A Distributed Active Perception Strategy for Source Seeking and Level Curve Tracking
TTHE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1
A Distributed Active Perception Strategy for Source Seeking and LevelCurve Tracking
Said Al-Abri,
Member, IEEE, and Fumin Zhang,
Senior Member, IEEE
Abstract —Algorithms for multi-agent systems to locate asource or to follow a desired level curve of spatially distributedscalar fields generally require sharing field measurements amongthe agents for gradient estimation. Yet, in this paper, we proposea distributed active perception strategy that enables swarms ofvarious sizes and graph structures to perform source seekingand level curve tracking without the need to explicitly esti-mate the field gradient or explicitly share measurements. Theproposed method utilizes a consensus-like Principal ComponentAnalysis perception algorithm that does not require explicitcommunication in order to compute a local body frame. Thisbody frame is used to design a distributed control law whereeach agent modulates its motion based only on its instantaneousfield measurement. Several stability results are obtained within asingular perturbation framework which justifies the convergenceand robustness of the strategy. Additionally, efficiency is validatedthrough various computer simulations and robots implementa-tion in -D scalar fields. The active perception strategy leveragesthe available local information and has the potential to be used invarious applications such as modeling information propagationin biological and robotic swarms. I. I
NTRODUCTION
An important problem in swarm robotics is the deploymentof multiple robots in order to achieve source seeking orlevel-curve tracking behavior in a scalar field. In sourceseeking problems, agents are tasked with finding the locationthat minimizes or maximizes the scalar field while in level-curve tracking, agents are tasked with tracking a trajectorythat achieves a constant field value. The field can representenvironmental characteristics such as chemical concentrations,light intensities, or heat. These two problems have variousapplications including environmental monitoring, source sig-nal localization, exploration, hazardous regions mapping, andsearch and rescue [19], [21], [24], [29], [34].In the literature, previous efforts to solve the dual problemsof source seeking and level-curve tracking have relied onfield gradient and Hessian estimation [6], [8], [17], [35], [40],[41], [43], [47], extremum seeking control [12], [13], [25],[28], sliding-mode control [32], [38] , and weighted consensuslaws [16]. Most of the aforementioned control strategies relyeither on sharing measurements via communication channels,requiring specific spatial formations, or apply only to certainsizes and structures of interacting graphs. Communication-based solutions are particularly difficult to implement due toharsh environmental conditions that prevent networking suchas in underwater mobile sensor networks [26].
Said Al-Abri and Fumin Zhang are at School of Electrical and ComputerEngineering, Georgia Institute of Technology, Atlanta, GA 30332, E-mail: { saidalabri,fumin } @gatech.edu . In this paper, we propose a distributed strategy composedof two layers for perception and control. In the perceptionlayer, each agent uses the relative positions of its neighborsto learn a geometric body frame. In the control layer, eachagent modulates its motion based on the body frame anda locally measured environmental field value. The interplaybetween the two layers results in an indirect active perceptionof the spatial gradient of the environmental property wherethe controlled behavior of the agents enhances the informationcontents of the instantaneous measurements of the field andrelative positions. This strategy enables swarms of varioussizes and graph structures to perform collective source seekingand level curve tracking of scalar fields without the need toexplicitly estimate the field gradient or explicitly exchangefield measurements.For source seeking, in [35], the field gradient is assumedto be known and agents velocities are designed to climb thegradient and move in a desired formation. In [8], agents arerequired to form a circular formation, and then exchange fieldmeasurements via a communication channel to estimate andclimb the gradient. Without knowing the global positions ofthe agents, an algorithm is developed in [17], however, itincorporates explicit sharing of field measurements to estimatethe gradient. A different gradient-based strategy is presented in[40] where agents autonomously split into multiple subgroupsthat each steer towards a source. Alternatively, extremum-based source seeking techniques are developed for a singlevehicle in -D space in [12], and in -D space in [13]. The pro-posed method relies on a constant forward velocity and designsthe angular velocity in order to achieve extremum seekingbehavior [4]. Although the approach is simple to implement,the vehicle needs a relatively long distance to travel until theperformance improves. A multi-agent extremum-based sourceseeking is developed in [25] and [28], however, the agents needto exchange some estimated parameters. In [16], a strategy isdeveloped for a large number of robots with a complete graphbased on a weighted consensus and a Gaussian perturbation.Although it is independent of communication, agents bouncerandomly in all directions leading to a slow and impracticalmovement towards the source.For the level curve tracking, in [47] and [43], the fieldgradient is assumed to be known. Alternatively, the algorithmsin [41] and [6] rely on communicating field measurementsand maintaining prescribed formations to estimate the fieldgradient. In [42], a cooperative control law is designed for twoagents such that one agent estimates the field gradient and theother one tracks the plume front. Independent of gradientestimation, algorithms are designed in [9], [10] for a -agent system but require communicating field measurements.A discontinuous sliding mode control law is designed in [38] ©2020 IEEE a r X i v : . [ ee ss . S Y ] F e b HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2 and [32] for an agent to track the level curve, independentof both gradient estimation and measurement communication,while a formation law is designed to spread the agents acrossthe level curve.Inspired by a school of fish seeking darker areas [7], theSpeeding-Up and Slowing-Down (SUSD) strategy is devel-oped for source seeking without gradient estimation in [45]and [44] for -D and -D environments, respectively. TheSUSD strategy requires a common motion direction that canbe computed locally and without explicit communication onlyfor -agent and -agent systems in -D and -D, respectively.Differently, in [2] we used a leader-follower consensus-on-asphere to obtain the common motion direction where agentsare assumed to be able to measure the velocity directions oftheir neighbors.The primary novelty of this work is in utilizing the OjaPrincipal Component Analysis (PCA) flow [5], [37], [46]to agree on a frame to describe the shape of the swarm,which we call a body frame. The PCA flow works as aconsensus law, however, its input is the covariance of therelative positions. This is different than the common consensuslaw which requires the headings of neighbors as an input[30]. Since the relative positions are locally measured, thenthe PCA flow achieves consensus without requiring the agentto exchange data among them.The main challenge this paper solves is to design a motiondirection that all agents compute locally without commu-nication. This motion direction converges to the negativedirection of the field gradient without estimating it. We solvethis challenge by the PCA perception algorithm which, bycapturing changes on the spatial shape and orientation ofthe swarm, represents an indirect feedback signal of how thefield is affecting the motions of other agents. Additionally,by autonomously tuning the velocity along the components ofthe body frame, the strategy achieves both source seeking andlevel curve tracking with a single control law.The first contribution of this paper is utilizing a PCA per-ception algorithm on relative positions to achieve a consensusin the body frame. The second contribution is a distributedcontrol law that accomplishes both missions of source seekingand level curve tracking. The third contribution is derivingthe information dynamics, not only for source seeking andlevel curve tracking but for general control laws. The fourthcontribution is obtaining input-to-state stability results withina singular perturbation framework analysis for (1) the conver-gence of the SUSD search direction to the negative gradientdirection for source seeking and level curve tracking undercomplete graphs, (1) the convergence of the SUSD searchdirection to the negative gradient direction for source seekingunder incomplete graphs (3) the convergence of the swarmto the source location under complete and incomplete graphs,and to the desired level curve under complete graphs. Theseresults reflect robust convergence to both the source locationunder complete and incomplete graphs and to the desiredlevel curve under complete graphs. The last contribution isvalidating the proposed strategy for various source seekingand level curve tracking behaviors through simulations andexperiments. The experiments are conducted using the Georgia Teach Robotarium [39] and the Georgia Teach MiniatureAutonomous Blimps [11].Preliminary results of this paper appear in our two confer-ence papers [1], [3]. In these two papers, we only consideredcomplete graphs, and in the convergence analysis, we ignoredthe higher-order terms of the field in both the dynamicsderivation and convergence analysis. Significantly, in thispaper, we derive the information dynamics for the general caseof incomplete graphs and for a generic control law withinthe distributed active perception algorithm. Additionally, weconsider the nonlinearities of the field which allows us to refinethe convergence neighborhood around the desired equilibrium.While in the conference papers we only proved that the SUSDdirection converges to the negative gradient direction, in thispaper, we also prove that the swarm converges to either thesource location or the desired level curve. Furthermore, theconference papers do not include any experimental results.The proposed distributed active perception strategy offersa new method that leverages the available local informationand enables robots with limited resources to perform variousswarming activities. The PCA body frame might be usedto design control laws for purposes other than the sourceseeking and level curve tracking. An important insight weanalytically show in this paper is that the field measurementsare communicated via the distributed perception algorithm.This might be useful in modeling information propagation inbiological and robotic swarms [14], [20], [31], [33].The rest of the paper is organized as follows. The problem isformulated in Section II. Then the distributed active perceptionstrategy is presented in Section III. In Section IV, we derive theinformation dynamics which are used in Section V for stabilityanalysis. Finally, simulation and experimental results are givenin and Section VI, and concluding remarks and suggestions forfuture work are provided in Section VII.II. P ROBLEM F ORMULATION
A. Preliminaries
In this paper, we consider a swarm of M agents describedby an undirected visibility graph, G ⊆ V × E where V is theset of all agents, and E is the set of all edges. An undirectededge ( i, j ) ∈ E exists if both agents can sense the relativepositions of each other. A graph is connected if, for each i, j ∈ V , there exists a sequence of edges connecting the i -thand j -th agents. If each agent shares an edge with all otheragents, then the graph is complete , otherwise, it is incomplete .The neighboring set of i is defined by N i = { j | ( i, j ) ∈ E} .Additionally, if for each agent the neighboring set N i is fixed,the graph is static, otherwise, it is dynamic. We consider thefollowing assumption about the graph. Assumption II.1. G is undirected and connected. This assumption is to simplify the convergence analysis.However, the design in Section III is applicable to a broaderclass of graphs which will be supported by simulation results.Additionally, we will show in Section IV that connectivity isimplicitly guaranteed when the graph is complete.Let r i ∈ R be the position of the i -th agent in a -D space. HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 3
Assumption II.2.
Each i -th agent knows the relative positions ( r j − r i ) of all its neighbors, j ∈ N i . In practice, robots can be equipped with sensors to measurethe relative positions of their neighbors with respect to theirlocal frame, which is less challenging than requiring the globalpositions [36].Furthermore, suppose there exists a scalar field z : R → R .The analytical expression of the field function z is not known,but each agent can only measure its value z ( r i ) at its currentposition r i ( t ) . Assumption II.3.
The field is assumed to be real analytic,time-invariant and bounded, i.e. ≤ z ( r i ) ≤ z max , and has aunique minimum at the source location r where z ( r ) = 0 . The field being real analytic implies that it is smooth andat any location it can be approximated by a Taylor expansion.We require the smoothness of the field because later we aregoing to make the speed of each agent to be proportional to thefield measurement. However, some non-smooth fields might betransformed into smooth fields using, for example, Stochasticmodeling, as in [44]. Additionally, we require the use of Taylorapproximation for the information dynamics derivations andthe convergence analysis.Consider z d ∈ R to be a desired level curve field value,where a level curve is the set { r | z ( r ) = z d , ∀ r ∈ R } . Assumption II.4.
The desired level curve { r | z ( r ) = z d } isconnected. Finally, let the motion of each agent be described by ˙ r i = d r i dt = u i , i = 1 , · · · , M, (1)where u i is a local control law to be designed. B. Problem Statement
Problem 1.
Without explicitly estimating the field gradient andwithout explicitly communicating field measurements, designthe local control law u i , such that the swarm autonomouslysteers towards either the source location r , or the desiredlevel curve { r | z ( r ) = z d , ∀ r ∈ R } , and keeps tracking it. Problem 2.
Analyze the convergence and robustness of theproposed strategy.
Remark 1.
Note that in this paper we assume the measure-ments to be noiseless. However, early versions of SUSD, asin [45], show that it is robust against noisy measurements.We will consider analyzing the robustness of the proposedcontrol laws in this paper against noisy measurements forfuture works.
III. T HE D ISTRIBUTED A CTIVE P ERCEPTION S TRATEGY
The proposed strategy is composed of two layers for percep-tion and control as illustrated in Fig. 1. In the perception layer,each agent learns from the relative positions a time-varyinglocally computed body frame. The motion of each agent isdesigned in the control layer based on the perceived bodyframe and the locally measured environmental field value. The interplay between the two layers results in an active perceptionof the spatial gradient of the environmental property where themotions of the agents enhance the information contents of theinstantaneous measurements of the field and relative positions.In what follows, we first present the perception algorithm andthen the distributed control law.Fig. 1: The two layers of the active perception strategy. In theperception layer, agent learns the PCA body frame compo-nents ( q , n ) . Along these components agent modulate itsmotion based only on the instantaneous field measurement. A. The PCA Perception Algorithm
Principal Component Analysis (PCA) is a statistical methodthat computes directions of maximum (minimum) variation ofa data set [22]. Given a covariance matrix of a data set, itseigenvector corresponding to the maximum (minimum) eigen-value represents the direction of the maximum (minimum)variance of the data, with the eigenvalue giving the variance ofthe data along that direction. For example, in Fig. 2, the dataset of the positions of the agent i and its neighbors (the spatialshape) has a maximum variance along the PC1 direction, anda minimum variance along the PC2 direction.Fig. 2: The blue dash lines are the edges of the connectivitygraph. The shape that agent i observes is composed of itselfand the neighbors from which agent i learns the directionsof the largest and smallest variances, PC1 and PC2. Acrossthese variances, agent i forms the body frame ( q i , n i ) .For each agent, consider the set of positions defined by H i = N i (cid:83) { i } . Then the position covariance matrix C i ( t ) ∈ R × observed by each agent is constructed locally as C i ( t ) = (cid:88) k ∈H i (cid:0) r k ( t ) − r c,i ( t ) (cid:1)(cid:0) r k ( t ) − r c,i ( t ) (cid:1) (cid:124) , (2)where r c,i = M i (cid:80) k ∈H i r k is the center of the swarm as seenby agent i , and M i = |H i | .Let the PCA body-frame of agent i be ( q i ( t ) , n i ( t )) , where q i ( t ) and n i ( t ) are orthonormal vectors in R that representthe principal components of the covariance matrix C i ( t ) ,corresponding to the largest and smallest eigenvalues, λ qi and λ ni , respectively. HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 4
Let ( ˆ q i , ˆ n i ) be an estimate of the true PCA body-frame ( q i ( t ) , n i ( t )) , which is given by the Oja PCA flow [5], [46] d ˆ q i dτ = ( I − ˆ q i ˆ q (cid:124) i ) C i ( t ) ˆ q i , ˆ n i = R ˆ q i , (3)where R is a ◦ counterclockwise rotation matrix. Observethat we use the argument τ instead of t to emphasize thatfor any given covariance matrix C i ( t ) at time instant t , agent i runs (3) at a different time scale τ . Since each agent mayhave different neighbors, as in the case when the graph isincomplete, then each agent may obtain different principaldirections. Observe that the PCA model (3) is scalable tographs with an arbitrary number of agents and structures. Assumption III.1.
For each agent, λ qi (cid:54) = λ ni . This is to ensure the eigenvectors q i and n i are orthogonalwhich is a requirement for mathematical correctness of thederived dynamics and convergence results obtained in thispaper. Otherwise, such as when the agents are evenly dis-tributed on a circle, the covariance matrix may have multiplesolutions where some of them may not be orthogonal. Due tosensing errors on measuring relative positions, it is unlikelyto have λ qi = λ ni . Additionally, since we design ˆ n i = R ˆ q i in (3), then q i and n i are ensured to be orthogonal even if λ qi = λ ni . Moreover, at each time t , we initialize (3) with q i ( τ ) = q i ( t − dt ) so to ensure that the obtained solution isnot arbitrary when λ qi = λ ni . B. The Distributed Control Law
Given the body frame ( q i ( t ) , n i ( t )) obtained by (3), wepropose the control law u i ( t ) = k ( z i ( t ) − z d ) n i ( t ) + k q i ( t ) , (4)where z i ( t ) and z d are measured and desired field values,respectively. The parameters k , k ∈ R are positive tuningparameters. To intuitively explain the control law (4), wesimulate it in Fig. 3 for a -agent system in a scalar field.In this example, q i is along the line-of-sight between the twoagents, and n i is perpendicular to the line-of-sight. When z d = 0 and k = 0 , then agent i speeds up or slowsdown along the direction n i depending on the local fieldmeasurement z i ( t ) . Since the two agents are moving in thesame direction at different speeds, then eventually they steertowards the minimum of the field.On the other hand, when z d (cid:54) = 0 and k = 0 , the two agentsapproach the level curve { r | z ( r ) = z ( r ) = z d } . Finally,when z d (cid:54) = 0 and k (cid:54) = 0 , then the first term k ( z i ( t ) − z d ) n i ( t ) steers the 2-agent system towards the desired levelcurve, while the second term k q i ( t ) moves the swarm alongthe level curve. Note that the first term changes its sign as thesign of ( z i ( t ) − z d ) changes, which stabilize the agents at thelevel curve. Additionally, the gains k and k determine thetracking speed and accuracy. In particular, small k comparedto k leads to slow tracking, but high accuracy, and vice versa.A pseudocode description for the distributed active per-ception strategy is given in Algorithm 1. The terminationcondition for source seeking could be to terminate when z i ( t ) < ¯ z where ¯ z is a satisfactory threshold. For the level curve tracking, we may terminate after a certain desired time.All the steps in the algorithm are locally computed withoutexplicit communication of any values. Algorithm 1
The Distributed Strategy while termination condition is not met, do for each agent do compute covariance matrix using (2) compute principal components using (3) update motion using (4) end for end whileRemark 2. The control law (4) has three main differences thanthe source seeking laws in our previous works [2], [44], [45].First, it solves a combined task of source seeking and levelcurve tracking. Second, it does not require a rigid prescribedshape formation. Third, it utilizes a PCA perception algorithmto define a locally computed body frame, which previously isobtained by a consensus-on-a sphere law [30].
Remark 3.
Using a consensus law, i.e. ˙ q i = ( I − q i q (cid:124) i ) (cid:80) j ∈N i q j requires agents to share their q i via acommunication channel. Interestingly, the PCA flow (3), i.e. d q i dτ = ( I − q i ( τ ) q (cid:124) i ( τ )) C i q i ( τ ) , can also be viewed as aconsensus law. However, its input is the local covariance C i ,not the neighboring headings (cid:80) j ∈N i q j .Therefore, agents canagree on a common direction by only observing local posi-tions, without requiring them to communicate. As the agentschange their speed due to the encountered field values, thePCA consensus value changes. Thus agents indirectly receivethe effects of the field values on their neighbors by the changeof the PCA value calculated locally. This cannot be achievedby using the classical heading average consensus where theconsensus value is independent of the agents’ positions. Infact, we will prove later that the PCA consensus value undersome conditions aligns with the field gradient. Remark 4.
Extra terms can be added in (4) to, for example,maintain prescribed formations or avoid collisions. In the ex-perimental results, we show examples of formation terms andtheir effects on the performance. However, we don’t considerthem in the convergence analysis for the sake of simplicity.Additionally, collision avoidance may be guaranteed usinglower-level controllers such as barrier functions as in [18].
IV. T HE I NFORMATION D YNAMICS
Let z dc,i = z c,i − z d where z c,i is the field measurementat the local center r c,i = M i (cid:80) l ∈H i r l . Define N c,i = ∇ z ( r c,i ) / (cid:107)∇ z ( r c,i ) (cid:107) to be a unit-length vector along thedirection of the field gradient at the local center r c,i . Then,using (1) and (4), we obtain ˙ z dc,i = (cid:107)∇ z c,i (cid:107) M i (cid:88) l ∈H i [ k ( z l − z d ) (cid:104) N c,i , n l (cid:105) + k (cid:104) N c,i , q l (cid:105) ] . (5)Observe that z dc,i → if and only if z c,i → z d for the levelcurve tracking, or z c,i → for the source seeking. However, HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 5 (a) z d = 0 , k = 0 (b) z d (cid:54) = 0 , k = 0 (c) z d (cid:54) = 0 , k (cid:54) = 0 Fig. 3: The blue arrows are the velocities which turn red at the end time. The circular curves are the field level curves.to analyze the convergence of the origin z dc,i = 0 of (5), weneed the dynamics of the two principle directions ( ˙ q l , ˙ n l ) =( d q l dt , d n l dt ) for l = 1 , · · · , M . Remark 5.
The dynamics ( d q l dt , d n l dt ) for l = 1 , · · · , M aredifferent from ( d ˆ n l dτ , d ˆ q l dτ ) given by the PCA flow (3), whichdescribes the dynamics of learning the body frame from agiven covariance matrix C l ( t ) at time instant t . In what follows we derive the dynamics of the body frame ( ˙ q l , ˙ n l ) first for a general control law u l , and then forthe proposed control law (4). We start the derivation withincomplete graphs and then we consider complete graphs.These dynamics eventually will be used in Section V for theconvergence analysis of the proposed strategy.Let the covariance matrix seen by agent i be given by (2)with λ ni and λ qi be the smallest and largest eigenvalues of C i ,corresponding to the eigenvectors n i and q i , respectively. Thefollowing result presents the dynamics of the PCA body framefor a general control law, u i . Lemma IV.1.
Let Assumption III.1 holds. Then, under incom-plete graphs, and when the agents move according to (1), thedynamics of the body frame are ˙ n i = − κ q i , ˙ q i = κ n i , (6) where κ = λ qi − λ ni (cid:80) k ∈H i (cid:104) q i , u k (cid:105)(cid:104) r k − r c,i , n i (cid:105) + λ qi − λ ni (cid:80) k ∈H i (cid:104) q i , r k − r c,i (cid:105)(cid:104) u k , n i (cid:105) . See proof in Section VIII. It is interesting to observe thatthe actions of the neighbors in u k are present in (6) not dueto communication, but due to the distributed active perceptionalgorithm where the body frame is obtained via the PCA (3).We then derive the dynamics of the body frame under theproposed control law (4). Lemma IV.2.
Let Assumption III.1 holds. Then, using themotion dynamics (1) along with the control law (4), thedynamics of the body frame for source seeking and level curvetracking with general incomplete graphs are ˙ n i = − k λ qi − λ ni w (cid:124) i q i q i − λ qi − λ ni σ i q i , (7) ˙ q i = k λ qi − λ ni w (cid:124) i q i n i + 1 λ qi − λ ni σ i n i , (8) where for z dk = z k − z d and z dc,i = z c,i − z d , w i = (cid:88) k ∈H i ( z dk (cid:104) n k , n i (cid:105) − z dc,i )( r k − r c,i ) , (9) σ i = k (cid:88) k ∈H i ( z dk (cid:104) n k , q i (cid:105) − z c,i ) (cid:104) r k − r c,i , n i (cid:105) + k (cid:88) k ∈H i [ (cid:104) q k , n i (cid:105)(cid:104) r k − r c,i , q i (cid:105) + (cid:104) q k , q i (cid:105)(cid:104) r k − r c,i , n i (cid:105) ] . (10)See proof in Section VIII. Note that σ i → as (cid:104) n i , n k (cid:105) → ∀ k ∈ H i , i.e. as when the local PCA vectors align.To find the relationship between n i and the local gradient ∇ z c,i , we approximate the measurement z k = z ( r k ) by Taylorexpansion with respect to the center r c,i . Since according toAssumption II.3 the field function z is analytic, then we canwrite z k − z c,i = (cid:104) r k − r c,i , ∇ z c,i (cid:105) + ν k , (11)where ∇ z c,i = ∇ z ( r c,i ) is the local gradient in the vicinityof the center r c,i observed by agent i , and ν k = O(cid:107) r k − r c,i (cid:107) represents the higher-order terms. Note that for the field tobe real analytic as required in Assumption II.3, the distance (cid:107) r k − r c,i (cid:107) has to be small enough such that z ( r k ) = (cid:80) ∞ a =0 z ( a ) ( r c,i ) a ! ( r k − r c,i ) a , where z ( a ) ( r c,i ) is the a − th orderderivative of z at location ( r c,i ) . Otherwise, ν k in (11) containsa residual term in addition to the higher-order derivatives ofthe field function. Corollary IV.3.
Let Assumption II.3 and Assumption III.1hold. Then, using the motion dynamics (1) along with thecontrol law (4), the dynamics of the body frame for sourceseeking with incomplete graphs are ˙ n i = − k (cid:107)∇ z c,i (cid:107) λ qi λ qi − λ ni (cid:104) N c,i , q i (cid:105) q i − ˆ ν i q i − E i q i , (12) ˙ q i = + k (cid:107)∇ z c,i (cid:107) λ qi λ qi − λ ni (cid:104) N c,i , q i (cid:105) n i + ˆ ν i n i + E i n i , (13) where N c,i = ∇ z c,i / (cid:107)∇ z c,i (cid:107) , ˆ ν i = ( k / ( λ qi − λ ni ) (cid:80) k ∈H i ν k (cid:104) r k − r c,i , q i (cid:105) is due to the nonlinearity ofthe field, and E i = ( k / ( λ qi − λ ni )) (cid:80) k ∈H i z k [ (cid:104) n k , q i (cid:105)(cid:104) r k − r c,i , n i (cid:105) +( (cid:104) n k , n i (cid:105)− (cid:104) r k − r c,i , q i (cid:105) ] is due to the mismatchon the local PCA components. See proof in Section VIII. Observe that this result is only forsource seeking. For the level curve tracking, a similar result
HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 6 can be obtained with a different E i . We omit this case as inthis paper we don’t analyze the convergence of level curvetracking when the graph is incomplete.When the graph is complete, we view the entire swarm as asuper agent and define ∇ z c = ∇ z ( r c ) to be the field gradientat the center of the swarm. Then, we obtain the followingresult. Lemma IV.4.
Let Assumption II.3 and Assumption III.1 hold.Then, using the motion dynamics (1) along with the controllaw (4), the dynamics of the body frame for source seekingand level curve tracking with complete graphs are ˙ n = − k (cid:107)∇ z c (cid:107) λ q λ q − λ n (cid:104) N c , q (cid:105) q − ˆ ν q , (14) ˙ q = + k (cid:107)∇ z c (cid:107) λ q λ q − λ n (cid:104) N c , q (cid:105) n + ˆ ν n , (15) where N c = ∇ z c (cid:107)∇ z c (cid:107) , and ˆ ν = k λ q − λ n (cid:80) Mk =1 ν k (cid:104) r k − r c , q (cid:105) . See proof in Section VIII. Note that, although when thegraph is complete, it is not enough to substitute E i = 0 in Corollary IV.3 to obtain the result in
Lemma IV.4 , as
Corollary IV.3 is valid only for source seeking.Since n and q are orthonormal, we can write qq (cid:124) = I − nn (cid:124) . Hence, we can reform (14) as ˙ n = − k λ q λ q − λ n (cid:107)∇ z c (cid:107) ( I − nn (cid:124) ) N c − ˆ ν q . (16)Note that the second term in (16) vanishes when ν k = ν forall agents. i.e. the field is linear, or when the agents are at thesame level curve. Remark 6.
The first term in (16) represents a consensus-on-asphere control law [30]. This is interesting since although weare explicitly applying (4) with (3), the direction n is implicitlytracking the negative direction of the gradient − N c . We conclude this section by the following result
Lemma IV.5.
Let Assumption II.3 and Assumption III.1 hold.Suppose the graph is complete and let the motion dynamicsof each agent be described by (1). Then, for both sourceseeking and level curve tracking, λ n ( t ) < λ q ( t ) = λ q ( t ) ,where λ q ( t ) is the initial maximum variance of the spatialdistribution of the agents. See proof in Section VIII. Since λ q is the largest varianceof the swarm, then Lemma IV.5 implies that variance of theswarm is bounded by the initial variance λ q ( t ) . Conse-quently, the connectivity of the graph is ensured even withouta formation controller.V. C ONVERGENCE A NALYSIS
In this section, we first obtain conditions under which weprove that (A) for complete graphs, the SUSD direction n con-verges to the negative gradient direction − N c for both sourceseeking and level curve tracking, and (B) for incompletegraphs, the SUSD direction n i of each agent converges to thenegative gradient direction − N c,i for source seeking.Then, weprovide conditions under which we prove that (C) for completegraphs, trajectories z ( r c ) − z d for level curve tracking, or of z ( r c ) for source seeking, are ultimately bounded. Finally, forincomplete graphs, in (D) we provide conditions under whichtrajectories of z ( r i,c ) for the source seeking are ultimatelybounded.Recall that the PCA flow (3) runs in the time scale τ , whilethe control law (4) runs in the time scale t . That is, for eachtime instance t , each agent runs (3) for some time τ . Letthe relationships between the control time t , and the PCAperception time τ be dtdτ = (cid:15) , where (cid:15) ∈ (0 , . This impliesthat τ = t − t (cid:15) , where τ = 0 . Using this relationship, theperception and control dynamics in the singular perturbationframework are ˙ r i = k ( z i − z d ) n i + k q i , ∀ i, (17) ˙ n i = g ( · ) , (18) ˙ q i = g ( · ) , (19) (cid:15) ˙ˆ q i = ( I − ˆ q i ˆ q (cid:124) i ) C i ˆ q i , ∀ i. (20)where g ( · ) and g ( · ) are the general information dynamicsequations given by (7) and (8). The control dynamics (17)-(19)are viewed as a slow system whereas the perception dynamics(20) are viewed as a fast system. A. Convergence of the SUSD Direction for Complete Graphsfor Source Seeking and Level Curve Tracking
We view the swarm as one body where its individualsare moving in the same direction but with different speedsdepending on their field measurements. Define θ = 1 + (cid:104) N c , n (cid:105) , (21)where θ → when n → − N c . i.e. when the swarm speeds upor slows down in the negative direction of the field gradient.Additionally, define ψ = 1 − (cid:104) q , ˆ q (cid:105) , (22)where ψ → when ˆ q → q , i.e when the PCA perceptionalgorithm converges to the exact eigenvector of the covariancematrix C . We then obtain the coerced slow and fast systems ˙ θ = k (cid:107)∇ z c (cid:107) λ q λ q − λ n θ ( θ −
2) + δ, (23) (cid:15) ˙ ψ = − ( λ q − λ n ) ψ (1 − ψ )(2 − ψ ) + (cid:15)η, (24)where δ is viewed as an input disturbance due to the nonlinear-ity of the field, and η represents the interconnection betweenthe coerced slow and fast systems. They are defined by δ = − k λ q − λ n ϑ (cid:104) N c , q (cid:105) + (cid:104) n , ˙ N c (cid:105) ,η = ± k λ q − λ n (cid:16) ϑ ± (cid:107)∇ z c (cid:107) λ q (cid:112) θ (2 − θ ) (cid:17)(cid:112) ψ (2 − ψ ) , (25)where ϑ = (cid:80) Mk =1 ν k (cid:104) r k − r c , q (cid:105) , and (cid:104) n , ˙ N c (cid:105) = (cid:107)∇ z c (cid:107) n (cid:124) ( I − N c N (cid:124) c ) ∇ z c ( k ( z a − z d ) n + k q ) , where z a is the averagefield measurement and ∇ z c is the hessian matrix of the field. Proof. of (23) and (24).
To derive (23), we take the timederivative of (21) and apply (16) of
Lemma
IV.4 for ˙ n . On HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 7 the other hand, we derive (24) by the following steps. By theChain rule, d ˆ q dτ = (cid:15) d ˆ q dt , or d ˆ q dt = (cid:15) d ˆ q dτ . Hence dψdτ = (cid:15) dψdt = − (cid:15) (cid:104) d q dt , ˆ q (cid:105) − (cid:104) q , d ˆ q dτ (cid:105) . (26)From (3), we obtain (cid:104) q , d ˆ q dτ (cid:105) = (1 − ψ ) (cid:16) λ q − (cid:104) ˆ q , C ˆ q (cid:105) (cid:17) . (27)Write ˆ q = (cid:104) ˆ q , q (cid:105) q + (cid:104) ˆ q , n (cid:105) n . Hence (cid:104) ˆ q , C ˆ q (cid:105) = λ q (1 − ψ ) + λ n ψ (2 − ψ ) . (28)Substituting (28) into (27) yields (cid:104) q , d ˆ q dτ (cid:105) = ( λ q − λ n ) ψ (1 − ψ )(2 − ψ ) . (29)On the other hand, using (15), we obtain (cid:104) d q dt , ˆ q (cid:105) = k λ q − λ n (cid:16) (cid:107)∇ z c (cid:107) λ q (cid:104) N c , q (cid:105) + ϑ (cid:17) (cid:104) n , ˆ q (cid:105) , (30)where ϑ = (cid:80) Mk =1 ν k (cid:104) r k − r c , q (cid:105) , and (cid:104) n , ˆ q (cid:105) = ± (cid:112) ψ (2 − ψ ) .Substituting (29) and (30) in (26), we obtain dψdτ = (cid:15) dψdt = − ( λ q − λ n ) ψ (1 − ψ )(2 − ψ ) + (cid:15)η, (31)where η is as defined by (25).We first let (cid:15) = 0 in (23) and (24) to analyze the stabilityof the resulting decoupled reduced and boundary systems (32)and (37), respectively. Then, we analyze the stability of thecoupled system of (23) and (24) by deriving (cid:15) ∗ ∈ (0 , suchthat for all (cid:15) ≤ (cid:15) ∗ , some of the stability results of the reducedand boundary systems, (32) and (37), hold for the coupledsystem.
1) Stability of the Reduced System:
The coerced reducedsystem is given by ˙ θ = − k (cid:107)∇ z c (cid:107) λ q λ q − λ n θ (2 − θ ) + δ = f ( t, θ, δ ) . (32)Note that, when θ ∈ { , } , then n = ± N c which impliesthat (cid:104) N c , q (cid:105) = 0 and n (cid:124) ( I − N c N (cid:124) c ) = 0 . Hence δ vanishesat the equilibria θ ∈ { , } . Additionally observe that δ = 0 when ∇ z c = 0 and ν k = 0 , ∀ k , i.e. when the field is linear.The following result describes the stability of the origin ofthe reduced system. Theorem V.1.
Consider the reduced system (32). Supposethere exists a lower bound µ > such that (cid:107)∇ z ( r c ) (cid:107) > µ .Then the equilibrium θ = 0 of the unforced system f ( t, θ, is asymptotically stable in which whenever θ (0) ∈ [0 , ,then θ ( t ) → as t → ∞ . Furthermore, for an inputdisturbance satisfying | δ | ≤ k (cid:15) λ q λ q − λ n µ , where (cid:15) ∈ (0 , ,the equilibrium θ = 0 of forced system f ( t, θ, δ ) is locallyinput-to-state stable.Proof. Consider the domain D = { θ | θ ∈ [0 , } i.e. (cid:104) N c , n (cid:105) (cid:54) = 1 . Let V : D → R be a Lyapunov candidatefunction defined by V = θ − θ , (33) where V = 0 if and only if θ = 0 . Additionally, V → ∞ as θ → . For the unforced system f ( t, θ, , we obtain ˙ V = − k (cid:107)∇ z c (cid:107) λ q λ q − λ n V ≤ . (34)Since ˙ V = 0 if and only if θ = 0 , then the origin of the un-forced system f ( t, θ, is asymptotically stable. Additionally, ˙ V → −∞ as θ → . This along with the fact that V → ∞ whenever θ → and (cid:107)∇ z c (cid:107) > µ > , implies that D is aforward invariant set, and thus θ ∈ [0 , for all t .For the forced system f ( t, θ, δ ) , we obtain ˙ V ≤ − − (cid:15) ) k µ λ q λ q − λ n V , ∀| θ | ≥ ρ ( | δ | ) , (35)where ρ ( | δ | ) = 1 − (cid:113) − ( λ q − λ n ) | δ | k (cid:15) λ q µ is a class K function inthe domain [0 , k (cid:15) λ q λ q − λ n µ ] . Let α ( | θ | ) = α ( | θ | ) = | θ | −| θ | which are class K functions that satisfy: α ( | θ | ) ≤ V ( θ ) ≤ α ( | θ | ) . Therefore, using Definition 3.3 of local input-to-statestability in [15], and according to
Theorem 4.19 in [23], theorigin of the forced system f ( t, θ, δ ) is locally input-to-statestable.In Theorem V.1 we showed that the set { θ | θ ∈ [0 , } is forward invariant. The following result shows that therestricted set { θ | θ ∈ [0 , } is also forward invariant. Thisresult will be required later in Section V-C. Corollary V.2.
Consider the reduced system (32). Supposethere exists a lower bound µ > such that (cid:107)∇ z ( r c ) (cid:107) > µ .Then the equilibrium θ = 0 of the unforced system f ( t, θ, is asymptotically stable in which whenever θ (0) ∈ [0 , ,then θ ( t ) → as t → ∞ . Furthermore, for an inputdisturbance satisfying | δ | ≤ k (cid:15) λ q λ q − λ n µ , where (cid:15) ∈ (0 , ,the equilibrium θ = 0 of the forced system f ( t, θ, δ ) is locallyinput-to-state stable.Proof. If we modify V in (33) to be V = θ − θ , where V : [0 , → R , then we can show that ˙ V satisfies (35).Hence, using the same argument in proving Theorem V.1 ,we conclude that the origin of of the forced system f ( t, θ, δ ) is locally input-to-state stable and the set { θ | θ ∈ [0 , } isforward invariant. Lemma V.3.
The set where θ ∈ [ ρ ( | δ | ) , is not empty isdefined by B = { r c |(cid:107)∇ z ( r c ) (cid:107) > µ } where µ = | ϑ | + (cid:113) | ϑ | + 4 (cid:15) λ q ( λ q − λ n )( | z a − z d | + k k ) (cid:107)∇ z c (cid:107) (cid:15) λ q , (36)in which ϑ is as defined in (25). See proof in Section VIII.Observe that Theorem V.1 implies that wherever the swarmis in a landscape where (cid:107)∇ z ( r c ) (cid:107) > µ , then the SUSDdirection n follows the negative gradient direction − N . Sinceaccording to Assumption II.3 the field has a unique minimum,then the bound µ defines a neighborhood around the sourcelocation r where the magnitude of the gradient (cid:107)∇ z ( r c ) (cid:107) For more details about the definitions of class K functions, the reader isreferred to Definition 4.2 of [23].
HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 8 is dominated by the higher-order terms. Once the swarm isinside this neighborhood, then n may not track − N . Withouta termination policy in the Algorithm, the swarm may passthe source. Fortunately, Lemma IV.5 shows that the swarm isguaranteed to switch between n and q and hence the swarmsteers back to the set (cid:107)∇ z ( r c ) (cid:107) > µ . Remark 7.
Note that µ is a sufficient bound, and if weconsider only source seeking, then we simply substitute k = 0 and z d = 0 in (36). Intuitively, since ϑ = O ( (cid:107) r k − r c (cid:107) ) ,then µ decreases as the spatial size of the swarm shrinks.Moreover, when the swarm is more spatially distributed then ( λ q − λ n ) decreases which reduce the effect of the Hessianterm.2) Stability of the Boundary System: By setting (cid:15) = 0 in(31), we obtain the boundary system dψdτ = − ( λ q − λ n ) ψ (1 − ψ )(2 − ψ ) . (37)Observe that in (37), λ q and λ n are constants with respectto the time scale τ . Additionally, system (37) is at equilib-rium when ψ ∈ { , , } . The desired equilibrium ψ = 0 corresponds to ˆ q = q , and the undesired ψ = 1 and ψ = 2 correspond to ˆ q = ± n and ˆ q = − q , respectively. We havethe following result for origin of the boundary system Theorem V.4.
The origin of the boundary system (37) isasymptotically stable uniformly in λ q and λ n , in which when-ever at τ = 0 , ψ (0) ∈ [0 , , then ψ → as τ → ∞ .Proof. Let D = { ψ ∈ R | ψ ∈ [0 , } which is equivalent to < (cid:104) q , ˆ q (cid:105) ≤ . Then let V ( ψ ) : D → R be a Lyapunovcandidate function defined by V = ψ − ψ (38)where V ≥ and V = 0 if and only if ψ = 0 . Furthermore, V → ∞ as ψ → . Using (37), we obtain dV dτ = − ( λ q − λ n )(2 − ψ ) V ≤ , (39)where in D , dV dτ = 0 if and only if ψ = 0 . Furthermore,since from Assumption
III.1 ( λ q − λ n ) (cid:54) = 0 , then dV dτ → −∞ as ψ → . This along with the fact that V → ∞ whenever ψ → implies that D is a forward invariant set, and thus ψ ( τ ) ∈ [0 , for all τ . Let W ( ψ ) = W ( ψ ) = V whichimplies that W ( ψ ) ≤ V ≤ W ( ψ ) . Consequently, accordingto Theorem 4.9 in [23], we can conclude that the equilibrium ψ = 0 of the boundary system (37) is asymptotically stable,uniformly in λ q and λ n .
3) Stability of the Unforced Coupled System:
Let (cid:2) ˙ θ (cid:15) ˙ ψ (cid:3) (cid:124) = h ( t, (cid:15), θ, ψ, δ ) where h is as given by (23) and(24). The following theorem establishes a sufficient range for (cid:15) such that the stability of the origin of the decoupled unforcedsystem h ( t, , θ, ψ, still holds for the origin of the coupledunforced system h ( t, (cid:15), θ, ψ, . Theorem V.5.
Consider the coupled system given by (23) and(24). Suppose the shape covariance eigenvalues satisfy < χ ≤ λ q − λ n ≤ χ < ∞ , where χ and χ are constants.Furthermore, assume that (cid:15) < (cid:15) ∗ where (cid:15) ∗ = 2(1 − d ) χ dk µ (cid:96) χ , (40) in which d ∈ (0 , and (cid:96) ≥ are constants. Con-sider the neighborhood B = { r c |(cid:107)∇ z ( r c ) (cid:107) > µ } where µ > is given by (36). Then the origin of the unforcedsystem h ( t, (cid:15), θ, ψ, is uniformly asymptotically stable inwhich whenever θ (0) ∈ [0 , and ψ (0) ∈ [0 , − (cid:96) ) , then ( θ ( t ) , ψ ( t )) → (0 , as t → ∞ .Proof. Construct the domain D = D ∪ D = { [ θ, ψ ] (cid:124) | θ ∈ [0 , , ψ ∈ [0 , − (cid:96) ) } . Let V : D → R be a Lyapunovcandidate function defined as V = (1 − d ) V + dV , where d ∈ (0 , is a constant and V and V are as defined in (33)and (38), respectively.In the proof of Theorem
V.1, we showed that, given theclass K functions α ( | θ | ) = α ( | θ | ) = | θ | −| θ | , α ( | θ | ) ≤ V ( θ ) ≤ α ( | θ | ) , and ˙ V ≤ − k (cid:107)∇ z c (cid:107) λ q λ q − λ n V . Similarly,in the proof of Theorem
V.4, we showed that, given the class K functions W ( ψ ) = W ( ψ ) = ψ − ψ , W ( ψ ) ≤ V ( ψ ) ≤ W ( ψ ) , and dV dτ ≤ − ( λ q − λ n )(2 − ψ ) V .For the interconnected system, since (cid:15) (cid:54) = 0 , ˙ V now includesthe second indefinite term in (31). That is: ˙ V = − (cid:15) L + Q , (41)where L = ( λ q − λ n )(2 − ψ ) V is a continuous positivedefinite function in the domain D , and Q = η (1 − ψ ) isan indefinite function due to the mismatch between q and ˆ q .Using |(cid:104) N c , q (cid:105)| = (cid:112) θ (2 − θ ) , and after many manipulations,we obtain Q ≤ hφ ( θ ) φ ( ψ ) where h = 2 k µ √ (cid:96) χ χ , φ ( θ ) = (cid:113) θ − θ , and φ ( ψ ) = (cid:113) ψ (2 − ψ )1 − ψ . Additionally, from(35) , we write ˙ V ≤ − k µ φ ( θ ) . Similarly, from (39), wewrite ˙ V ≤ − χ φ ( ψ ) . Let x = (cid:2) φ ( θ ) φ ( ψ ) (cid:3) (cid:124) Finally, weobtain ˙ V ( x ) ≤ − x (cid:124) Λ x , Λ = (cid:20) k µ (1 − d ) dh dh dχ (cid:15) (cid:21) , (42)where Λ is a positive definite matrix for all (cid:15) < (cid:15) ∗ where (cid:15) ∗ is as given by (40). Hence, according to Theorem 5.1 in[27], the origin of unforced system h ( t, (cid:15), θ, ψ, is uniformlyasymptotically stable. B. Convergence of the SUSD Direction for Source Seekingunder Incomplete Graphs
Similar to the complete graph case, we first define θ i = 1 + (cid:104) N c,i , n i (cid:105) , (43)where θ i → when n i → − N c,i . i.e. when n i convergesto the negative direction of the local field gradient, N c,i = ∇ z c,i (cid:107)∇ z c,i (cid:107) . Additionally, define ψ i = 1 − (cid:104) q i , ˆ q i (cid:105) , (44)where ψ i → when ˆ q i → q i ∀ i , i.e when all the local PCAperception algorithms converge to the exact eigenvectors of HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 9 the local covariance matrices C i . Using Corollary
IV.3, andsimilar to the procedure of deriving (23), (24), and (25), weobtain the coerced slow and fast systems for incomplete graphs ˙ θ i = k (cid:107)∇ z c,i (cid:107) λ qi λ qi − λ ni θ i ( θ i −
2) + δ i , i = 1 , · · · , M, (45) (cid:15) ˙ ψ i = − ( λ qi − λ ni ) ψ i (1 − ψ i )(2 − ψ i ) + (cid:15)η i (46)where δ i and η i are defined by δ i = −(cid:104) N c,i , q i (cid:105)E i − k λ qi − λ ni ϑ i (cid:104) N c,i , q i (cid:105) + (cid:104) n i , ˙ N c,i (cid:105) ,η i = ± k λ qi − λ ni (cid:16) ϑ i + E i ± (cid:107)∇ z c,i (cid:107) λ qi (cid:112) θ i (2 − θ i ) (cid:17) · (cid:112) ψ i (2 − ψ i ) , (47)where E i , as defined in Corollary
IV.3, accounts forthe mismatch between the local PCA components.Additionally, ϑ i = (cid:80) k ∈H i ν k (cid:104) r k − r c,i , q i (cid:105) and (cid:104) n i , ˙ N c,i (cid:105) =( k / (cid:107)∇ z c,i (cid:107) ) n (cid:124) i ( I − N c,i N (cid:124) c,i ) ∇ z i (1 /M i ) (cid:80) k ∈H i z k n k ,where ∇ z i is the local hessian matrix of the field around r c,i .
1) The Reduced System:
The coerced reduced system isgiven by ˙ θ i = k (cid:107)∇ z c,i (cid:107) λ qi λ qi − λ ni θ i ( θ i −
2) + δ i . (48)Observe that, when θ i = 0 , , then n i = ± N c,i which impliesthat (cid:104) N c,i , q i (cid:105) = 0 and n (cid:124) i ( I − N c,i N (cid:124) c,i ) = 0 . Hence, δ i in (47) vanishes at the equilibria of (32). The term, E i , isdetermined by the graph structure, which decreases as theconnectivity of the static graph increases.Define θ = [ θ , · · · , θ M ] (cid:124) , and δ = [ δ , · · · , δ M ] (cid:124) . Let ˙ θ = [ f ( θ , δ ) , · · · , f M ( θ M , δ M )] (cid:124) where f i ( t, θ i , δ i ) is asdefined by (48). Proposition V.6.
Consider the reduced system (48). Supposethere exists a lower bound µ > such that (cid:107)∇ z ( r c,i ) (cid:107) > µ , ∀ i . Then the equilibrium θ = of the unforced system f ( t, θ , ) is asymptotically stable in which whenever ∀ i , θ i (0) ∈ [0 , , then θ i ( t ) → as t → ∞ . Furthermore,for an input disturbance satisfying (cid:107) δ (cid:107) ≤ k (cid:15) λµ , where λ = min i λ qi λ qi − λ ni , (cid:15) ∈ (0 , , then the origin of forced system f ( t, θ , δ ) is locally input-to-state stable.Proof. Consider the domain D = { θ | (cid:107) θ (cid:107) ∈ [0 , √ M ) } ≡{ θ |∀ i, θ i ∈ [0 , } . Let V : D → R be a Lyapunovcandidate function defined by V = M (cid:88) i =1 θ i − θ i , (49)where V = 0 if and only if θ = . Additionally, V → ∞ whenever any or all θ i → . For the unforced system f ( t, θ , ,we obtain ˙ V = − k M (cid:88) i =1 (cid:107)∇ z c,i (cid:107) λ qi λ qi − λ ni θ i − θ i ≤ , (50)where ˙ V = 0 if and only if θ = . Then the originof the unforced system f ( t, θ , ) is asymptotically stable. Additionally, ˙ V → −∞ whenever any or all θ i → . Thisalong with the fact that V → ∞ whenever any or all θ i → ,implies that D is a forward invariant set and thus trajectoriesstart inside it will never go outside it.For the forced system f ( t, θ , δ ) , let (cid:15) ∈ (0 , be aconstant, then we obtain ˙ V ≤ − k (1 − (cid:15) ) λµ V , ∀(cid:107) θ (cid:107) > ρ ( (cid:107) δ (cid:107) ) , (51)where λ = min i λ qi λ qi − λ ni and ρ ( (cid:107) δ (cid:107) ) = √ M − (cid:115) M − Mk (cid:15) λµ (cid:107) δ (cid:107) (52)is a class K function in the domain (cid:107) δ (cid:107) ∈ [0 , k (cid:15) λµ ] .Since it is assumed that (cid:107) δ (cid:107) ≤ k (cid:15) λµ , then (cid:107) δ (cid:107) ≤ √ M ,and hence the set (cid:107) θ (cid:107) ∈ [ ρ ( (cid:107) δ (cid:107) ) , √ M ) is not empty. Let α ( (cid:107) θ (cid:107) ) = (cid:107) θ (cid:107) , and α ( (cid:107) θ (cid:107) ) = M (cid:107) θ (cid:107) √ M −(cid:107) θ (cid:107) which, in thedomain D , are class K functions that satisfy: α ( (cid:107) θ (cid:107) ) ≤ V ( θ ) ≤ α ( (cid:107) θ (cid:107) ) . Therefore, using Definition 3.3 of localinput-to-state stability in [15], and according to
Theorem 4.19 in [23], the origin of the forced system f ( t, θ , δ ) is locallyinput-to-state stable.Using a similar procedure of deriving (36), we can obtainthe sufficient bound µ = min i µ ,i , where µ ,i = | ϑ i + E i | + (cid:112) | ϑ i + E i | + 4 (cid:15) z a,i λ qi ( λ qi − λ ni ) (cid:107)∇ z i,c (cid:107) (cid:15) λ qi , (53)in which ϑ i and E i are as defined in (47).In Proposition V.6 we showed that the set { θ |∀ i, θ i ∈ [0 , } is forward invariant. The following result shows thatthe restricted set { θ |∀ i, θ i ∈ [0 , } is also forward invariant.This result will be required later in Section V-D. Corollary V.7.
Consider the reduced system (48). Supposethere exists a lower bound µ > such that (cid:107)∇ z ( r c,i ) (cid:107) > µ , ∀ i . Then the equilibrium θ = of the unforced system f ( t, θ , ) is asymptotically stable in which whenever ∀ i , θ i (0) ∈ [0 , , then θ i ( t ) → as t → ∞ . Furthermore,for an input disturbance satisfying (cid:107) δ (cid:107) ≤ k (cid:15) λµ , where λ = min i λ qi λ qi − λ ni , (cid:15) ∈ (0 , , then the origin of forced system f ( t, θ , δ ) is locally input-to-state stable.Proof. If we modify V in (49) to be V = (cid:80) Mi =1 2 θ i − θ i , where V : [0 , → R , then we can show that ˙ V satisfies (51).Hence, using the same argument in proving Proposition V.6 ,we conclude that the origin of of the forced system f ( t, θ, δ ) is locally input-to-state stable and the set { θ |∀ i, θ i ∈ [0 , } is forward invariant.
2) The Boundary System:
Setting (cid:15) = 0 in (46) with dψ i dτ = (cid:15) dψ i dt , we obtain the boundary system dψ i dτ = − ( λ qi − λ ni ) ψ i (1 − ψ i )(2 − ψ i ) , ∀ i. (54) HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 10
Proposition V.8.
The origin of the boundary system (54) isasymptotically stable uniformly in all λ qi and λ ni , in whichwhenever at τ = 0 , all ψ i (0) ∈ [0 , , then all ψ i → as τ → ∞ .Proof. Let D = { ψ |∀ i, ψ i ∈ [0 , } where ψ i = 1 − (cid:104) q i , ˆ q i (cid:105) .Then let V ( ψ ) : D → R be a Lyapunov candidate functiondefined by V = M (cid:88) i =1 ψ i − ψ i , (55)where V ≥ and V = 0 if and only if ψ i = 0 , ∀ i .Furthermore, V → ∞ as any ψ i → . Using (54), we obtainobtain dV dτ = − M (cid:88) i =1 ( λ qi − λ ni ) ψ i (2 − ψ i )(1 − ψ i ) ≤ . (56)where in D dV dτ = 0 if and only if ψ i = 0 , ∀ i . Furthermore,since from Assumption
III.1 ( λ qi − λ ni ) (cid:54) = 0 , ∀ i , then dV dτ → −∞ whenever any ψ i → . This along with thefact that V → ∞ whenever any ψ i → implies that D is a forward invariant set, and thus all ψ i ( τ ) ∈ [0 , for all τ . Let ψ = [ ψ , · · · , ψ M ] and W ( ψ ) = W ( ψ ) = V whichimplies that W ( ψ ) ≤ V ≤ W ψ ) . Consequently, accordingto Theorem 4.9 in [23], we can conclude that the equilibrium ψ = 0 of the boundary system (54) is asymptotically stable,uniformly in all λ qi and λ ni .
3) The Coupled System:
Let θ = [ θ , · · · , θ M ] (cid:124) , ψ =[ ψ , · · · , ψ M ] (cid:124) , and δ = [ δ , · · · , δ M ] (cid:124) . Define the coupledsystem (cid:2) ˙ θ (cid:15) ˙ ψ (cid:3) (cid:124) = h ( t, (cid:15), θ , ψ , δ ) where ˙ θ i and ˙ ψ i are asdefined by (45) and (46), respectively. Proposition V.9.
Consider the coupled system given by (45)and (46). Suppose that the shape covariance eigenvaluessatisfy ∀ i , < χ ≤ λ qi − λ ni ≤ χ < ∞ , where χ and χ are constants. Consider the neighborhood B = { r c,i | <µ ≤ |(cid:107)∇ z ( r c,i ) (cid:107) ≤ ¯ µ < ∞} ∀ i , where µ is a lower boundgiven by (53) and ¯ µ is a finite upper bound. Furthermore,assume that (cid:15) < (cid:15) ∗ where (cid:15) ∗ = 2(1 − d ) µ χ dk ¯ µ (cid:96) χ , (57) in which d ∈ (0 , and (cid:96) ≥ are constants. Then the origin ofthe unforced system h ( t, (cid:15), θ , ψ , ) is uniformly asymptoticallystable in which whenever, ∀ i , θ i (0) ∈ [0 , and ψ i (0) ∈ [0 , − (cid:96) ) , then ( θ ( t ) , ψ ( t )) → ( , ) as t → ∞ .Proof. Consider the domain D = D ∪ D = where D = { θ |∀ i, θ i ∈ [0 , } and D = { ψ |∀ i, ψ i ∈ [0 , − (cid:96) ) } . Let V : D → R be a Lyapunov candidate function for the coupledsystem and defined as V = (1 − d ) V + dV where d ∈ (0 , ,and V and V are as defined by (49) and (55), respectively.Since (cid:15) (cid:54) = 0 , then using (46), we obtain ˙ V = − (cid:15) L + Q , (58)where L = (cid:80) Mi =1 ( λ qi − λ ni ) ψ i (2 − ψ i )(1 − ψ i ) is a continuous positivedefinite function in the domain D , and Q = (cid:80) Mi =1 η i (1 − ψ i ) is an indefinite function. Following similar procedure of provingTheorem V.5, and after many manipulations, we show Q ≤ h √ V (cid:112) ¯ V where h = 2 k √ (cid:96) χ ¯ µ χ , and ¯ V = (cid:80) i ψ i (2 − ψ i )1 − ψ i .Additionally, we obtain ˙ V ≤ − k µ ( √ V ) , and ˙ V ≤− χ ( (cid:112) ¯ V ) . Let x = (cid:104) √ V (cid:112) ¯ V (cid:105) (cid:124) Finally, we obtain ˙ V ( x ) ≤ − x (cid:124) Λ x , Λ = (cid:20) − d ) k µ dh dh dχ (cid:15) (cid:21) , (59)where Λ is a positive definite matrix for all (cid:15) < (cid:15) ∗ where (cid:15) ∗ is as given by (57). Hence, according to Theorem 5.1 in[27], the origin of unforced system h ( t, (cid:15), θ , ψ , is uniformlyasymptotically stable. C. Convergence of the Swarm Source Seeking and Level CurveTracking under Complete Graphs
In this section, we study the convergence of the swarm toeither the source location or desired level curve. In particular,we analyze the trajectory of the field measurement at thecenter of the swarm. Define z dc = z c − z d . This implies that z dc = 0 if and only if z c = z d . Taking the time derivative, ˙ z dc = ˙ z c = (cid:104)∇ z c , ˙ r c (cid:105) . But, using (17) for the complete graphcase, ˙ r c = M (cid:80) k [ k ( z k − z d ) n + k q ] = k ( z a − z d ) n + k q where z a = M (cid:80) k z k is the average field measurement.Note that , using (11), z a = M (cid:80) k z k = M (cid:80) Mk =1 [ z c + (cid:104)∇ z c , r k − r c (cid:105) + ν k ] = z c + ν where ν = M (cid:80) Mk =1 ν k and M (cid:80) Mk =1 (cid:104)∇ z c , r k − r c (cid:105) = 0 . That is the difference betweenthe average and center measurements equals to the average ofhigher-order terms. Then we obtain ˙ z dc = k (cid:107)∇ z c (cid:107) ( z dc + ν ) (cid:104) N , n (cid:105) + k (cid:107)∇ z c (cid:107)(cid:104) N , q (cid:105) , (60)where ν = z a − z c . Note that, even when (cid:104) N , n (cid:105) = ± which implies (cid:104) N , q (cid:105) = 0 , z dc = 0 is not an equilibrium to(60) due to the existence of ν . In the following, we present aboundedness result for the trajectory z dc ( t ) . Theorem V.10.
Suppose (cid:107)∇ z ( r c ) (cid:107) > µ where µ > isa constant. Furthermore, suppose − ≤ (cid:104) N , n (cid:105) ≤ − (cid:15) and | ν | ≤ ¯ ν , where (cid:15) ∈ (0 , and ¯ ν > are constants. Then, thesolutions of (60) are uniformly ultimately bounded.Proof. Let V : R → R be a Lyapunov candidate functiondefined by V = ( z dc ) , where V = 0 if and only if z dc = 0 .Then we obtain ˙ V ≤ − k µ (cid:15) (1 − (cid:15) ) V , ∀| z dc | ≥ k ¯ ν + k (cid:112) − (cid:15) k (cid:15) , (61)where ˙ V = 0 if and only if z dc = 0 . Let α ( | z dc | ) = α ( | z dc | ) = | z dc | be class K functions. Then α ( | z dc | ) ≤ V ≤ α ( | z dc | ) .Therefore, according to Theorem 4.18 in [23], the trajectoriesof the system (60) are uniformly ultimately bounded.Note that, (61) implies that the z dc trajectories of (60) willconverge to a strip around the desired level curve and the stripis defined by { r c || z dc | ≤ k ¯ ν + k √ − (cid:15) k (cid:15) } . If we only considersource seeking, i.e. k = 0 and z dc = z c , then the z c trajectories HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 11 of (60) will converge to a neighborhood around the sourcelocation defined by { r c || z dc | ≤ ¯ ν(cid:15) } . Remark 8.
Theorem V.10 can be viewed as a general result. Inparticular, if each agent moves according to ˙ r i = k ( z ( r i ) − z d ) v + k v ⊥ , then as long as − ≤ (cid:104) N , v (cid:105) ≤ − (cid:15) , the centerof the swarm will converge to a neighborhood of the sourcelocation or to a strip along the desired level curve. The sizeof these neighborhoods is determined by k , k , and ¯ ν . Thedirection v could be along the exact negative gradient, oralong a sub-gradient direction, i.e. v satisfies z ( r i ) − z ( r c ) ≥(cid:104) v , r i − r c (cid:105) . In this paper, v is given by the PCA direction n which is proved in Corollary
V.2 to satisfy the requiredassumption − ≤ (cid:104) N , n (cid:105) ≤ − (cid:15) < .D. Convergence of the Swarm for Source Seeking underIncomplete Graphs Using (17) for z d = 0 and k = 0 , we obtain ˙ r c,i = k z a,i n i + k M i (cid:88) j ∈N i z j ( n j − n i ) , (62)where z a,i = (1 /M i ) (cid:80) k ∈H i z i , r c,i = (1 /M i ) (cid:80) k ∈H i r k ,and M i = |H i | = 1 + |N i | . Let ν c,i = M i (cid:80) k ∈H i ν k . Hence,using (11) ν c,i = z a,i − z c,i , i.e. the difference between thelocal average and center measurements of the field. Let e i = 1 M i (cid:88) j ∈N i z j ( n j − n i ) . (63)Then, substituting (63) into (62), we obtain ˙ z c,i = k (cid:107)∇ z c,i (cid:107) ( z c,i + ν c,i ) (cid:104) N c,i , n i (cid:105) + k (cid:107)∇ z c,i (cid:107)(cid:104) N c,i , e i (cid:105) , (64)Define z c = [ z c, , · · · , z c,M ] (cid:124) . Then let ˙ z c =[ ˙ z c, , · · · , ˙ z c,M ] (cid:124) , where for each agent ˙ z c,i is given by(64). In the following, we present a boundedness result forthe trajectory z c ( t ) . Proposition V.11.
Suppose for each agent, (cid:107)∇ z c,i (cid:107) > µ , − ≤ (cid:104) N c,i , n i (cid:105) ≤ − (cid:15) , | ν c,i | ≤ ¯ ν , and (cid:104) N c,i , e i (cid:105) ≤ ¯ e , where µ > , (cid:15) > , ¯ ν > and ¯ e > are constants. Then, thesolutions of (64) are uniformly ultimately bounded.Proof. Let V : R → R be a Lyapunov candidate functiondefined by V = (cid:80) Mi =1 z c,i = (cid:107) z c (cid:107) , where V = 0 if andonly if z c = , i.e. z c,i = 0 ∀ i . Then we obtain ˙ V ≤ − k µ (cid:15) (1 − (cid:15) ) V , ∀(cid:107) z c (cid:107) ≥ (¯ ν + ¯ e ) (cid:15) √ M (65)where ˙ V = 0 if and only if z c = , and M is the total numberof agents. Let α ( (cid:107) z c (cid:107) ) = α ( (cid:107) z c (cid:107) ) = (cid:107) z c (cid:107) which areclass K that satisfy: α ( (cid:107) z c (cid:107) ) ≤ V ≤ α ( (cid:107) z c (cid:107) ) . Therefore,according to Theorem 4.18 in [23], the trajectories of ˙ z c areuniformly ultimately bounded.Note that the assumption that − ≤ (cid:104) N c,i , n i (cid:105) ≤ − (cid:15) < is satisfied by Corollary
V.7. This result implies that,as seen from (65), the trajectories of z c , and hence theagents, converge to a neighborhood defined by { r c |(cid:107) z ( r c ) (cid:107) ≥ (¯ ν +¯ e ) (cid:15) √ M } . The bound ¯ e , as seen from (63), is determinedby the connectivity of the graph. A more connected graphincreases the chance that the local PCA directions n i beingaligned with each other.VI. S IMULATION AND E XPERIMENTAL R ESULTS
In this section, we validate the proposed model throughcomputer simulation and physical experiments. We used tworobotic platforms: the Georgia Tech Robotarium mobile robots[39], and the flying Georgia Tech Miniature AutonomousBlimps [11]. In what follows, we first present the sourceseeking results and then the level curve tracking results.In all simulations and experiment, we set (cid:15) = 0 . in(20). This means that we run the PCA flow (3) for a time τ = dt(cid:15) where dt = 0 . is the step time used to update(4). Additionally, for convex fields, we use the field function z ( r i ) = (cid:107) r i (cid:107) . For non-convex fields, we use the field function z ( r i ) = 2 − exp( − ( r i − a ) (cid:124) S ( r i − a )) − exp( − ( r i − b ) (cid:124) A (cid:124) S A ( r i − b )) + (cid:107) r i (cid:107) , where a = [1 , (cid:124) , b = [0 , − (cid:124) , S = 0 . (cid:20) (1 / √
30) 00 1 (cid:21) , S = 0 . (cid:20) / √ (cid:21) and A = ( √ / (cid:20) −
11 1 (cid:21) . A. Source Seeking1) Simulation Results:
We simulated
Algorithm 1 in virtualscalar convex and non-convex -D fields for swarms thathave complete and incomplete connectivity graphs. In all thesimulations, we set k = 1 in (4). Additionally, the sourceis located at the origin. In all of the following figures, boldblue discs represent the agents and the blue arrows indicatethe direction n i , where they turned to red color at the endof the simulation. The lines connecting the agents representthe edges of the network and the pink paths represent thetrajectories of the agents. The contour lines represent thelevel curves of the field. In Fig. 4, swarms of agents (left)Fig. 4: Swarms of agents (left) and agents (right) incomplete graphs.and agents (right) in a complete graph are used to locatea convex field starting from different initial positions. Aspredicted by Theorem V-A3 , the two swarms successfullysteered towards the source even though the -agent swarmwas initially heading towards the positive direction of thefield gradient. Additionally, as predicted by Lemma IV.5 , thevariance λ q is constant while λ n is varying.In the -agent swarm in the left of Fig. 5, the connectivitygraph is incomplete. Hence each agent applies the PCA flow HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 12
Fig. 5: Swarms of agents (left) and agents (right) inincomplete graphslocally resulting in different n i , as it clear by the blue arrows atthe initial time. Nevertheless, as predicted by Proposition V.9 ,the swarm steers towards the source and each n i convergesto the negative direction of the local gradient N c,i . However,since initially for some agents (cid:104) n i , N c,i (cid:105) > , the swarmsdisperse significantly. To save the connectivity of the graphwhen it is incomplete, in the -agent swarm in the right ofFig. 5, modify the control law (4) as u i ( t ) = k ( z i ( t ) − z d ) n i + k q i + k f v i, q , (66)where k f is a constant gain, and v i, q is designed as follows v i, q = (cid:88) j ∈N i ( (cid:104) r j − r i , q i (cid:105) − d ij ) (cid:104) r j − r i , q i (cid:105) q i , (67)which is to maintain a desired distance d ij only along the q i direction [1]. As shown in the right of Fig. 5, each n i converges to − N c,i , but also, the agents in the swarm keepclose to each other due to (67).Fig. 6: Complete network of agents in a non convex fieldAs suggested by (16), n changes faster when ( λ q − λ n ) issmall. To justify this, in Fig. 6, a swarm of agents and acomplete graph is simulated in a non-convex field. In the left,we added (66) to maintain a distance only along q . However,in the right we maintain a distance along both directions byadding v i, n to (66), where v i, n is obtained by replacing q by n in (67). Although the two swarms start at the same location,since the one in the right maintained smaller ( λ q − λ n ) , itsteered faster towards the source than the one in the left whichtook a long distance to turn. This intuitively reveals the effectof the different formation schemes. In particular, a swarm witha larger spatial distribution encodes more diverse informationabout the field and hence the swarm steers faster towards thesource.
2) Experimental Results:
We tested
Algorithm 1 usingunicycle mobile robots at the Georgia Teach Robotarium[39]. The diameter of each robot is about . m and the Fig. 7: A -agent swarm in an incomplete graphFig. 8: A -agent swarm in convex (left) and non-convex fields(right)dimensions of the experimental space are about m × m .Since these robots do not have light sensors, we used virtualfields where we projected their level curves on the surfacefor a visualization purpose. In Fig. 7, we show the results ofa -robot system in a convex field. The graph is complete,however, without formation. Alternatively, we show in Fig. 8the results of a -robot system in a complete graph and withformation. The two swarms start at the same location, however,since the field is convex in the left, the swarm approachesthe source and quickly settle. However, since the field in non-convex in the right, the swarm requires more time to eventuallysettle at the source.Alternatively, we installed light sensors in the Georgia TechMiniature Blimps [11] and then performed source seeking ina physical light field. The diameter of each blimp is about . m and the dimensions of the experimental space are about m × m. To make the minimum at the source, we inverted thefield by using z i instead of z i . Snapshots of two experimentsare shown in Fig. 9 and Fig. 10, where initially in the former (cid:104) N , n (cid:105) < , and in the latter (cid:104) N , n (cid:105) > . Additionally, weFig. 9: The two blimps initially have (cid:104) N , n (cid:105) < .presented in Fig. 11 the trajectories of these two experiments, HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 13
Fig. 10: The two blimps initially have (cid:104) N , n (cid:105) > .where the one on the lest corresponds to Fig. 9, and the otherone corresponds to Fig. 10. The trajectories are colored basedon the light intensity where the black diamond is the sourcelocation and the red discs are the starting locations. Despitemany messing measurements, the blimps are able to locate thesource.Fig. 11: Trajectories of the two experiments of Fig. 9 (left)and Fig. 10 (right). The trajectories are colored based on thelight intensity. The red discs represent the starting points andthe diamond is the light source location. B. Level Curve Tracking
In what follows, we set z d = 2 and (cid:15) = 0 . . Additionally,we set k = 2 , k = 0 . .
1) Simulation Results:
In Fig. 12, a -agent system witha line graph is simulated in a convex field (left) and anon-convex field (right). Alternatively, a -agent system issimulated in a convex field Fig. 13, where in the left thegraph is incomplete and static, while in the right the graphis dynamic. In the dynamic graph, each agent chooses theclosest three agents as its neighbors at each instant of time.Despite the graphs are arbitrary, the system is able to trackthe desired level curve.Fig. 12: a -agent system with a line graph in a convex field(left) and a non-convex field (right).
2) Experimental Results:
In Fig. 14, we implemented thelevel curve tracking algorithm using four Robotarium robots ina virtual non-convex field. The robots are connected by a linegraph. Despite the lack of formation control and the simplicityof the control law, the swarm is able to track the desired levelcurve smoothly. VII. C
ONCLUSION
In this paper, we proposed a distributed active perceptionstrategy for source seeking and level curve tracking. Using Fig. 13: a -agent system with a static incomplete graph (left)and a dynamic graph (right).Fig. 14: Four robots in a non convex field.the body frame obtained by PCA perception, we designed adistributed control law that enables swarms of various sizesand graph structures to perform collective source seekingand level curve tracking of scalar fields without the needto explicitly estimate the field gradient or explicitly sharemeasurements among the agents. We obtained several stabilityresults in a singular perturbation framework justifying therobustness and convergence of the algorithms. The simulationand experimental results suggest the efficiency and generalityof the proposed model. In the future, we will design controllaws for different swarm applications within the proposedframework of the distributed active perception. ACKNOWLEDGEMENTS
S. Al-Abri and F. Zhang were supported by ONR grantsN00014-19-1-2556 and N00014-19-1-2266; NSF grants OCE-1559475, CNS-1828678, and S&AS-1849228; NRL grantsN00173-17-1-G001 and N00173-19-P-1412 ; and NOAA grantNA16NOS0120028.VIII.
PROOFS OF THE INFORMATION DYNAMICS
Proof of Lemma IV.1.
The covariance matrix satisfies C i n i = λ ni n i , and C i q i = λ qi q i . Since with Assumption III.1, theeigenvalues and eigenvectors of C i are uniquely defined, thentheir derivatives exist. Thus, taking the derivative, we obtain ˙ C i n i + C i ˙ n i = ˙ λ ni n i + λ ni ˙ n i , (68) ˙ C i q i + C i ˙ q i = ˙ λ qi q i + λ qi ˙ q i . (69)Inner product both sides of (68) with the eigenvector q i , andboth sides of (69) with the eigenvector n i (cid:104) q i , ˙ C i n i (cid:105) + (cid:104) q i , C i ˙ n i (cid:105) = ˙ λ ni (cid:104) q i , n i (cid:105) + λ ni (cid:104) q i , ˙ n i (cid:105) , (70) (cid:104) n i , ˙ C i q i (cid:105) + (cid:104) n i , C i ˙ q i (cid:105) = ˙ λ qi (cid:104) n i , q i (cid:105) + λ qi (cid:104) n i , ˙ q i (cid:105) . (71)Since C i is symmetric, then ˙ C i is also symmetric. Thisimplies that (cid:104) q i , C i ˙ n i (cid:105) = (cid:104) C i q i , ˙ n i (cid:105) = λ qi (cid:104) q i , ˙ n i (cid:105) . Similarly, HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 14 (cid:104) n i , C i ˙ q i (cid:105) = (cid:104) C i n i , ˙ q i (cid:105) = λ ni (cid:104) n i , ˙ q i (cid:105) . Using these along withthe fact that (cid:104) q i , n i (cid:105) = (cid:104) n i , q i (cid:105) = 0 , we obtain from (70) and(71) (cid:104) q i , ˙ n i (cid:105) = − λ qi − λ ni (cid:104) q i , ˙ C i n i (cid:105) , (72) (cid:104) n i , ˙ q i (cid:105) = 1 λ qi − λ ni (cid:104) q i , ˙ C i n i (cid:105) . (73)Since n i and q i are orthonormal, then we can write ˙ n i = (cid:104) q i , ˙ n i (cid:105) q i , ˙ q i = (cid:104) n i , ˙ q i (cid:105) n i . (74)Substituting (72) and (73) in (74) ˙ n i = − λ qi − λ ni (cid:104) q i , ˙ C i n i (cid:105) q i (75) ˙ q i = 1 λ qi − λ ni (cid:104) q i , ˙ C i n i (cid:105) n i . (76)Taking the derivative of the covariance (2), we obtain ˙ C i = (cid:88) k ∈H i [( ˙ r k − ˙ r c,i )( r k − r c,i ) (cid:124) + ( r k − r c,i )( ˙ r k − ˙ r c,i ) (cid:124) ] . (77)But using (1) ˙ r k − ˙ r c,i = u k − M i (cid:88) l ∈H i u l = 1 M i (cid:88) l ∈H i ( u k − u l ) . (78)Hence, substituting (78) in (77) ˙ C i = 1 M i (cid:88) k ∈H i (cid:88) l ∈H i ( u k − u l )( r k − r c,i ) (cid:124) + 1 M i (cid:88) k ∈H i ( r k − r c,i ) (cid:88) l ∈H i ( u k − u l ) (cid:124) = (cid:88) k ∈H i (cid:2) u k ( r k − r c,i ) (cid:124) + (cid:0) r k − r c,i ) u (cid:124) k ] , (79)where we used the fact that M i (cid:80) k ∈H i (cid:80) l ∈H i u k ( r k − r c,i ) (cid:124) = (cid:80) k ∈H i u k ( r k − r c,i ) (cid:124) , M i (cid:80) k ∈H i (cid:80) l ∈H i u l ( r k − r c,i ) (cid:124) = ( M i (cid:80) l ∈H i )( (cid:80) k ∈H i ( r k − r c,i ) (cid:124) ) = 0 , M i (cid:80) k ∈H i ( r k − r c,i ) (cid:80) l ∈H i u (cid:124) k = (cid:80) k ∈H i ( r k − r c,i ) u (cid:124) k , and M i (cid:80) k ∈H i ( r k − r c,i ) (cid:80) l ∈H i u (cid:124) l = 0 . Substituting (79) in(75) and (76) leads to the desired result (6). Proof of Lemma IV.2.
Substituting the control law (4) in (79),we obtain ˙ C i = (cid:88) k ∈H i (cid:104) ( k z dk n k + k q k )( r k − r c,i ) (cid:124) + ( r k − r c,i )( k z dk n k + k q k ) (cid:124) (cid:105) , (80)where z dk = z k − z d . Let z dc,i = z c,i − z d . Then, we add − k (cid:80) k ∈H i z dc,i n i ( r k − r c,i ) (cid:124) = k (cid:80) k ∈H i z dc,i ( r k − r c,i ) q (cid:124) i to (80) to obtain ˙ C i = (cid:88) k ∈H i (cid:104)(cid:0) k [ z dk n k − z dc,i n i ] + k q k (cid:1) ( r k − r c,i ) (cid:124) + ( r k − r c,i ) (cid:0) k [ z dk n k − z dc,i q i ] + k q k (cid:1) (cid:124) (cid:105) , (81) which is used to derive (cid:104) n i , ˙ C i q i (cid:105) = (cid:88) k ∈H i [ k ( z dk (cid:104) n k , n i (cid:105) − z dc,i ) + k (cid:104) q k , n i (cid:105) ] (cid:104) r k − r c,i , q i (cid:105) +[ k ( z dk (cid:104) n k , q i (cid:105) − z dc,i ) + k (cid:104) q k , q i (cid:105) ] (cid:104) r k − r c,i , n i (cid:105) . (82)Finally, we obtain (7) and (8) by first substituting (82) in(75) and (76), and then applying (9) and (10) for w i and σ i ,respectively. Proof of Corollary IV.3.
Since we this lemma is about sourceseeking only, then we have k = 0 and z dc,i = z c,i . Substituting z c,i from (11) into (9), we obtain w i = C i ∇ z c,i + (cid:88) k ∈H i [ z k ( (cid:104) n k , n i (cid:105) −
1) + ν k ]( r k − r c,i ) , (83)where C i ∇ z c,i = (cid:80) k ∈H i ( r k − r c,i )( r k − r c,i ) (cid:124) ∇ z c,i . Thenwe obtain k λ qi − λ ni (cid:104) w i , q i (cid:105) = k λ qi λ qi − λ ni (cid:107)∇ z c,i (cid:107)(cid:104) N c,i , q i (cid:105) + ˆ ν i − k λ qi λ qi − λ ni (cid:88) k ∈H i z k ( (cid:104) n k , n i (cid:105) − (cid:104) r k − r c,i q i (cid:105) , (84)where ˆ ν i = ( k / ( λ qi − λ ni ) (cid:80) k ∈H i ν k (cid:104) r k − r c,i , q i (cid:105) . On theother hand, when k = 0 , then using (10), we obtain λ qi − λ ni σ i = k λ qi − λ ni (cid:88) k ∈H i z k (cid:104) n k , q i (cid:105)(cid:104) r k − r c,i , n i (cid:105) , (85)where we use the fact that (cid:80) k ∈H i z c,i (cid:104) r k − r c,i , n i (cid:105) = z c,i n (cid:124) i (cid:80) k ∈H i ( r k − r c,i ) = 0 . Finally, substituting (84)and (85) into (7) and (8), and defining E i = ( k / ( λ qi − λ ni )) (cid:80) k ∈H i z k [ (cid:104) n k , q i (cid:105)(cid:104) r k − r c,i , n i (cid:105) + ( (cid:104) n k , n i (cid:105) − (cid:104) r k − r c,i , q i (cid:105) ] , we obtain the claimed (12) and (13). Proof of Lemma IV.4.
We first proof that ˙ n = − k λ q − λ n w (cid:124) qq (86) ˙ q = + k λ q − λ n w (cid:124) qn , (87)where w = M (cid:88) k =1 ( z k − z c )( r k − r c ) . (88)When the graph is complete, then each agent computes thesame covariance matrix C i = C = M (cid:88) k =1 ( r k − r c )( r k − r c ) (cid:124) , (89)where all the agents see the same center r c = M (cid:80) Mk =1 r k .This implies that n i = n k = n , and q i = q k = n for all i, k .Hence, (cid:104) n i , q k (cid:105) = 0 , and (cid:104) q i , q k (cid:105) = 1 for all i, k , and thus σ i = (cid:88) k ∈H i [ − k z dc,i (cid:104) r k − r c,i , n i (cid:105) + k (cid:104) r k − r c,i , n i (cid:105) ] = 0 , (90) HE IEEE TRANSACTIONS ON AUTOMATIC CONTROL 15 where we used the fact that (cid:80) k ∈H i ( r k − r c,i ) = 0 . Addition-ally, since z dk − z dc = z k − z d − M (cid:80) Mk =1 ( z k − z d ) = z k − z c ,and (cid:104) n k , n i (cid:105) = 1 , then (9) we obtain (88). Finally, using (90),and substituting (88) in (7) and (8) yields the claimed (86)and (87).Substituting ( z k − z c ) of (11) in (88) yields w = M (cid:88) k =1 [ (cid:104) r k − r c , ∇ z c (cid:105) ( r k − r c ) + ν k ( r k − r c )] . (91)But, (cid:80) Mk =1 (cid:104) r k − r c , ∇ z c (cid:105) ( r k − r c ) = (cid:80) Mk =1 ( r k − r c )( r k − r c ) (cid:124) ∇ z c = C ∇ z c . Hence w = C ∇ z c + M (cid:88) k =1 ν k ( r k − r c ) . (92)Finally, using ˆ ν = k λ q − λ n (cid:80) Mk =1 ν k (cid:104) r k − r c , q (cid:105) , and substitut-ing (92) in (86) and (87) yields the claimed (14) and (15). Proof of Lemma IV.5.
By the definition of PCA [22], λ n = argmin u u (cid:124) Cu is the smallest variance, and λ q = argmax u u (cid:124) Cu is the largest variance. This implies that bydefinition λ n ( t ) ≤ λ q ( t ) . What remains is to show that ˙ λ q = 0 . Taking the time derivative of λ q = q (cid:124) Cq = (cid:80) Mi =1 (cid:104) r i − r c , q (cid:105) , we obtain ˙ λ q = 2 M (cid:88) i =1 (cid:104) r i − r c , q (cid:105) ( (cid:104) ˙ r i − ˙ r c , q (cid:105) + (cid:104) r i − r c , ˙ q (cid:105) ) (93)But, for a complete graph (cid:104) ˙ r i − ˙ r c , q (cid:105) = k ( z di − z da ) (cid:104) n , q (cid:105) = 0 ,where z a = M (cid:80) Mi =1 z i is the average field measurement.Since from (15) (cid:104) q , ˙ q (cid:105) = 0 , then ˙ λ q = 2 M (cid:88) i =1 (cid:104) r i − r c , q (cid:105)(cid:104) r i − r c , ˙ q (cid:105) = 2 λ q (cid:104) q , ˙ q (cid:105) = 0 . (94)Similarly, taking the time derivative of λ n = n (cid:124) Cn = (cid:80) Mi =1 (cid:104) r i − r c , n (cid:105) , and using ˙ r i − ˙ r c = k ( z i − z a ) n , weobtain (cid:104) ˙ r i − ˙ r c , n (cid:105) = k ( z i − z a ) (cid:104) n , n (cid:105) = k ( z i − z a ) .Additionally, since from (7) (cid:104) n , ˙ n (cid:105) = 0 , we obtain (cid:80) Mi =1 (cid:104) r i − r c , n (cid:105)(cid:104) r i − r c , ˙ n (cid:105) = 2 (cid:104) Cn , ˙ n (cid:105) = 2 λ n (cid:104) n , ˙ n (cid:105) = 0 . Therefore ˙ λ n = 2 k (cid:80) Mi =1 ( z i − z a ) (cid:104) r i − r c , n (cid:105) . Additionally, using(11), z a = M (cid:80) Mk =1 z k = z c + M (cid:80) Mk =1 (cid:104) r k − r c , ∇ z c (cid:105) + M (cid:80) Mk =1 ν k = z c + M (cid:80) Mk =1 ν k , where we used the fact (cid:80) Mk =1 (cid:104) r k − r c , ∇ z c (cid:105) = 0 . Hence ˙ λ n = 2 k M (cid:88) i =1 ( z i − z c ) (cid:104) r i − r c , n (cid:105) , (95)where we used the fact that (cid:80) Mi =1 (cid:80) Mk =1 ν k (cid:104) r i − r c , n (cid:105) = (cid:80) Mk =1 ν k (cid:80) Mi =1 (cid:104) r i − r c , n (cid:105) = 0 . Finally, using (11), we canfurther write ˙ λ n = 2 k λ n (cid:107)∇ z c (cid:107)(cid:104) N c , n (cid:105) + 2 k M (cid:88) i =1 ν i (cid:104) r i − r c , n (cid:105) . (96)Note that, from (96), λ n will increase or decrease, hence theshape will stretch or shrink along the n direction, dependingon the signs of (cid:104) N c , n (cid:105) and (cid:80) Mi =1 ν i (cid:104) r i − r c , n (cid:105) . However, if it increases, it will do so only up to λ n = λ q . At this point, since λ n = argmin u u (cid:124) Cu and λ q = argmax u u (cid:124) Cu , then the PCAflow will interchange n and q and hence the swarm performsa turn of at most ◦ . Consequently, λ q ( t ) ≤ λ q ( t ) . Proof of Lemma V.3.
Since ρ ( | δ | ) is a real number, then wemust have (cid:113) − ( λ q − λ n ) | δ | k (cid:15) λ q µ ≥ which implies that (cid:107)∇ z c (cid:107) >µ ≥ ( λ q − λ n ) | δ | k (cid:15) λ q . But | δ | ≤ | δ | + | δ | ≤ k λ q − λ n | ϑ | + k | z a − z d | + k (cid:107)∇ z c (cid:107) (cid:107)∇ z c (cid:107) , where ϑ = (cid:80) k ν k (cid:104) r k − r c , q (cid:105) and ∇ z c is the Hessian matrix. Then we must have (cid:107)∇ z c (cid:107) > (cid:15) λ q | ϑ | + ( λ q − λ n )( k | z a − z d | + k ) k (cid:15) λ q (cid:107)∇ z c (cid:107) (cid:107)∇ z c (cid:107) . Solving this inequality yieldsthe desired result (36). R EFERENCES[1] Said Al-Abri, Sean Maxon, and Fumin Zhang. Integrating a PCAlearning algorithm with the SUSD strategy for a collective sourceseeking behavior. In ,pages 2479–2484. IEEE, 2018.[2] Said Al-Abri, Wencen Wu, and Fumin Zhang. A gradient-free three-dimensional source seeking strategy with robustness analysis.
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Said Al-Abri is currently a Postdoctoral Scholarat the School of Electrical and Computer Engi-neering at Georgia Institute of Technology workingunder the supervision of Professor Fumin Zhang. Heearned his Ph.D. degree from the Georgia Institute ofTechnology on August 2019, M.S. degree from theUniversity of Central Florida on May 2013, and B.S.degree from the Sultan Qaboos University on June2009, all in Electrical and Computer Engineering.He is currently conducting research on bio-inspireddistributed autonomy and optimization algorithms.He received the Outstanding ECE Graduate Teaching Assistant Award atGeorgia Tech in 2019, the Best Three-Minute Presentation at AmericanControl Conference 2019, and the Best Poster Award at the CoordinatedScience Laboratory Student Symposium, 2019.