2012 American Control Conference Fairmont Queen Elizabeth, Montréal, Canada June 27-June 29, 2012
Distributed Command Governor Strategies for Constrained Coordination of Multi-Agent Networked Systems F. Tedesco, A. Casavola, E. Garone. Abstract— In this paper a novel distributed FeedBack Command Governor (FB-CG) supervision strategy is presented for multi-agent networked systems subject to pointwise-in-time coordination constraints on the overall network evolutions. Specifically, a sequential distributed strategy where only one agent at a time is allowed to manipulate its own reference signal is fully detailed and its stability, feasibility and viability (liveness) properties rigorously proved. It is shown that, unlike the centralized case, liveness (the absence of deadlock situations) can be ensured only if the constraints satisfy a Constraints Qualification (CQ) condition. Several results and open problems in the liveness analysis are discussed and a counterexample is also provided for showing that for multi-input agents case the issue requires further investigations.
I. I NTRODUCTION The problem here addressed is the extension of the distributed supervision and coordination FeedForward Command Governor (FF-CG) strategies, determined in [1] and [2] for networked multi-agent systems in the presence of bounded persistent disturbances and subject to pointwise-intime coordination constraints, to the more general class of FeedBack Command Governor (FB-CG) schemes. FF-CG solutions [3] are mainly characterized by the fact that, unlike the standard FB-CG approach [4], their actions computation is not based on the current measure or estimate of the state and are generated by maintaining constant the FFCG commands for several sampling steps, so as to enforce the system evolutions to stay close to space of steady-state feasible equilibria. Although this peculiarity of the FF-CG schemes make them an attractive solution for distributed frameworks because it alleviates the need to make the entire aggregate state, or substantial parts of it, known to all agents at each time instant, their tracking and coordination performance are sub-optimal when fast-varying reference signals are of interest and especially when bounded persistent disturbances are present. In this paper we present a distributed supervision strategy based on the FB-CG ideas [4] that represents a particularly interesting solution in large-scale problems. The distributed context is depicted in Figure 1, where the supervisory task is distributed amongst many master agents which are assumed to be able to communicate each other. A. Casavola and F. Tedesco are with the Dipartimento di Elettronica, Informatica e Sistemistica, Universit`a degli Studi della Calabria, Via Pietro Bucci, Cubo 42-c, Rende (CS), 87036, Italy. E. Garone is with Universit´e Libre de Bruxelles, 50 Av. F.D. Roosvelt, B-1050 Brussels, Belgium {casavola, ftedesco}@deis.unical.it,
[email protected] 1 This paper was supported by the Belgian Network DYSCO (Dynamical Systems, Control and Optimization) funded by the Interuniversity Attraction Poles Program, initiated by the Belgian State, Science Policy Office.
978-1-4577-1096-4/12/$26.00 ©2012 AACC
There, each master station is in charge of supervising and coordinating one specific slave system via a data network. In particular, r i , gi , xi , yi and ci represent respectively: the nominal references, the applied references, the states, the performance-related and the coordination-related outputs of the slave systems. In such a context, the supervision task can be expressed as the requirement of satisfying some tracking performance, viz. y i ≈ ri , whereas the coordination task consists of enforcing some pointwise-in-time constraints ci ∈ Ci and/or f (c1 , c2 , ...., cN ) ∈ C on each slave system and/or on the overall network evolutions. To this end, each i − th supervisor is in charge of modifying the nominal local references ri into the feasible ones g i , when the tracking of the nominal references would produce constraint violations and hence loss of coordination. The proposed distributed approach differs from that presented in [1] and [2] because here the state is assumed to be available (with some time-delay due to the network latency) at the master agent sites and may be exploited for the distributed CG actions computation. g1 r1
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Multi-agent architectures
A sequential distributed scheme is presented and its stability, feasibility and viability (liveness) properties fully investigated. It is important to remark that the introduction of the state in the play, unlike in [1] and [2], introduces many new technical challenges for the development of distributed schemes which cannot be overlooked and deserve carefully analysis. In this respect, this paper extends and makes clear several theoretical aspects of this sequential distributed scheme, not directly derivable from [1] and [2]. Novel insight to the liveliness analysis discussed in [2] is provided here and remaining open issues discussed. In particular, a final counterexample is also presented that show that the notion of viability used, although instrumental for deriving simple numerical procedures for checking the liveness property and effective for single-input agents, is not enough general in multi-input cases.
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II. S YSTEM DESCRIPTION AND P ROBLEM F ORMULATION Let us consider a set of N subsystems A = {1, . . . , N }, each one being a LTI closed-loop dynamical system regulated by a local controller which ensures stability and good closedloop performance in linear regimes when the constraints are not active (small-signal regimes when the coordination is effective). Let the i-th closed-loop subsystem be described by the following discrete-time model xi (t+1)=Φii xi (t)+Gigi (t) +Gdidi (t)+ Φij xj (t) j∈A−{i} (1) y (t) =Hiy xi (t) i c d ci (t) =Hi x(t) + Li g(t) + Li d(t)
where: t ∈ ZZ+ , xi ∈ IRni is the state vector (which includes the controller states under dynamic regulation), g i (t) ∈ IRmi the CG action, which, if no constraints were present, would essentially coincide with the reference r i (t) ∈ IRmi . The vector di (t) ∈ IRndi is an exogenous bounded disturbance satisfying di (t) ∈ Di , ∀t ∈ ZZ+ with Di a specified convex and compact set such that 0 ndi ∈ Di ; yi (t) ∈ IRm the output, viz. a performance related signal. Moreover, the following c matrices are defined Finally, c i ∈ IRni represents the local constrained vector which has to fulfill the set-membership constraint (2) ci (t) ∈ C i , ∀t ∈ ZZ+ , i C being a convex and compact polytopic set. It is worth pointing out that, in order to possibly characterize global (coupling) constraints amongst states of different subsystems, the vector ci in (1) is allowed to depend on the aggregate state and manipulable reference vectors x = [x T1 , . . . , xTN ]T ∈ N T T T IRn , with n = ∈ IRm , i=1 ni , and g = [g 1 , . . . , gN ] N nd T T T with m = i=1 mi , d = [d1 , . . . , dN ] ∈ IR , with nd = N T T T ∈ i=1 ndi . Moreover, we denote by r = [r 1 , . . . , rN ] m m T T T T IR , y = [y1 , . . . , yN ] ∈ IR and c = [c1 , . . . , cTN ]T ∈ N c c IRn , with nc = i=1 ni , the other relevant aggregate vectors. The overall system arising by the composition of the above N subsystems can be described as x(t + 1) = Φx(t) + Gg(t) + Gd d(t) y(t) = H y x(t) (3) c(t) = H c x(t) + Lg(t) + Ld d(t)
where Φ = [Φij ]i,j=1,...,N G = diag(G1 , ..., GN ), y Gd = diag(Gd1 , ..., GdN ), H y = diag(H1y , ..., HN ), H c = c T T ) ] , L = [(L1 )T , ..., (LN )T ]T , and Ld = [(H1c )T , ..., (HN [(Ld1 )T , ..., (LdN )T ]T . It is further assumed that A1. The overall system (3) is asymptotically stable. A2. System (3) is off-set free i.e. H y (In − Φ)−1 G = Im . Roughly speaking, the CG design problem is that of locally determining, at each time instant t and for each master agent i ∈ A associated to each subsystem, a suitable reference signal gi (t) which is the best feasible approximation of ri (t) and such that its application never produces constraint violations, i.e. ci (t) ∈ C i , ∀t ∈ ZZ+ , i ∈ A. A centralized solution A centralized solution of the above stated CG design problem have been proposed in [7]. There, at each time
instant t, the overall CG action g(t) is determined as a function of the current overall reference r(t) and measured state x(t) (4) g(t) := g(r(t), x(t)) such that g(t) is the best feasible approximation of r(t) under c(t) ∈ C, where C ⊆ {C 1 × ... × C N } is the global admissible region. The solution is based on the following arguments: by linearity, one is allowed to separate the effects of the initial conditions and inputs from those of disturbances, e.g. x(t) = x(t)+˜ x(t), where x(t) is the disturbance-free component and x ˜(t) depends only on the disturbances. Then, in the sequel we adopt the following notation xg := (In − Φ)−1 Gg y g := Hy (In − Φ)−1 Gg (5) cg := Hc (In − Φ)−1 Gg + Lg for the disturbance-free equilibrium solutions of (3) to a constant command g(t) ≡ g, with g ∈ IR m . Consider next the following set recursion C0 Ck
C∞
:= := .. . :=
C ∼ Ld D Ck−1 ∼ Hc Φk−1 Gd D (6) ∞ \
Ck
k=0
where, for given sets A, E ⊂ R n , A ∼ E ⊂ A is a proper restriction of A denoted in the literature as the P-difference between sets: A ∼ E := {x ∈ IRn : x + e ∈ A, ∀e ∈ E} (7) In [6] it has been proved that the sets C k are nonconservative restrictions of C such that c(t) ∈ C ∞ , ∀t ∈ ZZ+ , implies that c(t) ∈ C, ∀t ∈ ZZ+ . In the above references, it has also been shown that the sets Ck are the largest restrictions of C such that c¯(t) ∈ Ck implies c(t+i) ∈ C, for any d(t+i) ∈ D, ∀i ∈ {0, 1, ..., k − 1}. In particular, c¯(t) ∈ C ∞ , ∀t ∈ ZZ+ , implies that c(t) ∈ C, ∀t ∈ ZZ+ . Thus, one is allowed to consider the disturbance-free system evolutions only and adopt a “worst case” approach. For reasons which have been clarified in [4], it is convenient to introduce the following sets for a given δ>0 (8) C δ := C∞ ∼ Bδ 2m δ Wδ := g ∈ IR : cg ∈ C (9) where Bδ is a ball of radius δ centered at the origin and Wδ represents the set of all commands whose corresponding steady-state solutions cg satisfy the constraints with margin δ > 0. The idea behind the CG approach is to select at each time instant t an action g(t) amongst all vectors of a suitable state depending subset of W δ , each vector of which, if constantly applied as a command to the system from t onwards, would give rise to system evolutions without constraint violations. Such a CG command is applied only for one sampling period after which a new state is measured and the procedure is repeated at the next time instant t + 1 on the basis of the new measurement x(t + 1). Then, if we consider the following family of constant virtual command sequences (10) g(·) = {g(k) ≡ g ∈ Wδ , ∀k ∈ ZZ+ }
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the disturbance-free state x ¯(t) and c-vector evolutions emanating from x ¯(t) under a whatever constant command g are given by k−1 x ¯(k, x(t), g) := Φk x(t) + i=0 Φk−i−1 Gg (11) c¯(k, x(t), g) := Hc x¯(k, x(t), g) + Lg where c¯(k, x(t), g) has to be understood as the disturbancefree virtual evolution at the virtual time instant k (opposite to the real time t) of the constrained vector c, from the initial condition x(t) (applied at virtual time zero) under the constant command g. Finally, for any k¯ ≥ 0 we define the convex and closed set of admissible virtual sequences ¯ ¯ = Wδ ∩ G(x, k) V(x, k) (12) where ¯ = g ∈ IRm : c¯(k, x, g) ∈ C ¯ , ∀k ∈ ZZ+ . (13) G(x, k) k+k Note that the case k¯ = 0 is the standard one and describes the set of all constant virtual commands g ∈ W δ whose corresponding c-evolutions starting at time k = 0 from the current state x satisfy the constraints ∀k ∈ ZZ + . As it will be clarified in the next section, the necessity of introducing k¯ > 0 comes out for handling situations where time-delays and/or communication latency occur (see [5] for details). Therefore, provided that V(¯ x(t), 0) is nonempty, the centralized CG action can be chosen as the solution of the following constrained optimization problem (14) gˆ(t)=arg min g − r(t) 2Ψg g∈V(x(t),0)
Ψg = ΨTg
T
where > 0m and v Ψ := v Ψv. Such a solution represents the best feasible approximation of r(t) which, if constantly applied from t onwards never produces constraints violation. III. D ISTRIBUTED S EQUENTIAL CG (S-CG) Here we introduce a distributed CG scheme based on the above centralized CG approach. We assume the agents (the Master nodes in Fig. 1) connected via a communication network. Such a network is modeled with a communication graph: an undirected graph G = (A, B), where A denotes the set of the N subsystems and B ⊂ A × A the set of edges representing the communication links amongst agents. More precisely, the edge (i, j) belongs to B if and only if the agents governing the i-th and the j-th subsystems are able to directly share information within one sampling time. The communication graph is assumed to be connected, i.e. for each couple of agents i ∈ A, j ∈ A there exists at least one sequence of edges connecting i and j, with the minimum number of edges connecting the two agents denoted by d i,j . The set of all agents with a direct connection with the i-th agent will be referred to as the Neighborhood of the i-th agent Ni = {j ∈ A : di,j = 1}. We will assume that each agent acts as a gateway in redistributing data amongst the other, no directly connected, agents. Then, at each time instant t, each i-th agent is aware of the following vectors: T (t − di,N )]T ξi (t − 1)=[g1T (t − di,1 ), . . . , giT (t − 1), . . . , gN T T T ϑi (t − 1)=[x1 (t − di,1 ), . . . , xi (t), . . . , xN (t − di,N )]T As a consequence, the most recent common information regarding the measurement of the overall state available to each agent is x(t − dmax,i ) where dmax,i = maxj∈A di,j .
Next, let assume, without loss of generality, that the sequence H = {1, 2, ..., N − 1, N } is a Hamiltonian cycle defined on G. The idea behind the approach is that only one agent per decision time is allowed to manipulate its local command signal g i (t) while all others are instructed to keep applying their commands. After a new CG computation, the agent in charge transmits its local command and state to the next updating agent that is necessarily a neighbor. Such a polling policy implies that, eventually after a preliminary initialization cycle, at each time instant t, the following information are available to the generic i-th ”agent in charge” • the history of the aggregate vectors applied in the last N steps g(t − N + j), j = {1, . . . , N − 1} • the measurement of the state at time t − d max,i , i.e. x(t − dmax,i ) By exploiting this information, is possible for it to compute the estimation x ˆ(t) of the current free-disturbance state at time t by means of the following recursions x ˆ(t − dmax,i ) = x(t − dmax,i ) (15) x ˆ(k + 1) = Φˆ x(k) + Gg(k), k = t − dmax,i , ..., t Then, by setting the parameter k¯ in (12) equal to the timedelay dmax,i , we can formulate the following distributed S-CG algorithm: Sequential-CG Algorithm (S-CG) - i-th Agent AT EACH TIME t 1.1 IF(t mod N ) == i 1.1.1 RECEIVE g(t − 1) AND x(t − 1) FROM THE PREVI OUS AGENT IN THE CYCLE H 1.1.2 SOLVE gi (t) = arg min(wi ) wi − ri (t) 2Ψi SUBJECT TO : T [g1T (t−1),...,wiT,...,gN (t−1)]T∈ V(ˆ x(t), dmax,i) (16) 1.1.3 APPLY g i (t) T 1.1.4 UPDATE g(t) = [g 1T (t−1), ..., giT (t), ..., gN (t−1)]T 1.2 ELSE 1.2.1 APPLY g i (t) = gi (t − 1) 1.3 TRANSMIT ALL THE INFORMATION TO THE NEIGH BORS
where Ψi > 0 are weighting matrices, t mod N is the remainder of the integer division t/N . 2 In order to present properties of the above algorithm let us introduce some definitions, notations and assumptions. Definition 1 (Admissible direction) - Let a convex set S ⊂ IRm and a point g ∈ S. The vector v ∈ IR m represents an ¯>0 admissible direction for g ∈ S if there exists a real λ ¯ such that (g + λv) ∈ S, λ ∈ [0, λ]. 2 Definition 2 (Decision Set for the i-th agent) - The Decision Set ViS (g) for the i-th agent at a point g ∈ S represents the set of all admissible directions bealong which the agent could move in longing to IR m i updating its action when all other agents hold their commands unvaried, viz. V iS (g) := {d ∈ IRmi : [0T1 ,. . ., 0Ti−1 , dT , 0Ti+1 ,. . ., 0TN ]T is an admissible direction for g ∈ S}. 2
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Definition 3 (Viability property) - A point g ∈ S is said to be T T ] ∈ ”viable” if, for any admissible direction v = [v 1T , ..., vN N IRm , vi ∈ IRmi with i=1 mi = m, at least one sub-vector 2 vi = 0 exists such that vi ∈ ViS (g). Definition 4 (Pareto Optimal Solution) - Let vectors r i , i = 1, 2, ..., N be given. Consider the following multi-objective problem: ming [ subject
g1 − r1 2Ψ1 , . . . , gi − ri 2Ψi , . . . , gN T T to g = [g1T,...,giT ,...,gN ] ∈ Wδ
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(17) A solution g ∈ Wδ is a Pareto Optimal solution of the optimization problem (17) if there not exist g ∈ W δ , such that: gi − ri 2Ψi ≤ gi∗ − ri 2Ψi ∀i ∈ {1, . . . , N } and 2 gj − rj 2Ψj < gi∗ − ri 2Ψi , j ∈ A . ∗
down
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The above definitions are instrumental to characterize deadlock situations that, unlike the centralized solution, may exist in this decentralized scheme when the same constraint set W δ of the centralized scheme is used. The rationale is that by acting one agent at a time, certain viable paths existing in the centralized scheme are precluded and the agents could get stuck indefinitely without getting a Pareto Optimum. In order to clarify such matters, next Figg. 2-3 depict different viable and no-viable situations for points on the border of Wδ .
Fig. 2. Two-agent cases. Each agent selects its command along a different axis. a) A no viable point: in this case the point g(t) is not viable. In fact both agents cannot change their local command without violating boundaries; b)Viable points: in this case, for each red vertex at least one out of the two agents can move inside Wδ . In fact, each admissible direction v = [v1 , v2 ]T at each vertex is such that either λ1 [v1 , 0]T , λ1 > 0 or λ2 [0, v2 ]T , λ2 > 0 is admissible.
In order to avoid this deadlock situations we have to introduce the following assumption for the points belonging to the border of W δ A3. Each point belonging to ∂(W δ ) is viable, ∂(Wδ ) denoting the border of W δ . In [2], the characterization of viable points, a computable way of checking if the viability property A3 is satisfied for a generic polyhedral set W δ and a geometrical method allowing one to compute suitable inner approximations of Wδ satisfying A3 have been presented, along with a proof that all internal points of W δ are viable. Some new insight and open issues on the liveness analysis are given in the next section. Finally the following properties can be shown to hold under A3 for the above stated S-CG scheme.
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Fig. 3. A three-agent non viable case with agents A = {1, 2, 3} aimed at minimizing respectively g1 − r1 , g2 − r2 and g3 − r3 . At the point g′ , although agents 1 and 2 are locked (green arrows represent not admissible directions). However, no deadlock occurs because agent 3 could move along the red arrow in the Sdown region. In the present situation g′ represents a Nash equilibrium for the S-CG algorithm because the viable direction (red arrow) of agent 3 corresponds to increase its cost. In fact, its distance from the reference r3 in Sup increases. Nevertheless g′ is not a Pareto-Optimal solution because all points g∗ lying over the segment connecting g′ and r are such that gi∗ − ri < gi′ − ri , i = 1, 2, 3..
Theorem 1: Let assumptions A1-A3 be fulfilled. Consider system (3) along with the distributed S-CG selection rule (16) and let V(x(t), d max ) be non empty at time t = 0. Then: 1) for each agent i ∈ A, at each decision time t, the minimizer in (16) uniquely exists and can be obtained by locally solving a convex constrained optimization problem; 2) the overall system acted by the agents implementing the S-CG policy never violates the constraints, i.e. c(t) ∈ C for all t ∈ ZZ+ ; T T 3) whenever r(t)≡[ r 1T ,. . . ,rN ] , ∀t, with ri a constant setT (t)]T point, the sequence of solutions g(t) = [g 1T (t), . . . , gN asymptotically converges to a Pareto-Optimal stationary (constant) solution of (17), which is given by r whenever r ∈ Wδ , or by any other Pareto-Optimal solution rˆ ∈ W δ otherwise. 2 Proof For space reason such a proof is here omitted. Anyway it can be found in [8]. IV. C ONSTRAINTS Q UALIFICATION AND V IABLE A PPROXIMATIONS In this section, some aspects of the viability property (Definition (3)) discussed in [2] will be recalled and some open issues pointed out. The numerical test presented in [2], here recalled for the sake of the understanding, checks a finite number of points belonging to ∂(W δ ) in order to verify Assumption A3. Such a test is necessary and sufficient to state that: if a point of ∂(Wδ ) is viable according to Definition (3) then no deadlocks can take place under the stated distributed S-CG policy.
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where aTj and bj denote the rows of the matrix A and, respectively, vector b. A further very challenging issue already discussed in [2] is represented by the achievement of arbitrarily accurate multibox inner approximations of the original constraint set which satisfy the viability property. A method was presented in [2], directly adaptable to the present context, which is proved to satisfy Assumption A3 in the case that all agents have monodimensional decision sets, viz. m i = 1, ∀i ∈ A and N = m. Numerical simulations suggest that this method makes the control strategy to converge to a Pareto-optimum also in the case of multi-dimensional problems m i ≥ 1, ∀i ∈ A. The investigation of this conjecture is a current research argument whose partial results are hereafter discussed. In particular, we will detail that it is possible to prove that such a claim cannot be given in terms of the viability Definition (3) because it is possible to build a counterexample (Example 1 of the next subsection). This fact encourages the investigation the less restrictive notions of viability, such as the one given in Definition (6). In order to fully understand the counterexample, the key ideas of the multi-box approximation method are here briefly recalled. Such an approximation is obtained by partitioning the original non viable set S by means of an arbitrary number of boxes obtaining a multi-box inner approximation M(S) ⊂ S, as depicted in Fig. IV for a specific two-dimensional case. Then, the inner-approximating set is computed as the convex hull of M(S), say it S ′ := co{M(S)}. Such a set is viable by construction and enjoys the following properties: Proposition 1: Let S ′ ⊂ IRm be expressed as the intersection of |J | inequalities Ag ≤ b. Then, for each point g of the border of S ′ , say it ∂(S ′ ), the next two properties hold true
1) at g there exist m admissible directions aligned to the ¯ vi ∈ axes, i.e. g + [01 , . . . , λvi , . . . , 0m ]T ∈ S ′ , λ ∈ [0, λ], ′ {−1, 1}, ∀i ∈ A, ∀g ∈ ∂(S ). 2) for all s ∈ IRm such that A(g + s) ≤ b, the following condition is satisfied A(g + [01 , ..., si , ..., 0m ]T ) ≤ b
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for at least an index i ∈ A. 2 In [2] it has been proved that the viability of S ′ is ensured by the second item of Proposition 1. Then, in our context, a no viable polyhedron W δ can be always approximated with a viable multi-box polyhedron W δ′ and the S-CG problem recast as follows
gi (t) = arg mingi gi − ri (t) 2Ψi subject to : T (t−τ )]T∈ V ′ (ˆ x(t), dmax,i ) g(t) = [g1T (t−τ ),...,giT ,...,gN (20) where ¯ := W ′ ∩ G(x, k). ¯ V ′ (x, k) (21) δ Substantially, the set Wδ′ is used in the place of W δ . 9 8 7 6 5 g2
Lemma 1: Let the polyhedral S ⊂ IR m be given and expressed as Ag ≤ b, A ∈ IR|J |×m , b ∈ IR|J | . Consider also a generic point g ′ ∈ S and the set J ′ := {j ∈ J : aTj g ′ = bj }. Then, g ′ is viable iff the following test fails. T T ] ∈ Test - Find, if there exists, a vector w = [w 1T , ..., wN m IR such that A(g ′ + w) ≤ b aT[0T , ..., wiT , ..., 0TN ]T >0, for all i ∈ A, (18) j 1 for at least one j ∈ J ′
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Fig. 4. Multi-inner box approximation by means of 91 boxes of a not viable polyhedron S. According to [2], it is a collection of full-dimensional boxes such that i) the intersection between any two boxes is not full-dimensional; ii) the union of all boxes in M(S) is contained in S.
A. Analysis in multi-dimensional cases Goal of this subsection is to show two interesting examples of use of the above presented multi-box inner approximation technique in the case of systems with multi-dimensional decision sets. The first is the above mentioned counterexample showing the fact that the convex hull of the multi-box approximation set may fail to be viable in general multidimensional problems, when the viability notion of Definition 3 is adopted. The second example is instead provided in order to remark the fact that this method may however be used as a ”first attempt” to obtain an approximated viable set of a constrained region, because in many cases it is a successful approach. 1) Example 1: Consider a two-agent case each one acting on two-dimensional local space (m 1 = m2 = 2) with m = 4. We are looking for two hyper-planes
T a 1 g = b1 aT2 g = b2
for which the Test of Lemma 1 successes. Because we assume that such hyper-planes arise from the multi-box approximating procedure, we consider that they represent faces of the convex hull of such an approximation which enjoy the properties stated in Proposition 1. In order to simplify our search, without any loss of generality, we look for a direction w having always the same coordinate in all axes, i.e. w = [v, v, v, v]T where v ∈ IR. The success of the Test in Lemma 1 suggests us that w and the coefficient vectors a1 and a2 have to jointly satisfy the following
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conditions
8 (a11 + a21 + a31 + a41 )v < 0 > > 4 3 2 1 > (a > 2 + a2 + a2 + a2 )v < 0 > < 2 1 (a1 + a1 )v > 0 (a31 + a41 )v < 0 > > > > > (a12 + a22 )v < 0 : (a32 + a42 )v > 0
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In fact, according to Lemma 1, the latter indicates that any g lying on the considered hyperplane is not viable. Moreover, we have to add further constraints in order to impose that such hyperplane belongs to the convex hull of a multiinner box approximation and then satisfies Proposition 1. In particular, we impose that the first component of w satisfies point (2) of Proposition 1. Furthermore, we have also to consider the following additional conditions a11 v < 0 a12 v < 0
(23)
Because the other components of w satisfy point (1) of Proposition 1, the following constraints have to be included in our search 8 2 < (a1 v < 0 ∧ a22 v < 0) ∨ (−a21 v < 0 ∧ −a22 v < 0) (a3 v < 0 ∧ a32 v < 0) ∨ (−a31 v < 0 ∧ −a32 v < 0) : 14 (a1 v < 0 ∧ a42 v < 0) ∨ (−a41 v < 0 ∧ −a42 v < 0)
(24)
Observe that several solutions for the system of inequalities (22)-(24) exist and may be represented by the following compact form 8 1 a > 12 > a1 > −a11 > > > > 0 a31 < −a11 − a21 ∧ 0 < a41 < −a11 − a21 − a31 ∧ a12 < 0∧1 < C B 0 < a22 < −a32 ∧ a32 < 0 ∧ −a32 < a42 < −a12 − a22 C B > > − a32 ∧ v > 0 ∨ a31 > 0∧ C B > > > A @4 1 2 3 2 1 > 0 ∧ 0 < a2 < −a2 ∧ > : a1 < −a31 − a1 1− a1 2∧ a11 < (0 < a2 ≤ −a2 − a2 ∧ −a32 < a42 < 0 ∧ v > 0) (25)
where ∧ and ∨ represents logic AND and, respectively, OR conditions. As a conclusion, from the success of the Test of Lemma 1 is not possible to guarantee that the multibox approximation is viable in this simple multi-dimensional case.
8 1 < (a1 w1 < 0 ∧ a21 w1 < 0) ∨ (−a12 w1 < 0 ∧ −a12 w1 < 0) (a3 w2 < 0 ∧ a32 w22 < 0) ∨ (−a31 w22 < 0 ∧ −a32 w22 < 0) : 41 23 (a1 w2 < 0 ∧ a42 w23 < 0) ∨ (−a41 w23 < 0 ∧ −a42 w23 < 0)
V. C ONCLUSIONS In this paper, a distributed CG scheme has been proposed for the supervision of dynamically coupled interconnected linear systems subject to local and global constraints and used for solving constrained coordination problems in networked control systems. A sequential distributed strategy has been proposed and its stability, feasibility and liveness properties analyzed in full details. These results are encouraging and stimulate further researches on the topic, especially towards reducing conservativeness and extending the proposed methodology to more general and complex large-scale examples, eventually in the presence of communication time-delays and packet losses. Novel insights on the liveliness analysis of the scheme have been provided and still open issues requiring further investigations discussed. In particular, the approximation procedure presented in [2] able to achieve arbitrarily accurate viable inner approximations of the original constraint set for mono-dimensional problems has been adapted to the S-CG algorithm and it has been shown, via counter-examples, that its direct application to multi-dimensional problems does not lead to viable sets. In particular this issue remains on open problem for future research works. R EFERENCES
2) Example 2: As a second scenario, we consider the same problem of Example 1 with one of the two agents acting on a scalar decision space, i.e. m 1 = 1 and m2 = 3. In this case we look for two hyper-planes and an admissible direction w = [w1 , w2 ], w1 ∈ IR, w2 ∈ IR3 that satisfy the Test of Lemma 1 or, equivalently, the following system of 8 T inequalities a1 w < 0 > > > > > aT2 w < 0 > < 2 1 a1 w2 + a31 w22 + a41 w23 < 0 > a22 w21 + a32 w22 + a42 w23 > 0 > > 1 > > > a11 w1 > 0 : a2 w1 < 0.
(26)
Furthermore, an additional constraint of type (23) has to be added for the satisfaction of property (2) of Proposition 1, 2 1 that is a1 w2 < 0 a22 w21 < 0
(28)
Please note that any attempt to solve the system of inequalities fails because the first inequalities of (28) and the latter two inequalities of (26) are incompatible. Then, in this case, two hyper-planes belonging to the convex hull of a multi-box region are always viable and the multibox inner approximation succeeds in making the resulting approximated region viable.
(27)
and further constraints of type (24) are needed in order to take into account point (1) of Proposition 1
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[1] Casavola A. , Garone E. , Tedesco F. , Distributed Coordination-byConstraint Strategies for Multi-agent Networked Systems, CDC 2011, Orlando, USA. [2] Casavola A. , Garone E. , Tedesco F., A Liveliness Analysis of a Distributed Constrained Coordination Strategy for Multi-Agent Linear Systems, CDC 2011, Orlando, USA. [3] E. Garone, F. Tedesco A. Casavola “Sensorless Supervision of LTI dynamical sytems: The Feed Forward Command Governor”, Automatica, in press, 2011. [4] A. Bemporad, A. Casavola and E. Mosca, “Nonlinear Control of Constrained Linear Systems via Predictive Reference Management”, IEEE Trans. Automat. Control, Vol. 42, pp. 340-349, 1997. [5] A. Casavola and Franz`e, “Dealing with Time-Delay in Coordination by Constraints Methods”, First International Workshop on Control over Communication Networks - CONCOM 2007, Ciprus, April 2007. [6] E.G. Gilbert, I. Kolmanovsky and K. Tin Tan. “Discrete-time Reference Governors and the Nonlinear Control of Systems with State and Control Constraints”. Int. Journ. on Robust and Nonlinear Control, 5, pp. 487-504, 1995. [7] A. Casavola, E. Mosca and D. Angeli. “Robust command governors for constrained linear systems”. IEEE Trans. Automat. Control, 45, pp. 2071-2077, 2000. [8] F. Tedesco “Distributed Command Governor Strategies for Multiagent Dynamical Systems” DEIS-UNICAL, PhD Thesis, University of Calabria, 2011, http://tedescof.wordpress.com/publication/. [9] M. Kvasnica, P. Grieder, M. Baotic, M. Morari “Multi-Parametric Toolbox (MPT)” Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, 2004, Volume 2993/2004, 121-124.
