3.3 Asynchronous information consensus in distributed con-
3.3.1 Problem statements for asynchronous discrete-time system
From MAS point of view, our control system is characterized as a discrete-time system, in which the information flow among controllers is directed and the communication topology may be time-dependent (also calledswitching topology) [94]. As the subsystem states change over time, each controller attempts to reach an agreement with others on some values deduced from their states. We formulate a consensus problem for the agreement of all controllers by introducing the information states of controllers. The information state vector of a controller records the instantaneous values of the information shared by all controllers and itself. The asynchronous consensus protocol is eventually investigated to solve the convergence of information states to the newest values as shared by its neighbors (an example is presented in Fig. 3.6). By
3.3 Asynchronous information consensus in distributed control of irrigation channels | 65
Fig. 3.6 An example of consensus problem in distributed control of irrigation channels is used to illustrate the asynchronous information consensus of the leader-switching system. In this example, the controller,C2, applies a computed action, u2, to the subsystem, which changes the value of its information state,χ2.C2actually becomes the leader. The consensus process is started byC2for providing the convergence of information elementχ2of all controllers.
accomplishing the consensus processes, the information state vectors of all controllers must contain the same values asymptotically. In this chapter, we are interested in the consensus problem of an asynchronous discrete-time MAS-like system under fixed or time-varying topology in the presence of communication issues such as time delays [15, 111].
Assumptions 3.3.1. Regularity assumptions[28].
The convergence analysis of asynchronous consensus process requires making some commu- nication assumptions (introduced in [28], Section II as regularity assumptions) such as:
1. any delay is bounded (see [94], section X.A for quantitatively bounded delay), only a finite number of updating instants can occur within any finite time interval;
2. a controller does not fail to be updated or is updated infinitely often;
3. all controllers can access their own states and at least one of these controllers can achieve it without delay.
The basic idea for a consensus protocol design is to impose similar dynamics for updating information states of all controllers [113]. Solving a consensus problem consists in designing an update law so that the information states of all controllers converge to common values.
We consider a control system ofncontrollers,C={C1,C2, . . . ,Ci, . . . ,Cn}, indexed using an index set N ={1,2, . . . ,n}. The information state of controllerCi, i∈N, is defined as:
χi = [χ1i χ2i . . . χni ]T. All controllers share a common state space χi ∈Rn×1. Lett0<
t1<ã ã ã<ta< . . . be event-based discrete-time instants [151, 60] when an event (e.g., applying an action) occurs in the control system provoking a change in information states of controllers.
We assume that, at the time instantta, the controllerCiapplies the control action which changes the information state: χi (ta). The controllerCi becomes then the actual leader and starts the consensus processes executed on other controllers (i.e., followers) so that the change is updated in their information states. The updating instants for asynchronous information exchanged among controllers are denoted byτ0<τ1<ã ã ã<τu< . . .. By accomplishing the consensus process, the consensus vector defined as:
χ(τ) = [χ1(τ) . . . χ i(τ) . . . χn(τ)]T = [χi1(τ) . . . χii(τ) . . . χin(τ)]T (3.2) converges to the same valueχ i(τ0) =χii(ta)for all elements. An example of the asynchronous framework used for the convergence of the information exchanged among controllers is illustrated in Fig. 4.5, which shows different time instants and intervals.
For the control system, we consider alocalized consensus protocol [28], in which each controller updates its information state using latest-known state values of its neighbors and itself such that [94]:
χ i(τu+1) =χi(τu) +λui(τu) (3.3) where:λ >0 is step-size andui(τu)is a linear consensus protocol defined by:
ui(τu) =
n
∑
j=1
Ai j(χ j (τu)−χ i(τu)) (3.4) in whichAi j, i,j∈Nrepresents neighboring information flow (i.e., the magnitude ofAi j(τ)can represent time-varying relative confidence of the controllerCiin the information shared by the controllerCjat timeτ or the relative reliability of information exchange links between them).
Using the protocol Eq(s). (3.4), the dynamics of discrete-time consensus process Eq(s). (3.3) whileλ =1 can be described by:
χ i(τu+1) =χi (τu) +
n
∑
j=1
Ai j(χ j (τu)−χi(τu))
= (1−
n
∑
j=1,j̸=i
Ai j)χ i(τu) +
n
∑
j=1,j̸=i
Ai jχ j (τu)
=Aiiχ i(τu) +
n
∑
j=1,j̸=i
Ai jχ j (τu) =
n j=1∑
Ai jχ j (τu)
(3.5)
where, we setAii= (1−∑nj=1,j̸=iAi j).
Whereas numerous asynchronous models have been proposed and successfully applied to some practical problems [53, 32, 8], the asynchronous model proposed in [129] is one of the prevalent models in asynchronous theory [28]. According to [129, 28], the time-varying topology and communication delays can be integrated into the asynchronous consensus protocol 3.4, for
3.3 Asynchronous information consensus in distributed control of irrigation channels | 67 updating information states of a controllerCi, as follows:
χi (τu+1) =
∑nj=1Ai j(τu)χ j (τu−di j(τu)), i∈S(τu)
χ i(τu), otherwise
(3.6)
where: fori,j∈N,χ i(τ0) is initial information state of controllerCi; Ai j(τ)is neighboring information flow between controllersCi andCj and Aii = (1−∑nj=1,j̸=iAi j); di j(τ) are time- varying communication delays; S(τ)is the set of already updated controllers at the instantτ.
The Eq(s). (3.6) is also called the asynchronous consensus protocol in the literature [28].
The matricesA(τ) = [Ai j(τ)]arenon-negative stochastic matrices, which have an eigenvalue equal to 1 and a corresponding eigenvector equal to 1n= [1 1 . . . 1]T, that is:A(τ)1n=1n,∀τ.
The matricesA(τ)represent thetime-varying communication topologiesof the control system and are principally explored in the convergence analysis of the asynchronous consensus problem.
The global information consensus is asymptotically achieved for all controllers when the information states of all controllers tend to the initial value of the actual leader. Assuming that the leader isCi: χi(τ0) =χii(ta)and for any initial values of followers: χ j (τ0),(j̸=i), the asymptotic achievement of the consensus means:
∀j∈N, ||χ j (τ)−χ i(τ0)|| →0, asτ→τ+∞. (3.7) Formally, the Assumptions 3.3.1 used in the convergence analysis of asynchronous consensus can be formulated as follows:
Assumptions 3.3.2. Communication assumptions[129, 28].
1. 0≤di j(τ)≤Dmax≤+∞,∀i,j∈N,∀τ, where:Dmax is a constant (regulated delays);
2. S+∞τ=τcS(τ) =C,∀τc(admissible updating sets);
3. Aii>0,∀i∈N, and∃i∈N:dii(τ) =0,∀τ;
4. χi(τ0)>0,∀i∈Nto avoid considering the trivial consensus point[0 0. . .0].
Under the Assumptions 3.3.2, we investigate the reachability of a consensus while the information exchange among controllers is in asynchronous mode. Concretely, we search the conditions for information states of controllers to converge to the consensus point, depending on the initial state of the root node. Normally, asynchronous consensus points are determined depending on the set of updated controllers, delays, and initial states of controllers [28]. The convergence analysis is based on the properties of the matrixArepresenting the communication topologies of the control system.