Dynamics–Decoupling Control for Strings
of Heterogenous Nonlinear Autonomous Agents
Abstract
We introduce a distributed control architecture for a class of heterogeneous, nonlinear dynamical agents moving in the “string” formation, while guaranteeing trajectory tracking, collision avoidance and the preservation of the formation’s topology. Each autonomous agent uses information and relative measurements only with respect to its predecessor in the string. The performance of the scheme is independent of the number of agents in the network and also on the agent’s relative position in the network. The scalability is a consequence of the “decoupling” of a certain bounded approximation of the closed–loop equations, which allows the regulation and controller design (at each agent) to be done individually, in a completely decentralized manner. A practical method for compensating communication induced delays is also presented. Numerical examples illustrate the effectiveness and the main features of the proposed approach.
I Introduction
Practical algorithms for distributed control of dynamically coupled systems are needed in many diverse applications raging from formation control of autonomous mobile agents [Ali, PATRU], synchronization of local clocks offsets or phase differences between (neighboring) coupled oscillators [Mallada] or synchronous generators in power networks, sensor networks, load balancing [SASE], distributed agreement algorithms, cooperative control of multirobot systems [CINCI], opinion dynamics etc. In the specific setting of autonomous agents, the intricacies of dynamical coupling are not caused by the structure of the plant but rather by: (i) the structure of the cost functional resulting from the definition of the regulated measurements (e.g. in formation control  the interagent spacing distances defining the topology of the formation) and (ii) the coupling induced in the entailing feedback loop (with the distributed controller). The subsequent controller design problem is further complicated by the constraints to be imposed on the sensing and communications radii^{1}^{1}1By communications radius we mean the number of hops associated to an agent in the communications graph. The communications graph designates for each agent the neighbors with which it is able to communicate. of each agent.
The goal of the distributed control scheme is for all the agents to attain certain types of global collaborative behavior, such as their outputs/states reaching agreement in a precisely defined metric. In modern control parlance this class of objectives have been dubbed consensus [OPT, NOUA] and synchronization [ZECE, UNSPE, DOISPE, TREISPE, PAISPE] problems. In the existing literature there is no clear demarcation line for this terminology (consensus versus synchronization), therefore we refer to [SAPTE, Section 1], [Rick2016, Section 1] for a useful discussion. It is worth mentioning that synchronization is a far more ambitious objective than classical reference tracking [Kwakernaak, Chapter 5.4], not only due to the distributed nature of the problem but also because the reference signals are never explicitly available to all agents, while in certain scenarios the synchronization trajectory is not even assigned beforehand and therefore there is no explicit reference to be tracked [Rick2016].
In this paper we deal with a group of heterogenous, nonlinear dynamical agents that solve a conventional agreement task (velocities matching in our case) but to which we append a specific set of constraints linking the individual states of two adjacent agents (in our case their positions in space). It turns out that the inclusion of such constraints render existing methods in distributed synchronization (e.g. distributed output regulation) inapplicable directly. The constraints relate to the interspacing distance between two agents, defined as the difference of their individual positions in space. In distancebased formation control^{2}^{2}2When referring to the position in space of an autonomous agent within a group of agents, the position must be defined with respect to an inertial system of reference common to all agents, e.g. a Global Positioning System. Note that the methods described here do not employ positioning systems, relying exclusively on measurements of relative distances between agents (acquired using for example onboard lidars). such constraints on the interspacing distances between neighboring agents arise naturally when: (i) defining the formation’s steadystate topology and (ii) when framing the collision avoidance requirement, via restrictions on interagent distances in the transitory regime. For illustrative simplicity we will only look at the string graph, while our method handles the heterogeneity of the formation and the interagent communications time delays, overlooking applications in the automotive industry. For consistency, we will refer to this setting as a synchronization problem.
Existing results in distributed synchronization (or distributed agreement) for Linear and Time Invariant (LTI) dynamics rely on observerbased distributed controllers [SAPTE, CINSPE, SAISPE], while the results from [SAPTISPE, OPTISPE, NOUASPE] remove the necessity of communicating the internal states of the local observers/subcontrollers and rely solely on the communication of the agents’ outputs. The results [2ZECI, 2ZECISIUNU, 2ZECISIDOI] for the heterogenous case, rely on the existence of a (virtual) exosystem generating the reference trajectory, while [2ZECISITREI] outlines the intrinsic connections between the set of possible agreement trajectories and the sharing of all agents of certain “common dynamics”. The authors’ recent results in [TAC2016] provide a solution in the more ambitious setting of a distributed \mathcal{H}_{2}/\mathcal{H}_{\infty} disturbances attenuation problem for the string graph, encompassing heterogenous agents and communications induced time delays. In this context, the current paper can be looked at as an extension to nonlinear control of the novel ideas for LTI dynamics from [TAC2016].
Theoretical advancements for nonlinear agents are in an incipient phase and have only been available more recently. The Lyapunov function approach in [2ZECISISAPTE] is based on differential inequalities. The results in [2ZECISIPATRU] pertaining to weakly minimum phase nonlinear agents are viable only under a passivity hypothesis, while those in [2ZECISICINCI, 2ZECISISASE] pertain to globally Lipshitzlike conditions on the nonlinearities and leaderfollowing networks. The reference [2ZECISIOPT] brings forward a necessary condition but no controller synthesis procedure while in [2ZECISINOUA] the agreement objective can only be set to a constant. Very recent results applicable to more complicated nonlinear dynamics include feedforward schemes [3ZECI] or are set up as cooperative output regulation problems e.g. [3ZECISITREI, 3ZECISIPATRU, 3ZECISIUNU, 3ZECISIDOI] and the references within, among which [3ZECISITREI, 3ZECISIPATRU] deal with leaderfollowing networks. A notable feature of [3ZECISIUNU, 3ZECISIDOI] and especially [Rick2016] is that the subcontroller corresponding to an agent can be designed independently to all other subcontrollers. The downside of [3ZECISIUNU, 3ZECISIDOI] is the requirement of full state information exchange among agents, requirement entirely circumvented in [Rick2016] in a generic setting.
IA Motivation and Scope of Work
The references above deal with the standard setup in which agents must achieve agreement of certain, prespecified variables from each agent’s own statespace. For the class of problems treated in this paper, this takes the form of guaranteeing velocity matching in the steadystate, irrespective of the velocity profile of the leader (whose trajectory represents the reference for the entire formation) and which is seen as an adversarial player. However, the problem statement is further complicated by the inclusion of constraints that impose a substantial “coupling” between individual state variables of distinct agents, where these states represent the positions of agents^{3}^{3}3See also footnote 2 on page one.. These constraints cast on the relative distances between two neighboring agents render the existing methods referred above inapplicable directly^{4}^{4}4Existing results [Baillieul, AndersonRigid, MorseRigid] on graph rigidity show how easy it is for these types of constraints to cause certain variations of this formulation of the distancebased formation control problem to become not wellposed. That happens when in an effort to preserve the topology of the formation, the controller encounters simultaneously conflicting constraints.. However, the constraints are needed in order to frame sufficient conditions for: (i) collision avoidance in the transitory regimes and (ii) topology preservation of the formation in steadystate (i.e. the interspacing distances between agents must converge asymptotically to certain prespecified, constant values).
For high performance displacementbased formation control [formsurv, Section 6], global positioning systems are not viable due to their relative large latencies and problematic reliability. The absence of a global coordinate system (combined with the fact that we avoid the use of accelerometers^{5}^{5}5Longitudinal accelerometers are notoriously unreliable for applications in the automotive industry.) requires that the agents must rely only on real time measurements of relative variables with respect to their neighbors, with all the difficulties such schemes entail, including the fact that collision avoidance and topology preservation cannot be reduced to a cooperative, output regulation problem (as those referred above).
IB Contributions of the Paper
Our controller’s architecture is borrowed from platoon control literature^{6}^{6}6The conceptual architecture behind such distributed control schemes have been dubbed Cooperative Adaptive Cruise Control in the platooning control parlance [Ploeg, TAC2016]. and is conceptually different from the aforementioned methods (e.g. [3ZECISITREI, 3ZECISIPATRU, Rick2016] and the references within). Unlike [3ZECISITREI, 3ZECISIPATRU] it doesn’t require exchange of internal states (plant internal states or controller states) among agents. In turn, each agent needs to transmit its control action only to its immediate follower in the string. Furthermore, the design of each subcontroller, (“local” to an agent ) can be done in a completely independent manner  feature which is known to be especially challenging in distributed synchronization (see [Rick2016] and the references within for a comprehensive discussion in a related setting). Indeed, solely the knowledge of the dynamical model of the immediate predecessor is required for the local subcontroller at each agent, but once this is made available the regulation and controller design (at each agent) is done individually, in a completely decentralized manner.
Perhaps the most appealing feature of the proposed scheme is a particular dynamic “decoupling” of a certain bounded approximation of the closed–loop equations, entailing that individual, local analyses of the closed–loop stability at each agent will in turn guarantee the aggregated stability of the entire formation. This entails a complete scalability with respect to: (i) the number of agents in the string and (ii) the same performance irrespective of the relative position in formation (front or back of the string).
By comparison to our method, the main result in [SCL2012] is restricted to an undirected topology of the distributed controller, with stringent requirements involving: (i) the transmission of the exact state of the leader to many agents in the formation (virtual leaders) and (ii) the necessity of high control gains (see the last paragraph in [Sabau2017, page 1] for a more detailed discussion).
Overall, our scheme improves on existing results in the following essential aspects:

The agent dynamics are permitted to be heterogenous as long as they are nonlinear, globally Lipschitz.

The agents achieve the synchronization of their velocities in the steadystate, while guaranteeing collision avoidance.

The scheme guarantees steadystate topology preservation. Very recent results [Nader] are able to achieve this but only for identical singleintegrators, exploiting an adaptation of the CuckerSmale type nonlinear controllers [CuckerSmale2007]. Collision avoidance is obtained in [CuckerDong2011] for singleintegrators but without topology preservation.

The distributed controller determines a “dynamic decoupling” of the closed loop, rendering the same performance independent of the number of agents or the relative position in formation. (The Lyapunov function guaranteeing the closedloop stability of the entire formation is actually the sum of “local” Lyapunov functions, proper to each agent. This decoupling is also the root cause of the feature stated at the next point.)

Completely independent regulation and controller design at each agent, under the sole requirement that each agent knows the dynamical model of its predecessor^{7}^{7}7This aspect is essential when dealing with merging/exiting of agents, since it allows only local reconfigurations (at the merging agent or at the follower of the exiting agent) without the need to reconfigure the control scheme for the entire formation..

We provide a simple, practical method for the efficient compensation of the notoriously detrimental (communications induced) timedelays [Rick2014], at the expense of a negligible loss in performance.
IC Paper Organization
The paper is organized as follows: in Section II we introduce the general framework and problem formulations. Section III provides a preliminary description of the novel distributed control architecture introduced in this work along with a first glimpse at the closedloop dynamics “decoupling” featured by the control scheme. Section IV contains the main result as it delineates the guarantees for stability, velocity matching, collision avoidance and topology preservation. Finally, Section V outlines a practical delays compensation mechanism while Section VI provides an illustrative numerical example, worked out on an actual dynamical model for road vehicles.
II General Framework and Problem Statement
The notation being used is fairly standard throughout the literature, for example the derivatives \frac{d}{dt}{z}(t) with respect to the time variable are sometimes denoted by \dot{{z}}(t). Also, throughout the paper it will become apparent from the context when the time argument (t) is being omitted for the sake of brevity. The notation a\overset{def}{=}b means that the left hand side quantity a is defined to be the right hand side quantity b.
Definition II.1
The \sigma–norm of a vector x is defined as
\x\_{\sigma}\overset{def}{=}\frac{1}{\sigma}\Big{[}\sqrt{1+\x\_{2}^{2}}1% \Big{]}  (1) 
where \sigma is a strictly positive constant. Note that (1) is a class \mathcal{K}_{\infty} function of \x\_{2}^{2} and is differentiable everywhere.
Definition II.2
A set \Omega is said to be forward invariant with respect to an equation, if any solution x(t) of the equation satisfies: x(0)\in\Omega\Longrightarrow x(t)\in\Omega,\>\forall t>0.
Definition II.3
Artificial Potential Function (APF). The function V_{k,k1}(\cdot) is a class C^{1}, nonnegative, radially unbounded function of \z\_{\sigma} satisfying the following properties:
(i) V_{k,k1}(\z\_{\sigma})\rightarrow\infty as (\z\_{\sigma})\rightarrow 0,
(ii) V_{k,k1}(\z\_{\sigma}) has a unique minimum, which is attained at \z\_{2}=\delta_{k}, with \delta_{k} being a positive constant.
IIA Distributed Trajectory Tracking in the String Formation
We consider a heterogeneous group of n+1 agents (e.g. autonomous road vehicles) moving along the same (positive) direction of a roadway, with the origin at the starting point of the leader. The dynamical model for the agents, relating the control signal u_{k}(t) of the k–th vehicle to its position y_{k}(t) is given by
\dot{{y}}_{k}(t)={v}_{k}(t),\quad\dot{{v}}_{k}(t)={f_{k}}({v}_{k}(t))+{u}_{k}(% t)\ ;  (2a)  
y_{k}(0)=\sum_{j=0}^{k}\ell_{j},\quad v_{k}(0)=0.  (2b) 
where v_{k}(t) is the instantaneous speed of the k–th agent, u_{k}(t) is its command signal and \ell_{k} is the initial interspacing distance between the k–th agent and its predecessor in the string. Throughout the sequel we will use the notation
y_{k}=G_{k}\star u_{k}  (3) 
to denote (especially for the graphical representations) the input–output operator G_{k} of the dynamical system from (2a), with the initial conditions (2b).
Assumption II.4
The index “0” is reserved for the leader agent, the first agent in the string. This situation leads to exactly n interagent distances, which are part of the regulated measurements.
In the rest of the paper it will become apparent from the context that we often omit the time argument (t), for the sake of brevity. Let us further define
{{z}}_{k}\overset{def}{=}{y}_{k1}{y}_{k},\quad{{z}}_{k}^{v}\overset{def}{=}{% v}_{k1}{v}_{k}\quad\text{for}\quad 1\leq k\leq n,  (4) 
to be the interspacing and relative velocity error signals respectively (with respect to the predecessor in the string). By differentiating the first equation in (4) it follows that \dot{{z}}_{k}(t)={{z}}_{k}^{v}(t), therefore implying that constant interspacing errors (in steady state) are equivalent with zero relative velocity errors and also allowing to write the following time evolution for the relative velocity error of the k–th vehicle
\dot{{z}}_{k}^{v}={f_{k1}}({v}_{k1}){f_{k}}({v}_{k})+{u}_{k1}{u}_{k}.  (5) 
III A Practical Distributed Control Architecture
After five decades of consistent academic efforts and hundreds of references on the subject, it turned out that control of a string of mere double integrators might well be the epitome of the difficulties typical to distributed control, since it suffers from all pitfalls one might have expected from more general and complex dynamical networks, e.g. performance is in general dependant on the number of agents in the string and on their relative position in formation and is highly sensitive to communications delays.
We introduce a novel control architecture featuring a highly beneficial “decoupling” property of the closed–loop dynamics, that resolves the troubling nested interdependencies of the regulated measurements. We consider non–linear controllers built on the socalled Artificial Potential Functions (APF), in particular we will look at control laws of the type
\begin{split}\displaystyle{u}_{k}&\displaystyle={u}_{k1}+{\beta_{k}}({v}_{k1% }{v}_{k})\nabla_{{y}_{k}}V_{k,k1}(\{y}_{k1}{y}_{k}\_{\sigma})\\ &\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad{f}_{k}({v}_{k})+{% f}_{k1}({v}_{k})\end{split}  (6) 
with k\geq 1, where each of the V_{k,k1}(\cdot) functions is an Artificial Potential Function [SCL2012, Definition 7], with \beta_{k} being a proportional gain to be designed for supplemental performance requirements. With the notation from (4), the control policy (6) for the k–th agent becomes
\begin{split}\displaystyle{u}_{k}&\displaystyle={u}_{k1}+{\beta_{k}}z_{k}^{v}% \nabla_{{y}_{k}}V_{k,k1}(\{z}_{k}\_{\sigma})\\ &\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad{f}_{k}({v}_{k})+{% f}_{k1}({v}_{k}).\end{split}  (7) 
Note that the distributed control laws rely only on information locally available to each agent, since it can further be written as the sum of the following two components: firstly, the control signal u_{k1}(t) of the preceding agent, which is received onboard the k–th agent via wireless communications (e.g. digital radio) along with the function {f}_{k1}(\cdot) characterizing the predecessor’s dynamical model. Secondly, the local component, which we denote with
u_{k}^{\ell}\overset{def}{=}{\beta_{k}}{z}_{k}^{v}\nabla_{{y}_{k}}V_{k,k1}(% \{z}_{k}\_{\sigma}){f}_{k}({v}_{k})+{f}_{k1}({v}_{k})  (8) 
and which is based solely on a high accuracy speedometer for measuring v_{k}(t) ^{8}^{8}8For automotive applications, high accuracy speedometers are affordable and widely available. On the contrary, longitudinal accelerometers are notoriously unreliable and only used for purposes extraneous to navigation, such as the triggering of airbags in a collision event. and on the measurements (4), locally available to the k–th agent (acquirable for instance via onboard LIDAR sensors^{9}^{9}9For automotive applications, commercially available affordable and high accuracy “dot” LIDARs have latencies well under 1\mus. Given the typical speeds of road vehicles, this implies that a numerical differentiation of the interspacing distance z_{k}(t) in order to obtain the relative speed z_{k}^{v}(t) is feasible via a high sampling frequency.). Thus, the control law at the k–th agent reads:
u_{k}=u_{k1}+u_{k}^{\ell}. 
In Figure 1, we denoted with K_{k} the input–output operator from z_{k},z_{k}^{v} and v_{k} respectively to u_{k}^{\ell} of the kth subcontroller from (8), namely
u_{k}^{\ell}=K_{k}\star\big{(}z_{k},\>z_{k}^{v},\>v_{k}\big{)}.  (9) 
The resulted control architecture for any two consecutive agents (k\geq 2) can be pictured as in Figure 1. For all practical purposes, the existence of a time delay on each of the feedforward links u_{k}, with 1\leq k\leq(n1) must be taken into account. For readability, these time delays are figuratively denoted by e^{\theta s} in Figure 1 (the Laplace transform of a delay of \theta seconds), representative to the situation in which the delayed version u_{k}(t\theta) version of the u_{k}(t) signal is received on board of the (k+1) agent. In applications, these delays are caused by the physical limitations of the wireless communications system used for the implementation of the feedforward link , entailing a \theta time delay at the receiver. For automotive applications the standard digital radio communications systems (included in Figure 1) are DSRC^{10}^{10}10\quad IEEE 802.11p  Dedicated Short Range Communications.
Remark III.1
Without the assumption of interagent communications delays, one might argue that information from the leader propagates instantaneously to all the agents in formation, via a relay mechanism (from each agent to its successor) and consequently the resulted distributed scheme doesn’t employ local, but rather global information from the leader. It is known that precisely this type of time delays can drastically alter the performance control architectures based on such relay schemes [rick].
Remark III.2
For the illustrative simplicity of the exposition, we look first at the scenario in which there are no time–delays induce by the interagent (wireless) communication of information, such as the predecessor’s control signal u_{k1}. A “synchronization” mechanism that can cope with the timevarying communications induced time–delays will be addressed in Section LABEL:delaycompensation.
IIIA A First Glance at the Closed–Loop Dynamics Decoupling
The control policy (7) entails a highly beneficial “decoupling” feature of the closed–loop dynamics at each agent, as illustrated next. Firstly, note that by plugging (7) into (5) we obtain the following closed–loop error equations at the k–th agent:
\dot{{z}}_{k}^{v}={{f}_{k1}({v}_{k1})f_{k1}}({v}_{k}){\beta_{k}}{z}_{k}^{% v}+\nabla_{{y}_{k}}V_{k,k1}(\z_{k}\_{\sigma}).  (10) 
The following result will be instrumental in the sequel. Consider the following Lyapunov candidate functions:
\begin{split}\displaystyle L_{k}\big{(}z_{k}(t),z_{k}^{v}(t)\big{)}\overset{% def}{=}\frac{1}{2}\Big{(}V_{k,k1}(\{{z}}_{k}(t)\_{\sigma})+\\ \displaystyle\quad\quad\quad\quad\quad\quad\quad+{z}_{k}^{v}{}^{\top}(t){z}_{k% }^{v}(t)\Big{)},\ \text{with}\;1\leq k\leq n.\end{split}  (11) 
Lemma III.3
The derivative of the Lyapunov candidate function L_{k}(\cdot,\cdot) introduced in (11) along the trajectories of (5) and (7) is given by
\begin{split}\displaystyle\frac{d}{dt}L_{k}(z_{k}(t),z_{k}^{v}(t))&% \displaystyle={z}_{k}^{v}{}^{\top}(t)\Big{(}f_{k1}\big{(}v_{k1}(t)\big{)}f_% {k1}\big{(}v_{k}(t)\big{)}\Big{)}\\ &\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad{\beta_{k}}{z}_{k}^{v}{% }^{\top}(t){z}_{k}^{v}(t)\>,\end{split}  (12) 
and does not depend on the choice of the APFs V_{k,k1}(\cdot).
Proof:
Differentiating the APF V_{k,k1}(\cdot) at the k–th agent with respect to time, yields
\begin{split}&\displaystyle\frac{d}{dt}V_{k,k1}(\{y}_{k1}{y}_{k}\_{\sigma% })={(\dot{{y}}_{k1}\dot{{y}}_{k})}^{\top}\times\\ &\displaystyle\big{(}\nabla_{{y}_{k1}}V_{k,k1}(\{y}_{k1}{y}_{k}\_{\sigma% })\nabla_{{y}_{k}}V_{k,k1}(\{y}_{k1}{y}_{k}\_{\sigma})\big{)}\end{split}  (13) 
and by employing the anti–symmetrical property of APFs [SCL2012, pp. 197] : \nabla_{{y}_{k}}V_{k,k1}(\{z}_{k}\_{\sigma})=\nabla_{{y}_{k1}}V_{k,k1}(% \{z}_{k}\_{\sigma}) we get that
\frac{d}{dt}V_{k,k1}(\{z}_{k}\_{\sigma})=2\>{\dot{{z}}_{k}}{}^{\top}\>% \nabla_{{y}_{k}}V_{k,k1}(\{z}_{k}\_{\sigma}).  (14) 
Therefore from (11) it follows that
\begin{split}\displaystyle\frac{d}{dt}L_{k}\big{(}z_{k}(t),z_{k}^{v}(t)\big{)}% &\displaystyle={z}_{k}^{v}{}^{\top}\dot{{z}}_{k}^{v}{z}_{k}^{v}{}^{\top}% \nabla_{{y}_{k}}V_{k,k1}(\{z}_{k}\_{\sigma})\\ &\displaystyle\hskip34.143307pt={z}_{k}^{v}{}^{\top}\big{(}\dot{{z}}_{k}^{v}% \nabla_{{y}_{k}}V_{k,k1}(\{z}_{k}\_{\sigma})\big{)}\\ &\displaystyle\hskip34.143307pt\overset{(\ref{secundo})}{=}{z}_{k}^{v}{}^{% \top}\big{(}{f}_{k1}({v}_{k1}){f}_{k1}({v}_{k})){\beta_{k}}{z}_{k}^{v}% \big{)}\\ &\displaystyle\hskip34.143307pt={z}_{k}^{v}{}^{\top}\big{(}{f}_{k1}({v}_{k1% }){f}_{k1}({v}_{k})\big{)}{\beta_{k}}{z}_{k}^{v}{}^{\top}{z}_{k}^{v}\end{split} 
IV Decoupling control design
The following result is the main result of this Section, as it delineates a “decoupling” property of the closed–loop dynamics, achieved by the type (7) control policy along with: (i) closedloop stability, (ii) velocity matching, (iii) collision avoidance and (iv) formation topology preservation. Specifically, assuming that the acceleration of the leader vehicle becomes zero after a finite period of time (i.e. v_{0}(t) reaches a steadystate) then the following theorem holds:
Theorem IV.1
If the functions f_{k}(\cdot) with 0\leq k\leq n from (2a) satisfy the Lipshitz–like condition [SCL2012, Assumption 1]
\left(v_{2}v_{1})^{\top}\big{(}f_{k}(v_{2})f_{k}(v_{1})\big{)}\right\leq% \alpha_{k}\v_{2}v_{1}\_{2}^{2},\quad\forall\>v_{1},\>v_{2}  (15) 
then for any of the type (7) control laws, such that \beta_{k}>\alpha_{k1}, the following hold:
(A) Given the Lyapunov function L_{k}(\cdot,\cdot) from (11), local to the kth agent,
then for any real constant c>0 the sub–level sets \Omega_{c}^{k}\overset{def}{=}\{(z_{k},z_{k}^{v})L_{k}(z_{k},z_{k}^{v})\leq c\} of L_{k}(\cdot,\cdot) are compact and they represent forward invariant sets for the local closed–loop dynamics (10) of the k–th agent.
(B) The control laws (7) guarantee velocity matching in the steadystate i.e. \displaystyle\lim_{t\rightarrow\infty}\z_{k}^{v}(t)\=0 and collision avoidance in the transient regime, i.e. there exists \eta_{c}>0 such that
\z_{k}(t)\_{2}>\eta_{c},\;\forall t\geq 0. 
(C) The controller (6) guarantees the formation’s topology preservation in the steadystate, i.e.
\lim_{t\rightarrow\infty}\z_{k}(t)\_{2}=\delta_{k} 
where \delta_{k} is a prespecified real, positive value.
Proof:
(A) We show that for any real c>0 the local sub–level sets \Omega_{c}^{k}\overset{def}{=}\{(z_{k},z_{k}^{v})\ L_{k}(z_{k},z_{k}^{v})\leq c\} of L_{k}(\cdot,\cdot) are compact. Note that L_{k}(z_{k},z_{k}^{v})<c implies that \{z}_{k}^{v}\<2c and V_{k,k1}(\z_{k}\_{\sigma})<2c. Since V_{k,k1}(\cdot) is radially unbounded this implies that \z_{k}\_{\sigma} is bounded and consequently \z_{k}\_{2} is bounded. Therefore \Omega_{c}^{k}\subset{\bf R}^{2\text{dim}(y_{k})} is a bounded set^{11}^{11}11Here \text{dim}(y_{k}) denotes the dimension of the y_{k}(t) vector valued function of time, which in general may be greater than one.. Moreover due to the continuity of \\cdot\_{\sigma} and of L_{k}(\cdot), one obtains that \Omega_{c}^{k} is a closed set. Precisely \Omega_{c}^{k} is the preimage of a closed set through a continuous function. In the Banach space {\bf R}^{2\text{dim}(y_{k})} it therefore holds that \Omega_{c}^{k} is closed and bounded thus \Omega_{c}^{k} is compact. Furthermore, Lemma III.3 and the Lipschitz–like assumption (15) on all f_{k}(\cdot) implies that
\frac{d}{dt}L_{k}\big{(}z_{k}(t),z_{k}^{v}(t)\big{)}\leq(\alpha_{k1}\beta_{k% }){z}_{k}^{v}{}^{\top}(t){z}_{k}^{v}(t),\ \forall k 
along the trajectories of (10). Therefore it suffices to choose the controller gain \beta_{k}>\alpha_{k1} in order to guarantee that along the trajectories of (10) it holds that \displaystyle\frac{d}{dt}L_{k}\big{(}z_{k}(t),z_{k}^{v}(t)\big{)}<0 and also that \displaystyle\Omega_{c}^{k} is a forward invariant set for the “decoupled” closed–loop system (10), local to the kth agent.
(B) From the properties of the APF (Definition II.3) it follows that V_{k,k1}(\z_{k}\_{\sigma})\rightarrow\infty as \z_{k}\_{2}\rightarrow 0, i.e. \forall c>0,\ \exists\eta_{c}>0 such that
V_{k,k1}(\z_{k}\_{\sigma})>c,\ \forall\ \z_{k}\_{2}<\eta_{c}.  (16) 
Let c_{k}=\min_{r\geq 0}V_{k,k1}(r)>0. It follows from (16) that for any positive c>c_{k} one has that
V_{k,k1}(\z_{k}\_{\sigma})\leq c\mbox{ implies }\z_{k}\_{2}\geq\eta_{c}.  (17) 
Note that an increase of c is correlated with a corresponding decrease of \eta_{c}. Next, let us fix c=2L_{k}(z_{k}(0),z_{k}^{v}(0)). From point (A) above it follows that for \beta_{k}>\alpha_{k1} it holds that \Omega_{c}^{k} is a forward invariant set with respect to (10) and consequently L_{k}(z_{k}(t),z_{k}^{v}(t))\leq\frac{c}{2},\ \forall t\geq 0. This implies via (11) that V_{k,k1}(\z_{k}(t)\_{\sigma})<c,\ \forall t\geq 0 which in turn yields c>c_{k} and so from (16) we conclude that
\z_{k}\_{2}>\eta_{c},\ \forall t\geq 0. 
It is noteworthy that \eta_{c} is implicitly defined by c which in turn depends on the initial conditions (z_{k}(0),z_{k}^{v}(0)).
(C) Given L_{k}(\cdot,\cdot) as introduced in (11), it is claimed that the string formation’s steady–state configuration is attained at the minimum of the following formationlevel Lyapunov function, defined as
L(z(t),z^{v}(t))\overset{def}{=}\frac{1}{2}\sum_{k=1}^{n}L_{k}(z_{k}(t),z_{k}^% {v}(t))  (18) 
where z(t),z^{v}(t) are the aggregated vectors of the regulated measurements for the entire formation, obtained by adequately stacking the local measurements z_{k}(t),z_{k}^{v}(t) of the agents:
z(t)\overset{def}{=}\left[\begin{array}[]{cccc}z_{1}(t)&z_{2}(t)&\dots&z_{n}(t% )\cr\omit\span\omit\span\omit\span\@@LTX@noalign{ }\omit\\ \end{array} 