Modular Feedback Control of Networked Systems by Clustering: A Drinking Water Network Case Study

processes Article Modular Feedback Control of Networked Systems by Clustering: A Drinking Water Network Case Study José María Maestre 1,* , Francisco Lopez-Rodriguez 1, Francisco Javier Muros 1 and Carlos Ocampo-Martinez 2   Citation: Maestre, J.M.; Lopez-Rodriguez, F.; Muros, F.J.; Ocampo-Martinez, C. Modular Feedback Control of Networked Systems by Clustering: A Drinking Water Network Case Study. Processes 2021,9, 389. https://doi.org/ 10.3390/pr9020389 Academic Editors: Pastora Isabel Vega Cruz and Ramón Vilanova Arbós Received: 31 December 2020 Accepted: 10 February 2021 Published: 20 February 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Systems and Automation Engineering, University of Seville, 41092 Seville, Spain; [email protected] (F.L.-R.); [email protected] (F.J.M.) 2Automatic Control Department, Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Universitat Politècnica de Catalunya, 08028 Barcelona, Spain; [email protected] *Correspondence: pepemaestr[email protected]; Tel.: +34-652804804 Abstract: This article presents a method based on linear matrix inequalities (LMIs) for designing a modular feedback control law, whose synthesis guarantees the system stability, while switching to different network topologies. Such stability is achieved by means of a common Lyapunov function to all network admissible configurations. Several mechanisms to relieve the computational burden of this methodology in large-scale systems are also presented. To assess its applicability, the modular controller is tested on a real case study, namely the Barcelona drinking water network (DWN), and its performance is compared with that of other control strategies, showing the effectiveness of the proposed approach. Keywords: modular control; clustering; coalitional control; distributed control; water systems; drinking water networks (DWNs) 1. Introduction In recent years, distributed control architectures have gained relevance due to the multiple advances in information and communication technologies [ 1 ]. In general, distributed control systems are characterized by independent and interacting subsystems governed by controllers that exchange information to get additional performance by coordinating their control actions. This approach is useful in many practical problems that cannot be addressed from a centralized perspective, e.g., due to the sheer size of the system and/or limits in the information exchange between controllers [2]. In this context, we are interested in the design of feedback controllers that respect the constraints imposed by the communication topology. To this end, several methods have been proposed in the literature. For example, in [ 3 ], a gradient method for considering sparsity constraints in linear quadratic regulators is implemented by a game-theoretic algorithm. In [ 4 ], communication constraints are imposed via sparsity-promoting penalty functions on the cardinality of the communication links used in the control architecture. In [ 5 ], the notion of quadratic invariance of a set of sparsity or delay constraints on the feedback controller is introduced as a means to guarantee a convex design problem. Also, a similar strategy was recently used in [ 6 ] for large-scale systems. Linear matrix inequality (LMI)-based approaches for feedback controllers can also be found in the literature. For instance, a design method based on LMIs for distributed linear systems is proposed in the context of coalitional control first in [ 7 ] and later in [ 8 ]. Under this framework, communication links remain enabled as long as they provide a significant performance increase, and otherwise they are disconnected. As a consequence, controllers are grouped dynamically into disjoint cooperating sets, which are called coalitions. In this way, a trade-off between performance and communication burden is obtained at the cost of a more complex implementation. In particular, the design method seeks a feedback gain K with as many Processes 2021,9, 389. https://doi.org/10.3390/pr9020389 https://www.mdpi.com/journal/processes Processes 2021,9, 389 2 of 18 zero entries as possible. Some other works also propose LMI-based methodologies [9–11] to design sparse feedback matrices. Similarly, H∞ techniques are used in [ 12 ] to achieve a structured feedback control law. It is also remarkable that these design constraints can also be considered to compute other classical controllers by using a feedback gain, e.g., a proportional-integral controller for an irrigation canal is calculated in [ 13 ] in this manner. Here, we apply the modular control foundations presented in [ 14 ], which rely on the principles introduced in [ 7 , 8 ]. The key idea of this methodology is that the feedback controllers used in the different topologies are created considering a common template composed of pieces associated with the different communication links in the network. Hence, if a link is disabled the corresponding elements in the controller are merely replaced by zeros without affecting the rest of the elements. As a consequence, given two different network topologies, the nonzero elements of the corresponding feedback controllers share the same values. Therefore, the feedback matrix shall easily change its internal configuration depending on the topology to be controlled. Furthermore, the modular control law guarantees the stability by means of a common Lyapunov function to all network topologies. Undoubtedly, this control strategy can be of interest for the aforementioned framework of coalitional control, where the information structure of the system plays an essential role [ 15 – 19 ]. Natural application fields for this type of online partitioning approach are, e.g., traffic [ 20 ], water [21–23] , cellular [ 24 , 25 ] and power networks [ 26 ], and renewable energy generation systems [ 27 , 28 ]. In particular, the approach presented here may be suitable to be combined with the game-theoretic methods in [ 29 – 31 ], which are also based on LMIs, and with plug-and-play control capabilities [ 32 – 34 ]. Also, controllers designed with the modular structure mentioned above are interesting for applications where communications between controllers can fail, e.g., due t o packet losses and jamming attacks [ 35 – 37 ]. The rationale is simple: any missing information from a neighbor can be considered to be a disabled communication link, which allows applying the results of this article. Likewise, model predictive control (MPC)-based works could use the proposed controller as stabilizing terminal feedback, especially for coupling-dependant clustering architectures, e.g., in [38–40], and also including most of the coalitional works commented before. The main contribution of this paper is a methodology to design a feedback gain suitable for the control of networked systems by clustering [ 14 ], which is applied here to complex large-scale systems. Certainly, modular controller synthesis for large-scale schemes is challenging due to the computational complexity. Different ways to tackle the computational explosion while guaranteeing the system stability are shown in this article. In particular, an approach based on clustering of agents has been implemented in the Barcelona drinking water network (DWN) as a case study, where the model deployed in [ 41 – 43 ] has been partitioned into a network of eight agents following [ 44 ]. In fact, the need for systematic methods to achieve the partitioning objective has gained importance recently, with partitioning schemes based on graph theory [ 40 , 42 , 44 – 48 ], states and inputs estimation [ 49 ], social network algorithms [ 50 ], genetic algorithms [ 51 ], and PageR- ank [ 22 , 52 ]. Specific partitioning techniques applied to large-scale water systems as the proposed case study can be found in [ 42 – 44 , 50 , 53 – 55 ]. Indeed, control applications to water distribution systems are becoming more common, with recent contributions in the monitoring and control of valves leakage [ 56 ], pumps speed [ 57 ], pressure management by clustering [23], or pump scheduling [58]. The outline of the rest of the paper is organized as follows. Section 2presents the problem statement. Section 3introduces the concept of modular control, providing an LMI-based design method together with some properties of interest. Section 4proposes different techniques to tackle the computational explosion for large-scale systems. Section 5 introduces the application of modular feedback control to the Barcelona DWN case study. Finally, concluding remarks are given in Section 6. Processes 2021,9, 389 3 of 18 2. Problem Formulation We consider an overall discrete linear time-invariant system, which is composed of a set N={1, 2, . . . , N}of subsystems characterized as xi(k+1) = Aiixi(k) + Biiui(k) + ˆ di(k), ˆ di(k) = ∑j6=iAijxj(k) + ∑j6=iBijuj(k),(1) where xi∈Rnxi and ui∈Rnui , with i= 1, . . . , n , are the states and inputs of each subsystem i∈ N , respectively, with nxi and nui denoting the size of vectors xi and ui . Matrices Aii ∈Rnxi×nxi and Bii ∈Rnxi×nui refer to the state and input-to-state matrices, and ˆ di∈Rnxi represents the influence of the neighboring states and inputs in the update of xi . Finally, matrices Aij ∈Rnxi×nxj and Bij ∈Rnxi×nuj map the states and inputs of subsystem j∈ N into the states of subsystem i, respectively. The goal of the subsystems in Nis to minimize the following stage cost: `i(k) = xT i(k)Qixi(k) + uT i(k)Riui(k), (2) where Qi∈Rnxi×nxi and Ri∈Rnui×nui are positive semi-definite and definite constant weighting matrices, respectively. From a global viewpoint, the overall dynamics are simply described by xN(k+1) = ANxN(k) + BNuN(k), (3) where subscript N emphasizes that all system vectors and matrices come from the aggregation of local subsystems, i.e., xN= [xi]i∈N , uN= [ui]i∈N , AN= [Aij]i,j∈N , and BN= [Bij]i,j∈N . For convenience, we will respectively denote by nx and nu the number of states and controls of the overall system. Please note that in the global model there are no neighbors disturbances, because mutual interactions are already included in (3). Likewise, the stage cost of the overall system can be expressed as a function of states and inputs of the corresponding subsystems, i.e., `N(k) = xT N(k)QNxN(k) + uT N(k)RNuN(k), (4) where QN=diag(Qi)i∈N and RN=diag(Ri)i∈N . 2.1. Modular Control Law and Communication Constraints A linear feedback controller is proposed here to minimize the cost of the system while steering it towards the origin. Also, it must be designed taking into account constraints in the information flows due to the communication network that connects the subsystems. In particular, we assume that the network is described by a directed graph ( N , L ), where N is a set of subsystems and L is a set of unidirectional links given by L⊆LN={{i , j}|{i , j} ⊆ N , i6=j} . For convenience, we will define link {i , j} , or simply lij , as an arrow that goes from j to i and vice versa, with i , j∈ N , to stress that i receives information from j . Likewise, we assume that links allow only direct communication, i.e., agent i receives information from j only if they are directly connected by lij , although this assumption can be relaxed if needed. For representation simplicity, we symbolize two links in opposite directions with a double arrow. In this work, it is also considered that some communication links might be disabled to reduce the communication burden, due to, e.g., jamming attacks. To this end, let us define Λ as the topology given by the set of active links in L . Given that there are L=|L| links, it is possible to define a set T={ΛDC , Λ1 , Λ2 , . . . , Λ2L−2 , ΛL} composed of 2 L topologies. Please note that we have introduced a slightly different notation for two special topologies, namely ΛDC and ΛL , which correspond to the decentralized topology (all links are disabled) and the full communication topology (all links are enabled), respectively. Processes 2021,9, 389 4 of 18 Since the feedback controller must be designed taking into account the constraints in the information flows imposed by topology Λ , a superscript will be added to stress this fact so that uN=KΛxN , with Λ∈ T . More specifically, the control law must be suitable for any Λ∈ T . Unlike [ 8 ], where different feedback gains are calculated for each possible topology, here we propose a controller composed of blocks associated with the links of the communication network [ 14 ]. To illustrate this idea, Figure 1shows a system composed of four agents that communicate using 12 directed links, which are associated with the nondiagonal elements of the corresponding modular feedback controller for the full communication topology ΛL, which is described by KΛL=    K11 K12 K13 K14 K21 K22 K23 K24 K31 K32 K33 K34 K41 K42 K43 K44    . (5) However, if the topology changes and becomes that of Figure 2, say Λex , with five links omitted with respect to ΛL, the control law becomes KΛex =    K11 0K13 K14 K21 K22 0 0 0K32 K33 0 K41 K42 K43 K44    , (6) where all nonzero entries have the same value that they had in KΛL . Hence, modular feedback controller KΛL provides us with a family of control laws that can be adapted to different communication topologies by simply making zero the elements that correspond to disabled links. Please note that this modular structure adds robustness to the feedback laws designed through the proposed approach even if the system partitioning has been poorly performed because it rearranges the partitioning online starting from the set of atomic components in which the system is divided. Hence, an inadequate initial partitioning is not an issue because if two subsystems need to cooperate, the controller will promote the cooperation as long as the benefits expected are greater than the corresponding cooperation efforts. Figure 1. Nondiagonal entries of a modular feedback controller for a control system composed of four agents connected by 12 unidirectional links. Figure 2. Example topology Λex where five directed links of ΛL in Figure 1have been disconnected. Processes 2021,9, 389 5 of 18 Finally, some definitions regarding parents-children topologies are introduced below and illustrated in Figure 3. Figure 3. Topologies derived from N= 3 agents and L= 3 links. The coloured edges join the topologies that are related through the parents-children relations. Definition 1. A set of ascendant topologies coming from a given topology Λi∈ T , with |Λi|<L , can be defined as SΛi={Λj∈ T | Λi⊂Λj,|Λj|≥|Λi|+1}. (7) A topology Λj∈SΛiis a parent of topology Λiif |Λj|=|Λi|+1holds. Definition 2. A set of descendant topologies coming from a given topology Λi∈ T , with |Λi|>1 can be defined as ¯ SΛi={Λj∈ T | Λj⊂Λi,|Λj|≤|Λi| − 1}. (8) A topology Λj∈¯ SΛiis a child of topology Λiif |Λj|=|Λi| − 1is fulfilled. 2.2. Stability Before addressing the controller design procedure, it is necessary to give some remarks regarding stability. Since the network topology might change at any time, there is a need for guaranteeing the stability of the closed-loop system despite the switchings between the corresponding control laws KΛ , with Λ∈ T . To deal with this issue, a common Lyapunov function f(xN(k)) = xT N(k)PxN(k) is designed for all feedback controllers KΛ . In particular, let P∈Rnx×nx be a positive definite matrix that will also provide us with a bound on the cost-to-go of the closed loop system, i.e., xT N(k)PxN(k)≥ ∞ ∑ t=k `N(t). (9) In [ 8 ], it is shown that the Lyapunov function is a bound on the cost-to-go of the closed loop system if the following inequality holds: xT N(k)PxN(k)≥`N(k) + xT N(k+1)PxN(k+1). (10) In particular, (9) can be derived from (10) by telescope summation. Processes 2021,9, 389 6 of 18 3. Modular Controller Design In this section, we provide a theorem and a lemma for the modular controller design with a stage cost defined by QN and RN following (4) . To this end, we consider that the system is decomposed into a set N of subsystems connected by means of a set L of directed communication links that give rise to a set Tof different topologies. Theorem 1. Let a system be described by (3) with discrete-time linear dynamics given by AN and BN . If there exist matrices W=WT=diag(Wi)i∈N , where Wi∈Rnxi×nxi , and Y∈Rnu×nx such that the following constraint is satisfied, for all Λ∈ T :      W WAT N+YT ΛBT NWQ1/2 NYT ΛR1/2 N ANW+BNYΛW0 0 Q1/2 NW0I0 R1/2 NYΛ0 0 I      >0, (11) with YΛ,ij =Yij if link lij is activated, i.e., if lij ∈Λ , and YΛ,ij = 0otherwise, then there exists a modular controller that provides a family of feedback control laws KΛ=YΛW−1, which stabilizes the system for topologies Λ∈ T . Also, a common Lyapunov function f(xN(k)) = xT N(k)PxN(k) that provides a bound on the cost-to-go is generated, with P =W−1. Proof. The iterative application of the Schur complement [ 59 ] in a backwards manner together with the proposed variable change allows us to transform LMI (11) into (10), which guarantees that stability for the cost-to-go must decrease at each time step. Then, a telescope summation of this inequality from t=k to infinity gives (9), which allows us to use the Lyapunov function to get a bound on the cost-to-go. The constraints imposed on YΛ and W guarantee that KΛ satisfies the communication restrictions imposed by topology Λ , which require KΛ,ij = 0, ∀i , j such that lij /∈Λ . Here, let us recall that KΛ=YΛW−1 . Since YΛ,ij = 0 if lij /∈Λ , and W has a diagonal block structure that is to be inherited by its inverse matrix, i.e., W−1 ij = 0 for any i6=j , then by the properties of matrix multiplication it holds KΛ,ij = 0 if link lij /∈Λ . Finally, as YΛ,ij always has the same value for topologies Λ∈ T when lij = 1, otherwise being zero, this fact forces block KΛ,ij to be the same for all these topologies, hence providing the desired modular structure. Remark 1. Matrix W must be a block sparse/diagonal matrix to preserve the modular features. for instance , let us assume that block KΛ,34 is associated with link l34 , being KΛ,34 the product of the 3rd-row of YΛ and the 4th-column of W−1 . The only way to assure that KΛ,34 is non-null for a given YΛ , when link l34 is enabled, is to shape W as a block diagonal matrix, i.e., W =WT=diag(Wi)i∈N . Solving the set of LMIs defined by Theorem 1provides us with a matrix P=W−1 and matrices KΛ=YΛW−1 , where KΛ,ij maps the contribution of the state of agent j to the control action of agent i . Hence, if a link lij ∈ L is activated, block KΛ,ij 6= 0; otherwise KΛ,ij = 0. It must be noticed that the set of LMIs is solved for matrices W and Y=YΛL , i.e., there are only two unknown matrices to be found. This fact represents a major difference with respect to [ 8 ], where different WΛ and YΛ were obtained for each topology Λ . It is also important to remark that all feedback matrices KΛ are associated with the same Lyapunov matrix P , what means that the closed-loop stability is guaranteed for any topology Λ∈ T , so that switchings between the different control laws can be performed without compromising the stability of the closed-loop system. A necessary and sufficient condition for Theorem 1to have a solution are given by the lemma below. Lemma 1. If there exists a feasible solution of (11) for topology Λi , then topologies Λj in SΛi⊆ T , i.e., the set of ascendants of Λi(see Definition 1), admit a modular solution. Processes 2021,9, 389 7 of 18 Corollary 1. The family of LMIs of Theorem 1has a feasible solution if and only if W and YΛ in (11) can be found for ΛDC, i.e., for the decentralized case. Remark 2. The condition in Corollary 1is necessary because any solution satisfying Theorem 1 must provide a feedback law for the decentralized topology, i.e., KΛDC . Hence, without a solution for ΛDC there is no solution for the overall problem. Furthermore, it is a sufficient condition because W and YΛDC constitute a feasible solution for the rest of topologies, which solve the very same LMI constraint with additional degrees of freedom due to the additional nonzero elements in the corresponding YΛ . Hence, the more demanding constraints are those of the decentralized topology, which requires stable global dynamics although each subsystem uses only local state information. Consequently, any set of local feedback controllers leading to overall stable dynamics could generate a modular controller applying Theorem 1. Design Method To design the controller, we solve max W,YΛ tr(W), (12) subject to (11) for all Λ∈ T . Then, it is enough to take KΛ=YΛW−1 . Please note that the maximization of the trace of W is an indirect manner of minimizing that of P=W−1 , hence minimizing the cost-to-go of the closed-loop system. Once the modular controller is designed, it is straightforward to find new bounds on the cost-to-go tailored to each topology, i.e., PΛ , if needed, e.g., for a rapid topology selection in a coalitional control system. The fact that a common Lyapunov function exists guarantees that switchings can be performed without endangering the system stability. 4. Dealing with Computational Burden LMI (11) is solved simultaneously for all network topologies Λ∈ T . Notice that there exists a YΛ declared for every network topology, where each block YΛ,ij corresponds to an enabled link lij. Therefore, the number of decision variables is defined by ∑ i∈N nxi×(nxi+1) 2+nu×nx, (13) being independent of the number of topologies considered. Nevertheless, the computational burden of solving LMIs does not scale linearly with the number of topology constraints [ 60 ], which might render the problem unfeasible for practical use. For example, in our experiments, we have been able to apply this method to systems with thousands of topologies, which in the most conservative case corresponds to a modest number of links, e.g., in the range 10–20. To overcome this issue, we provide some plausible strategies: 1. Exploiting Convexity: Since constraint (11) is convex, any convex combination of solutions is also a solution. Given that W is common for all topologies, new solutions are generated by simply combining the results for matrices YΛ corresponding to different topologies. Likewise, it is straightforward to check that the same holds for KΛ. With this idea in mind, the problem can be simplified by solving only a subset of topologies that can be used to generate the rest of the topologies. For instance, it is possible to solve the decentralized topology, and then topologies with only one active link, i.e., instead solving 2 L LMIs the problem is reduced to the resolution of L+ 1 LMIs, thus avoiding the combinatorial explosion. Interestingly, the resulting feedback solution will preserve the values for the common elements in the feedback gains combined. Also, the values of the noncommon elements are easy to calculate due to the block structure of the modular controller (recall that blocks corresponding to a link become zero when the link is disabled). This strategy can be very convenient for hierarchical control, where an upper control layer can compute a convex combination of KΛ to generate a feedback gain for the Processes 2021,9, 389 8 of 18 desired topology, e.g., by searching for the combination that maximizes the trace of W (or minimizes the trace of P). 2. Ascendants replaced by descendants: Any topology provides a feasible solution for all its ascendants (see Definition 1). Thus, it is possible to remove topologies from set T and simply use one of its descendants (Definition 2) instead to reduce the number of constraints of the problem. 3. Branch-and-bound-like approaches: Those topologies that degrade the most the performance of the controller are removed, e.g., those that decrease the most the trace of W . Please note that given a topology, its trace of W , i.e., when the LMI is solved for this specific topology, is a lower bound for all its descendants. This fact is used to compute a modular controller only for those topologies that provide the best performance according to the aforementioned criterion. 4. Topology clustering: For large-scale systems, clustering the different agents/subsystems in super agents can be an interesting approach to handle the computational burden. For instance, the agents clustering could be achieved by different algorithms in the line of those described in [ 28 , 40 , 42 , 44 ], where the partitioning is performed by preselecting those sets of agents that are not highly interrelated with other agents and may separately work well offline prior to proceed with the control law implementation. 5. Case Study: Barcelona DWN In this article, we analyze the Barcelona drinking water network (DWN), which is a complex large-scale system managed by the public entity Aguas de Barcelona (AGBAR), S.A. This system supplies water to the metropolitan area of Barcelona and it is fed by the Ter and Llobregat rivers using regulated dams with an overall capacity of 600 hm 3 . Besides the rivers, some additional underground wells also contribute to an overall inflow of around 7 m3/s, which becomes potable by four drinking water treatment plants. The Barcelona DWN can be broken down in two layers. The first (upper) layer consists of a transport network, which connects the water treatment plants with reservoirs distributed across the metropolitan area. The second (lower) layer is the distribution network, which in turn is subdivided in subnetworks that guarantee the water supply from the reservoirs to the consumers. We have selected this case study to prove that the proposed approach can be applied to a real large-scale problem. In particular, we focus here on the transport network. Hence, the subnetworks within the distribution network are considered as demand sectors that will be characterized by a scheduled pattern and taken as disturbances by the control system. 5.1. Barcelona DWN Description The Barcelona DWN model [ 41 , 43 ] is depicted in Figure 4and consists of nx= 63 tanks, nu= 114 actuators (75 pumps and 39 valves), nn= 17 junction nodes and nd= 88 sectors of water demand, which are considered as known disturbances. The system can be modeled using flow-based differential-algebraic equations, which interrelate water levels x at tanks, controlled pipe flows u, and demands d. In particular, we have: 1. Water tanks differences equations xi(k+1) = xi(k) + ∆t ∑ i qin,i(k)−∑ j qout,j(k)!, (14) where xi is the water level in tank i , and qin,i and qout,j are respectively the i -th and j -th inflows and outflows in m 3/s . The aggregation of all differences equations allows us to reformulate the problem as xN(k+1) = ANxN(k) + BNuN(k) + BpdN(k), (15) Processes 2021,9, 389 9 of 18 with dN∈Rnd being the disturbances vector, Bp∈Rnx×nd , and where the rest of variables are defined as introduced in Section 2. 2. Mass-balance constraints imposed by the nodes ∑ i qin,i(k) = ∑ j qout,j(k), (16) with qin,i and qout,j defined as before. Equation (16) can be rewritten in matrix form and considering the known disturbances as EuuN(k) + EddN(k) = 0, (17) where Eu∈Rnn×nu and Ed∈Rnn×nd respectively deal with the flows associated with the control variables and those corresponding to the water demands. 3. Bounds on inputs, i.e., umin ≤uN(k)≤umax, (18) where the values umin and umax are the upper and lower limits of the different actuators at the DWN, respectively. 4. Bounds on states at tanks, i.e., xmin ≤xN(k)≤xmax, (19) being xmin and xmax respectively the minimum and maximum levels at the water tanks . 5.2. Control Variables Parameterization Equation (17) relates the control variables with the measured disturbances in the system nodes. Based on [ 61 ], let us assume that rank(Eu) = rank(Ed) = nn , with nn≤nd≤nu , i.e., some components of uN(k) are not longer independent and can be parameterized as a function of the known disturbances. To this end, let us recast (17), as EuEduN(k) dN(k)=0. (20) At this point, we consider a linear transformation ˆ P∈Rnu×nu that allows us to perform the Gauss-Jordan elimination, i.e., Euˆ P=InnM1,M1∈Rnn×nu−nn, (21) which yields EuEdP=InnM1M2,M2∈Rnn×nd, with P=ˆ P0 0Inn. (22) Now, it is possible to utilize Pto reformulate (20), obtaining EuEdP PTuN(k) dN(k)=0=⇒Euˆ P Edˆ PTuN(k) dN(k)=0. (23) Then, in order to separate the effects of the dependent and independent control inputs, we define vN(k) = ˆ PTuN(k) = ¯ uN(k) ˆ uN(k), (24) reaching the following expression equivalent to (20): InnM1M2  ¯ uN(k) ˆ uN(k) dN(k) =0=⇒¯ uN(k) = −M1ˆ uN(k)−M2dN(k). (25) Processes 2021,9, 389 16 of 18 Institutional Review Board Statement: Not Applicable. Informed Consent Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. References 1. Hespanha, J.P.; Naghshtabrizi, P.; Xu, Y. A survey of recent results in networked control systems. Proc. IEEE 2007 ,95, 138–162. [CrossRef] 2. 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