Event-triggered dynamic output feedback control for ... - IEEE Xplore

www.ietdl.org Published in IET Control Theory and Applications Received on 25th March 2013 Revised on 5th September 2013 Accepted on 23rd October 2013 doi: 10.1049/iet-cta.2013.0253

ISSN 1751-8644

Event-triggered dynamic output feedback control for networked control systems Xian-Ming Zhang, Qing-Long Han Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton, QLD 4702, Australia E-mail: [email protected]

Abstract: This study is concerned with the event-triggered control for networked control systems via dynamic output feedback controllers (DOFCs). The output measurement signals of the physical plant are sampled periodically. An output-based discrete event-triggering mechanism is introduced to choose those only necessary sampled-data packets to be transmitted through a communication network for controller design. Under this event-triggering mechanism, the resultant closed-loop system is first modelled as a linear system with an interval time-varying delay. Then a novel stability criterion is established by employing the Lyapunov–Krasovskii functional approach. Based on this stability criterion, a new sufficient condition is derived to codesign both the desired DOFCs and the event-triggering parameters. Finally, a satellite control system is taken to show the effectiveness of the proposed method.

1

Introduction

Networked control systems (NCSs) have received considerable attention during the past decade due to the fact that the signal transmission between control system components is completed through a communication network rather than point-to-point wirings. The introduction of a communication network into a control system can bring several advantages such as reduction of costs and ease of installation and maintenance. However, it also causes some issues on account of limited network bandwidth, usually leading to performance degradation or even instability of an NCS. In recent years, much effort has been made on stability analysis and control synthesis for NCSs, and one can refer to [1–7] and references therein. In an NCS with a continuous-time physical plant, suppose that the signals of the physical plant are first sampled periodically and then released to a communication network for transmission to a controller node. When the sampling period is small, the number of sampled signal packets to be transmitted will increase considerably, which possibly leads to an overloaded communication network and a congested network traffic. As a result, longer network-induced delays and more packet dropouts will frequently occur, inevitably degrading system performance or making an NCS unstable. On the other hand, in some certain time intervals when there is little fluctuating of the measurement signals of the physical plant, a number of sampled signal packets are unnecessary to be transmitted for control design. If those unnecessary signal packets are discarded directly, then a good quality of service of the communication network can be ensured and thus a satisfactory system performance of the NCS can be guaranteed in premise. Therefore how to choose those necessary signal packets to be transmitted for 226 © The Institution of Engineering and Technology 2014

controller design is a significant topic in the application of NCSs. Recently, an event-triggering mechanism has been proposed in the digital implementation of real-time control systems, by which the control task is executed only if a state-dependent triggering condition is violated. The conspicuous advantage of the event-triggering mechanism is that it can significantly reduce the number of control task executions while retaining a satisfactory closed-loop performance. In recent years, the event-triggering mechanism has been applied to NCSs to save the limited communication resources, and the readers can be referred to [8–17]. However, a great number of results are based on such an assumption that the full states of the plant can be measured [18–20]. However, when the states of the plant are not available for feedback, those aforementioned results are thus inapplicable. In order to extend the event-triggering mechanism to the case that the states of the plant cannot be measurable, very recently, output-based event-triggered control has been proposed to investigate the dynamic output feedback control for NCSs. By introducing output-dependent event-triggering conditions, the stability and passivity of the NCS were studied in [21, 22], respectively. Nevertheless, on the one hand, since the defined event-triggering conditions in [21, 22] depend closely on the instant system output, the system output needs to be monitored continuously; on the other hand, desirable dynamic output feedback controllers (DOFCs) are difficult to be designed but are presumed to be known a priori, which limits the application scope of the obtained results. This paper focuses on the event-triggered dynamic output feedback control for an NCS. First, an output-based discretetime event-triggering condition is introduced to choose those necessary signal packets to be transmitted for controller IET Control Theory Appl., 2014, Vol. 8, Iss. 4, pp. 226–234 doi: 10.1049/iet-cta.2013.0253

www.ietdl.org design. Since this triggering condition is related only to the system output in discrete instants, there is no need to monitor the system output continuously. Second, the closed-loop system subject to an event-triggering condition is modelled as a linear system with an interval time-varying delay. By employing the Lyapunov–Kraovskii functional approach, a novel stability criterion is formulated. Based on this stability criterion, both suitable DOFCs and event-triggering parameters can be co-designed provided that a set of linear matrix inequalities (LMIs) are satisfied. Finally, a satellite system is used to show the effectiveness of the proposed method. Notation: The notation throughout this paper is standard. The symbol ‘’ denotes the symmetric term in a symmetric matrix.

2

Problem formulation

The problem of event-triggered dynamic output feedback control considered in this paper is shown in Fig. 1, where each component is stated in the following. 2.1

The plant and the sampler

The plant is a continuous-time linear system described by

x˙ (t) = Ax(t) + Bu(t), x(0) = x0 y(t) = Cx(t)

(1)

where x(t) ∈ Rn and y(t) ∈ Rm are the system state vector and measurement output vector, respectively; the system matrices A, B and C are known real matrices with compatible dimensions; and x0 is the initial condition. The sampler samples the measurement output signal y(t) every other h seconds. The sampled signal y(lh) (l ∈ {1, 2, . . .}) together with its time stamp l is encapsulated into a sampled-data packet (l, y(lh)) for the transmission through a communication network. 2.2

Event-trigger

The event-trigger consists of a register and a comparator. The register stores information on the last released datapacket (ik , y(ik h)) (ik ∈ {1, 2, . . .}). The comparator is used to check if the current sampled-data packet (ik + j, y((ik + j)h)) ( j ∈ {1, 2, . . .}) satisfies the following triggering condition 1 2

1 2

[ y(ik h + jh) − y(ik h)]2 ≤ ε y(ik h)2

(2)

where ε > 0 is a threshold and > 0 is a weighting matrix. Once the triggering condition (2) is violated, the current sampled-data packet is immediately released and transmitted to the zero-order-hold (ZOH) through a communication network; otherwise, it is discarded off right away. Clearly,

DOFC

u (t )

Plant

y (t )

~ y (t )

ZOH

the next release time instant ik+1 h of the event-trigger can be determined by 1 ik+1 h = ik h + min jh| 2 [ y(ik h + jh) − y(ik h)]2 j≥1 1 (3) > ε 2 y(ik h)2 Remark 1: The triggering condition in (2) is an outputbased discrete event-triggering condition. Compared with the output-based continuous-time event-triggering conditions in [21, 22], there is no need to introduce extra hardware to monitor the instantaneous measurement output signal y(t). It is worth pointing out that the idea of the discrete eventtriggering mechanism was proposed in [14, 15, 23] to deal with the state-feedback control. This paper can be regarded as an extension of the discrete event-triggering mechanism to the dynamic output feedback control. 2.3

The network

In this paper, suppose that the data transmission over the network is in a single-packet manner. Under the above discrete event-triggering mechanism, a number of sampleddata packets will be discarded, which means that the network loads will be greatly reduced. Thus, a good quality of service of the network can be possibly ensured. Hereafter, it is assumed that no packet dropouts and no packet disorders will occur during the data transmission over the network, but network-induced delays are unavoidable. 2.4

The ZOH

The ZOH is set to be event-driven. Once a certain data packet arrives at the ZOH, the ZOH immediately sends information on the packet to the controller to be designed. Denoted by tk (k = 1, 2, . . .) the time instant at which the data packet (ik , y(ik h)) arrives at the ZOH. Then it is clear that t1 < t2 < · · · < tk < · · · because of the assumption that packet dropouts and packet disorders do not occur. As a result, one can see clearly that the packet (ik , y(ik h)) is delayed by τk := tk − ik h in the transmission through a communication network. For convenience of presentation, denote τm := min{τk |k = 1, 2, . . .} and τM := max{τk | k = 1, 2, . . .}. 2.5

The DOFC

From the previous description, the input signal y˜ (t) of the DOFC to be designed is given by y˜ (t) = y(ik h),

t ∈ [tk , tk+1 )

Let ρk := min{ j|tk + jh ≥ tk+1 , j = 0, 1, 2, . . .}

y(ik h)

Sampler

Event-trigger

Fig. 1 Diagram for event-triggered dynamic output feedback control for an NCS IET Control Theory Appl., 2014, Vol. 8, Iss. 4, pp. 226–234 doi: 10.1049/iet-cta.2013.0253

(5)

Then ρk ≥ 1 because tk < tk+1 . The interval [tk , tk+1 ) can thus be written as [tk , tk+1 ) =

ρk

y (lh )

Network

(4)

Ij

(6)

j=1

where Ij = [tk + ( j − 1)h, tk + jh),

j = 1, 2, . . . , ρk − 1

Iρk = [tk + (ρk − 1)h, tk+1 ) 227 © The Institution of Engineering and Technology 2014

www.ietdl.org Now, we define two functions η(t) and e(t) on [tk , tk+1 ) as ⎧ t ∈ I1 t − ik h, ⎪ ⎪ ⎪ ⎨t − ik h − h, t ∈ I2 η(t) := (7) . .. . ⎪ ⎪ . . ⎪ ⎩ t − ik h − (ρk − 1)h, t ∈ Iρk ⎧ y(ik h) − y(ik h), t ∈ I1 ⎪ ⎪ ⎪ ⎨y(ik h) − y(ik h + h), t ∈ I2 e(t) := (8) .. .. ⎪ ⎪ . . ⎪ ⎩ y(ik h) − y(ik h + (ρk − 1)h), t ∈ Iρk It is clear that t ∈ [tk , tk+1 )

(9)

Thus, we have a new expression of y˜ (t), which is related to the error function e(t) and the delayed measurement output y(t − η(t)). The delay η(t) satisfies τm ≤τk ≤ η(t) < h + τk ≤ h + τM

(10)

In this paper, we are interested in designing a stabilising DOFC of the following form x˙ c (t) = AK xc (t) + BK xc (t − η(t)) + CK y˜ (t) (11) u(t) = DK xc (t), tk ≤ t < tk+1 where AK , BK , CK and DK are constant real matrices to be determined. Remark 2: The idea of defining two functions η(t) and e(t) is taken from [14, 15]. However, by introducing the minimum integer ρk in (5), the derivative of two functions is much easier than that in [14, 15], where two cases are discussed. The illustration of η(t) and e(t) by a diagram can be seen in [18]. 2.6 The problem of event-triggered dynamic output feedback control Set x˜ (t) := col{x(t), xc (t)}. Then the resultant closed-loop system of the plant (1) with the controller (11) is given as x˙˜ (t) = A0 x˜ (t) + A1 x˜ (t − η(t)) + A2 e(t), tk ≤ t < tk+1 (12)

A0 :=

In this section, we will present a stability criterion for the closed-loop system (12) subject to (13) by employing the Lyapunov–Krasovskii functional approach. Proposition 1: For given scalars h > 0, τm ≥ 0, τM (≥ τm ), the closed-loop system (12) subject to (13) is asymptotically stable if there exist ε > 0, real matrices > 0, P > 0, Qi > 0, Ri > 0 (i = 1, 2) and S of appropriate dimensions such that

y˜ (t) = y(ik h) = e(t) + y(t − η(t)),

where

3 Stability analysis of the event-triggered closed-loop system

A BDK , 0 AK

A1 :=

0 CK C

0 , BK

A2 :=

0 CK

eT (t)e(t) ≤ ε 2 [e(t) + CE x˜ (t − η(t))]T

ϒ1 :=

11

(13)

12 0, τm and τM satisfying τM ≥ τm ≥ 0, co-design (ε, ) and (AK , BK , CK , DK ) such that the resultant closedloop system (12) subject to (13) is asymptotically stable. 228 © The Institution of Engineering and Technology 2014

S ≥ 0, R2

t

and e(t) satisfies the following constraint for tk ≤ t < tk+1 × [e(t) + CE x˜ (t − η(t))]

R2

−τm

t+θ

+ (η¯ − τm )

t−τm t−η¯

x˜ T (s)Q2 x˜ (s) ds

x˙˜ T (s)R1 x˙˜ (s) ds dθ −τm t −η¯

x˜˙ T (s)R2 x˙˜ (s) ds dθ

t+θ

Taking the time derivative along the trajectory of system (12) yields V˙ (t, x˜ (t)) = V˙ 1 (t, x˜ (t)) + V˙ 2 (t, x˜ (t)) + V˙ 3 (t, x˜ (t))

(17)

IET Control Theory Appl., 2014, Vol. 8, Iss. 4, pp. 226–234 doi: 10.1049/iet-cta.2013.0253

www.ietdl.org where

(18)

Proposition 1 presents a stability condition for the eventtriggered closed-loop system (12) subject to (13). In the next section, we will focus on solving the problem of eventtriggered dynamic output feedback control on the basis of Proposition 1.

(19)

4

V˙ 1 (t, x˜ (t)) = 2˜xT (t)P[A0 x˜ (t) + A1 x˜ (t − η(t)) + A2 e(t)] ¯ 2 x˜ (t − η) ¯ V˙ 2 (t, x˜ (t)) = x˜ T (t)Q1 x˜ (t) − x˜ T (t − η)Q + x˜ T (t − τm )(Q2 − Q1 )˜x(t − τm ) V˙ 3 (t, x˜ (t)) = x˙˜ T (t)[τm2 R1 + (η¯ − τm )2 R2 ]x˙˜ (t) t x˙˜ T (s)R1 x˙˜ (s) ds − τm t−τm

− (η¯ − τm )

t−τm

x˜˙ T (s)R2 x˙˜ (s) ds

t−η¯

(20)

Use Jensen inequality to obtain t t−τm

≤ −[˜x(t) − x˜ (t − τm )]T R1 [˜x(t) − x˜ (t − τm )] Since

R2 S S T R2

− (η¯ − τm )

(21)

≥ 0, it follows that

t−τm t−η¯

Note that the controller gain matrices AK , BK , CK and DK cannot be directly computed based on Proposition 1 due to the fact that they are coupled with the Lyapunov matrix P in the matrix inequality ϒ1 < 0 in (14). In order to solve out the controller gain matrices, we first state and establish the following result. Proposition 2: For given scalars h > 0, τm > 0, and τM (≥ τm ), consider the closed-loop system (12) subject to (13). The following two conditions are equivalent

x˜˙ T (s)R1 x˙˜ (s)ds

− τm

Event-triggered DOFC design

x˜˙ T (s)R2 x˙˜ (s) ds

(i) There exists a scalar ε > 0 and real matrices > 0, P > 0, Qi > 0, Ri > 0 (i = 1, 2) and S of compatible dimensions such that the matrix inequalities in (14) are satisfied. (ii) There exists a scalar ε > 0 and real matrices > 0, ˜ i > 0, R˜ i > 0 (i = 1, 2), S˜ and Wj X > 0, Y > 0, Q ( j = 1, . . . 4) of compatible dimensions such that

X S˜ ≥ 0, Z := R˜ 2

21 22 ϒ2 := 0 such that V˙ (t, x˜ (t)) < −ξ T (t)ξ(t) ≤ − x˜ T (t)˜x(t). Therefore one can conclude that system (12) is asymptotically stable if the matrix inequalities in (14) hold. IET Control Theory Appl., 2014, Vol. 8, Iss. 4, pp. 226–234 doi: 10.1049/iet-cta.2013.0253

I > 0, Y

Proof: (i) ⇒ (ii): Suppose that there exist real matrices > 0, P > 0, Qi > 0, Ri > 0 (i = 1, 2) and S such that R2 RS2 ≥0 and (14) are satisfied. Now partition P as P = NYT YN1 , where Y , N , Y1 ∈ Rn×n . It is clear that Y > 0 and Y1 > 0. If N is singular, then there exists a sufficient 00 > 0 such that N + 0 I is non-singular and , P + 0 I 00 I , Q1 , Q2 , R1 , R2 , S satisfies (14). Thus, without loss generality, we suppose that N is non-singular. Introduce X > 0 such that Y1 = N T (Y − X −1 )−1 N . Clearly, Y − X −1 > 0 because of Y1 > 0 and non-singular 229 © The Institution of Engineering and Technology 2014

www.ietdl.org N . By Schur complement, Y − X −1 > 0 ⇐⇒ Z > 0, where Z is defined in (26). Moreover, we have that

X I I Y = P (28) N −1 (I − YX ) 0 0 NT Set

X I , N −1 (I − YX ) 0

J1 :=

J2 :=

I Y 0 NT

Proposition 3: For given scalars h > 0, τm > 0, and τM (≥ τm ), the closed-loop system (12) subject to (13) is asymptotically stable if there exist real matrices > 0, X > 0, ˜ i > 0, R˜ i > 0 (i = 1, 2), S˜ and Wj ( j = 1, . . . , 4) of Y > 0, Q appropriate dimensions such that (26). Moreover, the DOFC (11) is equivalent to (34) with ⎧ −1 ⎨A = (W4 − YAX − YBW1 )(I − YX ) −1 B = (W3 − W2 CX )(I − YX ) ⎩ C = W2 , D = W1 (I − YX )−1

(29)

Then J1 and J2 are non-singular. Denote T := diag{J1 , J1 , J1 , J1 , I , I , J2 , J2 } and

⎧ −1 ⎨W1 := DK N (I − YX ), W2 := NCK W := W2 CX + NBK N −1 (I − YX ) ⎩ 3 W4 := YAX + YBW1 + NAK N −1 (I − YX )

(30)

(31)

Performing a congruence transformation on ϒ1 defined in (14) by a non-singular matrix T and after some simple algebraic manipulations, we obtain that T T ϒ1 T = ϒ2

(32)

where ϒ2 is defined in (26). Therefore, if there exist real matrices > 0, P > 0, Qi > 0, Ri > 0 (i = 1, 2) and S such that (14) is satisfied, then there exist real matrices > 0, ˜ i = J1T Qi J1 , R˜ i = J1T Ri J1 (i = 1, 2), S˜ = X > 0, Y > 0, Q J1T SJ1 and W1 , W2 , W3 , W4 defined in (31) such that the matrix inequalities in (26) are satisfied. (ii) ⇒ (i): Suppose that there exist > 0, X > 0, Y > ˜ i , R˜ i (i = 1, 2), S˜ and Wj ( j = 1, . . . , 4) such that the 0, Q matrix inequalities in (26) are satisfied. Then it is clear that the matrix I − YX is invertible because of Z > 0. Let N ∈ Rn×n be a non-singular matrix and define two non-singular matrices J1 and J2 given in (29). Denote P = J2 J1−1 , S = ˜ 1−1 , Qi = J1−T Q ˜ i J1−1 , Ri = J1−T R˜ i J1−1 and J1−T SJ ⎧ −1 −1 ⎨AK = N (W4 − YAX − YBW1 )(I − YX ) N −1 −1 (33) B = N (W3 − W2 CX )(I − YX ) N ⎩ K CK = N −1 W2 , DK = W1 (I − YX )−1 N Then it is easy to verify that P > 0 and ϒ1 = T −T ϒ2 T −1 < 0. That is, there exist real matrices > 0, P = J2 J1−1 > 0, ˜ i J1−1 , Ri = J1−T R˜ i J1−1 (i = 1, 2) and S = J1−T SJ ˜ 1−1 Qi = J1−T Q such that (14) is satisfied. The proof is thus completed. It is shown from Proposition 2 that Proposition 1 is equivalent to the condition (ii). Moreover, from the proof of Proposition 2, one can see clearly that, if the condition (ii) is true, then the stabilising DOFC (11) is readily obtained, whose parameter matrices are given explicitly in (33). However, there is an unknown matrix N in (33). In the sequel, we will show that the DOFC (11) with (AK , BK , CK , DK ) is algebraically equivalent to the one with (A , B, C , D ) = (NAK N −1 , NBK N −1 , NCK , DK N −1 ). In fact, performing an irreducible linear transformation xc (t) = N −1 xˆ c (t) on the state in (11) yields x˙ˆ c (t) = A xˆ c (t) + B xˆ c (t − η(t)) + C y˜ (t) (34) u(t) = D xˆ c (t), tk ≤ t < tk+1 Clearly, the DOFC (34) is algebraically equivalent to (11). To summarise, we have the following result. 230 © The Institution of Engineering and Technology 2014

(35)

Clearly, an explicit expression on the controller gain matrices is given in Proposition 3. Thus, provided that a solution of the matrix variables (X , Y , W1 , W2 , W3 , W4 ) to a set of matrix inequalities in (26) can be worked out, then a desired DOFC (34) with (35) is readily obtained. However, it is not easy to directly solve out the matrix variables X , Y and Wj ( j = 1, . . . , 4) because of the non-linear terms such as ˜ −1 (ε2 )−1 , Z R˜ −1 1 Z and Z R2 Z existing in (26). In order to deal with the non-linear terms, a cone-complementary linearisation (CCL) algorithm [24] is regarded as a non-conservative approach, by which one can convert the non-convex feasibility problem described by Proposition 3 into a non-linear minimisation problem subject to LMIs. However, the CCL algorithm usually leads to high computational complexity. As a tradeoff between conservatism and computational complexity, one approach to dealing with the non-linear terms is based on the fact that −(ε2 )−1 ≤ − 2ε −1 I ,

˜ −Z R˜ −1 1 Z ≤ R1 − 2Z,

˜ −Z R˜ −1 2 Z ≤ R2 − 2Z In the situation, an LMI-based condition can be readily formulated, which is given in the following. Proposition 4: For given scalars h > 0, τm > 0, and τM (≥ τm ), the event-triggered dynamic output feedback control problem is solvable if there exist a scalar ε˜ > 0 and real ˜ i > 0, R˜ i > 0 (i = 1, 2), S˜ matrices > 0, X > 0, Y > 0, Q and Wj ( j = 1, . . . , 4) of appropriate dimensions such that

R˜ 2

S˜ ≥ 0, R˜ 2

Z=

X

I > 0, Y

21

22 0 is a certain constant. Without such a constraint, solving the LMIs described in (36) usually yields a threshold ε very IET Control Theory Appl., 2014, Vol. 8, Iss. 4, pp. 226–234 doi: 10.1049/iet-cta.2013.0253

www.ietdl.org close to zero, say ε = 10−5 . It can be seen from (2) that when the threshold ε is approaching to zero, the event-triggering scheme possibly reduces to a time-triggering scheme, leading to the worst result that all sampled-data packets will be transmitted without saving any communication resources. Thus, in Section 5, where simulation is made, such a constraint ε ≥ ε0 , correspondingly, ε˜ ≤ ε˜ 0 in Proposition 4, is used, where ε˜ 0 > 0 is a certain constant. Note that from Propositions 4, τm and τM , which are the lower and upper bounds of the network-induced delay τk (k = 1, 2, . . .), are explicitly required in the proposed codesign approach. One natural question is how τm and τM are related to the event-triggering condition (2)? We now give an answer to this question from two aspects. On the one hand, when the threshold ε and the weighting matrix are given, suitable DOFCs can be designed by Proposition 4. In this case, the event-triggering condition is given a priori. Consequently, the event-triggering condition is not related to τm and τM . On the other hand, when co-designing the event-triggering parameters (ε, ) and DOFCs, it is clear from Proposition 4 that there exist some relations between event-triggering parameters (ε, ) and τm and τM . This relation is implicitly included in the LMIs in (36), which is difficult to be disclosed. As for how to estimate the values of τm and τM , one can refer to [25].

5

An application to a satellite control system

Suppose that the plant in Fig. 1 is a satellite control system [26]. A sketch of the satellite control system is shown in Fig. 2, where it is assumed that two masses are connected by a spring with torque constant k and viscous damping constant d. The equations of motion from Fig. 2 are given

Fig. 2

by

J1 θ¨1 + d(θ˙1 − θ˙2 ) + k(θ1 − θ2 ) = Tc J2 θ¨2 + d(θ˙2 − θ˙1 ) + k(θ2 − θ1 ) = 0

(37)

where Tc is the control torque, J1 and J2 are inertias. Let x = col{θ2 , θ˙2 , θ1 , θ˙1 } and u = Tc . Then the state-space representation of (37) can be given by (1) with ⎡

0 ⎢− k J2 A=⎢ ⎣ 0

k J1

C=

0 1

1 − Jd2 0

0 k J2

d J1

0 0

1 1

0 0

0 − Jk1

0

⎤

⎥ ⎥, 1 ⎦ d − J1 d J2

⎡ ⎤ 0 ⎢0⎥ B = ⎣0⎦, 1 J1

In this paper, suppose that J1 = J2 = 1, k = 0.09 and d = 0.0219. Then the eigenvalues of A are −0.0219 + 0.4237j, −0.0219 − 0.4237j, 0, and 0. Thus, the above system is unstable. In the following, we will discuss the co-design approach based on Propositions 4 and Remark 4 to design suitable DOFCs of the form (11) to stabilise the satellite system (37). For this purpose, it is assumed that the sampling period h is 100 millisecond (ms) and that the network-induced delay is bounded from below by τm = 20 ms and from above by τM = 600 ms. The initial condition of the satellite system is taken as x0 = col{0.2, −0.3, 0.3, −0.2}. First, we co-design the event-triggering parameters (ε, ) and the DOFC (AK , BK , CK , DK ) by Proposition 4 without extra constraints imposed on ε˜ . Applying Proposition 4, it is found that the event-triggered dynamic output feedback control problem is solvable, and the obtained event-triggering parameters (ε, ) and the DOFC (AK , BK , CK , DK ), denoted

Sketch of the satellite control system


231 © The Institution of Engineering and Technology 2014

www.ietdl.org

Fig. 3

State responses under the DOFC1

Fig. 4

by DOFC1, are given as

ε = 0.0013,

342.4817 −13.7430 = −13.7430 322.9744

(38)

and AK , BK , CK and DK are given at the bottom of the page. Under the DOFC1, Fig. 3 depicts the state responses of the satellite system. It is clear that the satellite system is stabilised by the DOFC1. Now, we turn to the obtained event-triggering condition (2) with (38). An important performance index to measure an event-triggering scheme is the transmission rate (TR) of the sampled-data packets, which is defined as TR =

the number of transmitted sampled-data packets the total number of sampled-data packets

It is clear that on the time interval [0,100s], there are 1000 sampled-data packets in all. Simulation shows that under the event-triggering condition (2) with (38), all the 1000 sampled-data packets are transmitted, which means that the TR on the time interval [0,100s] is 100%, no any communication resources being saved.

State responses under the DOFC2

Second, as stated in Remark 4, we impose a constraint ε˜ ≤ ε˜ 0 on ε˜ in Proposition 4 to co-design the event-triggering parameters (ε, ) and the DOFC (AK , BK , CK , DK ). Set ε˜ 0 = 8.6. Then applying Proposition 4 yields a feasible solution to the problem of event-triggered dynamic output feedback control. The obtained event-triggering parameters (ε, ) and the DOFC (AK , BK , CK , DK ), denoted by DOFC2, are given as

ε = 0.1183,

=

2.0666 0.3006

0.3006 4.6047

(39)

and AK , BK , CK and DK are given at the bottom of the page. Under the event-triggering condition (2) with (39), only 323 data packets are sent to the ZOH through a communication network, which means that the TR on [0,100s] is 32.3%. Thus, 67.7% communication resources can be saved while the closed-loop stability can be ensured. Figs. 4 and 5 depict the state responses of the satellite system and the release time intervals between any two consecutive release instants, respectively.

⎧ ⎡ ⎤ 0.7565 0.5653 0.8839 0.2654 ⎪ ⎪ ⎪ ⎪ ⎢−0.6218 −0.2544 ⎪ 0.1256 −0.0803⎥ ⎪ ⎥ ⎪ AK = ⎢ ⎪ ⎣ ⎪ 1.3840 0.7343 2.0986 0.8884 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ −8.6310 −4.9808 −13.2900 −4.2976 ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ −0.5551 −0.0801 −0.1416 −0.0562 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎨BK = ⎢−0.0928 −0.0162 −0.0683 −0.0160⎥ ⎣ ⎦ −0.1688 −0.0618 −0.6436 −0.1055 DOFC1: ⎪ ⎪ −0.0708 −0.0157 −0.1061 −0.0201 ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ 34.9749 −40.0414 ⎪ ⎪ ⎪ ⎢ 1.5786 ⎪ −6.1526 ⎥ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪CK = ⎣−46.7916 −4.8722 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ −3.9604 −4.0370 ⎪ ⎪ ⎪ ⎩ DK = 0.0239 0.0132 0.0343 0.0109 232 © The Institution of Engineering and Technology 2014


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6

1

Release time intervals

5

2

4

3

3

4 5

2

6 1

7 0 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Fig. 5

8

Release instants and time intervals under the DOFC2 9

6

Conclusion

10

The event-triggered dynamic output feedback control for NCSs has been addressed. A discrete event-triggering mechanism has been introduced to determine whether or not a sampled signal packet will be transmitted for controller design. The Lyapunov–Krasovskii functional approach has been employed to formulate a novel stability criterion for the resultant closed-loop system. Based on this stability criterion, some sufficient condition has been derived on the existence of suitable DOFC and proper event-triggering parameters. A satellite control system has been used to show the validity of the proposed method.

11 12 13 14 15 16

7

Acknowledgments 17

This work was supported in part by the Australian Research Council Discovery Projects under Grant DP1096780, and the Research Advancement Awards Scheme Programme (January 2010 – December 2012) at Central Queensland University, Australia.

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⎧ ⎡ ⎤ 0.1121 0.0055 0.1130 0.0051 ⎪ ⎪ ⎪ ⎪ ⎢ 0.4936 ⎪ 0.0242 0.4979 0.0224 ⎥ ⎪ ⎥ ⎪ AK = 104 × ⎢ ⎪ ⎣ ⎪ 0.0241 0.0012 0.0243 0.0011 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ −3.5796 −0.1759 −3.6104 −0.1623 ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ −0.0513 −0.0022 −0.0514 −0.0023 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎨BK = ⎢−0.0246 −0.0011 −0.0246 −0.0011⎥ ⎣ ⎦ −0.0656 −0.0029 −0.0661 −0.0029 DOFC2: ⎪ ⎪ −0.0233 −0.0010 −0.0233 −0.0010 ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ −0.2142 −3.6153 ⎪ ⎪ ⎪ ⎢−0.0826 −1.7724⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪CK = ⎣−1.9691 −3.7787⎦ ⎪ ⎪ ⎪ ⎪ ⎪ −0.0169 −1.7056 ⎪ ⎪ ⎪ ⎩ DK = 1.8273 0.0898 1.8430 0.0828 IET Control Theory Appl., 2014, Vol. 8, Iss. 4, pp. 226–234 doi: 10.1049/iet-cta.2013.0253


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