 𝑞 ∈ {1, 2, 3}, the following LMI holds:

[

𝐴

𝑇

𝑞

𝑃 + 𝑃𝐴

𝑞+ 𝑄 𝑃𝐹

𝑞

∗ −𝑄

] < 0. (14)

Proof. Choose Lyapunov functional 𝑉(𝑥𝑡) = 𝑥

𝑇

(𝑡)𝑃𝑥(𝑡) +

∫

𝑡

𝑡−𝜏

𝑥

𝑇

(𝑠)𝑄𝑥(𝑠)𝑑𝑠. Taking derivative of 𝑉(𝑥𝑡) along the solu-

tion of system (12) leads to

𝑉 (𝑥

𝑡) = 𝑥

𝑇

(𝑡) (𝐴

𝑇

𝑞

𝑃 + 𝑃𝐴

𝑞) 𝑥 (𝑡) + 2𝑥

𝑇

(𝑡) 𝑃𝐹

𝑞𝑥 (𝑡 − 𝜏)

+ 𝑥

𝑇

(𝑡) 𝑄𝑥 (𝑡) − 𝑥

𝑇

(𝑡 − 𝜏)𝑄𝑥 (𝑡 − 𝜏)

= 𝜉

𝑇

[

𝐴

𝑇

𝑞

𝑃 + 𝑃𝐴

𝑞+ 𝑄 𝑃𝐹

𝑞

∗ −𝑄

] 𝜉 < 0,

(15)

where 𝜉(𝑡) = [𝑥𝑇(𝑡) 𝑥𝑇(𝑡 − 𝜏)]𝑇

. Thus 𝑉(𝑥𝑡) → 0 as 𝑡 →

+∞. Uniformly globally asymptotical stability is guaranteed.

Corollary 2. If the controller parameters 𝐾𝑞and 𝐾

𝑖𝑞(𝑞 ∈

{1, 2, 3}) are chosen such that the condition of Theorem 1 issatisfied, then the output of the temperature control systemFigure 4 can track the reference signal 𝑢(𝑡) asymptotically.

Theorem 1 gives the sufficient condition to guarantee thestability of system (12) by common Lyapunov functional,

4 Mathematical Problems in Engineering

while the obtained LMI condition is delay-independent,which is usually conservative. Next we will give LMI condi-tion depending on the delay bound 𝜏 to guarantee the stabilityof system (1).

Theorem3. The switched delayed system (12) is asymptoticallystable if there exist symmetric positive definite matrices 𝑃 =

𝑃

𝑇

> 0, 𝑄 = 𝑄𝑇 ≥ 0, and 𝑍 = 𝑍𝑇 > 0, a symmetric semiposi-tive definite matrix 𝑋 = [

𝑋11 𝑋12

𝑋

𝑇

12𝑋22

], and any appropriatelydimensioned matrices 𝑌 and 𝑁 such that for all 𝑞 ∈ {1, 2, 3},the following LMIs hold:

Φ =

[

[

[

[

Φ

11Φ

12𝜏𝐴

𝑇

𝑞

𝑍

Φ

𝑇

12

Φ

22𝜏𝐹

𝑇

𝑞

𝑍

𝜏𝑍𝐴

𝑞𝜏𝑍𝐹

𝑞−𝜏𝑍

]

]

]

]

< 0,

Ψ =

[

[

[

[

𝑋

11𝑋

12𝑌

𝑋

𝑇

12

𝑋

22𝑁

𝑌

𝑇

𝑁

𝑇

𝑍

]

]

]

]

≥ 0,

(16)

where

Φ

11= 𝑃𝐴

𝑞+ 𝐴

𝑇

𝑞

𝑃 + 𝑌 + 𝑌

𝑇

+ 𝑄 + 𝜏𝑋

11,

Φ

12= 𝑃𝐹

𝑞− 𝑌 + 𝑁

𝑇

+ 𝜏𝑋

12,

Φ

22= −𝑁 −𝑁

𝑇

− 𝑄 + 𝜏𝑋

22.

(17)

Proof. The main argument is based on Theorem 2 in [10].Choose Lyapunov functional as follows:

𝑉 (𝑥

𝑡) = 𝑥

𝑇

(𝑡) 𝑃𝑥 (𝑡) + ∫

𝑡

𝑡−𝜏

𝑥

𝑇

(𝑠) 𝑄𝑥 (𝑠) 𝑑𝑠

+ ∫

0

−𝜏

∫

𝑡

𝑡+𝜃

��

𝑇

(𝑠) 𝑍�� (𝑠) 𝑑𝑠 𝑑𝜃.

(18)

Combining Theorem 2 in [10] and conditions in Theorem 3,we have that 𝑉(𝑥

𝑡) is a common Lyapunov functional for

switched delayed system (12). Thus system (12) is asymptoti-cally stable.

Corollary 4. If the controller parameters 𝐾𝑞and 𝐾

𝑖𝑞(𝑞 ∈

{1, 2, 3}) are chosen such that the condition of Theorem 3 issatisfied, then the output of the temperature control systemFigure 4 can track the reference signal 𝑢(𝑡) asymptotically.

4. Experiment

Figure 4 is the illustration of the experiment. By the ascend-ing curve method, the parameters of the temperature systemof the thermostated oven are measured as follows: 𝐾

1= 1,

𝑇

1= 420, 𝐾

2= 1.4, 𝑇

2= 424, 𝐾

3= 1.4, and 𝑇

3= 426.

The corresponding PI controller parameters are chosen as𝐾

𝑝1= 8.7, 𝐾

𝑖1= 0.30, 𝐾

𝑝2= 9, 𝐾

𝑖2= 0.32, 𝐾

𝑝3= 9.2,

and𝐾𝑖3= 0.35.

0 500 1000 1500 2000 2500 300050

100

150

200

250

300

350

Time (s)

ActualSetpoint

Col

umn

tem

pera

ture

(∘C)

30∘C/min

20∘C/min

5∘C/min

Figure 5: Experiment of the control thermostated oven tempera-ture.

Then we have

𝐴

1= [

0 1

0 −

1

420

] , 𝐹

1= [

0 0

−

1

1400

−

29

1400

] ,

𝐶

1= − [0.3 8.7] ,

𝐴

2= [

0 1

0 −

1

424

] , 𝐹

2= [

0 0

−

7

6625

−

63

2120

] ,

𝐶

2= − [0.448 12.6] ,

𝐴

3= [

0 1

0 −

1

426

] , 𝐹

3= [

0 0

−

49

42600

−

161

5325

] ,

𝐶

3= − [0.49 12.88] .

(19)

Applying Corollary 4, it is concluded that the switcheddelayed system is stable. Figure 5 shows the practical trackingcurve of the temperature control system of the thermostatedoven. It can be seen from Figure 5 that the temperaturecontrol system can track the reference accurately.

5. Conclusion

In this paper, we address the temperature tracking problemof the gas chromatograph.Wemodel the temperature controlsystem into a switched delayed system. By the commonLyapunov functional technique, stability of the temperaturecontrol system is derived and the PI controller parameterscan be given based on the LMIs conditions. An experiment isgiven to illustrate the effectiveness of the proposed criterion.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Mathematical Problems in Engineering 5

Acknowledgment

This work was supported by the National Natural Sci-ence Foundation of China under Grant nos. 61174058 and61325014.

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