Decentralized Control of Constrained Linear Systems via Assume-Guarantee Contracts
aa r X i v : . [ m a t h . O C ] A p r Decentralized Control of Constrained Linear Systems viaAssume-Guarantee Contracts
Weixuan Lin † Eilyan Bitar † Abstract —We consider the decentralized control of a discrete-time, linear system subject to exogenous disturbances andpolyhedral constraints on the state and input trajectories. Theunderlying system is composed of a finite collection of dynam-ically coupled subsystems, where each subsystem is assumedto have a dedicated local controller. The decentralization ofinformation is expressed according to sparsity constraints onthe state measurements that each local controller has accessto. In this context, we investigate the design of decentralizedcontrollers that are affinely parameterized in their measure-ment history. For problems with partially nested informationstructures, the optimization over such policy spaces is knownto be convex. Convexity is not, however, guaranteed undermore general (nonclassical) information structures in which theinformation available to one local controller can be affectedby control actions that it cannot access or reconstruct. Withthe aim of alleviating the nonconvexity that arises in suchproblems, we propose an approach to decentralized controldesign where the information-coupling states are effectivelytreated as disturbances whose trajectories are constrained totake values in ellipsoidal contract sets whose location, scale,and orientation are jointly optimized with the underlying affinedecentralized control policy. We establish a natural structuralcondition on the space of allowable contracts that facilitates thejoint optimization over the control policy and the contract setvia semidefinite programming.
I. I
NTRODUCTION
We investigate the design of affine decentralized con-trol policies for stochastic discrete-time, linear systems thatevolve over a finite horizon, and are subject to polyhedralconstraints on the state and input trajectories. The computa-tional tractability of such problems depends in part on theirinformation structures [1], [2]. In particular, a decentralizedcontrol problem is said to have a nonclassical informationstructure if the information available to one controller canbe affected by the control actions of another that it cannotaccess or reconstruct. Under such information structures, thecalculation of optimal decentralized control policies is knownto be computationally intractable, because of the incentive forcontrollers to communicate with each other via the actionsthey undertake—the so called signalling incentive [1]–[3].To complicate matters further, there may be hard constraintscoupling the local actions and states of different controllersthat must be jointly enforced without explicit communication.In this paper, we address these challenges by relaxing therequirement that the decentralized controller be optimal withrespect to the broad family of all causal policies, and instead
Supported in part by NSF grants ECCS-1351621 and IIP-1632124, andthe Holland Phillips Trust. † Weixuan Lin ( [email protected] ) and Eilyan Bitar( [email protected] ) are with the School of Electrical andComputer Engineering, Cornell University, Ithaca, NY, 14853, USA. search for suboptimal decentralized controllers that can beefficiently computed via convex programming methods.
Related Literature:
There is a related literature thatleverages on techniques derived from tube-based model pre-dictive control (MPC) to facilitate the design of decentralizedcontrollers for constrained dynamical systems [4]–[14]. Typi-cally, these approaches rely on a decomposition of the decen-tralized control problem into a collection of decoupled localcontrol problems by treating the coupling states and inputsassociated with each subsystem’s “neighbors” as independentexogenous disturbances that are assumed to take values inthe given state and input constraint sets. Given the resultingcollection of decoupled local control problems, centralizedMPC methods can be applied to compute local control poli-cies that are guaranteed to be feasible for each sub-problem.Although decentralized control policies calculated accordingto such decomposition methods are guaranteed to be feasiblefor the full problem, they may result in behaviors that areoverly conservative in terms of the cost they incur for anumber of reasons.
First , the treatment of the coupling statesand inputs as independent disturbances ignores the potentialdynamical coupling between these variables.
Second , theover approximation of the coupling state and input trajectorysets by their corresponding state and input constraint setswill likely be very loose for many problem instances. Moreimportantly, the over approximation of the coupling state andinput trajectory sets in this manner ignores the fact that thesesets depend on the control policy being used to regulatethe system, and, therefore, neglects the possibility of co-optimizing their specification with the control policy.
Contribution:
We provide a computationally tractablemethod to calculate control policies that are guaranteed tobe feasible for constrained decentralized control problemswith nonclassical information structures. Loosely speaking,the proposed approach seeks to neutralize the nonconvexityarising from the informational coupling between subsystemsby treating the information-coupling states as disturbanceswhose trajectories are “assumed” to take values in a certain“contract” set. To ensure the satisfaction of this assumption,we impose a contractual constraint on the control policythat “guarantees” that the information-coupling states thatit induces indeed belong to said contract set. Naturally,this approach yields an inner approximation of the originaldecentralized control design problem, where the conservatismof the resulting approximation depends on the specificationof the contract set. To limit the extent of the suboptimalitythat may result, we formulate a semi-infinite program to co-optimize the decentralized control policy with the location,cale, and orientation of an ellipsoidal contract set. Weestablish a condition on the set of allowable contracts thatfacilitates the joint optimization of the control policy and thecontract set via semidefinite programming.We note that there are several recent papers appearingin the literature that investigate a similar approach to de-centralized control design via the co-optimization of controlpolicies and contract sets [15], [16]. Importantly, the tech-niques developed in these papers only permit the scaling andtranslation of a base contract set when co-optimizing it withthe control policy. To the best of our knowledge, the methodproposed in this paper provides the first systematic approachto co-optimize the control policy with the location, scale,and orientation of the contract set, expanding substantiallythe family of contracts that can be efficiently optimized over.
Notation:
Let R and R + denote the sets of real andnon-negative real numbers, respectively. Given a collection ofvectors x , . . . , x N , we let ( x , . . . , x N ) denote their vectorconcatenation in ascending order of their indices. Given anindex set J ⊆ { , . . . , N } , we let x J denote the vectorconcatenation of the vectors x j for j ∈ J in ascending orderof their indices. Given a sequence { x ( t ) } and time indices s ≤ t , we let x s : t = ( x ( s ) , x ( s + 1) , . . . , x ( t )) denote itshistory from time s to time t . Given a block matrix A , welet [ A ] ij denote its ( i, j ) -th block. We denote the trace of asquare matrix A by Tr ( A ) . We denote the Minkowski sumof two sets S , T ⊆ R n by S ⊕T := { x + y | x ∈ S , y ∈ T } .II. P ROBLEM F ORMULATION
A. System Model
Consider a discrete-time, linear time-varying system con-sisting of N dynamically coupled subsystems whose dynam-ics are described by x i ( t + 1) = N X j =1 ( A ij ( t ) x j ( t ) + B ij ( t ) u j ( t )) + w i ( t ) , (1)for i = 1 , . . . , N . We denote the local state , local input ,and local disturbance associated with each subsystem i attime t by x i ( t ) ∈ R n ix , u i ( t ) ∈ R n iu , and w i ( t ) ∈ R n ix ,respectively. The system is assumed to evolve over a finitetime horizon T , and the initial condition is assumed to bea random vector with known probability distribution. In thesequel, we will work with a more compact representation ofthe full system dynamics given by x ( t + 1) = A ( t ) x ( t ) + B ( t ) u ( t ) + w ( t ) . Here, we denote by x ( t ) := ( x ( t ) , .., x N ( t )) ∈ R n x , u ( t ) := ( u ( t ) , .., u N ( t )) ∈ R n u , and w ( t ) :=( w ( t ) , .., w N ( t )) ∈ R n x the full system state, input, anddisturbance at time t . The dimensions of the system stateand input are given by n x := P Ni =1 n ix and n u := P Ni =1 n iu ,respectively.The input and disturbance trajectories are related to thestate trajectory according to x = Bu + Lw, (2) where x , u , and w denote the system state, input, anddisturbance trajectories, respectively. They are defined by x := ( x (0) , . . . , x ( T )) ∈ R N x , N x := n x ( T + 1) , (3) u := ( u (0) , . . . , u ( T − ∈ R N u , N u := n u T, (4) w := ( w ( − , w (0) , . . . , w ( T − ∈ R N x , (5)where the initial component w ( − of the system disturbancetrajectory is given by w ( −
1) = x (0) . This notational con-vention will help simplify the specification of disturbance-feedback control policies in the sequel. B. Disturbance Model
We model the disturbance trajectory w as a zero-meanrandom vector whose support is an ellipsoid given by W := (cid:8) z ∈ R N x (cid:12)(cid:12) z ⊤ Σ − z ≤ (cid:9) , (6)where the shape parameter Σ ∈ R N x × N x is assumed to besymmetric and positive definite. We let M := E [ ww ⊤ ] de-note the second moment matrix of the disturbance trajectory w . The matrix M is guaranteed to be positive definite andfinite-valued, as the support of w is assumed to be an ellipsoidwith a non-empty interior. C. System Constraints
We consider a general family of polyhedral constraints onthe state and input trajectories of the form F x x + F u u + F w w ≤ g ∀ w ∈ W , (7)where F x ∈ R m × N x , F u ∈ R m × N u , F w ∈ R m × N x , g ∈ R m are assumed to be given. Note that such constraints maycouple states and inputs across subsystems and time periods. D. Information Structure
We consider information structures that are specified ac-cording to sparsity constraints on the state measurementsthat each controller has access to. Specifically, we encodethe pattern according to which information is shared betweensubsystems with a directed graph G I = ( V, E I ) , which werefer to as the information graph of the system. Here, thevertex set V = { , . . . , N } assigns a distinct vertex i toeach subsystem i . Additionally, we include the directed edge ( i, j ) ∈ E I if and only if subsystem j has access to subsystem i ’s local state at each time t . We let V − I ( i ) denote the in-neighborhood of each subsystem i ∈ V in the informationgraph G I .Each subsystem is assumed to have access to the entirehistory of its local information up until and including time t .More formally, we define the local information available toeach subsystem i at time t as z i ( t ) := { x tj | ( j, i ) ∈ E I } . (8)The local control input to each subsystem i is restricted to bea causal function of its local information. That is, the localinput to subsystem i at time t is of the form u i ( t ) = γ i ( z i ( t ) , t ) , (9) The matrices B and L are specified in Appendix A. here γ i ( · , t ) is a measurable function of the local informa-tion z i ( t ) . We define the local control policy for subsystem i as γ i := ( γ i ( · , , . . . , γ i ( · , T − . We refer to thecollection of local control policies γ := ( γ , . . . , γ N ) as the decentralized control policy , which relates the state trajectory x to the input trajectory u according to u = γ ( x ) . Finally,we let Γ denote the set of all decentralized control policiesrespecting the information constraints encoded in Eq. (9). E. Decentralized Control Design
We consider the following family of constrained decentral-ized control design problems:minimize E (cid:2) x ⊤ R x x + u ⊤ R u u (cid:3) subject to γ ∈ Γ u = γ ( x ) x = Bu + LwF x x + F u u + F w w ≤ g ∀ w ∈ W . (10)Here, the cost matrices R x ∈ R N x × N x and R u ∈ R N u × N u are assumed to be symmetric and positive semidefinite.The tractability of the decentralized control design problem(10) depends on the nature of the information structure. Inparticular, if the information structure is partially nested ,then problem (10) can be equivalently reformulated (via theYoula parameterization) as a convex program in the spaceof disturbance feedback policies [17]. If, on the other hand,the information structure is nonclassical (i.e., not partiallynested), then problem (10) is known to be computationallyintractable, in general [2], [18], [19].III. I NFORMATION D ECOMPOSITION
The primary difficulty in solving decentralized controldesign problems stems from the informational coupling thatemerges when a subsystem’s local information is affectedby prior control actions that it cannot access or reconstruct.With the aim of isolating the effects of these actions on theinformation available to each subsystem, we propose an in-formation decomposition that partitions the local informationavailable to each subsystem into a partially nested subset (i.e.,an information subset that is unaffected by control actionspreviously applied to the system) and its complement. Thisinformation decomposition enables an equivalent reformula-tion of the decentralized control design problem where thecontrol policy is expressed as an explicit function of thesystem disturbance and the so called information-coupling states. This reformulation will serve as the foundation forthe contract-based approach to decentralized control designproposed in Section IV.
A. Decomposition of Local Information
We decompose the local information available to eachsubsystem according to a partition of its in-neighbors in theinformation graph G I . More specifically, for each subsystem i ∈ V , we let N ( i ) ⊆ V − I ( i ) denote the set of in-neighboring subsystems whose local statemeasurements contain information that is unaffected by theprior control actions of any subsystem. This requirement issatisfied if the local information of subsystem i is such thatit permits the reconstruction of all states and control actionsdirectly affecting the local states of all subsystems belongingto N ( i ) . We denote the complement of this set by C ( i ) := V − I ( i ) \ N ( i ) for each subsystem i ∈ V .With the goal of providing an explicit characterization ofthese sets, we first provide a characterization of the physicalcoupling between different subsystems as reflected by theblock sparsity patterns of the system matrices A and B . Wedescribe this coupling in terms of a pair of directed graphs, G A := ( V, E A ) and G B := ( V, E B ) , whose edge sets aredefined according to E A := { ( j, i ) ∈ V × V | ∃ t = 0 , . . . , T − s.t. A ij ( t ) = 0 } ,E B := { ( j, i ) ∈ V × V | ∃ t = 0 , . . . , T − s.t. B ij ( t ) = 0 } . We let V − A ( i ) and V − B ( i ) denote the in-neighborhoods asso-ciated with each node i ∈ V in G A and G B , respectively.Building on these representations, we have the followingdefinition that formalizes the class of information decompo-sitions considered in this paper. For each subsystem i ∈ V ,define the set N ( i ) := { j ∈ V − I ( i ) | (11), (12) are satisfied } , where the above conditions are given by V − A ( j ) ⊆ V − I ( i ) (11)and [ k ∈ V − B ( j ) V − I ( k ) ⊆ V − I ( i ) . (12)Condition (11) requires that subsystem i has access to allstates that directly affect subsystem j ’s state through thesystem dynamics. Condition (12) requires that subsystem i has access to the local information of each subsystemwhose control actions directly affect subsystem j ’s state. Thisensures that subsystem i is able to reconstruct all controlactions that directly affect subsystem j ’s state. Collectively,conditions (11) and (12) can be interpreted as a requirementon the local nesting of information , in the sense that if j ∈ N ( i ) , then subsystem i is assumed to have access toall states and control actions that directly affect subsystem j ’s state through the state equation. As a result, subsystem i can explicitly reconstruct the local disturbance w j ( t − acting on any subsystem j ∈ N ( i ) based only on its localinformation z i ( t ) as follows: w j ( t −
1) = x j ( t ) − X k ∈ V − A ( j ) A jk ( t − x k ( t − − X k ∈ V − B ( j ) B jk ( t − u k ( t − . The local states of subsystems not belonging to N ( i ) ,on the other hand, may contain information that can beinfluenced by prior control actions. We refer to these statess the information-coupling states associated with subsystem i at time t , denoting them by x C ( i ) ( t ) where C ( i ) := V − I ( i ) \ N ( i ) . The collection of information-coupling states across all sub-systems are denoted by the x C ( t ) ∈ R n C x , where C := [ i ∈ V C ( i ) . (13)The trajectory of information-coupling states is denoted by x C := ( x C (0) , . . . , x C ( T )) ∈ R N C x , where N C x := n C x ( T + 1) . Finally, it will be notationallyconvenient to express the mapping from the state trajectory x to its subvector x C in terms of the projection operator Π C : R N x → R N C x , where x C = Π C x . Remark 1 (Partially Nested Information) . It can be shownthat the given information structure is partially nested if andonly if the set of information coupling states is empty, i.e., C = ∅ . It is well known that such information structurespermit the equivalent reformulation of problem (10) as aconvex optimization problem in the space of disturbance-feedback control policies.B. Control Input Reparameterization The proposed information decomposition suggests a nat-ural reparameterization of the control policy in terms of thefollowing equivalent information set.
Lemma 1 (Equivalence of Information) . Define the informa-tion set ζ i ( t ) according to ζ i ( t ) := { x tj | j ∈ C ( i ) } ∪ { w − t − j | j ∈ N ( i ) } . The sets z i ( t ) and ζ i ( t ) are functions of each other for eachsubsystem i and time t . The proof of Lemma 1 is omitted, as it mirrors that of[20, Lemma 1]. Lemma 1 suggests the following equivalentreparameterization of the local control input: u i ( t ) = φ i ( ζ i ( t ) , t ) , (14)where φ i ( · , t ) is a measurable function of its arguments. Welet φ i := ( φ i ( · , , . . . , φ i ( · , T − and φ := ( φ , . . . , φ N ) denote the reparameterized control policy associated witheach subsystem i ∈ V and the full system, respectively. Witha slight abuse of notation, we express the input trajectoryinduced by the reparameterized control policy φ as u = φ ( w, x C ) . Finally, we denote by Φ the set of reparameterized decentral-ized control policies that respect the information constraintsencoded in Eq. (14). The reparameterization of the control input according toEq. (14) results in the following equivalent reformulation ofthe original decentralized control problem (10):minimize E (cid:2) x ⊤ R x x + u ⊤ R u u (cid:3) subject to φ ∈ Φ u = φ ( w, x C ) x = Bu + LwF x x + F u u + F w w ≤ g ∀ w ∈ W . (15)Clearly, problem (15) remains nonconvex, in general, if theset of information-coupling subsystems is nonempty, i.e., C 6 = ∅ . In Section IV, we construct a convex inner approximationto problem (15) where the information-coupling states areassumed to behave as disturbances with bounded support,and the control policy is constrained in a manner that ensuresthe consistency between the assumed and actual behaviors ofthe information-coupling states.IV. D ECENTRALIZED C ONTROL D ESIGN VIA C ONTRACTS
In this section, we construct a convex inner approximationof the decentralized control design problem (15) via theintroduction of an assume-guarantee contractual constraint onthe information-coupling states x C . We do so by introducinga surrogate information structure in which the information-coupling states are modeled as fictitious disturbances that are“assumed” to take values in a “contract” set. To “guarantee”the satisfaction of this assumption, we impose a contractualconstraint on the control policy requiring that the actualinformation-coupling states induced by the control policybelong to the contract set. Given a fixed contract set, theresulting problem is a convex disturbance-feedback controldesign problem, whose feasible policies are guaranteed to befeasible for problem (15). A. Surrogate Information
We associate a fictitious disturbance v i ( t ) ∈ R n ix witheach subsystem i ∈ V and time t = 0 , . . . , T . We let v ∈ R N x denote the corresponding fictitious disturbancetrajectory induced by these individual elements, which wemodel as a random vector whose support V ⊂ R N x isassumed to be a convex and compact set. We also assumethat the fictitious disturbance trajectory v is independent ofthe system disturbance trajectory w .Letting the collection of fictitious disturbances serve assurrogates for the information-coupling states, we define the surrogate local information for subsystem i as e ζ i ( t ) := { v tj | j ∈ C ( i ) } ∪ { w − t − j | j ∈ N ( i ) } . Given a decentralized control policy φ ∈ Φ , the surrogatelocal information induces a surrogate control input for eachsubsystem i defined according to e u i ( t ) := φ i ( e ζ i ( t ) , t ) . Additionally, the surrogate input trajectory induced by thesurrogate information structure is given by e u := φ ( w, v C ) , where v C := Π C v . . Surrogate Dynamics The treatment of the information coupling states as fic-titious disturbances induces a surrogate system state thatevolves according to the following surrogate state equation: e x i ( t + 1) = X j ∈ V \C ( i ) A ij ( t ) e x j ( t ) + X j ∈C ( i ) A ij ( t ) v j ( t )+ N X j =1 B ij ( t ) e u j ( t ) + w i ( t ) , (16)where e x i ( t ) denotes the surrogate state of subsystem i attime t . We require that the initial condition of the surrogatesystem equal that of the true system, i.e., e x i (0) = x i (0) foreach subsystem i . Moving forward, it will be convenient toexpress the surrogate state dynamics in terms of trajectoriesas follows: e x = e B e u + e Lw + e Hv C , (17)where the matrices e B , e L , and e H are defined in Appendix A.We close this subsection with a lemma that establishesconditions for the equivalence between the surrogate andactual state trajectories. We omit the proof, as it directlyfollows from the definition of the surrogate state equation(17). Lemma 2.
Let u ∈ R N u and w ∈ R N x . It holds that x = Bu + Lw if and only if x = e Bu + e Lw + e Hx C .C. Assume-Guarantee Contracts Thus far, we have treated the information-coupling statesas fictitious disturbances that are assumed to take values ina given set V C . Leveraging on concepts grounded in assume-guarantee reasoning [21], [22], we guarantee the satisfactionof this assumption by imposing a contractual constraint onthe control policy, which ensures that it induces information-coupling states that belong to V C . We formalize the notionof an assume-guarantee contract in the following definition. Definition 1 (Assume-Guarantee Contract) . A control policy φ ∈ Φ is said to satisfy the assume-guarantee contract speficied in terms of the contract set V C ⊆ R N C x if Π C e x ∈ V C ∀ ( w, v C ) ∈ W × V C , where e x = e Bφ ( w, v C ) + e Lw + e Hv C . Here, the set V C is referred to as a contract set , as itspecifies the set that the information-coupling states areboth assumed and required to belong to. The satisfactionof the assume-guarantee contract guarantees that the sur-rogate information-coupling states e x C := Π C e x belong tothe contract set. In the following lemma, we show that the actual information-coupling states that result under the policy u = φ ( w, x C ) are guaranteed to belong to the contract set ifthe assume-guarantee contract is satisfied. Lemma 3.
Let φ ∈ Φ be a control policy that satisfies theassume-guarantee contract specified in terms of the contractset V C ⊆ R N C x . It follows that Π C x ∈ V C for all w ∈ W ,where x = Bφ ( w, x C ) + Lw . The proof of Lemma 3 is omitted due to space constraints.In the following proposition, we provide an inner approxima-tion of the decentralized control design problem (15) via theintroduction of an assume-guarnatee contractual constraint.Its proof is omitted, as it follows directly from Lemma 3.
Proposition 1.
Let φ ∈ Φ be a feasible control policy forthe following problem:minimize E (cid:2)e x ⊤ R x e x + e u ⊤ R u e u (cid:3) subject to φ ∈ Φ e u = φ ( w, v C )Π C e x ∈ V C e x = e B e u + e Lw + e Hv C F x e x + F u e u + F w w ≤ g ∀ ( w, v C ) ∈ W × V C , (18) It follows that φ is also feasible for problem (15) . Problem (18) is a convex disturbance feedback controldesign problem, given a fixed contract set V C . The choiceof the contract set does, however, play an important role indetermining the performance of the control policies that itgives rise to. In Section V, we develop a systematic approachto enable the joint optimization of the contract set with thecontrol policy via semidefinite programming.V. P OLICY -C ONTRACT O PTIMIZATION
In this section, we provide a semidefinite programming-based method to co-optimize the design of the decentralizedcontrol policy together with the contract set that constrainsits design. As part of the proposed approach, we considera restricted family of control policies that are affinely pa-rameterized in both the disturbance and fictitious disturbancehistories. We also parameterize the fictitious disturbanceprocess as a causal affine function of a given (primitive)disturbance process—an approach that is similar in natureto the class of parameterizations that have been recentlystudied in the context of robust optimization with adjustableuncertainty sets [23]. As one of our primary results in thissection, we identify a structural condition on the family ofallowable contract sets that permits the inner approximationof the resulting policy-contract optimization problem as asemidefinite program.
A. Affine Control Policies
We restrict our attention to affine decentralizeddisturbance-feedback control policies of the form e u i ( t ) = u oi ( t ) + X j ∈N ( i ) t − X s = − Q wij ( t, s + 1) w j ( s )+ X j ∈C ( i ) t X s =0 Q vij ( t, s ) v j ( s ) , (19)for t = 0 , . . . , T − and i = 1 , . . . , N . Here, u oi ( t ) denotesthe open-loop control input, and the matrices Q wij ( t, s + 1) and Q vij ( t, s ) denote the feedback control gains. The affineontrol policy specified in Eq. (19) can be expressed in termsof trajectories as e u = u o + Q w w + Q v v, (20)where the gain matrices Q w and Q v are both T × ( T + 1) block matrices, whose ( t, s ) -th block is defined according to [ Q w ( t, s )] ij = ( Q wij ( t, s ) if j ∈ N ( i ) and t ≥ s, otherwise , (21) [ Q v ( t, s )] ij = ( Q vij ( t, s ) if j ∈ C ( i ) and t ≥ s, otherwise . (22)for i, j = 1 , . . . , N . We let Q N and Q C denote the matrixsubspaces respecting the block sparsity patterns specifiedaccording to Eqs. (21) and (22), respectively. B. Affine Parameterization of the Fictitious Disturbance
We focus our analysis on fictitious disturbances that areexpressed according to affine transformations of a primitivedisturbance . Such a parameterization yields contract sets thathave adjustable location, scale, and orientation. Specifically,we let the random vector ξ denote the primitive disturbancetrajectory , which is assumed to be an i.i.d. copy of the sys-tem disturbance trajectory w . We parameterize the fictitiousdisturbance trajectory affinely in the primitive disturbance as v := v + Zξ. (23)Here, the parameters v ∈ R N x and Z ∈ R N x × N x can beadjusted to control the shape of the resulting contract set V C ,which takes the form of V C = Π C ( v ⊕ Z W ) . (24)Throughout the paper, we will restrict our attention to trans-formations (23) in which the matrix parameter Z is bothlower triangular and invertible. We denote the set of all suchmatrices by Z ⊂ R N x × N x .The specification of the fictitious disturbance according toEq. (23) induces the following the surrogate control input: e u = u o + Q v v + Q w w + Q v Zξ. (25)We eliminate the bilinear terms in Eq. (25) through thefollowing the change of variables: u := u o + Q v v and Q ξ := Q v Z. (26)This change of variables gives rise to a reparameterization ofthe surrogate input trajectory as e u = u + Q w w + Q ξ ξ, (27)where the matrix Q ξ ∈ R N u × N x must satisfy the sparsityconstraint Q ξ Z − ∈ Q C in order to ensure the satisfaction of the original sparsityconstraint that Q v ∈ Q C . The parameterization of the contract set and control policyin this manner permits their co-optimization as follows:minimize E (cid:2)e x ⊤ R x e x + e u ⊤ R u e u (cid:3) subject to Q w ∈ Q N , Q ξ ∈ R N u × N x , Z ∈ Z u ∈ R N u , v ∈ R N x ,Q ξ Z − ∈ Q C v = v + Zξ e u = u + Q w w + Q ξ ξ e x = e B e u + e Lw + e Hv C Π C e x ∈ Π C ( v ⊕ Z W ) F x e x + F u e u + F w w ≤ g ∀ ( w, ξ ) ∈ W , (28)where W := W × W . Problem (28) is a nonconvexsemi-infinite program, where the nonconvexity is due to thesparsity constraint on the matrix Q ξ Z − and the contractualconstraint on the affine control policy. In what follows, weprovide convex inner approximations of these constraints,which yield an inner approximation of problem (28) as asemidefinite program. C. Restricting the Contract Set
In what follows, we introduce an additional restriction onthe set of allowable matrix parameters Z that guarantees theinvariance of the subspace Q C under multiplication by suchmatrices. This permits the equivalent reformulation of thebilinear constraint Q ξ Z − ∈ Q C as Q ξ ∈ Q C .Specifically, we require that the matrix Z be of the form Z = λI − Y, (29)where λ ≥ is scalar parameter and Y ∈ R N x × N x is a ( T + 1) × ( T + 1) strictly block lower triangular matrix ofthe form Y = Y (1 ,
0) 0 ... . . . . . . Y ( T, · · · Y ( T, T −
1) 0 . (30)Furthermore, each block of the matrix Y is an N × N blockmatrix, whose ( i, j ) -th block is of dimension n ix × n jx . Weimpose an additional restriction on the structure of the matrix Y in the form of sparsity constraints (that reflect the patternof informational coupling between subsystems) on each ofits blocks.More specifically, we encode the pattern of informationalcoupling between subsystems according to a directed graph G C := ( V, E C ) , whose directed edge set E C is defined as E C := { ( j, i ) ∈ E I | j ∈ C ( i ) } . We let V + C ( i ) denote the out-neighborhood of a node i ∈ V in the graph G C . Using this graph, we impose a sparsityconstraint on each block of the matrix Y of the form: [ Y ( t, s )] ij = 0 if V + C ( i ) * V + C ( j ) (31)or all i, j = 1 , . . . , N , and t, s = 0 , . . . , T . We let Y ( G C ) denote the subspace of all matrices that respect the sparsityconstraints implied by Eqs. (30) and (31).We have the following result establishing the invarianceof the subspace Q C under multiplication by matrices Y ∈Y ( G C ) . Lemma 4. If Q ∈ Q C and Y ∈ Y ( G C ) , then QY ∈ Q C Proof:
The sparsity constraint QY ∈ Q C is satisfied if thematrix Q ( t, s ) Y ( s, r ) satisfies the sparsity constraint [ Q ( t, s ) Y ( s, r )] ij = 0 ∀ i / ∈ V + C ( j ) for all times r, s, t satisfying ≤ r < s ≤ t ≤ T − .We prove this claim by showing that [ Q ( t, s ) Y ( s, r )] ij = 0 implies i ∈ V + C ( j ) . The condition that [ Q ( t, s ) Y ( s, r )] ij = 0 implies that there exists k ∈ V such that the blocks [ Q ( t, s )] ik and [ Y ( s, r )] kj are both nonzero. The fact that [ Q ( t, s )] ik is nonzero implies that i ∈ V + C ( k ) , as the matrix Q satisfies Q ∈ Q C . The fact that [ Y ( s, r )] kj is nonzero impliesthat V + C ( k ) ⊆ V + C ( j ) , as the matrix Y satisfies Y ∈ Y ( G C ) .The desired result follows. (cid:4) We have the following result as an immediate consequenceof Lemma 4.
Lemma 5.
Let Y ∈ Y ( G C ) and λ ∈ [1 , ∞ ) . It follows that Q C = (cid:8) Q ξ ( λI − Y ) − | Q ξ ∈ Q C (cid:9) . It follows from Lemma 5 that the constraint Q ξ Z − ∈ Q C is equivalent to Q ξ ∈ Q C if Z = λI − Y , where Y ∈ Y ( G C ) and λ ≥ . D. Semidefinite Programming Approximation
To lighten notation, we write the surrogate state trajectory e x more compactly as e x = x + P w w + P ξ ξ, where x := e Bu + e H Π C v , P w := e BQ w + e L , and P ξ := e BQ ξ + e H Π C ( λI − Y ) .We first address the robust linear constraints in prob-lem (28). The following result provides an equivalent re-formulation as second-order cone constraints. Its proof isomitted, as it is an immediate consequence of the identity sup w ∈W c ⊤ w = k Σ / c k for all c ∈ R N x . Lemma 6.
The semi-infinite constraint F x e x + F u e u + F w w ≤ g for all ( w, ξ ) ∈ W is satisfied if and only if (cid:13)(cid:13)(cid:13) Σ / e ⊤ i ( F x P w + F u Q w + F w ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) Σ / e ⊤ i ( F x P ξ + F u Q ξ ) (cid:13)(cid:13)(cid:13) ≤ e ⊤ i ( g − F x x − F u u ) , i = 1 , . . . , m, (32) where e i is the i th standard basis vector in R m . We now address the nonconvexity that stems from thecontractual constraint in problem (28). First, notice thatthe contractual constraint is equivalent to the following setcontainment constraint Π C (cid:0) x ⊕ P w W ⊕ P ξ W (cid:1) ⊆ Π C ( v ⊕ Z W ) . (33) The set containment constraint (33) amounts to requiring thatthe Minkowski sum of two ellipsoids be contained withinanother ellipsoid. It follows from [24][Theorem 4.2] that thisclass of set containment constraints can be approximatedfrom within by a quadratic matrix inequality. Through anapplication of Schur’s Lemma, one can approximate theresulting quadratic matrix inequality from within by a linearmatrix inequality. We summarize the resulting inner approx-imation in the following lemma. Lemma 7.
The set containment constraint (33) is satisfied ifthere exists a scalar β ∈ [0 , λ ] such that Π C ( x − v ) = 0 , (34) Π C e ΣΠ ⊤C Π C P w Π C P ξ P w ⊤ Π ⊤C β Σ − P ξ ⊤ Π ⊤C λ − β )Σ − (cid:23) , (35) where e Σ = λ Σ − Y Σ − Σ Y ⊤ . By applying Lemmas 5–7, one can approximate the non-convex semi-infinite program (28) from within as the follow-ing finite-dimensional semidefinite program.
Proposition 2.
Each feasible solution to the followingsemidefinite program is feasible for problem (28) :minimize Tr (cid:16) P ξ ⊤ R x P ξ M + P w ⊤ R x P w M (cid:17) + Tr (cid:16) Q w ⊤ R u Q w M + Q ξ ⊤ R u Q ξ M (cid:17) + x ⊤ R x x + u ⊤ R u u subject to Q w ∈ Q N , Q ξ ∈ Q C , Y ∈ Y ( G C ) ,u ∈ R N u , v, x ∈ R N x , λ, β ∈ R + ,P w , P ξ ∈ R N x × N x ,λ ≥ max { , β } ,x = e Bu + e H Π C vP w = e BQ w + e LP ξ = e BQ ξ + e H Π C ( λI − Y ) (32) , (34) , (35) . (36)The decision variables for problem (36) are the matrices Q w , Q ξ , Y , P w , P ξ , the vectors u , v , x , and the scalars λ and β . Problem (36) is a convex inner approximationof the reformulated decentralized control design problem(15), in the sense that each feasible solution of problem(36) can be mapped to a feasible affine control policy forproblem (15) via the change of variables specified in (26).The decentralized control policies that this approximationgives rise to are suboptimal, in general. Bounds on theirsuboptimality, however, can be efficiently calculated usinginformation-based convex relaxations [25].VI. C ONCLUSION
We provide a method to compute feasible control policiesfor constrained decentralized control design problems byleveraging on the concept of assume-guarantee contracts.t the heart of this approximation is the treatment ofinformation-coupling states as fictitious disturbances that are“assumed” to take values in a contract set. We “guaran-tee” the inclusion of the information-coupling states in thecontract set by imposing an assume-guarantee contractualconstraint on the control policy. The introduction of suchassume-guarantee contracts gives rise to an inner approxi-mation of the decentralized control design problem, whosequality depends on the specification of the contract set. Weprovide a method of co-optimizing the decentralized controlpolicy with the location, scale, and orientation of the contractset via semidefinite programming.R
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Define the matrices e A ( t ) and e H ( t ) according to e A ij ( t ) = ( A ij ( t ) if j ∈ V \ C ( i ) , otherwise, e H ij ( t ) = A ij ( t ) − e A ij ( t ) , where i, j ∈ V . The matrices ( B, L ) in Eq. (2) and thematrices ( e B, e L ) in Eq. (17) are defined according to B := A B (0) 0 A B (0) A B (1) 0 ... . . . . . .... . . . A T B (0) A T B (1) · · · · · · A TT B ( T − , e B := e A B (0) 0 e A B (0) e A B (1) 0 ... . . . . . .... . . . e A T B (0) e A T B (1) · · · · · · e A TT B ( T − ,L := A A A ... . . . A T A T · · · A TT , e L := e A e A e A ... . . . e A T e A T · · · e A TT , where A ts := Q t − r = s A ( r ) and e A ts := Q t − r = s e A ( r ) for s < t ,and A tt = e A tt = I . Additionally, the matrix e H in Eq. (17) isdefined as e H := H Π ⊤C , where H := e A e H (0) 0 e A e H (0) e A e H (1) 0 ... . . . . . .... . . . e A T e H (0) e A T e H (1) · · · · · · e A TT e H ( T −
1) 0