www.ietdl.org Published in IET Control Theory and Applications Received on 30th March 2014 Revised on 22nd May 2014 Accepted on 12th June 2014 doi: 10.1049/iet-cta.2014.0330
Special Issue on Recent Developments in Networked Control and Estimation ISSN 1751-8644
Stochastic adaptive event-triggered control and network scheduling protocol co-design for distributed networked systems Hao Xu1 , Avimanyu Sahoo2 , Sarangapani Jagannathan2 1 College
of Science and Engineering,Texas A&M University-Corpus Christi, Corpus Christi,TX, USA of Electrical and Computer Engineering, Missouri University of Science andTechnology, Rolla, MO, USA E-mail:
[email protected] 2 Department
Abstract: In a distributed ‘networked control system’ (NCS), multiple physical systems or agents are connected to their corresponding controllers through a shared packet-switched communication network. For such distributed NCS, periodic sampled controller design is unsuitable to handle packet-switched closed-loop control systems and a novel stochastic optimal adaptive event-sampled controller scheme is proposed in the application layer for each physical system or agent expressed as an uncertain linear dynamic system. Lyapunov stability analysis will be utilised to derive the event trigger condition. In addition, a network scheduling protocol is also required for such NCS. Traditional network scheduling protocols are unsuitable for such NCS since the behaviour of the physical systems is disregarded during the protocol design. Therefore in this study, a novel distributed network scheduling protocol via cross-layer approach is developed to improve the performance of distributed NCS by minimising an overall system cost function which consists of the information collected from both the event-triggered controller for each physical system in the application layer and the distributed scheduling protocol from the network layer. It will be demonstrated that the proposed co-design approach will not only allocate the network resources efficiently but also it will improve the performance of the overall distributed NCS. Simulation results are included to demonstrate the effectiveness of the proposed cross-layer co-design.
1
Introduction
In 1990s, information revolution brought shared wireless network as the new communication option of choice to connect several distributed systems and their corresponding remote controllers. This new type of system referred to as ‘networked control system’ (NCS) [1–4] renders enormous benefits such as saving installation cost, while increasing adaptability, reliability and usability. However, it also brings several challenging issues because of the presence of the packet switched communication network. Recently, a cross layer design approach to accommodate the control [5] and network protocols [6] has been proposed for such systems. The presence of a shared packet switched communication network facilitates the functioning of several such NCS leading to an overall distributed NCS wherein a physical system is viewed as an agent. In such a distributed multi-agent NCS [7], the control and networking schemes for each agent have to be designed by considering the linkage between them. Incorporating the effects of a packet switched wireless network, Xia et al. in [1] proposed a networked control scheme to maintain the stability of the physical system part of the distributed multiagent NCS. By contrast in [2], from the wireless network protocol side, authors evaluated the performance of widely used IEEE 802.11protocol for distributed NCS. IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2253–2265 doi: 10.1049/iet-cta.2014.0330
However, most of these methods [1, 2] have not considered the interaction between the control system and wireless network protocol during the design process. A revolutionary scheme for distributed multi-agent NCS should utilise the real-time interaction between the application and network layers in order to optimise the performance of both the control system and wireless network components. From the well-known ‘open system interconnection’ architecture [8], the control system in a NCS belongs to the application layer whereas the wireless network protocol is included in the network and data-link layers. Therefore considering the interaction among different layers properly, a cross-layer design [9–12] is necessary. In [9], Srivastava and Motani demonstrated that the crosslayer design can attain performance gains by exploiting the dependence among protocol layers when compared with the traditional individual layered designs. However, the cross-layer design is implemented for data-link and physical layers [10] wherein the application layer is neglected. For a distributed multi-agent NCS, the control design at the application layer and the network protocol design for communication at the data-link layer have to be considered together. In the data-link layer, a distributed scheduling scheme is critical for a wireless network protocol design [13]. Compared with the traditional centralised scheduling [14], the main advantage of distributed scheduling is that it does not 2253 © The Institution of Engineering and Technology 2014
www.ietdl.org require a centralised processor to deliver the schedules after collecting information from all the wireless network links in the network. According to IEEE 802.11 standard [13], ‘carrier sense multiple access’ (CSMA) protocol is introduced to schedule wireless network links in a distributed manner where a communication link wishing to transmit does so only if it does not hear an on-going transmission from the network. Moreover, Jiang and Walrand [15] derived a throughput-optimal distributed scheduling algorithm and proven that even distributed scheduling scheme can still achieve the throughput maximisation. However, since a random access scheme is widely used in most CSMA-based distributed scheduling [15, 16] schemes, they focus on improving data-link layer performance alone which in turn can affect the performance of control system in the application layer. Thus, these protocols [15, 16] are unsuitable since they can degrade the performance of the overall distributed multi-agent NCS. Meanwhile, for the application layer, control of such NCS has to be carefully dealt with since existing periodically sampled control schemes are unsuitable to handle both the continuous dynamics of the physical system and the transmission of state vector over the wireless network. Periodic sampling and transmission can lead to network congestion causing degradation in performance and instability of the closed-loop system. The practical NCS requires eventtriggered control system design wherein the system output vector is transmitted to the controller in an aperiodic manner reducing the amount of transmitted information over the network. Non-periodic transmission also reduces computation whereas it creates challenges in terms of stability and convergence. Specifically, the optimal adaptive event triggered controller design is necessary for each NCS as it is quite challenging in the presence of uncertain system dynamics [17]. Although Wang and Lemmon in [18] evaluated the stability of event-triggered control for a distributed NCS, the scheme is not optimal. The ‘adaptive dynamic programming’ (ADP) technique by using Q-learning [19], on the other hand, has been utilised to obtain optimal control of uncertain linear dynamic systems in a forward-in-time manner instead of traditional backward-in-time ‘Riccati equation’-based optimal control when system dynamics are known. Inspired by the ADP framework, this paper proposes a novel cross-layer scheme for each agent with uncertain dynamics in a distributed multi-agent NCS which includes an optimal adaptive eventtriggered controller design in the application layer and a distributed scheduling scheme for the wireless network in the data-link layer. First a novel stochastic optimal adaptive event triggered control scheme for each agent by using Q-learning is introduced in the presence of network delays and packet losses. Lyapunov analysis is utilised to derive the event trigger condition. Next, a distributed scheduling protocol is presented and the cross layer design is verified by using a simulation example. The main contributions of this paper include (1) a novel optimal adaptive event-triggered control scheme designed in a forward-in-time manner without using the knowledge of agent dynamics that is represented as an uncertain linear discrete-time system, (2) a distributed network scheduling
scheme via cross-layer approach which improves the performance of the multi-agent NCS by minimising the cost function from the application layer for the linear system and maximising the network throughput from the datalink layer, and (3) Lyapunov stability analysis of the event triggered control system design and (4) performance analysis of the overall system. The proposed event-triggered scheme utilising event-based sampling saves network resources.
2
Background
The basic structure of a distributed multi-agent NCS is described in Fig. 1, where numerous agents/systems communicate to their corresponding remote controllers through a shared wireless network such as an IEEE 802.1x. This type of systems is quite different than the traditional interconnected system where geographically located subsystems are connected physically via interconnected terms. It is clear that the shared wireless network will affect the performance of the closed-loop control system of each agent. For example, the shared wireless network when congested because of an inadequate scheduling, the stability of the distributed multi-agent NCS cannot be maintained since the agent information cannot be communicated to their respective controllers successfully and could experience excessive delays. Therefore a novel cross-layer design is necessary for distributed multi-agent NCS where the design incorporates the control and distributed scheduling simultaneously. Without loss of generality, in this paper numerous coupled homogeneous agents are considered as part of distributed multi-agent NCS. Next, a brief description in the context of a multi-agent dynamics is introduced. Consider the overall distributed multi agent NCS described by a linear time-invariant continuous-time system as x˙ l (t) = Al x(t) + Bl ul (t), y(t) = Cl x(t), ∀l = 1, 2, . . . , N with dynamics Al , Bl , Cl for the lth agent, and Ts being the periodic sampling interval. For the optimal controller design, the communication network delay for each agent is ¯ s, assumed to be unknown but bounded above as τl ≤ dT ∀l = 1, . . . , N , a stringent delay bound that needs to be ensured by utilising a proper data rate in the communication network. Furthermore, incorporating the network delay and packet losses, the lth NCS agent dynamics can be represented [20] as a stochastic time-varying system that is described by (see (1)) where ula,k is the control input received by the lth NCS agent actuator at the time instant kTs , ul,k is the control input computed by the lth NCS agent controller at the time instant kTs . Note, the time index k has to be semipositive if k < 0 ul,k = 0. The γl,k represents the stochastic variable representing the packet losses for the lth agent at the time instant kTs which follows the Bernoulli distribution with P(γl,k = 1) = γ¯l , Al,s , Bl1,k , . . . , Bl d,k ¯ , Cl denote the augment agent dynamics resulting by incorporating the communication network delay and packet losses at the time instant kTs . Defining the augmented state vector to be zl,k = [(xl,k )T (ul,k−1 )T (ul,k−2 )T · · · (ul,k−d¯ )T ]T for the lth agent at the time instant kTs , the agent dynamics (1)
xl,k+1 = Al,s xl,k + Bl1,k ula,k−1 + Bl2,k ula,k−2 + · · · + Bl d,k ¯ ula,k−d¯ + Bl0,k ula,k yl,k = Cl xl,k ula,k = γl,k ul,k 2254 © The Institution of Engineering and Technology 2014
∀l = 0, 1, 2, . . . , N , ∀k = 0, 1, 2, . . .
(1)
IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2253–2265 doi: 10.1049/iet-cta.2014.0330
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Fig. 1 Distributed multi-agent NCS
can be rewritten as a linear stochastic time-varying system described by zl,k+1 = Al,zk zl,k + Bl,zk ul,k , ∀l = 0, 1, 2, . . . , N ,
yl,k = Cl,z zl,k ,
∀k = 0, 1, 2, . . .
(2)
with the time-varying augmented system matrices are detailed in [21]. The pair (Al,zk , Bl,zk ) is considered to be controllable and (Al,zk , Cl,z ) is observable with the observability index being N [21], then (2) can be expressed in the input–output form as ycl,k+1 = Ql,k ycl,k + Fl,k ul,k ,
∀l = 1, 2, . . . , N
(3)
T T T T ¯ l,k−1,k−N u¯ l,k−1,k−N = [ul,k−1 where ycl,k = [¯yl,k,k−N +1 u +1 ] , T T T T T T T ul,k−2 · · · ul,k−N ] , y¯ l,k−1,k−N = [yl,k−1 yl,k−2 · · · yl,k−N ] , and Ql,k , Fl,k are defined in [21]. After incorporating the random delays and packet losses from the network, the continuoustime linear system becomes uncertain stochastic linear discrete-time system in the input–output form as shown in (3). Note that the order of system (3) and the bound for Fl,k should be known as it is needed in the proof but not for the controller design. After generating the agent representation, the controller design is introduced next. It is well known that eventtriggered control [6, 22–24] can bring in significant benefits to the multi-agent NCS in terms of saving computational cost and reducing the network traffic enormously. To harvest these benefits in the multi-agent NCS, the optimal eventtriggered control framework is considered in this paper over stabilising event-triggered control design [18]. Therefore a novel optimal adaptive event-triggered control is introduced next.
3 Optimal adaptive event-triggered control of multi-agent NCS First, a brief background on the general event-triggered control system is introduced. Subsequently, optimal adaptive ‘zero-order-hold’ (ZOH) event-triggered control scheme, IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2253–2265 doi: 10.1049/iet-cta.2014.0330
which is derived for multi-agent NCS by using inputs and measured outputs data, can maintain stability even when the system dynamics are unknown and in the presence of eventbased sampling of system output and control input vector. Without loss of generality, the lth NCS agent is selected to describe the optimal adaptive event-triggered control as follows. In the literature, the event triggered control system design is introduced by using the ZOH and therefore will be referred to as ZOH based event triggered control scheme. 3.1
Event-triggered control system
Recently, event-triggered control of linear system has been a topic of significant interest for NCS because of its network benefits [18]. In Fig. 2, the basics structure of a ‘ZOH’ based event-triggered control system is shown. Compared with a traditional control system, a trigger mechanism is included at the sensor node to decide when to sense and transmit the system output information yl,k over the communication network to the controller. For the sake of simplicity, a NCS agent is considered as a linear discrete-time system. According to (2) and (3), the lth agent at the time instant ‘k’ is described as ycl,k+1 = Ql.k ycl,k + Fl.k ul,k . Moreover, a ZOH event-triggered controller will hold the last received system state vector until a new output vector is received. In order to improve the performance of the ZOH event-triggered system, the optimal control approach [16] has to be incorporated into the even-triggered control design. However, since network imperfections can bring uncertainty into the system dynamics (2), a novel ADP-based stochastic optimal adaptive event triggered control [18] scheme will be introduced as follows. 3.2
Optimal adaptive ZOH event-triggered control
In this subsection, first, a value function is introduced and estimated adaptively. Then, the optimal ZOH event-triggered control scheme is developed by utilising the estimated value function. 3.2.1 Value function setup: According to optimal control [16], the stochastic value function for the lth agent 2255 © The Institution of Engineering and Technology 2014
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Fig. 2 ZOH event-triggered system
can be represented over infinite horizon as Vl,k =
∞
c T cT c (ycT l,k W l,z y l,k + ul,k S l,z ul,k ) = E {y l,k Pl,k y l,k }
(4)
τ ,γ
k=0
where W l,z , S l,z are positive semidefinite and positive definite matrices, respectively, Pl,k ≥ 0 is the solution to the stochastic Riccati equation (SRE) for the lth agent with E {•} is the expectation operator (i.e. mean value) of
τ ,γ
c {ycT l,k Pl,k y l,k }.
Then, the Hamiltonian for the lth agent can be represented as H (zl,k , ul,k ) = r(ycl,k , ul,k ) + V (ycl,k+1 , ul,k+1 ) − V (ycl,k , ul,k ) (5)
c T with r(ycl,k , ul,k ) = ycT l,k W l,z y l,k + ul,k S l,z ul,k is a one-step costto-go function. Utilising the standard optimal control theory [16], the stochastic optimal control signal of the lth agent at time instant kTs can be solved by minimising the stochastic value function (3), (ul,k )∗ = argmin Vl,k , ∀l = 1, 2, . . . , N , ∀k = 1, 2, . . ., which yields ∗ ∗ c T = Kl,k yl,k = −[S l,z + E (Fl,k Pl,k+1 Fl,k )]−1 ul,k τ ,γ
×E
τ ,γ
T (Fl,k Pl,k+1 Ql,k )ycl,k
is the Kronecker product quadratic polynomial basis vector of lth agent and h¯ l,k = vec(H¯ l,k ) with the vector function acting on a square matrices thus yielding a column vector. Note that the vec(•) function is constructed by stacking the columns of the matrix into one column vector with the offdiagonal elements which can be combined as Hl,mn + Hl,nm [18]. Next, according to [16, 18], the optimal control gain of the lth agent can be represented by using Hl matrix as T T K ∗l,k = −[S l,z + E (Fl,k Pl,k+1 Fl,k )]−1 E (Fl,k Pl,k+1 Ql,k ) τ ,γ
=
τ ,γ
uu −1 ¯ uz −(H¯ l,k ) Hl,k
(8)
Therefore if H l matrix is solved for the lth agent, the stochastic optimal control gain can be obtained by utilising (5). However, since the system dynamics are unknown, H l cannot be solved directly. Similar to [18], a novel adaptive estimator will be proposed to estimate value function, the H l , and the optimal control gain. 3.2.2 Model-free online tuning of value function in event-triggered scheme: Recall ZOH event-triggered control scheme in Section 3.1 and the novel ADP technique in [21], the value function can be estimated whereas an event is triggered as
(6)
Remark 1: It is important to note that computing optimal control input using (3) requires the system dynamics which are not available in practice. On the other hand, the stochastic optimal action dependent value function with event sampled output vector can be expressed as T T T ]Hl,k [(ycl,k )T ul,k ] } = h¯ Tl,k χ¯ l,k Vl,k = E {[(ycl,k )T ul,k
(7)
Vˆ l,i = E {[(ycl,i )T yTl,i ]Hˆ l,i [(ycl,i )T ycl,i ]T } = hˆ¯ Tl,i χ¯ l,i
(9)
τ ,γ
where iTs is the triggering time instant with Ts denoting the sampling time. According to traditional optimal control theory [16], the Bellman equation is represented in terms of the value function as Vl,i+1 − Vl,i + r(ycl,i , ul,i ) = 0
(10)
τ ,γ
However, (10) cannot be guaranteed whereas the estimated Hˆ l,i is utilised. After incorporating the estimated Hˆ l,i into the
with (see equation at the bottom of the page)
H¯ l,k
zz H¯ l,k = E (Hl,k ) = uz τ ,γ H¯ l,k
χl,k =
[(ycl,k )T uT (ycl,k )]T
⎤ ⎡W + E [(Q )T P E [(Ql,k )T Pl,k+1 Fl,k ] l,z l,k l,k+1 Ql,k ] zu H¯ l,k τ ,γ τ ,γ ⎦, =⎣ uu S l,z + E [(Ql,k )T Pl,k+1 Ql,k ] H¯ l,k E [(Fl,k )T Pl,k+1 Ql,k ] τ ,γ
and
τ ,γ
χ¯ l,k
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IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2253–2265 doi: 10.1049/iet-cta.2014.0330
www.ietdl.org Bellman equation, (10) can be presented as Vˆ l,i+1 − Vˆ l,i + r(ycl,i , ul,i ) = el,i+1
(11)
where el,i+1 is referred to as the temporal difference error [21] in the reinforcement learning literature. Moreover, substituting (9) into (11), (11) can be represented as el,i+1 = r(ycl,i , ul,i ) + Vˆ l,i+1 − Vˆ l,i = r(ycl,i , ul,i )
+ hˆ¯ Tl,i+1 χ¯ l,i
(12)
with χ¯ l,i = χ¯ l,i+1 − χ¯ l,i . Subsequently, the parameter update law, Hˆ l,i , for the lth agent can be introduced as 1
(13)
Since the parameter estimation error is defined as h˜¯ l,i = h¯ l − hˆ¯ l,i , the parameter estimation error dynamics can be expressed as h˜¯ l,i+1 = h˜¯ l,i − αl
T χ¯ l,i el,i+1 χ¯ l,iT χ¯ l,i + 1
(14)
Remark 2: It is observed that the estimated value function will no longer be updated whereas the system output vector converges to zero. It can be seen as a persistency of excitation (PE) [25] requirement for the input to the value function estimator whereas the output vector must be persistently existing long enough for the estimator to learn the value function. The PE condition is well known in adaptive control literature and can be ensured by adding exploration noise [16]. This PE condition is satisfied only when there is enough number of triggered events such that adaptive estimator generates the H matrix. This PE condition can be viewed as ‘exploration’ in the reinforcement learning literature. Next, the estimation of optimal adaptive ZOH eventtriggered control is derived. According to [21], the optimal ZOH event-triggered control can be obtained by minimising the value function. Recall (8), the optimal control gain can be designed by using the estimated parameter Hˆ l,i as uˆ l,i = Kˆ l,i ycl,i = −(Hˆ l,iuu )−1 Hˆ l,iux ycl,i
(15)
Obviously, the optimal adaptive ZOH event-triggered control gain can be obtained in terms of Hˆ l,i matrix, which is solved by estimating value function with event sampled output vector. It is important to note that proposed design (13) and (15) not only relaxes the requirement of the agent dynamics but also eliminates the value and policy iterations. Next, the event triggering condition is derived. Since the controller will hold the latest received system output vector, ycl,i , 0 < i ≤ k at the time instant kTs , the measurement ZOH error el,k or also referred to as event trigger error can be represented in terms of the output vector as ZOH = ycl,k − ycl,i , el,k
(17)
= r(ycl,i , ul,i ) + hˆ¯ Tl,i+1 (χ¯ l,i+1 − χ¯ l,i ) = r(ycl,i , ul,i )
hˆ¯ l,i+1 = hˆ¯ l,i +
ZOH ycl,k+1 = (Ql,k + Fl,k Kk )ycl,k − Fl,k Kl,k el,k
0 event is initiated c c yl,k − yl,i event is not initiated. Next, to maintain the stability of the closed-loop ZOH event-triggered control system, a novel event-trigger condition [18] can be represented as ZOH with el,k =
+ hˆ¯ Tl,i+1 χ¯ l,i+1 − hˆ¯ Tl,i+1 χ¯ l,i
T χ¯ l,i el,i+1 αl χ¯ l,iT χ¯ l,i +
vector to the controller. Upon receiving the output vector, the controller generates the new control input and resets ZOH to zero. Moreover, the ZOHthe event trigger error el,k based event-triggered closed-loop agent dynamics can be represented as
0 0, it indicates that lth NCS agent is Obviously, when Jl,k allowed to transmit over the network yields additional benefit. Otherwise, scheduling lth NCS agent will degrade the S performance. Therefore when Jl,k > 0, this NCS agent can be considered as scheduled. It is important to note that there are multiple NCS systems in a multi-agent NCS (i.e. M coupled NCS agents) eligible to transmit, and probably the cost function of several subsystems are higher than zero which indicates that all of these NCS agents have to be scheduled. However, according to the literature on networking [13], only one NCS agent can access the network. In order to improve the performance of the network and recall to (22), the efficient scheduling policy should minimise the total cost function given by 1 1 ∗ = min (27) S π JkS Jl,k l∈G k
where G k is the set of agents with positive value of cost S function at the time instant kTs i.e. Jl,k > 0 for l ∈ G k . Obviously, for centralised scheduler design, the efficient schedule has to select an agent that has the minimum value S of 1/Jl,k . However, in the centralised scheduling scheme, S requires significant inforfinding the minimum value 1/Jl,k mation from every agent which might overload the network. Therefore the novel distributed scheduling scheme is needed to solve this drawback. In this paper, the main idea of scheduling algorithm demonstrated in Fig. 4 is to separate the transmission time of different systems by using back-off interval (BI) [30] which is designed based on a related cost function in distributed manner. To solving efficient scheduling problem (26), the BI can be designed as BIl,k = ς × (e−(1/Jl,k ) + nl,k ),
for l ∈ G k
(28)
with ς is the scaling factor and nl,k is a random variable satisfying the Gaussian distribution (i.e. nl,k ∼ L ∗ N (0, σ 2 )), 2259 © The Institution of Engineering and Technology 2014
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Fig. 5 Novel efficient distributed scheduling scheme
and L = minl,j∈Gk (e−(1/Jl,k ) − e−(1/Jj,k ) ) is the range of the random value nl,k . Next, the proposed novel distributed scheduling algorithm is introduced (see Fig. 5). Remark 3: Since each NCS agent decides its schedule by only using local information from the application and data-link layers, the proposed novel cross-layer scheduling scheme is distributed.
Remark 5: Fairness is an important factor to evaluate the performance of scheduling schemes. For proposed distributed scheduling algorithm, an index given by 2 M ∞ T T Ri / (xi,j Qi xi,j + ui,j Ri ui,j ) + βi Ri FI =
Remark 4: Compared with other distributed scheduling schemes [14–16], the proposed algorithm generates the BI intelligently by optimising a cost function instead of selecting it at random as in [14–16], which can be considered as the main contribution of this distributed scheduling algorithm. Next, the effectiveness of the proposed novel distributed scheduling is shown in Theorem 2. Theorem 2 (Efficient distributed scheduler performance): Given the multiple-agent NCS and adaptive optimal event-triggered control scheme given by (15), the proposed distributed scheduling scheme selects the agent with smallest cost function value since it has the shortest BI and highest priority to access the shared communication network. In addition, the proposed algorithm can render best performance schedule for each agent in the overall NCS. 2260 © The Institution of Engineering and Technology 2014
Proof: Refer to the appendix.
i=1
M∗
M i=1
j=0
Ri /
∞
2 T (xi,j Qi xi,j
+
T ui,j Ri ui,j )
+ β i Ri
j=0
is defined to measure the fairness among different agents. Until now, the novel distributed scheduling is proposed to improve the performance of both application and data-link layers since cost function includes information from both layers. Combining with the proposed optimal adaptive ZOH event-triggered control scheme in Section 3, the proposed cross-layer approach is stated in the theorem. Theorem 3 (Performance of the cross-layer design): Consider an agent with its dynamics represented as a linear discrete-time system in a multi-agent NCS (2), the adaptive estimator (13) and estimated optimal control input IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2253–2265 doi: 10.1049/iet-cta.2014.0330
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c and Fl,M , Kˆ l,k defined in Theorem 1. In addition, the 1, yl,k−i proposed algorithm can render best schedules for each agent of the overall NCS. Next, to evaluate the proposed scheme, the following simulation example has been considered.
10 Regulation errors
(15). Let u0 (ycl,k ) be an initial stabilising control policy for the agent (3). Let adaptive estimator parameter hˆ¯ l,0 be initialised in D ⊂ n and the proposed distributed scheduling scheme given in Theorem 2 selects the NCS agent with smallest cost function value since it has the shortest BI and highest priority to access the shared communication network. Furthermore, there exists positive constant αl such that the closed-loop event-triggered augmented system output vector, ycl,k , and adaptive estimator parameter estimation errors hˆ¯ l,k , are AS in the mean square [26] provided the agent outputs are transmitted to the controller and the adaptive estimator parameters are updated by using (13) through the violation of ZOH c event-trigger condition defined by el,k ≤ σl,k yl,k where
2 c 2 2 σl,k = (1 − 3ρ)/3Fl,M [yl,k−i + Kˆ l,k ], with 0 <
0. Therefore utilising the developed event-triggering condition, the closed loop system will be AS in the mean square [26] under subcase 1. Next, subcase 2 (i.e. event is triggered at time (k − 1)Ts ) is analysed. Subcase 2: Event is triggered at time (k − 1)Ts . Consider the Lyapnuov function candidate as Lcl = E {(ycl,k )T (ycl,k )} + τ ,γ
2 2 E {h˜¯ Tl,k−1 h˜¯ l,k−1 }, where = 4(χ¯ l,M + 1)Fl,M 2M ξM2 /αl (1 − τ ,γ
αl ). According to the update law of adaptive estimator and Cauchy–Schwartz inequality, the first difference of the selected Lyapunov function candidate can be represented as (see (33)) where 0 < [Ql,k Fl,k ] ≤ M , history information 0 < [ycl,k−1 ul,k−1 ] < ξM and 0 < χ¯ l,k−1 < χ¯ M . Using
Lcl = E {(ycl,k+1 )T (ycl,k+1 )} − E {(ycl,k )T (ycl,k )} + E {h˜¯ Tl,k h˜¯ l,k } − E {h˜¯ Tl,k−1 h˜¯ l,k−1 } τ ,γ
= E
τ ,γ
τ ,γ
{Ql,k ycl,k
+
∗ Fl,k ul,k
−
τ ,γ
∗ Fl,k ul,k
+ Fl,k ul,k − Fl,k ul,k +
τ ,γ
ZOH 2 Fl,k ul,k }
− E {ycl,k 2 } + E {h˜¯ Tl,k h˜¯ l,k } − E {h˜¯ Tl,k−1 h˜¯ l,k−1 } τ ,γ
τ ,γ
τ ,γ
2 2 ZOH 2 ≤ −(1 − 3ρ)ycl,k 2 + 3Fl,M Kˆ l,k 2 el,k + E {h˜¯ Tl,k h˜¯ l,k } − E {{h˜¯ Tl,k−1 h˜¯ l,k−1 } E {K˜ l,k 2 ycl,k 2 } + 3Fl,M τ ,γ
τ ,γ
2 2 ≤ −[(1 − 3ρ) − 3Fl,M Kˆ l,k 2 σ 2 ]ycl,k 2 + 3Fl,M 2M ξM2 h˜¯ l,k−1 2 − αl (1 − αl )
τ ,γ
h˜¯ l,k−1 2 2 χ¯ l,M +1
2 2 ≤ −[(1 − 3ρ) − 3Fl,M Kˆ l,k 2 σ 2 ]ycl,k 2 − Fl,M 2M ξM2 h˜¯ l,k−1 2
(33) IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2253–2265 doi: 10.1049/iet-cta.2014.0330
2263 © The Institution of Engineering and Technology 2014
www.ietdl.org developed event-triggering condition(2), (5) can be represented as
Proof of Theorem 2: Assume the lth NCS agent has the smallest cost function value (i.e. 1/Jl,k = min(1/Ji,k )), i∈G k
then we have Jl,k > Ji,k for any i = G k , i = l. Therefore 2 Lcl ≤ −[(1 − 3ρ) − 3Fl,M Kˆ l,k 2 σ 2 ]ycl,k 2
−
2 Fl,M 2M ξM2 h˜¯ l,k−1 2
−
2 Fl,M 2M ξM2 h˜¯ l,k−1 2
≤ −(1 − 3ρ)(1 −
e−(1/Ji,k ) < e−(1/Jl,k )
)ycl,k 2
τ ,γ
2 2 E {h˜¯ Tl,k h˜¯ l,k }, where = 4(κl2 χ¯ l,M + 1)Fl,M 2M ξM2 /αl (1 − τ ,γ
αl ). According to the update law of adaptive estimator and Cauchy–Schwartz inequality, the first difference of the selected Lyapunov function candidate can be represented as (see (35)) By using the developed event-triggering condition (31), (35) can be represented as
Next, for any i = G k , i = l, the BI can be expressed as BIl,k = ς × (e−(1/Jl,k ) + nl,k ) < ς × [e−(1/Jl,k ) + min (e−(1/Jj,k ) − e−(1/Jl,k ) )] l,j∈[1,M ]
−(1/Jl,k )
< ς × (e
sj
Jk
Since 0 < < 1, 0 < ρ < (1/3), the first difference of the selected Lyapunov function candidate, Lcl , is negative definite, that is Lcl < 0 whereas the Lyapunov function candidate is positive definite, that is Lcl > 0. Therefore by utilising the developed event-triggering condition, the closed loop system will also be AS under case 2. According to (A.3), (A.5) and (A.7), the proposed event-triggering condition and adaptive optimal ZOH event-triggered control can ensure the closed-loop system to be AS in the mean square [26].
(38)
Hence, BIl,k < BIi,k for any i = G k , i = l. Based on the proposed distributed scheduling algorithm, lth NCS agent can be scheduled to use share the communication network because of its shortest BI. Next, the cost-efficient of proposed scheme will be proven by using the contradiction method. Assume that there exists another jth agent, and scheduling it can render better performance than scheduling lth agent even the lth agent has shortest BI (i.e. cost function value sj 1/Jksl < 1/Jk , but BIl,k < BIj,k ). According to the definition sj of cost function (19), Jk can be defined as
2 Lcl ≤ −[(1 − 3ρ) − 3Fl,M Kˆ l,k 2 σ 2 ]ycl,k 2
2 ≤ −(1 − 3ρ)(1 − )ycl,k 2 − Fl,M 2M ξM2 h˜¯ l,k 2 (36)
+ e−(1/Jj,k ) − e−(1/Jl,k ) ) < ς
× e−(1/Ji,k ) < ς × (e−(1/Ji,k ) + ni,k ) < BIi,k
1 2 − Fl,M 2M ξM2 h˜¯ l,k 2
(37)
(34)
Since 0 < < 1, 0 < ρ < (1/3), the first difference of selected Lyapunov function candidate, Lcl , is negative definite, that is Lcl < 0 whereas the Lyapunov function candidate is positive definite, that is Lcl > 0. Therefore utilising the developed event-triggering condition, the closed loop system will also be AS in the mean square [26] under subcase 2. Next, case 2 (i.e. event is triggered at time kTs ) is analysed. Case 2: Event is not triggered at time kTs . Consider c T c ) (yl,k )} + the Lyapnuov function candidate as Lcl = E {(yl,k
i = G k , i = l
M 1 1 − = Jj,k J i=1 i,k
j = 1, 2, . . . , M
(39)
Since BIl,k < BIj,k is given in assumption above, we have 1/Jl,k < 1/Jj,k by using (28) and (9). Meanwhile, the cost function of j th NCS agent can be derived as 1 sj
Jk
M M 1 1 1 1 1 − − = < = sl J J J J J j,k j,k i,k l,k k i=1 i=1 (40) sj
It is important to note that 1/Jk > 1/Jksl in (40) contradicts sj the assumption 1/Jk < 1/Jksl . By contradiction, there is no other system which can obtain a better performance than scheduling lth agent which has the shortest BI. In the other words, proposed distributed scheduling scheme can render
Lcl = E {(ycl,k+1 )T (ycl,k+1 )} − E {(ycl,k )T (ycl,k )} + E {h˜¯ Tl,k+1 h˜¯ l,k+1 } − E {h˜¯ Tl,k h˜¯ l,k } τ ,γ
τ ,γ
τ ,γ
τ ,γ
∗ ∗ ZOH 2 = E {Ql,k ycl,k + Fl,k ul,k − Fl,k ul,k + Fl,k ul,k − Fl,k ul,k + Fl,k ul,k } − E {ycl,k 2 } + E {h˜¯ Tl,k+1 h˜¯ l,k+1 } − E {h¯˜ Tl,k h¯˜ l,k } τ ,γ
τ ,γ
τ ,γ
τ ,γ
2 2 ZOH 2 ≤ −(1 − 3ρ)ycl,k 2 + 3Fl,M Kˆ l,k 2 el,k + E {h˜¯ Tl,k+1 h˜¯ l,k+1 } − E {h˜¯ Tl,k h˜¯ l,k } E {K˜ l,k 2 ycl,k 2 } + 3Fl,M τ ,γ
τ ,γ
2 2 ≤ −[(1 − 3ρ) − 3Fl,M Kˆ l,k 2 σ 2 ]ycl,k 2 + 3Fl,M 2M ξM2 h˜¯ l,k−1 2 − αl (1 − αl )
τ ,γ
h˜¯ l,k 2 2 2 κl χ¯ l,M +
1
2 2 ≤ −[(1 − 3ρ) − 3Fl,M Kˆ l,k 2 σ 2 ]ycl,k 2 − Fl,M 2M ξM2 h˜¯ l,k 2
(35)
2264 © The Institution of Engineering and Technology 2014
IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2253–2265 doi: 10.1049/iet-cta.2014.0330
www.ietdl.org the best performance by scheduling the lth agent with the shortest BI. On the other hand, any system with negative cost function value, 1/Ji,k < 0, should not contend the shared communication network resource since it will degrade the performance. Next, the proof is given in detail. Assume the pth agent with a negative cost function, 1/Jp,k < 0 is scheduled, then the cost function with this decision can be expressed as
IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2253–2265 doi: 10.1049/iet-cta.2014.0330
1 1 1 1 − < sp = Jk J Jp,k J i=1 i,k i=1 i,k M
=
M
1 (no pair is scheduled) Jk
(41)
Therefore scheduling a NCS agent with negative cost function will degrade the performance.
2265 © The Institution of Engineering and Technology 2014
