# Temperature Control of Gas Chromatograph Based on Switched Delayed System Techniques

Authors: Xiao-Liang Wang; Ming-Xu Zhang; Kun-Zhi Liu; Xi-Ming Sun
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

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


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## Body

## 1. Introduction

Gas chromatograph can separate the mixture by using chromatographic column and then the components of the mixture can be analyzed qualitatively. At present, gas chromatograph has been widely used in medicine, food safety, petrochemical [1], environmental science [2], and many other fields.However, with gas chromatograph applied to process analysis [3], quality testing [4], environment online monitoring [5], and sudden emergency monitoring, the contradiction between non-real-time measurement and the demand of the real-time measurement in various fields is becoming more and more obvious. Therefore, the application of gas chromatograph into the field of measure is restricted. Recently, the contradiction is solved partly by improving the speed of temperature programming of chromatographic column or by improving the column flow velocity, as well as by reducing the chromatographic column inner diameter [6]. Among these approaches, the first method, that is, by improving the speed of temperature programming of chromatographic column, seems more effective. As demonstrated in the paper [7], the analysis time can be shortened to 10 percent by improving the speed of temperature programming. Since the temperature of the chromatograph column affects directly the gas chromatograph column efficiency, separation selectivity, and the sensitivity and the stability of detector, therefore the accurate temperature control for the thermostated oven is very important and is our main attention in this paper. In general, the thermostated oven works at 0°C~400°C. Since heating process of the thermostated oven is essentially a heat transfer process, time delay phenomena are inevitable. In the meanwhile, the parameters of the thermostated oven system model change with the variation of the temperature. Therefore, the controller design and stability analysis for this kind of system are complicated extremely, and to the best of the authors' knowledge, there are few works available in the existing literature till now. In this paper, in order to track the reference temperature signal, a switching controller is introduced, whose parameters can change with the variation of the temperature. We model the temperature control system as a switched delayed system [8]. Based on such a switched delayed system model, stability of gas chromatograph can be analyzed by the common Lyapunov functional [9], and the PI controller parameters can be given such that the temperature of the gas chromatograph tracks the reference signal asymptotically. An experiment is given to illustrate the effectiveness of the stability.


## 2. Modeling Based on Switched Delayed System

Gas chromatograph consists of several parts as shown in Figure1. The mixture to be detected is firstly gasified and then goes into the chromatographic column through injector. The temperature programming of the thermostated oven is executed by the electrical control equipment. The model can be described by the following transfer function:
(1)G(s)=Ke-τs1+Ts,
where T and K are, respectively, the constant parameters and τ is the transmission delays. These parameters can be obtained by analyzing ascending curve as shown in Figure 2. Specifically, K=y(∞)/ΔU, where y(∞) is the steady state value of the step response and ΔU is the difference of a given step signal. PI controller is adopted to control the temperature system as follows:
(2)P(t)=Kpe(t)+Ki∫0t‍e(t)dt,
where Kp is the proportion coefficient, Ki is the integral coefficient, and e(t) is the error between the reference and output. Figure 3 is the control block diagram.Figure 1
Structure diagram of gas chromatograph.Figure 2
Illustration of ascending curve method.Figure 3
Signal flow graph.The transfer function of the whole system can be given as follows:(3)E(s)U(s)=1-KKpse-τs+KKie-τsTs2+s+KKpse-τs+KKie-τs.
Let M(s)/U(s)=1/(Ts2+s+KKpse-τs+KKie-τs) and Z(s)/M(s)=KKpse-τs+KKie-τs, where M is intermediate variable; then, we have
(4)u=Tm¨(t)+m˙(t)+KKpm˙(t-τ)+KKim(t-τ),z=KKpm˙(t-τ)+KKim(t-τ),e=u-z.
Set x1=m and x2=m˙; then, the system’s state space equation can be written as follows:
(5)x˙1=x2,x˙2=-1Tx2-KKiTx1(t-τ)-KKpTx2(t-τ)+1Tu,e=u-KKpx2(t-τ)-KKix1(t-τ),
where x1 and x2 are the system state. Denote x=[x1x2]T and x(t-τ)=[x1(t-τ)x2(t-τ)]T; then, (5) can be reformulated as follows:
(6)x˙=[010-1T]x+[00-KKiT-KKpT]x(t-τ)+[01T]u,e=u-[KKiKKp]x(t-τ).Setu(t)=r. Let x¯=x-cr, where c=[1/KKi0]; then, we have
(7)x-˙=[010-1T]x¯+[00-KKiT-KKpT]x¯(t-τ),e=-[KKiKKp]x¯(t-τ).When the thermostated oven temperature varies from 0°C to 120°C, the system model is given as follows:(8)G(s)=K1e-τs1+T1s.
The corresponding PI controller parameters are Kpi and Ki1. The obtained temperature control subsystem is given as follows:
(9)x-˙=[010-1T1]x¯+[00-K1Ki1T1-K1Kp1T1]x¯(t-τ),e=-[K1Ki1K1Kp1]x¯(t-τ).When the thermostated oven temperature varies from 120°C to 260°C, the parameters of the thermostated oven temperature system areK2 and T2. The corresponding PI controller parameters are Kp2 and Ki2. The obtained temperature control subsystem is given as follows:
(10)x-˙=[010-1T2]x¯+[00-K2Ki2T2-K2Kp2T2]x¯(t-τ),e=-[K2Ki2K2Kp2]x¯(t-τ).When the thermostated oven temperature varies from 260°C to 400°C, the parameters of the temperature system areT3 and K3 and 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:
(11)x-˙=[010-1T3]x¯+[00-K3Ki3T3-K3Kp3T3]x¯(t-τ),e=-[K3Ki3K3Kp3]x¯(t-τ).We model the whole temperature control system to be a switched delayed system as follows:(12)x˙(t)=Aq(t)x(t)+Fq(t)x(t-τ),(13)e(t)=Cq(t)x(t-τ),
where x∈R2 is the state, τ≥0 is the delay, q(t):R≥0→{1,2,3}, Aq=[010-1/Tq], Fq=[00-KqKiq/Tq-KqKpq/Tq], and Cq=-[KqKiqKqKpq]. The switched delayed system consists of three subsystems. Denote the continuous function space from [-τ,0] to R2 by Cτ. Let Ω1={(ϕ1,ϕ2)T∈Cτ∣0≤KKiϕ1(-τ)+KKpϕ2(-τ)≤120}, Ω2={(ϕ1,ϕ2)T∈Cτ∣120<KKiϕ1(-τ)+KKpϕ2(-τ)≤260}, and Ω3={(ϕ1,ϕ2)T∈Cτ∣260  <  KKiϕ1(-τ)+KKpϕ2(-τ)≤400}. The system (12) is switched to subsystem q∈{1,2,3} when xt∈Ωq, where xt is defined as xt(s)=x(t+s),s∈[-τ,0].


## 3. Stability Analysis

Next, we will give a theorem guaranteeing the uniformly globally asymptotical stability of system (12).Theorem 1.
The switched delayed system (12) is uniformly globally asymptotically stable for any large delay τ and for any switching signal q(t) if there exist positive definite matrices P and Q such that for all q∈{1,2,3}, the following LMI holds:
(14)[AqTP+PAq+QPFq*-Q]<0.Proof.
Choose Lyapunov functionalV(xt)=xT(t)Px(t)+∫t-τt‍xT(s)Qx(s)ds. Taking derivative of V(xt) along the solution of system (12) leads to
(15)V˙(xt)=xT(t)(AqTP+PAq)x(t)+2xT(t)PFqx(t-τ)+xT(t)Qx(t)-xT(t-τ)Qx(t-τ)=ξT[AqTP+PAq+QPFq*-Q]ξ<0,
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 parametersKq and Kiq(q∈{1,2,3}) are chosen such that the condition of Theorem 1 is satisfied, then the output of the temperature control system Figure 4 can track the reference signal u(t) asymptotically.Figure 4
Temperature control system.Theorem1 gives the sufficient condition to guarantee the stability of system (12) by common Lyapunov functional, while the obtained LMI condition is delay-independent, which is usually conservative. Next we will give LMI condition depending on the delay bound τ to guarantee the stability of system (1).Theorem 3.
The switched delayed system (12) is asymptotically stable if there exist symmetric positive definite matrices P=PT>0, Q=QT≥0, and Z=ZT>0, a symmetric semipositive definite matrix X=[X11X12X12TX22], and any appropriately dimensioned matrices Y and N such that for all q∈{1,2,3}, the following LMIs hold:
(16)Φ=[Φ11Φ12τAqTZΦ12TΦ22τFqTZτZAqτZFq-τZ]<0,Ψ=[X11X12YX12TX22NYTNTZ]≥0,
where
(17)Φ11=PAq+AqTP+Y+YT+Q+τX11,Φ12=PFq-Y+NT+τX12,Φ22=-N-NT-Q+τX22.Proof.
The main argument is based on Theorem  2 in [10]. Choose Lyapunov functional as follows:
(18)V(xt)=xT(t)Px(t)+∫t-τt‍xT(s)Qx(s)ds+∫-τ0‍∫t+θt‍x˙T(s)Zx˙(s)dsdθ.
Combining Theorem  2 in [10] and conditions in Theorem 3, we have that V(xt) is a common Lyapunov functional for switched delayed system (12). Thus system (12) is asymptotically stable.Corollary 4.
If the controller parametersKq and Kiq(q∈{1,2,3}) are chosen such that the condition of Theorem 3 is satisfied, then the output of the temperature control system Figure 4 can track the reference signal u(t) asymptotically.


## 4. Experiment

Figure4 is the illustration of the experiment. By the ascending curve method, the parameters of the temperature system of the thermostated oven are measured as follows: K1=1, T1=420, K2=1.4, T2=424, K3=1.4, and T3=426. The corresponding PI controller parameters are chosen as Kp1=8.7, Ki1=0.30, Kp2=9, Ki2=0.32, Kp3=9.2, and Ki3=0.35.Then we have(19)A1=[010-1420],F1=[00-11400-291400],C1=-[0.38.7],A2=[010-1424],F2=[00-76625-632120],C2=-[0.44812.6],A3=[010-1426],F3=[00-4942600-1615325],C3=-[0.4912.88].
Applying Corollary 4, it is concluded that the switched delayed system is stable. Figure 5 shows the practical tracking curve of the temperature control system of the thermostated oven. It can be seen from Figure 5 that the temperature control system can track the reference accurately.Figure 5
Experiment of the control thermostated oven temperature.


## 5. Conclusion

In this paper, we address the temperature tracking problem of the gas chromatograph. We model the temperature control system into a switched delayed system. By the common Lyapunov functional technique, stability of the temperature control system is derived and the PI controller parameters can be given based on the LMIs conditions. An experiment is given to illustrate the effectiveness of the proposed criterion.



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