2011 Third International Conference on Measuring Technology and Mechatronics Automation
Performance comparison and feedback controller design of network controlled systems with continuous loss of states Weisong Tian, Chunjiang Qian Department of Electrical and Computer Engineering University of Texas at San Antonio, San Antonio San Antonio, Texas, USA Email:
[email protected],
[email protected]
Abstract—This paper considers the problem of global stabilization in network control systems (NCS) with continuous/consective loss of states. In this context, data packets loss is an important issue that must be adequately addressed. Compared with random data packets loss, continuous (arbitrary) data packets loss is much more challenging to deal with because it requires to consider the maximum degree of continuous loss of states that a system can tolerate. Specifically, we address three important issues, including (i) comparison of two NCS models, (ii) determination of longest continuous state loss which a NCS can tolerate, and (iii) development of a NCS controller under continuous state loss. Several illustrative examples are given as well.
Shouhuai Xu Department of Computer Science University of Texas at San Antonio, San Antonio San Antonio, Texas, USA Email:
[email protected]
Figure 1.
General structure of NCS, Model 1
Figure 2.
General structure of NCS, Model 2
Keywords-Network Control Systems, Continuous State Loss, Denial of Service (DoS) Attack, Network Security
I. I NTRODUCTION Network Control System (NCS) is a kind of feedback control systems whose control loops are closed through a digital communication network. NCS has many industrial applications in automobiles, aircrafts, biomedical, critical infrastructure and power grids. Compared with traditional point-to-point wiring, the use of the communication networks can reduce the cost of cables and power, simplify the installation and maintenance of the whole system, and imporve the scalability of systems. Cyber security is an important problem that need be considered in NCS. Compared with traditional point-to-point wiring, communication networks are open and unreliable especially when communicating via wireless channels and Internet. While highly cost-effective, it also makes NCS vulnerable to cyber attacks. In this paper, we consider the case where the system to be stabilized is one-dimensional, i.e. x˙ = ax + bu
period T , namely that continuous-time states x(t) are turned into discrete-time signals x(k), where k ∈ Z+ and Z+ denotes the set of all nonnegative integers. The most significant difference between the two structures is the use of different kinds of zero order hold (ZOH). In Fig.1, the controller will be stored in a buffer of the extended ZOH before a new controller signal is received. If the current controller is lost, the system will use the last received controller stored in the ZOH. During the sampling period, the controller will remain intact. However, in Fig.2, the controller will be set to zero when the packet is lost during transfer. Packets loss is said to be arbitrary if the values ik and jk are picked arbitrarily [3]. Arbitrary packets loss can be seen as a kind of Denial of Service (DoS) attack, by which the attacker tries to block the communication channel as long as possible [2]. Our motivation was to characterize the maximum window of lost packets the system can tolerate. Related work. Walsh et al. [1] considered a continuoustime plant with a continuous-time controller. The paper [5] gave sufficient conditions of network sampling rate under which the original non-networked system remains
(1)
where a and b are constants. Moreover, we are only interested in unstable systems, which implies a > 0. There are two common NCS models as shown in Fig.1 and Fig. 2. The two structures have much in common: The states of continuous-time system are sampled with the sampling 978-0-7695-4296-6/11 $26.00 © 2011 IEEE DOI 10.1109/ICMTMA.2011.493
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stable when the control loop is closed over the network. Later in [6], the authors provided a weaker condition to determine an acceptable range of sampling periods in a NCS. In the work of [7], researchers established a conservative stability result for NCS via a new Lyapunov function and a relaxed condition. For nonlinear systems, this issue is more challenging. Scholars [8] studied the problem of stabilizing a nonlinear continuous-time system via sampled encoded measurement of states under certain conditions. The paper [4] developed a feedback control with packets dropping network links for the structure of Model 2 for 1-D linear system. Recently in [9], the authors gave a rule to decide maximum sampling period for linear NCS, and provided a condition for detecting if the continuous-time controller can still stabilize the NCS in the presence of continuous states loss. Our contribution. In this paper we consider three problems in NCS. First, for system (1), we compare which of Model 1 or Model 2 would perform better. Second, if the sampled controller remains the same as in the continuous-time system, we determine the maximum length of continuous data packets loss a specific linear system can tolerate. Third, for a given sampling rate, we design a controller so that the system can survive as long as possible in the presence of continuous data packets loss. The rest of the paper is organized as follows: In Section II, we discuss the model choice for different linear systems, and give a method to determine the maximum length of tolerable continuous data packets loss. In section III, we develop an algorithm for designing network controller for the NCS which can make the NCS tolerate more continuous data packets loss. Examples are also presented in this section. We conclude the paper in Section IV.
Now we consider data packets loss. First we consider Model 1. Assume uk = −Kxk is already stored in ZOH, ˜ xk+1 = (A˜ − Bk)x k.
If xk+1 is lost, the controller will still be −Kxk . Thus 2T ˜ k+1 − BKx ˜ xk+2 = Ax eas dsΩ + 1)xk . (7) k =( Define 0 eas ds = Φ2 , then we have xk+2 = (Φ2 Ω+1)xk . Recursively, we know after the last received control input data at time step k, if there are m consecutive packets loss, the state at time step k + m can be presented as: xk+m = (Φm Ω + 1)xk (8) mT as where Φm = 0 e ds. Next we analyze Model 2. Assume xk or uk is lost in the system, then the control input is 0. Thus, ˜ k xk+1 = Ax
˜ k+1 = A˜2 xk . xk+2 = Ax Thus we can conclude that, after the last received control input data at time step k, if there are m consective packets loss, the state at time step k + m is: xk+m = A˜m xk
xk+m+n = (ΦΩ + 1)n (Φm Ω + 1)xk ˜m
where A
(11)
, and mT
eas dsΩ + 1 =
0
1 maT e Ω + 1. a
(12)
For Model 2, we have xk+m+n = (ΦΩ + 1)n A˜m xk .
(13)
For unstable linear systems with a ≥ 1, because Ω = a−bK is designed to be negative to stablize the system, we have 1 maT e < emaT . Ω + 1 a
(2)
(3)
Based on the above relation, we conclude that, for a system with fixed sampling period, Model 1 performs better than Model 2, which is stated in the following theorem. Theorem 2.1: For a certain number of consectively received packets, Model 1 can tolerate more packets of consective loss than Model 2 with respect to the same controller. According to the above discussion, for a given linear system, we are able to estimate the relations between m and
a aT b (e
If there is no data loss in the network, we can conclude that xk+n = (ΦΩ + 1)n xk
=e
maT
Φm Ω + 1 =
˜ = where A˜ = e , B − 1), and uk = −Kxk . By T as defining Φ = 0 e ds and Ω = a − bK, (3) can be rewritten as (4) xk+1 = (ΦΩ + 1)xk . aT
(10)
Let m represent the number of consective packets that are lost, and n be number of received controller packets. For Model 1, we have
The system is equivalent to the following system : ˜ k + Bu ˜ k xk+1 = Ax
(9)
suppose xk+1 is also lost during transferring. At time step k + 2, we have
For controllable continuous-time linear system (1), we can design a control gain K such that a − bK is asymptotically stable. However, computer control signals are discrete thus in real applications we use a constant controller during each sampling period. In other words, we have the following closed-loop system, ∀t ∈ [kT, (k + 1)T ].
0
2T
II. M ODEL C HOICES IN THE P RESENCE OF C ONSECTIVE S TATES L OSS
x˙ = ax − bKxk ,
(6)
(5)
842
n under a proper sampling period by applying the following restrictions for Model 1 and Model 2 systems respectively:
4
0
x
|(ΦΩ + 1)n (Φm Ω + 1)m | < 1, |(ΦΩ + 1)n A˜m | < 1 (14)
−2 −4 0
Remark 2.1: Note that the more unstable the linear system is (a >> 1), the better performance Model 1 offers. Remark 2.2: For 0 < a < 1, Model 2 may be advantageous, but we may still be able to stabilize Model 1 by modifying the value of Ω. Now, we consider the tolerable maximum degree of consectkve state loss between sensor and controller, and/or between controller and actuators for a ≥ 1. Note that the controller is generated by the control law of the original continuous-time system. Theorem 2.2: For 1-D linear system (1), with sampling period T , the maximum degree of consective data loss m can be calculated by: ln(Ω − 2a) − ln Ω (15) m= aT
100
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Model 2(Zero input)
x
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Figure 3. Comparison between Model 1 (7 received packets can recover from 10 steps of continuous losses) and Model 2 (7 received packets can only recover from 8 steps of continuous losses).
where r ≤ 1 and C is a constant. Equation (18) guarantees that by using our newly designed network controller, the required number of received data packets will not be dramatically increased in the presence of continuous states loss. Let ΦΩ + 1 = ε, and a − bK = −c, where c > 0. We assume if there is no state loss, the hybrid system will be stable, then we have |ε| < 1. According to (11), in order to guarantee that the NCS is still stable in the presence of consective state loss, the relation between n and m is:
1 1 aT (e − 1), Φm = (enaT − 1) a a when the new controller is lost in the network, the controller stored in ZOH is still able to stabilize the system if Φm satisfies this condition: ln(Ω − 2a) − ln Ω (16) |Φm Ω + 1| < 1 ⇒ m < aT Remark 2.3: For a 1-D linear system, there are n received samples (not necessarily consective) and m continuous losses in a finite length of time steps. We can determine whether the controller can stabilize the system in this period using |(ΦΩ + 1)n (Φm Ω + 1)| < 1 Φ=
|(ΦΩ + 1)n (Φm Ω + 1)| < 1 c a a c or |ε|n {1 + − [ + 1 − ε ]m } < 1 (19) a a c c One sufficient condition for ensuring Equation (19) is 1 c |ε|n (1 + ) < a 2 a 1 nc a |ε| [ + 1 − ε ]m < a c c 2 From (20) and (21), we obtain: 1 c n < lg 2(1 + )/ lg a |ε| c a a 1 n < (lg 2 + m lg( + 1 − ε ))/ lg a c c |ε|
Example 2.1: Consider the following system: (17)
(20) (21)
(22) (23)
From (23), if we want to make n < rm + C, where r ≤ 1 and C is a constant, we would like to find the parameter c such that lg( ac + 1 − ε ac ) ≤1 (24) 1 lg |ε|
with sample period T = log10 23 . for Model 1, by using Theorem 2.2, we can compute several sets of m and n. For example, we choose n = 7, m = 10. For the same system with Model 2, the same n can only tolerate 8 steps of continuous losses. The comparison is shown in Fig. 3.
where
III. N ETWORK C ONTROLLER D ESIGN
c c − eaT . (25) a a To solve c, we have to consider two cases. Case 1: ε > 0. Now (24) can be re-written as: a a a ε2 − ( + 1)ε + 1 ≥ 0 ⇒ ( ε − 1)(ε − 1) ≥ 0 c c c ε = ΦΩ + 1 = 1 +
Now we present a controller design method such that the resulting controller exhibit the following characteristics: the necessary number of received packets, n, and the tolerated number of consective packets loss, m, satisfies: n < rm + C
50
6
where Ω = a − bK is negative. Proof. As we discussed above, the length of continuous loss of control input data (namely m) has a tight relation with the sampling period. In Equation (11),
x˙ = x + u, u = −2x
Model 1(ZOH)
2
(18) 843
Because the assumption |ε| < 1, ε − 1 < 0 is confirmed. So, it requires ac ε − 1 ≤ 0, which implies that ε ≤ ac . Thus (25) becomes c c a c 1 + − eaT ≤ , ⇒ c ≥ aT (26) a a a e On the other hand, ε > 0 gives the upper-bound: c c a (27) ε = 1 + − eaT > 0 ⇒ c < aT a a e −1 Thus we have: a a (28) ≤ c < aT eaT e −1 Case 2: ε < 0. Now (24) can be re-written as a a (29) ε2 − ( + 1)ε − 1 ≤ 0. c c Taking advantage of (25), we have:
500
x
−500 −1000 −1500 0
50
100
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250 time
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6
2
x 10
"Over−weighted" controller
x
1 0 −1 −2 0
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Figure 4. Comparison between proper NCS controller (7 received packets can recover 30 consective packets losses) and ‘ ‘over-weighted controller” (7 received packets cannot recover from ≥5 packets of consective packets losses).
R EFERENCES
a(e−aT + 1) c≤ (30) eaT − 1 to guarantee n < rm + C, where r ≤ 1. Similarly, ε < 0 gives the lower-bound: c a c (31) ε = 1 + − eaT < 0 ⇒ c > aT a a e −1 Thus we can choose the controller as:
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a(e−aT + 1) a < c ≤ (32) eaT − 1 eaT − 1 Note that ε = 0 makes xk+1 = 0 directly. Theoretically, it still stabilizes the NCS. In summary, we obtain the following result for choosing choosing controller: Theorem 3.1: To guarantee the following relation:
[4] C. N. Hadjicostis and R. Touri, Feedback control utlizing packet dropping network links. Proc. of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, December 2002, pages: 1205-1210. [5] J. Zhang, M. S. Branicky, and S. Phillips, Stability of networked control systems. IEEE Control Systems Magazine, February 2001, pages:84-99.
n < rm + C for |r| ≤ 1, the controller gain K should satisfy the following condition: a(1 + eaT ) a(e−aT + eaT ) ≤K≤ aT be b(eaT − 1)
New NCS controller
0
[6] F. Lian, J. Moyne and D. Tilbury, Network Design Consideration for Distributed Control Systems. IEEE Transactions on Control Systems Technology, vol.10(2), March 2002, pages:297-207.
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[7] L. Hu, T. Bai, P. Shi and Z. Wu, Sampled-data control of networked linear control systems. Automatica, vol.43, 2007, pages:903-911.
Example 3.1: In Example 2.1, under the controller uk = −2xk , every 7 received packets can recover as many as 10 packets of consective data packets loss. Applying Theorem 3.1 to Example 2.1, we obtain a range of available controller gains: 1.8385 < K ≤ 9.5485. Let’s set K = 8 as an example. To illustrate the usefulness of our theorem, we take K = 10, which is beyond our tolerable maximum number of packets loss in this example. Fig.4 shows the comparison between using proper controller and using “over-weighted” controller.
[8] D. Liberzon and J. P. Hespanha, Stabilization of Nonlinear Systems With Limited Information Feedback. IEEE Transactions on Automatic Control, vol.50(6), June 2005, pages:910915. [9] C. Qian, W. Tian and S. Xu, Stabilization of Nonlinear Systems With Limited Information Feedback. Proc. of 2010 IEEE/ASME International Conference on Mechatronics and Embedded Systems and Applications (MESA), Qingdao, China, July 2010, pages:556-570.
ACKNOWLEDGMENT This work was supported in part by the Collaborative Research Seed Grant from the University of Texas at San Antonio. 844
