A New Encounter Between Leader-Follower Tracking and Observer-Based Control: Towards Enhancing Robustness against Disturbances
AA New Encounter Between Leader-Follower Tracking andObserver-Based Control: Towards Enhancing Robustnessagainst Disturbances
Chuan Yan a , Huazhen Fang a, ∗ a Department of Mechanical Engineering, University of Kansas, Lawrence, KS, 66045 USA
Abstract
This paper studies robust tracking control for a leader-follower multi-agent system (MAS) subject to disturbances. Achallenging problem is considered here, which di ff ers from those in the literature in two aspects. First, we considerthe case when all the leader and follower agents are a ff ected by disturbances, while the existing studies assume onlythe followers to su ff er disturbances. Second, we assume the disturbances to be bounded only in rates of changerather than magnitude as in the literature. To address this new problem, we propose a novel observer-based distributedtracking control design. As a distinguishing feature, the followers can cooperatively estimate the disturbance a ff ectingthe leader to adjust their maneuvers accordingly, which is enabled by the design of the first-of-its-kind distributeddisturbance observers. We build specific tracking control approaches for both first- and second-order MASs and provethat they can lead to bounded-error tracking, despite the challenges due to the relaxed assumptions about disturbances.We further perform simulation to validate the proposed approaches. Keywords:
Multi-agent system, observer-based control, distributed control, unknown disturbances.
1. Introduction
A leader-follower multi-agent system (MAS) refers to an MAS in which a group of follower agents performdistributed control while interchanging information with their neighbors to collectively track the state of a leader agent.A large body of work has been developed recently to deal with the tracking control design under diverse challengingsituations, e.g., complex dynamics, communication delays, noisy measurements, switching topologies, and limitedenergy budget, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and the references therein. However, aproblem that has received inadequate attention to date is the case when the agents are subjected to disturbances. In areal world, disturbances can result from unmodeled dynamics, change in ambient conditions, inherent variability ofthe dynamic process, and sensor noises. They can cause degradation and even failure of tracking control if not welladdressed.A lead is taken in [3] with the study of disturbance-robust leader-follower tracking. It presents a distributed controldesign that achieves tracking with a bounded error when magnitude-bounded disturbances a ff ect the followers. Thisnotion is extended in [19] to make the followers a ff ected by disturbances enter a bounded region centered aroundthe leader in finite time. Another finite-time tracking control approach is o ff ered in [20], where the sliding modecontrol technique is used to suppress the e ff ects of disturbances. It is noted that, while the control designs in theseworks yield robustness, they are based on upper bounds of the disturbances. By contrast, a di ff erent way is to capturethe disturbances by designing some observers and then adjust the control run based on the disturbance estimation.Obtaining an explicit knowledge of disturbances, this approach can advantageously reduce conservatism in controland thus enhance the tracking performance further. In [21, 22], disturbance observers are developed and integrated ∗ Corresponding author. E-mail address: [email protected].
Preprint submitted to Systems & Control Letters May 14, 2019 a r X i v : . [ c s . S Y ] M a y nto tracking controllers such that a follower can estimate and o ff set the local disturbance interfering with its dynamicsduring tracking. The results in both studies point to the e ff ectiveness of disturbance observers for improving trackingaccuracy — for instance, the tracking errors can approach zero despite non-zero disturbances under certain conditions.However, other than these two, there are no more studies on this subject to the best of our knowledge. This leavesmany problems still open. Meanwhile, the potential of the disturbance-observer-based approach is still far frombeing fully explored. It is noteworthy that observer-based tracking control has been investigated in a few works,e.g., [3, 19, 23, 24, 25], but observers in these studies are meant to infer various state or input variables rather thandisturbances.In this study, we uniquely focus on an open problem: can we enable distributed tracking control when not onlythe followers but also the leader are a ff ected by unknown disturbances and when only the rates of change of the dis-turbances are bounded? The state of the art, e.g., [3, 19, 20, 21, 22], generally considers that disturbances plaguejust the followers and that they are bounded in magnitude or approach fixed values as time goes by. The leader’s dy-namics, however, can also involve disturbances from a practical viewpoint. For example, consider an MAS composedof a few mobile ground robots, the changes in the slope of the road act as disturbances on every robot including theleader. The same can be said for the wind a ff ecting a group of unmanned aerial vehicles. Such disturbances are moredi ffi cult to be rejected because the leader cannot measure them and share the information with any of the followers.Therefore, the tracking performance may be damaged when this occurs. Furthermore, it is usually desirable to relaxthe assumptions about disturbances to enhance the practical robustness of the control design. In [19, 20, 21, 22], the disturbances are assumed to be bounded in magnitude. However, we wish to require the disturbances to bebounded in rates of change. This relaxation will be realistically beneficial for dealing with large disturbances but alsopresent more complexity to capture and suppress the disturbances. It must be pointed out that the observer designsin [3, 19, 20, 21, 22, 23, 24, 25] cannot be extended to deal with the considered problem, due to the more challengingpresence and nature of the disturbances. Hence, a solution is still absent from the literature.To address the above problem, we develop a novel observer-based distributed tracking control framework, whichis the main contribution of this paper. Di ff erent from the previous studies, this framework builds on the notion thata follower can gain a real-time awareness of not only its own but also the leader’s dynamics through distributedestimation. We hence design a set of new observers and particularly, distributed disturbance observers that, for thefirst time, can enable the followers to collectively infer the disturbance a ff ecting the leader. We perform the observer-based tracking control design for both first- and second-order MASs. We then conduct theoretical analysis of theproposed approaches. We show that, even though disturbances are imposed on all the agents, the tracking errors arestill upper bounded (bounded-error tracking) as long as the rates of change of the disturbances are bounded. Further,the tracking errors will approach zero (zero-error tracking) if the disturbances converge to certain fixed points. Wefinally present simulations to validate the proposed approaches.The rest of this paper is organized as follows. Section 2 introduces some preliminaries. Section 3 considers aleader-follower MAS with first-order dynamics, develops an observer-based distributed tracking control approach,and analyzes its performance rigorously. Section 4 proceeds to study the second-order MASs and develops a moresophisticated tracking control approach. Numerical simulation is o ff ered in Section 5 to illustrate the e ff ectiveness ofthe proposed results. Finally, Section 6 gathers our concluding remarks.
2. Preliminaries
This section introduces notation and basic concepts about graph theory and nonsmooth analysis.
The notation used throughout this paper is standard. The n -dimensional Euclidean space is denoted as R n . Fora vector, (cid:107) · (cid:107) denotes the 1-norm, and (cid:107) · (cid:107) stands for the 2-norm. The notation represents a column vector ofones. We let diag( . . . ) and det( · ) represent a block-diagonal matrix and the determinant of a matrix, respectively. Theeigenvalues of an N × N matrix are λ i ( · ) for i = , , . . . , N . The minimum and maximum eigenvalues of a real,symmetric matrix are denoted as λ ( · ) and ¯ λ ( · ). Matrices are assumed to be compatible for algebraic operations if theirdimensions are not explicitly stated. A C k function is a function with k continuous derivatives.2 .2. Graph Theory We use a graph to describe the information exchange topology for a leader-follower MAS. First, consider a net-work composed of N independent followers, and model the interaction topology as an undirected graph. The followergraph then is expressed as G = ( V , E ), where V = { , , · · · , N } is the vertex set and E ⊆ V × V is the edge setcontaining unordered pairs of vertices. A path is a sequence of connected edges in a graph. The follower graph is con-nected if there is a path between every pair of vertices. The neighbor set of agent i is denoted as N i , which includesall the agents in communication with it. The adjacency matrix of G is A = [ a i j ] ∈ R N × N , which has non-negativeelements. The element a i j > i , j ) ∈ E , and moreover, a ii = i ∈ V . For the Laplacian matrix L = [ l i j ] ∈ R N × N , l i j = − a i j if i (cid:44) j and l ii = (cid:80) k ∈N i a ik . The leader is numbered as vertex 0 and can send informationto its neighboring followers. Then, we have a graph ¯ G , which consists of graph G , vertex 0 and edges from the leaderto its neighbors. The leader is globally reachable in ¯ G if there is a path in graph ¯ G from every vertex i to vertex 0. Toexpress the graph ¯ G more precisely, we denote the leader adjacency matrix associated with ¯ G by B = diag( b , . . . , b N ),where b i > i and b i = Lemma 1. [26, Lemma 1.1] The Laplacian matrix L ( G ) has at least one zero eigenvalue, and all the nonzero eigen-values are positive. Furthermore, L ( G ) has a simple zero eigenvalue and all the other eigenvalues are positive if andonly if G is connected. Lemma 2. [27, Lemma 4] The matrix H = L + B is positive stable if and only if vertex 0 is globally reachable in ¯ G .2.3. Nonsmooth Analysis Consider the following discontinuous dynamical system˙ x = f ( x ) , x ∈ R n , x (0) = x ∈ R n , (1)where f ( x ) : R n → R n is defined almost everywhere (a.e.). In other words, it is defined everywhere for x ∈ R n \ W ,where W is a subset of R n of Lebesgue measure zero. Moreover, f ( x ) is Lebesgue measurable in an open region andlocally bounded. A vector variable x ( · ) ∈ R n is a Filippov solution of (1) on [ t , t ] if x ( · ) is absolutely continuous on[ t , t ] while for almost all t ∈ [ t , t ], satisfying the following di ff erential inclusion:˙ x ∈ K [ f ]( x ) (cid:44) (cid:92) δ> (cid:92) µ ( M ) = co { f ( B ( x , δ ) \ M ) } , (2)where (cid:84) µ ( M ) = represents the intersection over all sets M of Lebesgue measure zero, co( · ) denotes the closure of aconvex hull and B ( x , δ ) denotes an open ball of radius δ centered at x . Let V ( x ) : R n → R be a locally Lipschitzcontinuous function. Its Clarke’s generalized gradient is given by ∂ V ( x ) (cid:44) co (cid:26) lim i →∞ ∇ V ( x i ) | x i → x , x i (cid:60) Ω V ∪ M (cid:27) , where ∇ V ( x ) is the conventional gradient, and Ω V denotes a set of Lebesgue measure zero which includes all pointswhere ∇ V ( x ) does not exist. Moreover, the set value of the derivative of V associated with (1) is defined as L ˙ V ( x ) = { a ∈ R }|∃ v ∈ K [ f ]( x ) such that ζ · v = a , ∀ ζ ∈ ∂ V ( x ) } . The following lemma will be used later.
Lemma 3. [28, Page 32] Let x ( t ) : [ t , t ] → R n be a Filippov solution of (2) . Let V ( x ) be a locally Lipschitz andregular function. Then d / dt ( V ( x ( t ))) exists a.e. and dV ( x ( t )) / dt ∈ L ˙ V ( x ) a.e.2.4. Assumptions Throughout this paper, we consider a leader-follower MAS with N + u i and simultaneously a ff ected by an external disturbance f i for i = , , . . . , N . We make the following assumptions. Assumption 1.
The input u ∈ C has a bounded first-order derivative, satisfying | ˙ u | ≤ w, where w is unknown. Assumption 2.
The external disturbance f i for i = , , . . . , N has a bounded first-order derivative, i.e., (cid:107) ˙ f N × (cid:107) ≤ q and (cid:13)(cid:13)(cid:13)(cid:13)(cid:104) ˙ f ˙ f · · · ˙ f N (cid:105) (cid:62) (cid:13)(cid:13)(cid:13)(cid:13) ≤ q , where q , q ≥ . . First-Order Leader-Follower Tracking This section studies first-order leader-follower tracking with disturbances. We develop an observer-based controlapproach, pivoting the design on a set of observers to make a follower aware of the leader’s and its own disturbances.We further analyze the closed-loop stability of the proposed approach.
Consider an MAS with N + x i = u i + f i , x i , u i , f i ∈ R , i = , , . . . , N , (3)where x i is the position, u i the control input equivalent to the velocity maneuver, and f i the unknown disturbance.Suppose that Assumptions 1-2 hold. Here, the objective is to design a distributed control law for u i such that eachfollower can control its dynamics to track the leader’s trajectory via exchanging information with its neighbors. Remark 1.
Compared with previous studies, the problem setting here is more generic and applicable to a wide rangeof practical scenarios. Below, we outline a comparison with [3, 19, 20, 21, 22], which are the main references abouttracking control with disturbances and henceforth referred to as the existing literature. First, this work considers aninput-driven leader, while the leader is usually assumed to be input-free in the literature. Assumption 1 only requiresthe leader’s input to be bounded in rate of change (with the bound unknown), which can be easily satisfied sincepractical actuators only allow limited ramp-ups. Second, Assumption 2 imposes disturbances on all the leader andfollower agents, while the literature assumes only followers to be a ff ected by disturbances. Note that the case whena disturbance is inflicted on the leader is nontrivial. This is because the leader’s disturbance is very di ffi cult to bedetermined by the followers, especially in a distributed network where many followers cannot directly interchangeinformation with the leader. Further, the disturbances are assumed to have only bounded rates of change rather thanbounded magnitude as required in the literature. This can be greatly useful for dealing with very large disturbances.From the comparison, we conclude that the considered problem is less restrictive than the predecessors, which stillremains an open challenge. • Given the above problem setting, we propose an observer-based tracking control approach. The development be-gins with the design of a distributed linear continuous controller for a follower (say, follower i ). It crucially incorpo-rates the estimation of three unknown variables, u , f and f i , enabling follower i to maneuver through simultaneouslyemulating the input and disturbance driving the leader and o ff setting the local disturbance. We subsequently constructthree observers to achieve the estimation to be integrated with the controller.Considering follower i , we propose to design its controller as follows: u i = − k (cid:88) j ∈N i a i j ( x i − x j ) + b i ( x i − x ) + ˆ u , i + ˆ f , i − ˆ f i , (4)where k > f , i and ˆ u , i are follower i ’s respective estimates of the leader’s disturbance f andinput u , and ˆ f i is follower i ’s estimate of its own disturbance f i . Note that b i > i ’s neighborand b i = − (cid:80) j ∈N i a i j ( x i − x j ) − b i ( x i − x ) is employed to drive follower i approaching theleader; the term ˆ u , i + ˆ f , i ensures that follower i applies maneuvers consistent with the leader’s input and disturbance;the term − ˆ f i is used to cancel the local disturbance. For this controller, we build a series of observers as shown belowto estimate u , f and f i , respectively.To begin with we propose an observer as follows to obtain ˆ u , i , which is based on the design in [16]:˙ˆ u , i = − (cid:88) j ∈N i a i j (ˆ u , i − ˆ u , j ) − b i (ˆ u , i − u ) − d i · sgn (cid:88) j ∈N i a i j (ˆ u , i − ˆ u , j ) + b i (ˆ u , i − u ) , ˙ d i = τ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ∈N i a i j (ˆ u , i − ˆ u , j ) + b i (ˆ u , i − u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5a)(5b)4or i = , , . . . , N , where d i is the observer gain and τ i > ffi cient. For (5a), the leading term on theright-hand side, − (cid:80) j ∈N i a i j (ˆ u , i − ˆ u , j ) − b i (ˆ u , i − u ), is used to make ˆ u , i approach u ; the signum function sgn( · ) isaimed to overcome the e ff ects of u ’s first-order dynamics, i.e., ˙ u , and ensure the convergence of ˆ u , i to u . Note thatthe observer gain, d i , is adaptively adjusted through (5b). As such, a reasonable gain can be determined even if theupper bound of ˙ u , w , is unknown (see Assumption 1).The following disturbance observer is proposed for follower i to estimate f :˙ z f , i = − b i z i − b i x − (cid:88) j ∈N i a i j ( ˆ f , i − ˆ f , j ) − b i u , ˆ f , i = z f , i + b i x , (6a)(6b)where z f , i is the internal state. The design of (6) is inspired by [29], in which a centralized disturbance observer isdeveloped for a single plant. Here, transforming the original design, we build the distributed observer as above suchthat follower i can estimate f in a distributed manner.The last observer, designed as follows, enables follower i able to infer the disturbance f i inherent in its owndynamics: ˙ z f , i = − lz f , i − l x i + u i , ˆ f i = z f , i + lx i . (7a)(7b)Here, l > z f , i is this observer’s internal state.Combining (4)-(7), we obtain a complete description of an observer-based distributed tracking controller. Next,we will analyze its closed-loop stability. Define e u , i = ˆ u , i − u , which is the input estimation error. According to (5), the closed-loop dynamics of e u , i canbe written as ˙ e u , i = − b i e u , i − (cid:88) j ∈N i a i j ( e u , i − e u , j ) − d i · sgn (cid:88) j ∈N i a i j ( e u , i − e u , j ) + b i e u , i − ˙ u . Further, let us concatenate e u , i for i = , , . . . , N and define e u = (cid:104) e u , e u , · · · e u , N (cid:105) (cid:62) . The dynamics of e u can beexpressed as ˙ e u = − He u − D · sgn( He u ) − ˙ u , (8)where H = B + L and D = diag( d , d , . . . , d N ). It is noted that the signum-function-based term at the right-handside of (8) is discontinuous, measurable and locally bounded. Hence, there exists a Filippov solution to (8), which isrepresented by a di ff erential inclusion as follows:˙ e u ∈ K (cid:2) − He u − D · sgn( He u ) − ˙ u (cid:3) . The following lemma characterizes the convergence of e u . Lemma 4.
If Assumption 1 holds, then lim t →∞ | e u , i | = , (9) for i = , , . . . , N. Proof:
By Lemmas 1 and 2, H is positive definite. For (8), consider¯ V ( e u ) = e (cid:62) u He u , ˜ V = N (cid:88) i = ( d i − β ) τ i , β ≥ w , as noted. We then take V = ¯ V ( e u ) + ˜ V as a Lyapunov functional candidate. For the set-valued Liederivative of ¯ V ( e u ), we have L ˙¯ V = K (cid:104) − e (cid:62) u H e u − e (cid:62) u HD · sgn( He u ) − e (cid:62) u H ˙ u (cid:105) = K (cid:104) − N (cid:88) i = d i (cid:88) j ∈N i a i j (ˆ u , i − ˆ u , j ) + b i (ˆ u , i − u ) (cid:62) · sgn (cid:88) j ∈N i a i j (ˆ u , i − ˆ u , j ) + b i (ˆ u , i − u ) − e (cid:62) u H e u − e (cid:62) u H ˙ u (cid:105) ≤ − N (cid:88) i = d i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ∈N i a i j (ˆ u , i − ˆ u , j ) + b i (ˆ u , i − u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − e (cid:62) u H e u + w (cid:107) He u (cid:107) by the fact that K [ f ] = { f } if f is continuous. Invoking Lemma 3, we obtain that ˙¯ V ∈ L ˙¯ V . Then, the derivative of V is given by ˙ V = ˙¯ V + ˙˜ V = ˙¯ V + N (cid:88) i = ( d i − β ) ˙ d i τ i ≤ − N (cid:88) i = d i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ∈N i a i j (ˆ u , i − ˆ u , j ) + b i (ˆ u , i − u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − e (cid:62) u H e u + w (cid:107) He u (cid:107) + N (cid:88) i = ( d i − β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ∈N i a i j (ˆ u , i − ˆ u , j ) + b i (ˆ u , i − u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − e (cid:62) u H e u − ( β − w ) (cid:107) He u (cid:107) . It is noted that e (cid:62) u H e u ≥
0. This, in addition to the fact that there always exists a β such that β ≥ w by Assumption 1,ensures ˙ V ≤
0. As a result, V ( e u ) is nonincreasing, which implies that e u and d i are bounded. From (5b), it followsthat d i is monotonically increasing, indicating that d i should converge to some finite value. In the meantime, since V is nonincreasing and lower-bounded by zero, it should approach a finite limit. Defining s ( t ) = (cid:82) t e (cid:62) u ( τ ) H e u ( τ ) d τ , wehave s ( t ) ≤ V (0) − V ( t ) by integrating ˙ V ( e u ) ≤ − e (cid:62) u H e u . Hence, s ( t ) will also approach a finite limit. Due to theboundedness of e u and ˙ e u , ¨ s is also bounded. This implies that ˙ s is uniformly continuous. By Barbalat’s Lemma [30],˙ s ( t ) → t → ∞ , indicating that e u → t → ∞ . • Now, consider the distributed observer for f . Define e f , i = ˆ f , i − f , which is follower i ’s estimation error for f .Using (6), the dynamics of e f , i is given by˙ e f , i = − b i e f , i − (cid:88) j ∈N i a i j ( ˆ f , i − ˆ f , j ) − ˙ f . Then, defining e f = (cid:104) e f , e f , · · · e f , N (cid:105) (cid:62) , we have˙ e f = − He f − ˙ f . (10)The following lemma reveals the upper boundedness of e f under Assumption 2. Lemma 5.
If Assumption 2 holds, then (cid:107) e f ( t ) (cid:107) ≤ (cid:107) e f (0) (cid:107) + q λ ( H ) , t > , lim t →∞ (cid:107) e f ( t ) (cid:107) ≤ q λ ( H ) . (11a)(11b)6 roof: Consider the Lyapunov function candidate V ( e f ) = e (cid:62) f e f for (10). According to Assumption 2, we have˙ V ( e f ) = − e (cid:62) f He f − e (cid:62) f ˙ f ≤ − λ ( H ) (cid:107) e f (cid:107) + (cid:107) e f (cid:107)(cid:107) ˙ f (cid:107) ≤ − λ ( H ) (cid:107) e f (cid:107) + q (cid:107) e f (cid:107) . The above inequality can be rewritten as ˙ V ≤ − λ ( H ) V + √ q (cid:112) V . It then follows that (cid:112) V ( t ) ≤ (cid:112) V (0) e − λ ( H ) t + √ q λ ( H ) (cid:16) − e − λ ( H ) t (cid:17) ≤ (cid:112) V (0) + √ q λ ( H ) . (12)Then, (11a) can result from (12) because √ V = √ (cid:107) e f (cid:107) . Meanwhile, for the first inequality in (12), taking the limitsof both sides as t → ∞ would yield (11b). • For the estimation of f i , define the error as e f , i = ˆ f i − f i and further the vector e f = (cid:104) e f , e f , . . . e f , N (cid:105) (cid:62) .By (7), the dynamics of e f is governed by ˙ e f = − le f − ˙ f , (13)where ˙ f = (cid:104) ˙ f ˙ f · · · ˙ f N (cid:105) (cid:62) . The next lemma shows that the error e f is bounded under Assumption 2. Its proof issimilar to that of Lemma 5 and thus omitted here. Lemma 6.
If Assumption 2 holds, then (cid:107) e f ( t ) (cid:107) ≤ (cid:107) e f (0) (cid:107) + q l , t > t →∞ (cid:107) e f ( t ) (cid:107) ≤ q l . With the above results , we are now in a good position to characterize the properties of the tracking error.Define follower i ’s tracking error as e i = x i − x , and put together e i for i = , , . . . , N to form the vector e = (cid:104) e e · · · e N (cid:105) (cid:62) . Using (3) and (4), it can be derived that the dynamics of e can be described as˙ e = − kHe + e f + e u − e f . (14)The following theorem provides a key technical result. Theorem 1.
Suppose that Assumptions 1 and 2 hold. Then, (cid:107) e ( t ) (cid:107) ≤ (cid:107) e (0) (cid:107) + (cid:107) e f (0) (cid:107) + (cid:107) e u (0) (cid:107) + (cid:107) e f (0) (cid:107) + q λ ( H ) + q l k λ ( H ) , lim t →∞ (cid:107) e (cid:107) ≤ q λ ( H ) + q l k λ ( H ) . (15a)(15b) Proof:
Take the Lyapunov function candidate V ( e ) = e (cid:62) e for (14). Consider its derivative:˙ V = − ke (cid:62) He + e (cid:62) e f + e (cid:62) e u − e (cid:62) e f ≤ − k λ ( H ) (cid:107) e (cid:107) + (cid:107) e (cid:107) · (cid:107) e f (cid:107) + (cid:107) e (cid:107) · (cid:107) e u (cid:107) + (cid:107) e (cid:107) · (cid:107) e f (cid:107) , where λ ( H ) >
0. Equivalently, we have˙ V ≤ − k λ ( H ) V + √ (cid:107) e f (cid:107) + (cid:107) e u (cid:107) + (cid:107) e f (cid:107) ) (cid:112) V . (cid:112) V ( t ) ≤ (cid:112) V (0) e − k λ ( H ) t + √ (cid:107) e f ( t ) (cid:107) + (cid:107) e u ( t ) (cid:107) + (cid:107) e f ( t ) (cid:107) )2 k λ ( H ) (1 − e − k λ ( H ) t ) ≤ (cid:112) V (0) + √ (cid:107) e f ( t ) (cid:107) + (cid:107) e u ( t ) (cid:107) + (cid:107) e f ( t ) (cid:107) )2 k λ ( H ) , which, based on Lemmas 4-6, indicates (15a)-(15b). • Theorem 1 shows that the proposed observer-based controller can make each follower track the leader withbounded position errors despite the disturbances. Therefore, we can say that the influence of the disturbances ise ff ectively suppressed and that tracking is achieved in a practically meaningful manner. Remark 2.
For the proposed controller, the tracking performance will be further improved if the disturbances satisfysome stricter conditions. In particular, it is noteworthy that perfect or zero-error tracking can be attained if thedisturbances see their rates of change gradually settle down to zero, i.e., ˙ f i ( t ) → as t → ∞ for i = , , . . . , N. Theproof can be developed following similar lines as above and is omitted here. •
4. Second-Order Leader-Follower Tracking
This section considers leader-follower tracking control for a second-order MAS. An agent’s dynamics now in-volves position, velocity, acceleration and disturbance: (cid:40) ˙ x i = v i , x i ∈ R , ˙ v i = u i + f i , v i ∈ R , (16)for i = , , . . . , N . Here, x i is the position, v i the velocity, u i the acceleration input, and f i the disturbance. Still, agent0 is the leader, and the others are followers numbered from 1 to N . We continue to apply Assumptions 1-2 here and setthe objective of making the followers achieve bounded-error tracking of the leader in the presence of the disturbances.For a general problem formulation, we further assume that no velocity sensor is deployed on the leader andfollowers. Hence, there are no velocity measurements throughout the tracking process. The absence of the velocityinformation, together with the unknown disturbances, makes the tracking control problem more complex than in thefirst-order case, thus requiring a substantial sophistication of the observer-based control approach in Section 3. Here,we will custom build an observer-based tracking controller and develop new velocity and disturbance observers.Consider follower i . We propose the following distributed controller: u i = − k (cid:88) j ∈N i a i j ( x i − x j ) + b i ( x i − x ) − (ˆ v i − ˆ v , i ) + ˆ u , i + ˆ f , i − ˆ f i , (17)where k > u , i , ˆ v , i , ˆ f , i , ˆ v i and ˆ f i are, respectively, follower i ’s estimates of u , v , f , v i and f i . The terms − (cid:80) j ∈N i a i j ( x i − x j ) − b i ( x i − x ) and − (ˆ v i − ˆ v , i ) are used to enable the follower to track the leaderin both position and velocity; the term ˆ u , i + ˆ f , i is used to make the follower steer itself with a maneuvering inputclose to the combined input and disturbance driving the leader; the term − ˆ f i is meant to o ff set the local disturbance.With the above controller structure, we need to construct observers that can obtain the needed estimates. First, itis noted that the input observer of u in (5) can be applied here without any change, and its convergence property asshown in Lemma 4 also holds in this case. Then, we develop the following observer such that follower i can estimatethe leader’s unknown velocity: ˙ z v , i = − b i z v , i − b i x − (cid:88) j ∈N i a i j (ˆ v , i − ˆ v , j ) + ˆ f , i + ˆ u , i , ˆ v , i = z v , i + b i x , (18a)(18b)8here z v , i is the internal state of this observer. Follower i ’s observer for the leader’s disturbance is then proposed as˙ z f , i = − z f , i − ˆ v , i − ˆ u , i , ˆ f , i = z f , i + ˆ v , i . (19a)(19b)The next observer enables follower i to estimate its own velocity:˙ z v , i = − lz v , i − l x i + ˆ f i + u i , ˆ v i = z v , i + lx i . (20a)(20b)Here, l > z v , i the internal state. The final observer is aimed to allow follower i toinfer its local disturbance. It is designed as ˙ z f , i = − z f , i − ˆ v i − u i , ˆ f i = z f , i + ˆ v i , (21a)(21b)where z f , i is the internal state.From above, a complete observer-based distributed tracking controller can be built by putting together the controllaw (17) and the observers in (5) and (18)-(21). The following theorem is the main result about the closed-loopstability of the proposed controller. Theorem 2.
Suppose that Assumptions 1-2 hold and apply the proposed distributed tracking controller given in (5) and (17) - (21) to the MAS in (16) . Then, there exist δ > and (cid:15) > such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:34) x i ( t ) − x ( t ) v i ( t ) − v ( t ) (cid:35)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ δ, t > , lim t →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:34) x i ( t ) − x ( t ) v i ( t ) − v ( t ) (cid:35)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:15), (22a)(22b) for i = , , . . . , N. Proof:
The proof is organized into three parts. Part a) proves that the coupled observers in (18)-(19) yield bounded-error estimation of v and f ; Part b) shows that the observers in (20)-(21) lead to bounded errors when estimating v i and f i ; finally, based on Parts a) and b), Part c) demonstrates the upper boundedness of the position and velocitytracking errors when the proposed controller is applied. Part a):
Define the estimation errors of the observers in (18)-(19) as e v , i = ˆ v , i − v and e f , i = ˆ f , i − f . Accordingto (16) and (18)-(19), their dynamics can be written as˙ e v , i = − b i e v , i − (cid:88) j ∈N i a i j (ˆ v , i − ˆ v , j ) + ˆ u , i − u + ˆ f , i − f , ˙ e f , i = − b i e v , i − (cid:88) j ∈N i a i j (ˆ v , i − ˆ v , j ) − ˙ f . Defining e v f = (cid:104) e v , . . . e v , N e f , . . . e f , N (cid:105) (cid:62) , we have˙ e v f = Q e v f + (cid:96) , (23)where Q = (cid:34) − H I − H (cid:35) , (cid:96) = (cid:34) e u − ˙ f (cid:35) . The characteristic polynomial of Q is given bydet( sI − Q ) = det (cid:32)(cid:34) sI + H − IH sI (cid:35)(cid:33) = det( s I + Hs + H ) = N (cid:89) i = (cid:104) s + λ i ( H ) s + λ i ( H ) (cid:105) .
9s is seen from above, the poles of Q is stable since H is positive definite. Then, there must exist a positive definitematrix P such that P Q + Q (cid:62) P = − I . For (23), take a Lyapunov function V ( e v f ) = e (cid:62) v f P e v f . Consider its derivative:˙ V = − e (cid:62) v f ( P Q + Q (cid:62) P ) e v f + e (cid:62) v f P (cid:96) ≤ − (cid:107) e v f (cid:107) + (cid:107) e v f (cid:107)(cid:107) P (cid:107)(cid:107) (cid:96) (cid:107)≤ − (cid:107) e v f (cid:107) + (cid:113) e u (0) + q (cid:107) P (cid:107)(cid:107) e v f (cid:107) , where the fact suggested by Lemma 4 that e u exponentially decreases to zero is used. The above inequality can bewritten equivalently as ˙ V ≤ − α V + β (cid:112) V , with α = / ¯ λ ( P ) and β = (cid:113) e u (0) + q ) (cid:107) P (cid:107) /λ ( P ). Hence, (cid:112) V ( t ) ≤ (cid:112) V (0) e − α t + β α (1 − e − α t ) ≤ (cid:112) V (0) + β α . (24)It then follows from (24) that (cid:107) e v f ( t ) (cid:107) ≤ (cid:115) ¯ λ ( P ) λ ( P ) (cid:107) e v f (0) (cid:107) + β α = (cid:115) ¯ λ ( P ) λ ( P ) (cid:107) e v f (0) (cid:107) + ¯ λ ( P ) (cid:113) e u (0) + q ) (cid:107) P (cid:107) λ ( P ) , lim t →∞ (cid:107) e v f ( t ) (cid:107) ≤ √ λ ( P ) q (cid:107) P (cid:107) λ ( P ) . Part b):
Consider the observers for v i and f i . Define their respective estimation errors as e v , i = ˆ v i − v i and e f , i = ˆ f i − f i . Their dynamics can be described as ˙ e v , i = − le v , i + e f , i , ˙ e f , i = − le v , i − ˙ f i . Defining e v f = (cid:104) e v , . . . e v , N e f , . . . e f , N (cid:105) (cid:62) , we have˙ e v f = Q e v f + (cid:96) , where Q = (cid:34) − lI I − lI (cid:35) , (cid:96) = (cid:104) − ˙ f − ˙ f · · · − ˙ f N (cid:105) (cid:62) . Following similar lines to Part a), we can obtain that e v f is upper bounded: (cid:107) e v f ( t ) (cid:107) ≤ (cid:115) ¯ λ ( P ) λ ( P ) (cid:107) e v f (0) (cid:107) + β α = (cid:115) ¯ λ ( P ) λ ( P ) (cid:107) e v f (0) (cid:107) + √ λ ( P ) q (cid:107) P (cid:107) λ ( P ) , lim t →∞ (cid:107) e v f ( t ) (cid:107) ≤ √ λ ( P ) q (cid:107) P (cid:107) λ ( P ) , where P is a positive definite matrix satisfying P Q + Q (cid:62) P = − I .10 art c): Based on Parts a) and b), now let us move on to analyze the tracking performance under the controllerin (17). Note that the position and velocity tracking errors are governed by˙ x i − ˙ x = v i − v , ˙ v i − ˙ v = − k (cid:88) j ∈N i a i j ( x i − x j ) + b i ( x i − x ) − ( v i − v ) − (ˆ v i − v i ) + (ˆ v , i − v ) + ˆ f , i − f + f i − ˆ f i + ˆ u , i − u , for i = , , . . . , N . Define e = (cid:104) x − x · · · x N − x v − v · · · v N − v (cid:105) (cid:62) . Then, ˙ e = Q e + (cid:96) , (25)where Q = (cid:34) I − k H − I (cid:35) , (cid:96) = (cid:34) − e v + e v + e f − e f + e u (cid:35) . The characteristic polynomial of Q isdet( sI − Q ) = det (cid:32)(cid:34) sI − IkH sI + I (cid:35)(cid:33) = det( s I + sI + kH ) = n (cid:89) i = (cid:104) s + s + k λ i ( H ) (cid:105) . It is seen from above that Q is stable because H is positive definite and k >
0. If Q is stable, there exists a positivedefinite matrix P such that P Q + Q (cid:62) P = − I . Define V ( e ) = e (cid:62) P e for (25). Then,˙ V = − e (cid:62) ( P Q + Q (cid:62) P ) e + e (cid:62) P (cid:96) ≤ − (cid:107) e (cid:107) + (cid:107) e (cid:107) · (cid:107) P (cid:107) · (cid:107) (cid:96) (cid:107)≤ − (cid:107) e (cid:107) + ( √ (cid:107) e v f (cid:107) + √ (cid:107) e v f (cid:107) + (cid:107) e u (cid:107) ) · (cid:107) P (cid:107) · (cid:107) e (cid:107) . It can be rewritten as ˙ V ≤ − α V + β (cid:112) V , where α = / ¯ λ ( P ) and β = (cid:16) (cid:107) e v f (cid:107) + (cid:107) e v f (cid:107) + √ (cid:107) e u (cid:107) (cid:17) (cid:107) P (cid:107) /λ ( P ). Therefore, we have (cid:112) V ≤ (cid:112) V (0) e − α t + β α (1 − e − α t ) ≤ (cid:112) V (0) + β α . It then follows that e satisfies (cid:107) e ( t ) (cid:107) ≤ (cid:115) ¯ λ ( P ) λ ( P ) (cid:107) e (0) (cid:107) + β α ≤ (cid:115) ¯ λ ( P ) λ ( P ) (cid:107) e (0) (cid:107) + ¯ λ ( P ) (cid:107) P (cid:107) λ ( P ) (cid:115) ¯ λ ( P ) λ ( P ) (cid:107) e v f (0) (cid:107) + √ λ ( P ) q (cid:107) P (cid:107) λ ( P ) + (cid:115) ¯ λ ( P ) λ ( P ) (cid:107) e v f (0) (cid:107) + λ ( P ) (cid:113) e u (0) + q ) (cid:107) P (cid:107) λ ( P ) + √ (cid:107) e u (0) (cid:107) , lim t →∞ (cid:107) e ( t ) (cid:107) ≤ ¯ λ ( P ) (cid:107) P (cid:107) λ ( P ) √ λ ( P ) q (cid:107) P (cid:107) λ ( P ) + √ λ ( P ) q (cid:107) P (cid:107) λ ( P ) . (26)(27)11y (26)-(27), there exist δ and (cid:15) such that (22a)-(22b) hold. This completes the proof. • Theorem 2 reveals that, for a second-order MAS, the proposed observer-based controller can enable a follower totrack the leader with bounded position and velocity errors when the disturbances are bounded in rates of change. Suchan e ff ectiveness is mainly attributed to the proposed observers, through which a follower can estimate the disturbanceand velocity variables for tracking control. Further, similar to Remark 2, the tracking error e i ( t ) → t → ∞ if˙ f i ( t ) →
5. NUMERICAL STUDY
This section presents an illustrative simulation example to validate the proposed results. We consider a second-order MAS consisting of one leader and five followers, which share a communication topology shown in Figure 1(a).Vertex 0 is the leader, and vertices numbered from 1 to 5 are followers. The leader will only send information updatesto follower 1, which is its only neighbor. The followers maintain bidirectional communication with their neighbors.For the topology graph, the edge-based weights are set to be unit for simplicity. The corresponding Laplacian matrix L is then given as follows: L = − − − − − − − − − − . Based on the communication topology, the leader adjacency matrix is B = diag(1 , , , , l = k = .
5, respectively. The initial conditions include x (0) = (cid:104) − − (cid:105) (cid:62) , v (0) = (cid:104) − − (cid:105) (cid:62) . Further, u ( t ) = − . π t ) , f ( t ) = − cos(0 . π t ) , f ( t ) = (cid:104) . . . . . (cid:105) (cid:62) t . Note that the disturbances enforced on the followers are bounded in rates of change but linearly diverge through time.This extreme setting is used to illustrate the e ff ectiveness of disturbance rejection here. Apply the observer-basedcontrol approach in Section 4 to the MAS. The observer-based control approach in Section 4 is applied, with thesimulation results outlined in Figure 1. Figure 1(b) and 1(c) demonstrate that the followers maintain bounded positionand velocity tracking errors, which is in agreement with the results in Theorem 2. It is shown in Figure 1(d) that theobserver for u can gradually achieve accurate estimation through time. This is because the leader can send u toits neighbor follower i , and with the implicit information propagation, the other followers can eventually estimate u precisely. By comparison, the estimation of v , f , v i and f i is less accurate, as is seen in Figures 1(e)-1(h), becausethere is no measurement of them available. However, the di ff erences or estimation errors are still bounded, matchingthe expectation as suggested by the theoretical analysis.
6. Conclusion
MASs have attracted significant research interest in the past decade due to their increasing applications. In thispaper, we have studied leader-follower tracking for the first- and second-order MASs with unknown disturbances.Departing from the literature, we have considered a much less restrictive setting about disturbances. Specifically,disturbances can be applied to all the leader and followers and assumed to be bounded just in rates of change. Thisconsiderably relaxes the usual setting that only followers are a ff ected by magnitude-bounded disturbances. To solvethis problem, we have developed observer-based tracking control approaches, which particularly included the designof novel distributed disturbance observers for followers to estimate the leader’s unknown disturbance. We have provedthat the proposed approaches can enable bounded-error tracking in the considered disturbance setting. Simulationresults further demonstrated the e ff ectiveness of the proposed approaches.The proposed framework and methodology provide an ample scope for future work. They can be extended to moresophisticated MASs, such as those with nonlinear dynamics, multiple leaders, or a directed communication topology.12n addition, the communication protocols or malicious cyber attacks are of growing importance for the design offuture MASs. It will be an interesting question to leverage the proposed observer-based framework to address thesechallenges. References [1] G. Wen, Z. Peng, A. Rahmani, Y. Yu, Distributed leader-following consensus for second-order multi-agent systems with nonlinear inherentdynamics, International Journal of Systems Science 45 (9) (2014) 1892–1901.[2] G. Shi, Y. Hong, Global target aggregation and state agreement of nonlinear multi-agent systems with switching topologies, Automatica45 (5) (2009) 1165–1175.[3] Y. Hong, G. Chen, L. Bushnell, Distributed observers design for leader-following control of multi-agent networks, Automatica 44 (3) (2008)846–850.[4] Z. Li, Z. Duan, G. Chen, L. Huang, Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint, IEEETransactions on Circuits and Systems I: Regular Papers 57 (1) (2010) 213–224.[5] W. Zhu, D. Cheng, Leader-following consensus of second-order agents with multiple time-varying delays, Automatica 46 (12) (2010) 1994–1999.[6] F. A. Yaghmaie, F. L. Lewis, R. Su, Leader-follower output consensus of linear heterogeneous multi-agent systems via output feedback, in:Decision and Control (CDC), 2015 IEEE 54th Annual Conference on, IEEE, 2015, pp. 4127–4132.[7] C. Ma, T. Li, J. Zhang, Consensus control for leader-following multi-agent systems with measurement noises, Journal of Systems Scienceand Complexity 23 (1) (2010) 35–49.[8] M. Nourian, P. E. Caines, R. P. Malham´e, M. Huang, Mean field lqg control in leader-follower stochastic multi-agent systems: Likelihoodratio based adaptation, IEEE Transactions on Automatic Control 57 (11) (2012) 2801–2816.[9] M. Ji, A. Muhammad, M. Egerstedt, Leader-based multi-agent coordination: Controllability and optimal control, in: American ControlConference, 2006, IEEE, 2006, pp. 1358–1363.[10] A. K. Bondhus, K. Y. Pettersen, J. T. Gravdahl, Leader / follower synchronization of satellite attitude without angular velocity measurements,in: Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05. 44th IEEE Conference on, IEEE, 2005, pp. 7270–7277.[11] M. Defoort, A. Polyakov, G. Demesure, M. Djemai, K. Veluvolu, Leader-follower fixed-time consensus for multi-agent systems with unknownnon-linear inherent dynamics, IET Control Theory & Applications 9 (14) (2015) 2165–2170.[12] J. Wu, Y. Shi, Consensus in multi-agent systems with random delays governed by a markov chain, Systems & Control Letters 60 (10) (2011)863–870.[13] D. V. Dimarogonas, P. Tsiotras, K. J. Kyriakopoulos, Leader–follower cooperative attitude control of multiple rigid bodies, Systems & ControlLetters 58 (6) (2009) 429–435.[14] S. Mou, M. Cao, A. S. Morse, Target-point formation control, Automatica 61 (2015) 113 – 118.[15] C. Yan, H. Fang, H. Chao, Energy-aware leader-follower tracking control for electric-powered multi-agent systems, Control EngineeringPractice 79 (2018) 209–218.[16] C. Yan, H. Fang, Observer-based distributed leader-follower tracking control: A new perspective and results, International Journal of Con-trolIn press.[17] C. Yan, H. Fang, Leader-follower tracking control for multi-agent systems based on input observer design, in: Proceedings of AmericanControl Conference, 2018, pp. 478–483.[18] C. Yan, H. Fang, H. Chao, Battery-aware time / range-extended leader-follower tracking for a multi-agent system, in: Proceedings of AnnualAmerican Control Conference, 2018, pp. 3887–3893.[19] S. Li, H. Du, X. Lin, Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics, Automatica 47 (8) (2011)1706–1712.[20] Y. Zhang, Y. Yang, Y. Zhao, G. Wen, Distributed finite-time tracking control for nonlinear multi-agent systems subject to external disturbances,International Journal of Control 86 (1) (2013) 29–40.[21] W. Cao, J. Zhang, W. Ren, Leader–follower consensus of linear multi-agent systems with unknown external disturbances, Systems & ControlLetters 82 (2015) 64–70.[22] J. Sun, Z. Geng, Y. Lv, Z. Li, Z. Ding, Distributed adaptive consensus disturbance rejection for multi-agent systems on directed graphs, IEEETransactions on Control of Network Systems In press.[23] Y. Zhao, Z. Duan, G. Wen, Y. Zhang, Distributed finite-time tracking control for multi-agent systems: An observer-based approach, Systems& Control Letters 62 (1) (2013) 22–28.[24] Y. Zhao, Z. Duan, G. Wen, G. Chen, Distributed finite-time tracking for a multi-agent system under a leader with bounded unknown acceler-ation, Systems & Control Letters 81 (2015) 8–13.[25] H. Du, S. Li, P. Shi, Robust consensus algorithm for second-order multi-agent systems with external disturbances, International Journal ofControl 85 (12) (2012) 1913–1928.[26] W. Ren, Y. Cao, Distributed Coordination of Multi-Agent Networks: Emergent Problems, Models, and Issues, Springer Science & BusinessMedia, 2010.[27] J. Hu, Y. Hong, Leader-following coordination of multi-agent systems with coupling time delays, Physica A: Statistical Mechanics and itsApplications 374 (2) (2007) 853–863.[28] F. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons New York, NY, USA, 1983.[29] J. Yang, S. Li, X. Yu, Sliding-mode control for systems with mismatched uncertainties via a disturbance observer, IEEE Transactions onIndustrial Electronics 60 (1) (2013) 160–169.[30] H. K. Khalil, Nonlinear Systems, Prentice-Hall, New Jersey, 1996. a) Time (s) -80-60-40-200204060 x i V S x Node 0(Leader)Node 1(Follower)Node 2(Follower)Node 3(Follower)Node 4(Follower)Node 5(Follower) (b)
Time (s) -15-10-5051015 v i V S v Node 0(Leader)Node 1(Follower)Node 2(Follower)Node 3(Follower)Node 4(Follower)Node 5(Follower) (c)
Time (s) -4-3-2-10123 u i V S u u ˆ u , ˆ u , ˆ u , ˆ u , ˆ u , (d) Time (s) -15-10-5051015 v , i V S v v ˆ v , ˆ v , ˆ v , ˆ v , ˆ v , (e) Time (s) -50510 ˆ f , i V S f f ˆ f , ˆ f , ˆ f , ˆ f , ˆ f , (f) Time (s) -15-10-5051015 ˆ v i ˆ v ˆ v ˆ v ˆ v ˆ v (g) Time (s) ˆ f i V S f i f ˆ f f ˆ f f ˆ f f ˆ f f ˆ f (h)Figure 1: Second-order MAS tracking control: (a) communication topology; (b) leader’s and followers’ position trajectory profiles; (c) leader’sand followers’ velocity profiles; (d) leader’s acceleration profile and the estimation by each follower; (e) leader’s velocity profile and theestimation by each follower; (f) leader’s disturbance profile and the estimation by each follower; (g) followers’ estimation of their own velocities;(h) followers’ disturbance profiles and the estimation on their own.(h)Figure 1: Second-order MAS tracking control: (a) communication topology; (b) leader’s and followers’ position trajectory profiles; (c) leader’sand followers’ velocity profiles; (d) leader’s acceleration profile and the estimation by each follower; (e) leader’s velocity profile and theestimation by each follower; (f) leader’s disturbance profile and the estimation by each follower; (g) followers’ estimation of their own velocities;(h) followers’ disturbance profiles and the estimation on their own.