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Interconnections of dissipative systems and distributed economic MPC
6th IFAC Conference on Nonlinear Model Predictive Control 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Available online at www.sciencedirect.com 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Madison, WI, USA, August 19-22, 2018
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IFAC PapersOnLine 51-20 (2018) 86–91
Interconnections of dissipative systems Interconnections of dissipative systems Interconnections of dissipative systems and distributed Interconnections of economic dissipativeMPC systems and distributed economic MPC and distributed economic MPC and distributed economic MPC Philipp N. Köhler ∗∗ Matthias A. Müller ∗∗ Frank Allgöwer ∗∗
Philipp N. Köhler ∗ Matthias A. Müller ∗ Frank Allgöwer ∗ Philipp N. Köhler Matthias A. Müller Frank Allgöwer ∗ N. Köhler ∗ Matthias A. Müller ∗ Frank Allgöwer ∗ Philipp ∗ Institute for Systems Theory and Automatic Control, ∗ Institute for Systems Theory and Automatic Control, Institute for SystemsofTheory and Germany Automatic Control, University Stuttgart, University Stuttgart, ∗ Institute for Systemsof Theory and Germany Automatic Control, University of Stuttgart, Germany (e-mail: {koehler,mueller,allgower}@ist.uni-stuttgart.de). (e-mail: {koehler,mueller,allgower}@ist.uni-stuttgart.de). University of Stuttgart, Germany (e-mail: {koehler,mueller,allgower}@ist.uni-stuttgart.de). (e-mail: {koehler,mueller,allgower}@ist.uni-stuttgart.de). Abstract: Abstract: The The interconnection interconnection of of dynamically dynamically decoupled decoupled subsystems, subsystems, each each exhibiting exhibiting aa Abstract: The interconnection of dynamically decoupled subsystems, each a certain dissipativity property considered in the context of economic MPC, is investigated. certain dissipativity property considered in the context of economic MPC, is exhibiting investigated. Abstract: The ofinterconnection ofis dynamically decoupled subsystems, each exhibiting a certain dissipativity property considered in the context of economic MPC, is investigated. Interconnection the subsystems by means of their cost functions being separable in a Interconnection of the subsystems is by means ofcontext their cost functionsMPC, being isseparable in a certain dissipativity property considered in the of economic investigated. Interconnection of the subsystems is by means of their cost functions being separable in a purely local economic and a coupling cost term. For certain classes of quadratic interconnection purely local economic a couplingiscost classes of quadratic interconnection Interconnection ofconditions theand subsystems by term. meansFor of certain their cost functions being separable in a purelywe local economic and a coupling cost term. For certain classes of quadratic interconnection costs provide on the interconnection structure under which the overall system system costs we provide conditions on the interconnection structure under which the overall purely local economic and a coupling cost Moreover, term. For certain classes of quadratic interconnection costs we provide conditions on the interconnection structure under which the overall system exhibits the same dissipativity property. we apply a non-iterative distributed MPC exhibits the same conditions dissipativity property. Moreover, we apply aunder non-iterative distributed MPC costs weto provide onsystem the interconnection structure which the overall optimal system exhibits the same dissipativity property. Moreover, we apply a stability non-iterative distributed MPC scheme the interconnected which yields asymptotic of the overall scheme to the interconnected system which yields asymptotic stability of the overall optimal exhibits thethe same dissipativity property. Moreover, wethe apply a stability non-iterative distributed MPC scheme to interconnected system which yields asymptotic of the overall optimal steady state by exploiting the structural properties of system interconnection. steady state byinterconnected exploiting the structural properties of the system interconnection. scheme to the system which yields asymptotic stability of the overall optimal steady state by exploiting the structural properties of the system interconnection. © 2018,state IFACby (International of Automatic Control) Hosting by interconnection. Elsevier Ltd. All rights reserved. steady exploiting Federation the structural properties of the system Keywords: Keywords: Decentralized Decentralized and and distributed distributed control, control, Economic Economic MPC, MPC, Modeling Modeling and and decision decision Keywords: Decentralized and distributed control, Economic MPC, Modeling and decision making in complex systems, Multiagent systems. making in complex systems, Multiagent systems. Keywords: Decentralized and distributed control, Economic MPC, Modeling and decision making in complex systems, Multiagent systems. making in complex systems, Multiagent systems. 1. only 1. INTRODUCTION INTRODUCTION only few few results results are are available available in in the the literature. literature. In In Köhler Köhler 1. INTRODUCTION only few results are available in the literature. In Köhler et al. (2016), convergence to an initially unknown overall et al.few (2016), convergence to an initially unknown overall 1. INTRODUCTION only results are available in the literature. In Köhler et al. (2016), convergence to an initially unknown overall optimal steady state is enforced by average constraints. Distributed model predictive control (MPC) has recently optimal steady state is enforced by average constraints. Distributed model predictive control (MPC) has recently et al.approach (2016), convergence toal. an(2012) initially unknown overall optimal steady state is enforced by average constraints. The of Driessen et and Lee and Distributed model predictive control (MPC) has recently gained much attention as a flexible tool for control of approach ofstate Driessen et al. (2012) and Leeconstraints. and Angeli Angeli gained muchmodel attention as a control flexible (MPC) tool forhas control of The optimal steady is enforced by average The approach of Driessen et al. (2012) and Lee and Angeli Distributed predictive recently (2014) is based on a dissipativity assumption on the gained attention as a flexiblesystems tool for control of (2014) is based on a dissipativity assumption on the overall complex,much large-scale, interconnected such as, e.g., e.g., overall complex, large-scale, interconnected systems such as, The approach of Driessen et al. (2012) and Lee and Angeli (2014) is based on a dissipativity assumption on the overall gained much attention as a flexible tool for control of system and an iterative distributed MPC scheme. Thereby, complex, large-scale, interconnected as, e.g., electric grids grids or interacting interacting vehicles.systems While such enjoying the system and an iterative distributed MPC scheme. Thereby, electric or vehicles. While enjoying the (2014) isand based on a dissipativity assumption onproblem the overall system an iterative distributed MPC scheme. Thereby, complex,grids large-scale, interconnected systems such as, e.g., the centralized economic MPC optimization is electric or interacting vehicles. While enjoying the advantages of classical MPC such as satisfaction of certhe centralized economic MPC optimization problem is advantages of or classical MPC such as While satisfaction of cerand an iterative distributed MPC scheme. Thereby, the centralized economic MPC optimization problem is electric gridsof interacting enjoying the system solved distributedly, which yields stability and convergence advantages classical MPCvehicles. such as satisfaction of certain constraints and incorporation of some performance solved distributedly, which yields stability and convergence tain constraints and incorporation of some performance the centralized economic MPC optimization problem is solved distributedly, which yields stability and convergence advantages of classical MPCfunction, such as satisfaction of cerresults when iterated until convergence in each timestep. tain constraints and incorporation of it some performance criterion in terms of a cost does not rely on results when iterated until convergence in each timestep. criterion in termsand of aincorporation cost function,of itsome doesperformance not rely on solved distributedly, which yields stability and convergence results when iterated until convergence in each timestep. tain constraints criterion terms a cost function, it doesreduces not rely on On the contrary, in this work we follow a bottom-up central in unit and ofhence hence favors scalability, comaacriterion central unit and favors scalability, reduces comiterated in each timestep. On thewhen contrary, in until this convergence work we follow a bottom-up terms a cost function, it does not A rely on results amunication central in unit and ofhence favors scalability, reduces comrequirements and is less error prone. large On the contrary, in this work weof follow a bottom-up approach and derive dissipativity the overall system munication requirements and is less error prone. A large approach and derive dissipativity of the overall system a central and hence favors scalability, reduces munication requirements and is less error prone. A comlarge variety of unit distributed MPC schemes has been proposed in approach On the contrary, inthe this work subsystems weoffollow a bottom-up and derive dissipativity the overall system from dissipativity of isolated under certain variety of distributed MPC schemes has been proposed in from dissipativity of the isolated subsystems under certain munication requirements and is less error prone. A large variety of distributed MPCsystem schemes has been proposed in approach the literature literature for different different setups, see Christofides Christofides and derive dissipativity of the overall system from dissipativity of the isolated subsystems under certain conditions on the interconnection structure. In particular, the for system setups, see variety of distributed MPC schemes has been proposed in conditions on the interconnection structure. In particular, the literature for different system setups, see Christofides et al. (2013); Maestre and Negenborn (2013); Müller and from dissipativity of the isolated subsystems under certain the interconnection structure. In particular, we consider dynamically decoupled subsystems interconet (2013); Maestre and system Negenborn (2013); Müller and conditions consideron decoupled subsystems intercontheal. literature for for different setups, see Christofides et al. (2013); Maestre and Negenborn (2013); Müller and we Allgöwer (2017) an overview. Most of these schemes conditions ondynamically thetheir interconnection structure. In particular, we consider dynamically decoupled subsystems interconnected through cost functions involving other sysAllgöwer (2017) for an overview. Most of these schemes nected through their cost functions involving other syset al.concerned (2013); Maestre Negenborn (2013); Müller and we consider dynamically decoupled subsystems interconAllgöwer (2017) for stabilization anand overview. Most ofsetpoint, these schemes are with of some which nected through theirthe cost functions involving other systems’ states. Hence, main focus of this work is not to are concerned with stabilization of some setpoint, which Allgöwer (2017) for an overview. Most of these schemes tems’ states. Hence, the main focus of this work is not to are concerned with of someInstead, setpoint, might not always always be stabilization the main main objective. objective. in which many nected through their cost functions involving other systems’ states. Hence, the main focus of this work is not to develop another distributed economic MPC scheme, but might not be the Instead, in many develop another distributed economic MPC scheme, but are concerned with stabilization of some setpoint, which might not always be the main objective. Instead, in many practical applications, applications, optimal optimal operation operation with with respect respect to to tems’ states. Hence, the maineconomic focus ofthe this work is notbut to develop another distributed MPC scheme, rather to reveal connections between interconnection practical toanother reveal connections between the interconnection mightreal notapplications, always be theoptimal main objective. Instead, in many practical operation with or respect to rather some performance criterion such as profit working develop distributedkind economic MPC scheme, subbut rather to reveal connections between the interconnection structure of the considered of locally dissipative some real performance criterion such as profit or working of the considered kind of locally subpractical applications, optimal operation with respect to structure some real performance criterion such as profit or working costs is desired. desired. Allowing for an an arbitrary cost function rather toand reveal connections between the dissipative interconnection structure of the considered kind of locally dissipative subsystems dissipativity of the resulting overall system. costs is Allowing for arbitrary cost function some real performance criterion such as profit or working systems and dissipativity of the resulting overall system. costs desired. Allowing an arbitrary cost function which isneed need not be be positivefor definite with respect respect to some some structure ofthis the considered kind of locally dissipative subsystems and dissipativity of the resulting overall system. Note that is in the spirit of investigating the inputwhich not positive definite with to that this is in the spirit of resulting investigating thesystem. inputcosts isneed desired. Allowing an arbitrary cost economic function which not be positivefor definite with respect to some Note steady state is the distinctive feature of so called systems and dissipativity of the overall Note that this is in the spirit of investigating the inputoutput-interconnection of dissipative systems as the steady state not is the featurewith of so respect called economic dissipative systems the as in in the which need bedistinctive positive to some steady state is the distinctive feature so called may, economic MPC. Consequently, optimaldefinite system ofoperation operation e.g., output-interconnection Note that thisofisWillems in the of ofMoreover, investigating inputoutput-interconnection ofspirit dissipative systems as derived in the seminal work (1972). having MPC. Consequently, optimal system may, e.g., seminal work of Willems (1972). Moreover, having derived steady state is the distinctive feature of so called economic MPC. Consequently, optimal system operation may, e.g., output-interconnection be at some steady state or at some periodic orbit. of dissipative systems as in the seminal work of Willems (1972). Moreover, having derived conditions for dissipativity of the overall system, we show be at some steady state or at some periodic orbit. forofdissipativity of theMoreover, overall system, we show MPC. Consequently, optimal operation may, e.g., conditions be at some steady state or at system some periodic orbit. seminal work Willems (1972). having derived for dissipativity of the overall system, how the interconnection structure can beshow exEconomic MPC has been studied extensively for single single conditions how the considered considered interconnection structure canwe exbe at someMPC steadyhas state or at some periodic orbit. Economic been studied extensively for conditions for dissipativity of the overall system, webe show how the considered interconnection structure can be exploited by applying a non-iterative, sequential distributed Economic MPC hascertain been studied extensively for single systems, and under dissipativity properties, operploited by applying a non-iterative, sequential distributed systems, and under certain dissipativity properties, operhow the considered interconnection structure can be exploited by applying a non-iterative, sequential distributed Economic MPC has been studied extensively for single MPC scheme proposed in Müller et al. (2012) for a stabisystems, under certain dissipativity ation at at and steady state can be be shown to to properties, be optimal, optimal,operand MPC scheme proposed in Müller et al. (2012) for a stabiation steady state can shown be and ploited by applying a non-iterative, sequential distributed MPC scheme proposed in Müller et al. (2012) for a stabisystems, under certain dissipativity operlizing MPC setting to the economic setup at hand. ation at and steady state can be shownsteady to properties, be state optimal, and moreover, stability of the optimal can be MPC setting to the economic setup at hand. moreover, stability of can the be optimal steady canand be lizing proposed in Müller et al. (2012) for a stabilizing scheme MPC setting to the economic setup at hand. ation at steady state shown toDiehl be state optimal, moreover, stability of the optimal steady state be MPC established for the the closed loop (see e.g. et al. al. can (2011); This work is structured as follows. In Section 2, established for closed loop (see e.g. Diehl et (2011); lizing MPC setting to the economic setup at hand. This work is structured as follows. In Section 2, a a more more moreover, stability of the optimal steady state can be established the closed loopet(see DiehlGrüne et al. (2011); Amrit et et al. al.for(2011); (2011); Angeli al. e.g. (2012); (2013); This work is structured as follows. In Section 2, a given. more precise system description and problem statement is Amrit Angeli et al. (2012); Grüne (2013); precise system description and problem statement is given. established for the closed loop (see e.g. Diehl et al. (2011); Amrit al. Allgöwer (2011); Angeli et al. (2012); inGrüne (2013); This Müller etand and (2017)). However, the context context work is we structured asand follows. Indissipativity Section 2, isapresermore precise system description problem statement given. In Section 3, provide conditions for Müller Allgöwer (2017)). However, in the Section 3, wedescription provide conditions for dissipativity preserAmrit etand al. Allgöwer (2011); Angeli et ofal. (2012); inGrüne (2013); In Müller (2017)). However, the systems, context of distributed economic MPC interconnected precise system and problem statement isFinally, given. In Section 3, we provide conditions for dissipativity preservation for specific interconnection cost functions. of distributed economic MPC of interconnected systems, for 3, specific interconnection cost functions. Finally, Müller and Allgöwer in the systems, context vation of distributed economic(2017)). MPC ofHowever, interconnected In Section wewe provide conditions for dissipativity preservation for specific interconnection cost functions. Finally, in Section 4, investigate sequential distributed eco of distributed economic MPC of interconnected systems, in Section 4, we investigate sequential distributed ecoauthors thank the German Research Foundation (DFG) for The vation for specific interconnection cost functions. Finally, in Section 4, we investigate sequential distributed economic model predictive control of the considered class of The authors thank the German Research Foundation (DFG) for nomic model predictive control of the considered class of support of this work within grant AL 316/11-1 and within the The authors German Foundation (DFG) the for in Section 4, predictive wegiving investigate sequential distributed ecosupport of thisthank work the within grantResearch AL 316/11-1 and within nomic model control of the considered class of systems, before some concluding remarks in Sec systems, before giving some concluding remarks in SecCluster ofofExcellence in Simulation Technology (EXC 310/2) at the The authors thank the German Research Foundation (DFG) for support this work within grant AL 316/11-1 and within nomic model predictive control of the considered of Cluster of Excellence in Simulation Technology (EXC 310/2) at the systems, before giving some concluding remarks class in Section 5. University of Stuttgart. support of this work within grant AL 316/11-1 and within the tion 5. Cluster of Excellence in Simulation Technology (EXC 310/2) at University of Stuttgart. systems, before giving some concluding remarks in Section 5. Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart. tion 5. University of Stuttgart.
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Notation. The interconnection structure between subsystems is described by a directed graph G = {V, E} with vertices V = {1, . . . , P } representing the subsystems and edges E = {(i, j) ∈ V × V} representing the directed interconnection of systems. Denote the set of all neighbors of subsystem i by Ni := {j ∈ V | (i, j) ∈ E} and let di := |Ni | denote its cardinality. Similarly, denote the set of undirected neighbors of subsystem i as N i := {j ∈ V | (i, j) ∈ E ∨ (j, i) ∈ E}. A directed graph G is said to be weakly connected if there is a path between every pair of vertices when neglecting the edge orientation. Denote by B ∈ {−1, 0, 1}|V|×|E| the incidence matrix of a directed graph G with elements Bve equalling 1 if vertex v is the head of edge e, −1 if v is the tail of e, and 0 otherwise. For a thorough presentation of algebraic graph theory we refer the reader to Godsil and Royle (2013). A function ρ : Rn → R is positive semidefinite if ρ(0) = 0 and ρ(x) ≥ 0 for all x ∈ Rn . The function ρ : Rn → R is positive definite if ρ(0) = 0 and ρ(x) > 0 for all x = 0. A matrix A = A ∈ Rn×n is positive (semi)definite if the quadratic form x Ax is positive (semi)definite, and we write A 0 (A 0). The operator ⊗ denotes the standard Kronecker product, diag(·) denotes the block diagonal matrix built from the respective arguments, In ∈ Rn×n denotes the n × n-identity matrix, and 1n ∈ Rn denotes the all ones vector of dimension n.
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subsystem i, i.e., x−i = [x i1 , . . . , xid1 ] , where {i1 , . . . , idi } is an ordered sequence of the elements of the set Ni , that is, i1 < · · · < idi . Such coupling costs are found, e.g., in classical multiagent settings representing a common objective, or in the case of subsystems sharing a common resource. Accordingly, the stage cost of the overall system is given as P
i (xi , ui , x−i ). (4)
(x, u) = i=1
The separable structure of the cost functions considered here reflects the main theme of this work, namely studying the interconnection of local subsystems with local properties. For each subsystem i, denote by Z∗i the set of locally optimal steady state and input pairs, and hence Z∗i = {(xsi , usi ) ∈ Zsi | ii (xsi , usi ) ≤ ii (xi , ui )∀(xi , ui ) ∈ Zsi } . Definition 1. (Angeli et al. (2012)). A control system as (2) is dissipative with respect to a supply rate s : Z → R if there exists a non-negative function λ : X → R, a positive semidefinite function ρ : X → R and x ¯ ∈ X such that λ(f (x, u)) − λ(x) ≤ s(x, u) − ρ(x − x ¯) (5) for all (x, u) ∈ Z. If there exists ρ positive definite, then the system is said to be strictly dissipative. Throughout the following, we assume that each subsystem i considered individually, i.e., neglecting the coupling cost ij , has the following local dissipativity property. Assumption 2. Each subsystem i ∈ I[1,P ] is dissipative on Zi with respect to the supply rate si (x, u) := ii (xi , ui ) −
ii (xsi , usi ) for all (xsi , usi ) ∈ Z∗i , i.e., the following inequality holds for all (xi , ui ) ∈ Zi λi (fi (xi , ui )) − λi (xi ) (6) ≤ ii (xi , ui ) − ii (xsi , usi ) − ρi (xi − xsi ).
2. SYSTEM DESCRIPTION AND PROBLEM STATEMENT Consider a set of P dynamically decoupled systems, with each of the systems obeying the discrete-time nonlinear dynamics xi (t + 1) = fi (xi (t), ui (t)), xi (0) = xi0 (1) where xi (t) ∈ Xi ⊂ Rn denotes the state and ui (t) ∈ Ui ⊂ Rm the control input to system i, i ∈ I[1,P ] := {1, 2, . . . , P }, and fi : Xi × Ui → Xi continuous. The set of admissible state-input pairs is denoted by Zi ⊆ Xi × Ui . Denote by Zsi the set of all admissible steady states for each system i, i.e., Zsi := {(xi , ui ) ∈ Zi | xi = fi (xi , ui )}. For the overall system, the state and input variables are written nP in stacked vector form, i.e., x = [x and 1 , . . . , xP ] ∈ R mP s u = [u1 , . . . , uP ] ∈ R , and similarly for x and us . The corresponding overall system dynamics result in x(t + 1) = f (x(t), u(t)), x(0) = x0 (2) with (x(t), u(t)) ∈ Z := Z1 × · · · × ZP and f (x, u) = [f1 (x1 , u1 ) , . . . , fP (xP , uP ) ] . Accordingly, the set of admissible steady states for the overall system is Zs = Zs1 × · · · × ZsP . Note that until this point, the subsystems i are completely decoupled.
This dissipativity assumption is explicitly formulated for the case of the optimal steady state (xsi , usi ) not being unique. Note that in the case of multiple optimal steady states, ρi can only be positive semidefinite. Remark 3. Note that for local tracking costs, i.e., cost functions ii that are positive definite w.r.t. (xsi , usi ), strict dissipativity is trivially fulfilled with λi = 0. Hence, all our following results on system interconnections also hold for local tracking costs as a special case of the considered setup with general economic local cost functions. As discussed in the introduction, the assumed (strict) dissipativity of a subsystem with respect to the particular supply rate of Assumption 2 has strong implications on optimality of steady state operation (c.f. Müller et al. (2015)), and yields (asymptotic) stability results for the closed-loop system under economic model predictive control (Amrit et al., 2011; Angeli et al., 2012). Given a set of locally dissipative systems in the sense of Assumption 2, in this work we investigate specific cost interconnections
ij and derive conditions under which the local (strict) dissipativity properties are preserved for the overall system with overall cost function (4). Hence, we aim at providing (strict) dissipativity of the overall system (2), i.e., verifying that there exists a storage function Λ : X → R and ρ˜ : X → R (at least) positive semidefinite, such that Λ(f (x, u)) − Λ(x) ≤ s˜(x, u) − ρ˜(x − x ¯) (7)
Coupling between the subsystems is introduced through the economic stage cost functions i each system is trying to minimize. In particular, we consider separable cost functions of the following form
ij (xi , xj ), (3)
i (xi , ui , x−i ) = ii (xi , ui ) + j∈Ni
with the continuous local economic cost function ii : Rn × Rm → R of subsystem i, and the coupling cost
ij : Rn × Rn → R induced by the coupling between subsystem i and its neighbors. By x−i we denote the collection of states of all subsystems that are neighbors of
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the overall optimal steady state directly appears in the coupling cost. Hence, interconnection costs of such a structure may rather appear in a tracking MPC application, and is introduced here for the sake of a clear presentation. ˜ ∈ Lemma 6. Let Assumption 5 hold. If there exists Q nP ×nP R such that for the interconnection weights in (11) Q11 Q12 · · · Q1P Q21 Q22 ˜ Q Q := .. ... .
holds for all (x, u) ∈ Z with supply rate s˜(x, u) :=
P i=1
i (xi , ui , x−i ) − i (¯ xi , u ¯i , x ¯−i )
(8)
and (¯ x, u ¯) ∈ Zs being an (optimal) steady state of the overall system. The special separable structure of the overall system and the local dissipativity properties suggest to use as a storage function of the overall system the P sum of local storage functions, i.e., Λ(x) = i=1 λi (xi ). According to Assumption 2, we obtain Λ(f (x, u))−Λ(x) ≤ P s i=1 si (xi , ui ) − ρi (xi − xi ). In order to deduce the desired dissipativity inequality for the overall system (7) P with interconnection costs, the inequality i=1 si (xi , ui )− ρi (xi − xsi ) ≤ s˜(x, u) − ρ˜(x − x ¯) needs to be satisfied for all (x, u) ∈ Z, for any (xsi , usi ) ∈ Z∗i and for (¯ x, u ¯) ∈ Zs , which expands to P −ii (xsi , usi ) − ρi (xi − xsi ) (9)
QP 1 QP P ˜ holds with Q positive semidefinite, then the overall system (2) with stabilizing coupling costs (11) is dissipative with respect to the supply rate (8) for x ¯ = xs with s s ∗ ˜ (xi , ui ) ∈ Zi . Moreover, if there exists Q 0, then the overall system is strictly dissipative.
Proof. Consider (¯ x, u ¯) = (xs , us ), choose ρ˜(x − xs ) := s ˜ s (x − x ) Q(x − x ), and note that this quadratic form is i=1 positive (semi)definite. Plugging this into (9) yields P P ≤ −ii (¯ xi , u ¯i ) + ij (xi , xj ) − ij (¯ xi , x ¯j ) − ρ˜(x − x ¯). ρi (xi − xsi ) + ij (xi , xj ) − ρ˜(x − xs ) 0≤ i=1
j∈Ni
i=1
Therefore, this inequality is the main object of interest throughout this work and will be investigated for specific system interconnections by means of ij . We summarize the preceding elaboration in the following lemma. Lemma 4. Let Assumption 2 hold, and let (¯ x, u ¯ ) ∈ Zs be a steady state of the overall system (2). If there exists a positive semidefinite function ρ˜ : X → R such that (9) is satisfied for all (x, u) ∈ Z, then the overall system (2) is dissipative with supply rate s˜(x, u) := P xi , u ¯i , x ¯−i ). Moreover, (¯ x, u ¯) is an i=1 i (xi , ui , x−i ) − i (¯ optimal steady state of the overall system. If there exists ρ˜ positive definite, then the overall system is strictly dissipative.
j∈Ni s
˜ − xs ) = (x − x ) Q(x − x ) − (x − xs ) Q(x ˜ − xs ), = (x − xs ) (Q − Q)(x s
which is satisfied due to the assumed positive (semi)defi˜ Hence, all conditions of Lemma 4 are niteness of Q − Q. fulfilled which yields the claim. Besides the purely algebraic characterization of the cost interconnection structure as in Lemma 6, the interconnection weight Qij can be seen as a weight assigned to the (i, j)-edge of the graph G. If we define QD := diag(Q11 , . . . , QP P ) then the matrix Q − QD can be interpreted as the matrix weighted adjacency matrix of the interconnection graph G. Matrix weighted graphs constitute a non-trivial extension of classical (scalar) weighted graphs and have only recently been investigated in the literature, see Tuna (2016); Trinh et al. (2018). For the sake of a simple presentation clearly revealing the implications of certain graph properties, in the following we restrict ourselves to classic scalar weighted graphs.
3. QUADRATIC COUPLING COSTS AND DISSIPATIVITY PRESERVATION In the remainder of this work, we focus on quadratic cost interconnections ij . Moreover, we assume that the local dissipativity Assumption 2 holds with ρi being quadratic: Assumption 5. Let Assumption 2 hold for (10) ρi (xi − xsi ) = (xi − xsi ) Qii (xi − xsi ) with Qii ∈ Rn×n 0 for all i ∈ I[1,P ] .
Hence, in the following we consider the weighted graph G q := {V, E, W}, which corresponds to the graph G augmented by non-negative weights qij ∈ R assigned to each edge, hence with the according set of weights W := {qij ∈ R, qij ≥ 0 ∀ (i, j) ∈ E}.
Despite restricting the class of considered cost coupling functions and slightly strengthening the local dissipativity assumption, the local economic stage cost functions ii as well as the system dynamics fi are treated arbitrary and nonlinear throughout this work.
3.2 Diffusive coupling cost In this section, we consider diffusive coupling costs, typically found in multiagent agreement or synchronization applications, of the form ij (xi , xj ) = (xi − xj ) Qij (xi − xj ) (12) with positive semidefinite interconnection weights Qij = n×n Q given by the weighted graph G q . Hence, as ij ∈ R discussed above, for simplicity we specialize the interconnection weights in (12) to Qij = qij In for (i, j) ∈ E. Similarly, we assume the following for the weights of the quadratic form in (10) of Assumption 5 (and hence specialize Assumption 2 further).
3.1 Stabilizing coupling cost In this section, we consider “stabilizing” coupling costs of the form (11) ij (xi , xj ) = (xi − xsi ) Qij (xj − xsj )
with interconnection weights Qij ∈ Rn×n , Qij = 0 for (i, j) ∈ / E. Note that such a cost formulation does not make much sense in an economic MPC framework, since
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Assumption 7. Let Assumption 2 hold for (13) ρi (xi − xsi ) = (xi − xsi ) Qii (xi − xsi ) with Qii = qii In , qii ≥ 0 for all i ∈ I[1,P ] . Remark 8. The restriction to identity matrices certainly further restricts the class of cost interconnection structure. However, we intentionally chose this radical simplification to clearly highlight implications from graph theory.
holds for all (x, u) ∈ Z. For the case of none of the systems being strictly dissipative, hence QD = 0, the above inequality is trivially fulfilled for ρ˜(x−¯ x) := 0 thanks to positive semidefiniteness of Ln . This yields dissipativity of the overall system and hence concludes the proof for this special case. For the case of at least one subsystem i being strictly dissipative, i.e., Qii 0, we first do a completion of squares to obtain
Before stating the main results of this section, we exploit the above simplifications, rewrite the diffusive cost coupling of the overall system using a compact graph theoretic notation, and draw some immediate consequences. For the diffusive coupling cost in (12) we obtain P P ×P j∈Ni ij (xi , xj ) = x (L ⊗ In )x with L ∈ R i=1 being the weighted Laplacian of the interconnection graph G q . The Laplacian is given by L = BW B with B the incidence matrix of G q and W ∈ R|E|×|E| the diagonal matrix of weights with entries Wee = qij if vertex i is the tail and vertex j is the head of edge e. For the graph G q being weakly connected, the Laplacian L is positive semidefinite, with a single 0 eigenvalue and according eigenvector 1P . In the following we assume the graph G q to be weakly connected. In the case of a graph consisting of several disconnected components, all our following results hold for each of the separate subgraphs. Note that due to Assumption 7, QD = qD ⊗ In with qD := diag(q11 , . . . , qP P ). Thanks to the properties of the Kronecker product, the matrix (QD + L ⊗ In ) = (qD + L) ⊗ In has the same eigenvalues as (qD + L) but with multiplicity n, and for brevity we write L ⊗ In =: Ln .
x (QD + Ln )x − 2x QD xs + xs QD xs
= (x − x ˜) (QD + Ln )(x − x ˜) + c˜
with x ˜ = (QD + Ln )−1 QD xs and c˜ = xs QD xs − xs QD x ˜. Observe that (QD + Ln )−1 = ((qD + L) ⊗ In )−1 is indeed invertible since qD + L is positive definite: Positive definiteness of qD +L results from L being positive semidefinite with ker L = 1P and qD 0 being diagonal with at least one strictly positive diagonal element qii and 1 ˜ = (QD + Ln )−1 QD xs = P qD 1P > 0. Hence, we obtain x s s s x = 1P ⊗ x1 , since QD (1P ⊗ x1 ) = (QD + Ln )(1P ⊗ xs1 ) by (L ⊗ In )(1P ⊗ xs1 ) = (L1P ) ⊗ (In xs1 ) = 0, and c˜ = 0. Consequently, for xs = x ¯ and QD + Ln being positive ˜ 0 such that (x − x definite, there exists Q ¯) (QD + ˜ −x ¯) ≥ (x − x ¯) Q(x ¯) := ρ˜(x − x ¯), and hence (9) Ln )(x − x is satisfied with this choice of ρ˜. For each of the subsystems being strictly dissipative with respect to the same steady state, the above result is not surprising but confirms that the diffusive cost coupling indeed preserves the local strict dissipativity properties as expected. For the case of only some of the subsystems being strictly dissipative, the result is more interesting by also making the non-strict dissipative subsystems “find” the overall optimal steady state in an economic MPC application, see Section 4.
Systems with identical optimal steady states. First, we consider the case in which the intersection of the sets of local optimal steady states with the consensus subspace is nonempty, i.e., {Z∗1 × · · · × Z∗P } ∩ {(x, u) ∈ Z | xi = ¯ ∗ = ∅. Obviously, for the case of xj ∀ (i, j) ∈ E} =: Z at least one subsystem i being strictly dissipative, this intersection is a singleton. Hence, all systems share at least one local optimal steady state. Please note that any overall ¯ ∗ lies on the consensus steady state (¯ x, u ¯) from the set Z subspace and hence is of the form x ¯ = 1P ⊗ α with (α, βi ) ∈ Z∗i for all i ∈ I[1,P ] . ¯ ∗ is Lemma 9. Let Assumption 7 hold. If the set Z nonempty, then the overall system (2) with diffusive coupling costs (12) is dissipative with respect to the supply ¯ ∗ . Moreover, if at least one rate (8) for any (¯ x, u ¯) ∈ Z subsystem i ∈ I[1,P ] is strictly dissipative, i.e., Qii 0, then the overall system is strictly dissipative.
Systems with different optimal steady states. In the following, we consider as a second case the interconnection of subsystems which do not share a local optimal steady ¯ ∗ = ∅. Hence we cannot expect to show dissipastate, i.e., Z tivity of the overall system with respect to a steady state P on the consensus manifold by using Λ(x) = i=1 λi (x). Roughly speaking, in the following we determine, by the tools established above, a steady state of the overall system, which trades off the local systems’ (strict) dissipativity and thereby yields dissipativity of the overall system with respect to this steady state. In order to determine such a “compromise steady state”, some additional information on the shape of the local economic cost functions ii for steady states in the vicinity of the local optimal steady state is required. In this work, this information is given in terms of the following assumption. Assumption 10. For (xsi , usi ) ∈ Z∗i the local dissipativity inequality in Assumption 2 holds with equality for all steady states (¯ xi , u ¯i ) ∈ Zsi , i.e., ii (xsi , usi ) + ρi (¯ xi − xsi ) = ii (¯ xi , u ¯i ). Lemma 11. Let Assumptions 7 and 10 hold, and let at least one subsystem i ∈ I[1,P ] be strictly dissipative. If x, u ¯) ∈ Zs with x ¯ = there exists u ¯ ∈ RmP such that (¯ −1 s s s (QD + Ln ) QD x for any (xi , ui ) ∈ Z∗i , then the overall system (2) with diffusive coupling costs (12) is strictly dissipative with respect to the supply rate (8). Moreover, (¯ x, u ¯) is the optimal steady state of the overall system.
Proof. In order to prove the claim, we apply Lemma 4. Hence, we need to provide a positive semidefinite function ρ˜ and verify (9) for the given interconnection cost functions. Since all subsystems share at least one common local optimal steady state, in the following we consider without loss of generality the local optimal steady state x, u ¯), hence (xsi , usi ) ∈ Z∗i such that xs = 1P ⊗ (xs , us ) = (¯ s x1 . Note that ij (¯ xi , x ¯j ) = 0, and verification of (9) reduces to showing that P (xi − xsi ) Qii (xi − xsi ) + (xi − xj ) Qij (xi − xj ) i=1
89
j∈Ni
= x (QD + Ln )x − 2x QD xs + xs QD xs ≥ ρ˜(x − x ¯)
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Proof. Again, we employ Lemma 4 and hence aim at verifying (9). First, we reuse the arguments from the proof of the above lemma in order to show that P (xi − xsi ) Qii (xi − xsi ) + (xi − xj ) Qij (xi − xj ) i=1
the overall optimal steady state. Note that the algorithm proposed in Müller et al. (2012) is a cooperative scheme in the sense that each subsystem i also considers its influence on its neighbors by incorporating the cost terms ji in its MPC optimization problem.
j∈Ni
The local MPC optimization problem of subsystem i ∈ I[1,P ] is given as follows ˜ −i (t)) (14) min J¯i (xi (t), ui (t), x
= (x − x ˜) (QD + Ln )(x − x ˜) + c˜
with (QD + Ln ) 0, x ˜ = (QD + Ln )−1 QD xs =: x ¯, and c˜ = xs QD xs −xs QD x ¯. Plugging this into (9), it remains to show that (x − x ¯) (QD + Ln )(x − x ¯) + c˜ − ρ˜(x − x ¯) ≥
P i=1
−ii (xsi , usi ) + ii (¯ xi , u ¯i ) +
ui (t)
subject to xi (0|t) = xi (t) (15a) xi (k + 1|t) = fi (xi (k|t), ui (k|t)) (15b) ¯i (15c) xi (N |t) = x xi (k|t), ui (k|t) ∈ Zi for k = 0, . . . , N − 1 (15d) ˜ −i (t)) := with prediction horizon N ∈ I>0 , J¯i (xi (t), ui (t), x N −1 (x (k|t), u (k|t)) + (x (k|t), x i j (k|t)) + i k=0 ii i j∈N i ij 1 ji (xj (k|t), xi (k|t)), and ui (t) = {ui (0|t), . . . , ui (N − 1|t)} the predicted input, and xi (t) = {xi (0|t), . . . , xi (N |t)} ˜ −i (t) the resulting predicted state sequences. As before, x denotes the stacked state sequences of all neighbors of subsystem i. We denote the optimizer of the above MPC optimization problem by u0i (t) and the according optimal state trajectory by x0i (t). Given feasible input and according state sequences ui (t) and xi (t) at time t, a feasible input and according state sequence to the MPC optimization problem at time t+1 is given by the sequences ˆ i (t+1) and x ˆ i (t+1) obtained from shifting ui (t) and xi (t) u and appending the steady state and input (¯ xi , u ¯i ) ∈ Zsi , respectively. The main steps of the algorithm are sketched below, for details we refer the reader to (Müller et al., 2012, Section 3.2). Algorithm 1. (Sketch of Algo. 2 in Müller et al. (2012)).
ij (¯ xi , x ¯j )
j∈Ni
= (¯ x − xs ) QD (¯ x − xs ) + x ¯ Ln x ¯, where the last equality follows from Assumption 10. Finally we show that these remaining terms equal c˜, since (¯ x − xs ) QD (¯ x − xs ) + x ¯ Ln x ¯ − c˜ = −xs QD x ¯+x ¯ (QD + −1 s Ln )¯ x = 0 by using x ¯ = (QD + Ln ) QD x . As before, for ˜ 0 such QD + Ln being positive definite, there exists Q ˜ that (x− x ¯) (QD +Ln )(x− x ¯) ≥ (x− x ¯) Q(x− x ¯) := ρ˜(x− x ¯) and hence (9) is satisfied for this choice of ρ˜. 4. DISTRIBUTED ECONOMIC MPC WITH SEPARABLE COST FUNCTIONS In this section, we investigate distributed economic model predictive control of the overall system (2) subject to the overall cost function (4) by locally solving an MPC problem for each subsystem i ∈ I[1,P ] . In the previous section, we established conditions on the cost interconnection under which the overall system is (strictly) dissipative. Hence, under the conditions of Lemma 6, 9 and 11 for strict dissipativity of the overall system, centralized economic MPC yields asymptotic stability of the overall optimal steady state, c.f. Amrit et al. (2011); Angeli et al. (2012); Grüne and Pannek (2017). Also the distributed economic MPC approach taken by Lee and Angeli (2014) relies on strict dissipativity of the overall system and guarantees asymptotic stability of the overall optimal steady state by solving the centralized problem iteratively via distributed optimization in each timestep. In contrast, in the following we show how the structural properties of the specific system interconnection in terms of separable cost functions and local dissipativity properties studied in the work at hand can be exploited in order to overcome the “centralization requirement”. In particular, we employ the sequential cooperative distributed MPC algorithm proposed by (Müller et al., 2012, Section 3.2), where the control goal was stabilization of a set, and hence, both the local cost functions ii and the resulting cost for the overall system were assumed to be positive definite with respect to the set to be stabilized. In the following we show that this algorithm can be applied for distributed economic MPC of the considered class of systems, namely dynamically decoupled subsystems as in (1) interconnected by a cost function of the separable structure as in (3) subject to arbitrary local economic cost functions ii . Moreover, if the overall system exhibits the strict dissipativity property which we established from local dissipativity in the foregoing section, the algorithm yields asymptotic stability of
˜ −i (t) = x ˆ −i (t). (1) At time t, all subsystems initialize x (2) Sequentially from i = 1 to P , each system solves its MPC optimization problem (14)–(15) and sends u0i (t) and x0i (t) to its neighbors j ∈ N i , which update ˜ −j (t) by replacing x ˆ i (t) with x0i (t). x (3) Each subsystem i ∈ I[1,P ] applies ui (t) = u0i (0|t). (4) Set t := t + 1 and go to Step (1). Theorem 12. Let the conditions of Lemma 6, 9 or 11 be satisfied for strict dissipativity, suppose the local MPC optimization problems (14)–(15) are initially feasible for all i ∈ I[1,P ] , and let (¯ x, u ¯) ∈ Zs be the optimal steady state of the overall system. Then the overall closed-loop system resulting from application of Algorithm 1 is asymptotically stable with respect to x ¯. We only sketch the necessary changes to the proof of Theorem 2 in Müller et al. (2012) required due to the economic cost formulation employed in the MPC optimization problems (14)–(15). Proof. (Sketch) As customary in economic MPC (c.f. Diehl et al. (2011); Amrit et al. (2011); Angeli et al. (2012)), we introduce the local rotated cost functions ˜ii (xi , ui ) := ii (xi , ui ) − ii (¯ xi , u ¯i ) + λi (xi ) − λi (fi (xi , ui )) ≥ ρi (xi − x ¯i ) exploiting the local dissipativity property of subsystem i. The rotated cost functional of subsystem i is then given N −1 ˜ ˜ −i (t)) := by J˜i (xi (t), ui (t), x k=0 ii (xi (k|t), ui (k|t)) + 1
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For notational simplicity let ij ≡ 0 for (i, j) ∈ / E.
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5. CONCLUSIONS
and the according rotated co¯ (x (t), ui (t), x ˜ −i (t)) := operative cost functional is J˜ N −1 i i ˜ ˜ −i (t)) + k=0 Ji (xi (t), ui (t), x xj (k|t), xi (k|t)). j∈N i ji (˜ Note that the original optimization problem (14)–(15) ¯i have the as well as (14)–(15) with J¯i replaced by J˜ same feasible sets and optimizers, compare Angeli et al. (2012). Hence we may analyze stability of the MPC closed-loop system by considering the latter rotated MPC optimization problems. Now consider the func˜ i (t), u0 (t), x0 (t)) and use V˜ (t) = tions V˜i (t) := J(x i −i P ˜ (t) as a Lyapunov function candidate. By apV i i=1 plication of Algorithm 1 we obtain, thanks to locally optimizing the cooperative cost functionals sequentially, P ˆ i (t), x ˆ −i (t)) following exactly the V˜ (t) ≤ i=1 J˜i (xi (t), u same arguments as in Müller et al. (2012). This yields j∈Ni ij (xi (k|t), xj (k|t))
V˜ (t + 1) − V˜ (t) =
P i=1
91
In this work, we investigated the interconnection of dynamically decoupled subsystems by means of a coupling cost, where each of the subsystems is assumed to be dissipative with respect to a certain supply rate commonly considered in economic MPC. We provided conditions for a certain class of interconnection cost functions resulting in a structured dissipativity property for the overall system. Moreover, we showed how these structural properties can be exploited in a non-iterative distributed (economic) MPC scheme providing asymptotic stability of the optimal overall steady state. REFERENCES Amrit, R., Rawlings, J.B., and Angeli, D. (2011). Economic optimization using model predictive control with a terminal cost. Annual Reviews in Control, 35(2), 178–186. Angeli, D., Amrit, R., and Rawlings, J.B. (2012). On average performance and stability of economic model predictive control. IEEE Transactions on Automatic Control, 57(7), 1615–1626. Christofides, P.D., Scattolini, R., de la Pena, D.M., and Liu, J. (2013). Distributed model predictive control: A tutorial review and future research directions. Computers & Chemical Engineering, 51, 21–41. Diehl, M., Amrit, R., and Rawlings, J.B. (2011). A lyapunov function for economic optimizing model predictive control. IEEE Transactions on Automatic Control, 56(3), 703–707. Driessen, P.A.A., Hermans, R.M., and van den Bosch, P.P.J. (2012). Distributed economic model predictive control of networks in competitive environments. In Proc. IEEE Conference on Decision and Control, 266–271. Maui, HI, USA. Godsil, C. and Royle, G.F. (2013). Algebraic graph theory, volume 207. Springer Science & Business Media. Grüne, L. and Pannek, J. (2017). Nonlinear Model Predictive Control: Theory and Algorithms. Communications and Control Engineering. Springer, Switzerland. Grüne, L. (2013). Economic receding horizon control without terminal constraints. Automatica, 49(3), 725–734. Köhler, P.N., Müller, M.A., and Allgöwer, F. (2016). A distributed economic MPC scheme for coordination of self-interested systems. In Proc. American Control Conference, 889–894. Boston, USA. Lee, J. and Angeli, D. (2014). Cooperative economic model predictive control for linear systems with convex objectives. European Journal of Control, 20(3), 141 – 151. Maestre, J.M. and Negenborn, R.R. (2013). Distributed model predictive control made easy, volume 69. Springer Science & Business Media. Müller, M.A. and Allgöwer, F. (2017). Economic and distributed model predictive control: Recent developments in optimizationbased control. SICE Journal of Control, Measurement, and System Integration, 10(2), 39–52. Müller, M.A., Angeli, D., and Allgöwer, F. (2015). On necessity and robustness of dissipativity in economic model predictive control. IEEE Transactions on Automatic Control, 60(6), 1671–1676. Müller, M.A., Reble, M., and Allgöwer, F. (2012). Cooperative control of dynamically decoupled systems via distributed model predictive control. International Journal of Robust and Nonlinear Control, 22(12), 1376–1397. Trinh, M.H., Nguyen, C.V., Lim, Y.H., and Ahn, H.S. (2018). Matrix-weighted consensus and its applications. Automatica, 89, 415 – 419. Tuna, S.E. (2016). Synchronization under matrix-weighted laplacian. Automatica, 73, 76–81. Willems, J.C. (1972). Dissipative dynamical systems part i: General theory. Archive for rational mechanics and analysis, 45(5), 321– 351.
λi (fi (xi (t), ui (t)) − λi (xi (t))
xi , u ¯i , x ¯−i ) − i (xi (t), ui (t), x−i (t)) + i (¯ ≤ −˜ ρ(x − x ¯),
where the last inequality follows from dissipativity of the overall system. Hence, asymptotic stability of x ¯ follows.
As discussed above, the cooperative nature of the distributed MPC Algorithm 1 requires each subsystem i to consider its influence on the cost of its neighbors, i.e., the cooperative cost functional J¯i including ji in the local MPC optimization problems (14)–(15). In some applications, e.g., for privacy reasons or for a more straight forward implementation, it would be favourable to locally only consider each systems own cost function i . Due to solving the local MPC optimization problems in a distributed non-iterative way, however, the proof of asymptotic stability of the overall optimal steady state (also for tracking costs) as shown before does in general not hold anymore. Though, in the case of the diffusive quadratic cost interconnection considered in the work at hand being symmetric, we can provide the following result. Corollary 13. Let the conditions of Lemma 9 or 11, and of Theorem 12 hold. Suppose G q is symmetric, i.e., (i, j) ∈ E ⇔ (j, i) ∈ E, with symmetric weights, i.e., Qij = Q ij for all (i, j) ∈ E. Then asymptotic stability of the optimal overall steady state follows for Algorithm 1 executed in a non-cooperative fashion, i.e., considering −1 ˜ −i (t)) = N only Ji (xi (t), ui (t), x k=0 ii (xi (k|t), ui (k|t)) + (x (k|t), x (k|t)) as the objective function in the ij i j j∈Ni local MPC optimization problems (14)–(15). Proof. Thanks to the assumed symmetry it is observed that the non-cooperative cost functional equals the cooperative cost functional formulated for cost interconnection ¯ ij = 1 Qij for all (i, j) ∈ E. Hence, analysis of weights Q 2 Algorithm 1 executed in a non-cooperative fashion equals the above analysis of Algorithm 1 for halved interconnection weights, which finally requires to show that ρ˜(x − x ¯) given by Lemma 9 or 11 is positive definite also for halved interconnection weights. This is true since QD + Ln 0 ⇒ QD + 12 Ln 0, and hence (x − x ¯) (QD + 1 ˜ ¯) ≥ ρ˜(x − x ¯) := (x − x ¯) Q(x − x ¯) holds for a 2 Ln )(x − x ˜ matrix Q positive definite. 93
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IFAC PapersOnLine 51-20 (2018) 86–91
Interconnections of dissipative systems Interconnections of dissipative systems Interconnections of dissipative systems and distributed Interconnections of economic dissipativeMPC systems and distributed economic MPC and distributed economic MPC and distributed economic MPC Philipp N. Köhler ∗∗ Matthias A. Müller ∗∗ Frank Allgöwer ∗∗
Philipp N. Köhler ∗ Matthias A. Müller ∗ Frank Allgöwer ∗ Philipp N. Köhler Matthias A. Müller Frank Allgöwer ∗ N. Köhler ∗ Matthias A. Müller ∗ Frank Allgöwer ∗ Philipp ∗ Institute for Systems Theory and Automatic Control, ∗ Institute for Systems Theory and Automatic Control, Institute for SystemsofTheory and Germany Automatic Control, University Stuttgart, University Stuttgart, ∗ Institute for Systemsof Theory and Germany Automatic Control, University of Stuttgart, Germany (e-mail: {koehler,mueller,allgower}@ist.uni-stuttgart.de). (e-mail: {koehler,mueller,allgower}@ist.uni-stuttgart.de). University of Stuttgart, Germany (e-mail: {koehler,mueller,allgower}@ist.uni-stuttgart.de). (e-mail: {koehler,mueller,allgower}@ist.uni-stuttgart.de). Abstract: Abstract: The The interconnection interconnection of of dynamically dynamically decoupled decoupled subsystems, subsystems, each each exhibiting exhibiting aa Abstract: The interconnection of dynamically decoupled subsystems, each a certain dissipativity property considered in the context of economic MPC, is investigated. certain dissipativity property considered in the context of economic MPC, is exhibiting investigated. Abstract: The ofinterconnection ofis dynamically decoupled subsystems, each exhibiting a certain dissipativity property considered in the context of economic MPC, is investigated. Interconnection the subsystems by means of their cost functions being separable in a Interconnection of the subsystems is by means ofcontext their cost functionsMPC, being isseparable in a certain dissipativity property considered in the of economic investigated. Interconnection of the subsystems is by means of their cost functions being separable in a purely local economic and a coupling cost term. For certain classes of quadratic interconnection purely local economic a couplingiscost classes of quadratic interconnection Interconnection ofconditions theand subsystems by term. meansFor of certain their cost functions being separable in a purelywe local economic and a coupling cost term. For certain classes of quadratic interconnection costs provide on the interconnection structure under which the overall system system costs we provide conditions on the interconnection structure under which the overall purely local economic and a coupling cost Moreover, term. For certain classes of quadratic interconnection costs we provide conditions on the interconnection structure under which the overall system exhibits the same dissipativity property. we apply a non-iterative distributed MPC exhibits the same conditions dissipativity property. Moreover, we apply aunder non-iterative distributed MPC costs weto provide onsystem the interconnection structure which the overall optimal system exhibits the same dissipativity property. Moreover, we apply a stability non-iterative distributed MPC scheme the interconnected which yields asymptotic of the overall scheme to the interconnected system which yields asymptotic stability of the overall optimal exhibits thethe same dissipativity property. Moreover, wethe apply a stability non-iterative distributed MPC scheme to interconnected system which yields asymptotic of the overall optimal steady state by exploiting the structural properties of system interconnection. steady state byinterconnected exploiting the structural properties of the system interconnection. scheme to the system which yields asymptotic stability of the overall optimal steady state by exploiting the structural properties of the system interconnection. © 2018,state IFACby (International of Automatic Control) Hosting by interconnection. Elsevier Ltd. All rights reserved. steady exploiting Federation the structural properties of the system Keywords: Keywords: Decentralized Decentralized and and distributed distributed control, control, Economic Economic MPC, MPC, Modeling Modeling and and decision decision Keywords: Decentralized and distributed control, Economic MPC, Modeling and decision making in complex systems, Multiagent systems. making in complex systems, Multiagent systems. Keywords: Decentralized and distributed control, Economic MPC, Modeling and decision making in complex systems, Multiagent systems. making in complex systems, Multiagent systems. 1. only 1. INTRODUCTION INTRODUCTION only few few results results are are available available in in the the literature. literature. In In Köhler Köhler 1. INTRODUCTION only few results are available in the literature. In Köhler et al. (2016), convergence to an initially unknown overall et al.few (2016), convergence to an initially unknown overall 1. INTRODUCTION only results are available in the literature. In Köhler et al. (2016), convergence to an initially unknown overall optimal steady state is enforced by average constraints. Distributed model predictive control (MPC) has recently optimal steady state is enforced by average constraints. Distributed model predictive control (MPC) has recently et al.approach (2016), convergence toal. an(2012) initially unknown overall optimal steady state is enforced by average constraints. The of Driessen et and Lee and Distributed model predictive control (MPC) has recently gained much attention as a flexible tool for control of approach ofstate Driessen et al. (2012) and Leeconstraints. and Angeli Angeli gained muchmodel attention as a control flexible (MPC) tool forhas control of The optimal steady is enforced by average The approach of Driessen et al. (2012) and Lee and Angeli Distributed predictive recently (2014) is based on a dissipativity assumption on the gained attention as a flexiblesystems tool for control of (2014) is based on a dissipativity assumption on the overall complex,much large-scale, interconnected such as, e.g., e.g., overall complex, large-scale, interconnected systems such as, The approach of Driessen et al. (2012) and Lee and Angeli (2014) is based on a dissipativity assumption on the overall gained much attention as a flexible tool for control of system and an iterative distributed MPC scheme. Thereby, complex, large-scale, interconnected as, e.g., electric grids grids or interacting interacting vehicles.systems While such enjoying the system and an iterative distributed MPC scheme. Thereby, electric or vehicles. While enjoying the (2014) isand based on a dissipativity assumption onproblem the overall system an iterative distributed MPC scheme. Thereby, complex,grids large-scale, interconnected systems such as, e.g., the centralized economic MPC optimization is electric or interacting vehicles. While enjoying the advantages of classical MPC such as satisfaction of certhe centralized economic MPC optimization problem is advantages of or classical MPC such as While satisfaction of cerand an iterative distributed MPC scheme. Thereby, the centralized economic MPC optimization problem is electric gridsof interacting enjoying the system solved distributedly, which yields stability and convergence advantages classical MPCvehicles. such as satisfaction of certain constraints and incorporation of some performance solved distributedly, which yields stability and convergence tain constraints and incorporation of some performance the centralized economic MPC optimization problem is solved distributedly, which yields stability and convergence advantages of classical MPCfunction, such as satisfaction of cerresults when iterated until convergence in each timestep. tain constraints and incorporation of it some performance criterion in terms of a cost does not rely on results when iterated until convergence in each timestep. criterion in termsand of aincorporation cost function,of itsome doesperformance not rely on solved distributedly, which yields stability and convergence results when iterated until convergence in each timestep. tain constraints criterion terms a cost function, it doesreduces not rely on On the contrary, in this work we follow a bottom-up central in unit and ofhence hence favors scalability, comaacriterion central unit and favors scalability, reduces comiterated in each timestep. On thewhen contrary, in until this convergence work we follow a bottom-up terms a cost function, it does not A rely on results amunication central in unit and ofhence favors scalability, reduces comrequirements and is less error prone. large On the contrary, in this work weof follow a bottom-up approach and derive dissipativity the overall system munication requirements and is less error prone. A large approach and derive dissipativity of the overall system a central and hence favors scalability, reduces munication requirements and is less error prone. A comlarge variety of unit distributed MPC schemes has been proposed in approach On the contrary, inthe this work subsystems weoffollow a bottom-up and derive dissipativity the overall system from dissipativity of isolated under certain variety of distributed MPC schemes has been proposed in from dissipativity of the isolated subsystems under certain munication requirements and is less error prone. A large variety of distributed MPCsystem schemes has been proposed in approach the literature literature for different different setups, see Christofides Christofides and derive dissipativity of the overall system from dissipativity of the isolated subsystems under certain conditions on the interconnection structure. In particular, the for system setups, see variety of distributed MPC schemes has been proposed in conditions on the interconnection structure. In particular, the literature for different system setups, see Christofides et al. (2013); Maestre and Negenborn (2013); Müller and from dissipativity of the isolated subsystems under certain the interconnection structure. In particular, we consider dynamically decoupled subsystems interconet (2013); Maestre and system Negenborn (2013); Müller and conditions consideron decoupled subsystems intercontheal. literature for for different setups, see Christofides et al. (2013); Maestre and Negenborn (2013); Müller and we Allgöwer (2017) an overview. Most of these schemes conditions ondynamically thetheir interconnection structure. In particular, we consider dynamically decoupled subsystems interconnected through cost functions involving other sysAllgöwer (2017) for an overview. Most of these schemes nected through their cost functions involving other syset al.concerned (2013); Maestre Negenborn (2013); Müller and we consider dynamically decoupled subsystems interconAllgöwer (2017) for stabilization anand overview. Most ofsetpoint, these schemes are with of some which nected through theirthe cost functions involving other systems’ states. Hence, main focus of this work is not to are concerned with stabilization of some setpoint, which Allgöwer (2017) for an overview. Most of these schemes tems’ states. Hence, the main focus of this work is not to are concerned with of someInstead, setpoint, might not always always be stabilization the main main objective. objective. in which many nected through their cost functions involving other systems’ states. Hence, the main focus of this work is not to develop another distributed economic MPC scheme, but might not be the Instead, in many develop another distributed economic MPC scheme, but are concerned with stabilization of some setpoint, which might not always be the main objective. Instead, in many practical applications, applications, optimal optimal operation operation with with respect respect to to tems’ states. Hence, the maineconomic focus ofthe this work is notbut to develop another distributed MPC scheme, rather to reveal connections between interconnection practical toanother reveal connections between the interconnection mightreal notapplications, always be theoptimal main objective. Instead, in many practical operation with or respect to rather some performance criterion such as profit working develop distributedkind economic MPC scheme, subbut rather to reveal connections between the interconnection structure of the considered of locally dissipative some real performance criterion such as profit or working of the considered kind of locally subpractical applications, optimal operation with respect to structure some real performance criterion such as profit or working costs is desired. desired. Allowing for an an arbitrary cost function rather toand reveal connections between the dissipative interconnection structure of the considered kind of locally dissipative subsystems dissipativity of the resulting overall system. costs is Allowing for arbitrary cost function some real performance criterion such as profit or working systems and dissipativity of the resulting overall system. costs desired. Allowing an arbitrary cost function which isneed need not be be positivefor definite with respect respect to some some structure ofthis the considered kind of locally dissipative subsystems and dissipativity of the resulting overall system. Note that is in the spirit of investigating the inputwhich not positive definite with to that this is in the spirit of resulting investigating thesystem. inputcosts isneed desired. Allowing an arbitrary cost economic function which not be positivefor definite with respect to some Note steady state is the distinctive feature of so called systems and dissipativity of the overall Note that this is in the spirit of investigating the inputoutput-interconnection of dissipative systems as the steady state not is the featurewith of so respect called economic dissipative systems the as in in the which need bedistinctive positive to some steady state is the distinctive feature so called may, economic MPC. Consequently, optimaldefinite system ofoperation operation e.g., output-interconnection Note that thisofisWillems in the of ofMoreover, investigating inputoutput-interconnection ofspirit dissipative systems as derived in the seminal work (1972). having MPC. Consequently, optimal system may, e.g., seminal work of Willems (1972). Moreover, having derived steady state is the distinctive feature of so called economic MPC. Consequently, optimal system operation may, e.g., output-interconnection be at some steady state or at some periodic orbit. of dissipative systems as in the seminal work of Willems (1972). Moreover, having derived conditions for dissipativity of the overall system, we show be at some steady state or at some periodic orbit. forofdissipativity of theMoreover, overall system, we show MPC. Consequently, optimal operation may, e.g., conditions be at some steady state or at system some periodic orbit. seminal work Willems (1972). having derived for dissipativity of the overall system, how the interconnection structure can beshow exEconomic MPC has been studied extensively for single single conditions how the considered considered interconnection structure canwe exbe at someMPC steadyhas state or at some periodic orbit. Economic been studied extensively for conditions for dissipativity of the overall system, webe show how the considered interconnection structure can be exploited by applying a non-iterative, sequential distributed Economic MPC hascertain been studied extensively for single systems, and under dissipativity properties, operploited by applying a non-iterative, sequential distributed systems, and under certain dissipativity properties, operhow the considered interconnection structure can be exploited by applying a non-iterative, sequential distributed Economic MPC has been studied extensively for single MPC scheme proposed in Müller et al. (2012) for a stabisystems, under certain dissipativity ation at at and steady state can be be shown to to properties, be optimal, optimal,operand MPC scheme proposed in Müller et al. (2012) for a stabiation steady state can shown be and ploited by applying a non-iterative, sequential distributed MPC scheme proposed in Müller et al. (2012) for a stabisystems, under certain dissipativity operlizing MPC setting to the economic setup at hand. ation at and steady state can be shownsteady to properties, be state optimal, and moreover, stability of the optimal can be MPC setting to the economic setup at hand. moreover, stability of can the be optimal steady canand be lizing proposed in Müller et al. (2012) for a stabilizing scheme MPC setting to the economic setup at hand. ation at steady state shown toDiehl be state optimal, moreover, stability of the optimal steady state be MPC established for the the closed loop (see e.g. et al. al. can (2011); This work is structured as follows. In Section 2, established for closed loop (see e.g. Diehl et (2011); lizing MPC setting to the economic setup at hand. This work is structured as follows. In Section 2, a a more more moreover, stability of the optimal steady state can be established the closed loopet(see DiehlGrüne et al. (2011); Amrit et et al. al.for(2011); (2011); Angeli al. e.g. (2012); (2013); This work is structured as follows. In Section 2, a given. more precise system description and problem statement is Amrit Angeli et al. (2012); Grüne (2013); precise system description and problem statement is given. established for the closed loop (see e.g. Diehl et al. (2011); Amrit al. Allgöwer (2011); Angeli et al. (2012); inGrüne (2013); This Müller etand and (2017)). However, the context context work is we structured asand follows. Indissipativity Section 2, isapresermore precise system description problem statement given. In Section 3, provide conditions for Müller Allgöwer (2017)). However, in the Section 3, wedescription provide conditions for dissipativity preserAmrit etand al. Allgöwer (2011); Angeli et ofal. (2012); inGrüne (2013); In Müller (2017)). However, the systems, context of distributed economic MPC interconnected precise system and problem statement isFinally, given. In Section 3, we provide conditions for dissipativity preservation for specific interconnection cost functions. of distributed economic MPC of interconnected systems, for 3, specific interconnection cost functions. Finally, Müller and Allgöwer in the systems, context vation of distributed economic(2017)). MPC ofHowever, interconnected In Section wewe provide conditions for dissipativity preservation for specific interconnection cost functions. Finally, in Section 4, investigate sequential distributed eco of distributed economic MPC of interconnected systems, in Section 4, we investigate sequential distributed ecoauthors thank the German Research Foundation (DFG) for The vation for specific interconnection cost functions. Finally, in Section 4, we investigate sequential distributed economic model predictive control of the considered class of The authors thank the German Research Foundation (DFG) for nomic model predictive control of the considered class of support of this work within grant AL 316/11-1 and within the The authors German Foundation (DFG) the for in Section 4, predictive wegiving investigate sequential distributed ecosupport of thisthank work the within grantResearch AL 316/11-1 and within nomic model control of the considered class of systems, before some concluding remarks in Sec systems, before giving some concluding remarks in SecCluster ofofExcellence in Simulation Technology (EXC 310/2) at the The authors thank the German Research Foundation (DFG) for support this work within grant AL 316/11-1 and within nomic model predictive control of the considered of Cluster of Excellence in Simulation Technology (EXC 310/2) at the systems, before giving some concluding remarks class in Section 5. University of Stuttgart. support of this work within grant AL 316/11-1 and within the tion 5. Cluster of Excellence in Simulation Technology (EXC 310/2) at University of Stuttgart. systems, before giving some concluding remarks in Section 5. Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart. tion 5. University of Stuttgart.
2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 88 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 88 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 88 10.1016/j.ifacol.2018.10.179 Copyright © 2018 IFAC 88
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Notation. The interconnection structure between subsystems is described by a directed graph G = {V, E} with vertices V = {1, . . . , P } representing the subsystems and edges E = {(i, j) ∈ V × V} representing the directed interconnection of systems. Denote the set of all neighbors of subsystem i by Ni := {j ∈ V | (i, j) ∈ E} and let di := |Ni | denote its cardinality. Similarly, denote the set of undirected neighbors of subsystem i as N i := {j ∈ V | (i, j) ∈ E ∨ (j, i) ∈ E}. A directed graph G is said to be weakly connected if there is a path between every pair of vertices when neglecting the edge orientation. Denote by B ∈ {−1, 0, 1}|V|×|E| the incidence matrix of a directed graph G with elements Bve equalling 1 if vertex v is the head of edge e, −1 if v is the tail of e, and 0 otherwise. For a thorough presentation of algebraic graph theory we refer the reader to Godsil and Royle (2013). A function ρ : Rn → R is positive semidefinite if ρ(0) = 0 and ρ(x) ≥ 0 for all x ∈ Rn . The function ρ : Rn → R is positive definite if ρ(0) = 0 and ρ(x) > 0 for all x = 0. A matrix A = A ∈ Rn×n is positive (semi)definite if the quadratic form x Ax is positive (semi)definite, and we write A 0 (A 0). The operator ⊗ denotes the standard Kronecker product, diag(·) denotes the block diagonal matrix built from the respective arguments, In ∈ Rn×n denotes the n × n-identity matrix, and 1n ∈ Rn denotes the all ones vector of dimension n.
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subsystem i, i.e., x−i = [x i1 , . . . , xid1 ] , where {i1 , . . . , idi } is an ordered sequence of the elements of the set Ni , that is, i1 < · · · < idi . Such coupling costs are found, e.g., in classical multiagent settings representing a common objective, or in the case of subsystems sharing a common resource. Accordingly, the stage cost of the overall system is given as P
i (xi , ui , x−i ). (4)
(x, u) = i=1
The separable structure of the cost functions considered here reflects the main theme of this work, namely studying the interconnection of local subsystems with local properties. For each subsystem i, denote by Z∗i the set of locally optimal steady state and input pairs, and hence Z∗i = {(xsi , usi ) ∈ Zsi | ii (xsi , usi ) ≤ ii (xi , ui )∀(xi , ui ) ∈ Zsi } . Definition 1. (Angeli et al. (2012)). A control system as (2) is dissipative with respect to a supply rate s : Z → R if there exists a non-negative function λ : X → R, a positive semidefinite function ρ : X → R and x ¯ ∈ X such that λ(f (x, u)) − λ(x) ≤ s(x, u) − ρ(x − x ¯) (5) for all (x, u) ∈ Z. If there exists ρ positive definite, then the system is said to be strictly dissipative. Throughout the following, we assume that each subsystem i considered individually, i.e., neglecting the coupling cost ij , has the following local dissipativity property. Assumption 2. Each subsystem i ∈ I[1,P ] is dissipative on Zi with respect to the supply rate si (x, u) := ii (xi , ui ) −
ii (xsi , usi ) for all (xsi , usi ) ∈ Z∗i , i.e., the following inequality holds for all (xi , ui ) ∈ Zi λi (fi (xi , ui )) − λi (xi ) (6) ≤ ii (xi , ui ) − ii (xsi , usi ) − ρi (xi − xsi ).
2. SYSTEM DESCRIPTION AND PROBLEM STATEMENT Consider a set of P dynamically decoupled systems, with each of the systems obeying the discrete-time nonlinear dynamics xi (t + 1) = fi (xi (t), ui (t)), xi (0) = xi0 (1) where xi (t) ∈ Xi ⊂ Rn denotes the state and ui (t) ∈ Ui ⊂ Rm the control input to system i, i ∈ I[1,P ] := {1, 2, . . . , P }, and fi : Xi × Ui → Xi continuous. The set of admissible state-input pairs is denoted by Zi ⊆ Xi × Ui . Denote by Zsi the set of all admissible steady states for each system i, i.e., Zsi := {(xi , ui ) ∈ Zi | xi = fi (xi , ui )}. For the overall system, the state and input variables are written nP in stacked vector form, i.e., x = [x and 1 , . . . , xP ] ∈ R mP s u = [u1 , . . . , uP ] ∈ R , and similarly for x and us . The corresponding overall system dynamics result in x(t + 1) = f (x(t), u(t)), x(0) = x0 (2) with (x(t), u(t)) ∈ Z := Z1 × · · · × ZP and f (x, u) = [f1 (x1 , u1 ) , . . . , fP (xP , uP ) ] . Accordingly, the set of admissible steady states for the overall system is Zs = Zs1 × · · · × ZsP . Note that until this point, the subsystems i are completely decoupled.
This dissipativity assumption is explicitly formulated for the case of the optimal steady state (xsi , usi ) not being unique. Note that in the case of multiple optimal steady states, ρi can only be positive semidefinite. Remark 3. Note that for local tracking costs, i.e., cost functions ii that are positive definite w.r.t. (xsi , usi ), strict dissipativity is trivially fulfilled with λi = 0. Hence, all our following results on system interconnections also hold for local tracking costs as a special case of the considered setup with general economic local cost functions. As discussed in the introduction, the assumed (strict) dissipativity of a subsystem with respect to the particular supply rate of Assumption 2 has strong implications on optimality of steady state operation (c.f. Müller et al. (2015)), and yields (asymptotic) stability results for the closed-loop system under economic model predictive control (Amrit et al., 2011; Angeli et al., 2012). Given a set of locally dissipative systems in the sense of Assumption 2, in this work we investigate specific cost interconnections
ij and derive conditions under which the local (strict) dissipativity properties are preserved for the overall system with overall cost function (4). Hence, we aim at providing (strict) dissipativity of the overall system (2), i.e., verifying that there exists a storage function Λ : X → R and ρ˜ : X → R (at least) positive semidefinite, such that Λ(f (x, u)) − Λ(x) ≤ s˜(x, u) − ρ˜(x − x ¯) (7)
Coupling between the subsystems is introduced through the economic stage cost functions i each system is trying to minimize. In particular, we consider separable cost functions of the following form
ij (xi , xj ), (3)
i (xi , ui , x−i ) = ii (xi , ui ) + j∈Ni
with the continuous local economic cost function ii : Rn × Rm → R of subsystem i, and the coupling cost
ij : Rn × Rn → R induced by the coupling between subsystem i and its neighbors. By x−i we denote the collection of states of all subsystems that are neighbors of
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the overall optimal steady state directly appears in the coupling cost. Hence, interconnection costs of such a structure may rather appear in a tracking MPC application, and is introduced here for the sake of a clear presentation. ˜ ∈ Lemma 6. Let Assumption 5 hold. If there exists Q nP ×nP R such that for the interconnection weights in (11) Q11 Q12 · · · Q1P Q21 Q22 ˜ Q Q := .. ... .
holds for all (x, u) ∈ Z with supply rate s˜(x, u) :=
P i=1
i (xi , ui , x−i ) − i (¯ xi , u ¯i , x ¯−i )
(8)
and (¯ x, u ¯) ∈ Zs being an (optimal) steady state of the overall system. The special separable structure of the overall system and the local dissipativity properties suggest to use as a storage function of the overall system the P sum of local storage functions, i.e., Λ(x) = i=1 λi (xi ). According to Assumption 2, we obtain Λ(f (x, u))−Λ(x) ≤ P s i=1 si (xi , ui ) − ρi (xi − xi ). In order to deduce the desired dissipativity inequality for the overall system (7) P with interconnection costs, the inequality i=1 si (xi , ui )− ρi (xi − xsi ) ≤ s˜(x, u) − ρ˜(x − x ¯) needs to be satisfied for all (x, u) ∈ Z, for any (xsi , usi ) ∈ Z∗i and for (¯ x, u ¯) ∈ Zs , which expands to P −ii (xsi , usi ) − ρi (xi − xsi ) (9)
QP 1 QP P ˜ holds with Q positive semidefinite, then the overall system (2) with stabilizing coupling costs (11) is dissipative with respect to the supply rate (8) for x ¯ = xs with s s ∗ ˜ (xi , ui ) ∈ Zi . Moreover, if there exists Q 0, then the overall system is strictly dissipative.
Proof. Consider (¯ x, u ¯) = (xs , us ), choose ρ˜(x − xs ) := s ˜ s (x − x ) Q(x − x ), and note that this quadratic form is i=1 positive (semi)definite. Plugging this into (9) yields P P ≤ −ii (¯ xi , u ¯i ) + ij (xi , xj ) − ij (¯ xi , x ¯j ) − ρ˜(x − x ¯). ρi (xi − xsi ) + ij (xi , xj ) − ρ˜(x − xs ) 0≤ i=1
j∈Ni
i=1
Therefore, this inequality is the main object of interest throughout this work and will be investigated for specific system interconnections by means of ij . We summarize the preceding elaboration in the following lemma. Lemma 4. Let Assumption 2 hold, and let (¯ x, u ¯ ) ∈ Zs be a steady state of the overall system (2). If there exists a positive semidefinite function ρ˜ : X → R such that (9) is satisfied for all (x, u) ∈ Z, then the overall system (2) is dissipative with supply rate s˜(x, u) := P xi , u ¯i , x ¯−i ). Moreover, (¯ x, u ¯) is an i=1 i (xi , ui , x−i ) − i (¯ optimal steady state of the overall system. If there exists ρ˜ positive definite, then the overall system is strictly dissipative.
j∈Ni s
˜ − xs ) = (x − x ) Q(x − x ) − (x − xs ) Q(x ˜ − xs ), = (x − xs ) (Q − Q)(x s
which is satisfied due to the assumed positive (semi)defi˜ Hence, all conditions of Lemma 4 are niteness of Q − Q. fulfilled which yields the claim. Besides the purely algebraic characterization of the cost interconnection structure as in Lemma 6, the interconnection weight Qij can be seen as a weight assigned to the (i, j)-edge of the graph G. If we define QD := diag(Q11 , . . . , QP P ) then the matrix Q − QD can be interpreted as the matrix weighted adjacency matrix of the interconnection graph G. Matrix weighted graphs constitute a non-trivial extension of classical (scalar) weighted graphs and have only recently been investigated in the literature, see Tuna (2016); Trinh et al. (2018). For the sake of a simple presentation clearly revealing the implications of certain graph properties, in the following we restrict ourselves to classic scalar weighted graphs.
3. QUADRATIC COUPLING COSTS AND DISSIPATIVITY PRESERVATION In the remainder of this work, we focus on quadratic cost interconnections ij . Moreover, we assume that the local dissipativity Assumption 2 holds with ρi being quadratic: Assumption 5. Let Assumption 2 hold for (10) ρi (xi − xsi ) = (xi − xsi ) Qii (xi − xsi ) with Qii ∈ Rn×n 0 for all i ∈ I[1,P ] .
Hence, in the following we consider the weighted graph G q := {V, E, W}, which corresponds to the graph G augmented by non-negative weights qij ∈ R assigned to each edge, hence with the according set of weights W := {qij ∈ R, qij ≥ 0 ∀ (i, j) ∈ E}.
Despite restricting the class of considered cost coupling functions and slightly strengthening the local dissipativity assumption, the local economic stage cost functions ii as well as the system dynamics fi are treated arbitrary and nonlinear throughout this work.
3.2 Diffusive coupling cost In this section, we consider diffusive coupling costs, typically found in multiagent agreement or synchronization applications, of the form ij (xi , xj ) = (xi − xj ) Qij (xi − xj ) (12) with positive semidefinite interconnection weights Qij = n×n Q given by the weighted graph G q . Hence, as ij ∈ R discussed above, for simplicity we specialize the interconnection weights in (12) to Qij = qij In for (i, j) ∈ E. Similarly, we assume the following for the weights of the quadratic form in (10) of Assumption 5 (and hence specialize Assumption 2 further).
3.1 Stabilizing coupling cost In this section, we consider “stabilizing” coupling costs of the form (11) ij (xi , xj ) = (xi − xsi ) Qij (xj − xsj )
with interconnection weights Qij ∈ Rn×n , Qij = 0 for (i, j) ∈ / E. Note that such a cost formulation does not make much sense in an economic MPC framework, since
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Assumption 7. Let Assumption 2 hold for (13) ρi (xi − xsi ) = (xi − xsi ) Qii (xi − xsi ) with Qii = qii In , qii ≥ 0 for all i ∈ I[1,P ] . Remark 8. The restriction to identity matrices certainly further restricts the class of cost interconnection structure. However, we intentionally chose this radical simplification to clearly highlight implications from graph theory.
holds for all (x, u) ∈ Z. For the case of none of the systems being strictly dissipative, hence QD = 0, the above inequality is trivially fulfilled for ρ˜(x−¯ x) := 0 thanks to positive semidefiniteness of Ln . This yields dissipativity of the overall system and hence concludes the proof for this special case. For the case of at least one subsystem i being strictly dissipative, i.e., Qii 0, we first do a completion of squares to obtain
Before stating the main results of this section, we exploit the above simplifications, rewrite the diffusive cost coupling of the overall system using a compact graph theoretic notation, and draw some immediate consequences. For the diffusive coupling cost in (12) we obtain P P ×P j∈Ni ij (xi , xj ) = x (L ⊗ In )x with L ∈ R i=1 being the weighted Laplacian of the interconnection graph G q . The Laplacian is given by L = BW B with B the incidence matrix of G q and W ∈ R|E|×|E| the diagonal matrix of weights with entries Wee = qij if vertex i is the tail and vertex j is the head of edge e. For the graph G q being weakly connected, the Laplacian L is positive semidefinite, with a single 0 eigenvalue and according eigenvector 1P . In the following we assume the graph G q to be weakly connected. In the case of a graph consisting of several disconnected components, all our following results hold for each of the separate subgraphs. Note that due to Assumption 7, QD = qD ⊗ In with qD := diag(q11 , . . . , qP P ). Thanks to the properties of the Kronecker product, the matrix (QD + L ⊗ In ) = (qD + L) ⊗ In has the same eigenvalues as (qD + L) but with multiplicity n, and for brevity we write L ⊗ In =: Ln .
x (QD + Ln )x − 2x QD xs + xs QD xs
= (x − x ˜) (QD + Ln )(x − x ˜) + c˜
with x ˜ = (QD + Ln )−1 QD xs and c˜ = xs QD xs − xs QD x ˜. Observe that (QD + Ln )−1 = ((qD + L) ⊗ In )−1 is indeed invertible since qD + L is positive definite: Positive definiteness of qD +L results from L being positive semidefinite with ker L = 1P and qD 0 being diagonal with at least one strictly positive diagonal element qii and 1 ˜ = (QD + Ln )−1 QD xs = P qD 1P > 0. Hence, we obtain x s s s x = 1P ⊗ x1 , since QD (1P ⊗ x1 ) = (QD + Ln )(1P ⊗ xs1 ) by (L ⊗ In )(1P ⊗ xs1 ) = (L1P ) ⊗ (In xs1 ) = 0, and c˜ = 0. Consequently, for xs = x ¯ and QD + Ln being positive ˜ 0 such that (x − x definite, there exists Q ¯) (QD + ˜ −x ¯) ≥ (x − x ¯) Q(x ¯) := ρ˜(x − x ¯), and hence (9) Ln )(x − x is satisfied with this choice of ρ˜. For each of the subsystems being strictly dissipative with respect to the same steady state, the above result is not surprising but confirms that the diffusive cost coupling indeed preserves the local strict dissipativity properties as expected. For the case of only some of the subsystems being strictly dissipative, the result is more interesting by also making the non-strict dissipative subsystems “find” the overall optimal steady state in an economic MPC application, see Section 4.
Systems with identical optimal steady states. First, we consider the case in which the intersection of the sets of local optimal steady states with the consensus subspace is nonempty, i.e., {Z∗1 × · · · × Z∗P } ∩ {(x, u) ∈ Z | xi = ¯ ∗ = ∅. Obviously, for the case of xj ∀ (i, j) ∈ E} =: Z at least one subsystem i being strictly dissipative, this intersection is a singleton. Hence, all systems share at least one local optimal steady state. Please note that any overall ¯ ∗ lies on the consensus steady state (¯ x, u ¯) from the set Z subspace and hence is of the form x ¯ = 1P ⊗ α with (α, βi ) ∈ Z∗i for all i ∈ I[1,P ] . ¯ ∗ is Lemma 9. Let Assumption 7 hold. If the set Z nonempty, then the overall system (2) with diffusive coupling costs (12) is dissipative with respect to the supply ¯ ∗ . Moreover, if at least one rate (8) for any (¯ x, u ¯) ∈ Z subsystem i ∈ I[1,P ] is strictly dissipative, i.e., Qii 0, then the overall system is strictly dissipative.
Systems with different optimal steady states. In the following, we consider as a second case the interconnection of subsystems which do not share a local optimal steady ¯ ∗ = ∅. Hence we cannot expect to show dissipastate, i.e., Z tivity of the overall system with respect to a steady state P on the consensus manifold by using Λ(x) = i=1 λi (x). Roughly speaking, in the following we determine, by the tools established above, a steady state of the overall system, which trades off the local systems’ (strict) dissipativity and thereby yields dissipativity of the overall system with respect to this steady state. In order to determine such a “compromise steady state”, some additional information on the shape of the local economic cost functions ii for steady states in the vicinity of the local optimal steady state is required. In this work, this information is given in terms of the following assumption. Assumption 10. For (xsi , usi ) ∈ Z∗i the local dissipativity inequality in Assumption 2 holds with equality for all steady states (¯ xi , u ¯i ) ∈ Zsi , i.e., ii (xsi , usi ) + ρi (¯ xi − xsi ) = ii (¯ xi , u ¯i ). Lemma 11. Let Assumptions 7 and 10 hold, and let at least one subsystem i ∈ I[1,P ] be strictly dissipative. If x, u ¯) ∈ Zs with x ¯ = there exists u ¯ ∈ RmP such that (¯ −1 s s s (QD + Ln ) QD x for any (xi , ui ) ∈ Z∗i , then the overall system (2) with diffusive coupling costs (12) is strictly dissipative with respect to the supply rate (8). Moreover, (¯ x, u ¯) is the optimal steady state of the overall system.
Proof. In order to prove the claim, we apply Lemma 4. Hence, we need to provide a positive semidefinite function ρ˜ and verify (9) for the given interconnection cost functions. Since all subsystems share at least one common local optimal steady state, in the following we consider without loss of generality the local optimal steady state x, u ¯), hence (xsi , usi ) ∈ Z∗i such that xs = 1P ⊗ (xs , us ) = (¯ s x1 . Note that ij (¯ xi , x ¯j ) = 0, and verification of (9) reduces to showing that P (xi − xsi ) Qii (xi − xsi ) + (xi − xj ) Qij (xi − xj ) i=1
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j∈Ni
= x (QD + Ln )x − 2x QD xs + xs QD xs ≥ ρ˜(x − x ¯)
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Proof. Again, we employ Lemma 4 and hence aim at verifying (9). First, we reuse the arguments from the proof of the above lemma in order to show that P (xi − xsi ) Qii (xi − xsi ) + (xi − xj ) Qij (xi − xj ) i=1
the overall optimal steady state. Note that the algorithm proposed in Müller et al. (2012) is a cooperative scheme in the sense that each subsystem i also considers its influence on its neighbors by incorporating the cost terms ji in its MPC optimization problem.
j∈Ni
The local MPC optimization problem of subsystem i ∈ I[1,P ] is given as follows ˜ −i (t)) (14) min J¯i (xi (t), ui (t), x
= (x − x ˜) (QD + Ln )(x − x ˜) + c˜
with (QD + Ln ) 0, x ˜ = (QD + Ln )−1 QD xs =: x ¯, and c˜ = xs QD xs −xs QD x ¯. Plugging this into (9), it remains to show that (x − x ¯) (QD + Ln )(x − x ¯) + c˜ − ρ˜(x − x ¯) ≥
P i=1
−ii (xsi , usi ) + ii (¯ xi , u ¯i ) +
ui (t)
subject to xi (0|t) = xi (t) (15a) xi (k + 1|t) = fi (xi (k|t), ui (k|t)) (15b) ¯i (15c) xi (N |t) = x xi (k|t), ui (k|t) ∈ Zi for k = 0, . . . , N − 1 (15d) ˜ −i (t)) := with prediction horizon N ∈ I>0 , J¯i (xi (t), ui (t), x N −1 (x (k|t), u (k|t)) + (x (k|t), x i j (k|t)) + i k=0 ii i j∈N i ij 1 ji (xj (k|t), xi (k|t)), and ui (t) = {ui (0|t), . . . , ui (N − 1|t)} the predicted input, and xi (t) = {xi (0|t), . . . , xi (N |t)} ˜ −i (t) the resulting predicted state sequences. As before, x denotes the stacked state sequences of all neighbors of subsystem i. We denote the optimizer of the above MPC optimization problem by u0i (t) and the according optimal state trajectory by x0i (t). Given feasible input and according state sequences ui (t) and xi (t) at time t, a feasible input and according state sequence to the MPC optimization problem at time t+1 is given by the sequences ˆ i (t+1) and x ˆ i (t+1) obtained from shifting ui (t) and xi (t) u and appending the steady state and input (¯ xi , u ¯i ) ∈ Zsi , respectively. The main steps of the algorithm are sketched below, for details we refer the reader to (Müller et al., 2012, Section 3.2). Algorithm 1. (Sketch of Algo. 2 in Müller et al. (2012)).
ij (¯ xi , x ¯j )
j∈Ni
= (¯ x − xs ) QD (¯ x − xs ) + x ¯ Ln x ¯, where the last equality follows from Assumption 10. Finally we show that these remaining terms equal c˜, since (¯ x − xs ) QD (¯ x − xs ) + x ¯ Ln x ¯ − c˜ = −xs QD x ¯+x ¯ (QD + −1 s Ln )¯ x = 0 by using x ¯ = (QD + Ln ) QD x . As before, for ˜ 0 such QD + Ln being positive definite, there exists Q ˜ that (x− x ¯) (QD +Ln )(x− x ¯) ≥ (x− x ¯) Q(x− x ¯) := ρ˜(x− x ¯) and hence (9) is satisfied for this choice of ρ˜. 4. DISTRIBUTED ECONOMIC MPC WITH SEPARABLE COST FUNCTIONS In this section, we investigate distributed economic model predictive control of the overall system (2) subject to the overall cost function (4) by locally solving an MPC problem for each subsystem i ∈ I[1,P ] . In the previous section, we established conditions on the cost interconnection under which the overall system is (strictly) dissipative. Hence, under the conditions of Lemma 6, 9 and 11 for strict dissipativity of the overall system, centralized economic MPC yields asymptotic stability of the overall optimal steady state, c.f. Amrit et al. (2011); Angeli et al. (2012); Grüne and Pannek (2017). Also the distributed economic MPC approach taken by Lee and Angeli (2014) relies on strict dissipativity of the overall system and guarantees asymptotic stability of the overall optimal steady state by solving the centralized problem iteratively via distributed optimization in each timestep. In contrast, in the following we show how the structural properties of the specific system interconnection in terms of separable cost functions and local dissipativity properties studied in the work at hand can be exploited in order to overcome the “centralization requirement”. In particular, we employ the sequential cooperative distributed MPC algorithm proposed by (Müller et al., 2012, Section 3.2), where the control goal was stabilization of a set, and hence, both the local cost functions ii and the resulting cost for the overall system were assumed to be positive definite with respect to the set to be stabilized. In the following we show that this algorithm can be applied for distributed economic MPC of the considered class of systems, namely dynamically decoupled subsystems as in (1) interconnected by a cost function of the separable structure as in (3) subject to arbitrary local economic cost functions ii . Moreover, if the overall system exhibits the strict dissipativity property which we established from local dissipativity in the foregoing section, the algorithm yields asymptotic stability of
˜ −i (t) = x ˆ −i (t). (1) At time t, all subsystems initialize x (2) Sequentially from i = 1 to P , each system solves its MPC optimization problem (14)–(15) and sends u0i (t) and x0i (t) to its neighbors j ∈ N i , which update ˜ −j (t) by replacing x ˆ i (t) with x0i (t). x (3) Each subsystem i ∈ I[1,P ] applies ui (t) = u0i (0|t). (4) Set t := t + 1 and go to Step (1). Theorem 12. Let the conditions of Lemma 6, 9 or 11 be satisfied for strict dissipativity, suppose the local MPC optimization problems (14)–(15) are initially feasible for all i ∈ I[1,P ] , and let (¯ x, u ¯) ∈ Zs be the optimal steady state of the overall system. Then the overall closed-loop system resulting from application of Algorithm 1 is asymptotically stable with respect to x ¯. We only sketch the necessary changes to the proof of Theorem 2 in Müller et al. (2012) required due to the economic cost formulation employed in the MPC optimization problems (14)–(15). Proof. (Sketch) As customary in economic MPC (c.f. Diehl et al. (2011); Amrit et al. (2011); Angeli et al. (2012)), we introduce the local rotated cost functions ˜ii (xi , ui ) := ii (xi , ui ) − ii (¯ xi , u ¯i ) + λi (xi ) − λi (fi (xi , ui )) ≥ ρi (xi − x ¯i ) exploiting the local dissipativity property of subsystem i. The rotated cost functional of subsystem i is then given N −1 ˜ ˜ −i (t)) := by J˜i (xi (t), ui (t), x k=0 ii (xi (k|t), ui (k|t)) + 1
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For notational simplicity let ij ≡ 0 for (i, j) ∈ / E.
2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018
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5. CONCLUSIONS
and the according rotated co¯ (x (t), ui (t), x ˜ −i (t)) := operative cost functional is J˜ N −1 i i ˜ ˜ −i (t)) + k=0 Ji (xi (t), ui (t), x xj (k|t), xi (k|t)). j∈N i ji (˜ Note that the original optimization problem (14)–(15) ¯i have the as well as (14)–(15) with J¯i replaced by J˜ same feasible sets and optimizers, compare Angeli et al. (2012). Hence we may analyze stability of the MPC closed-loop system by considering the latter rotated MPC optimization problems. Now consider the func˜ i (t), u0 (t), x0 (t)) and use V˜ (t) = tions V˜i (t) := J(x i −i P ˜ (t) as a Lyapunov function candidate. By apV i i=1 plication of Algorithm 1 we obtain, thanks to locally optimizing the cooperative cost functionals sequentially, P ˆ i (t), x ˆ −i (t)) following exactly the V˜ (t) ≤ i=1 J˜i (xi (t), u same arguments as in Müller et al. (2012). This yields j∈Ni ij (xi (k|t), xj (k|t))
V˜ (t + 1) − V˜ (t) =
P i=1
91
In this work, we investigated the interconnection of dynamically decoupled subsystems by means of a coupling cost, where each of the subsystems is assumed to be dissipative with respect to a certain supply rate commonly considered in economic MPC. We provided conditions for a certain class of interconnection cost functions resulting in a structured dissipativity property for the overall system. Moreover, we showed how these structural properties can be exploited in a non-iterative distributed (economic) MPC scheme providing asymptotic stability of the optimal overall steady state. REFERENCES Amrit, R., Rawlings, J.B., and Angeli, D. (2011). Economic optimization using model predictive control with a terminal cost. Annual Reviews in Control, 35(2), 178–186. Angeli, D., Amrit, R., and Rawlings, J.B. (2012). On average performance and stability of economic model predictive control. IEEE Transactions on Automatic Control, 57(7), 1615–1626. Christofides, P.D., Scattolini, R., de la Pena, D.M., and Liu, J. (2013). Distributed model predictive control: A tutorial review and future research directions. Computers & Chemical Engineering, 51, 21–41. Diehl, M., Amrit, R., and Rawlings, J.B. (2011). A lyapunov function for economic optimizing model predictive control. IEEE Transactions on Automatic Control, 56(3), 703–707. Driessen, P.A.A., Hermans, R.M., and van den Bosch, P.P.J. (2012). Distributed economic model predictive control of networks in competitive environments. In Proc. IEEE Conference on Decision and Control, 266–271. Maui, HI, USA. Godsil, C. and Royle, G.F. (2013). Algebraic graph theory, volume 207. Springer Science & Business Media. Grüne, L. and Pannek, J. (2017). Nonlinear Model Predictive Control: Theory and Algorithms. Communications and Control Engineering. Springer, Switzerland. Grüne, L. (2013). Economic receding horizon control without terminal constraints. Automatica, 49(3), 725–734. Köhler, P.N., Müller, M.A., and Allgöwer, F. (2016). A distributed economic MPC scheme for coordination of self-interested systems. In Proc. American Control Conference, 889–894. Boston, USA. Lee, J. and Angeli, D. (2014). Cooperative economic model predictive control for linear systems with convex objectives. European Journal of Control, 20(3), 141 – 151. Maestre, J.M. and Negenborn, R.R. (2013). Distributed model predictive control made easy, volume 69. Springer Science & Business Media. Müller, M.A. and Allgöwer, F. (2017). Economic and distributed model predictive control: Recent developments in optimizationbased control. SICE Journal of Control, Measurement, and System Integration, 10(2), 39–52. Müller, M.A., Angeli, D., and Allgöwer, F. (2015). On necessity and robustness of dissipativity in economic model predictive control. IEEE Transactions on Automatic Control, 60(6), 1671–1676. Müller, M.A., Reble, M., and Allgöwer, F. (2012). Cooperative control of dynamically decoupled systems via distributed model predictive control. International Journal of Robust and Nonlinear Control, 22(12), 1376–1397. Trinh, M.H., Nguyen, C.V., Lim, Y.H., and Ahn, H.S. (2018). Matrix-weighted consensus and its applications. Automatica, 89, 415 – 419. Tuna, S.E. (2016). Synchronization under matrix-weighted laplacian. Automatica, 73, 76–81. Willems, J.C. (1972). Dissipative dynamical systems part i: General theory. Archive for rational mechanics and analysis, 45(5), 321– 351.
λi (fi (xi (t), ui (t)) − λi (xi (t))
xi , u ¯i , x ¯−i ) − i (xi (t), ui (t), x−i (t)) + i (¯ ≤ −˜ ρ(x − x ¯),
where the last inequality follows from dissipativity of the overall system. Hence, asymptotic stability of x ¯ follows.
As discussed above, the cooperative nature of the distributed MPC Algorithm 1 requires each subsystem i to consider its influence on the cost of its neighbors, i.e., the cooperative cost functional J¯i including ji in the local MPC optimization problems (14)–(15). In some applications, e.g., for privacy reasons or for a more straight forward implementation, it would be favourable to locally only consider each systems own cost function i . Due to solving the local MPC optimization problems in a distributed non-iterative way, however, the proof of asymptotic stability of the overall optimal steady state (also for tracking costs) as shown before does in general not hold anymore. Though, in the case of the diffusive quadratic cost interconnection considered in the work at hand being symmetric, we can provide the following result. Corollary 13. Let the conditions of Lemma 9 or 11, and of Theorem 12 hold. Suppose G q is symmetric, i.e., (i, j) ∈ E ⇔ (j, i) ∈ E, with symmetric weights, i.e., Qij = Q ij for all (i, j) ∈ E. Then asymptotic stability of the optimal overall steady state follows for Algorithm 1 executed in a non-cooperative fashion, i.e., considering −1 ˜ −i (t)) = N only Ji (xi (t), ui (t), x k=0 ii (xi (k|t), ui (k|t)) + (x (k|t), x (k|t)) as the objective function in the ij i j j∈Ni local MPC optimization problems (14)–(15). Proof. Thanks to the assumed symmetry it is observed that the non-cooperative cost functional equals the cooperative cost functional formulated for cost interconnection ¯ ij = 1 Qij for all (i, j) ∈ E. Hence, analysis of weights Q 2 Algorithm 1 executed in a non-cooperative fashion equals the above analysis of Algorithm 1 for halved interconnection weights, which finally requires to show that ρ˜(x − x ¯) given by Lemma 9 or 11 is positive definite also for halved interconnection weights. This is true since QD + Ln 0 ⇒ QD + 12 Ln 0, and hence (x − x ¯) (QD + 1 ˜ ¯) ≥ ρ˜(x − x ¯) := (x − x ¯) Q(x − x ¯) holds for a 2 Ln )(x − x ˜ matrix Q positive definite. 93