Barrier Function-based Collaborative Control of Multiple Robots under Signal Temporal Logic Tasks
11 Barrier Function-based Collaborative Control ofMultiple Robots under Signal Temporal Logic Tasks
Lars Lindemann and Dimos V. Dimarogonas
Abstract —Motivated by the recent interest in cyber-physicaland autonomous robotic systems, we study the problem ofdynamically coupled multi-agent systems under a set of signaltemporal logic tasks. In particular, the satisfaction of each ofthese signal temporal logic tasks depends on the behavior of adistinct set of agents. Instead of abstracting the agent dynamicsand the temporal logic tasks into a discrete domain and solvingthe problem therein or using optimization-based methods, wederive collaborative feedback control laws. These control laws arebased on a decentralized control barrier function condition thatresults in discontinuous control laws, as opposed to a centralizedcondition resembling the single-agent case. The benefits of ourapproach are inherent robustness properties typically present infeedback control as well as satisfaction guarantees for continuous-time multi-agent systems. More specifically, time-varying controlbarrier functions are used that account for the semantics of thesignal temporal logic tasks at hand. For a certain fragment ofsignal temporal logic tasks, we further propose a systematic wayto construct such control barrier functions. Finally, we showthe efficacy and robustness of our framework in an experimentincluding a group of three omnidirectional robots.
Index Terms —Control barrier functions, formal methods-based control, multi-agent systems, autonomous systems.
I. I
NTRODUCTION
A multi-agent system is a collection of independent agentswith individual actuation, computation, sensing, and decisionmaking capabilites. Compared to single-agents systems, ad-vantages are scalability with respect to task complexity, ro-bustness to agent failure, and better overall performance. Col-laborative control of multi-agent systems deals with achievingtasks such as consensus [1], formation control [2], connectivitymaintenance [3], and collision avoidance [4] (see [5] for anoverview). A recent trend has been to extend beyond thesestandard objectives and to consider more complex task spec-ifications by using temporal logics. Towards this goal, bothsingle-agent systems [6]–[8] as well as multi-agent systems[9]–[13] have been considered by using linear temporal logic(LTL). Most of these works require a discrete abstraction of theagent dynamics to then employ computationally costly graphsearch methods. Signal temporal logic (STL) [14], as opposedto LTL, allows to impose tasks with strict deadlines and offersa closer connection to the agent dynamics by the introductionof robust semantics [15], [16], hence offering the benefit
L. Lindemann ([email protected]) and D. V. Dimarogonas ([email protected])are with the Division of Decision and Control Systems, School of ElectricalEngineering and Computer Science, KTH Royal Institute of Technology,Stockholm, Sweden.This work was supported in part by the Swedish Research Council (VR),the European Research Council (ERC), the Swedish Foundation for StrategicResearch (SSF), the EU H2020 Co4Robots project, and the Knut and AliceWallenberg Foundation (KAW). of not necessarily relying on an abstraction of the system.Recent control methods for STL tasks then consider discrete-time systems and result, even for single-agent systems, incomputationally costly mixed integer linear programs [17]–[19]. Control approaches for the non-deterministic setup, stillin discrete time, have been presented in [20], while learning-based approaches appeared in [21], [22]. An initial approach toobtain satisfaction guarantees for continuous-time multi-agentsystems under a fragment of STL tasks has been presented inour previous work [23]. Such continuous-time guarantees havealso appeared for single-agent systems in [24] where, however,a possibly non-convex optimization problem is to be solved.Verification of safe sets for dynamical systems has been an-alyzed by the notion of barrier functions, which are also calledbarrier certificates. The construction of such barrier functionsfor polynomial systems using sum of squares programminghas been presented in [25]. For control systems and based onthe notion of barrier functions, control barrier functions havefirst been presented in [26] to guarantee the existence of acontrol law that renders a desired safe set forward invariant.The authors in [27] present control barrier functions tailordfor safe robot navigation, while [28] presents decentralizedcontrol barrier functions for safe multi-robot navigation. Firstrobustness considerations of control barrier functions haveappeared in [29]. Nonsmooth and time-varying control barrierfunctions have been proposed in [30] and [31], respectively. Asimilar work by the authors of [30] recently proposed hybridnonsmooth control barrier functions [32]. In case that suchcontrol barrier functions can not be found, safety kernels canbe calculated. Safety kernels are subsets of the safe set thatcan be rendered invariant by an active set invariance controlmethod [33]. Control barrier functions have also been usedto control systems under temporal logic tasks. For single-agent systems, our previous work in [34] has established aconnection between the semantics of an STL task and time-varying control barrier functions, while [35] considers finite-time control barrier functions for LTL tasks. Although both[34] and [35] deal with achieving finite-time attractivity (see[36] for a definition), the underlying problem definitions differdue to the quantitative, in time and space, nature of STL tasks.Furthermore, [35] provides upper bounds on the time when aregion specified by a static control barrier function is reached,while time-varying control barrier functions provide genericfreedom to shape the level sets of a control barrier function ateach point in time. Following ideas of [34], we have presenteda collaborative feedback control law for multi-agent systems in[37] where distinct sets of agents are considered and each suchset is subject to an STL task; [37] also presents a procedure a r X i v : . [ ee ss . S Y ] F e b to construct control barrier functions for fragments of STLtasks. In contrast to [37], the work in [38] considers multi-agent systems under possibly conflicting local, i.e., individual,tasks, and deals with finding least violating solutions, so thatthe problem definitions of [37] and [38] are different.In this paper, we consider dynamically coupled multi-agentsystems under a set of STL tasks. The satisfaction of eachtask depends on a distinct set of agents. With respect to thissetup, the contributions of this paper are threefold. Assumingthe existence of control barrier functions that account forthe semantics of the STL tasks according to [34], we firstpresent a collaborative feedback control law that guaranteesthe satisfaction of all STL tasks. This control law is basedon a decentralized control barrier function condition. It turnsout, as argued in the technical section of this paper, thatthis control law is discontinuous so that Filippov solutionsand nonsmooth analysis have to be considered. Second, wepresent an optimization-based approach to construct controlbarrier functions for a fragment of STL tasks. Third, weprovide an experiment that shows the efficacy and robust-ness of the presented framework. Compared to optimization-based techniques, such as the MILP formulation in [17], themotivation for control barrier function-based techniques isto obtain robust feedback control laws that directly provideSTL satisfaction guarantees for continuous-time systems. Thispaper is an extension of [37]. We here additionally present anexperiment of a group of three omnidirectional robots, whilewe also provide important proofs that are not included in [37].We also motivate in detail why a discontinuous control law isobtained as opposed to the case where a centralized controlbarrier function condition is used, resembling the single-agent case. We further extend [37] by constructing controlbarrier functions that induce a linear instead of an exponentialtemporal behavior (as explained in detail in the paper). Theadvantages of this are shorter computation times to constructthe control barrier functions as well as practical benefits suchas making it less likely to experience input saturations.Section II states preliminaries and the problem formulation,while our proposed problem solution is stated in Sections IIIand IV. The experiment using three omnidirectional robots ispresented in Section V followed by conclusions in Section VI.II. P RELIMINARIES AND P ROBLEM F ORMULATION
True and false are (cid:62) and ⊥ , while R and R ≥ are the set ofreal and non-negative real numbers; R d is the d -dimensionalreal vector space. Scalars and column vectors are depictedas non-bold letters v and bold letters v , respectively. TheEuclidean, sum, and max norm of v are (cid:107) v (cid:107) , (cid:107) v (cid:107) , and (cid:107) v (cid:107) ∞ ,respectively. Let be a vector of appropriate size containingonly zeros. An extended class K function α : R → R is alocally Lipschitz continuous and strictly increasing functionwith α (0) = 0 . The partial derivatives, here assumed to berow vectors, of a function b ( v , t ) : R d × R ≥ → R evaluatedat ( v ∗ , t ∗ ) are ∂ b ( v ∗ ,t ∗ ) ∂ v := ∂ b ( v ,t ) ∂ v (cid:12)(cid:12)(cid:12) v = v ∗ t = t ∗ and ∂ b ( v ∗ ,t ∗ ) ∂t := ∂ b ( v ,t ) ∂t (cid:12)(cid:12)(cid:12) v = v ∗ t = t ∗ . For two sets S and S , the Minkowski sumis defined as S ⊕ S := { v + v | v ∈ S , v ∈ S } . A. Discontinuous Systems and Nonsmooth Analysis
Consider ˙ v = f ( v , t ) where f : R d × R ≥ → R d is locallybounded and measurable. We consider Filippov solutions [39]to this system and define the Filippov set-valued map F [ f ]( v , t ) := co { lim i →∞ f ( v i , t ) | v i → v , v i / ∈ N ∪ N f } where co denotes the convex closure; N f denotes the set ofLebesgue measure zero where f ( v , t ) is discontinuous, while N denotes an arbitrary set of Lebesgue measure zero. AFilippov solution to ˙ v = f ( v , t ) is an absolutely continuousfunction v : [ t , t ] → R d that satisfies ˙ v ( t ) ∈ F [ f ]( v , t ) foralmost all t ∈ [ t , t ] . Due to [40, Prop. 3] it holds that thereexists a Filippov solution to ˙ v = f ( v , t ) if f : R d × R ≥ → R d is locally bounded and measurable. For switched systems withstate-dependent switching, existence of Filippov solutions isdiscussed in [41]. The switching mechanism of the switchedsystem presented in Section III is time-dependent so that[40, Prop. 3] can still be applied. Consider a continuouslydifferentiable function b ( v , t ) so that Clarke’s generalizedgradient of b ( v , t ) coincides with the gradient of b ( v , t ) [40,Prop. 6], denoted by ∇ b ( v , t ) := (cid:104) ∂ b ( v ,t ) ∂ v ∂ b ( v ,t ) ∂t (cid:105) . The set-valued Lie derivative of b ( v , t ) with respect to F [ f ]( v , t ) at ( v , t ) is then defined as L F [ f ] b ( v , t ) := {∇ b ( v , t ) (cid:2) ζ T (cid:3) T | ζ ∈ F [ f ]( v , t ) } . According to [42, Thm. 2.2], it holds that ˙ b ( v ( t ) , t ) ∈ L F [ f ] b ( v ( t ) , t ) for almost all t ∈ [ t , t ] . Let ˆ L F [ f ] b ( v , t ) := (cid:8) ∂ b ( v ,t ) ∂ v ζ | ζ ∈ F [ f ]( v , t ) (cid:9) , the set-valued Lie derivative is then equivalent to L F [ f ] b ( v , t ) = ˆ L F [ f ] b ( v , t ) ⊕ (cid:8) ∂ b ( v ,t ) ∂t (cid:9) . Lemma 1:
Consider ˙ v = f ( v , t ) + f ( v , t ) where f : R d × R ≥ → R d and f : R d × R ≥ → R d are locally boundedand measurable. It then holds that L F [ f + f ] b ( v , t ) ⊆ ˆ L F [ f ] b ( v , t ) ⊕ ˆ L F [ f ] b ( v , t ) ⊕ (cid:110) ∂ b ( v , t ) ∂t (cid:111) . Proof:
Applying the definition of L F [ f ] b ( v , t ) gives L F [ f + f ] b ( v , t ):= {∇ b ( v , t ) (cid:2) ζ T (cid:3) T | ζ ∈ F [ f + f ]( v , t ) }⊆ {∇ b ( v , t ) (cid:2) ζ T (cid:3) T | ζ ∈ F [ f ]( v , t ) ⊕ F [ f ]( v , t ) } = (cid:110) ∂ b ( v , t ) ∂ v ζ | ζ ∈ F [ f ]( v , t ) ⊕ F [ f ]( v , t ) (cid:111) ⊕ (cid:110) ∂ b ( v , t ) ∂t (cid:111) = ˆ L F [ f ] b ( v , t ) ⊕ ˆ L F [ f ] b ( v , t ) ⊕ (cid:110) ∂ b ( v , t ) ∂t (cid:111) . where we used the fact that F [ f + f ]( v , t ) ⊆ F [ f ]( v , t ) ⊕ F [ f ]( v , t ) due to [43, Thm. 1]. B. Signal Temporal Logic (STL)
Signal temporal logic [14] is based on predicates µ thatare obtained after evaluation of a continuously differentiablepredicate function h : R d → R as µ := (cid:62) if h ( v ) ≥ and µ := ⊥ if h ( v ) < for v ∈ R d . We consider, in this paper,an STL fragment that is recursively defined as ψ ::= (cid:62) | µ | ψ (cid:48) ∧ ψ (cid:48)(cid:48) (1a) φ ::= G [ a,b ] ψ | F [ a,b ] ψ | ψ (cid:48) U [ a,b ] ψ (cid:48)(cid:48) | φ (cid:48) ∧ φ (cid:48)(cid:48) (1b)where ψ (cid:48) , ψ (cid:48)(cid:48) denote formulas of class ψ in (1a), whereas φ (cid:48) , φ (cid:48)(cid:48) denote formulas of class φ in (1b). Note that ¬ µ can beencoded in (1a) by defining ¯ µ := ¬ µ and ¯ h ( v ) := − h ( v ) .The operators ¬ , ∧ , G [ a,b ] , F [ a,b ] , and U [ a,b ] denote thenegation, conjunction, always, eventually, and until operatorswith a ≤ b < ∞ . Formulas of class ψ in (1a) are non-temporal(Boolean) formulas whereas formulas of class φ in (1b) aretemporal formulas. Let ( s , t ) | = φ denote the satisfactionrelation, i.e., if a signal s : R ≥ → R d satisfies φ at time t ; φ is satisfiable if ∃ s : R ≥ → R d such that ( s , | = φ . Fora given s : R ≥ → R d , the STL semantics [14] of the fragmentin (1) are recursively defined by: ( s , t ) | = µ iff h ( s ( t )) ≥ , ( s , t ) | = ψ (cid:48) ∧ ψ (cid:48)(cid:48) iff ( s , t ) | = ψ (cid:48) ∧ ( s , t ) | = ψ (cid:48)(cid:48) , ( s , t ) | = G [ a,b ] ψ iff ∀ ¯ t ∈ [ t + a, t + b ] , ( s , ¯ t ) | = ψ , ( s , t ) | = F [ a,b ] ψ iff ∃ ¯ t ∈ [ t + a, t + b ] s.t. ( s , ¯ t ) | = ψ , and ( s , t ) | = ψ (cid:48) U [ a,b ] ψ (cid:48)(cid:48) iff ∃ ¯ t ∈ [ t + a, t + b ] s.t. ( s , ¯ t ) | = ψ (cid:48)(cid:48) ∧ ∀ t ∈ [ t, ¯ t ] , ( s , t ) | = ψ (cid:48) .Robust semantics [16, Def. 3] are recursively defined by ρ µ ( s , t ) := h ( s ( t )) , ρ ¬ µ ( s , t ) := − ρ µ ( s , t ) ,ρ ψ (cid:48) ∧ ψ (cid:48)(cid:48) ( s , t ) := min( ρ ψ (cid:48) ( s , t ) , ρ ψ (cid:48)(cid:48) ( s , t )) ,ρ G [ a,b ] ψ ( s , t ) := min ¯ t ∈ [ t + a,t + b ] ρ ψ ( s , ¯ t ) ,ρ F [ a,b ] ψ ( s , t ) := max ¯ t ∈ [ t + a,t + b ] ρ ψ ( s , ¯ t ) ,ρ ψ (cid:48) U [ a,b ] ψ (cid:48)(cid:48) ( s , t ) := max ¯ t ∈ [ t + a,t + b ] min( ρ ψ (cid:48)(cid:48) ( s , ¯ t ) , min t ∈ [ t, ¯ t ] ρ ψ (cid:48) ( s , t )) , and determine how robustly a signal s satisfies φ at time t . Itholds that ( s , t ) | = φ if ρ φ ( s , t ) > [15, Prop. 16]. C. Control Barrier Functions encoding STL tasks
Our previous work [34] has established a connection be-tween a function b : R d × R ≥ → R (later shown to be a validcontrol barrier function) and the STL semantics of φ given in(1b). In particular, if this function is according to [34, StepsA, B, and C], then, for a given signal s : R ≥ → R d with b ( s ( t ) , t ) ≥ for all t ≥ , it holds that ( s , | = φ . Let C ( t ) := { v ∈ R d | b ( v , t ) ≥ } so that equivalently s ( t ) ∈ C ( t ) for all t ≥ implies ( s , | = φ . The conditions in [34, Steps A, B, and C] aresummarized next. To encode conjunctions contained in φ ,a smooth approximation of the min-operator in the robustsemantics is used. For p functions b l : R d × R ≥ → R where l ∈ { , . . . , p } , let b ( v , t ) := − η ln (cid:0) (cid:80) pl =1 exp( − η b l ( v , t )) (cid:1) with η > . Note that min l ∈{ ,...,p } b l ( v , t ) ≈ b ( v , t ) wherethe accuracy of this approximation increases with η , i.e., lim η →∞ − η ln (cid:16) p (cid:88) l =1 exp( − η b l ( v , t )) (cid:17) = min l ∈{ ,...,p } b l ( v , t ) . Regardless of the choice of η , we have − η ln (cid:16) p (cid:88) l =1 exp( − η b l ( v , t )) (cid:17) ≤ min l ∈{ ,...,p } b l ( v , t ) (2) which is useful since b ( v , t ) ≥ implies b l ( v , t ) ≥ for each l ∈ { , . . . , p } , i.e., the conjunction operator can be encoded.Let the predicate funtion h l ( v ) correspond to the predicate µ l . In Steps A and B, we illustrate the main idea for singletemporal operators, i.e., only one always, eventually, or untiloperator is contained in φ . Step A) Single temporal operators in (1b) without conjunctions G [ a,b ] µ p = 1 , ∀ t (cid:48) ∈ [ a, b ] , b ( v , t (cid:48) ) ≤ h ( v ) F [ a,b ] µ p = 1 , ∃ t (cid:48) ∈ [ a, b ] s.t. b ( v , t (cid:48) ) ≤ h ( v ) µ U [ a,b ] µ p = 2 , ∃ t (cid:48) ∈ [ a, b ] s.t. b ( v , t (cid:48) ) ≤ h ( v ) , ∀ t (cid:48)(cid:48) ∈ [0 , t (cid:48) ] , b ( v , t (cid:48)(cid:48) ) ≤ h ( v ) Note that (2) ensures satisfaction of µ U [ a,b ] µ if b ( s ( t ) , t ) ≥ for all t ∈ [ a, b ] . In Step B, we generalize theresults from Step A, but now with conjunctions of predicatesinstead of a single predicate. Let ψ := µ ∧ . . . ∧ µ ˜ p and ψ := µ ˜ p +1 ∧ . . . ∧ µ ˜ p +˜ p where ˜ p , ˜ p ≥ . Step B) Single temporal operators in (1b) with conjunctions. G [ a,b ] ψ p = ˜ p , ∀ t (cid:48) ∈ [ a, b ] , ∀ l ∈ { , . . . , ˜ p } , b l ( v , t (cid:48) ) ≤ h l ( v ) F [ a,b ] ψ p = ˜ p , ∃ t (cid:48) ∈ [ a, b ] , ∀ l ∈ { , . . . , ˜ p } s.t. b l ( v , t (cid:48) ) ≤ h l ( v ) ψ U [ a,b ] ψ p = ˜ p +˜ p , ∃ t (cid:48) ∈ [ a, b ] , ∀ l (cid:48) ∈ { ˜ p +1 , . . . , ˜ p +˜ p } s.t. b l (cid:48) ( v , t (cid:48) ) ≤ h l (cid:48) ( v ) and ∀ t (cid:48)(cid:48) ∈ [0 , t (cid:48) ] , ∀ l (cid:48)(cid:48) ∈{ , . . . , ˜ p } , b l (cid:48)(cid:48) ( v , t (cid:48)(cid:48) ) ≤ h l (cid:48)(cid:48) ( v ) In Step C), conjunctions of single temporal operators areconsidered. The conditions on b ( v , t ) are a straightforwardextension of Steps A and B. For instance, consider G [ a ,b ] ψ ∧ F [ a ,b ] ψ ∧ ψ U [ a ,b ] ψ . Let p = 3 where b ( v , t ) , b ( v , t ) ,and b ( v , t ) are associated with G [ a ,b ] ψ , F [ a ,b ] ψ , and ψ U [ a ,b ] ψ and constructed as in Steps A and B.Similar to [34], a switching mechanism is introducedand we integrate o l : R ≥ → { , } into b ( v , t ) := − η ln (cid:0) (cid:80) pl =1 o l ( t ) exp( − η b l ( v , t )) (cid:1) ; p is again the total num-ber of functions b l ( v , t ) obtained from Steps A, B, and Cand each b l ( v , t ) corresponds to either an always, eventually,or until operator with a corresponding time interval [ a l , b l ] .We remove single functions b l ( v , t ) from b ( v , t ) when thecorresponding always, eventually, or until operator is satisfied.For each temporal operator, the associated b l ( v , t ) is removedat t = b l , i.e., o l ( t ) = 1 if t < b l and o l ( t ) := 0 if t ≥ b l .We denote the switching sequence by { s := 0 , s , . . . , s q } with q ∈ N as the total number of switches. This sequence isknown due to knowledge of [ a l , b l ] . At time t ≥ s j we have s j +1 := argmin b l ∈{ b ,...,b p } ζ ( b l , t ) where ζ ( b l , t ) := b l − t if b l − t > and ζ ( b l , t ) := ∞ otherwise. We further requirethat each function b l ( v , t ) is continuously differentiable so that b ( v , t ) is continuously differentiable on R d × ( s j , s j +1 ) . D. Coupled Multi-Agent Systems
Consider M agents modeled by an undirected graph G :=( V , E ) where V := { , . . . , M } while E ∈ V × V indicatescommunication links. Let x i ∈ R n i and u i ∈ R m i be statesand inputs of agent i . Also let x := (cid:2) x T . . . x M T (cid:3) T ∈ R n with n := n + . . . + n M . The dynamics of agent i are ˙ x i = f i ( x i , t ) + g i ( x i , t ) u i + c i ( x , t ) (3) where f i : R n i × R ≥ → R n i , g i : R n i × R ≥ → R n i × m i ,and c i : R n × R ≥ → R n i are locally Lipschitz continuous; c i ( x , t ) may model given dynamical couplings such as thoseinduced by a mechanical connection between agents; c i ( x , t ) may also describe unmodeled dynamics or disturbances. Weassume that f i ( x i , t ) and g i ( x i , t ) are only known by agent i and c i ( x , t ) is bounded, but otherwise unknown so that noknowledge of x and c i ( x , t ) is required by agent i for thecontrol design. In other words, there exists a known C ≥ such that (cid:107) c i ( x , t ) (cid:107) ∞ ≤ C for all ( x , t ) ∈ R n × R ≥ . Assumption 1:
The function g i ( x i , t ) has full row rank forall ( x i , t ) ∈ R n i × R ≥ . Remark 1:
Assumption 1 implies m i ≥ n i . Since c i ( x , t ) is not known by agent i , the system (3) is, however, not feedback equivalent to ˙ x i = u i . Canceling f i ( x i ) may alsoinduce high control inputs, while we derive a minimum normcontroller in Section III-B. Assumption 1 allows to decouplethe construction of control barrier functions from the dynamicsof the agents as discussed in Section IV. In other words,for a function b ( x , t ) it holds that ∂ b ( x ,t ) ∂ x i g i ( x i , t ) = if and only if ∂ b ( x ,t ) ∂ x i = . We note that most of thestandard multi-agent literature assume simplified dynamicsto deal with the complexity of the problem at hand. Colli-sion avoidance, consensus, formation control, or connectivitymaintenance can be achieved through a secondary controller f u i . Let V u i ⊆ V be a set of agents that induce dynamicalcouplings , and let x u i := (cid:104) x j T . . . x j |V u i | T (cid:105) T and n u i := n j + . . . + n j |V u i | for j , . . . , j |V u i | ∈ V u i . By using u i := g i ( x i , t ) T ( g i ( x i , t ) g i ( x i , t ) T ) − f u i ( x u i , t ) + v i the dynamics ˙ x i = f i ( x i , t ) + f u i ( x u , t ) + g i ( x i , t ) v i + c i ( x , t ) resemble (3)if f u i : R n u i × R ≥ → R n i is locally Lipschitz continuous. E. Problem Formulation
Consider K temporal formulas φ , . . . , φ K of the form (1b)and let the satisfaction of φ k for k ∈ { , . . . , K } depend onthe set of agents V k ⊆ V . This means that knowledge of thesolutions x i : R ≥ → R n i to (3) for i ∈ V k is sufficient toevaluate if φ k is satisfied. Assume further that V ∪ . . . ∪V K = V and that the sets of agents V , . . . , V K ∈ V are disjoint, i.e., V k ∩V k = ∅ for all k , k ∈ { , . . . , K } with k (cid:54) = k . Thereare hence no formula dependencies between agents in V k andagents in V k , although these agents may still be dynamicallycoupled through c i ( x , t ) . The formula dependencies need tobe in accordance with the graph topology of G as follows. Assumption 2:
For each φ k with k ∈ { , . . . , K } , it holdsthat ( i , i ) ∈ E for all i , i ∈ V k . Problem 1:
Consider K formulas φ k of the form (1b).Derive a decentralized control law u i for each agent i ∈ V sothat, for each Filippov solution x : R ≥ → R n to (3) under u i , < r ≤ ρ φ ∧ ... ∧ φ K ( x , where r is maximized.III. B ARRIER F UNCTION - BASED C ONTROL S TRATEGIES
We first motivate why the decentralized multi-agent caserequires a discontinuous control law, while the centralizedcase, resembling a single-agent formulation, permits contin-uous control laws [38, Coroll. 1]. For j , . . . , j |V k | ∈ V k , define ¯ x k := (cid:2) x j T . . . x j |V k | T (cid:3) T ∈ R ¯ n k with ¯ n k := n j + . . . + n j |V k | . Note that, for agents in V k , the stackedagent dynamics of the elements in (3) are ˙¯ x k = ¯ f k (¯ x k , t ) + ¯ g k (¯ x k , t )¯ u k (¯ x k , t ) + ¯ c k ( x , t ) with ¯ u k (¯ x k , t ) explicitly depending on ¯ x k and t and where ¯ f k (¯ x k , t ) := (cid:104) f j ( x j , t ) T . . . f j |V k | ( x j |V k | , t ) T (cid:105) T , ¯ g k (¯ x k , t ) := diag (cid:0) g j ( x j , t ) , . . . , g j |V k | ( x j |V k | , t ) (cid:1) , ¯ u k (¯ x k , t ) := (cid:104) u j (¯ x k , t ) T . . . u j |V k | (¯ x k , t ) T (cid:105) T , ¯ c k ( x , t ) := (cid:104) c j ( x , t ) T . . . c j |V k | ( x , t ) T (cid:105) T . The function ¯ c k ( x , t ) may dynamically couple some or evenall agents. Let ¯ b k : R ¯ n k × R ≥ → R denote the control barrierfunction corresponding to φ k and accounting for Steps A, B,and C. For the stacked agent dynamics, the centralized controlbarrier function condition (see, e.g., [38, eq. (6)]) is ∂ ¯ b k (¯ x k , t ) ∂ ¯ x k ( ¯ f k (¯ x k , t ) + ¯ g k (¯ x k , t )¯ u k (¯ x k , t ))+ ∂ ¯ b k (¯ x k , t ) ∂t ≥ − α k (¯ b k (¯ x k , t )) + (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x k , t ) ∂ ¯ x k (cid:13)(cid:13)(cid:13) C (4)where α k : R → R is an extended class K function and wherewe use the constant C since it holds that (cid:107) ¯ c k ( x , t ) (cid:107) ∞ ≤ C ,which follows since (cid:107) ¯ c k ( x , t ) (cid:107) ∞ = max i ∈V k (cid:107) c i ( x , t ) (cid:107) ∞ ≤ C since (cid:107) c i ( x , t ) (cid:107) ∞ ≤ C for each i ∈ V k by assumption. Notethat ∂ ¯ b k (¯ x k ,t ) ∂t ≥ − α k (¯ b k (¯ x k , t )) will hold if ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k = asensured by the control barrier function construction proposedin Section IV and by virtue of Lemma 4; The solution of (4)admits a continuous and bounded control law ¯ u k (¯ x k , t ) [38,Coroll. 1]. Remark 2:
There are two ways to compute and implement ¯ u k (¯ x k , t ) from (4): 1) Each agent i ∈ V k solves (4) andapplies the portion u i (¯ x k , t ) of ¯ u k (¯ x k , t ) , or 2) Inequality (4)is solved by one agent i ∈ V k that sends the portions u j (¯ x k , t ) of ¯ u k (¯ x k , t ) to the agents j ∈ V k \{ i } . The drawbacks are thatat least one agent needs to know the dynamics of each otheragent, i.e., ¯ f k (¯ x k , t ) and ¯ g k (¯ x k , t ) , and, for a large number ofagents, (4) may contain a large number of decision variables,equal to the dimension of ¯ u k (¯ x k , t ) . The second approachalso requires more communication and lacks robustness sincea malfunctioning agent i results in a halt of the whole system.We, as opposed to Remark 2, propose the decentralization of(4), and hence of the control input computation, such that eachagent computes its own control input u i (¯ x k , t ) alleviating theabove issues. Each agent solves its own decentralized controlbarrier function condition so that their conjunction implies (4).A straightforward idea is to let each agent i ∈ V k solve ∂ ¯ b k (¯ x k , t ) ∂ x i ( f i ( x i , t ) + g i ( x i , t ) u i (¯ x k , t )) ≥− D i (cid:16) ∂ ¯ b k (¯ x k , t ) ∂t + α k (¯ b k (¯ x k , t )) (cid:17) + (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x k , t ) ∂ x i (cid:13)(cid:13)(cid:13) C (5)where the weight D i := |V k | distributes (4) equally to eachagent. Note that other weights D i could be imagined, as long as (cid:80) i ∈V k D i = 1 similarly to [28]. We remark that we show,in the proof of Theorem 2, why (5) for each i ∈ V k implies(4). With D i := |V k | , however, the obtained control law mayinduce problems when the gradients ∂ ¯ b k (¯ x k ,t ) ∂ x i become equal tothe zero vector. In particular, assume that ∂ ¯ b k (¯ x k ,t ) ∂ x i = while ∃ j ∈ V k \ { i } such that ∂ ¯ b k (¯ x k ,t ) ∂ x j (cid:54) = , then (5) for agent i may not be feasible and hence not imply (4) (note in this casethat ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k (cid:54) = ). Severely critical, it can be seen that itmay happen that (cid:107) u i ( x i , t ) (cid:107) → ∞ as ∂ ¯ b k (¯ x k ,t ) ∂ x i → if ∃ j ∈V k \ { i } such that ∂ ¯ b k (¯ x k ,t ) ∂ x j → v for v (cid:54) = . Consequently, aweight function D i : R ¯ n k × R ≥ is needed and, as it will turnout, this weight function will be discontinuous. Remark 3:
Local Lipschitz continuity for barrier functions-based control laws has been proven in [27, Thm. 3] underthe “relative degree one condition”. For (5), this condition isequivalent to ∂ ¯ b k (¯ x k ,t ) ∂ x i (cid:54) = , which does not hold in general,so that discontinuities in the control law can be expectedas analyzed in the proof of Theorem 2. For (4), note thatsituations where ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k = are taken into account in theproof of [38, Coroll. 1], ensuring continuity of the control law.Section III-A extends [34] and [38, Coroll. 1] to obtaina centralized control barrier function condition for multi-agent systems with discontinuous control laws. Section III-Buses these results and proposes a control law, based on adecentralized control barrier function condition, that solvesProblem 1. Sections III-A and III-B assume the existence ofthe functions ¯ b k (¯ x k , t ) that satisfy Steps A, B, and C. InSection IV, we present a procedure to construct such ¯ b k (¯ x k , t ) . A. A Centralized Control Barrier Function Condition forMulti-Agent Systems with Discontinuous Control Laws
The results in this section are derived without the needfor Assumption 1. The functions ¯ b k : R ¯ n k × R ≥ → R are continuously differentiable on R ¯ n k × ( s kj , s kj +1 ) where { s k := 0 , s k , . . . , s kq k } are the associated switching sequencesas discussed in Section II-C. Similarly, define C k ( t ) := (cid:8) ¯ x k ∈ R ¯ n k | ¯ b k (¯ x k , t ) ≥ (cid:9) . For a particular k ∈ { , . . . , K } , let x : [ t kj , t kj +1 ] → R n be a Filippov solution to (3) under the control laws u i (¯ x k , t ) where t kj := s kj . We distinguish between t kj +1 and s kj +1 sincewe want to ensure closed-loop properties over [ s kj , s kj +1 ] , whileFilippov solutions may only be defined for t kj +1 < s kj +1 . Definition 1 (Control Barrier Function):
The function ¯ b k : R ¯ n k × R ≥ → R is a candidate control barrier function (cCBF)for [ s kj , s kj +1 ) if, for each ¯ x k ( s kj ) ∈ C k ( s kj ) , there exists anabsolutely continuous function ¯ x k : [ s kj , s kj +1 ) → R ¯ n k suchthat ¯ x k ( t ) ∈ C k ( t ) for all t ∈ [ s kj , s kj +1 ) . A cCBF ¯ b k (¯ x k , t ) for [ s kj , s kj +1 ) is a valid control barrier function (vCBF) for [ s kj , s kj +1 ) and for (3) under locally bounded and measurablecontrol laws u i (¯ x k , t ) if the following holds. For each i ∈ V k with c i : R n × R ≥ → R n i such that (cid:107) c i ( x , t ) (cid:107) ∞ ≤ C , x ( t kj ) ∈ R n with ¯ x k ( t kj ) ∈ C k ( t kj ) implies, for each Filippovsolution x : [ t kj , t kj +1 ] → R n to (3) under the control laws u i (¯ x k , t ) with t kj = s kj , that ¯ x k ( t ) ∈ C k ( t ) for all t ∈ [ t kj , min( t kj +1 , s kj +1 )) .Note that the definition of a vCBF does not require that t kj +1 ≥ s kj +1 . In the remainder, we consider open sets D k ∈ R ¯ n k such that D k ⊃ C k ( t ) for all t ∈ [ s kj , s kj +1 ) . Theorem 1:
Assume that ¯ b k (¯ x k , t ) is a cCBF for [ s kj , s kj +1 ) .If each u i (¯ x k , t ) is locally bounded and measurable and ifthere exists an extended class K function α k such that min L F [ ¯ f k +¯ g k ¯ u k ] ¯ b k (¯ x k , t ) ≥− α k (¯ b k (¯ x k , t )) + (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x k , t ) ∂ ¯ x k (cid:13)(cid:13)(cid:13) C (6)for all (¯ x k , t ) ∈ D k × ( s kj , s kj +1 ) , then ¯ b k (¯ x k , t ) is a vCBFfor [ s kj , s kj +1 ) and for (3) under u i (¯ x k , t ) . Proof:
Note first that (6) implies min L F [ ¯ f k +¯ g k ¯ u k ] ¯ b k (¯ x k , t ) ⊕ ∂ ¯ b k (¯ x k , t ) ∂ ¯ x k ¯ c k ( x , t ) ≥− α k (¯ b k (¯ x k , t )) (7)since − ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k ¯ c k ( x , t ) ≤ (cid:107) ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k (cid:107) (cid:107) ¯ c k ( x , t ) (cid:107) ∞ ≤(cid:107) ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k (cid:107) C by H¨older’s inequality (recall also that (cid:107) ¯ c k ( x , t ) (cid:107) ∞ ≤ C since (cid:107) c i ( x , t ) (cid:107) ∞ ≤ C for each i ∈ V k ). Assume next that ¯ x k ( t kj ) ∈ C k ( t kj ) and considerFilippov solutions x : [ t kj , t kj +1 ] → R n to (3) underthe control laws u i (¯ x k , t ) with t kj = s kj , which areensured to exist since f i ( x i , t ) , g i ( x i , t ) , c i ( x , t ) , and u i (¯ x k , t ) are locally bounded and measurable. Notehence that ˙¯ b k (¯ x k ( t ) , t ) ∈ L F [ ¯ f k +¯ g k ¯ u k +¯ c k ] ¯ b k (¯ x k ( t ) , t ) foralmost all t ∈ ( t kj , min( t kj +1 , s kj +1 )) and consequentlyalso for almost all t ∈ [ t kj , min( t kj +1 , s kj +1 )) . Dueto (7) it holds that min L F [ ¯ f k +¯ g k ¯ u k ] ¯ b k (¯ x k ( t ) , t ) ⊕ ∂ ¯ b k (¯ x k ( t ) ,t ) ∂ ¯ x k ¯ c k ( x ( t ) , t ) ≥ − α k (¯ b k (¯ x k ( t ) , t )) and accordingto Lemma 1, we have min L F [ ¯ f k +¯ g k ¯ u k +¯ c k ] ¯ b k (¯ x k ( t ) , t ) ≥ min {L F [ ¯ f k +¯ g k ¯ u k ] ¯ b k (¯ x k ( t ) , t ) ⊕ ˆ L F [¯ c k ] ¯ b k (¯ x k ( t ) , t ) } =min L F [ ¯ f k +¯ g k ¯ u k ] ¯ b k (¯ x k ( t ) , t ) ⊕ ∂ ¯ b k (¯ x k ( t ) ,t ) ∂ ¯ x k ¯ c k ( x ( t ) , t ) since ˆ L F [¯ c k ] ¯ b k (¯ x k ( t ) , t ) = { ∂ ¯ b k (¯ x k ( t ) ,t ) ∂ ¯ x k ¯ c k ( x ( t ) , t ) } , i.e., asingleton, due to [43, Thm. 1] and since ¯ b k (¯ x k , t ) iscontinuously differentiable. It then holds that ˙¯ b k (¯ x k ( t ) , t ) ≥ min L F [ ¯ f k +¯ g k ¯ u k +¯ c k ] ¯ b k (¯ x k ( t ) , t ) ≥ − α k (¯ b k (¯ x k ( t ) , t )) . By[30, Lem. 2], it follows that ¯ b k (¯ x k ( t ) , t ) ≥ for all t ∈ [ t kj , min( t kj +1 , s kj +1 )) . Remark 4:
We use D k to obtain a similar notion ofrobustness as discussed in [29]. If ¯ x k ( t ) ∈ D k \ C k ( t ) , notethat ˙¯ b k (¯ x k ( t ) , t ) ≥ − α k (¯ b k (¯ x k ( t ) , t )) > for almost all t ∈ [ s kj , s kj +1 ] since ¯ b k (¯ x k ( t ) , t ) < , which is an importantproperty in the experimental setup in Section V.To guarantee satisfaction of φ k , Filippov solutions x : [ t :=0 , t ] → R n need to be defined for t ≥ max( s q , . . . , s Kq ) sothat we require C k ( t ) to be compact. This requirement is notrestrictive and can be achieved by considering φ k ∧ φ bd k insteadof φ k , as assumed in the remainder, where φ bd k := G [0 , ∞ ) µ bd k with h bd k (¯ x k ) := D k − (cid:107) ¯ x k (cid:107) for a suitably selected D k ≥ . Corollary 1:
Let ¯ b k (¯ x k , t ) satisfy the conditions in Steps A,B, and C for φ k and be a cCBF for each [ s kj , s kj +1 ) . Let each u i (¯ x k , t ) be locally bounded and measurable. If, for each k ∈ { , . . . , K } , ¯ u k (¯ x k , t ) is such that (6) holds for all (¯ x k , t ) ∈ D k × ( s kj , s kj +1 ) , then it follows that ( x , | = φ ∧ . . . ∧ φ K for each Filippov solution to (3) under u i (¯ x k , t ) . Proof:
Note that ¯ b k (¯ x k , t ) is piecewise continuous in t with discontinuities at times s kj . The set C k ( t ) is non-decreasing at these switching times s kj , i.e., lim τ → − s kj C k ( τ ) ⊆ C k ( s kj ) where lim τ → − s kj C k ( τ ) denotes the left-sided limitof C k ( t ) at t = s kj . This follows due to the switchingmechanism and, in particular, the function o l ( t ) as explained inSection II-C. It is hence sufficient to ensure forward invarianceof C k ( t ) for each [ s kj , s kj +1 ) separately since ¯ x k ( s kj +1 ) ∈ C k ( s kj +1 ) if ¯ x k ( t ) ∈ C k ( t ) for all t ∈ [ s kj , s kj +1 ) . Dueto Theorem 1, it follows that ¯ x k ( t ) ∈ C k ( t ) for all t ∈ [ t , min( t , s , . . . , s K )) . Note that C k ( t ) ⊆ { ¯ x k ∈ R ¯ n k | D k −(cid:107) ¯ x k (cid:107) ≥ } and that C k ( t ) ⊂ D k . Consequently, there existsa compact set D (cid:48) k ⊂ D k so that ¯ x k ( t ) ∈ C k ( t ) implies ¯ x k ( t ) ∈ D (cid:48) k . This means that x ( t ) remains in a compact set D (cid:48) × . . . × D (cid:48) K , which implies t ≥ min( s , . . . , s K ) by [39,Ch. 2.7]. The same reasoning can be applied for consecutivetime intervals. By the conditions imposed on ¯ b k (¯ x k , t ) in StepsA, B, and C, it follows that each Filippov solution satisfies (¯ x k , | = φ k since ¯ b k (¯ x k ( t ) , t ) ≥ for all t ∈ [ s k , s kq ] sothat ( x , | = φ ∧ . . . ∧ φ K follows. B. Collaborative Control Laws based on a DecentralizedControl Barrier Function Condition
In this section, we again assume that Assumption 1 holds.We first analyze cases where ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k ¯ g k (¯ x k , t ) = T . Thesecases mean that ¯ b k (¯ x k , t ) , although possibly being a cCBFfor [ s kj , s kj +1 ) , may not be a vCBF for [ s kj , s kj +1 ) and for (3)under any control law ¯ u k (¯ x k , t ) since (6) may fail to hold. Dueto Assumption 1, it holds that the nullspace of ¯ g k (¯ x k , t ) T isempty, i.e., ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k ¯ g k (¯ x k , t ) = T if and only if ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k = . To take care of these cases, we define ω k (¯ x k , t ) := ∂ ¯ b k (¯ x k , t ) ∂t + α k (¯ b k (¯ x k , t )) (8) B kj := (cid:110) (¯ x k , t ) ∈ D k × ( s kj , s kj +1 ) | ∂ ¯ b k (¯ x k , t ) ∂ ¯ x k = (cid:111) and pose the following assumption. Assumption 3:
For some (cid:15) k > , it holds that ω k (¯ x k , t ) ≥ (cid:15) k for each (¯ x k , t ) ∈ B kj .Assumption 3 will be addressed in Section IV in Lemma 4an the intuition is that ω k (¯ x k , t ) ≥ (cid:15) k > ensures that (6) canbe satisfied by a proper choice of α k even if ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k = .From now on, assume further that D k is bounded. Theorem 2:
Let ¯ b k (¯ x k , t ) satisfy the conditions in Steps A,B, and C for φ k , be a cCBF for each [ s kj , s kj +1 ) , and satisfyAssumption 3. If, for each k ∈ { , . . . , K } , each agent i ∈ V k applies the control law u i (¯ x k , t ) := u i where u i is given by argmin u i ∈ R mi u Ti u i (9a)s.t. ∂ ¯ b k (¯ x k , t ) ∂ x i ( f i ( x i , t ) + g i ( x i , t ) u i ) ≥− D i (¯ x k , t ) ω k (¯ x k , t ) + (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x k , t ) ∂ x i (cid:13)(cid:13)(cid:13) C, (9b) with D i (¯ x k , t ) := (cid:13)(cid:13) ∂ ¯ b k (¯ x k,t ) ∂ x i (cid:13)(cid:13) (cid:80) v ∈V k (cid:13)(cid:13) ∂ ¯ b k (¯ x k,t ) ∂ x v (cid:13)(cid:13) if (cid:80) v ∈V k (cid:13)(cid:13) ∂ ¯ b k (¯ x k ,t ) ∂ x v (cid:13)(cid:13) (cid:54) = 01 otherwise,then it follows that ( x , | = φ ∧ . . . ∧ φ K for each Filippovsolution to (3) under u i (¯ x k , t ) . Proof:
The proof can be found in the appendix.The load sharing function D i (¯ x k , t ) shares the centralizedcontrol barrier function condition (6) among agents by meansof the decentralized control barrier function condition (9b).Computation of u i is hence decentralized so that smalleroptimization problems can be solved without the requirementthat an agent knows ¯ f k (¯ x k , t ) and ¯ g k (¯ x , t ) . The optimizationprogram (9) is a computationally tractable convex quadraticprogram with m i decision variables and agents need noknowledge of x and c i ( x , t ) . Also, the desired robustness isobtained, e.g., even if an agent i ∈ V k malfunctions, the otheragents in V k \ { i } will still work towards satisfying φ k .IV. C ONTROL B ARRIER F UNCTION C ONSTRUCTION
The construction of ¯ b k (¯ x k , t ) is the same for each φ k .For readability reasons, we hence omit the index k andconsider instead φ and b ( x , t ) with x ∈ R n . To enforce theconditions in Steps A, B, and C, we will consider a function γ l : R ≥ → R that is associated with the predicate function h l ( x ) and the predicate µ l . Let h opt l := sup x ∈ R n h l ( x ) forwhich it has to hold that h opt l ≥ . Otherwise, i.e, if h opt l < , µ l is not satisfiable. We aim at satisfying φ with robustness r ∈ R ≥ , i.e., ρ φ ( x , ≥ r , and proceed in two steps (Steps1 and 2). Note that Steps A, B, and C lead to a function b ( x , t ) := − η ln (cid:0) (cid:80) pl =1 o l ( t ) exp( − η b l ( x , t )) (cid:1) where each b l ( x , t ) is associated either with an eventually ( F [ a l ,b l ] µ l ) oran always ( G [ a l ,b l ] µ l ) formula. Recall that an until operatoris encoded in Steps A, B, and C as the conjunction of analways and an eventually operator. We present in Step 1 howto construct b ( x , t ) when φ := F [ a l ,b l ] µ l or φ := G [ a l ,b l ] µ l where µ l does not contain any conjunctions, i.e., p = 1 . InStep 2, we explain how to construct b ( x , t ) in the more generalcase when φ contains conjunctions, i.e., p > . Step 1)
Consider φ := G [ a l ,b l ] µ l or φ := F [ a l ,b l ] µ l and let t ∗ l := (cid:40) b l if F [ a l ,b l ] µ l a l if G [ a l ,b l ] µ l , (10)which reflects the requirement that µ l has to hold at least oncebetween [ a l , b l ] for F [ a l ,b l ] µ l (here this time instant is chosento be t ∗ l := b l ) or at all times within [ a l , b l ] for G [ a l ,b l ] µ l (indicated by t ∗ l := a l ). It is assumed that b l > . Otherwise,i.e., b l = 0 , satisfaction of φ would purely depend on theinitial condition of the system. Next, choose r ∈ (cid:40) (0 , h opt l ) if t ∗ l > , h l ( x (0))] if t ∗ l = 0 where the second case is explained as follows: if h l ( x (0)) < r and t ∗ l = 0 , there does not exist a signal x : R ≥ → R n withan initial condition x (0) such that ρ φ ( x , ≥ r . Let now b l ( x , t ) := − γ l ( t ) + h l ( x ) . t h l ( x ( t ) , t ) γ l ( t )0 1 2 3 4 5 6 7 8 9 101 − − Fig. 1: The functions γ l ( t ) (dashed line) and h l ( x ( t ) , t ) (solidline) for φ := G [7 . , ( (cid:107) x (cid:107) < with r := 0 . and acandidate trajectory x : R ≥ → R n satisfying φ .In [37], γ l ( t ) is an exponential function. The drawback is,from a practical point of view, that larger control inputs mayoccur as compared to the case where γ l ( t ) is a linear function.We aim to avoid this and define the piecewise linear function γ l ( t ) := (cid:40) γ l, ∞ − γ l, t ∗ l t + γ l, if t < t ∗ l γ l, ∞ otherwise.The switching sequence is now { s = 0 , s = b l } since p = 1 and it holds that γ l ( t ) is continuous on ( s , s ) . Weremark that γ l ( t ) is continuously differentiable on ( s , s ) if φ = F [ a l ,b l ] µ l , while γ l ( t ) is only piecewise continuouslydifferentiable on ( s , s ) if φ = G [ a l ,b l ] µ l . In fact, in thelatter case, γ l ( t ) is only continuously differentiable on ( s , t ∗ l ) and on ( t ∗ l , s ) . This, however, does not affect the theoreticalresults derived in Corollary 1 and Theorem 2. To see this, notethat [ s , s ) = [ s , t ∗ l ) ∪ [ t ∗ l , s ) and consider the modifiedswitching sequence { ¯ s := s , ¯ s = t ∗ l , ¯ s := s } . Now, thesame guarantees given in Corollary 1 and Theorem 2 applyfor b l ( x , t ) under the modified switching sequence. Next, let γ l, ∈ (cid:0) − ∞ , h l ( x (0)) (cid:1) (11a) γ l, ∞ ∈ (cid:0) max( r, γ l, ) , h opt l (cid:1) (11b)so that ≤ b l ( x (0) , and b l ( x (0) , ≤ h l ( x (0)) − r if t ∗ l = 0 so that a satisfaction with a robustness of r ispossible. By the choice of γ l, ∞ it is ensured that b l ( x ( t (cid:48) ) , t (cid:48) ) ≤ h l ( x ( t (cid:48) )) − r for all t (cid:48) ≥ t ∗ l . Hence, if now b l ( x ( t (cid:48) ) , t (cid:48) ) ≥ for all t (cid:48) ≥ t ∗ l , then it follows that h l ( x ( t (cid:48) )) − r ≥ , whichimplies h l ( x ( t (cid:48) )) ≥ r leading to ρ φ ( x , ≥ r by the choiceof t ∗ l and r . We note that γ l ( t ) is a non-decreasing function.By these construction rules, it is straightforward to concludethat b l ( x , t ) is a cCBF for [¯ s , ¯ s ) and for [¯ s , ¯ s ) . Example 1:
Consider the formula φ := G [7 . , ( (cid:107) x (cid:107) < that yields h l ( x ) := 5 − (cid:107) x (cid:107) and h opt l = 5 so that wecan choose r := 0 . . Assume the initial condition x (0) := (cid:2) (cid:3) T so that h l ( x (0)) = − . . We select t ∗ l := 7 . , γ l, := − . , and γ l, ∞ := 0 . in accordance with (10) and(11). Recall that b l ( x , t ) := − γ l ( t ) + h l ( x ) and note that b l ( x ( t ) , t ) ≥ for all t ≥ is equivalent to h l ( x ( t )) ≥ γ l ( t ) for all t ≥ . This leads to ρ φ ( x , > r , i.e., ( x , | = φ , bythe construction of γ l ( t ) as illustrated in Fig. 1. Step 2)
For p > , a more elaborate procedure is needed.Recall that b ( x , t ) := − η ln (cid:0) (cid:80) pl =1 o l ( t ) exp( − η b l ( x , t )) (cid:1) . Let, similarly to Step 1, b l ( x , t ) := − γ l ( t ) + h l ( x ) with γ l ( t ) according to (11). We further pose the following assumption. Assumption 4:
Each predicate function contained in φ ,denoted by h l ( x ) : R n → R with l ∈ { , . . . , p } , is concave.Concave predicate functions h l ( x ) contain the class oflinear functions as well as functions that express, for instance,reachability tasks using predicates such as (cid:107) x − p (cid:107) ≤ (cid:15) for p ∈ R n and (cid:15) ∈ R ≥ . Assumption 4 is needed to formallyshow that b ( x , t ) is a cCBF and a vCBF (Lemmas 3 and 4)relying on the fact that b ( x , t (cid:48) ) is concave in x as proven next. Lemma 2:
Let Assumption 4 hold. Then, for a fixed t (cid:48) , b ( x , t (cid:48) ) is concave. Proof:
For a fixed t (cid:48) , η b l ( x , t (cid:48) ) is concave. Due to [44,Sec. 3.5] it holds that exp( − η b l ( x , t (cid:48) )) is log-convex. It alsoholds that a sum of log-convex functions is log-convex. Hence, − η ln (cid:0) (cid:80) pl =1 o l ( t ) exp( − η b l ( x , t (cid:48) )) (cid:1) is concave.Compared to Step 1, it is now not enough to select γ l, as in(11a) to ensure b ( x (0) , ≥ due to (2). To see this, consider b ( x , t ) := − η ln (cid:0) exp( − η b ( x , t )) + exp( − η b ( x , t )) (cid:1) . If b ( x (0) , > and b ( x (0) , > (which is both ensuredby (11a)), then it does not neccessarily hold that b ( x (0) , ≥ depending on the value of η . Therefore, η now needs tobe selected sufficiently large. Note again that increasing η increases the accuracy of the approximation used for conjunc-tions. More importantly, γ l, ∞ , which has to be selected accord-ing to (11b), and r need to be selected so that for all t ∈ [ s , s q ] there exists x ∈ R n so that b ( x , t ) ≥ . Define next γ := (cid:2) γ , . . . γ p, (cid:3) T and γ ∞ := (cid:2) γ , ∞ . . . γ p, ∞ (cid:3) T thatcontain the parameters γ l, and γ l, ∞ for each eventually- andalways-operator encoded in b l ( x , t ) . Let ξ , . . . , ξ q ∈ R n anddefine ξ := (cid:2) ξ T . . . ξ qT (cid:3) T . As argued in Section III-A, C ( t ) := { x ∈ R n | b ( x , t ) ≥ } needs to be compact. This isrealized by including b p +1 ( x , t ) := D −(cid:107) x (cid:107) and o p +1 ( t ) := 1 into b ( x , t ) := − η ln (cid:0) (cid:80) p +1 l =1 o l ( t ) exp( − η b l ( x , t )) (cid:1) for asuitably selected D . Select η , r , D , γ , and γ ∞ accordingto the solution of the following optimization problem argmax η,r,D, γ , γ ∞ , ξ r (12a)s.t. b ( x (0) , ≥ χ (12b) lim τ → s − j b ( ξ j , τ ) ≥ χ for each j ∈ { , . . . , q } (12c) γ l, as in (11a) for each l ∈ { , . . . , p } (12d) γ l, ∞ as in (11b) for each l ∈ { , . . . , p } (12e) η > and r > and D > . (12f)where χ ≥ is a given parameter. Note that lim τ → s − j b ( ξ j , τ ) can easily be evaluated since o l ( t ) is piecewise continuous. Remark 5:
The optimization problem (12) is nonconvex.An MILP formulation such as in [17] provides, for discrete-time systems, an open-loop control sequence that needs to beiteratively solved online in order to get a feedback control law.We obtain in (12), which can be solved offline, a control barrierfunction that can be used, in a provably correct manner, toobtain a continuous feedback control law as in (9). Comparedto [37], we observed faster computation times due to the use ofpiecewise linear functions γ l ( t ) instead of exponential ones. If maximization of r is not of interest, then a feasibility programwith the constraints in (12b)-(12f) can be solved instead.Denote the modified switching sequence by { ¯ s :=0 , ¯ s , . . . , ¯ s ¯ q = s q } where ¯ q denotes the number of switcheswith ¯ q ≥ q . More formally, let t := { a l , . . . , a l ¯ p , b , . . . , b p } where, for a l ∈ { a l , . . . , a l ¯ p } , the corresponding b l ( x , t ) en-codes an always operator, i.e., ¯ p denotes the number of b l ( x , t ) in b ( x , t ) that encode an always operator. At time t ≥ ¯ s j , wedefine ¯ s j +1 := argmin t ∗ ∈ t ζ ( t ∗ , t ) with ζ ( t ∗ , t ) := t ∗ − t if t ∗ − t > and ζ ( t ∗ , t ) := ∞ otherwise. We now show that b ( x , t ) is a cCBF for each [¯ s j , ¯ s j +1 ) . Lemma 3:
Let Assumption 4 hold. Then the function b ( x , t ) obtained by the solution of (12) is a cCBF for each [¯ s j , ¯ s j +1 ) . Proof:
Feasibility of (12) implies that C ( t ) is non-emptyfor all t ∈ [ s , s q ] . This follows due to (12b), (12c), and since b ( x , t ) is non-increasing in t for all t ∈ [ s j , s j +1 ) by (12d)-(12e), which implies C ( t ) ⊇ C ( t ) for s j ≤ t < t < s j +1 .For b ( x , t ) to be a cCBF for [¯ s j , ¯ s j +1 ) , there needs to exist anabsolutely continuous function x : [¯ s j , ¯ s j +1 ) → R n for each x ( s j ) ∈ C ( s j ) such that x ( t ) ∈ C ( t ) for all t ∈ [¯ s j , ¯ s j +1 ) .Since b ( x , t ) is concave in x for each fixed t , it holds that allsuperlevel sets of b ( x , t ) are convex [44, Sec. 3.1.6] and hence C ( t ) is connected. Since ∂ b ( x ,t ) ∂t is finite, the existence of anabsolutely continuous function x : [¯ s j , ¯ s j +1 ) → R n such that b ( x ( t ) , t ) ≥ for all t ∈ [¯ s j , ¯ s j +1 ) follows.Lemma 3 has shown that b ( x , t ) is a cCBF, while we nextshow that α can be selected such that b ( x , t ) is a vCBF. Lemma 4:
Assume that (12) is solved for χ > , then α can be selected such that b ( x , t ) satisfies Assumption 3. Proof:
Concavity of b ( x , t ) in x implies that, for each t (cid:48) ∈ [ s j , s j +1 ) , x ∗ t (cid:48) := argmax x ∈ R n b ( x , t (cid:48) ) is such that x ∗ t (cid:48) ∈ C ( t (cid:48) ) (recall that χ > ) with b ( x ∗ t (cid:48) , t (cid:48) ) > b ( x , t (cid:48) ) for all x (cid:54) = x ∗ t (cid:48) . Furthermore, ∂ b ( x (cid:48) ,t (cid:48) ) ∂ x = if and only if x (cid:48) := x ∗ t (cid:48) . It holds that b ( x ∗ t (cid:48) , t (cid:48) ) ≥ χ > for each t (cid:48) ∈ [ s , s q ] due to (12b) and (12c) so that b l ( x ∗ t (cid:48) , t (cid:48) ) ≥ χ > for each l ∈ { , . . . , p + 1 } with o l ( t (cid:48) ) = 1 . Next, note that thereexists a constant b max l for each l ∈ { , . . . , p + 1 } such that b l ( x ∗ t (cid:48) , t (cid:48) ) ≤ b max l for each t (cid:48) ∈ [ s , s q ] due to continuity of b l ( x , t ) on D × [ s , s q ] . Let b max := max( b max , . . . , b max p +1 ) sothat max( b ( x ∗ t (cid:48) , t (cid:48) ) , . . . , b p +1 ( x ∗ t (cid:48) , t (cid:48) )) ≤ b max and let ∆ l :=sup t ≥ | ∂ b l ( x ,t ) ∂t | = γ l, ∞ − γ l, t ∗ l . Hence, it follows that ∂ b ( x ∗ t (cid:48) , t (cid:48) ) ∂t = (cid:80) p +1 l =1 o l ( t (cid:48) ) exp( − η b l ( x ∗ t (cid:48) , t (cid:48) )) ∂ b l ( x ∗ t (cid:48) ,t (cid:48) ) ∂t (cid:80) p +1 l =1 o l ( t (cid:48) ) exp( − η b l ( x ∗ t (cid:48) , t (cid:48) ))= − (cid:80) p +1 l =1 exp( − η b l ( x ∗ t (cid:48) , t (cid:48) )) | ∂ b l ( x ∗ t (cid:48) ,t (cid:48) ) ∂t | (cid:80) p +1 l =1 exp( − η b l ( x ∗ t (cid:48) , t (cid:48) )) ≥ − exp( − ηχ )∆ l exp( − η b max ) =: ζ. where ζ is negative. If it is now guaranteed that ζ ≥ − α ( χ )+ (cid:15) ,it holds that ∂ b ( x ∗ t (cid:48) ,t (cid:48) ) ∂t ≥ − α ( b ( x ∗ t (cid:48) , t (cid:48) ) + (cid:15) for all t (cid:48) ∈ [ s , s q ] so that Assumption 3 holds. By the specific choice of α ( χ ) = κχ , we can select κ ≥ (cid:15) − ζχ such that this is the case.The intuition behind Lemma 4 is that χ > ensures that b ( x , t ) ≥ χ if ∂ b ( x ,t ) ∂ x = and that then choosing κ in α ( χ ) = κχ large enough guarantees that Assumption 3 holds.We combine the results from Sections III-B and IV. Theorem 3:
Consider the same assumptions as in The-orem 2. If each φ k additionally satisfies Assumption 4, ¯ b k (¯ x k , t ) is the solution of (12) for χ > , and α k ( χ ) := κχ with κ > (cid:15) − ζχ , then ρ φ k (¯ x k , ≥ r k > where r k is obtainedby the solution of (12) for each k ∈ { , . . . , K } . Proof:
Follows by Theorem 2 and Lemmas 3 and 4.V. E
XPERIMENTS
We consider three Nexus 4WD Mecanum Robotic Cars,which are equipped with low-level PID controllers that tracktranslational and rotational velocity commands. The state ofrobot i is x i := (cid:2) x i y i θ i (cid:3) T where p i := (cid:2) x i y i (cid:3) T denotes the two dimensional position while θ i denotes theorientation. For simplicity, we here assume that all statesare given in a global coordinate frame. Conversion fromlocal to global coordinate frames is performed by each robotwhere the local information is obtained by means of a mo-tion capture system. The considered dynamics are given by ˙ x i = u i + f u i ( x , t )+ c i ( x , t ) where f u i ( x , t ) describes induceddynamical couplings as discussed in Remark 1, here usedfor the purpose of collision avoidance. In particular, f u i ( x , t ) is a potential field inducing a repulsive force between tworobots when the distance between them is below . meters; c i ( x , t ) models disturbances such as those induced by thedigital implementation of the continuous-time control law orinaccuracies in the low-level PID controllers with C := 2 . Therobots are subject to φ := φ (cid:48) ∧ φ (cid:48)(cid:48) ∧ φ (cid:48)(cid:48)(cid:48) ∧ φ (cid:48)(cid:48)(cid:48)(cid:48) with φ (cid:48) := G [15 , ( (cid:107) p + p x − p (cid:107) ≤ (cid:15) ) φ (cid:48)(cid:48) := G [25 , ( (cid:107) p + p y − p (cid:107) ≤ (cid:15) ) ∧ F [30 , ( (cid:107) p − p B (cid:107) ≤ (cid:15) ) φ (cid:48)(cid:48)(cid:48) := F [40 , ( (cid:107) p − p C (cid:107) ≤ (cid:15) ) φ (cid:48)(cid:48)(cid:48)(cid:48) := F [50 , (cid:0) ( (cid:107) p − p A (cid:107) ≤ (cid:15) ) ∧ ( (cid:107) p + p x − p (cid:107) ≤ (cid:15) ) (cid:1) where (cid:15) := 0 . , p A := (cid:2) − . . (cid:3) T , p B := (cid:2) . . (cid:3) T , p C := (cid:2) . − . (cid:3) T , p x := (cid:2) . (cid:3) T , p y := (cid:2) − . (cid:3) T .The software implementation is available under [45] (alsoincluding a detailed description of f u i ( x , t ) ), written in C++ ,and embedded in the
Robot Operating System (ROS) [46].The quadratic program (9) is solved using
CVXGEN [47] ata frequency of Hz; b ( x , t ) corresponding to φ is obtainedoffline and in MATLAB by solving (12) using
YALMIP [48]with the ’fmincon option’. The calculation of b ( x , t ) took . seconds on an Intel Core i7-6600U with GB of RAMwithout maximizing r . In fact, increased oscillations in thecontrol input were observed when we decided to maximize r .The experimental result is shown in Figs. 2-4 as well as in[49] where we provide a video of the experiment. To illustrateRemark 4, we have intentionally chosen an initial condition x (0) of the robots that does not coincide with the initialcondition x (0) := used in (12) to construct b ( x , t ) . In Fig. 2,it is hence visible that initially b ( x (0) , ≈ − . . However,after approximately t ≈ sec, it holds that b ( x ( t ) , t ) ≥ and the robots have recovered from this situation. This is,in particular, a strength compared to our previous approach[23] where the control law would have not been definedin case of such a mismatch. Furthemore, Fig. 2 shows that Fig. 2: Barrier function evolution b ( x ( t ) , t ) . b ( x ( t ) , t ) ≥ for the rest of the experiment so that it can beconcluded that ( x , | = φ or, to be more precise, ρ φ ( x , ≥ r where r = 0 . was obtained by the solution of (12). Fig. 3shows the corresponding robot trajectories. As emphasized inSection III-B, the control law u i ( x , t ) is discontinuous. Thisis shown by plotting the x and y component of u ( x , t ) inFig. 4. We intentionally avoided to use an additional filteron u i ( x , t ) to smoothen the control input in order to showthe nature of the discontinuous control law. The low-levelPID controllers, however, filter u i ( x , t ) when applied to themotors of the robots. We further remark that using linearfunctions γ l ( t ) to construct b ( x , t ) as introduced in SectionIV compared to exponential ones as presented in [37] isbeneficial since input saturations are less likely to occur. Anexponential function γ l ( t ) would, for some t , induce highcontrol inputs, while for other t nearly no control actionwould be needed. A linear function γ l ( t ) distributes the neededcontrol action more uniformly over time and is hence moresuited for experiments. Finally, note that collisions are avoidedby the use of f u i ( x , t ) , especially in the first seconds where acollision would occur between robot and without induceddynamical couplings in f u i ( x , t ) . We remark that approachessuch as [17] are not applicable here. First of all, the inducedcomputational complexity does not allow to obtain the solutionto a mixed linear program in reasonable time; [17] alsodoes not allow for nonlinear predicate functions as requiredby φ . Existing approaches work with discrete-time systems.We, however, directly consider continuous-time systems andprovide continuous-time satisfaction guarantees.VI. C ONCLUSION
We have proposed a collaborative feedback control strategyfor dynamically coupled multi-agent systems under a set ofsignal temporal logic tasks. For each agent, we have firstderived a collaborative decentralized feedback control lawthat guarantees the satisfaction of all tasks. This control lawis discontinuous, hence Filippov solutions and nonsmoothanalysis is used, and based on the existence of a controlbarrier function that accounts for the semantics of the signaltemporal logic task at hand. We have then presented how a Fig. 3: Robot trajectories. The increasing color occupancyindicates the evolution of the robots as time progresses.Fig. 4: The x and y components u x and u y of u ( x , t ) plottedover time. Discontinuities and chattering are visible.control barrier function can be constructed for a fragmentof signal temporal logic tasks by solving an optimizationproblem. Finally, we have validated our theoretical results inan experiment including three omnidirectional robots.A PPENDIX P ROOF OF T HEOREM u i (¯ x k , t ) , and thatCorollary 1 can be applied. According to the assumptions, ¯ b k (¯ x k , t ) is a cCBF for each time interval [ s kj , s kj +1 ) and ¯ b k (¯ x k , t ) is again piecewise continuous in t . As argued inthe proof of Corollary 1, it is hence sufficient to look at eachtime interval [ s kj , s kj +1 ) separately. Next, define B kj,i := (cid:110) (¯ x k , t ) ∈ D k × ( s kj , s kj +1 ) | ∂ ¯ b k (¯ x k , t ) ∂ x i = (cid:111) \ B kj ¯ B kj,i := (cid:110) (¯ x k , t ) ∈ D k × ( s kj , s kj +1 ) | ∂ ¯ b k (¯ x k , t ) ∂ x i (cid:54) = (cid:111) We remark that B kj ∪ B kj,i ∪ ¯ B kj,i = D k × ( s kj , s kj +1 ) andthat B kj , B kj,i , and ¯ B kj,i are disjoint sets. To understand the details of the following proof, note that B kj and B kj,i can not beclosed sets (note that ( s kj , s kj +1 ) is open) and that informationregarding these sets being open or not is not available. Wewill, however, show and use the fact that ¯ B kj,i is open. Part 1 - Feasibility of (9) : We next show that (9) is alwaysfeasible and distinguish between three cases indicated by B kj , B kj,i , and ¯ B kj,i . It will turn out that u i (¯ x k , t ) may bediscontinuous on the boundaries of B kj , B kj,i , and ¯ B kj,i .Case 1 applies when (¯ x k , t ) ∈ B kj . This is equivalentto (¯ x k , t ) ∈ D k × ( s kj , s kj +1 ) such that ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k = (which is equivalent to (cid:80) v ∈V k (cid:107) ∂ ¯ b k (¯ x k ,t ) ∂ x v (cid:107) = 0 ) and im-plies ∂ ¯ b k (¯ x k ,t ) ∂ x i = ; (9b) reduces to ω k (¯ x k , t ) ≥ since D i (¯ x k , t ) = 1 so that (9b) is satisfied due to Assumption3. Hence u i (¯ x k , t ) = is the optimal solution to (9).Case 2 applies when (¯ x k , t ) ∈ B kj,i . This is equivalent to (¯ x k , t ) ∈ D k × ( s kj , s kj +1 ) such that ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k (cid:54) = (which isequivalent to (cid:80) v ∈V k (cid:107) ∂ ¯ b k (¯ x k ,t ) ∂ x v (cid:107) (cid:54) = 0 ) and ∂ ¯ b k (¯ x k ,t ) ∂ x i = .The optimal solution to (9) is again u i (¯ x k , t ) = since (9b)is trivially satisfied (note that D i (¯ x k , t ) = 0 ).Case 3 applies when (¯ x k , t ) ∈ ¯ B kj,i . This is equivalentto (¯ x k , t ) ∈ D k × ( s kj , s kj +1 ) such that ∂ ¯ b k (¯ x k ,t ) ∂ x i (cid:54) = sothat (9) is feasible. Note again that ∂ ¯ b k (¯ x k ,t ) ∂ x i (cid:54) = im-plies ∂ ¯ b k (¯ x k ,t ) ∂ x i g i ( x i , t ) (cid:54) = ; u i (¯ x k , t ) is locally Lipschitzcontinuous on int ( ¯ B kj,i ) where int ( · ) denotes the interior ofa set. This follows by virtue of [27, Thm. 3] and sinceall functions in (9) are locally Lipschitz continuous onint ( ¯ B kj,i ) . In particular, D i (¯ x k , t ) is locally Lipschitz continu-ous on int ( B kj,i ) ∪ int ( ¯ B kj,i ) and ∂ ¯ b k (¯ x k ,t ) ∂ x i , ∂ ¯ b k (¯ x k ,t ) ∂t , f i ( x i , t ) , g i ( x i , t ) , and α k (¯ b k (¯ x k , t )) are locally Lipschitz continuouson int ( B kj ) ∪ int ( B kj,i ) ∪ int ( ¯ B kj,i ) .The optimization problem (9) is hence always feasible and u i (¯ x k , t ) is locally Lipschitz continuous on int ( B kj ) , int ( B kj,i ) ,and int ( ¯ B kj,i ) . Part 2 - Existence of Filippov Solutions to (3) under u i (¯ x k , t ) : The control law u i (¯ x k , t ) may, as indicated above,be discontinuous; u i (¯ x k , t ) is, however, locally bounded andmeasurable on D k × ( s kj , s kj +1 ) as argued next. In particu-lar, we already know that u i (¯ x k , t ) is locally bounded onint ( B kj ) , int ( B kj,i ) , and int ( ¯ B kj,i ) due to being locally Lipschitzcontinuous on these domains. If we ensure that u i (¯ x k , t ) isalso locally bounded on the boundaries of B kj , B kj,i , and ¯ B kj,i , we can conclude that u i (¯ x k , t ) is locally bounded on D k × ( s kj , s kj +1 ) . Therefore, we next systematically investigatethe cases where (¯ x k , t ) is in { bd ( B kj ) ∩ bd ( B kj,i ) } \ bd ( ¯ B kj,i ) (Cases 1 or 2), { bd ( B kj ) ∩ bd ( ¯ B kj,i ) } \ bd ( B kj,i ) (Cases 1or 3), { bd ( B kj,i ) ∩ bd ( ¯ B kj,i ) } \ bd ( B kj ) (Cases 2 or 3), andbd ( B kj ) ∩ bd ( B kj,i ) ∩ bd ( ¯ B kj,i ) (Cases 1, 2, or 3) where bd ( · ) denotes the boundary of a set.When (¯ x k , t ) ∈ { bd ( B kj ) ∩ bd ( B kj,i ) } \ bd ( ¯ B kj,i ) , either (¯ x k , t ) ∈ B kj or (¯ x k , t ) ∈ B kj,i . Either way, due to continuity(recall that u i (¯ x i , t ) = in Case 1 and 2) there exists a neigh-borhood U ⊆ { B kj ∪ B kj,i } \ ¯ B kj,i around (¯ x k , t ) so that, foreach (¯ x (cid:48) k , t (cid:48) ) ∈ U , (cid:107) u i (¯ x (cid:48) k , t (cid:48) ) (cid:107) = . Consequently, u i (¯ x k , t ) is locally bounded on { bd ( B kj ) ∩ bd ( B kj,i ) } \ bd ( ¯ B kj,i ) .When (¯ x k , t ) ∈ { bd ( B kj ) ∩ bd ( ¯ B kj,i ) } \ bd ( B kj,i ) , either (¯ x k , t ) ∈ B kj or (¯ x k , t ) ∈ ¯ B kj,i . Note that ω k (¯ x k , t ) ≥ (cid:15) k if (¯ x k , t ) ∈ B kj due to Assumption 3. Recall also that ω k (¯ x k , t ) is continuous on D k × ( s kj , s kj +1 ) . By the definitionof continuity it follows that for a given (cid:15) k > (in this case,the (cid:15) k from Assumption 3) there exists a δ k > so thatfor each (¯ x (cid:48) k , t (cid:48) ) with (cid:107) (cid:2) ¯ x (cid:48) Tk t (cid:48) (cid:3) T − (cid:2) ¯ x Tk t (cid:3) T (cid:107) < δ k itholds that ω k (¯ x k , t ) − (cid:15) k < ω k (¯ x (cid:48) k , t (cid:48) ) < ω k (¯ x k , t ) + (cid:15) k so that consequently ω k (¯ x (cid:48) k , t (cid:48) ) ≥ . Hence, there exists aneighborhood U ⊆ { B kj ∪ ¯ B kj,i } \ B kj,i around (¯ x k , t ) sothat, for each (¯ x (cid:48) k , t (cid:48) ) ∈ U , either (cid:107) u i (¯ x (cid:48) k , t (cid:48) ) (cid:107) = (if (¯ x (cid:48) k , t (cid:48) ) ∈ B kj ∩ U ) or ω k (¯ x (cid:48) k , t (cid:48) ) ≥ (if (¯ x (cid:48) k , t (cid:48) ) ∈ ¯ B kj,i ∩ U ).For the latter case, i.e., (¯ x (cid:48) k , t (cid:48) ) ∈ ¯ B kj,i ∩U , note that a feasible(not necessarily optimal) and analytic control law for (9b) is u feas i (¯ x (cid:48) k , t (cid:48) ) := g i ( x (cid:48) i , t (cid:48) ) T G i ( x (cid:48) i , t (cid:48) ) − ( − f i ( x (cid:48) i , t (cid:48) ) + v feas i ) where x (cid:48) i is the corresponding element in ¯ x (cid:48) k , v feas i is explainedin the remainder, and where the inverse of G i ( x (cid:48) i , t (cid:48) ) := g i ( x (cid:48) i , t (cid:48) ) g i ( x (cid:48) i , t (cid:48) ) T exists due to Assumption 1 so that (cid:107) u feas i (¯ x (cid:48) k , t (cid:48) ) (cid:107) ≤ C g i C G i ( C f i + (cid:107) v feas i (cid:107) ) where C g i , C G i , and C f i are upper bounds on (cid:107) g i ( x i , t ) (cid:107) , (cid:107) G i ( x i , t ) (cid:107) , and (cid:107) f i ( x i , t ) (cid:107) that follow due to continuityof g i ( x i , t ) , G i ( x i , t ) , and f i ( x i , t ) on the bounded domain D k . Note especially that G i ( x i , t ) is upper bounded by theinverse of the smallest singular value of G i ( x i , t ) when themax matrix norm is used [50, Ch.5.6]. We next show howto select v feas i and that (cid:107) v feas i (cid:107) is also upper bounded. Using u feas i (¯ x (cid:48) k , t (cid:48) ) , (9b) reduces to ∂ ¯ b k (¯ x (cid:48) k , t (cid:48) ) ∂ x i v feas i ≥ (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x k , t ) ∂ x i (cid:13)(cid:13)(cid:13) C − D i (¯ x (cid:48) k , t (cid:48) ) ω k (¯ x (cid:48) k , t (cid:48) ) . (13)We select v feas i (¯ x (cid:48) k , t (cid:48) ) := sgn (cid:0) ∂ ¯ b k (¯ x (cid:48) k ,t (cid:48) ) ∂ x i (cid:1) T κ i where sgn ( · ) isthe element-wise sign operator so that (13) becomes (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x (cid:48) k , t (cid:48) ) ∂ x i (cid:13)(cid:13)(cid:13) (cid:16) κ i − C + ω k (¯ x (cid:48) k , t (cid:48) ) (cid:80) v ∈V k (cid:107) ∂ ¯ b k (¯ x (cid:48) k ,t (cid:48) ) ∂ x v (cid:107) (cid:17) ≥ . (14)In particular, it holds that (14) is satisfied if κ i := C (recall that ω k (¯ x (cid:48) k , t (cid:48) ) ≥ if (¯ x (cid:48) k , t (cid:48) ) ∈ ¯ B kj,i ∩ U ) sothat (cid:107) v feas i (¯ x (cid:48) k , t (cid:48) ) (cid:107) ≤ C . Consequently, (cid:107) u i (¯ x (cid:48) k , t (cid:48) ) (cid:107) ≤(cid:107) u feas i (¯ x (cid:48) k , t (cid:48) ) (cid:107) ≤ C g i C G i ( C f i + C ) and u i (¯ x k , t ) is locallybounded on { bd ( B kj ) ∩ bd ( ¯ B kj,i ) } \ bd ( B kj,i ) .When (¯ x k , t ) ∈ { bd ( B kj,i ) ∩ bd ( ¯ B kj,i ) } \ bd ( B kj ) , either (¯ x k , t ) ∈ B kj,i or (¯ x k , t ) ∈ ¯ B kj,i and a similar analysis can bemade as above. In particular, then there exists a neighborhood U ⊆ { B kj,i ∪ ¯ B kj,i } \ B kj around (¯ x k , t ) so that, for each (¯ x (cid:48) k , t (cid:48) ) ∈ U , either (cid:107) u i (¯ x (cid:48) k , t (cid:48) ) (cid:107) = (if (¯ x (cid:48) k , t (cid:48) ) ∈ B kj,i ∩ U )or (cid:80) v ∈V k (cid:107) ∂ ¯ b k (¯ x (cid:48) k ,t (cid:48) ) ∂ x v (cid:107) ≥ ν for some ν > (if (¯ x (cid:48) k , t (cid:48) ) ∈ ¯ B kj,i ∩ U ) since (¯ x (cid:48) k , t (cid:48) ) / ∈ B kj and again due to continuity. Inthe latter case, selecting κ i := C − ω k (¯ x (cid:48) k ,t (cid:48) ) ν satisfies (14). Thesame arguments as before then show that u i (¯ x k , t ) is locallybounded on { bd ( B kj,i ) ∩ bd ( ¯ B kj,i ) } \ bd ( B kj ) .When (¯ x k , t ) ∈ bd ( B kj ) ∩ bd ( B kj ) ∩ bd ( ¯ B kj,i ) , it can againbe shown that u i (¯ x k , t ) is locally bounded on bd ( B kj ) ∩ bd ( B kj,i ) ∩ bd ( ¯ B kj,i ) . The proof is straighforward using thesame arguments as in the previous discussion and omitted.It follows that u i (¯ x k , t ) is locally bounded on bd ( B kj ) ,bd ( B kj,i ) , and bd ( ¯ B kj,i ) . Since we have already concluded thatthe same holds on int ( B kj ) , int ( B kj,i ) , and int ( ¯ B kj,i ) , u i (¯ x k , t ) is consequently locally bounded on B kj ∪ B kj,i ∪ ¯ B kj,i = D k × ( s kj , s kj +1 ) . To see that u i (¯ x k , t ) is measureable, notethat B kj , B kj,i , and ¯ B kj,i are measurable sets. The product ofmeasurable functions is measurable and the indicator function(here used to indicate Cases 1, 2, and 3) defined on mea-surable sets is measurable so that u i (¯ x k , t ) is measurable.Consequently, the multi-agent system described by the stackeddynamics of each agent in (3) admits Filippov solutions x : [ t , t ] → R n from each initial condition in D × R ≥ where D := D j × . . . × D j |V k | for j , . . . , j |V k | ∈ V k . Part 3 - Application of Corollary 1:
For each ¯ b k (¯ x k , t ) , theindividual solutions u i (¯ x k , t ) to (9) result in (cid:88) i ∈V k ∂ ¯ b k (¯ x k , t ) ∂ x i ( f i ( x i , t ) + g i ( x i , t ) u i (¯ x k , t )) ≥ (cid:88) i ∈V k (cid:16) − D i (¯ x k , t ) ω k (¯ x k , t ) + (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x k , t ) ∂ x i (cid:13)(cid:13)(cid:13) C (cid:17) ⇔ ∂ ¯ b k (¯ x k , t ) ∂ ¯ x k ( ¯ f k (¯ x k , t ) + ¯ g k (¯ x k , t )¯ u k (¯ x k , t )) ≥− ω k (¯ x k , t ) + (cid:88) i ∈V k (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x k , t ) ∂ x i (cid:13)(cid:13)(cid:13) C ⇔ ∂ ¯ b k (¯ x k , t ) ∂ ¯ x k ( ¯ f k (¯ x k , t ) + ¯ g k (¯ x k , t )¯ u k (¯ x k , t )) ≥− ω k (¯ x k , t ) + (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x k , t ) ∂ ¯ x k (cid:13)(cid:13)(cid:13) C (15)where the last equivalence follows by the definition of the sumnorm.In our analysis below, it is crucial to note that ¯ B kj,i is open,which we show next. Denote by inv (cid:16) ∂ ¯ b k (¯ x k ,t ) ∂ x i ( O n i ) (cid:17) theinverse image of ∂ ¯ b k (¯ x k ,t ) ∂ x i under O n i where O := ( −∞ , ∪ (0 , ∞ ) . Now, inv (cid:16) ∂ ¯ b k (¯ x k ,t ) ∂ x i ( O n i ) (cid:17) is open since O n i is openand since the inverse image of a continuous function under anopen set is open [51, Prop. 1.4.4]. It then holds that ¯ B kj,i = { D k × ( s kj , s kj +1 ) } ∩ inv (cid:16) ∂ ¯ b k (¯ x k , t ) ∂ x i ( O n i ) (cid:17) is open since the intersection of open sets is open.We next show that (15) implies (6) so that Corollary 1can be applied for each ¯ b k (¯ x k , t ) . Since ¯ f k (¯ x k , t ) is lo-cally Lipschitz continuous, it follows that ˆ L F [ ¯ f k ] ¯ b k (¯ x k , t ) := (cid:8) ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k ¯ f k (¯ x k , t ) (cid:9) . For ˆ L F [¯ g k ¯ u k ] ¯ b k (¯ x k , t ) , we have to dis-tinguish between the aforementioned three cases. First notethat, if for each i ∈ V k we have (¯ x k , t ) ∈ ¯ B kj,i (Case3), then ˆ L F [¯ g k ¯ u k ] ¯ b k (¯ x k , t ) = (cid:8) ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k ¯ g k (¯ x k , t )¯ u k (¯ x k , t ) (cid:9) .This in particular follows since ¯ B kj,i is open so that u i (¯ x k , t ) as well as ¯ g k (¯ x k , t ) are locally Lipschitz continuous onint ( ¯ B kj,i ) = ¯ B kj . If for each i ∈ V k we have (¯ x k , t ) ∈ B kj,i (Case 2), then ˆ L F [¯ g k ¯ u k ] ¯ b k (¯ x k , t ) = { } since ∂ ¯ b k (¯ x k ,t ) ∂ x i = for each i ∈ V k . If for some agents (¯ x k , t ) ∈ ¯ B kj,i whilefor others (¯ x k , t ) ∈ B kj,i (i.e., a mix of Case 2 and Case 3),the resulting ˆ L F [¯ g k ¯ u k ] ¯ b k (¯ x k , t ) will still be a singleton. If wehave (¯ x k , t ) ∈ B kj (Case 1), then ˆ L F [¯ g k ¯ u k ] ¯ b k (¯ x k , t ) = { } since ∂ ¯ b k (¯ x k ,t ) ∂ ¯ x k = . Note that min S = S when S is asingleton and recall ω k (¯ x k , t ) in (8). Since ˆ L F [ ¯ f k ] ¯ b k (¯ x k , t ) , ˆ L F [¯ g k ¯ u k ] ¯ b k (¯ x k , t ) , and ∂ ¯ b k (¯ x k ,t ) ∂t are singletons, (15) is equiv-alent to min (cid:110) ˆ L F [ ¯ f k ] ¯ b k (¯ x k , t ) ⊕ ˆ L F [¯ g k ¯ u k ] ¯ b k (¯ x k , t ) ⊕ (cid:8) ∂ ¯ b k (¯ x k , t ) ∂t (cid:9)(cid:111) ≥ − α k (¯ b k (¯ x k , t )) + (cid:13)(cid:13)(cid:13) ∂ ¯ b k (¯ x k , t ) ∂ ¯ x k (cid:13)(cid:13)(cid:13) C. (16)Due to Lemma 1, L F [ ¯ f k +¯ g k ¯ u k ] ¯ b k (¯ x k , t ) ⊆ ˆ L F [ ¯ f k ] ¯ b k (¯ x k , t ) ⊕ ˆ L F [¯ g k ¯ u k ] ¯ b k (¯ x k , t ) ⊕ (cid:8) ∂ ¯ b k (¯ x k ,t ) ∂t (cid:9) sothat min L F [ ¯ f k +¯ g k ¯ u k ] ¯ b k (¯ x k , t ) ≥ min (cid:8) ˆ L F [ ¯ f k ] ¯ b k (¯ x k , t ) ⊕ ˆ L F [¯ g k ¯ u k ] ¯ b k (¯ x k , t ) ⊕ (cid:8) ∂ ¯ b k (¯ x k ,t ) ∂t (cid:9)(cid:9) . Consequently, (16) implies(6) and ( x , | = φ ∧ . . . ∧ φ K follows by Corollary 1.R EFERENCES[1] W. Ren and R. W. Beard, “Consensus seeking in multiagent systemsunder dynamically changing interaction topologies,”
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Lars Lindemann (S’17) was born in L¨ubbecke,Germany, in 1989. He received the B.Sc. degree inElectrical and Information Engineering and the B.Sc.degree in Engineering Management both from theChristian-Albrechts-University (CAU), Kiel, Ger-many, in 2014 and the M.Sc. degree in Systems,Control and Robotics from KTH Royal Instituteof Technology, Stockholm, Sweden, in 2016. SinceJune 2016, he is pursuing the Ph.D. degree at KTHRoyal Institute of Technology, Stockholm, Sweden.His current research interests include control theory,formal methods, multi-agent systems, and autonomous systems. He was aBest Student Paper Award Finalist at the 2018 American Control Conferenceand is a recipient of the Outstanding Student Paper Award of the 58th IEEEConference on Decision and Control.