Robust Consensus of High-Order Systems

Robust Consensus of High-Order Systems under Output Constraints: Application to Rendezvous of Underactuated UAVs Esteban Restrepo, Antonio Loria, Ioannis Sarras, Julien Marzat

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Esteban Restrepo, Antonio Loria, Ioannis Sarras, Julien Marzat. Robust Consensus of High-Order Systems under Output Constraints: Application to Rendezvous of Underactuated UAVs. 2021. hal- 03275331

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Robust Consensus of High-Order Systems under Output Constraints: Application to Rendezvous of Underactuated UAVs Esteban Restrepo Antonio Lor´ıa Ioannis Sarras Julien Marzat

Abstract— We address the output and state consensus On the other hand, in realistic settings, physical sys- problems for multi-agent high-order systems in feedback tems operate under multiple restrictions in the form of form under realistic conditions. First, under the premise output or state constraints such as limited range measure- that measurements may be of different kinds, we consider systems interconnected over undirected-topology ments/communication, physical limitations imposed by the networks as well as directed spanning-trees and directed actuators (input saturation) or the environment (collisions, cycles. Second, we assume that the systems may be sub- bounded workspace). In addition, such systems are constantly ject to multiple restrictions in the form of output or state subject to disturbances such as external inputs, modeling constraints, such as limited-range measurements, physical uncertainties, delays, etc. Thus, for consensus control laws limitations imposed by the actuators or by the environment, etc. In addition, we suppose that the systems may be sub- to be of practical use, constraints must be considered and ject to external disturbances, such as undesired forces or robustness with respect to disturbances must be guaranteed. modeling uncertainties. Under these conditions, we provide These pose significant challenges for control design and formal a control framework and a formal analysis that establishes analysis that we address in this paper. robust stability in the input-to-state sense. The former relies A good example of a scenario of cooperative systems in on a modified backstepping method and the latter on multi- stability theory. In addition, we show how our approach ap- which a plethora of difficulties appear naturally is that of ren- plies to meaningful problems of physical systems through dezvous control of unmanned-autonomous vehicles (UAVs). a case-study of interest in the aerospace industry: safety- First, the systems’ dynamics are clearly nonlinear and under- aware rendezvous control of underactuated UAVs subject actuated [15]. Therefore, the literature on consensus tailored to connectivity and collision-avoidance constraints. for linear low-order systems [1], [16] does not apply. Second, the measurements usually come from embedded relative- I.INTRODUCTION measurement sensors, such as cameras, LiDAR, ultrasound, etc. The use of such devices naturally imposes directed net- A. Problem description work topologies [17], which add difficulty to the consensus- Consensus control constitutes the basis of cooperative in- control design. Third, autonomous vehicles moving “freely” teraction for multi-agent systems [1], [2]. For instance, in in the workspace are prone to undesired collisions among the case of (robotic) vehicles the consensus objective is themselves; therefore, guaranteeing the safety of the system fundamental to achieve complex formation maneuvering tasks in the sense of inter-agent collision avoidance is a restriction [3]. Hence, the literature is rife with works addressing diverse that must be considered as well. A fourth difficulty stems from consensus problems for a variety of dynamical systems, such the use of on-board relative-measurement devices, which are as linear systems [4]–[6], relative-degree-one nonholonomic reliable only if used within a limited range. This translates vehicles [7], or second-order Euler-Lagrange systems [8], into guaranteeing that the UAVs do not drift “too far” apart [9], to mention a few. However, such models may fall short from their neighbors. Finally, UAVs are constantly subject to at representing many meaningful and complex engineering external undesired forces and they may also be affected by problems in which the input-output relationship imposes a modeling uncertainties, etc.; these constitute external distur- high relative degree (with respect to an output of interest) bances at different levels in the dynamic model. model. Consensus of high-order systems has been addressed, for instance, in [10]–[13]. A good example of high-order B. Literature review systems is that of certain autonomous vehicles —see [1], [2], [14], and Section V in this paper. The characteristics described above coin a realistic scenario of automatic control of multi-agent systems, not restricted to E. Restrepo, I. Sarras, and J. Marzat are with DTIS, ON- UAVs. Consensus under such conditions has been addressed ERA, Universite´ Paris-Saclay, F-91123 Palaiseau, France. E-mail: {esteban.restrepo, ioannis.sarras, julien.marzat}@onera.fr. A. Lor´ıa is in the literature, but to the best of our knowledge never with L2S, CNRS, 91192 Gif-sur-Yvette, France. E-mail: [email protected] simultaneously. E. Restrepo is also with L2S-CentraleSupelec,´ Universite´ Paris-Saclay, There are many articles that address the constrained con- Saclay, France. The work of A. Lor´ıa was supported by the French ANR via project HANDY, contract number ANR-18-CE40-0010 and by sensus problem for low-order systems interconnected over, CEFIPRA under the grant number 6001-A. both, undirected and directed topologies [16], [18]–[21]. Fewer 2 SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021

works, however, address the consensus control problem with field that “explodes” near the limits to avoid. That is, the constraints for high-order systems. In [22] a tracking con- control input as a function of the state grows unboundedly sensus controller is proposed for networked systems over as the vehicle approaches a specified region. This technique undirected graphs, but the constraints are considered on the is also reminiscent of potential/navigation functions used in synchronization error and not directly on the inter-agent rela- robot control [27], [32]. tive states. In [23] a synchronization control is designed using Now, because we consider systems in feedback form, the an adaptation of the prescribed performance framework in control design also appeals to the popular passivity-based order to achieve consensus over directed graphs with desired backstepping method [33]. However, since in the presence of bounds on the transient response. Nevertheless, as in [22], constraints this may involve successive derivatives of functions the prescribed-performance constraints are imposed on the with multiple saddle points, we use the command-filtered consensus error and not on the inter-agent relative states. backstepping approach [34], in which the successive deriva- A consensus control for high-order systems with constraints tives are approximated by linear strictly-positive-real filters. and interacting over strongly connected directed graphs is We describe our control-design framework for systems in presented in [24]. Yet, the constraints considered therein weigh feedback form in Section III. on each individual agent’s states (e.g., constraints on the In regards to the formal analysis, it is important to stress that velocity, the acceleration, etc.), and do not reflect inter-agent in contrast to the more-often used nodes-based graph modeling restrictions. approach, our approach is based on the edge-agreement frame- In the case of rendezvous of underactuated UAVs, only a work introduced in [29]. Since in this framework the consensus handful of works in the literature consider the problem under a problem is recast as one of stabilization of the origin, it set of realistic assumptions while not at the expense of formal constitutes a more natural setting for the consideration of inter- analysis. A distributed controller is proposed in [25] based on agent constraints. Furthermore, since inter-agent constraints prescribed-performance control that achieves formation track- may introduce undesired unstable equilibria within the edges’ ing with collision avoidance for multiple UAVs. Nonetheless, perspective, the closed-loop system is analyzed relying on the results therein apply only to undirected topologies. Based the theory of multi-stable systems [35], [36]. Based on the on the attitude and thrust extraction algorithm, the authors in latter, we establish asymptotic convergence of the multi-agent [26] solve the formation problem for multiple UAVs subject system to the consensus manifold, as well as robustness with to connectivity constraints, but only undirected topologies respect to external disturbances in the sense of practical-input- are considered and collision-avoidance constraints are not to-state stability. This property cannot be overestimated; it is addressed. In [8], [9], and [27] robust formation controllers stronger than mere convergence, which is more commonly are proposed based on the prescribed-performance-control [28] established in the literature —see, e.g., [24], [37]– [39], [40]. and edge-agreement frameworks [14], [29], guaranteeing, also, Indeed, input-to-state stability implies boundedness of the collision avoidance and connectivity maintenance. However, systems’ state trajectories and the satisfaction of the inter-agent in these references only fully-actuated Lagrangian systems output constraints may also be assessed, even in the presence interconnected over undirected-tree topologies are considered. of external disturbances. The same cannot be ascertained if in the absence of disturbances it is only known that the C. This paper’s contents in perspective errors converge. We provide a formal analysis of our control approach in Section IV. In this paper we propose a control method that applies to Beyond our main theoretical findings, in Section V we show nonlinear systems in feedback form and covers, simultane- how our framework may be applied to the rendezvous control ously, all aspects evoked above. The systems’ model and the problem for a group of underactuated UAVs, subject to con- problem formulation are described in detail in Section II. nectivity and collision avoidance constraints, using only local As the study of networked systems over undirected graphs information from its neighbors. This is a relevant problem to is better covered by the literature, our main statements are for- the aerospace industry that is motivated by the increasing in- mulated only for directed spanning trees and directed cycles. terest for safety-aware fleet deployment. It is fitting to mention These directed topologies present practical interests of their that similar problems have been studied in the literature, as for own as they appear naturally in leader-follower configurations instance in [25], [26], but only for systems interconnected over [1], [2] and in the context of cyclic pursuit [30]. Our control undirected graphs. A more detailed description of this problem approach, however, also applies to systems interconnected over and of our contributions relative to the literature is presented undirected graphs. To guarantee the systems’ safety as well in Section V as well. as the integrity of the topology through the maintenance of The paper is wrapped up with some Concluding remarks in connectivity, our controllers guarantee that output constraints Section VI and complementary technical Appendices. are respected. Essentially, these take the mathematical form of a lower and an upper bound on the norm of relative- error states, which are more natural inter-agent constraints II.MODELANDPROBLEMFORMULATION than those considered, e.g., in [22] and [23]. Then, the said We consider multi-agent nonlinear systems in feedback constraints are encoded in the control design via laws that are form with high relative degree with respect to an output of derived as the gradient of barrier Lyapunov functions [31]. interest. Plants that fit in this category include fully feedback- Loosely speaking, the control may be assimilated to a force linearizable systems, but also a number of instances of physical SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021 3

systems, such as robot manipulators [41], flying vehicles [42], predefined coordinates (as for instance in a formation control spacecrafts [43], flexible-joint manipulators [44], [45], etc. problem), in the presence of output constraints as given by the Now, for simplicity and without loss of generality, in the set D in (3). Mathematically, the consensus problem translates d d following theoretical development we consider that each agent into making xi,1 − xj,1 → zk, or equivalently, zk → zk in d n is modeled as a high-order system in normal form subject to the relative coordinates, where zk ∈ R denotes the desired additive disturbances. More precisely, we consider N multi- relative state between a pair of neighboring agents i and j. d variable systems in normal form and of relative degree ρ as Note that setting zk ≡ 0 corresponds to the output-consensus follows: problem. To address the consensus problem under constraints we use x˙ = x + θ (t), l ≤ ρ − 1 (1a) i,l i,l+1 i,l the edge-agreement framework for modeling of graphs [29]. x˙ i,ρ = ui + θi,ρ(t) (1b) In this framework the variables defined in (2) denote the states n of the interconnection arcs in the graph, instead of those of the where xi,l ∈ R , l ≤ ρ, i ≤ N, denotes the components of n nodes, which are more commonly used. This has the advantage the state of each agent, ui ∈ R is the control input, θi,l : n of recasting the consensus objective as the stabilization of the R≥0 7→ R is an essentially bounded function that represents origin in error coordinates. a disturbance, and xi,1 is considered an output of interest. Note that many systems in feedback form can be trans- Let us denote the so-called incidence matrix of a graph by N×M formed into (1) through different nonlinear control approaches, E ∈ R , which is a matrix with rows indexed by the such as feedback linearization, input transformation, etc. nodes and columns indexed by the edges. Its (i, k)th entry is [33], [46]. The systems (1) cover, for instance, second-order defined as follows: [E]ik := −1 if i is the terminal node of feedback linearizable systems in which the output-consensus edge ek, [E]ik := 1 if i is the initial node of edge ek, and [E] := 0 otherwise. Let x> = x> ··· x> ∈ nN be problem consists in driving all the positions xi,1 to a common ik 1 1,1 N,1 R the collection of the first states, i.e., l = 1, of all the agents constant value and all the velocities xi,2 to zero. This problem has been studied thoroughly in the literature [1], [2], [14], even of the system. Then, the edge states in (2) satisfy considering state constraints [16]. As mentioned previously, a > z := [E ⊗ In]x1. (4) concrete non-trivial example is considered in Section V. > > > > nM In terms of the systems’ interaction, it is assumed that each where z = [z1 ··· zk ··· zM ] ∈ R ,‘⊗’ denotes the agent has access only to local information from a limited Kronecker product, and In the identity matrix of dimension number of neighbors. This local interaction is represented n × n. Similarly, an error variable is given by by a graph, denoted G = (V, E), where the set of nodes > d V := {1, 2,...,N} corresponds to the labels of the agents z˜ = [E ⊗ In]x1 − z , (5) E ⊆ V2 M and the set of edges , of cardinality , represents where zd> = [zd> ··· zd>] ∈ nM . Now, as before, let each the communication between a pair of nodes —an edge e , h 1 i M R k x> = x> ··· x> ∈ nN l ∈ {2, . . . , ρ} k ≤ M, is an ordered pair (i, j) ∈ E indicating that agent j l 1,l N,l R , , be the collection has access to information from node i, via measurement or of each state of all the agents of the system. Then, in the error communication. edge coordinates, the consensus objective is that As mentioned in the Introduction, the exchange of informa- lim z˜(t) = 0 (6a) tion among the agents may be bidirectional or unidirectional. t→∞ undirected lim xl(t) = 0, ∀l ∈ {2, . . . , ρ}. (6b) A bidirectional communication is represented by an t→∞ graph while an unidirectional exchange is modeled using a directed graph. In this paper we consider multi-agent systems Remark 1: Note that for ρ = 2 the objectives in (6) are communicating over undirected graphs, or two classes of equivalent to the output consensus of second-order multi-agent directed graphs: spanning-trees or cycles. Furthermore, in systems as it is commonly studied in the literature, e.g., on realistic scenarios multi-agent systems are commonly subject robot manipulators, mobile robots, etc. • to inter-agent constraints that may be defined as a set of One of the advantages of considering the edge states rather restrictions on the system’s output. than the node’s is that it is possible to obtain an equivalent Without loss of generality, let the first component xi,1 be reduced system, easier to analyze using stability theory. Recall the output for each agent, i ≤ N and define the relative-output that, as observed in [29], using an appropriate labeling of the state as edges, the incidence matrix is expressed as

zk := xi,1 − xj,1 ∀k ≤ M, (i, j) ∈ E. (2) E = [ Et Ec ] (7) For each k ≤ M, let ∆ and δ be, respectively, the upper k k where E ∈ N×(N−1) denotes the full-column-rank in- and minimal distances, satisfying 0 ≤ δ < ∆ . Then, the set t R k k cidence matrix corresponding to an arbitrary spanning tree of inter-agent output constraints is deﬁned as N×(M−N+1) Gt ⊂ G and Ec ∈ R represents the incidence nM D := z ∈ R : δk < |zk| < ∆k, ∀ k ≤ M . (3) matrix corresponding to the remaining edges not contained in Gt. Moreover, deﬁning The control goal is for the agents to achieve output consensus with a non-zero displacement, centered at a point of non- R := [ IN−1 T ] (8) 4 SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021

with IN−1 denoting the N − 1 identity matrix, and T := increase of complexity of the control law due to the successive > −1 > Et Et Et Ec, one obtains an alternative representation of differentiation of the virtual controllers [33]. This problem is the incidence matrix of the graph given by emphasized by the fact that the said virtual control is designed as the gradient of a barrier function, which has multiple local E = EtR. (9) minima and is defined only in open subsets of the state The identity (9) is useful to derive a reduced-order dynamic space. Therefore, in order to bypass these technical obstacles, model —cf. [29]. Now, as in the latter, the error edges’ states inspired by the command filtered backstepping approach [34], may be split as we approximate the virtual inputs and their derivatives in each step of the backstepping design by means of command filters. > >> n(N−1) n(M−N+1) z˜ = z˜t z˜c , z˜t ∈ R , z˜c ∈ R (10) This is explained in detail farther below. where z˜t are the states corresponding to the edges of an arbitrary spanning tree Gt and z˜c denote the states of the A. Barrier Lyapunov function remaining edges, ∈ G\G . Thus, after (4), (6), and (10), t Barrier Lyapunov functions are reminiscent of Lyapunov denoting zd ∈ n(N−1) as the vector of desired relative t R functions, so they are positive definite, but their domain displacements corresponding to G , we obtain t of definition is restricted by design to open subsets of the > z˜ = R ⊗ In z˜t. (11) Euclidean space. Furthermore, they grow unbounded as zk approaches the boundary of their domain. We define them as Now, collecting the inputs of the multiple agents into the follows —cf. [31]. > > > nN vector u = u1 ··· uN ∈ R and the disturbances Definition 1 (Barrier Lyapunov function): Consider the > h > > i nN into θl = θ1,l ··· θN,l ∈ R , with l ∈ {1, . . . , ρ}, the system x˙ = f(x) and let J be an open set containing the reduced-order system’s equations read origin. A Barrier Lyapunov function is a positive definite, 1 ˙ > > function V : J → R≥0, x 7→ V (x), that is C , satisfies z˜t = Et ⊗ In x2 + Et ⊗ In θ1(t) (12a) ∂V (x) > x˙ l = xl+1 + θl(t), l ∈ {2, . . . , ρ − 1} (12b) ∇V (x)>f(x) := f(x) ≤ 0, ∂x x˙ ρ = u + θρ(t). (12c) and has the property that V (x) → ∞ and |∇V (x)| → ∞ as In these coordinates, output-consensus as defined in (6) is x → ∂J . achieved if the origin is asymptotically stabilized for the Now, akin to (3), for each k ≤ M, the inter-agent constraints reduced-order system (12). More precisely, consider the fol- in terms of the error coordinates are given by the set lowing problem. ˜ n d Robust consensus problem with output constraints: Consider Dk := {z˜k ∈ R : δk < |z˜k + zk| < ∆k}. a multi-agent system of agents with high relative-degree Then, for each k ≤ M, we define a barrier Lyapunov function dynamics given by (1), interacting over a connected undirected candidate W : D˜ → , of the form graph, a directed spanning tree or a directed cycle. Assume, in k k R≥0 addition, that the systems are subject to inter-agent constraints 1 W (˜z ) = |z˜ |2 + B (˜z + zd) , (13) that consist in the outputs being restricted to remain in the k k 2 k k k k set defined in (3). Under these conditions, find a distributed d where Bk(˜zk + zk) is a non-negative function that satisfies: dynamic controller with outputs ui, i ≤ N, that, in the d d d Bk(zk) = 0, ∇Bk(zk) = 0, and Bk(˜zk + zk) → ∞ as either absence of disturbances, i.e., θi,l ≡ 0, l ≤ ρ, i ≤ N, d d |z˜k + z | → ∆k or |z˜k + z | → δk. Therefore, the barrier achieves the objective (6) and renders the constraints set (3) k k Lyapunov function candidate (13) satisfies: Wk(˜zk) → ∞ as forward invariant, i.e., z(0) ∈ D implies that z(t) ∈ D for d d either |z˜k +z | → ∆k or |z˜k +z | → δk, or equivalently in the all t ≥ 0. Furthermore, in the presence of essentially bounded k k original edge coordinates, as either |zk| → ∆k or |zk| → δk. disturbances, that is θl 6≡ 0, ui renders the origin of (12) d Remark 2: Note that the term Bk(˜zk + z ) in (13) encodes practically input-to-state stable and the set D in (3) forward k the constraints on the original edge coordinates zk in terms of invariant. • the error z˜k. This may lead to imposing conservative feasibility conditions in terms of the initial conditions when using, e.g., III.CONTROLDESIGNFORCONSENSUSUNDEROUTPUT logarithmic barrier Lyapunov functions [47]. To overcome this CONSTRAINTS d limitation, B(˜zk + zk) may be defined as an integral barrier The robust consensus problem with output constraints pre- Lyapunov function [47] or as a weight recentered barrier viously formulated is solved using a distributed dynamic non- function [48], [49] —see also Section V. • linear controller. Its design follows a backstepping approach Remark 3: The functions defined in (13) are reminiscent that naturally exploits the normal form of the system. Hence, of scalar potential functions in constrained environments [32] we start by defining a virtual control law for (12a), using x2 and, as for the latter, the appearance of multiple critical points as input. Now, in order to account for the output constraints, is inevitable. Indeed, the gradient of the barrier Lyapunov the design of the first virtual input for (12a), is based on the function of the form (13), ∇Wk(˜zk), vanishes at the origin gradient of a barrier Lyapunov function. It is well known, and at an isolated saddle point separated from the origin — however, that the backstepping approach may lead to an see Appendix I. Therefore, when using the gradient of (13) for SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021 5

the control, the closed-loop system has multiple equilibrium which requires the derivative of the right-hand side of (14). points. This prevents us from using the classical stability Furthermore, a recursive procedure requires up to ρ − 2 ∗ tools for the analysis of the system. Such technical difficulty successive derivatives of x2, which posses significant technical is addressed using tools tailored for so-called multi-stable and numerical difficulties. Thus, to avoid the use of successive systems—see [35], [36], and Appendix II. • derivatives of ∇W (˜z) we approximate the derivatives of the ∗ virtual controls xl , with l ∈ {2, . . . , ρ − 1} by means of B. Control design for systems over directed graphs command filters. For simplicity, we use second-order systems defined as in the figure below —cf. [34]; Remark 4: For brevity, in the remainder of the paper we assume that the graph G representing the interaction topology ∗ ∗ x 2 is either a directed spanning tree or a directed cycle at the xl lf ω H (s) H (s) := n initial time. Nonetheless we stress that, with appropriate mod- 1 1 2 2 s + 2ωns + ωn ifications, the results in this section also hold for undirected connected topologies. • Fig. 1: Command filter used for implementation. The dirty derivative ∗ ∗ As mentioned earlier, the control design builds upon a of xl may be obtained using x˙ lf := sH1(s)xl which is equivalent to x˙ = H (s)x ˙ ∗. recursive backstepping approach. Consider, first, the edge lf 1 l subsystem (12a) with x2 as an input. In order to cope with the output constraints, a good choice of control law consists, as The virtual controls are considered as the inputs of a mentioned previously, in the gradient of a barrier function [21], command filter, with the outputs corresponding to the ap- [50], [51]. As it is defined here in edge coordinates, specific proximated signals and their derivatives, denoted xlf and x˙ lf , notations are introduced. respectively. The filters’ natural frequency, ωn > 0, is a Let us define the so-called in-incidence matrix E ∈ control parameter which is chosen large enough so that the N×M x x∗ R , whose elements are defined as follows—cf. [52]— approximation lf converges to the desired virtual control l in a faster time-scale than that of the system’s dynamics — : [E ]ik := −1 if i is the terminal node of edge ek and see Section IV-A. Moreover, the filters are designed with unit [E ]ik := 0 otherwise. Similarly the elements of so-called N×M DC gain and unit damping coefficient so that the tracking out-incidence matrix E⊗ ∈ R are defined as follows: of the virtual signals is guaranteed without overshoot. This [E ]ik := 1 if i is the initial node of edge ek and [E ]ik := 0 otherwise. Then, in the edge agreement framework the ensures that, in the slower time-scale of the systems’ dy- consensus control based on the gradient of the barrier function namics, the “filtered forms” act as the desired virtual signals, is given by1 corresponding to a classical backstepping control. Similarly, x˙ = H (s)x ˙ ∗ approximates x˙ ∗. ∗ lf 1 l l x2 := −c1[E ⊗ In]∇W (˜z), (14) Remark 5: For clarity, we use second-order command- where c1 is a constant that is positive by design and ∇W (˜z) is filters as defined in Fig. 1 above. However, the design is not the gradient of a barrier Lyapunov function for the multi-agent restricted to this particular choice. Indeed, other possibilities system, which is defined as include first-order low-pass filters [54] or high-order Levant X differentiators [26]. • W (˜z) = W (˜z ), k k (15) For the purpose of stability analysis, we write the command k≤M filters’ dynamics in state form. To that end, let the filters’ h i with Wk(˜zk) given in (13) for all k ≤ M. Indeed, the right- > > > 2nN variables be denoted as αl−1 := αl−1,1 αl−1,2 ∈ R , hand side of (14) qualifies as a consensus control law that for l ∈ {2, . . . , ρ}. Then, in state-space representation, the guarantees connectivity maintenance for first-order multi-agent command filters are written as systems z˜˙t = u interconnected over directed graphs —cf. [53]. ∗ So, defining x¯2 := x2 − x and using (14), Equation (12a) ∗ 2 α˙ l−1 = ωn [A ⊗ InN ] αl−1 + ωn [B ⊗ InN ] xl (19a) becomes2 > > > xlf x˙ lf = [C ⊗ InN ] αl−1, l ∈ {2, . . . , ρ}, (19b) z˜˙ = −c [E>E ⊗ I ]∇W (˜z) + [E> ⊗ I ] [¯x + θ ] . (16) t 1 t n t n 2 1 0 1 0 1 0 A := ,B := ,C := (19c) With aim at making x¯2 → 0 in (16), following a −1 −2 1 0 ωn backstepping-based design, we rewrite the second equation in and the initial conditions are set to α (0) = x∗(0) and (12b), i.e., with l = 2, in error coordinates x¯2 and we consider l−1,1 l α (0) = 0. x3 as an input. We have l−1,2 Thus, the virtual control input, starting with (18), are ˙ ∗ x¯2 = x3 − x˙ 2 + θ2. (17) redefined using the filter variables as follows. First, we redefine Hence, the natural virtual control law at this stage is ∗ x := −c2x˜2 +x ˙ 2f (20) ∗ ∗ 3 x3 = −c2x¯2 +x ˙ 2, c2 > 0, (18) where x˜2 := x2 − x2f and x˙ 2f = ωnα1,2. Hence, in contrast 1For undirected graphs this virtual control law takes the form x∗ = 2 to (17), from −c1[E ⊗ In]∇W (˜z), where E is the incidence matrix—cf. [21]. 2To avoid a cumbersome notation we write ∇W (˜z) in place of the more > ∗ ∗ appropriate spelling ∇W ( R ⊗ In z˜t). x˙ 2 − x˙ 2f = x3 − x˙ 2f + θ2 + x3 − x3 + α2,1 − α2,1, 6 SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021

∗ defining x˜3 := x3 − x3f and using (20) and α2,1 = x3f , we filters. Hence, when αl−1,1 = xl and αl−1,2 = 0, the obtain reduced slow system, corresponding to the actual system with ˙ ∗ x˜2 = −c2x˜2 +x ˜3 + (α2,1 − x3) + θ2. the backstepping control, effectively achieves consensus with respect of the output constraints. Then, owing to the fact that the system is in feedback form, The proposed controller and stability analysis offer some we define advantages with respect to previous results in the literature. ∗ xl := −cl−1x˜l−1 + ωnαl−2,2 − x˜l−2, l ∈ {4, . . . , ρ}, (21) Using the reduction of the edge system considering only the states of a spanning tree contained in the graph, we are able to where c , c are positive constants, and the tracking errors 2 l−1 analyze the system using Lyapunov and input-to-state stability are given by theory (in a multi-stability sense [35], [36]) of the reduced x˜l := xl − xlf = xl − αl−1,1, l ∈ {2, . . . , ρ} (22) slow system. The latter is significant since the reduced slow system corresponds to a high-order system in closed-loop —cf. Eq. (20). That is, the virtual controls x∗ starting from l with a classical backstepping controller solving the output- l = 3 are redesigned to steer x towards the filtered virtual l−1 constrained consensus problem. On the other hand, relying on input x . Finally, the actual control input is set to l−1 f the multi-stability framework for singularly perturbed systems, u = −cρx˜ρ + ωnαρ−2,2 − x˜ρ−1. (23) it is possible to establish not only convergence to the consensus manifold but also explicit robustness properties in the sense Remark 6: The system being in feedback form, the third of practical input-to-state stability with respect to bounded term on the right-hand side of (21) and (23) are feedback disturbances. passivation terms —cf. [55]. These terms, that come from the backstepping-as-recursive-feedback-passivation approach [56] are used to render the system (12) passive with respect to the C. Extension to partial- and full-state consensus with output yρ := xρ − xρf . • output constraints Thus, taking the derivative of the backstepping error vari- The control design methodology presented above can be ables defined in (22) and using the input (23), with (19)-(21), directly extended to consider consensus in all or part of the we obtain the closed-loop system high-order states xl. This is useful, for instance, in the context ˙ > of flocking in formation, attitude consensus, etc. z˜t = − c1[Et E ⊗ In]∇W (˜z) > ∗ Suppose that, besides achieving the output-consensus goal + [Et ⊗ In] [˜x2 + (α1,1 − x2) + θ1] (24a) as defined in (6a), it is additionally required to achieve ˙ ∗ x˜2 = − c2x˜2 +x ˜3 + (α2,1 − x3) + θ2 (24b) consensus of a number of states r ≤ ρ. For convenience, and ˙ ∗ without loss of generality, suppose that 3 < r < ρ. Then, akin x˜l = − clx˜l +x ˜l+1−x˜l−1 + (αl,1 − xl+1) + θl, to the edge transformation (4), we define the edge states ∀ l ∈ {3, . . . , ρ − 1} (24c) z := [E> ⊗ I ]x , l ∈ {2, . . . , r} (25) x˜˙ ρ = − cρx˜ρ−x˜ρ−1 + θρ (24d) l n l ∗ α˙ l = ωn [A ⊗ InN ] αl + ωn [B ⊗ InN ] xl+1, and, accordingly, let the objectives in (6) be replaced with ∀ l ∈ {1, . . . , ρ − 1}. (24e) lim z˜(t) = 0 (26a) t→∞ Consequently, solving the robust consensus problem with lim zl(t) = 0, ∀l ∈ {2, . . . , r} (26b) t→∞ output constraints comes to guaranteeing that z˜t(t), as part of lim xl(t) = 0, ∀l ∈ {r + 1, . . . , ρ}. (26c) the solution to Eqs. (24), tends to zero. More precisely, that t→∞ the control law (23), with (14) and (19)-(21), solves the robust Furthermore, let consensus problem with output constraints for system (1) is a > fact established by the following statement. z˜l :=zl − [E ⊗ In]xlf ∀ l ∈ {2, . . . , r} (27)

Theorem 1: Consider the system (1) in closed loop with x˜l :=xl − xlf ∀ l ∈ {r + 1, . . . , ρ}; (28) the dynamic controller defined by (23), together with (14) and (19)-(21). Then, the constraints set (3) is forward invariant. we recall that xlf = αl−1,1. Then, to achieve the new Moreover, if θi,l ≡ 0, l ≤ ρ, i ≤ N, (6) holds for almost all consensus objectives (26) the virtual control inputs, at each initial conditions satisfying z(0) ∈ D. Otherwise, if θl 6≡ 0, step, are redefined as the closed-loop system is almost-everywhere practically input- z∗ := [E> ⊗ I ]x∗, ∀ l ∈ {2, . . . , r} (29) > > > l n l to-state stable with respect to θ := [θ1 ··· θρ ] . ∗ ∗ For clarity of exposition, the proof of Theorem 1 is pre- with x2 as in (14), x3 := −c2[E ⊗ In]˜z2 + ωnα1,2, sented in Section IV, but in anticipation of the latter we ∗ x := −cl−1[E ⊗ In]˜zl−1 + ωnαl−2,2 − x˜l−2 underline the following. As mentioned earlier, part of the l control approach consists in choosing ωn sufficiently large, for all l ∈ {4, . . . , r}, and ∗ so that αl−1,1 → xl faster than the dynamics of the system. ∗ xr+1 := −cr[E ⊗In]˜zr +ωnαr−1,2 −[E ⊗In]˜zr−1, (30) Note that, setting := 1/ωn as a singular parameter in (39a), ∗ the closed-loop system (24) may be separated into two time xl := −cl−1x˜l−1 + ωnαl−2,2 − x˜l−2, l ∈ {r + 2, . . . , ρ}. scales, where the fast system corresponds to the command (31) SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021 7

Finally, the actual control input is set to Then, defining the coordinate transformation > u := −cρx˜ρ + ωnαρ−2,2 − x˜ρ−1. (32) > > > > α˜ := α˜1,1 α˜1,2 ··· α˜ρ−1,1 α˜ρ−1,2 = α − h(ξ), (37) Then, we have the following. and using (24) we obtain the singularly perturbed system Theorem 2: Consider the system (1) in closed loop with the ˙ > > dynamic controller defined by (32), together with (29)-(31) z˜t = − c1[Et E ⊗ In]∇W (˜z) + [Et ⊗ In] [˜x2 +α ˜1,1 + θ1] and (19). Then, the constraints set (3) is forward invariant. (38a) Moreover, if θ ≡ 0, l ≤ ρ, i ≤ N, the limits in (26) i,l x˜˙ = − c x˜ +x ˜ +α ˜ + θ (38b) hold for almost all initial conditions satisfying z(0) ∈ D. 2 2 2 3 2,1 2 ˙ Otherwise, if θl 6≡ 0, the closed-loop system is almost- x˜l = − clx˜l +x ˜l+1−x˜l−1 +α ˜l,1 + θl, l = 3, . . . , ρ − 1 everywhere practically input-to-state stable with respect to (38c) > > > θ := [θ1 ··· θρ ] . x˜˙ ρ = − cρx˜ρ−x˜ρ−1 + θρ (38d) Modulo the change to the reduced-edge coordinates—see ∂h(ξ) α˜˙ =A˜α˜ − ξ.˙ (38e) (11)—for the higher-order edge states, the proof is identical ∂ξ to the proof of Theorem 1, which is presented next. In turn, the reduced system ξ˙ = f(ξ, h(ξ), θ, 0) takes the form IV. PROOFOF THEOREM 1 ˙ > > The proof is organized in two main parts. In the first part, z˜t = − c1[Et E ⊗ In]∇W (˜z) + [Et ⊗ In] [˜x2 + θ1] (39a) we show how the closed-loop system (24) can be written as x¯˙ = − [H ⊗ InN ]¯x + θ,¯ (39b) a singularly-perturbed system with singular parameter := > > > ¯> > > 1/ωn and in which the fast systems correspond to the dynam- where x¯ := x˜2 ··· x˜ρ , θ := θ2 ··· θρ , and ics of the command filters and the slow system corresponds c −1 0 ··· 0  to the high-order dynamics of the original multi-agent system. 2  1 c3 −1 ··· 0  In the second part we analyze the stability and the robustness    ......  of the singularly perturbed system, using analysis tools multi- H :=  . . . . .  . (40)   stable systems theory [35], [36].  0 ... 1 cρ−1 −1 0 ... 0 1 cρ A. Singular perturbation model On the other hand, the boundary layer system, (dα/dτ˜ ) = > > > 2nN(ρ−1) > Define α := α1 ··· αρ−1 ∈ R , ξ := g(ξ, α˜ + h(ξ), θ, 0), with τ = t/ and with ξ considered as > > > n(ρN−1) > fixed, is z˜t x˜2 ··· x˜ρ ∈ R , and θ := [θ1 ··· θρ] ∈ nρN dα˜ R . Then, the filter subsystem (24e) can be rewritten as = A˜α˜ (41) dτ α˙ = ω A˜α − χ(ξ, α), A˜ := blockdiag{[A ⊗ I ]}, (33) n nN where A˜ is a Hurwitz matrix —see Eqs. (33) and (19c). ∗> > ∗> > ∗> >> χ(ξ, α) := x2 0 x3 0 ··· xρ 0 . Even though the system (38) appears to be in the familiar form (34), its analysis is stymied by the fact that the function Now, as mentioned in the previous section, with := 1/ω n ∇W vanishes at multiple separate equilibria. Therefore, we as the singular parameter, the closed-loop system (24) may be rely on perturbation theory for multi-stable systems, developed written in the singular-perturbation form in [35]. The stability analysis is provided below and some def- ξ˙ = f(ξ, α, θ, ) (34a) initions and statements from [35] are recalled in Appendix II, α˙ = g(ξ, α, θ, ). (34b) for convenience. Setting = 0 in (34) we obtain the so-called quasi-state model B. Stability and robustness analysis Denote z˜∗ ∈ n(N−1) as the vector containing the saddle ξ˙ = f(ξ, α, θ, 0) (35a) t R points of the barrier Lyapunov function for each edge (13). 0 = g(ξ, α, θ, 0) (35b) Then, the equilibrium points of subsystem (38a) are collected in which (35b) becomes an algebraic equation. Hence, the into a disjoint set, denoted by analysis of the singular perturbation model (34) is normally ∗ W := {0} ∪ {z˜t }, (42) conducted studying its dynamic properties in different time scales [46]. which is an acyclic W-limit set3 of (38a). This means that Denote αs = h(ξ) the unique root of the algebraic equation asymptotic stability of the origin of (38a) may be guaranteed, (35b), at best, almost everywhere in D, that is, for all initial conditions in D except for a set of measure zero corresponding to h > > > > h(ξ) = (−c1[E ⊗ In]∇W (˜z)) 0 − c2x˜2 0 ··· the domain of attraction of the unstable critical point. > >> −cρ−1x˜ρ−1 0 . 3In the settings of this paper an acyclic limit set corresponds to an invariant (36) set of isolated points in Euclidean space. See [35] for a complete definition. 8 SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021

Now, we first analyze the stability of (38) with respect to where rk is the kth column of R in (8). For the time-being, ρ−1 the set of equilibria WΘ := W × {0} . For this purpose assume that D (equivalently Dt) is forward invariant; this we use Theorem 3 in Appendix II, which is essentially a hypothesis is relaxed below. Then, for all z˜t ∈ Dt, Vz(˜zt) reformulation of [35, Theorem 2] adapted to the contents of satisfies 1 2 this paper. Theorem 3 establishes sufficient conditions for a |z˜t|W ≤ Vz(˜zt), (43) practical input-to-state multi-stability property to hold for a 2 ∗ singularly perturbed system with respect to WΘ and to a where |z˜t|W = min{|z˜t|, |z˜t − z˜t |}. Furthermore, from (39a), bounded external input θ, granted that the reduced system (39) the derivative of Vz is given by is input-to-state stable with respect to set WΘ and input θ ˙ > > Vz = −c1∇W (˜z) R Et E ⊗ In ∇W (˜z) and that the origin for (41) is globally asymptotically stable. > > > Therefore, the stability and robustness analysis is conducted +∇W (˜z) R Et ⊗ In [˜x2 + θ1] , (44) in the following steps: where R is defined in (8). Equation (44) holds for, both, 1) We show that the origin is asymptotically stable for the directed-spanning-tree and directed-cycle topologies. Next, we boundary layer system (41). analyze the two considered topologies separately. 2) Relying on the results on cascaded multi-stable systems Case 1 (Directed spanning tree). In this case we have G = in [36], we show that the reduced system (39) is input- Gt. Therefore, z = zt, E = Et, and R = IN−1. Hence, (44) to-state stable with respect to set WΘ and input θ. becomes 3) Using Theorem 3 in Appendix II, we prove that, for a ˙ > sufficiently small , the singularly perturbed system (38) Vz = −c1∇W (˜zt) Et E ⊗ In ∇W (˜zt) > > is practically input-to-state stable with respect to the set +∇W (˜zt) Et ⊗ In [˜x2 + θ1] (45) W × {0} and a bounded external input θ. Moreover, Θ Define Ls := 1 E>E + E>E , which is the symmetric using [35, Theorem 3], in the absence of disturbances, e 2 t t part of the so-called directed edge Laplacian E>E . As it is we show convergence to the set of equilibria. t shown in the proof of Proposition 1 in [53], Ls is positive 4) Using the practical input-to-state multi-stability prop- e definite. Therefore, applying Young’s inequality to the second erty, we establish almost-everywhere-practical-input-to- term in the right-hand side of (45), we have state stability of the origin of (38). Similarly, if θ ≡ 0, we establish convergence to the origin. ˙ 0 2 1 2 2 Vz ≤ − c1|∇Vz(˜zt)| + |x˜2| + |θ1| , (46) 5) Finally, we show that the output-constraints set defined 2γ h > i in (3) is forward invariant. 0 s γλmax(Et Et) where c := c1 λmin(L ) − is positive for Step 1) Since A˜ is Hurwitz by design, the origin α˜ = 0 is 1 e 2 a sufficiently small γ > 0, λ (Ls) > 0 is the smallest exponentially stable for the boundary-layer system (41). min e eigenvalue of Ls, and λ (E>E ) is the largest eigenvalue Step 2) Consider the reduced system (39). Note that it has e max t t of E>E . the form of a cascaded system, in which (39b) is the “driving” t t Case 2 (Directed cycle). Consider equation (44). Using the system and the “driven” system (39a) has multiple equilibria identity (9), we have given by the set W in (42). In order to prove input-to-state > stability of (39) with respect to set WΘ and input θ, as per V˙z = −c1∇W (˜z) E E ⊗ In ∇W (˜z) in [36], we need to show that (39a) is input-to-state stable > > +∇W (˜z) E ⊗ In [˜x2 + θ1] . (47) with respect to the set W and the inputs x˜2 and θ1 whereas the system (39b) is input-to-state stable with respect to θ¯. We Now, for a directed-cycle topology, the following identity start with the latter. follows —cf. [53]— Input-to-state stability with respect to θ¯ for the system (39b) > > > E E + E E = E E. (48) follows directly from Lyapunov theory since (39b) is a linear time-invariant system and −[H ⊗ InN ] is Hurwitz matrix, Therefore, using (48) and (9) again, we have since so is −H. c V˙ =− 1 ∇W (˜z) E>E + E>E ⊗ I ∇W (˜z) Consider, in turn, the reduced subsystem (39a). For this z 2 n > > > system we use the barrier Lyapunov function W (˜z) given by +∇W (˜z) R Et ⊗ In [˜x2 + θ1] (15). First note that W (˜z) consists in a sum of the barrier c1 =− ∇W (˜z) R>E>E R> ⊗ I ∇W (˜z) Lyapunov functions defined for each edge in the initial graph. 2 t t n > > > Yet, from the identity (11) it is possible to express W in terms +∇W (˜z) R Et ⊗ In [˜x2 + θ1] . (49) of the edges corresponding to a spanning tree contained in the graph. Hence, define the candidate Lyapunov function Denote ∂V (˜z ) > z t > Vz(˜zt) = W R ⊗ In z˜t . ∇Vz(˜zt) := = [R ⊗ In] ∇W ( R ⊗ In z˜t). (50) ∂z˜t On the other hand, for consistency in the notation, we define Then, applying Young’s inequality to the second term of the the constraint set (3) in terms of the edges of the spanning right-hand side of (49), we have tree zt as ˙ 0 2 1 2 2 n(N−1) > Vz ≤ − c1|∇Vz(˜zt)| + |x˜2| + |θ1| , (51) Dt := zt ∈ R : rk ⊗ In zk ∈ (δk, ∆k), ∀ k ≤ M 2γ SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021 9

> 0 λmin(Et Et) where c1 := 2 [c1 − γ] is positive for a sufficiently Assume that the state constraints are not respected. Therefore, > small γ > 0. Note that, λmin(Et Et) is positive since it is from continuity of the solutions, there exists a time T > 0 the smallest eigenvalue of the edge Laplacian of an undirected such that zt(T ) ∈ ∂Dt. Now, from the previous analysis of spanning tree —cf. [21]. the singularly perturbed system, we have that in the interval Thus, for either the spanning tree or the cycle case, the [0,T ), condition (91b) holds. Moreover, since α˜(0) = 0 by derivative of Vz(zt) satisfies design, the solutions of the filter error satisfy α˜(t) ≤ d2, for t ∈ [0,T ). Consider the derivative of the Lyapunov function 1 V˙ ≤ − c0 |z˜ |2 + |x˜ |2 + |θ |2 . (52) (43) along the trajectories of (38a), which satisfies z 1 t W 2γ 2 1 1 It follows from (43), (52), and Theorem 4 in Appendix II, that V˙ (˜z ) ≤ −c0 |∇V (˜z )|2 + |x˜ |2 + |θ |2 + |α˜|2 z t 1 z t 2γ 2 1 the subsystem (39a) is input-to-state stable with respect to the set of equilibria W, and the inputs x˜2 and θ1. Therefore, since α˜(t) ≤ d2, θ1(t) is bounded and the system Thus, after [36, Theorem 3.1], the reduced system (39) in (39b) is input-to-state stable, x˜2(t) is bounded, for all t ∈ is input-to-state stable with respect to WΘ and input θ. [0,T ). Then, we have Furthermore, W qualifies as a W-limit set for (39). Θ V˙ (˜z (t)) ≤ −c0 |∇V (˜z (t))|2 + d, ∀t ∈ [0,T ) Step 3) Since the reduced system (39) is input-to-state stable z t 1 z t with respect to WΘ and θ, and the origin of (41) is asymp- where d is a positive constant. By Definition 1 we have totically stable, it follows, after Theorem 3 in Appendix II, |∇V (˜zt(t))| → ∞ as zt(t) approaches the border of the that the singularly perturbed system (38) is practically input- constraints set, ∂Dt. Therefore, if |z˜t(t)| grows, there exists 2nN(ρ−1) ∗ ∗ to-state stable with respect to set WΘ ×{0} and input a time 0 < T < T such that V˙z(˜zt(T )) ≤ 0. The latter, θ. More precisely, for any pair of constants d1, d2 > 0, there in turn means that Vz(˜zt(t)) is bounded for all t ∈ [0,T ), exists an ∗ > 0 such that, for any ∈ (0, ∗], the solutions which contradicts the initial assumption that the constraints are of (38) satisfy not respected. By resetting the initial conditions, the previous reasoning can be repeated for t ≥ T . Therefore, the interval lim sup |ξ(t)|WΘ ≤ ηθ(kθk∞) + d2 (53a) t→∞ where Vz(˜zt(t)) is bounded can be extended to infinity. The t boundedness of Vz(˜zt(t)) means, based on the definition of |α˜(t)| ≤ β |α˜(0)|, + d , ∀ t ≥ 0, (53b) α 2 the barrier Lyapunov function, that the constraints are always respected or, equivalently, that the set Dt is forward invariant. provided that ˙ max{|ξ(0)|WΘ , |α˜(0)|, kθk∞, kθk∞} ≤ d1, V. FORMATION CONTROL OF COOPERATIVE THRUST-PROPELLED UAVSWITHOUTPUTCONSTRAINTS where kθk∞ := lim sup kθ(t)k and |ξ|WΘ := inf |ξ − a|. t→∞ a∈WΘ We assert above that systems in strict feedback form (1) Now consider the case in which the disturbance θ ≡ 0. From cover many models of physical systems, the most obvious property (53a) we may conclude that the origin of the reduced instance being that of fully feedback-linearizable (hence fully- system (39) is multi-stable with respect to WΘ. Therefore, actuated) systems, such as second-order Lagrangian systems from the latter and the exponential stability of the boundary with n degrees of freedom and n control inputs. In this layer system (41) all the assumptions of [35, Theorem 3] are section we solve a problem of robust consensus for multi- satisfied and the solutions of (38) satisfy agent constrained 3rd-order systems, which has relevance in certain robotics applications. lim |ξ(t)|WΘ =0 (54a) t→∞ The case-study consists in designing robust distributed lim |α˜(t)| =0. (54b) t→∞ controllers to solve the position-consensus-based formation of multiple thrust-propelled UAVs under a set of realistic Step 4) Since the critical point z˜∗ of the barrier function is t assumptions. It is assumed that the drones are equipped with a saddle point —see Appendix I, after [57, Proposition 11], it relative-measurement sensors, so the graph representing the follows that the region of attraction of the unstable equilibrium interaction between the agents is directed (either a directed z˜∗ has zero Lebesgue measure. Therefore, we conclude that t spanning tree of a directed cycle). Furthermore, we assume the bound in (53a) and the limit in (54a) are satisfied for the that the system is subject to inter-agent constraints of the form origin {ξ = 0}. More precisely, we have (3). These constraints come, for one part from the embedded

lim sup |ξ(t)| ≤ ηθ(kθk∞) + d2 measurements devices, which are reliable only if used within t→∞ a limited range. Hence, in order to maintain the connectivity and, for θ ≡ 0, of the graph, the UAVs must remain within a limited distance lim |ξ(t)| = 0. from their neighbors. On the other hand, in order to guarantee t→∞ the safety of the systems, inter-agent collision avoidance must Step 5) Up to this point we have assumed that the inter- also be ensured. Finally, we assume that the agents are subject agent constraints are satisﬁed for all time, that is, z(t) ∈ D to bounded time-varying disturbances generated, e.g., by wind for all t ≥ 0. Then, in order to prove the forward invariance gusts, aerodynamic effects or unmodeled dynamics, which are of the constraints set we proceed by contradiction as follows. common in applications involving cooperative UAVs. 10 SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021

In the following we present the model and the mathematical of non-predefined coordinates. More precisely, let the output formulation of the problem at hand. We emphasize that the of the multi-agent system be the relative position between pairs dynamic model of a UAV does not, a priori, fit in the class of connected agents. That is, in the way of (4)-(5), the edge of systems characterized by Eq. (1), but we show that using states and error-edge states are defined as a preliminary control loop it may be transformed into one > (57) with relative degree ρ = 3. The implementation of such z := [E ⊗ I3]p controller, however, imposes the incorporation of saturations and > d to the control method described in Section III-B. Thus, this z˜ = [E ⊗ I3]p − z , (58) meaningful case-study shows both the applicability and the > > > 3N d> versatility of our framework. Its efficacy is illustrated via brief respectively, where p = [p1 ··· pN ] ∈ R and z = d> d> 3M numerical simulations. [z1 ··· zM ] ∈ R are the relative displacements of the desired formation. Therefore, the control objective is that

A. UAV’s model and problem formulation lim z˜k(t) =0 ∀k ≤ M (59a) t→∞ lim vi(t) =0 ∀i ≤ N. (59b) t→∞ T3 Now, in accordance to the realistic setting described previ- T2 d ously, it is assumed that the robots are equipped solely with z3 . . relative-measurements sensors. On one hand, this naturally ∆2 leads to considering directed-topology graphs. Only directed- T δ3 1 d spanning-tree and directed-cycle topologies are considered but, ωy,1 z2 .. alluding to Remark 4, we stress that the following results also −→ apply to connected undirected graphs. On the other hand, the p1 B1 I sensors are reliable only if “neighboring” agents remain within . e1 a certain range and, in order to guarantee the safety of the ωx,1 ω z,1 systems, these must also avoid collisions with one another. e3 e2 Let ∆k denote the maximal distance between the nodes i Fig. 2: Group of thrust propelled vehicles and Inertial frame and j, such that the node j has access to information from the node i through the arc ek = (i, j). Similarly, let δk denote We consider a swarm of N UAVs as illustrated in Fig. 2; the minimal distance among neighbors such that collisions each vehicle’s motion being described by a so-called “mixed” are avoided. Then, the connectivity and collision-avoidance model that consists in a second-order Cartesian dynamics constraints are encoded by the constraints set given in (3). equation on E(3) and a first-order attitude kinematics equation We also stress that the UAV model (55)-(56) is a nonlinear on SO(3) —see, e.g., [15], [58], and [59]. This underactuated underactuated system. Indeed, the UAVs have six degrees model describes some commercial UAVs, which accept only of freedom (three-dimensional displacements and three rota- thrust and angular rates as the control inputs. It is also well- tions), but only four dimensions can be directly actuated using suited for control design and analysis since, in view of the the actual inputs Ti and ωi. To address this difficulty, some fully-actuated and passive nature of the attitude dynamics, hierarchical approaches have been reported in the literature, angular torques can be easily defined in order to track the using the natural cascaded structure of the UAVs’ dynamics designed angular rates. [15], [59]–[61]. Such designs have been used to solve the The model for the ith agent is given by the equations formation problem of swarms of UAVs interconnected through undirected and directed communication topologies —cf. [59], p˙i = vi (55a) [60], [62]–[65]. Nonetheless, none of these works address the Ti problem under inter-agent constraints, as stated next. v˙i = − Rie3 + ge3 + θi,2(t) (55b) mi Robust formation problem with output constraints: Consider a multi-agent system composed of N quadrotor UAVs with R˙ i = RiS(ωi), (56) underactuated dynamics described by (55)-(56). Let the inter- > where mi is the mass of the quadrotor, e3 = [0 0 1] is the actions of the vehicles be modeled by a connected undirected unitary vector in the vertical direction of the inertial frame I, graph, a directed spanning tree or a directed cycle. Moreover, 3 3 pi ∈ R and vi ∈ R are respectively the inertial position let the output inter-agent constraints be given by the set (3). and inertial velocity, Ri ∈ SO(3) is the rotation matrix Find distributed controllers Ti and ωi, i ≤ N, that, in the of the body-fixed frame Bi with respect to I, g is gravity absence of disturbances, that is, with θi,2 ≡ 0 for all i ≤ N, 3 acceleration, and θi,2 : R≥0 → R is an essentially bounded achieve the objective (59) and render the constraints set (3) disturbance. The inputs are the thrust force produced by the forward invariant, i.e., z(0) ∈ D implies that z(t) ∈ D for 3 propellers, Ti ∈ R, and the angular rate of the vehicle ωi ∈ R all t ≥ 0. Furthermore, in the presence of disturbances, that is in the body-fixed frame Bi —see Figure 2 for an illustration. θi,2 6≡ 0, the control law must render the formation practically The control goal is for the robots to achieve a predetermined input-to-state stable with respect to the disturbances and the formation in the three-dimensional space, centered at a point set D in (3) forward invariant. • SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021 11

h i In the following, we present the control approach that solves ⇐⇒ (T˙i + c3 Ti)Ri + TiRiS(ωi) e3 = −miνi. (65) the robust formation problem with output constraints. It is > based on a change of variables, inspired by the hierarchical Left-multiplying by (the full-rank rotation matrix) Ri , we see backstepping approach of [59], that allows us to consider that the dynamic equation (65) is equivalent to the underactuated system (55)-(56) as a system in the form > h ˙ i > (1) with ρ = 3. Then, a slightly modiﬁed version of the Tiωyi, −Tiωxi, Ti + c3 Ti = −miRi νi. (66) controller designed in Section III-B may be used in order to obtain the desired formation under the connectivity and > > Now let ν˜i := [˜νi,x ν˜i,y ν˜i,z] = Ri νi. Then, (66) holds collision-avoidance constraints and with the desired robustness if the angular rates are set to properties. > miν˜i,y miν˜i,x ωi = − ωzi , (67) B. Control approach Ti Ti The control architecture follows a hierarchical approach and the thrust is given by the update law that exploits the natural cascaded interconnection between the ˙ translational dynamics (55) and the rotational kinematics, (56) Ti = −c3 Ti − miν˜i,z. (68) —see Fig. 3. The design builds on the method proposed in Remark 7: Note that by transforming the UAV model (55) the previous sections, but we must start by applying an im- using (61)-(62), only the three translational dimensions are plementable feedback transformation to represent the system directly controlled. Therefore, only three of the four available (55) in the form (1). To that end, ﬁrst note that (55) may be inputs are needed to solve the formation problem. Indeed, note assimilated to a second-order integrator that from equation (66), the yaw component of the angular rate p˙i = vi (60a) ωzi is not needed for the control. Hence it may be considered v˙i = ζi + θi,2, (60b) as an additional degree of freedom and may be designed so that the vehicle follows a desired yaw trajectory. • with Ti Thus, applying the previous transformation, the underactu- ζi := − Rie3 + ge3. (61) ated system (55) may be rewritten in the form (1), as desired, mi that is, Nonetheless, the implementation of a virtual controller for

(60), through the input ζi, is subject to the possibility of p˙i = vi (69a) solving (61) for T , which is the actual control input. Because i v˙ = ζ + θ (69b) of the underactuation of (55), however, this is far from i i i 3 ζ˙ = u . (69c) acquired. Indeed, note from (61) that the virtual input ζi ∈ R i i cannot take an arbitrary value since T ∈ and its direction i R Furthermore, denoting v> = [v> ··· v> ] ∈ 3N and ζ> = are determined by the vehicle’s orientation, R . In order to 1 N R i [ζ> ··· ζ>] ∈ 3N , and using the edge transformation (58), overcome the underactuation, we solve equation (61) dynam- 1 N R the multi-agent system in the reduced error-edge coordinates ically, inspired by the distributed-backstepping approach in becomes [59]. More precisely, we design the angular rates ωi and an update law for the thrust Ti, so that ζi in (61) satisfies the ˙ > z˜t = Et ⊗ I3 v (70a) dynamic equation v˙ = ζ + θ2 (70b) ˙ ζi = ui, i ≤ N, (62) ζ˙ = u. (70c) 3 where ui ∈ R is a new input. Note that now the system defined by (60) and (62) has the form (1) with ρ = 3. Hence, The transformed system (70) is in the form of (12) with ρ = 3. To apply the robust control law (23), designed in in the sequel, in (62), ui is assumed to correspond to an output-constrained consensus control law designed as per the the previous section for the constrained-consensus problem of framework described in Section III. high-order systems, we define the backstepping error variables Differentiating (61) with respect to time, and using (56), the ˜ v˜ = v − vf and ζ = ζ − ζf . (71) left-hand side of (62) becomes v ζ T˙ T The filtered signals f and f are the outputs of command − i R e − i R S(ω )e = u . (63) filters given in (19), with inputs v∗ and ζ∗, respectively, m i 3 m i i 3 i i i corresponding to the desired virtual controllers given by 3 Then, for a given ui, we define νi ∈ as R ∗ c3 v := −c1[E ⊗ I3]∇W (˜z) (72) νi := ui − TiRie3, (64) m i and where c3 is a positive control gain. Next, replacing (64) into ∗ ζ := −c2v˜ + ωnα1,2, (73) (63), we obtain

1 h i c3 where ∇W (˜z) is the gradient of the barrier Lyapunov function − T˙iRi + TiRiS(ωi) e3 = νi + TiRie3 mi mi deﬁned in (15) and (13) with the weight recentered barrier 12 SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021

Preliminary control loop θ UAV attitude and thrust extraction i,2 + Ti ζ i + Translational Dynamic Eq. (61) ui ⊗ dynamics feedback ωi Ri transformation Rotational kinematics

˙ ∗ ∗ ζf,i, ζf,i ζi vf,i, v˙f,i vi Backstepping Filter Acceleration Filter Gradient pj control H1(s) control H1(s) control j ∈ Ni High-order output-constrained consensus control

Fig. 3: Block diagram of the hierarchical control approach. function given by Proof: The proof follows the same arguments as the ∆2 ∆2 proof of Theorem 1. First, following the same arguments used B (z ) =κ ln k − ln k in Section IV-A, the system (70) in closed-loop with (76) may k k 1,k ∆2 − |z |2 ∆2 − |zd|2 k k k k be written in singular-perturbation form |z |2 |zd|2 + κ ln k − ln k , ˙ > > 2,k 2 2 d 2 2 z˜t = − c1[Et E ⊗ In]∇W (˜z) + [Et ⊗ In] [˜v +α ˜1,1] |zk| − δk |zk| − δk (74) (78a) 2 δk 1 κ := , κ := . (75) v˜˙ = − sat(−c2v˜ + ωnα˜1,2) + ζ˜ +α ˜2,1 − ωnα˜1,2 + θ2 1,k d 2 d 2 2 2,k 2 d 2 |zk| (|zk| − δk) ∆k − |zk| (78b) This function and its gradient are equal to zero at the desired ˜˙ ˜ d d ζ = − c3 ζ − v˜ (78c) formation configuration, i.e., Bk(zk) = 0, ∇Bk(zk) = 0. Moreover it directly encodes the constraints in terms of the ˙ ˜ ∂h(ξ) ˙ > h > > ˜>i α˜ =Aα˜ − ξ, ξ = z˜t v˜ ζ . (78d) original edge-state zk, i.e., Bk(zk) → ∞ as either |zk| → ∆k ∂ξ or |zk| → δk —see Remark 2. Now, proceeding as in Section IV-B, in order to apply the Then, akin to (23), the new input u is given by singular-perturbation result for multi-stable systems derived in [35, Theorem 2], we need to show that the boundary layer u := −c ζ˜ + ω α − v.˜ (76) 3 n 2,2 system is asymptotically stable and that the reduced slow After the developments in Section III and Theorem 1, the system is input-to-state stable with respect to the set W ×{0}2 transformed controller (76), with (72)-(73) and (19), solves the and input θi,2, i ≤ N. robust formation problem with output constraints for system Since A˜ is Hurwitz, the origin for the boundary-layer system (69). However, a closer inspection shows that there is one (41) is exponentially stable. Now consider the reduced system more technical difficulty to circumvent. Note that, from (67), > > z˜˙t = −c1[E E ⊗ In]∇W (˜z) + [E ⊗ In]˜v (79a) the dynamic solution to the equation (61) is valid if and only t t v˜˙ = −sat(c2v˜) + ζ˜ + θ2 (79b) if Ti 6= 0. In order to address this additional constraint, we ˙ perform a control redesign which respects the control method ζ˜ = −c3 ζ˜ − v.˜ (79c) and the stability analysis in Sections III-B and IV. From (52) the subsystem (79a) is input-to-state stable with T 6= 0 Note that, from (61), the condition i is satisfied if the respect to the set W and to the input v˜. Next, consider ζ∗ ζ∗ 6= ge i ≤ N desired virtual control i satisfies i 3, for all . the subsystem (79b)-(79c). Let ε ∈ (0, 1) and define the ζ∗ u Therefore, the virtual control input is modified to Lyapunov function ζ∗ := (−c v˜ + ω α ) , sat 2 n 1,2 (77) (1 + c3 εu) 2 1 2 > V2(˜v, ζ˜) = |v˜| + |ζ˜| + εu ζ˜ v,˜ (80) N N 2 2 where sat( · ) is a saturation function R → R defined > which is positive definite. Its derivative along (79b)-(79c) element-wise, i.e., sat(s) = σ(s )> ··· σ(s )> , where, 1 N satisfies e.g., σ(s ) = sign(s ) min{|s |, ζ¯ }, with ζ¯ < g, or any i i i M M ˙ ˜ other odd monotonic function, bounded in absolute value. V2(˜v, ζ) ≤ − (1 + εuc3 − γv)|v˜|sat(c2|v˜|) 2 Then, we have the following. εu 2 2εu 2 − c3 − εu − |ζ˜| + |θ| . Proposition 1: For almost any initial conditions satisfying 1 + c3 εu 1 + c3 εu z(0) ∈ D, except for a set of measure zero, there exists ∗, (81) ∗ such that, for ∈ (0, ] where := 1/ωn, the control law Hence, choosing γv > 0 and εu > 0 small enough so that 2 (67)-(68) and (76), with (19), (72) and (77), solves the robust εu γ2 := c3 − εu − > 0 formation problem with output constraints for system (55). 1 + c3 εu SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021 13

and γ1 := 1 + εuc3 − γv > 0, we have It is also assumed that the UAVs are subject to a disturbance modeled as a smoothed vanishing step, that is, ˙ ˜ ˜ 2 2 V2(˜v, ζ) ≤ −γ1|v˜|sat(c2|v˜|) − γ2|ζ| + γ3|θ2| (82) > θi(t) = − σi(t) [1 1 0]  where γ3 := 2εu/(1 + c3 εu). The inequality (82) implies −0.6 tanh(2(t − 15)) − 1 if i ∈ {3, 5}  input-to-state stability of (79b)-(79c) with respect to the origin σi(t) = 0.6 tanh(2(t − 15)) − 1 if i = 2 and to θ2. Using [36, Theorem 3.1] we conclude that, for all  0 if i ∈ {1, 4, 6}. initial conditions ξ(0) such that zt(0) ∈ Dt and all essentially bounded inputs θ2, the reduced system (79) is input-to-state In Fig. 5 are illustrated the paths of each agent as well as the 2 stable with respect to WΘ := W × {0} and to the input θ2. final desired formation for the multi-agent system. In Fig. 6 Furthermore, WΘ qualifies as a W-limit set for (79). are presented the trajectories of the inter-agent distances. As Now, since the boundary layer system is exponentially can be seen from the figure, both connectivity and collision stable and the reduced system is input-to-state stable with avoidance constraints (dashed lines) are always respected, even respect to WΘ and θ2, using Theorem 1, we conclude that in the presence of the disturbance. Furthermore, as soon as the the controller (76), with (19), (72), and (77) solves the robust disturbance vanishes after 15 seconds, the agents converge to consensus problem with output constraints for system (70). the desired static formation. This, in turn, implies that the actual controllers (67)-(68) solve the robust formation problem with output constraints for multi- agent system (55).

−1 C. Simulation results In this section we illustrate the performance of the con- 0 [m]

troller (67)–(68) via a numerical example consisting in the z 1 rendezvous of six UAVs, subject to inter-agent collision avoidance and connectivity restrictions. It is assumed that the 2 measurement range of each agent is different and that they are equipped with proximity sensors. Under these conditions, −5 0 −3 it is natural to model the network using a directed topology −1 as in Fig. 4. It is assumed, however, that only at the initial −2 1 time the vehicles are interconnected, so the controller must 3 −4 5 preserve such connectivity. y [m] 7 x [m] The initial conditions and constraint parameters are presented in Table I. The desired formation corresponds to a Fig. 5: Paths of the agents. hexagon and is determined by the desired relative position vector zdk = (zdk,x, zdk,y, zdk,z), for each k ≤ 5, set to (1, 0.5, 0), (−1, 1.5, 0), (−1, 0.5, 0), (−2, 1, 0), (−1, 0.5, 0). 4

3 3 e2 e4 5 e1 [m] 1 2 | k 2 z 4 e3 e5 6 | 1 Fig. 4: Interaction topology: directed spanning tree 0 0 10 20 30 40 The saturation limit for the desired controller of the transla- 2 t [s] tional dynamics was set to ζ¯M = 7 m/s , the controller gains to c1 = 1, c2 = 0.8, c3 = 3, and the ﬁlter natural frequency Fig. 6: Distances between neighbor UAVs. The dashed lines repre- to ωn = 350 rad/s. We consider the mass of each drone to be sent the connectivity and collision avoidance constraints. mi = 0.4 kg.

TABLE I: Initial conditions and constraint parameters VI.CONCLUSIONS

Index px py pz vx vy vz ∆k δk We presented a control framework of broad applicability 1 2.4 0 -1 0.6 -0.8 0 2.5 0.2 to solve the consensus problem for systems of high relative 2 -0.58 -0.9 0 -0.3 0 0 3.4 0.2 degree in feedback form, subject to inter-agent constraints. 3 4 1.8 0 1.1 0.3 0 3.8 0.2 4 5 -2 0 0.1 0 0 3.5 0.2 The design is based on the gradient of barrier Lyapunov 5 -4.2 -0.45 0 0 0 0 3.7 0.2 functions and on the command ﬁltered backstepping approach. 6 -2 -4.2 2 -0.8 0 0 4.2 0.2 From a theoretical viewpoint, it is important to emphasize that 14 SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021

∗ d ∗ d beyond mere convergence to the consensus manifold, we also Now, since δk < |z˜k + zk| < ∆k, we have z˜k + zk 6= 0. d ∗ d established robustness in the sense of practical input-to-state Also, zk 6= 0. Therefore, %k(˜zk + zk) 6= 0. It follows that ∗ d ∗ multi-stability with respect to external disturbances. Moreover, z˜k + zk =: zk, which is the critical point expressed in d our results hold for both undirected and directed graphs. the original relative-position coordinates, is aligned with zk. Furthermore, we showed that our control framework is Furthermore, for each k ≤ M the barrier-Lyapunov function versatile in that it serves as basis for consensus control Wk may possess only two singular points belonging to {zk ∈ n design of systems that are not, a priori, in the assumed strict- R : δk < |zk| < ∆k}, but these must have opposite sign. ∗ d feedback form. In particular, we solved the rendezvous (open) Consequently, there exists a > 0 such that zk = −a zk or, problem for a group of UAVs subject to connectivity and equivalently, inter-agent collision-avoidance constraints. From a control- ∗ 1 %k(zk) = − < 0, (90) practice viewpoint, it is remarked that each agent’s control a input requires only its own velocity and orientation, as well as required. as the relative position of its neighbors. In that light, we d On the other hand, we have from (84) that %k(zk) = 1, so believe that our theoretical results may pave the way for d from (87), we can conclude that zk is a minimum. solving other consensus-based problems in realistic settings for different types of dynamical systems. Current research fo- APPENDIX II cuses on solving the consensus-tracking problem with limited STABILITY OF MULTIPLE INVARIANT SETS measurements and the generalization of our main theoretical statements to multiagent systems interconnected over arbitrary Terminology: A function γ : R≥0 → R≥0 is said to be of directed graphs (containing an underlying spanning tree). class K, if it is continuous, strictly increasing and zero at zero. If moreover γ(s) → ∞ as s → ∞, we say that γ ∈ K∞.A APPENDIX I function β : R≥0 × R≥0 7→ R≥0 is said to be of class KL CRITICALPOINTSOFTHEBARRIER LYAPUNOV FUNCTION if, for each fixed t, the function β(·, t) is of class K and for each fixed s, the function β(s, ·) is non-decreasing and tends First note that the gradient of the barrier Lyapunov function to zero as t → ∞. (13) with (74) may be written as The following results on practical input-to-state stability of ∂Wk(˜zk) d d d multi-stable systems is adapted from [35, Theorem 2] to the ∇Wk(˜zk) := = %k(˜zk + zk) z˜k + zk − zk, (83) ∂z˜k notation used in this paper. where Theorem 3: Consider a singularly perturbed system of the 2 κ1,k κ2,kδk form (38). Assume that: %k(sk) = 1 + 2 2 − 2 2 2 . (84) ∆k − |sk| |sk| (|sk| − δk) 1) the reduced system (39) is input-to-state stable with Then the Hessian of the barrier Lyapunov function reads respect to an acyclic W-limit set WΘ and an input θ; 2) the equilibrium α˜ = 0 of the boundary layer system (41) ∂ h i H (˜z ) := ∇W (˜z ) is globally asymptotically stable. k k ∂z˜ k k k Then, there exist a class KL function β and a class K d d d d> α ∞ =% (˜z + z )I + 2˜% (˜z + z ) z˜ + z z˜ + z ∗ k k k N k k k k k k k function ηθ and, for any pair d1, d2 > 0, there exists an > 0 (85) such that, for any ∈ (0, ∗], any essentially bounded function where θ(t), and any initial condition ξ(0) ∈ D × nN(ρ−1), and κ κ κ t R 1,k 2,k 2,k max{|ξ(0)| , |α˜(0)|, kθk , kθ˙k } ≤ d , it holds that %˜k(sk) := 2 + 2 − 4 . (86) WΘ ∞ ∞ 1 2 2 2 2 |sk| ∆k − |sk| |sk| − δk lim sup |ξ(t)|WΘ ≤ ηθ(kθk∞) + d2 (91a) From a straightforward computation, we see that the eigen- t→+∞ values of Hk(˜zk) are t |α˜(t)| ≤ βα |α˜(0)|, + d2. ∀t ≥ 0 (91b) d λi,k(˜zk)=%k(˜zk + zk), i ∈ {1, . . . , n − 1} (87a) d d d 2 λn,k(˜zk)=%k(˜zk + zk) + 2˜%k(˜zk + zk)|z˜k + zk| . (87b) For completeness we include the following elements on To show that z˜∗ is a saddle point it suffices to prove that at k input-to-state multi-stability adapted, respectively, from [36, least one of these eigenvalues is negative. From (84), (86), and d Definition 2.7] and [36, Theorem 2.8]. (75), it follows that for all δk < |z˜k + zk| < ∆k, 1 Definition 2: A C function V : M → R≥0 is a practical 2 2 2 κ1,k(∆k + |sk| ) ISS-Lyapunov function for a system x˙ = f(x, θ) if there exist %k(sk) + 2˜%k(sk)|sk| =1 + 2 2 2 (∆ − |s | ) K∞ functions η1, η, γ and q ≥ 0 such that, for all x ∈ M k k (88) 2 2 2 and all θ, the following holds: κ2,kδk(3|sk| − δk) + 2 2 2 2 > 1. |sk| (|sk| − δk) η1(|x|W ) ≤V (x) ∗ ∗ d > (92) Hence, we show that λi,k(˜zk) = %k(˜zk + zk) < 0. To that ∇V (x) f(x, θ) ≤ − η(|x|W ) + γ(|θ|) + q. end, note that from (83), since z˜∗ is a singular point of W , k k If (92) holds with q = 0, then V is said to be an ISS-Lyapunov ∗ d ∗ d d %k(˜zk + zk) z˜k + zk = zk. (89) function. SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021 15

Theorem 4: Consider a system x˙ = f(x, θ) and an acyclic [22] J. Liu, C. Wang, and Y. Xu, “Distributed adaptive output consensus W-limit set W. Then, system x˙ = f(x, θ) is input-to-state tracking for high-order nonlinear time-varying multi-agent systems with output constraints and actuator faults,” Journal of the Franklin Institute, stable with respect to input θ and to the set W if and only if vol. 357, no. 2, pp. 1090–1117, 2020. it admits an ISS-Lyapunov function. [23] C. P. Bechlioulis and G. A. Rovithakis, “Decentralized robust synchronization of unknown high order nonlinear multi-agent systems with prescribed transient and steady state performance,” IEEE Transactions REFERENCES on Automatic Control, vol. 62, no. 1, pp. 123–134, 2016. [24] J. Fu, G. Wen, Y. Lv, and T. Huang, “Barrier function based consensus [1] W. Ren and R. Beard, Distributed Consensus in Multi-vehicle Cooper- of high-order nonlinear multi-agent systems with state constraints,” in ative Control: Theory and Applications, 1st ed. Springer Publishing International Conference on Neural Information Processing. Springer, Company, Incorporated, 2010. 2019, pp. 492–503. [2] Y. Guo, Distributed cooperative control: emerging applications. John [25] J. Ghommam, L. F. Luque-Vega, and M. Saad, “Distance-based forma- Wiley & Sons, 2017. tion control for quadrotors with collision avoidance via lyapunov barrier International Journal of Aerospace Engineering [3] S. Zhao, “Affine formation maneuver control of multiagent systems,” functions,” , 2020. IEEE Transactions on Automatic Control, vol. 63, no. 12, pp. 4140– [26] Y. Zou and Z. Meng, “Distributed hierarchical control for multiple verti- 4155, 2018. cal takeoff and landing UAVs with a distance-based network topology,” International Journal of Robust and Nonlinear Control, vol. 29, no. 9, [4] F. Chen, W. Ren et al., “On the control of multi-agent systems: A pp. 2573–2588, 2019. survey,” Foundations and Trends® in Systems and Control, vol. 6, no. 4, [27] C. K. Verginis, A. Nikou, and D. V. Dimarogonas, “Robust formation pp. 339–499, 2019. control in se(3) for tree-graph structures with prescribed transient and [5] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and coopera- steady state performance,” Automatica, vol. 103, pp. 538 – 548, 2019. tion in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1, pp. [28] C. P. Bechlioulis and G. A. Rovithakis, “Prescribed performance adap- 215–233, Jan. 2007. tive control for multi-input multi-output affine in the control nonlinear [6] D. Mukherjee and D. Zelazo, “Robustness of consensus over weighted systems,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. digraphs,” IEEE Transactions on Network Science and Engineering, 1220–1226, 2010. 2018. [29] D. Zelazo, A. Rahmani, and M. Mesbahi, “Agreement via the edge [7] A. Dong and J. A. Farrell, “Cooperative control of multiple nonholo- Laplacian,” in 46th IEEE Conference on Decision and Control, New IEEE Transactions on Automat. Contr. nomic mobile agents,” , vol. 53, Orleans, LA, USA, 2007, pp. 2309–2314. no. 6, pp. 1434–1447, 2008. [30] J. A. Marshall, M. E. Broucke, and B. A. Francis, “Formations of [8] C. P. Bechlioulis and K. J. Kyriakopoulos, “Robust model-free formation vehicles in cyclic pursuit,” IEEE Transactions on Automatic Control, control with prescribed performance and connectivity maintenance for vol. 49, no. 11, pp. 1963–1974, Nov 2004. nonlinear multi-agent systems,” in 53rd IEEE Conference on Decision [31] K. P. Tee, S. S. Ge, and E. H. Tay, “Barrier Lyapunov functions for the and Control, 2014, pp. 4509–4514. control of output-constrained nonlinear systems,” Automatica, vol. 45, [9] A. Nikou, C. K. Verginis, and D. V. Dimarogonas, “Robust distance- no. 4, pp. 918–927, 2009. based formation control of multiple rigid bodies with orientation align- [32] E. Rimon and D. E. Koditschek, “Exact robot navigation using artificial ment,” IFAC-PapersOnLine, vol. 50, no. 1, pp. 15 458 – 15 463, 2017, potential functions,” IEEE Transactions on Robotics and Automation, 20th IFAC World Congress. vol. 8, no. 5, pp. 501–518, Oct. 1992. [10] X. Wang, G. Wang, S. Li, and K. Lu, “Composite sliding-mode [33] M. Krstic, P. V. Kokotovic, and I. Kanellakopoulos, Nonlinear and consensus algorithms for higher-order multi-agent systems subject to adaptive control design. John Wiley & Sons, Inc., 1995. disturbances,” IET Control Theory & Applications, vol. 14, no. 2, pp. [34] J. A. Farrell, M. Polycarpou, M. Sharma, and W. Dong, “Command 291–303, 2019. filtered backstepping,” IEEE Transactions on Automatic Control, vol. 54, [11] C.-C. Hua, K. Li, and X.-P. Guan, “Leader-following output consensus no. 6, pp. 1391–1395, 2009. for high-order nonlinear multiagent systems,” IEEE Transactions on [35] P. Forni and D. Angeli, “Perturbation theory and singular perturbations Automatic Control, vol. 64, no. 3, pp. 1156–1161, 2018. for input-to-state multistable systems on manifolds,” IEEE Transactions [12] H. Rezaee and F. Abdollahi, “Average consensus over high-order multia- on Automatic Control, vol. 64, no. 9, pp. 3555–3570, 2018. gent systems,” IEEE Transactions on Automatic Control, vol. 60, no. 11, [36] ——, “Input-to-state stability for cascade systems with multiple invariant pp. 3047–3052, 2015. sets,” Systems & Control Letters, vol. 98, pp. 97–110, 2016. [13] Z. Zuo, B. Tian, M. Defoort, and Z. Ding, “Fixed-time consensus [37] Y.-H. Lim and H.-S. Ahn, “Consensus under saturation constraints tracking for multiagent systems with high-order integrator dynamics,” in interconnection states,” IEEE Transactions on Automatic Control, IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 563–570, vol. 60, no. 11, pp. 3053–3058, 2015. 2017. [38] D. H. Nguyen, T. Narikiyo, and M. Kawanishi, “Robust consensus [14] M. Mesbahi and M. Egerstedt, Graph theoretic methods in multiagent analysis and design under relative state constraints or uncertainties,” networks, ser. Princeton series in applied mathematics. Princeton: IEEE Transactions Automatic Control, vol. 63, no. 6, pp. 1784–1790, Princeton University Press, 2010, oCLC: ocn466341412. June 2018. [15] M. Hua, T. Hamel, P. Morin, and C. Samson, “Introduction to feedback [39] V. T. Pham, N. Messai, D. H. Nguyen, and N. Manamanni, “Robust control of underactuated VTOL vehicles: A review of basic control formation control under state constraints of multi-agent systems in design ideas and principles,” IEEE Control Systems Magazine, vol. 33, clustered networks,” Systems & Control Letters, vol. 140, p. 104689, no. 1, pp. 61–75, 2013. 2020. [16] H. A. Poonawala and M. W. Spong, “Preserving strong connectivity [40] G. Wang, C. Wang, and X. Cai, “Consensus control of output- in directed proximity graphs,” IEEE Transactions Automatic Control, constrained multiagent systems with unknown control directions under a vol. 62, no. 9, pp. 4392–4404, Sep. 2017. directed graph,” International Journal of Robust and Nonlinear Control, [17] M. Santilli, P. Mukherjee, A. Gasparri, and R. K. Williams, “Distributed vol. 30, no. 5, pp. 1802–1818, 2020. connectivity maintenance in multi-agent systems with field of view [41] T.-S. Li, D. Wang, G. Feng, and S.-C. Tong, “A DSC approach interactions,” in 2019 American Control Conference (ACC), 2019, pp. to robust adaptive NN tracking control for strict-feedback nonlinear 766–771. systems,” IEEE transactions on systems, man, and cybernetics, part b [18] Z. Zhou and X. Wang, “Constrained consensus in continuous-time mul- (cybernetics), vol. 40, no. 3, pp. 915–927, 2009. tiagent systems under weighted graph,” IEEE Transactions on Automatic [42] B. Xu, D. Wang, Y. Zhang, and Z. Shi, “Dob-based neural control Control, vol. 63, no. 6, pp. 1776–1783, 2017. of flexible hypersonic flight vehicle considering wind effects,” IEEE [19] Y. Shang, “Resilient consensus in multi-agent systems with state con- Transactions on Industrial Electronics, vol. 64, no. 11, pp. 8676–8685, straints,” Automatica, vol. 122, p. 109288, 2020. 2017. [20] H. Chu, D. Yue, C. Dou, and L. Chu, “Consensus of multiagent systems [43] H. Du, M. Z. Chen, and G. Wen, “Leader–following attitude consensus with time-varying input delay and relative state saturation constraints,” for spacecraft formation with rigid and flexible spacecraft,” Journal of IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2020. Guidance, Control, and Dynamics, vol. 39, no. 4, pp. 944–951, 2016. [21] E. Restrepo, A. Lor´ıa, I. Sarras, and J. Marzat, “Stability and robustness [44] E. Nuno,˜ D. Valle, I. Sarras, and L. Basanez,˜ “Leader–follower and lead- of edge-agreement-based consensus protocols for undirected proximity erless consensus in networks of flexible-joint manipulators,” European graphs,” International Journal of Control, pp. 1–9, 2020. Journal of Control, vol. 20, no. 5, pp. 249–258, 2014. 16 SUBMITTED TO IEEE TRANS. AUTOMAT. CONTROL, APRIL 27, 2021

[45] G. Duan, “High-order fully actuated system approaches: Part ii. general- [64] A. Roza, M. Maggiore, and L. Scardovi, “Local and distributed ren- ized strict-feedback systems,” International Journal of Systems Science, dezvous of underactuated rigid bodies,” IEEE Transactions on Automatic pp. 1–18, 2020. Control, vol. 62, no. 8, pp. 3835–3847, 2017. [46] H. K. Khalil, Nonlinear systems; 2nd ed. Upper Saddle River, NJ: [65] H. Wang, “Second-order consensus of networked thrust-propelled ve- Prentice-Hall, 1996. hicles on directed graphs,” IEEE Transactions on Automatic Control, [47] K. P. Tee and S. S. Ge, “Control of state-constrained nonlinear systems vol. 61, no. 1, pp. 222–227, 2016. using integral barrier Lyapunov functionals,” in 2012 IEEE 51st IEEE Conference on Decision and Control (CDC). IEEE, 2012, pp. 3239– Esteban Restrepo obtained his BSc degree on Mechatron- 3244. ics engineering from Universidad EIA, Envigado, Colom- [48] C. Feller and C. Ebenbauer, “Weight recentered barrier functions and bia and from ENSAM, Paris, France in 2017. He obtained smooth polytopic terminal set formulations for linear model predictive his MSc in Robotc Systems Engineering from ENSAM, control,” in 2015 American Control Conference (ACC). IEEE, 2015, Paris, France in 2018, receiving the Silver Medal honorary pp. 1647–1652. distinction. He is currently pursuing the PhD degree in [49] D. Panagou, D. M. Stipanovic, and P. G. Voulgaris, “Distributed coor- Automatic Control at the University Paris-Saclay, France. dination control for multi-robot networks using Lyapunov-like barrier His research interests include multi-agent systems and functions,” IEEE Transactions on Automat. Contr., vol. 61, no. 3, pp. nonlinear control with application to autonomous robots 617–632, Mar. 2016. and aerospace systems. [50] M. Ji and M. Egerstedt, “Distributed coordination control of multiagent systems while preserving connectedness,” IEEE Transactions Robot., Antonio Lor´ıa obtained his BSc degree on Electronics vol. 23, no. 4, pp. 693–703, Aug. 2007. Engineering from ITESM, Monterrey, Mexico in 1991 and [51] D. Boskos and D. V. Dimarogonas, “Robustness and invariance of his MSc and PhD degrees in Control Engg. from UTC, connectivity maintenance control for multiagent systems,” SIAM J. on France in 1993 and 1996, respectively. He has the honour Control and Optimization, vol. 55, no. 3, pp. 1887–1914, 2017. of holding a researcher position at the French National [52] Z. Zeng, X. Wang, and Z. Zheng, “Nonlinear consensus under directed Centre of Scientific Research (CNRS) since January 1999 graph via the edge Laplacian,” in The 26th Chinese Control and Decision (Senior Researcher since 2006). He has co-authored over Conference (2014 CCDC). Changsha, China: IEEE, May 2014, pp. 250 publications on control systems and stability theory 881–886. and he served as AE for IEEE Trans. Automat. Control, [53] E. Restrepo, A. Lor´ıa, I. Sarras, and J. Marzat, “Robust consensus IEEE Trans. Control Syst. Techn., IEEE Control Syst. Lett., and the IEEE and connectivity-maintenance under edge-agreement-based protocols for Conf. Editorial Board, as well as other jounrals on Automatic Control. directed-spanning tree graph,” in 21st IFAC World Congress, 2020. Dr. Ioannis Sarras (M 09) was born in Athens, Greece, [54] W. Dong, J. A. Farrell, M. M. Polycarpou, V. Djapic, and M. Sharma, in 1982. He graduated from the Automation Engineer- “Command filtered adaptive backstepping,” IEEE Transactions on Con- ing Department of the Technological Education Institute trol Systems Technology, vol. 20, no. 3, pp. 566–580, 2011. (T.E.I.) of Piraeus, , Greece, (now University of West [55] P. V. Kokotovic´ and H. J. Sussmann, “A positive real condition for global Attica) in 2004 and received the Master of Research stabilization of nonlinear systems,” Syst. & Contr. Letters, vol. 13, no. 4, (M2R) in Control Theory from the University of Paul pp. 125–133, 1989. Sabatier (France) in 2006, with scholarship from the [56] R. Sepulchre, M. Jankovic,´ and P. V. Kokotovic,´ “Recursive designs and General Michael Arnaoutis Foundation. He obtained the feedback passivation,” in Systems and control in the twenty-first century. PhD degree with distinction in Automatic Control Systems Springer, 1997, pp. 313–326. from the university of Paris-Sud XI (France) in 2010 with [57] P. Monzon´ and R. Potrie, “Local and global aspects of almost global a scholarship from the Greek Scholarships Foundation. From 2010 to 2017 stability,” in Proceedings of the 45th IEEE Conference on Decision and he was a postdoctoral researcher with the Universit Libre de Bruxelles, the Control. IEEE, 2006, pp. 5120–5125. Laboratoire des Signaux et Systmes of the National Centre for Scientific [58] M. Hua, T. Hamel, P. Morin, and C. Samson, “A control approach Research (CNRS), the Department of Automatic Control of Ecole Suprieure for thrust-propelled underactuated vehicles and its application to vtol dlectricit (SUPELEC) and the Centre of Automatic Control & Systems (CAS) drones,” IEEE Transactions on Automatic Control, vol. 54, no. 8, pp. of the cole Nationale Suprieure des Mines de Paris (MINES ParisTech), and 1837–1853, 2009. ONERA-The French Aerospace Lab. During the period 2012-2013 he worked [59] D. Lee, “Distributed backstepping control of multiple thrust-propelled as a control engineer at the French Institute of Petroleum (IFPEn). Since 2017 vehicles on a balanced graph,” Automatica, vol. 48, no. 11, pp. 2971 – he has been a Research Engineer at ONERA-The French Aerospace Lab. He 2977, 2012. has been a visiting researcher for short and long term periods at the university [60] A. Abdessameud, “Formation control of VTOL-UAVs under directed of Grningen, the Technical University of Catalunia, the university of Seville and dynamically-changing topologies,” in 2019 American Control Con- and the Indian Institutes of Technology in Bombay and in Madras. He has ference (ACC), 2019, pp. 2042–2047. over 60 papers in peer-reviewed international journals and conferences where [61] S. Bertrand, N. Gunard, T. Hamel, H. Piet-Lahanier, and L. Eck, “A he has also served as a reviewer. His research activities are in the fields of hierarchical controller for miniature VTOL UAVs: Design and stability nonlinear control and observer designs for single- or multi-agent systems with analysis using singular perturbation theory,” Control Engineering Prac- emphasis on aerospace applications. tice, vol. 19, no. 10, pp. 1099 – 1108, 2011. [62] Y. Zou and Z. Meng, “Coordinated trajectory tracking of multiple Julien Marzat graduated as an engineer from ENSEM vertical take-off and landing UAVs,” Automatica, vol. 99, pp. 33 – 40, (INPL Nancy) in 2008 and completed his PhD Thesis in 2019. 2011 and Habilitation in 2019, both from University Paris- [63] Y. Zou, Z. Zhou, X. Dong, and Z. Meng, “Distributed formation Saclay. He is currently a Research Scientist at ONERA, control for multiple vertical takeoff and landing UAVs with switching where his research interests include guidance, control topologies,” IEEE/ASME Transactions on Mechatronics, vol. 23, no. 4, and fault diagnosis for autonomous robots and aerospace pp. 1750–1761, 2018. systems.