Hierarchical distributed model predictive control based on fuzzy negotiation

This is a repository copy of Hierarchical distributed model predictive control based on fuzzy negotiation in the Depósito de Investigación de la Universidad de Sevilla. Version: Author Accepted Version. Citation: Masero, E., Francisco, M., Maestre, J. M., Revollar, S., & Vega, P. (2021). Hierarchical distributed model predictive control based on fuzzy negotiation. Expert Systems with Applications, 176, 114836. DOI: https://doi.org/10.1016/j.eswa.2021.114836 To cite this publication, please use the final published version (if applicable). Please check the document version above. Copyright: Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy: Please contact us ([email protected]) and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim. Hierarchical distributed model predictive control based on fuzzy negotiation Eva Maseroa,˚, Mario Franciscob, Jos´e M. Maestrea, Silvana Revollarb, Pastora Vegac aDepartment of Systems Engineering and Automation , Escuela T´ecnica Superior de Ingenier´ıa, Universidad de Sevilla, Av. Camino de los Descubrimientos, s/n., 41092 Seville, Spain. bDepartment of Computing and Automation, Escuela T´ecnica Superior de Ingenier´ıa Industrial, Universidad de Salamanca, Av. Fernando Ballesteros, B´ejar, 37700 Salamanca, Spain. cDepartment of Computing and Automation, Facultad de Ciencias, Universidad de Salamanca, Plaza de la Merced, s/n, 37008 Salamanca, Spain. Abstract This work presents a hierarchical distributed model predictive control approach for multiple agents with cooperative negotiations based on fuzzy inference. Specifically, a fuzzy-based two-layer control architecture is proposed. In the lower control layer, there are pairwise negotiations between agents according to the couplings and the communication network. The resulting pairwise control sequences are sent to a coordinator in the upper control layer, which merges them to compute the final ones. Furthermore, conditions to guarantee feasibility and stability in the closed-loop system are provided. The proposed control algorithm has been tested on an eight-coupled tank plant via simulation. Keywords: Model predictive control, Hierarchical distributed control, Pairwise negotiations, Fuzzy logic, Multi-agent systems, Stability. 1. Introduction Centralized control methods generally achieve the best performance, for all the measurements are gathered at a single point where decisions are taken using full information. Nevertheless, there are compelling reasons to consider distributed control architectures. Firstly, for complex large-scale processes, it is merely not feasible to compute control actions in this manner due to timing constraints. Additionally, other systems such as traffic, power, and water networks are geographically spread, i.e., there are several independent entities with decision-making capabilities and possibly conflicting goals. In these situations, it might be necessary (or preferable) to provide each subsystem with a local controller with communication capabilities to attain a coordinated solution with the rest of the system. ˚Corresponding author. Email addresses: [email protected] (Eva Masero), [email protected] (Mario Francisco), [email protected] (Jos´e M. Maestre), [email protected] (Silvana Revollar), [email protected] (Pastora Vega) In general, it is better to carry out local controller negotiations in a proactive fashion for the sake of coordination, i.e., based on sequences of future states and inputs that provide local subsystems with future coupling information. For this reason, Model Predictive Control (MPC), a family of control methods that uses a system model to predict its evolution and calculates optimal control actions for a given horizon (Rossiter, 2003), has become a popular choice for distributed implementations. The MPC framework allows dealing explicitly with multiple variables, constraints, and disturbance information, which are very convenient features in this context. During the last decade, numerous Distributed Model Predictive Control (DMPC) schemes have been proposed with significant differences on issues such as system decomposition, coupling sources, and control goals, to name a few. Depending on the degree of cooperation, the DMPC method can be decentralized, non-cooperative, or cooperative. Hierarchical architectures are also used for coordination, e.g., to unify the control signals proposed by different agents for some interconnecting variables, with iterative and ‘price-based’ algorithms. See (Scattolini, 2009; Negenborn & Maestre, 2014) for reviews on this topic. However, coordination comes at a price in terms of communication overhead. For example, Maestre et al. (2015) show that some DMPC methods may require each agent to exchange thousands and even millions of floats per time instant to attain an optimal solution. Whether this is a limiting issue depends on the particular application, but it might be preferable to decrease the degree of optimality if the coordination burden is reduced. Numerous studies can be found in the literature about this research line, i.e., Berglind et al. (2012) and Mi et al. (2019) propose a self-triggered controller that decides the time to update the control input, thus reducing the communication costs. Other studies such as (Maestre et al., 2009; Xing et al., 2007) also focus on reducing the communication steps and power. Another approach to tackle excessive communication burden is fuzzy logic, which can be used as a tool for agent negotiations. The fuzzy logic theory was introduced by Zadeh (1973), allowing for the consideration of imprecise or uncertain information in the same way as human cognition and perception. One of the advantages of considering fuzzy logic in the negotiation framework is the possibility to characterize a suitable control action employing a large variety of criteria that, otherwise, would produce unwanted results or would be very difficult to quantify. For example, given a constraint, the control action can be considered ‘good’ or ‘bad’ concerning that constraint, instead of considering only a feasible or nonfeasible action. Other criteria that can be considered for assessing control actions in the fuzzy framework are economy, process safety, and environmental effects. The use of fuzzy logic as a negotiation tool is scarce in the literature. Kosonen (2003) presents a distributed traffic signal control system with a fuzzy inference system considering the economy, fluency, environment, and safety as imprecise criteria for negotiation. The results are discrete control actions for traffic lights. Sahebjamnia et al. (2016) propose a fuzzy Q-learning algorithm in the distributed control of chemical plants. Negotiation with fuzzy constraints for the planning and scheduling of supply chains is presented by Hsu et al. (2016). Other studies such as (Kowalczyk, 2002) and (Thibodeau et al., 2013) include decision-making process using fuzzy logic in the field of e-commerce, where this technique is popular. 2 This work proposes a two-layer hierarchical DMPC control architecture that uses fuzzy logic in the negotiation process. In the low-level control layer, agents negotiate in pairs according to the couplings and the possibilities offered by the communication network, assuming that the variables of interest that belong to other players will follow their current evolution. The pairwise negotiation is based on a modification of the work proposed by Maestre et al. (2011a), where a two-communication-step mechanism is considered to generate sub-optimal yet stable solutions with very low communication burden. One of the disadvantages of that method is the combinatorial explosion of possible inputs when considering two or more subsystems.In general, the more agents are involved in a negotiation, the more communication steps are required to attain an agreement. In this context, fuzzy negotiation presents a remarkable advantage over other methods since computing the final input does not require as many communication steps, but just merging all the inputs proposed by the involved agents. Likewise, the proposed scheme allows agents to preserve their state and even their internal model private, which can be useful in problems where agents are willing to cooperate but may have concerns in this regard. Thus, the original negotiation scheme is modified to deal with multi-agent systems by adding an extra fuzzy step that reduces the communication burden. Moreover, it is considered additional criteria in the decision-making process to soften the trajectory of control signals, which are prone to abrupt changes in the original method. As a result of the pairwise negotiations, a set of stabilizing sequences or proposals for the system is obtained. In particular, we consider three possible input trajectories for each subsystem: i) its shifted input (computed as an extension of the input applied at the previous time step), which guarantees stability in the closed-loop subsystem when feedback is selected properly; ii) its selfish input, which is the best control sequence locally computed; and iii) the altruistic input, which consists of the action wished and computed by its neighbor. This triplet seems a reasonable heuristic choice to obtain the final inputs via a fuzzification process, as proposed by Maestre et al. (2011a). Note that the referred method was compared with other well-known schemes in a four-tank benchmark, proving good performance despite its heuristic nature (Alvarado et al., 2011). Afterwards, the set of proposal is sent to the fuzzy-based supervisory layer, which merges them to obtain new control sequences for the system. Finally, stability is guaranteed for this fast decision-making scheme. Note that hybrid Fuzzy-MPC approaches are rare in the literature, except for (Francisco et al., 2019), a previous work where fuzzy negotiation was applied to a four-coupled-tank system, but without stability guarantees and limited to systems with only two agents. We would also like to stress that hierarchical-distributed architectures are common in this context. Likewise, some schemes rely on coordinators or supervisory layers (Zafra-Cabeza et al., 2011; Doan et al., 2014; Saad et al., 2018). The approach benefits are shown via simulation of a non-linear eight-coupled tanks system, designed as an extension of the quadruple-tank process (Johansson, 2000). The quadruple-tank process has significantly been used as a benchmark to analyze distributed MPC techniques (Alvarado et al., 2011), to test a robust tube-based MPC for tracking (Lim´on et al., 2010), to study multi-variable dead times (Shneiderman & Palmor, 2010), and for the stabilization of linear systems with delays (El Haoussi et al., 2011). 3 The rest of the article is organized as follows. Section 2 introduces the problem formulation and the control goal. In Section 3, the proposed control architecture, the control algorithm, and the fuzzy pairwise negotiation are detailed. Section 4 displays stability properties and the procedure used to design the controller. Section 5 illustrates the proposed fuzzy DMPC scheme and the fuzzy negotiation criteria employed in the eight-coupled tanks benchmark. The results obtained from the simulation of this plant are shown in Section 6. Finally, conclusions are summarized in Section 7. Notation: N0`and R`are, respectively, the sets of non-negative integers and positive real numbers. Rnrefers to an n-dimension Euclidean space. The scalar product of vectors a, b PRnis denoted as abJor a¨b. Given sets X,YĎRn, the Cartesian product is XˆYfitpx, yq:xPX, y PYu. If tXiuiPNis a family of sets indexed by N, then the Cartesian product is ŚiPNXifiX1ˆ ¨ ¨ ¨ ˆ XN“ tpx1, . . . , xNq:x1PXi, . . . , xNP XNu. Moreover, the Minkowski sum is X‘Yfitx`y:xPX, y PYu. The set subtraction operation is symbolized by z. The image of a set XĎRnunder a linear mapping A:RnÞÑ Rmis AXfitAx :xPXu. The integral of the squared error εis defined as ISE “ş8 0ε2dt. The Np´long future input sequence at time instant kis defined as Upkq“rupkq, upk`1q, . . . , upk`Np´1qsJ, and the optimal sequence is denoted as U˚pkq“ru˚pkq, u˚pk`1q, . . . , u˚pk`Np´1qsJ. 2. Problem Formulation Let the system be composed of a set N={1, 2, . . . , N}of input-coupled subsystems whose state evolution is xipk`1q “ Aixipkq ` Biiuipkq ` wipkq,(1) where kPN0`denotes the time instant; xiPRqiand uiPRriare, respectively, the state and input vectors of each subsystem iPN, constrained in the convex sets containing the origin in their interior XifitxiPRqi:Ax,ixiďbx,iuand UifituiPRri:Au,iuiďbu,iu, respectively; and AiPRqiˆqiand Bii PRqiˆriare matrices of proper dimensions. The measurable disturbances vector wiPRqirepresents the coupling with other subsystems j belonging to the set of neighbors NifitjPNztiu:Bij ‰0u, i.e., wipkq “ ÿ jPNi Bijujpkq,(2) where ujPRrjis the input vector of subsystem jPNi, and matrix Bij PRqiˆrjmodels the input coupling between iand j. Moreover, wiis bounded in a convex set WifiÀjPNiBijUj due to the system constraints. The neighborhood affected by agent iis defined as Mifi tjPNztiu:Bji ‰0u. From the global viewpoint, the overall system evolution can be aggregated as xNpk`1q “ ANxNpkq ` BNuNpkq,(3) 4 where AN“ rAijsi,jPNand BN“ rBijsi,jPNare, respectively, the state and input-to-state matrices of the overall system. Global state and input vectors are constrained in XN“ ŚiPNXiand UN“ŚiPNUi, respectively. Regarding mutual interaction, wNis implicitly included in (3). 2.1. Control objective The control objective is to drive the system towards the origin of the state space guaranteeing that constraints are satisfied, and minimizing the sum of the local cost functions. At each time instant k, the cost function of subsystem iPNis calculated based on the predicted trajectories of its states and inputs over a future window of length Np(the so-called prediction horizon): Ji`xipkq, Uipkq, Ujpkq˘“ Np´1 ÿ t“0 Li`xipk`tq, uipk`tq˘`Fi`xipk`Npq˘,(4) where Lip¨q is the stage cost function, and Fip¨q is the terminal cost function defined as Li`xipk`tq, uipk`tq˘“xipk`tqJQixipk`tq ` uipk`tqJRiuipk`tq, Fi`xipk`Npq˘“xipk`NpqJPixipk`Npq, with Qibeing a semi-positive definite matrix, and Ri, Pibeing positive-definite matrices. 3. Proposed Control Architecture Highly coupled agents communicate using a network that can be modeled as the indirect graph G“ pN,Lq, where Nis the set of agents, and Lis the set of bidirectional links LĎLN“ tti, ju:ti, ju Ď N, i ‰ju, i.e., a link lij PLconnects agents iand jproviding a bidirectional information flow. If two agents are connected by a communication link lij, they follow a pairwise cooperative scheme to find a consensus on their control sequences via a multi-layer fuzzy negotiation algorithm. The multi-layer fuzzy-based control architecture is shown in Fig. 1. The low-level control layer uses DMPC and fuzzy techniques to deal with the pairwise-agent negotiations assuming that interest variables that belong to other agents follow their current evolutions. Then, the sequences resulting from the pairwise negotiations are sent to a supervisory layer, which merges and fuzzifies them to compute the final control sequence for the whole system. 3.1. Low-level control layer A DMPC algorithm for multiple agents is performed by the lower control layer. Specifically, fuzzy-based negotiations are made in pairs considering the couplings with their neighboring subsystems, which are assumed to hold their current trajectories. To this end, it is 5 Fuzzy Negotiation Fuzzy Pairwise Negotiation Distributed MPC Low-level control layer Upper-level control layer Figure 1: The multi-layer fuzzy-based control architecture. used a shifted sequence of agent i, which is defined by adding Kixipk`Npqto the sequence chosen at the previous time step Uipk´1q: Us ipkq “ » — — — — — – uipk`1|k´1q uipk`2|k´1q . . . uipk`Np´1|k´1q Kixipk`Np|k´1q fi ffi ffi ffi ffi ffi fl “ » — — — — — – us ipkq us ipk`1q . . . us ipk`Np´2q us ipk`Np´1q fi ffi ffi ffi ffi ffi fl .(5) 3.1.1. Distributed MPC algorithm for multiple agents Algorithm 1 extends the DMPC scheme proposed by Maestre et al. (2011a) for Nsubsystems without combinatorial explosion. Algorithm 1 (Stage 1): Agent Proposals. At each time step k: 1. Firstly, agent imeasures its local state ˜xipkq. 2. Agent icalculates its shifted trajectory Us ipkq, and sends it to its neighbors. 3. Agent iminimizes its cost function considering that neighbor jPNiapplies its shifted trajectory Us jpkq. It is assumed that the rest of the neighboring subsystems lPNiztjufollows their current control trajectories Us lpkq. Specifically, agent isolves U˚ ipkq “ arg min UipkqJi`xipkq, Uipkq, Us jpkq, Us lpkq˘,(6) subject to xipk`t`1q “ Aixipk`tq ` Biiuipk`tq ` Bijujpk`tq ` ÿ lPNiztju Bilulpk`tq, 6 xipkq “ ˜xipkq, i PN, xipk`tq P Xi, t “0, . . . , Np´1, xipk`Npq P Ωi, uipk`tq P Ui, t “0, . . . , Np´1, ujpk`tq “ us jpk`tq, t “0, . . . , Np´1, ulpk`tq “ us lpk`tq, t “0, . . . , Np´1, where set Ωiis imposed as terminal state constraint of agent i. Details regarding the calculation of Ωiare given in Section 4. 4. Agent ioptimizes again its cost Jip¨q maintaining its optimal input sequence U˚ ipkq fixed to find the input sequence wished for its neighbor j, i.e., Uwi jpkq. Here, it is also assumed that subsystems lfollow their current trajectories. To this end, agent isolves Uwi jpkq “ arg min UjpkqJi`xipkq, U˚ ipkq, Ujpkq, Us lpkq˘,(7) subject to xipk`t`1q “ Aixipk`tq ` Biiuipk`tq ` Bijujpk`tq ` ÿ lPNiztju Bilulpk`tq, xipkq “ ˜xipkq, i PN xipk`tq P Xi, t “0, . . . , Np´1, xipk`Npq P Ωi, uipk`tq “ u˚ ipk`tq, t “0, . . . , Np´1, ujpk`tq P Uj, j PNit“0, . . . , Np´1, ulpk`tq “ us lpk`tq, t “0, . . . , Np´1. 5. Finally, agent isends Uwi jpkqto agent j, and receives Uwj ipkq. 3.1.2. Fuzzy pairwise negotiations Following Stage 1 of the proposed algorithm, the idea is to apply fuzzy negotiation to obtain a control solution that decreases the performance index and guarantees stability. To this end, each pair of agents ti, juhas control actions tU˚ ipkq, Us ipkq, Uwj ipkqu and tU˚ jpkq, Us jpkq, Uwi jpkqu. A fuzzy inference system for negotiation generates the final control actions Uf ipkqand Uf jpkq, considering some operational and economic constraints. Fig. 2 depicts the fuzzy negotiation scheme (Uddin & Rahman, 1999) that will be implemented to control the closed-loop system. The main three steps of the negotiation process are: 1) Fuzzification: It consists of converting a numerical variable into a linguistic variable. In this way, the imprecise process knowledge determined by the membership functions transforms a crisp numerical value into fuzzy degrees of membership for each linguistic variable (Raviraj & Sen, 1997). For example, the temperature value in a boiler can be characterized by the degree of ‘high’ and ‘low’. If more linguistic labels are available, the temperature can be classified by: ‘dangerous’, ‘high’, ‘medium’, and ‘low’. 7 Fuzzifier DefuzzifierRule Evaluator Data Base Fuzzy Set Rule System 𝑈𝑖 fOutput Ref.Error FUZZY NEGOTIATION DMPC {𝑈𝑖 ∗, 𝑈𝑖 s,𝑈𝑖 𝑤𝑗} − + Figure 2: Scheme of the low-level control layer. Let xbe the state, input, or other algebraic variable characterizing the process (e.g., residence time in a water tank, energy consumption, and temperature). If only two linguistic labels are available for describing the variable, two fuzzy sets T1and T2can be considered. Their membership functions µT1pxqand µT2pxq: µT1pxq “ $ ’ & ’ % 1 for xăa b´x b´afor aďxăb 0 for bďx , µT2pxq “ $ ’ & ’ % 0 for xăa a´x a´bfor aďxăb 1 for bďx ,(8) are, respectively, associated with the linguistic labels ‘high’ and ‘low’, as shown in Fig. 3. a b x 0 0.5 1 1.5 T1(x); T2(x) Figure 3: Membership functions (blue solid line: µT1pxq, and magenta dashed line: µT2pxq). The process knowledge can often allow for the definition of more than two linguistic labels. Although the methodology presented here is for three linguistic labels, its extension is straightforward. If three linguistic labels are considered, three fuzzy sets T3,T4, and T5 must be defined. Let ybe another state, input, or algebraic variable of the process, the 8 Theorem 1. Let the global matrices be AN“ rAijsi,jPNand BN“ rBijsi,jPN, and stage cost matrices QN“diagpQiqiPNand RN“diagpRiqiPN. If there are matrices HN“HJ N“ diagpHiqiPN, where HiPRqiˆqi, and YN“diagpYiqiPN, where YiPRriˆqiin such a way that it is held: » — — — – HNHNAJ N`YJ NBJ NHNQ1{2 NYJ NR1{2 N ANHN`BNYNHN0 0 Q1{2 NHN0I0 R1{2 NYN0 0 I fi ffi ffi ffi fl ě0,(22) then there exits a matrix PN“H´1 Nthat satisfies (21), and a feedback control matrix KN“ YNH´1 Nthat stabilizes the closed-loop system. Proof. Following Maestre et al. (2011b), the condition (22) is derived from the recursive application of the Schur’s complement to pAN`BNKNqJPNpAN`BNKNq ´ PN`QN`KJ NRNKNď0.(23) Secondly, it is proved that the closed-loop system is stable with the linear control law uN“KNxN. Pre- and post-multiplying (23) by xNpkqJand xNpkq, and then by ´1, it can be rewritten as xNpkqJPNxNpkq ´ xNpk`1qJPNxNpk`1q ě LN`xNpkq˘,(24) where xNpk`1q “ pAN`BNKNqxNpkq, lN`xNpkq˘“xNpkqJQNxNpkq ` xNpkqJKJ NRNKNxNpkq. Hence, VN`xNpkq˘“xNpkqJPNxNpkqis a Lyapunov function and stability is guaranteed. Moreover, it can be checked that (21) holds by telescope summation. The LMI (22) is solved maximizing the trace of HNto design the controller for the overall system. This fact leads to the minimization of the trace of PN“H´1 N, and, therefore, minimizing the cost-to-go. The resulting feedback matrix KN“YNH´1 Nis the gain of the controller. 4.3. Stability of the MPC controller In the previous section, the system’s stability with the corresponding KNhas been proved. Here, it also proved stability guarantees when the MPC controller is used. Assumption 4. There is a feasible overall control sequence at time instant k“0. Theorem 2. Given matrices KN, PNcomputed by maximizing the trace of HNsubject to LMI (22), and let Assumption 4 hold, the MPC controller can ensure recursive feasibility and closed-loop system stability. 15 Proof. Given a feasible control sequence at time step k: UNpkq “ ruNpkq, uNpk`1q, . . . , uNpk`Np´1qs,(25) the constraints xNpkq P XN,uNpkq P UN, and xNpk`Npq P ΩNare satisfied. A feasible shifted sequence can then be formed with the tail extension of (25) at time instant k`1: Us Npk`1q “ rus Npk`1q, us Npk`2q, . . . , us Npk`Npqs “ ruNpk`1q, uNpk`2q, . . . , KxNpk`Npqs.(26) Since xNpk`Npq P ΩN, the following equation holds at k`1: pAN`BNKNqxNpk`Npq P ΩN. Hence, under Assumption 4, and applying this procedure recursively, feasibility is held for all k. Furthermore, it is proved that the global cost decreases in time when using Us Nas a backup control sequence. To this end, the overall cost decreases from time step kto k`1 if it holds that JN`xNpk`1q, Us Npk`1q˘ďJN`xNpkq, UNpkq˘.(27) Substituting (15) in (27), and considering definitions (25) and (26), we get Np ř t“1 LN`xNpk`tq, us Npk`tq˘`FN`xNpk`Np`1q˘ď Np´1 ř t“0 LN`xNpk`tq, uNpk`tq˘`FN`xNpk`Npq˘. (28) Removing terms in common and rearranging, we have LN`xNpk`Npq, us Npk`Npq˘`FN`xNpk`Np`1q˘´FN`xNpk`Npq˘ ďLN`xNpkq, uNpkq(29) Note that the left side hand of (29) is lower than or equal to zero because it is imposed by design (24), and the right side hand is greater than or equal to zero since the stage cost is positive definite. Thus it is proved that (27) is satisfied using the shifted control sequence, and, consequently, the global cost decreases in time. Finally, note that the control algorithm includes condition (17) to ensure stability. If it is not satisfied with the fuzzified control action obtained in the negotiation, the shifted control signal is implemented, which has been proved to decrease the overall cost. Hence, the feedback controller can be seen as a backup control strategy to provide the proposed scheme with feasibility and stability guarantees. In a nutshell, stability holds in a straight forward manner when Us Nis applied, which is formed using a linear feedback conveniently designed using an LMI-based procedure. At each time instant, a convex combination of feasible sequences is also formed by using 16 fuzzy inference and then its performance is compared with that of the stabilizing sequence Us N. If the fuzzy sequence Uf Nresults in a decrease of the cost of the system, then it is adopted and implemented; otherwise Us Nis kept and applied. In either case, there is a feasible sequence available at the next time step that can be shifted to form the new Us N guaranteeing both feasibility and stability. Hence, the feedback provides us with a backup strategy that guarantees stability while the fuzzy proposal created at each time step can introduce improvements in the control sequence that optimize the system evolution according to the fuzzy tuning parameters 5. Case Study This section describes the eight-coupled tanks plant based on the quadruple tank process (Johansson, 2000) and specifies the non-linear model and the parameters used in the simulations. Furthermore, the fuzzy negotiation criteria are detailed, bearing in mind the goal and constraints of the problem. 5.1. Plant description The eight-coupled tanks plant is composed of eight interconnected tanks as depicted in Fig. 7. There are four upper tanks (3,4,7, and 8) that discharge flows into the lower ones (1,2,5, and 6), and these, in turn, into sinking tanks. Four pumps control the plant, and there are also six three-way valves γv, with vP t1,2, . . . , 6umanually operated that divide the input flows into two ways. The system is divided into N“4 subsystems: tanks 1 and 3 belong to subsystem 1; tanks 2 and 4 form subsystem 2; tanks 5 and 7 take part in subsystem 3; and the remaining tanks form subsystem 4. The control goal is to reach some target levels considering the operational cost and satisfying operational constraints. Hence, it is a multi-variable control problem with four outputs (h1,h2,h5, and h6) and four inputs (qa,qb,qc, and qd). ?3·qb 1 2 3 4 5 6 7 8 h1h2 h4 h3 h5 h7h8 h6 qaqbqcqd (1- ?1)·qa (1-?2-?3)·qb (1-?5)·qd (1-?4-?6)·qc ?6·qc ?1·qa?2·qb?4·qc?5·qd Subsystem 1 Subsystem 2 Subsystem 3 Subsystem 4 Figure 7: Schematic diagram of the eight-coupled tanks plant with the proposed subsystems. 17 5.2. Plant and control model The non-linear model of the plant can be obtained by applying mass balances and Bernoulli’s law. The following differential equations describe the system: S1 dh1 dt “a3a2gh3´a1a2gh1`γ1 qa 3600, S2 dh2 dt “a4a2gh4´a2a2gh2`γ2 qb 3600, S3 dh3 dt “ ´a3a2gh3` p1´γ2´γ3qqb 3600, S4 dh4 dt “ ´a4a2gh4` p1´γ1qqa 3600 `γ6 qc 3600, S5 dh5 dt “a7a2gh7´a5a2gh5`γ4 qc 3600, S6 dh6 dt “a8a2gh8´a6a2gh6`γ5 qd 3600, S7 dh7 dt “ ´a7a2gh7` p1´γ5qqd 3600 `γ3 qb 3600, S8 dh8 dt “ ´a8a2gh8` p1´γ4´γ6qqc 3600, (30) where hnis the water level of tank nP t1,2, . . . , 8u,Sn“13.89¨10´3m2is the corresponding cross-section (equal for all tanks), and an“50.265 ¨10´6m2is the cross-section of all outlet pipes. The parameter γvP r0,1swith vP t1,2, . . . , 6urefers to the opening of the six threeway valves (γ1, γ4=0.3, γ2, γ5=0.4, and γ3, γ6=0.1); g“9.8 m{s2is the gravity constant; and qmcorresponds to the flow given by pump mP ta,b,c,du. Let us define the operating point of tank level h0 nfor nP t1,2, . . . , 8uas h0 1“0.10, h0 2“ 0.15, h0 3“0.07, h0 4“0.03, h0 5“0.10, h0 6“0.15, h0 7“0.025,and h0 8“0.10 (units: meters), and the operating point of pumps’ flow rate q0 mfor mP ta,b,c,duas q0 a“0.142, q0 b“0.421, q0 c“0.424,and q0 d“0.140 (units: m3{h). The linear discrete-time state-space model is ¯xNpk`1q “ AN¯xNpkq ` BN¯uNpkq,(31) where the state vector becomes ¯xN“ rh1pkq ´ h0 1, . . . , h8pkq ´ h0 8sJ; the input vector ¯uN“ rqapkq ´ q0 a, . . . , qdpkq ´ q0 dsJ; and AN,BNare the corresponding matrices of the global system. Similarly, the dynamics of each subsystem iP t1,2,3,4uare ¯xipk`1q “ Ai¯xipkq ` Bii ¯uipkq ` ¯wipkq, ¯wipkq “ ř jPNi Bij ¯ujpkq,(32) 18 where ¯wipkqrepresents the coupling with its neighbours jPNi, and the corresponding matrices of subsystems are: A1“„0.8810 0.1325 0 0.8587, B11 “„0.0282 0, B12 “„0.0035 0.0464, A2“„0.9017 0.1922 0 0.7973, B21 “„0.0071 0.0626, B22 “„0.0380 0, B23 “„0.0010 0.0089, A3“„0.8810 0.2104 0 0.7753, B32 “„0.0011 0.0088, B33 “„0.0282 0, B34 “„0.0067 0.0530, A4“„0.9017 0.1126 0 0.8814, B43 “„0.0035 0.0564, B44 “„0.0380 0, Aij “„0 0 0 0, i ‰j@i, j PN. Moreover, states and inputs are constrained by ´h0 nă¯xnpkq ď 0.08,´q0 mă¯unpkq ď 0.4, @nP t1,2, . . . , 8u,@mP ta,b,c,du. Matrices QN“diagpQiqiPNand RN“diagpRiqiPNare, respectively, the corresponding constant weighting matrices with: Qi“„1 0 0 0,@iP t1,2,3,4u, R1“„0.002 0 0 0.2,1R2, R3“» – 0.2 0 0 0 0.2 0 0 0 0.2 fi fl, R4“„0.02 0 0 0.2. Furthermore, local feedback gain Ki, and weighting matrix of the terminal cost Pi, which 1The weighting parameter R11 “0.002 is intentionally different to have one controller out of tune and assess its impact on the proposed scheme. 19 have been calculated maximizing the trace of HNwith Theorem 1, are KJ 1“„´0.0567 0.0913 , P1“„4.6685 2.3288 2.3288 2.4292, KJ 2“„´0.2863 ´0.2331, P2“„5.1779 3.0510 3.0510 3.0250, KJ 3“„´0.0575 ´0.0922, P3“„4.7185 2.8848 2.8848 3.0013, KJ 4“„´0.2319 ´0.1585, P4“„5.2133 2.4217 2.4217 2.3883. A standard approach has been implemented with MPT of MATLAB (Herceg et al., 2013) to compute each agent’s RPI set. The corresponding invariant sets Ωiof all subsystems are depicted in Fig. 8. Figure 8: Invariant sets of the four subsystems. 5.3. Fuzzy negotiation specifications This section details the fuzzy negotiation criteria in the eight-coupled tanks system, taking into account the process constraints and operational costs. 1) Fuzzification: Three performance indicators have been considered for the fuzzy negotiation procedure: the residence time for water stored in the lower tanks, the current water level, and the pumping energy. They represent typical constraints for the plant, and, in this case, act as fuzzy criteria similarly to soft constraints. 20 i. Residence Time (RT ) is the time that water spends at each tank. At each time instant k, the RT for tank nP t1,2,5,6uis evaluated with level hnat the end of the prediction horizon Npas RT npkq “ Sn ana2ghnpk`Npq.(33) This criterion is particularly interesting if some chemical reactions take place in the tanks. Thus, a minimum residence time has to be imposed at each tank to fulfill the requirements for reactions. To this end, two fuzzy sets representing constraint satisfaction and violation are defined by the limits presented in Table 1. ii. Current water level (h) is essential to prevent overflow or excess of product storage. Although a hard constraint is also added to the DMPC to avoid spilling, the water height hnis considered in the fuzzy negotiation to improve process safety. Due to the plant setup and the distributed control framework, only lower-tank levels (h1, h2, h5, and h6) are evaluated because the upper ones are only influenced by their neighbors’ control actions. Three linguistic variables (‘low’, ‘medium’, and ‘high’) are considered for water levels. iii. Pumping Energy (P E) is defined for each pump mP ta,b,c,duas the average P E over the prediction horizon, and it is assumed to be proportional to the water flows provided by pumps, i.e., PEmpkq “ 0.04 3600Np Np ÿ t“1 qmpk`tq.(34) Note that the definition includes absolute flow values instead of flow increments. Finally, the total pumping energy is computed as PEpkq “ P Eapkq ` PEbpkq ` P Ecpkq ` P Edpkq,(35) and the fuzzy sets for the three criteria are determined by the values of Table 1, representing the limits and vertices of triangles that define the membership functions. Table 1: Summary of fuzzy sets limits for the three performance indicators. Performance indicators Limits P Em (kWh) hn (m) RTn (s) a1¨10´5{Np0.15 26 b5¨10´5{Np0.20 32 c10 ¨10´5{Np0.25 ´ 21 2) Rule evaluation: A set of fuzzy rules has been defined according to the process. In this plant, only three negotiation problems between subsystems t1,2u,t2,3u, and t3,4uare performed in the lower layer due to their physical distribution. A generic rule to assign the fitness for control signal qmcan be expressed as Rri: If RTnpk`Npqis Tr1,PEmpkqis Tr2, and hnpk`Npqis Tr3, then output qmis Tr4, where Rri represents the r-th rule of subsystem iP t1,2,3,4u,Tr1={‘high’, ‘low’}, Tr2={‘high’, ‘medium’, ‘low’},T r3={‘high’, ‘medium’, ‘low’}, and T r4={‘good’, ‘acceptable’, ‘bad’}are the linguistic variables that characterize the criteria RTn,PEm, hn, and qm, respectively. Here are some examples: R1i: If (RTnis ‘high’), (P Emis ‘low’), and (hnis ‘medium’), then (qmis ‘good’), R2i: If (RTnis ‘low’), (P Emis ‘high’), and (hnis ‘high’), then (qmis ‘bad’). Since the number of antecedents in each rule is 3, a full set of rules comprises Nr“18 rules. 3) Defuzzification: It provides the fitness of a control sequence for each rule Rri, which is given by αRri pqmq “ wRTn r¨µT r1pRTnq ¨ wP Em r¨µT r2pPEmq ¨ whn r¨µT r3phnq,(36) where µT r1pRTnq,µT r2pP Emq, and µT r3phnqare the fuzzy sets for each criterion considered in rule r; and wRTn r,wP Em r, and whn rare the considered weights that depend on the linguistic variables included in the rule. In this work, the weights, whose values only depend on their associated fuzzy sets, are independent of the rule and the subsystem where they are applied. Specifically, weights wRT ,wP E , and whare wRT “"w1if RT is T1(high) w2if RT is T2(low) , wP E “$ & % w3if PE is T3(high) w4if PE is T4(medium) w5if PE is T5(low) , wh“$ & % w6if his T6(high) w7if his T7(medium) w8if his T8(low) . Finally, the total fitness of the control signal qmfor the full set of Nrrules is Tαpqmq “ Nr ÿ r“1 αRri pqmq.(37) The specific weights of each linguistic label are defined in the results section. 22 6. Results Results of the proposed DMPC with fuzzy negotiation are here presented. The sampling time used in simulations is Ts“5 s, and the MPC controllers consider a prediction horizon Np“20. To show the benefits of the proposed Fuzzy DMPC, it is compared with the centralized MPC and DMPC using a cooperative game (Maestre et al., 2011a), considering the same tuning parameters for local controllers. The influence of the fuzzy rules is also evaluated. The weights considered for each performance indicator in the fuzzy rules are shown in Table 2. Table 2: Weights of the three performance indicators for several criteria. Weights Alternative Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 RTn Low (w1) 0.01 1 0.01 0.01 0.01 0.01 High (w2) 1 0.01 1 1 1 1 hm Low (w3) 0.01 0.01 1 0.02 1 1 Medium (w4) 0.5 0.5 0.5 0.5 0.5 0.5 High (w5) 1 1 0.02 1 0.02 0.02 P Em Low (w6) 1 1 1 1 0.02 0.9 Medium (w7) 0.5 0.5 0.5 0.5 0.5 0.5 High (w8) 0.01 0.01 0.01 0.01 0.9 0.02 0 200 400 600 800 1000 1200 Time [s] 8 10 12 14 16 18 20 22 Level [cm] Tanks level h1 h2 h5 h6 href (a) Fuzzy DMPC (Case 5). 0 200 400 600 800 1000 1200 Time [s] 8 10 12 14 16 18 20 22 Level [cm] Tanks level h1 h2 h5 h6 href (b) Fuzzy DMPC (Case 6). Figure 9: Comparison of tanks level evolution (solid line: Centralized MPC, dashed line: Fuzzy DMPC, and dotted line: DMPC cooperative game). 23 0 200 400 600 800 1000 1200 Time [s] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Flow rate [m3/h] Pump flow rates qa qb qc qd (a) Fuzzy DMPC (Case 5). 0 200 400 600 800 1000 1200 Time [s] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Flow rate [m3/h] Pump flow rates qa qb qc qd (b) Fuzzy DMPC (Case 6). Figure 10: Comparison of pumps flow rates (solid line: Centralized MPC, dashed line: Fuzzy DMPC, and dotted line: DMPC cooperative game). Table 3: Comparison of ISE and P E for the considered control techniques. Control techniques Fuzzy DMPC (Case 5) Fuzzy DMPC (Case 6) DMPC Coop. Game Centralized MPC Total ISE 0.7913 0.7335 0.7867 0.6171 Total P E 1.605 ¨10´21.597 ¨10´21.600 ¨10´21.601 ¨10´2 As displayed in Table 3, Fig. 9, and Fig. 10, centralized MPC is always the controller with the smallest Integral Squared Error (ISE) for tracking due to the availability of full plant information for prediction. On the other hand, fuzzy DMPC (Case 5) presents a higher ISE than DMPC with the cooperative game because the fuzzy rules favor high pumping energies (see Fig. 9 (a) and Fig. 10 (a)). On the contrary, fuzzy DMPC (Case 6) presents smaller ISE and P E than DMPC with the cooperative game because the fuzzy weights favor small PE, but still higher than centralized MPC (see Fig. 9 (b) and Fig. 10 (b)). Although the pumping energy is considered in absolute value in these simulations, increments could also be included as a performance indicator for the fuzzy negotiation. The conclusion is that fuzzy DMPC performance depends strongly on the weights of the considered fuzzy rules, giving the control system designer freedom to promote different behaviors. In this regard, the flexibility of the fuzzy approach simplifies the controller’s tuning to attain better Key Performance Indicators (KPIs). As for the standard MPC approaches, tuning parameters (typically Qi,Ri, and Np) are indirectly related to these KPIs, and hence, complicating the controller setup. In the rest of this section, the standard MPC controllers will remain fixed to assess the tuning of the fuzzy rules in our approach. 24