and references therein). The survey of recent progress in NCS, [153] have discussed various network conditions required for different control purposes (such as the minimum rate coding for stabilizability of linear systems in the presence of time-varying channel capacity, the critical packet loss condition for stability of the Kalman filter with intermittent observations, network topology for coordination of networked MASs, as well as event-based sampling for energy and communication efficiency). Another interesting overview on the theoretical development of NCSs under some challenging issues induced from communication networks is discussed in [157]. In recent years, there has been a renewed interest in consensus problems introduced in NCSs with communication constraints (see [76, 151, 141, 60, 74, 143, 114]). The consensus protocols presented in these works usually operate in a synchronous way. Although being relatively sparse, the asynchronous consensus of MASs has been discussed in some related works (e.g., [35, 105, 149, 28, 82, 5]). However, the updating problem of shared information (e.g., interaction among subsystems, coordination variables) among controllers considered as a consensus problem has been rarely discussed in the literature. As an example, the following sections will introduce a case study presenting DMPC scheme for irrigation channels and details of the associated consensus problem.
3.2 Distributed model predictive control of irrigation chan- nels
We consider now an irrigation channel with multiple reaches interconnected through hydraulic control structures such as submerged gates, mixed gates and spillways (see Fig. 1.2). The irrigation channel is provided some local controllers implemented for the control of water flow at some points by adjusting the settings of control structures. In decentralized control schemes, these local controllers operate without being conscious of the presence of other controllers or subsystems. They compute the control action based only on local information without taking into account the information exchanged with other controllers. An observable drawback of such a decentralized control scheme is the reduced performance at a system-wide level due to unexpected and unanticipated interactions among subsystems by local controllers. For improving the global system performance, a distributed control is adequate in which controllers consider a negotiation, a coordination or a cooperation among them by exchanging the necessary information. An example of the DMPC scheme for the control of irrigation channels is given in Fig. 3.1. For the distributed control scheme, we observe that the computational complexity and theoretical properties (e.g., feasibility, stability and robustness) depend closely on the choice of the prediction model, control objectives, and constraints.
Fig. 3.1 A DMPC scheme for the control of irrigation channels.
3.2.1 Prediction model
The modeling of an irrigation channel using the LB method is introduced in Section 1.3.1. The dynamics of LB model are represented by Eq(s). (1.13) and rewritten as follows:
f1(l,t+∆t) = f1(l,t) +1
τ(f1e−f1(l,t)) f2(l,t+∆t) = f2(l−v∆t,t) +1
τ(f2e−f2(l,t))−∆t2vF f3(l,t+∆t) = f3(l+v∆t,t) +1
τ(f3e−f3(l,t)) +∆t2vF
(3.1)
Accounting for the interactions of subsystems, the coordination or the cooperation of controllers requires the consideration of interaction variables or coordination variables from subsystem models. The following sections consider these variables suitable to be used for the coordination of controllers.
3.2.2 Definition and management of shared information
We consider here an irrigation channel comprisingnreaches interconnected by the submerged gates (as shown in Fig. 3.2). The DMPC control of this system is divided over ncontrollers, {C1,C2, . . . ,Ci, . . . ,Cn}. The information shared by controllers is chosen depending on the control setting and system modeling. For example, as [86] adopted deterministic discrete-time linear integral delay models for channel control, the exchange of the flow rate at the downstream of each reach is sufficient for neighbor local controllers to compute actions because the inflow of downstream reach is equal to the outflow of upstream one. For our case study, we assume to use the downstream configuration(see Fig. 3.2) in which the control actions apply on the upstream gate of each reach i, i∈ {1,2, . . . ,n}to adjust the gate opening, θ i (manipulated input), in order to regulate the downstream flow rate,Qi (controlled variables). Some measurements are
3.2 Distributed model predictive control of irrigation channels | 61 assumed to be available such as the water heights at upstream, (husi), and downstream, (hdsi ), of each gate, the flow rate of water through a gate,Qgi. To take into account the interactions among subsystems and the coordination of controllers, they need to share interaction variables. Using the prediction model Eq(s). (3.1) and the DMPC scheme shown in Fig. 3.3, we can easily identify that the necessary information exchanged by the neighbors of the controller i at control stepk, denoted by χini(τ),τ ∈[k,k+1], contains: {f1i-1(lLi−1,k),f2i-1(lLi−1,k),f3i+1(lus,k)}. Thus, the management of the shared information needs to separate neighbor setMifor each controller, Ci: Mi={Ci1,Ci2, . . . ,Cimi}into upstream neighbor sub-set,Ui, and downstream neighbor sub- set, Di. For simplicity, the information shared by controllerCi to its neighbors at timeτ is chosen as: χouti (τ) = [f3i(lus,τ) f1i(lLi,τ) f2i (lLi,τ)]T. In inputs, the controllerCi receives the information shared by its neighbors at timeτ as follows: χini(τ) = [χin|Ci
i1
(τ) χin|Ci
i2
(τ). . . χin|Ci
imi(τ)]T. We observe that the gate equations (Eq(s). (4.4)) permit the computation of these interaction variables, once the neighbor controllers share the flow rate through the gate at their position. Therefore, the flow rate through each gate (Qgi ) is an appropriate variable to be used for coordination among controllers. The coordination variables, Qgi , are considered in this chapter as the shared information for the consensus problem (see Fig. 3.4). In a control system of ncontrollers,{C1,C2, . . . ,Ci, . . . ,Cn}, the information vector of the controllerCiis defined by: χi = [χ1i χ2i . . . χni ]T, where: χji =Qgi
j ∈Ris the flow rate through gate j recorded by the controllerCi(see the example in Fig. 3.6). Other variables can appropriately be used as coordination or cooperation variables depending on distributed control schemes (an example of the cooperation of controllers is given in Chapter 4), but the corresponding consensus problems can be solved in the same manner. Timing for sharing information depends on the synchronization mechanism and the coordination method. The following section investigates the consensus while controllers exchange the information for coordination in an asynchronous way.
Fig. 3.2 A downstream configuration for the distributed control of an irrigation channel. The controllerCiis implemented for the control of water flowQiat the downstream end of a reachi by adjusting the openingθiof the gate at the upstream end.
Fig. 3.3 An example of DMPC scheme in which the controllers share interaction variables ({f1(lLi−1,k),f2(lLi−1,k),f3(lus,k)}) at timekfor action computation.
Fig. 3.4 An example of DMPC scheme in which the controllers share flow rates through their gates (Qg) for coordination.