Decentralized Consensus Based Control Methodology for Vehicle Formations in Air and Deep Space

Transcription

1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, hb5.5 Decentralized Consensus Based Control Methodology for Vehicle Formations in Air and Deep Space Miloš S. Stanković, Dušan M. Stipanović and Srdjan S. Stanković Abstract In this paper a new methodology is proposed for decentralized overlapping tracking control of autonomous vehicles in air and deep-space. he methodology is based on the application of recently proposed decentralized consensusbased state estimators,, 3, in conjunction with the globally LQ optimal state feedback control law. In the case of unmanned aerial vehicles (UAVs), a specific formation model is proposed and the globally optimal control law derived using an aggregation of the formation model. It is demonstrated experimentally that the whole control scheme is advantageous over similar decentralized control algorithms. In the case of deep-space formations, it is shown that a decentralized consensus based estimator can be constructed using specific nonsingular transformations from one realization of the observable part of the formation model to another. Experimental results demonstrate excellent properties of a control scheme based on this estimator in the case of a small number of available measurements. I. INRODUCION During the last decade there has been an increasing interest for conducting research in analysis and control of formations of autonomous unmanned vehicles. his interest has been highly motivated by numerous applications such as distributed sensing, transportation, space exploration, etc.. Recently, important results in this area have been presented in various publications (e.g., see 4, 5, 6, 7, 8, 9, and references reported therein). One of the main challenges in the analysis is that the problems considering multi-agent systems are still very much open and very difficult to solve, in general. Numerous important results have been obtained recently in the domain of consensus-based multiagent schemes, e.g. 5,,, 3,, ). In this paper we propose two novel approaches to the decentralized control of autonomous unmanned vehicles in air and deep-space based on the state estimation methodology proposed in,, 4, 3, which utilizes the first order consensus scheme. In the first part of the paper we start from a state model of unmanned aerial vehicles (UAVs), which includes distances with respect to the centroid of the observed vehicles and apply the consensus-based estimator for obtaining estimates of his work has been supported by the Boeing Company via the Information rust Institute, University of Illinois at Urbana-Champaign. M. S. Stanković who is the corresponding author is with the School of Electrical Engineering, Royal Institute of echnology, -44 Stockholm, Sweden; milsta@kth.se D. M. Stipanović is with the Department of Industrial and Enterprise Systems Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 68, USA; dusan@illinois.edu S. S. Stanković is with the Faculty of Electrical Engineering, University of Belgrade, Belgrade, Serbia; stankovic@etf.rs the state vector of the whole formation within each vehicle. he algorithm assumes a scalable communication network for the exchange of the state estimate vectors. Control signals in the vehicles are obtained through an LQ optimization process at the formation level. his task is solved by constructing a specific aggregation model of the formation, using the concept of the inclusion of performance indices 5. he whole control scheme, including the consensus based estimator and the global state feedback, possesses potentially important advantages over the existing decentralized control methods, especially in complex tracking tasks, at the expense of additional communications between the neighboring vehicles (compare with 6). he second part of the paper deals with the methodologically important problem of decentralized control of vehicle formations in deep-space, characterized by relative sensing, i.e. by the availability of relative spacecraft positions only, as in, e.g.,, 8. It is shown that, in this case, the observable part of the formation model can be transformed into the spaces of relative distances with respect to all the particular vehicles in the formation. Using this fact, an algorithm for the estimation of the observable part of the formation model is constructed, incorporating local estimators which utilize the locally available measurements and a consensus scheme containing transformations from one state space to another. he control signals within each spacecraft are generated using the LQ methodology and the corresponding transformations between the realizations of the observable part of the formation model. he unobservable part of the formation is controlled according to 8. Experimental results show that this truly decentralized and scalable scheme provides very good performance even in the case of an extremely reduced number of available measurements. II. DECENRALIZED CONSENSUS BASED CONROL OF UAV FORMAIONS In 6, a decentralized estimation and control strategy has been proposed for formations of UAVs, starting from a specific formation model and the definition of subsystems attached to individual vehicles. he design of the proposed algorithms has been based on the expansion / contraction paradigm and the inclusion principle 6, 5, 9,, 7. In this section, we shall show how the consensus based estimation algorithm, proposed in,, can be successfully applied to the control of formations of UAVs. he main idea is to design a consensus based estimator based on the subsystems identified in 6, and connect it to the globally LQ optimal state feedback control law, expecting that the //$6. AACC 366

2 resulting control scheme is superior to those obtainable by strictly decentralized strategies, especially in complex tracking tasks (general aspects of such control schemes are discussed in 4). A. Formation Model Consider a set of N vehicles moving in a plane, where the i-th vehicle is represented by the following linear double integrator model (written for one coordinate, for the sake of simpler notation) ż i = A v z i + B v u i = z i + u i, () i =,...,N, where z i = p i v i and u i are the state and the control input, with p i and v i being absolute positions and velocities (see 7 for possible physical interpretations of ()). he vehicle models () are coupled through the control inputs. We shall assume that the i-th vehicle is provided with the information about distances with respect to the vehicles whose indices belong to a set of indices of the sensed vehicles S i = {s i,...,s i m i }. Accordingly, we define x i = j S i α i jp j p i, x i = v i, () where αj i and j S i αj i = ; x i represents the distance between the i-th vehicle and a centroid of the set of vehicles selected by S i obtained by using a priori selected weights αj i. In the case of formation leaders, when S i =, we have x i = p i. herefore, ẋ i = j S i α i jv j v i = j S i α i jx j x i, ẋ i = u i, (3) i =,...,N, so that x i = v i = ṗ i. he above described set of N vehicles with their sensing indices and the corresponding weights can be considered as a directed weighted graph G in which each vertex represents a vehicle, and an arc with the weight αj i leads from vertex j to vertex i if j S i. Consequently, the weighted adjacency matrix G = G ij is an N N square matrix defined by G ij = αj i for j S i, and G ij = otherwise. We shall define the weighted Laplacian of the graph as L = L ij, L ij = G ij, i j, L ii = j α ij (e.g., see 5). Defining the vehicle states and control inputs as x i = and ui, i =,...,N, respectively, we obtain x i x i from (3) the following formation state model S : ẋ = Ax + Bu = (G I) A v x + I B v u, (4) where x and u are the formation state and control vectors defined as concatenations of the vehicle state and control vectors, while denotes the Kronecker s product. We shall assume that each vehicle has the information about the reference state trajectories r i = r i r i, so that the control task to be considered is the task of tracking the desired, possibly time varying references. B. Global LQ Optimal State Feedback We shall attach the following quadratic criterion to (4) J = (x Qx + u Ru)dt, (5) where Q and R > are appropriately defined matrices. he design of the state feedback gain minimizing J is faced with the problem that the model (4) is in general not completely controllable. Namely, one can directly observe that the part of the state vector x of (4) which corresponds to the relative positions x = x x N, satisfies the relation x = (I G)p, where p is the vector of absolute vehicle positions with respect to a reference frame. If the graph G has a spanning tree, the Laplacian L has one eigenvalue at the origin, and the rest in the open left-half plane, e.g. 3. his means that in the case when I G = L, we have ρ x =, where ρ is the left eigenvector of L corresponding to the zero eigenvalue 6. he controllability matrix does not have full rank since rank B AB = (N ) (having in mind that A = ). However, it is possible to observe that the system is in this case controllable for the admissible initial conditions for (4), which have to satisfy ρ x = for a real formation. Notice that in the case of no formation leader, i.e. when I G L, the matrix I G is nonsingular, provided G has a spanning tree. A way of solving the problem can be seen after applying to x a nonsingular transformation = ρ in which W W is a full rank matrix, such that ρ is linearly independent of the rows of W. It can be seen that S is controllable for all the admissible initial conditions, provided the formation model is controllable with respect to x a = Wx. Model for x a is in the form S a : x a = A a x a + B a u, (6) and represents an aggregation of S, having in mind that W is a full rank matrix 5,,. he system matrices satisfy then the aggregation conditions 5 WA = A a W, B a = WB. (7) In order to take care of optimality of S a, we shall attach to (6) the following criterion J a = (x a Q a x a + u R a u)dt. (8) he criterion J includes the criterion J a, i.e. J = J a, if W Q a W = Q, R = R a (9) (see 5, for a general discussion about the inclusion of performance indices). If one starts from J, an approximate solution to the posed optimization problem can be found by formulating J a using the approximate relation Q a = W + QW + (where W + denotes the pseudoinverse of W ) and solving the minimization problem of J a, taking S a as a constraint. If K a is obtained as the corresponding optimal feedback gain matrix, an implementable feedback 366

3 gain matrix K for S can be found simply by applying the relation K = K a W, since in this case the closed-loop system (S a,k a ) is an aggregation of the closed-loop system (S,K), and J = J a. Notice that W can be chosen in many different ways; also, for a given W, different choices of A a are possible. Extensions of the above reasoning to the tracking problem are straightforward using the results from, e.g., 3. C. Decentralized State Estimation We shall consider the problem of formation state estimation starting from a formation model involving all the intervehicle distances measured by the agents: S : x = Ā x + Bu, () where x = x c... x c N, x c i = (p s i p i ) (p s i mi p i ) vi, Ā = blockcol {Āc...Āc N }, Ā c Āc i i =, in which N k= (m k+) Ā c i is composed of N blocks in a block-row having dimensions m i (m k + ), k =,...,N. he only nonzero block is obtained for: ) k = i, when it has the form.., and for ) all k S i when it has m i (m i+) the form., in which is placed at the. m i (m k +) row index j satisfying s i j = k; similarly, B = blockcol { B c... B N c }, mi Bc i =. It is easy to verify that x = x, where = blockdiag{... N } and α i s α i i = i s i m i, so that S represents an aggregation of S; the system matrices are, consequently, related by Ā = A and B = B 5, 9. Following the basic general idea exposed in 6, we shall extract the following overlapping subsystems from S: S (i) : x (i) = Ā(i) x (i) + B (i) ū (i), () i =,...,N, where x (i) = v s v i s (p i m s i i p i ) (p s i mi p i ) vi, ū (i) = us i u mi (m i+) s i u m i i, Ā (i) =, in which Ā i has the form Ā i (mi +). I mi and I mi mi B (i) = mi m i mi. mi Models S (i) can be used for constructing local observers of Luenberger type, providing overlapping estimates ˆ x (i) of the subsystem states: E (i) : ˆ x (i) = Ā(i)ˆ x (i) + B (i) ū (i) + L (i) y (i) C (i)ˆ x (i), () where y (i) is the local measurement vector available to the i-th vehicle, C (i) a matrix defining mapping from the local state to the local output, and L (i) is the estimator gain. It should be noticed that the formulated local observers () can be implemented provided the control vectors ū (i) are appropriately defined. If the final goal is to get an estimate ˆ x of the whole state vector x of S, a consensus scheme can be introduced, enabling all the agents to get in each node reliable estimates on the basis of: () the local estimates ˆ x (i), and () communications between the nodes based on a decentralized strategy uniform for all the nodes. In an algorithm providing a solution to this problem has been proposed. If ˆ x i is the estimate of the whole formation state vector x generated by the i-th agent, the following consensus based estimator results now directly from (): E i : ˆ x i = A iˆ x i +B i U i +Σ N j=γ ij (ˆ x j ˆ x i )+L i (y (i) C iˆ x i ), j i (3) i =,...,N, where nonnegative diagonal matrices Γ ij contain the consensus parameters, while matrices A i, B i and L i are obtained from A (i), B (i) and L (i) in such a way that the nonzero elements of A i, B i and L i are equal to the nonzero elements of A (i), B (i) and L (i), but placed at the indices derived from the definition of S (i) and E (i) (see, 4). he consensus parameters are to be chosen on the basis of the network topology implied by the choice of the sensed vehicles and the relative importance of the nodes (see e.g. 3,, ). he algorithm is, in fact, based on a combination of decentralized overlapping estimators and a consensus scheme with matrix gains Γ ij, tending to make the local estimates as close as possible. Vector U i introduced in (3) represents an approximation of u achievable by the i-th agent based on the assumption that all the agents are supposed to know the global state feedback gain K in the optimal mapping u = Kx = K a Wx, where K a is obtained by minimizing J a in (8). herefore, we have that U i = K ˆ x i ; this means that the implemented control signal u i is the i-th component of the N-dimensional vector U i. Stability analysis of the resulting closed loop system represents a formidable task 4, 3. D. Experiments In this subsection we shall illustrate the above proposed method. A formation of four vehicles without a formation leader has been simulated. It has been assumed that the 366

4 reference actual imes.8 Fig.. Distance plots reference actual imes Fig.. Velocity plots second and the third vehicle observe the first, the fourth observes the second and the third, while the first vehicle observes the fourth one. he globally LQ optimal feedback gain has been found on the basis of Subsection II.B. he consensus based estimator, proposed in Subsection II.C has been implemented by each agent, assuming the adopted information flow between the agents. he consensus gains are all set to be the same, equal to. In Fig. and Fig. x-components of the distances and velocities of all four vehicles in the formation are depicted, assuming step distance reference change. It can be observed that, in our case, using the consensus based control structure, better performance is obtained compared to the case in 6, where the controllers are designed using the expansion / contraction paradigm and inclusion principle with local estimators. he system dynamics can be adjusted by the free parameters in the local estimators and in the global performance index. III. DECENRALIZED CONSENSUS BASED CONROL OF DEEP-SPACE FORMAIONS In this section we shall demonstrate how the control methodology using the consensus based estimator and the globally optimal control law, proposed in, 4, can be efficiently applied to deep-space formations, characterized by relative sensing. Here, the main problem is lack of observability of the formation model when the measurements are the relative positions, 7, 8, 4. It will be shown how to define the observable parts of the formation model connected to the individual spacecraft, and how to formulate subsystems according to the locally available measurements. hen, the decentralized consensus based estimator is formulated by incorporating nonsingular transformations between particular realizations. he state feedback control law is formulated according to 8. he whole decentralized control scheme provides an efficient tool for large formations with a small number of local measurements, without requirements for high network connectedness. A. Problem Formulation We shall start from the same basic model of N vehicles as above using (), i.e., we have Ż = A z Z + B z u, (4) where Z = z... zn = p v... p N v N, A z = blockdiag{a v,...,a v } and B z = blockdiag{b v,...,b v }. We shall assume that only relative positions r ij = p j p i, i,j =,...,N, i j are accessible to measurements. Such a situation is typical for formations of spacecraft in deepspace, 7, 8, 4. Denote by r the N(N )-dimensional vector containing all the measurements of relative positions, i.e., r = r... r N, where r i, i =,...,N, contains all the relative positions with respect to the i-th vehicle: r = r... r N,...,r N = r N,... r N,N. he output equation r = CZ, (5) effectively couples the spacecraft, where C =, C = C N = C. C N, C =, or, r i = C i Z, i =,...,N. As stated in 8, the main general obstacle to the control design of (4) is the fact that the state Z is not fully observable from the relative position measurements (both the position and velocity of the formation centroid cannot be determined from r). Using a similarity transformation Z = Z Z Z Z, we can obtain = A A Z A r = C Z Z Z + B B, u (6) (7) where the pair (C,A ) is observable. Notice that for the given model (4) we have the observability matrix O =, and the property A C CA z =. Having in mind the structure of C and CA z, we imme- z diately conclude that dim{z } = (N ). Obvious 3663

5 choices for Z are Z () Z () = = r, ṙ, r,n ṙ,n, r, ṙ, r,n ṙ,n,..., Z (N) = r N, ṙ N, r N,N ṙ N,N. It is also possible to observe that the vectors r i and r j are connected by a nonsingular transformation t ij, in such a way that r i = t ij r j ; t ij can be easily obtained from C i = t ij C j, having in mind the specific structure of C i, i =,...,N. Also, all possible states Z are connected by the corresponding nonsingular transformations. For the above given examples we have that Z (i) = ij Z (j), where ij can be easily obtained from t ij. For example, in the case when N = 3, we have that r = C Z = Z and r = C Z = Z, so that t = and =. B. State Feedback racking Control he observable part of the formation dynamics can be used for a formation controller design based on relative position measurements. Various methodologies can be applied to this problem. One of the possibilities is to use the LQ methodology, as in Section II. It should be noticed, however, that the observable part of the formation dynamics in (6) does not suffer here from uncontrollability, as in the case of the problem discussed in Section II. A reference tracking state feedback controller can be constructed by exploiting redundancy in the relative position reference command, according to 8. Essentially, the methodology is based on finding a matrix N r so that the resulting control signal generated by the state feedback is u = KZ KN r c r, (8) where K is obtained by a convenient state feedback design methodology, and c r is the constant relative position reference satisfying C Z = c r, as required. Furthermore, following 8, it can be shown that the control signal can be calculated in such a way that one can control Z and Z components of the formation state independently. C. Decentralized Consensus Based Estimator Estimation of the observable states Z can be performed in a centralized way (fusion center) as suggested in 8, using an observer of Luenberger type, in which the gain matrix can be chosen in a pre-selected way. One may also use a parallel scheme proposed in, 4, in which all the spacecraft perform estimation of the whole state vector Z on the basis of the same set of measurements. Communications between the nodes are here aimed at achieving amelioration of the dynamics of the closed-loop system. We propose in this subsection the application of the decentralized overlapping methodology based on consensus, 4, adapted to the problem under consideration. he novel scheme is based on local estimators connected to subsystems extracted on the basis of the locally available measurements and a consensus scheme involving specific transformation from one observable realization to another. he scheme is truly distributed, and requires in the whole only N linearly independent measurements available to different spacecraft. Also, the scheme possesses very good scalability: the number of nodes can be easily expanded. Assume that y (i) is the measurement vector available to the i-th spacecraft: it contains a subset of the distances r ij, j =,...,N, j i, determined by the set of indices of the sensed vehicles S i ( S i = ν i N ). We connect to the i-th spacecraft a subsystem S χ i characterized by the state vector χ i containing the pairs (r ij,ṙ ij ) selected by S i, the state matrix in the form A (i) = diag{a v,...,a v } and the input matrix in the form B (i) = col diag, both with ν i characteristic blocks. he control vector u (i) of S χ i contains u i, together with all u j, j S i. We connect to this subsystem a local estimator, analogous to the one presented in (), having the form ˆχ i = A (i) ˆχ i + B (i) u (i) + L (i) (y (i) C (i) ˆχ i ), (9) where C (i) is the appropriate output transformation selecting the measured distances from the whole state vector. However, the control methodology based on consensus, proposed in, 4 and described in Section II, cannot be applied here directly, having in mind that the subsystem state vectors χ i are not all parts of a unique predefined global state vector of the observable part of the system model. hese vectors are, in fact, parts of the states of local realizations of the observable parts of the system model, i.e. parts of Z (),...,Z(N). he problem of compatibility of the state estimates within an exchange between the spacecraft will be solved by adequately transforming the state vectors before sending. Namely, the definition of (9) implies that the i- th spacecraft pre-assumes Z (i) as the state vector of the observable part of the formation model. Consequently, if Ẑ(i) denotes its estimate, each estimate Ẑ(j), j =,...,N, j i communicated to the i-th spacecraft is to be multiplied by ij (defined above, subsection III A), before being put in combination with the estimate Ẑ(i) within the consensus scheme. Also, let Z () be the state vector of the realization of the observable part of the system model utilized in the calculation of the optimal feedback gain K explained above, and let i and i be the corresponding nonsingular transformations between Z () and Z (i), i =,...,N. he proposed consensus based algorithm receives then the form Ẑ (i) = A i Ẑ (i) + B i U i + Σ N j= j i Γ ij (Ẑ(j) Ẑ(i) ) + + L i (y (i) C i Ẑ (i) ), () i =,...,N, which is analogous to (3), but possesses, in fact, the following specific properties: 3664

6 p p reference actual p3 p p3 p imes Fig. 3. Relative positions of the spacecraft ) matrices Γ ij have the form C ij ij, where C ij is a nonnegative diagonal matrix chosen according to general considerations mentioned in Section II (see 3,, ), and ) the control signal U i has the form U i = K i Ẑ (i). Notice that the control signals used in the algorithm depend formally on the spacecraft; physically, each control signal u i is generated according to the physical formation model. Matrices A i, B i and C i are defined in the sam way as in (3). he overall control law contains, in fact, two parts: the one related to the observable part of the system, and the one related to the unobservable part of the system 8. he above described form of the tracking control law enables decoupling between these two parts, according to the scheme described in 8. A theoretical stability analysis of the closed loop system is here faced with even greater problems than in the case of the algorithm described in Section II. D. Experiments he above described algorithm is applied to a formation of three spacecraft, connected by a directed ring. It has been supposed that only two measurements are available (r, by the first spacecraft and r,3 by the second). he results of the application of the described algorithm are presented in Figure 3. hey confirm that the scheme represents an efficient tool for practice. he responses can be adjusted by the free parameters in the local estimators and the global criterion function. IV. CONCLUSION In this paper two decentralized tracking control schemes are proposed for formations of autonomous unmanned vehicles in air and deep space, based on the application of decentralized overlapping consensus based estimators. he first scheme, applied to UAVs, consists of a consensus based estimator and a globally LQ optimal state feedback. he second scheme, applied to deep-space formations characterized by relative positions measurements, has the same structure as far as the observable part of the system model is concerned. However, the estimator is in this case based on appropriate transformations from one local state space to another. Both schemes give very good performance even in the case of sparse measurements and large formations, without scalability problems. Further efforts are being oriented towards a more precise insight into the stability of the proposed schemes. REFERENCES S. S. Stanković, M. S. Stanković, and D. M. 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