Equivalent sample theory of networked control systems and its application

Journal of Systems Engineering and Electronics Vol. 19, No. 6, 2008, pp.1199–1202

Equivalent sample theory of networked control systems and its application∗ Ji Shunping1,2 , Lu Yuping1 & Wang Shipeng1 1. Coll. of Automation, Nanjing Univ. of Aeronautics and Astronautics, Nanjing 210016, P. R. China; 2. Sanjiang Univ., Nanjing 210012, P. R. China (Received July 13, 2007)

Abstract: The equivalent sample theory and its application in analysis of networked control system (NCS) are presented. After analyzing NCS’s scheduling in master-slave mode, the characteristics of time delay and sample are summarized. Looking on master station visiting the slave station as a special sample process, the theory of equivalent sample is presented. And based on it, the stability of a kind of NCS is analyzed. The criterion to determine the upper bound of transmission delay is introduced, which guarantees the stability. Finally, an example with simulation shows the availability and usability of this analysis method.

Keywords: networked control system, equivalent sample theory, stability.

1. Introduction Networked control system (NCS) is the control system whose control loops are closed via an industrial communication networks, as shown in Fig. 1. The main characteristic of NCS is that information exchange between different control parts (such as sensors, controllers, and actuators) is accomplished through an industrial network[1−5]. The main merit of NCS is the advantage of communication. But the network also brings the new problems to the control loop, which induces the time delay. Ray[1] studied NCS earlier.Walsh[2], Branicky [3−4] , Wei Zhang[3−4] and Lian Fengli[5] also established their NCS models, and then presented the analysis methods. The main ideas of their methods are that they considered the network-induced delay as a time delay process to the control loop. In this article the authors look on master station visiting, on the slave station, as a special sample process, and the theory of the equivalent sample is presented. Based on the theory, the criterion to determine the upper bound of transmission delay is introduced, which guarantees stability. Finally, an example with simulation shows the availability of this analysis

method.

Fig. 1

Components of NCS

It is assumed that the sensor is driven by time and the actuator is driven by the events. And the network is in master-slave mode.

2. Scheduling of NCS Usually, the network-induced delay can be divided into four parts: transmission delay time, station wait time, station internal delay time, and transmission ready delay. The transmission delay time (TTD ) is the time needed for a data packet transmission from the sending station to the receiving station. The station wait time (TWAIT ) is the time spent by a station from the end of sending data to the beginning of receiving the new data. The station internal delay time (TSID )

* This project was supported by the National Natural Science Foundation of China (90605007).

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Ji Shunping, Lu Yuping & Wang Shipeng

is the time needed for the station internal data preparations and calculations. And the transmission ready delay time (TTRD ) is the time needed for bit synchronization or trial to link. In master-slave mode, the master station communicates with each slave station in turn, which is called polling. As shown in Fig. 2, station 0 sends frame F1

Fig. 2

Scheduling of NCS in master-slave mode

In master-slave mode, the circle time (Tcircle ) can be calculated as Eq.(1) Tcircle =

m 

i i (TSID + TTRD +

i,j=1 i TTD +

j TSID

+

(1) j TTRD

+

j TTD )

where T i is the delay time caused by master station’s enquiry to slave station i, and T j is the delay time induced by the answer of slave station j. The value of transmission delay time (TTD ) in Eq.(1) may be expressed as TTD =

to station 1 for enquiry, and then station 1 sends frame F2 back to station 0 as response. Then station 0 queries station 2, and station 2 replies. When station 0 has communicated with all slaves, one polling circle is completed. Then one by one circles will begin as before. The dark area in Fig. 2 means that some stations occupy the bus.

8×n v

(2)

where n is the length of the frame and v is the baud rate. When the length of the frame is 8 bytes, a typical length of NCS data, and the baud rate is 1.5 Mbps, then the result of TTD is 0.042 7 ms. The value is rather small relative to Tcircle or controllerinduced delay (τck ), and it may be ignored in following analysis.

lent sample theory. To ensure the stability and quality, the controller (master) must visit the sensor and actuator (slave) unceasingly. The controller visits the k sensor periodically with the circle time Te1 . When k k Te1 is a constant for any k, Te1 can be expressed as Te1 . Now the authors consider the controller visiting the sensor as a sample process with the sample time k . Two aspects should be considered. The data Te1 locked in the sensor may not be fresh, which was sampled from the plant ΔT k time ago. It is obvious that 0  ΔT k  T as shown in Fig. 1. In the other hand, it will cost some time for the data sampled to transfer from the sensor to the controller, which induces the k time delay. But the delay time is not Te1 but TTD . k Here TTD should be changed to TTD S , which is the

3. Equivalent sample theory of NCS Equivalent sample of NCS is shown as Fig.3. Look on master station visiting the slave station as a special sample process, which is the main thinking of equiva-

Fig. 3

Equivalent sample of NCS

Equivalent sample theory of networked control systems and its application time delay of the kth sample with the sensor. The process of the controller visiting the sensor also can be seemed as an equivalent sample. The controller visits the actuator periodically with the circle time k k k Te2 . When Te2 is a constant for any k, Te2 can be expressed as Te2 . Now the authors look at the controller visiting the sensor with the view of equivalent sample. When the controller visits the actuator, the data in the actuator were refreshed. The circle time of this k process is Te2 . This process will also cost some time for the data sampled to transfer from the controller to the actuator, which induces the time delay. But the k delay time is not Te2 but TTD . Here TTD should be k changed to TTD a , which is the time delay of the kth sample with the actuator. With the equivalent sample, the NCS is transferred to a common computer control system. The sample k k time is not T of Fig. 1, but Te1 and Te2 in Fig. 3. There are four time delay parts in the system, which k k k is ΔT k , TTD S , TTD a and controller-induced delay τC . For determinate control arithmetic, all the four parts can be combined to one delay τ k k

τ =

τCk

k

+ ΔT +

k TTD S

+

k TTD a

(3)

k k In fact, TTD S and TTD a is very small, whose value is only several decade microseconds. T is far smaller k k than Te1 and Te2 , and its value is also several decade microseconds. But the value of τ k is several decade k k milliseconds, which is much larger than Te1 , Te2 and k T . So in the real system, τ can be expressed as τCk approximatively,

τ k ≈ τCk

(4)

k k When Te1 and Te2 are constants, and Te = Te1 = Te2 , this kind of NCS is the basic form of networked control systems, which can be seemed as the most common computer control system with the sample time Te . Many networks satisfy this condition, such as PROFIBUS and CONTROL NET in IEC61158. The next discussion will be focused on this kind of system.

4. Stability of NCS After equivalent sample, the NCS is transferred to a common computer control system, which is shown in Fig. 3. In order to analysis the stability of NCS with

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certain conditions of Te , Gp (s) and Gc (z), the time delay of controller may be ignored here, which is shown in Fig. 3 with the dashed line instead of delay block τ k. The transform function of closed loop in z domain is G(z) =

Gc (z)Gp (z) C(z) = R(z) 1 + Gc (z)Gp (z)

(5)

Then the characteristic equation is 1 + Gc (z)Gp (z) = 0

(6)

If the authors assume that z0 , z1 , z2 . . . and zn is the roots of Eq. (6), z0 , z1 , z2 . . . and zn are eigenvalues of the closed-loop control system. The system is stable if [6] |zi | < 1, i = 0, 1, 2, . . . , n

(7)

The stability is decided by Te , Gp (s) and Gc (z). If Gp (s) is determined, Te and Gp (s) must satisfy some conditions to insure the stability. The bigger Te may turn the system from stable status to instable status. In order to make a concrete analysis of concrete problems, the authors focus on the situation mentioned in Ref. [7], where the plant is Gp (s) =

s2

ω02 , + 2ξ0 ω0 s + ω02

0 < ξ0 < 1

(8)

and the controller is Gc (z) = K

(9)

and then eigenvalues are z1,2 = e−ξ0 ω0 Te e±

√

1+K−ξ 2 ω0 Te

(10)

When ξ0 , ω0 , Te and K adapt to   √  −ξ0 ω0 Te ± 1+K−ξ2 ω0 Te  e <1 e

(11)

the system is stable. For every Te , the authors can find the critical K to satisfy Eq. (11). Moreover they can find the critical Te to satisfy Eq. (11) for every K. As shown in Fig. 4, the area under the curve is the stable area of the system.

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Ji Shunping, Lu Yuping & Wang Shipeng [2] Walsh G C, Ye H, Bushnell L. Stability analysis of networked control systems. Proc. of American Control Conf., San Diego, 1999: 2876–2880. [3] Branicky M S, Phillips S M, Wei Zhang. Stability of networked control systems: Explicit analysis of delay. Proc. of American Control Conf., Chicago, 2000: 2352–2357. [4] Wei Zhang, Branicky M S, Phillips S M. Stability of networked control systems. IEEE Control Systems Magazine, 2001,21(1): 84–99. [5] Lian Fengli, Moyne James, Tilbury Dawn. Fig. 4

Time delay

modeling and sample time selection for networked control

Critical K with equivalent sample time Te

systems. Proc. of ASME-DSC, International Mechanical Engineering Congress and Exposition, New York, 2001,

5. Simulation and example

DSC: 5–6. [6] Franklin G F, Powell J D, Workman M L. Digital con

For example, the plant is

trol of dynamic systems. Third Edition. Addison-Wesley,

25 Gp (s) = 2 s + 8s + 25

(12)

[7] Zhu Xuemei. The effect of the gain and sampling interval

Then it is known that ω0 = 5, and ξ0 = 0.8. If the authors choose the circle time of networked Tcircle , the equivalent sample time is known to be Te . It is assumed that Te = 0.1S

1998:73–93.

(13)

which is a typical circle time of Profibus. The authors can get critical value of K = 7.387 6. If K < 7.387 6, the system is stable.

on the stability in a digital control system.

Journal of

Nanjing Normal University (Engineering and Technology), 2001, 1(3): 8–11 (in Chinese).

Ji Shunping was born in 1975. He is currently pursuing the doctoral degree at the College of Automation, Nanjing University of Aeronautics and Astronautics, China. He is interested in networked control systems. E-mail: [email protected]

6. Conclusions After analyzing NCS’s scheduling with the masterslave mode, the characteristics of time delay and sample are summarized. Subsequently, the equivalent sample theory is presented. Based on the theory, the stability of a kind of NCS is analyzed. The relationship of Te and Gc (z) is introduced, which guarantees the stability. Finally, the example with simulation shows the availability and usability of the equivalent sample theory.

nautics and Astronautics. He is now the president of the College of the Astronautics, Nanjing University of Aeronautics and Astronautics, China. His research interests are networked control systems and flight control technologies. Wang Shipeng was born in 1982. He is currently pursuing the M. S. degree at the College of Automation, Nanjing University of Aeronautics and Astro-

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nautics, China. He is interested in networked control systems.