Decentralized Event-Triggered Control for Sampled ...

2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013

Decentralized Event-triggered Control for Sampled-data Systems with Asynchronous Sampling Yanpeng Guan, Qing-Long Han∗ and Chen Peng Abstract— This paper proposes a new decentralized eventtriggered communication scheme based on asynchronous sampling. The event-triggered communication scheme does not depend on the full-order state of the system. Several spatially distributed sensor nodes are employed to collect the state data. Each node transmits the samples according to a subcommunication scheme. An L2 controller design method is developed for the decentralized event-triggered control system, which can be used to codesign of the decentralized eventtriggered communication scheme and L2 controller. A numerical example illustrates the merit and effectiveness of the proposed approach.

I. I NTRODUCTION Event-triggered control has received increasing interest in the past few years. It is shown in some experimental results that the average sampling/transmission interval can be increased by an event-triggered sampling technique and thus the limited resource may be saved [1]–[6]. Nowadays, decentralized control attracts renewed attention due to the appearance of reliable wireless network transmission and low cost microprocessors [7]. There are many large scale systems applications where a group of physically distributed sensors are employed to measure the system output. Several control problems for decentralized control systems have been investigated [8]–[11]. For efficient use of the limited transmission resources (e.g. battery power and/or network bandwidth), it is nature to introduce an event-triggered transmission mechanism into decentralized control implementation to reduce some unnecessary transmissions. A challenge to such decentralized event-triggered control is that the complete measured data of the system output is not available to any of the sensor nodes. To deal with this problem, effort has been made recently [12]–[15]. Based on centralized event-triggered control, a group of adjustable parameters is introduced to develop a set of sub-triggering conditions [12], which ensures that the error of the full system state is upper bounded by a centralized event-triggering threshold. It is noted that although in [12] This work was supported in part by the Australian Research Council Discovery Projects under Grant DP1096780, the Research Advancement Award Scheme (January 2010 - December 2012) at Central Queensland University, Australia; and the Natural Science Foundation of China under Grants 61074024 and 61273114. ∗ Corresponding author, Tel: +61 7 4930 9270; E-mail:

[email protected] Y. Guan and Q.-L. Han are with the Centre for Intelligent and Networked Systems and School of Engineering and Technology, Central Queensland University, Rockhampton, QLD 4702, Australia

[email protected] C. Peng is with the School of Mechatronic Engineering and Automation, Shanghai University, 200072, PR China [email protected]

978-1-4799-0176-0/$31.00 ©2013 AACC

each sensor can locally determine when to trigger the new local sampled data, all sampled data in the centralized controller need to be updated synchronously, in which case each sensor requires a single informing message sent from the controller. In [13], a practical stabilization of nonlinear systems under asynchronous updates is established by using a decentralized event-triggered control strategy, where each component of state is triggered for sampling when the local error exceeds a specific positive scalar. It is noted that in the aforecited results on event-triggered control, system outputs have to be measured continuously by some special real-time detection hardware and they are assumed to be sampled and transmitted at exactly the time instant when the event-triggered threshold is violated. This may pose a critical requirement for the hardware. Another issue is that when such a continuous event-triggered mechanism is applied in decentralized control with asynchronous updates, a separate term of a positive scalar need to be added in the event-triggered threshold condition (as in [12], [14]) to guarantee a nonzero inter-event time for each sensor node. By this way, it generally only leads to a practical stability of the system [12], where the asymptotical stability is achieved if the positive scalars could be adjusted online. In this paper, we propose a new decentralized eventtriggered communication scheme with asynchronous sampling based on periodic sampling as in [16]–[19]. Without loss of generality, each component of the system state is periodically sampled by a separate sensor. The current sampled data is used to update the controller only when the difference between the current sampled data and the latest transmitted one exceeds a pre-designed threshold. Based on the L2 stability analysis of the decentralized event-triggered control system, a design method is developed to obtain a desired L2 controller. By the design result, one can also get a group of feasible parameters for the decentralized eventtriggered communication scheme, which is illustrated in the simulation example. The main contribution of this paper is a decentralized event-triggered communication scheme, which has some advantages compared with the existing decentralized eventtriggered mechanisms. On the one hand, the real-time detection hardware is no longer needed since sampling and transmission mechanisms are comparatively separated in this paper. On the other hand, the separate positive scalar term is no longer required in the proposed event-triggered threshold compared with the results in [13], [14], which leads to that the asymptotical stability of the system can be easily achieved. Compared with the decentralized event-triggered

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control in [12], synchronous transmission is not required in this paper. Another contribution is an L2 controller design method, which can be used to codesign of the decentralized event-triggered communication scheme and L2 controller. The organization of this paper is as follows. Section II presents a decentralized event-triggered communication scheme and formulates the decentralized event-triggered L2 control problem. The L2 stability analysis is given for the decentralized event-triggered closed-loop system in Section III by using the Lyapunov-Krasovskii functional. An L2 controller design method is developed in Section IV. A simulation example is given to demonstrate the effectiveness of the proposed approach in Section V, and this paper is concluded in Section VI.

(t ) Plant

Sensor 1 Sensor 2

ETT 1 ETT 2

Sensor n

ETT n

Actuator

ZOH

Fig. 1.

Controller

A framework of decentralized event-triggered control

scheme tiki +1 h = tiki h + min+ {lh|(xi (tiki h + lh) − xi (tiki h))2 l∈Z

II. P ROBLEM F ORMULATION

> δi x2i (tiki h + lh)}

Consider the following linear time-invariant system x(t) ˙ = Ax(t) + Bu(t) + Bω ω(t), t ≥ 0

(1)

where x(t) ∈ Rn , u(t) ∈ Rm and ω(t) ∈ L2 [0, ∞) are the system state, control input and the exogenous disturbance, respectively; A, B and Bω are constant matrices with appropriate dimensions; and the initial condition of the system (1) is given by x(0) = x0 . As in [12]–[14], it is assumed in this paper that the system state x(t) can not be measured by a centralized sensor node. For clear elaboration, without loss of generality, each component of the system state is measured by one sensor from a group of physically distributed sensor nodes. The results in this paper can be generalized to the case where the sensors are grouped into several nodes as in [14]. The xi (t), the ith component of x(t), is periodically sampled by sensor i with a sampling period h > 0, i = 1, 2, . . . , n. All the n sensors are assumed to have the same sampling rate, but they do not have to start sampling at the same time. In this sense, the whole sampling process is asynchronous. Whether or not the current sampled data is to be transmitted through wireless network to a controller is decided by each separate sensor according to a pregiven communication scheme. It is noted that this kind of asynchronous sampling/transmission mechanism makes system (1) have a advantage of modeling many physical process in reality [7]. For efficient use of the transmission resources (e.g. battery power and/or network bandwidth), we propose a decentralized event-triggered communication scheme to reduce the transmission traffic load. As is shown in Figure 1, each sensor is collocated with an event-triggered transmitter which is used to determine whether or not the current sampled data is transmitted based on the error between the current sampled data and the latest transmitted one. The broadcast release time instants sequence of the ith event-triggered transmitter is denoted as {tiki h}∞ ki =1 , which is generated according to the following communication

(2)

where tiki h is the ki th transmission time instant of the ith transmitter; Z+ is the set of positive integers; δi > 0 is an adjustable factor to determine the threshold of the eventtriggered transmitter. Remark 1: The decentralized event-triggered sampling scheme in [13] is given as tiki +1 = min{t > tiki |(xi (t) − xi (tiki ))2 > ηi }

(3)

where ηi > 0 is introduced to guarantee a nonzero minimum inter-event time. However, the positive scalar term ηi usually only leads to a practical stability of the system, i.e. the trajectory finally stays in a small compact set. It is also shown in [13] that asymptotic stability could be achieved if the positive scalars could be appropriately adjusted online. In [14], a separate positive scalar term is also introduced in the sub-threshold to guarantee a nonzero minimum inter-event time, in addition to a measurement dependent term. It can be seen that this kind of separate positive scalar term is no longer needed in this paper. Remark 2: The decentralized event-triggered communication scheme (2) is employed to reduce some unnecessary data transmissions. Compared to decentralized event-triggered mechanisms presented in [13], [14], the event-triggered mechanism in this paper and the sampling are executed separately. Therefore, the real-time detection hardware is no longer needed. In this paper, we are interested in designing the following controller  T u(t) = K x(t1k1 h) x(t2k2 h) · · · x(tnkn h) , (4) t ∈ [tk h, tk+1 h) where K ∈ Rm×n is to be determined; and tk h = maxi=1,2,...,n {tiki h}, tk+1 h = mini=1,2,...,n {tiki +1 h}. For t ∈ [tk h, tk+1 h), let

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ηki (t)

= t − tk h − max{lh|tik h + lh ≤ t}

eik (t)

= xi (t − ηki (t)) − xi (tiki h),

l∈Z+

(5)

i = 1, 2, . . . , n.(6)

Then the control input u(t) in (4) can be rewritten as

Π14 = −Γ1 + Z1T BK

D1

D2

· · · Dn

n X Di x(t − ηki (t)) − ek (t)), t ∈ [tk h, tk+1 h) (7) u(t) = K(

Π22 = −Z2 − Z2T  Π24 = Γ1 Θ + Z2T BK D1



D2

· · · Dn



i=1

where Di = diag{0, · · · , 1, · · · , 0} | {z }

(8)

Π44 = −Γ2 + diag{δ1 D1 , δ2 D2 , . . . , δn Dn }. Proof. It follows from the decentralized event-triggered communication scheme (2) that

i

ek (t) = [e1k (t), e2k (t), . . . , enk (t)]T .

x(t) ˙ = Ax(t) + BK

n X

Di x(t − ηki (t)) − BKek (t)

i=1

+Bω ω(t),

eTk (t)ek (t)

(9)

The error-dependent closed-loop system can be obtained as

t ∈ [tk h, tk+1 h). (10)

We supplement the initial condition of the state on [−h, 0] as x(t) = ϕ(t), t ∈ [−h, 0], where ϕ(t) is a continuous function on [−h, 0] and ϕ(0) = x0 . The purpose in what follows is to design an L2 controller in the form of (4) such that (i) the closed-loop system (10) with w(t) = 0 is asymptotically stable; (ii) the L2 gain from w to x is less than a given scalar γ > 0, that is, under zero initial condition, kx(t)k2 < γ kω(t)k2 for any nonzero ω(t) ∈ L2 [0, ∞).

N2 − M 2

Γ2 = diag{M1 − N1 −

· · · Nn − M n

N1T , · · · , Mn

− Nn −

1 X 1 X

n X

Π12 = A Z2 + P −

Z1T

+

n X i=1

θi M i

1 X

αθ1 θ2 ···θn (t) 6= 0

(14)

θn =0

Γ(h − η1 (t), h − η2 (t), . . . , h − ηn (t)) =

1 X 1 X

···

θ1 =0 θ2 =0

×

1 X

αθ1 θ2 ···θn (t)Γ(θ1 h, θ2 h, . . . , θn h). (15)

θn =0

It follows from (11), (14) and (15) that (16)

By the similar method and (12), one can see that Π(η(t)) := Π(h − η1 (t), h − η2 (t), . . . , h − ηn (t)) < 0. (17) Choose the following Lyapunov-Krasovskii functional candidate Z t T V (t, x(t)) = x (t)P x(t) + xT (s)Qx(s)ds t−h

+

n X

(h − ηki (t))

i=1 Mi × ∗



xT (t) xT (t − ηki (t))

Ni − M i Mi − Ni − NiT





x(t) x(t − ηki (t))



.(18)

It follows from (16) that there exists a positive scalar κ > 0 such that V (t, x(t)) > κxT (t)x(t). Notice that η˙ ki (t) = 1. Taking the right derivative of V (t, x(t)) with respect to t along the trajectory of (10) yields V˙ (t, x(t)) = 2xT (t)P x(t) ˙ − xT (t − h)Qx(t − h) n X T (h − ηki (t))(xT (t)Mi x(t) ˙ +x (t)Qx(t) + 2



T

NnT }

i=1 i ηk (t))(NiT −

+x (t − Mi )x(t)) ˙ n X  T  x (t) xT (t − ηki (t)) − i=1 Mi × ∗

Mi

i=1

T

···

θ1 =0 θ2 =0

Θ = diag{θ1 I, θ2 I, . . . , θn I} Π11 = AT Z1 + Z1T A + Q + I −

(13)

with Di given by (8). Denote the left side of (11) and (12) respectively as Γ(θ1 h, θ2 h, . . . , θn h) and Π(θ1 h, θ2 h, . . . , θn h). By mathematical induction, one can find that there exist 2n functions αθ1 θ2 ···θn (t) ≥ 0 such that

where N1 − M 1

δi xT (t − ηi (t))Di x(t − ηi (t))

Γ(h − η1 (t), h − η2 (t), . . . , h − ηn (t)) > 0.

In this section, we will analyze the L2 stability of the decentralized event-triggered control closed-loop system (10). A sufficient condition will be given to guarantee the L2 stability of the closed-loop system. Theorem 1: Given a scalar γ > 0, with communication scheme (2), the closed-loop system (10) is finite-gain L2 stable from ω to x with again less than n γ, if there exist real matrices P > 0, Q > 0, Mi = MiT i=1 , {Ni }ni=1 , Z1 , Z2 with appropriate dimensions such that for all θi ∈ {0, 1}, i = 1, 2, . . . , n   Pn P + h i=1 θi Mi hΓ1 Θ > 0 (11) ∗ (hΓ2 − I)Θ + I   Π11 Π12 0 Π14 −Z1T BK Z1T Bω  ∗ Π22 0 Π24 −Z2T BK Z2T Bω     ∗  ∗ −Q 0 0 0   < 0 (12)  ∗  ∗ ∗ Π 0 0 44    ∗  ∗ ∗ ∗ −I 0 ∗ ∗ ∗ ∗ ∗ −γ 2 I 

≤

n X i=1

III. L2 S TABILITY A NALYSIS

Γ1 =



Ni − M i Mi − Ni − NiT

where V˙ (t, x(t)) = lim sups→0+ 6582



x(t) x(t − ηki (t))



(19)

V (t+s,x(t+s))−V (t,x(t)) . s

It can be seen that there exist real nonsingular matrices Z1 and Z2 such that n X Di x(t − ηki (t)) 2(xT (t)Z1T + x˙ T (t)Z2T )(Ax(t) + BK

It is easy to verify that Z˜ T Di Z˜ = Z˜ T Ei EiT Z˜ KDi Z˜ = Fi ,

i=1

−BKek (t) + Bω ω(t) − x(t)) ˙ = 0.

V˙ (t, x(t)) ≤ ξ T (t)Π(η(t))ξ(t) − xT (t)x(t) + γ 2 wT (t)w(t) (21) where Π(η(t)) is given in (17); and  ξ(t)= xT (t) x˙ T (t) xT (t − h) xT (t − ηk1 (t)) · · · xT (t − ηkn (t))

eTk (t) ω T (t)

It follows from (17) that V˙ (t, x(t)) ≤ −xT (t)x(t) + γ 2 ω T (t)ω(t).

T

.

(22)

Since V˙ (t, x(t)) is continuous in time t, for any integer k > 0, we have Z tk h Z tk h ˙ −xT (s)x(s) + γ 2 ω T (s)ω(s)ds. V (s, x(s))ds ≤ 0

0

It is clear that, under zero initial condition, Z ∞ Z ∞ T 2 x (s)x(s)ds ≤ γ ω T (s)ω(s)ds 0

where   ˜2 − M ˜2 · · · N ˜n − M ˜n ˜1 − M ˜1 N Γ˜1 = N ˜1 − N ˜1 − N ˜T, · · · , M ˜n − N ˜n − N ˜T} Γ˜2 = diag{M 1 n   P n T T T ˜ ˜ + Z˜ A − U Z˜ + h M Ω P θ i i i=1 ˜ Π11 = ∗ −Z˜ − Z˜ T     ˜ 1 + U T B F1 F2 · · · Fn −Γ ˜   Π13 = ˜ 1 Θ + B F1 F2 · · · Fn hΓ   Pn T T ˜T ˜ 14 = −U BPn i=1 Fi U Bω Z Π −B i=1 Fi Bω 0 ˜ 35 = diag{Z˜ T E1 , Z˜ T E2 , . . . , Z˜ T En } Π

(23)

holds for any nonzero ω(t) ∈ L2 [0, ∞). When ω(t) = 0, (22) becomes (24)

from which one can conclude the asymptotic stability of the closed-loop system (10). The proof is completed.  Remark 3: One can see from the L2 stability result that if a controller gain matrix K in (7) is pre-given as in [12]–[14], a group of parameters for the decentralized event-triggered communication scheme (2) can be obtained by Theorem 1 to reduce the transmission traffic to a large extent. This will be illustrated by a simulation example in Section V.

˜ 44 = diag{ν 2 I − ν Z˜ − ν Z˜ T , −γ 2 I, −I} Π ˜ 55 = diag{−δ˜1 , −δ˜2 , . . . , −δ˜n } Π Θ = diag{θ1 I, θ2 I, . . . , θn I} P˜ = diag{P˜ , P˜ , . . . , P˜ }

IV. L2 C ONTROLLER D ESIGN Based on the stability analysis result developed in the previous section, we are now in a position to design an L2 controller in the form of (4) such that the closed-loop system (10) is finite-gain L2 stable from ω to x with a gain less than γ. It can be seen from (12) that Z2 in Theorem 1 is nonsingular. Then we have Z1 = U Z2 with U = Z1 Z2−1 . Let Z˜ = Z2−1 . Partition Z˜ and K respectively as  T Z˜ = Z˜1T Z˜2T · · · Z˜nT (25)   (26) K = K1 K2 · · · Kn . Denote

Ei = Di



Fi = Ki Z˜i ,

1 1

··· 1

T

i = 1, 2, . . . , n.

(27) (28)

(30)

By some matrix congruence transformations and linearization techniques, we can obtain the following result. Theorem 2: Given a scalar γ > 0, the closed-loop system (10) is finite-gain L2 stable from ω to x with a gain ˜ ˜ lessn than γ, ifothere nexistonreal matrices P > 0, Q > n n T ˜ ˜ ˜ ˜ 0, Mi = Mi , Ni , {Fi }i=1 , Z and U with i=1 i=1 n on appropriate dimensions and real constants δ˜i > 0 , ν i=1 such that for all θi ∈ {0, 1}, i = 1, 2, . . . , n   P ˜i ˜ 1Θ hΓ P˜ + h ni=1 θi M ˜ 2 Θ + hP(I ˜ − Θ) > 0 (31) ∗ hΓ  ˜  ˜ 14 ˜ 13 Π 0 Π11 0 Π ˜  ∗ −Q 0 0 0     ∗ ˜ ˜ ∗ − Γ2 0 Π35    < 0 (32)  ∗ ˜ 44 ∗ ∗ Π 0  ˜ 55 ∗ ∗ ∗ ∗ Π

0

V˙ (t, x(t)) ≤ −xT (t)x(t)

Fi .

i=1

(20)

Taking (13), (19) and (20) into consideration together, one can get that

xT (t − ηk2 (t))

K Z˜ =

(29) n X

˜ + U T AZ˜ + Z˜ T AT U − Ω=Q

n X

˜ i. M

i=1

The controller parameter matrix K and the decentralized n event-triggered transmitter parameters {δi > 0}i=1 are respectively given by K=

n X i=1

Fi Z˜ −1 ,

δi = 1/δ˜i , i = 1, 2, . . . , n.

(33)

Remark 4: Theorem 2 gives a sufficient condition by which the decentralized event-triggered communication scheme and L2 controller may be designed simultaneously, which is demonstrated in the simulation example in the following section. This implies that Theorem 2 can serve as an effective tool for codesign of the decentralized eventtriggered communication scheme and L2 controller.

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Release interval

TABLE I AVERAGE RELEASE INTERVALS IN THE SIMULATION

The scheme in [13] The scheme in this paper

t¯1 0.121s 0.061s

t¯2 0.073s 0.154s

t¯3 0.056s 0.100s

t¯4 0.057s 0.118s

Consider the following Batch Reactor system [13], [20], [21]   1.380 −0.208 6.715 −5.676  −0.581 −4.290 0 0.675   x(t) x(t) ˙ =  1.067 4.273 −6.654 5.893  0.048 4.273 1.343 −2.104  T 0 5.679 1.136 1.136 + u(t). (34) 0 0 −3.146 0

0 0

B. Design of L2 Controller and Decentralized Eventtriggered Communication Scheme In this subsection, we include the following disturbance to illustrate the proposed design method. 

0

0 0

15

20

2 x x

1 0 0

5 10 15 Release time (Second)

3 4

20

Release time intervals of the system without disturbance.

State responses

x x

60

x

1 2 3 4

40 20 0 −20 0

Fig. 3.

Bω =

2

5

10 Time (Second)

15

20

δ4 = 0.5

which ensures that system (34) with controller (35) is asymptotically stable. For the same control system (34)-(35), Table I shows the average release time intervals by using the two different decentralized event-triggered communication schemes, both of which guarantee the asymptotical stability of the corresponding event-triggered closed-loop system. Compared with the result in [13], one can see that the average release time intervals in three sensors are increased. And the total average release time intervals in the four sensors are increased from 0.307s to 0.433s. Figure 2 shows the release time instants distribution in the four sensor nodes by decentralized eventtriggered communication scheme 2. The corresponding state response is plotted in Figure 3.

1 , 1+t

10

80

With this controller, by Theorem 1, we can get a group of parameters for decentralized event-triggered communication scheme (2) as

ω(t) =

5

x

For comparisons, we take the following state feedback controller which is given in [13]:   0.1006 −0.2469 −0.0952 −0.2447 K= . (35) 1.4099 −0.1966 0.0139 0.0823

δ3 = 0.4,

1

100

A. Average Release Intervals

δ2 = 0.9,

x

0.5

Fig. 2.

We will first use this example to compare the proposed approach in this paper with an existing decentralized eventtriggered control strategy presented in [13]. Then this unstable system will be used to demonstrate the effectiveness of the proposed L2 controller design approach.

δ1 = 0.1,

x 1

Release interval

V. S IMULATION E XAMPLE

1.5

1



.

(36)

State response of the closed-loop system without disturbance.

We choose U = 0.3I, the sampling period h = 10ms. By using Theorem 2, one can find that the system with controller (4) is finite-gain L2 stable with a gain less than γ = 20. A feasible solution to (31)-(32) leads to the following controller parameter matrix and the decentralized event-triggered communication scheme parameters   1.0923 −0.73439 −0.48149 −1.8008 K= 5.4158 0.24353 1.9154 −0.67662 δ1 = 0.015, δ2 = 0.014, δ3 = 0.014, δ4 = 0.015. With controller (4) and decentralized event-triggered communication scheme (2), the event-triggered broadcast release time intervals in all the 4 sensors is shown in Figure 4. The state response of closed-loop system is shown in Figure 5. Within the simulation period Ts = 20s, each component of the system state is sampled 2000 times while the numbers of transmitted measurements are 164, 136, 158, 139, respectively. It is shown that the average release time intervals in the 4 sensors are respectively t¯1 = 0.121s, ¯t2 =

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Release interval Release interval

R EFERENCES

2 x x

1 0 0

5

10

15

1 2

20

1.5 x 1

x

3 4

0.5 0 0 Fig. 4.

5 10 15 Release time (Second)

20

Release time intervals of the system with disturbance.

2

State responses

x 1.5

x

1

x x

0.5

1 2 3 4

0 −0.5 −1 −1.5 0 Fig. 5.

5

10 Time (Second)

15

20

State response of the closed-loop system with disturbance.

0.146s, ¯t3 = 0.126s, ¯ t4 = 0.143s, all of which are much larger than the sampling period h = 0.01s. Compared with the periodic transmission scheme, it is clear that a large proportion of the required transmission resources may be saved, which shows the effectiveness of the approach. VI. C ONCLUSION A decentralized event-triggered communication scheme based on asynchronous sampling is proposed in this paper. An input delay approach is employed to model and analyze the decentralized event-triggered control system. A feasible decentralized event-triggered communication scheme can be obtained by the L2 analysis result. An L2 controller design method is developed to get a desired controller by which the L2 stability of the decentralized event-triggered control closed-loop system is guaranteed. A numerical example shows the effectiveness of the proposed approach.

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