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.
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 Figure 1. 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: where and are, respectively, the constant parameters and is the transmission delays. These parameters can be obtained by analyzing ascending curve as shown in Figure 2. Specifically, , where is the steady state value of the step response and is the difference of a given step signal. PI controller is adopted to control the temperature system as follows: where is the proportion coefficient, is the integral coefficient, and is the error between the reference and output. Figure 3 is the control block diagram.
The transfer function of the whole system can be given as follows: Let and , where is intermediate variable; then, we have Set and ; then, the system’s state space equation can be written as follows: where and are the system state. Denote and ; then, (5) can be reformulated as follows:
Set . Let , where ; then, we have
When the thermostated oven temperature varies from 0°C to 120°C, the system model is given as follows: The corresponding PI controller parameters are and . The obtained temperature control subsystem is given as follows:
When the thermostated oven temperature varies from 120°C to 260°C, the parameters of the thermostated oven temperature system are and . The corresponding PI controller parameters are and . The obtained temperature control subsystem is given as follows:
When the thermostated oven temperature varies from 260°C to 400°C, the parameters of the temperature system are and and the parameters of the corresponding PI controller are and . The state space equation for the temperature control subsystem can be written as follows:
We model the whole temperature control system to be a switched delayed system as follows: where is the state, is the delay, , , , and . The switched delayed system consists of three subsystems. Denote the continuous function space from to by . Let , , and < . The system (12) is switched to subsystem when , where is defined as .
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 and for any switching signal if there exist positive definite matrices and such that for all , the following LMI holds:
Proof. Choose Lyapunov functional . Taking derivative of along the solution of system (12) leads to where . Thus as . Uniformly globally asymptotical stability is guaranteed.
Corollary 2. If the controller parameters and 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 asymptotically.
Theorem 1 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 to guarantee the stability of system (1).
Theorem 3. The switched delayed system (12) is asymptotically stable if there exist symmetric positive definite matrices , , and , a symmetric semipositive definite matrix , and any appropriately dimensioned matrices and such that for all , the following LMIs hold: where
Proof. The main argument is based on Theorem 2 in [10]. Choose Lyapunov functional as follows: Combining Theorem 2 in [10] and conditions in Theorem 3, we have that is a common Lyapunov functional for switched delayed system (12). Thus system (12) is asymptotically stable.
Corollary 4. If the controller parameters and 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 asymptotically.
4. Experiment
Figure 4 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: , , , , , and . The corresponding PI controller parameters are chosen as , , , , , and .
Then we have 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.
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China under Grant nos. 61174058 and 61325014.