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Decoupled–Dynamics Distributed Control for Strings of Nonlinear Autonomous Agents
arXiv:1602.04146v1 [cs.SY] 12 Feb 2016
S¸erban Sab˘au] , Irinel–Constantin Mor˘arescu? and Ali Jadbabaie†
I. I NTRODUCTION The problem of distributed control of strings of dynamical agents can be roughly identified as a particular case of flocking in multi-agent systems. The flocking problem has been intensively studied in the last decade for agents with linear [2], [3], [16], [13] or nonlinear [1], [17] dynamics. Generally speaking, two important features characterize the flocking behavior of autonomous agents: cohesion and collision avoidance. In multi-agent systems they are implemented as connectivity/topology preservation [19], [18], [15], [14] and collision avoidance [2], [3], [1], respectively. While these features are sufficient for reasonable flocking behavior, in the control of strings (also known as platooning in the literature) it is essential to add a supplementary requirement related to avoiding the amplification of oscillations through the formation, phenomenon known as string instability [6], [9], [7], [8]. Even if string instability is circumvented, supplemental problems may be caused by the so called accordion effect (or slinky effect) consisting of weakly attenuated oscillations and long settling times of the relative positions and velocities of the vehicles. For the case of LTI agents, a novel distributed control architecture that can guarantee string stability even for distance–based headways and perfect trajectory tracking (i.e. the complete elimination of the accordion effect) in the presence of bounded disturbances and communications delays has been recently proposed in [4]. ]
S¸erban Sab˘au is with the Electrical and Computer Engineering Dept., Stevens Institute of Technology, Hoboken, New Jersey,
U.S.A. email:
[email protected] ?
I.-C. Mor˘arescu is with Universit´e de Lorraine, CRAN, UMR 7039 and CNRS, CRAN, UMR 7039, 2 Avenue de la Forˆet
de Haye, 54506 Vandœuvre-l`es-Nancy, France. email:
[email protected] †
Ali Jadbabaie is with the Systems Engineering Dept., University of Pennsylvannia. email:
[email protected]. DRAFT
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For many important applications in networks of dynamical agents, the regulated signals in trajectory tracking or synchronization problems represent relative measurements such as interspacing distances, relative velocities (with respect to neighboring agents) or phase differences between (neighboring) coupled oscillators. Pre–specified subsets of these relative measurements are then considered to be the input signals for the distributed/decentralized controllers. This framework is adopted in the overwhelming majority of existing references, following on the classical control pradigm in which the scenario where the regulated signals coincide with the measurements available to the controller is considered ideal. The fundamental limitations of such distributed/decentralized control architectures have been extensively studied over the last decades. A. Scope of Work In this paper, we present preliminary results on a class of novel distributed control policies where the relative measurements (with respect to the neighboring agents) are used by the local sub–controllers in conjunction with the knowledge of the control actions of the subcontrollers at the neighboring agents.1 It turns out that the performance of the resulted distributed control schemes vastly outperforms the distributed architectures based solely on relative measurements. Furthermore, at the cost of allowing limited communications between neighboring sub–controllers, our design renders useless the expansion of the “sensing radius” of agents – which appears not to bring any supplemental performance. The dynamical models of the agents are taken to be non–linear and time invariant, satisfying a global Lipschitz–like condition. Such models represent an effective framework for describing the dynamics of the transmission block of road vehicles, from the break/throttle controls to the vehicle’s position on the roadway. For illustrative simplicity, we look only at the situation in which the homogeneous formation graph is a string, but the proposed scheme is susceptible to be adaptable to multi–tree (no self–loops) graphs, once that the relative errors are defined adequately in order to avoid the well–known formation rigidity problems. However, the string formation alone has important applications as it addresses the longstanding platooning problem which is paramount to the autonomous vehicles industry. 1
In many situations of practical interest the control actions of the neighbors can be transmitted using cheap and reliable
communications systems. Moreover, the output of a digital controller is in general a low data–rate signal. DRAFT
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B. Contributions Compared to the state–of–the–art e.g [1], [2], [3] our solution features several key advantages. Firstly, it guarantees stability, velocity matching (trajectory tracking) and collision avoidance even for directed topologies, as illustrated by the one–step–look–ahead controller reported here. Secondly, it achieves complete scalability with respect to the number of agents in formation [1] (which is a daunting requirement for platooning systems even in the case of LTI agents) and also with respect to the connectivity of the communications graph. The scalability is a structural feature of the proposed schemes, valid for the class of all stabilizing controllers, as it is not achieved (as one might expect) via the tuning of the local sub–controllers [5]. The most interesting attribute of the proposed solution (reported in detail in the sequel) is the complete “decoupling” of the closed–loop equations achieved by the distributed controllers. This allows the study of the closed–loop stability by performing an individual, local analysis of the closed–loops stability at each agent, which in turn guarantees the aggregated stability of the formation. This is also remarkably convenient in the controller tuning procedure, since the involved optimization effort can be broke down into sub–problems, local to each agent. II. G ENERAL F RAMEWORK A. Preliminaries Definition II.1. The σ–norm of a vector x is defined as hq i def 1 1 + kxk22 − 1 kxkσ = σ
(1)
with σ is a positive constant and is a class K∞ function of kxk22 and is differentiable everywhere. Definition II.2. A set Ω is said to be forward invariant with respect to an equation, if any solution x(t) of the equation satisfies x(0) ∈ Ω =⇒ x(t) ∈ Ω, ∀t > 0. Definition II.3. Artificial Potential Function (APF). The function Vk,k−1 (·) is a differentiable, nonnegative, radially unbounded function of kzk − zk−1 kσ , representing the σ -norm of the inter– spacing error between the k–th agent and its predecessor, the (k − 1)–st agent and satisfying the following properties: DRAFT
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(i) Vk,k−1 (kzk − zk−1 kσ ) → ∞ as (kzk − zk−1 kσ ) → 0, (ii) Vk,k−1 (kzk − zk−1 kσ has a unique minimum, which is attained at kzk − zk−1 kσ = δk , with δk being a positive constant. B. The Problem: Trajectory Tracking of the String Formation We consider a homogeneous group of n agents (e.g. autonomous road vehicles) moving along the same (positive) direction of a roadway. The dynamical model for the agents, relating the control signal uk (t) of the k–th vehicle to its position yk (t) on the roadway, is given by y˙ k = vk ,
v˙ k = f (vk ) + uk ;
yk (0) = −
k X
δj ,
(2a)
vk (0) = 0.
(2b)
j=0
where vk (t) is the instantaneous speed of the k–th agent, uk (t) is its command signal and δk is the initial interspacing distance between the k–th agent and its predecessor in the string. Throughout the sequel we will use the notation yk = Gk ? uk
(3)
to denote (especially for the graphic representations) the input–output operator Gk of the dynamical system from (2a), with the initial conditions (2b). Assumption II.4. The index “0” is reserved for the leader vehicle, the first vehicle in the string, for which we assume that there is no controller on board and consequently the command signal u0 (t) will represent a reference signal for the entire formation. We further define def
zk = yk − yk−1 − δk ,
def
zkv = vk − vk−1
for
1 ≤ k ≤ n,
(4)
to be the interspacing and relative velocity error signals respectively (with respect to the predecessor in the string), with δk being the desired constant inter-spacing policy for the k–th vehicle. By differentiating (4) it follows that z˙k (t) = zkv (t), therefore implying that constant interspacing errors (in steady state) are equivalent with zero relative velocity errors and also allowing to write the following time evolution for the relative velocity error of the k–th vehicle z˙kv = f (vk ) − f (vk−1 ) + uk − uk−1 .
(5) DRAFT
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III. A N OVEL D ISTRIBUTED C ONTROL A RCHITECTURE The objective of the control scheme in formation control problems is to achieve the synchronization of the trajectories of all agents in the formation with the trajectory of the leader agent (Assumption II.4). Such trajectory tracking must be achieved while ensuring zero (steady– state) errors of the regulated measures zkv (velocity matching) and while avoiding collisions, i.e. performing the needed longitudinal steering (brake/throttle) maneuvers that guarantee the avoidance of collision with the preceding vehicle. The inherent difficulty in platooning control is rooted in the nested nature of the interdependencies between the regulated signals. Specifically, the regulated errors (e.g. interspacing errors or relative velocity errors) at the k–th agent depend on the regulated errors of its predecessor (the (k − 1)–st agent) and so on, such that by a recursive argument – going through all the predecessors of the k–th agent – they ultimately depend on the trajectory of the leader vehicle, which represents the reference for the entire formation. The novel control architecture we introduce next features certain beneficial “decoupling” properties of the closed–loop dynamics that break through the pitfalls of the aforementioned nested interdependencies. The distributed control policies rely only on information locally available to each vehicle. For the scope of this paper, we consider non–linear controllers built on the socalled Artificial Potential Functions (APF) [2], [3], in particular we will look at control laws of the type u1 = −β1 (v1 − v0 ) − ∇y1 V1,0 (ky1 − y0 kσ ) uk = uk−1 − βk (vk − vk−1 ) − ∇yk Vk,k−1 (kyk − yk−1 kσ )
and for
(6a) k ≥ 2,
(6b)
where for all k ≥ 1, Vk,k−1 (·) is an Artificial Potential Function [1, Definition 7] while βk is a proportional gain to be designed for supplemental performance requirements. With the notation from (4), the control policy (6b) for the k–th vehicle becomes uk = uk−1 − βk zkv − ∇yk Vk,k−1 (kzk kσ ).
(7)
and it can further be written as the sum of the following two components: firstly, the control signal uk−1 of the preceding vehicle, which is received onboard the k–th vehicle via wireless communications (e.g. digital radio). Secondly, the local component, which we denote with def
u`k = −βk zk − ∇yk Vk,k−1 (kzkv kσ )
(8) DRAFT
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Fig. 1.
Distributed Controller Implementation
and which is based on the measurements (4) which are locally available to the k–th vehicle, as they can be acquired via onboard LIDAR sensors. Thus, the k–th control law reads: uk = uk−1 + u`k . By further denoting with Kk the input–output operator in (8) from zk and zkv respectively, to u`k , namely u`k = Kk ? zk , zkv ,
(9)
the resulted control architecture for any two consecutive vehicles (k ≥ 2) can be pictured as in Figure 1. In accordance with Assumption II.4 and (6a), the equivalent scheme for the first three vehicles (following the leader) in the string is given in Figure 2. Note that due to the assumed homogeneity of the formation, the wirelessly received predecessor control signal uk−1 appears unfiltered in (7). For the illustrative simplicity of the current Section, we look only at the delay free scenario. A (GPS–time based) synchronization mechanism, that can cope with the communications induced time–delays will addressed in a future work. IV. A F IRST G LANCE AT THE C LOSED –L OOP DYNAMICS D ECOUPLING The control policy (6) entails a highly beneficial “decoupling” feature of the closed–loop dynamics at each agent, as we simply illustrate next. Firstly, note that by plugging (6b) into (5) we obtain the following closed–loop error equations at the k–th agent: z˙kv = f (vk ) − f (vk−1 ) − βk zkv − ∇yk Vk,k−1 (kzk kσ ).
(10) DRAFT
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Fig. 2.
Distributed Controller Implementation
The following result will be instrumental in the sequel. Consider the following Lyapunov candidate functions: def 1 Lk zk (t), zkv (t) = Vk,k−1 (kyk (t) − yk−1 (t)kσ ) + zkv > (t)zkv (t) , with 1 ≤ k ≤ n. 2
(11)
Lemma IV.1. The differential of the Lyapunov candidate function Lk (·, ·) introduced in (11) along the trajectories of (2) and (6) is given by d Lk (zk (t), zkv (t)) = zkv > (t) f (vk (t)) − f (vk−1 (t)) − βk zkv > (t)zkv (t) , dt
(12)
and does not depend on the particular choice of the APFs Vk,k−1 (·). Proof: Differentiating the APF Vk,k−1 (·) at the k–th agent with respect to time, yields d Vk,k−1 (kyk − yk−1 kσ ) = (y˙k − y˙ k−1 )> ∇yk Vk,k−1 (kyk − yk−1 kσ ) − ∇yk−1 Vk,k−1 (kyk − yk−1 kσ ) dt (13) and by employing the anti–symmetrical property of APFs [1, pp. 197] : ∇yk Vk,k−1 (kyk − yk−1 kσ ) = −∇yk−1 Vk,k−1 (kyk − yk−1 kσ ) we get that d Vk,k−1 (kyk − yk−1 kσ ) = 2 z˙k > ∇yk Vk,k−1 (kyk − yk−1 kσ ) dt
(14)
therefore from (11) it follows that d Lk zk (t), zkv (t) = zkv > ∇yk Vk,k−1 (kyk −yk−1 kσ )+zkv > z˙kv = zkv > ∇yk Vk,k−1 (kyk −yk−1 kσ )+ z˙kv dt (10) v > = zk f (vk ) − f (vk−1 ) − βk zkv = zkv > f (vk ) − f (vk−1 ) − βk zkv > zkv
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A preliminary framework in which the advantages of the closed–lop decoupling become transparent is illustrated by the following result. The local stabilization at the k–th agent is achieved irrespective of the dynamics involved at any other agents in the string and without invoking any stability arguments for the entire formation. Proposition IV.2. Let the real function f (·) in (2a) be concave and differentiable, with its differential f 0 (·) upper-bounded by a given constant γ. Then, the controller (6) guarantees the stability of “ 0” as an equilibrium point of the decoupled system (10) as far as βk > γ. Proof: Let us notice first that f (·) concave and differentiable implies f (vk ) − f (vk−1 ) ≤ f 0 (vk−1 )(vk − vk−1 ) and f 0 (·) upper-bounded by a given constant γ yields f (vk ) − f (vk−1 ) ≤ γzkv . Therefore, along the trajectory of the closed–loop error system (10) the following holds z˙kv ≤ (γ − βk )zkv − ∇yk Vk,k−1 (kzk kσ ). Using Gronwall’s Lemma the stability of “ 0” as an equilibrium point of the decoupled system (10) is ensured by the stability of “ 0” as an equilibrium point for z˙kv = (γ − βk )zkv − ∇yk Vk,k−1 (kzk kσ ).
(15)
The derivative of the Lyapunov function Lk defined in (11) along the trajectory of (15) can be computed following the prof of Lemma (IV.1) as d Lk (zk (t), zkv (t)) = (γ − βk )zkv > (t)zkv (t) . dt The proof ends by remarking that βk > γ guarantees they
d L (z (t), zkv (t)) dt k k
< 0.
V. M AIN R ESULT A. A Glimpse at the State–of–the-art The problem considered in this paper can be rephrased as flocking problem with collision avoidance in multi-agent systems. The literature on this topic is very rich and considers directed or undirected, fixed or time-varying interconnection graphs. Our approach is based on the use of DRAFT
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an APFs for the local control laws, an approach borrowed from [2], [1]. It is worth noting that the results presented here are a consistent extension of the ones in [2], [1], from at least two perspectives. Firstly, we allow communications based on directed graphs, which is a more realistic scenario in the context of control of strings and secondly, we achieve a complete decoupling of the aggregated problem (for the entire formation) in sub-problems (local to each agent) that do not involve restrictive assumptions. By comparison, the main result in [1] imposes that all the minor matrices of the weighted Laplacian matrix associated with the interconnection graph are positive definite and lower bounded by αIn with α defined below in Theorem V.1. This practically requires the maximization of the eigenvalues of the weighted Laplacian matrix which can be interpreted as the maximization of the number of interconnections in the underlined graph (see [10]) together with the maximization of its diagonal elements (see Gerˇsgorin disk theorem [11]). It is worth noting that the first requirement can be interpreted as transmission of the exact state of the leader to many agents in the formation while the second requirement represents high local control gains. As we will show in the following our methodology needs only information from the predecessor without employing leader’s information. While the control gains are strictly related to the reactivity of the system (i.e. faster systems needs higher controller gains) our scheme does not require making the leader’s information available to almost all the vehicles in the string, rendering our approach more suited to practical platooning applications.
B. Main Result We will prove here that the novel control architecture introduced in Section III can overcome all the drawbacks enumerated above. It employs a directed scheme, a minimal information exchange and communications radius for all agents (each agent receives measurements and information only from its predecessor), without the need to insert exact leader information in formation (virtual leaders). The following result is the main result of this Section, as it delineates a “decoupling” property of the closed–loop dynamics, achieved by the (6) type control policy along with velocity matching and collision avoidance. Theorem V.1. Let function f (·) from (2a) satisfy the globally Lipshitz–like condition [1, As sumption 1] (v2 −v1 )> f (v2 )−f (v1 ) ≤ αkv2 −v1 k22 , for any two vectors v1 , v2 . If the controller DRAFT
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(6) is designed such that βk > α then the following hold: (A) Given the Lyapunov function Lk introduced in (11), local to the k-th agent, the sub–level def
sets Ωkc = {(zk , zkv )|Lk ≤ c, withc > 0} of Lk are compact and they represent forward invariant sets for the local closed–loop dynamics (10) of the k–th vehicle. (B) The controller (6) guarantees velocity matching and collision avoidance. Furthermore, given δk (·) depending on δk , such any pre–specified value δk > 0 there exists a family of APFs Vk,k−1
that min Vk,k−1 (z) = δk and the interspacing distance satisfies kzk (t)k ≥ δk , ∀t ≥ 0 while the z>0
steady–state of the interspacing distance satisfies limt→∞ kzk (t)k = δk . def
Proof: (A) We show that the local sub–level sets Ωkc = {(zk , zkv )| Lk ≤ c, with c > 0} of Lk are compact. Note that Lk < c implies that kzkv k < 2c and Vk,k−1 (kyk − yk−1 kσ ) < 2c. Since Vk,k−1 is radially unbounded this implies that kzk kσ is bounded and consequently kzk k2 is bounded. Therefore Ωkc ⊂ R2dim(yk ) is a bounded set. Moreover due to continuity of k · kσ and Lk one obtains that Ωkc is a closed set. Precisely Ωkc is the pre-image of a closed set through a continuous function. In the Banach space R2dim(yk ) it therefore holds that Ωkc is closed and bounded thus Ωkc is compact. Furthermore, point (A) and the Lipschitz–like [1, Assumption 1] on f (·) implies that d Lk (zk (t), zkv (t)) ≤ (α − βk )zkv > (t)zkv (t) dt along the trajectories of (10). Therefore it suffices to choose the controller gain βk > α in order d to guarantee that Lk < 0 along the trajectories of (10) and that Ωkc is a forward invariant set dt for the decoupled closed–loop system (10), local to the k-th vehicle. (B) From the properties of the APF it follows that Vk,k−1 (kyk − yk−1 kσ ) → ∞ when kyk − yk−1 k2 → 0. Consequently, ∀c > 0, ∃ηc > 0 such that Vk,k−1 (kyk − yk−1 k2 ) > c, ∀ kyk − yk−1 kσ < ηc .
(16)
def
Denote with c = 2Lk (zk (0), zkv (0)) and so for βk > α it holds that Ωkc is forward invariant yielding Lk (zk (t), zkv (t)) ≤ 2c , ∀t ≥ 0. This implies that Vk,k−1 (kyk − yk−1 kσ ) < c, ∀t ≥ 0 and using (16) we conclude that kyk − yk−1 k2 < ηc , ∀t ≥ 0. Finally, by employing LaSalle’s invariance principle we conclude that the Lyapunov function Lk converges asymptotically to its minimum (i.e.
d L dt k
= 0) and consequently zkv converges to 0 while Vk,k−1 (·) converges to its
unique minimum min Vk,k−1 (z) = δk , hence the proof. z>0
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Proposition V.2. Given Lk (·, ·) as introduced in (11), the string formation’s steady–state configuration is attained at the minimum of the following formation’s Lyapunov function: Xn def 1 Lk (zk (t), zkv (t)) L(z(t), z v (t)) = k=1 2
(17)
which coincides component–wise with the minima of the Lyapunov functions (11) local to the k-th def
agent. Furthermore, the level sets of L given by Ωc = {(z, z v )|L ≤ c, with c > 0} are compact and they represent forward invariant sets for the closed–loop dynamics of the entire formation, as given in (2) and (6) with 1 ≤ k ≤ n. Furthermore, velocity matching and vehicles’ collision avoidance are achieved, without the need for inserting exact leader information in the formation (virtual leaders) by respecting the interspacing security distance kzk (t)k ≥ δk , ∀t ≥ 0. Proof: It follows from the definition of (17) and Lemma IV.1 that along the trajectories of (2) and (6) one has XN XN d βk zkv > zkv . zkv > f (vk ) − f (vk−1 ) − L= k=1 k=1 dt
(18)
Let us notice that Ωc is a finite cartesian product of the compacts Ωkc , thus Ωc is compact. Furthermore, designing the local controllers as in Theorem V.1 it follows that Ωc is a forward invariant set for the closed–loop dynamics of the entire formation. Consequently, without the need for inserting exact leader information in the formation, one guarantees the velocity matching and moreover from point (B) in Theorem V.1 the vehicles respect the interspacing distance kzk (t)k ≥ δk , ∀t ≥ 0. R EFERENCES [1] J. Zhou and X. Wu and W. Yu and M. Small and J. Lu “Flocking of multi–agent dynamical systems based on pseudo–leader mechanism”, Systems & Control Let. Vol.61, 2012. (pp. 195–202). [2] H.G.Tanner and A.Jadbabaie and G.J. Pappas, “Stable Flocking of mobile agents, part I: fixed topology” Decision and Control (CDC), IEEE 42nd Annual Conference on, 2003, (pp. 2010–2015) [3] H.G.Tanner and A.Jadbabaie and G.J. Pappas, “Stable Flocking of mobile agents, part II: dynamic topology” Decision and Control (CDC), IEEE 42nd Annual Conference on, 2003, (pp. 2015–2020) [4] S.Sabau and C. Oara and S. Warnick and A. Jadbabaie “Optimal distributed control for platooning via sparse coprime factorizations”, to appear in IEEE Trans. Aut. Control, 2016. [5] P.G. Mehta and P. Barooah and J.P. Hespanha “Mistuning based control design to improve closed–loop stability margins of vehicular platoons”, IEEE Trans. on Aut. Control, Vol. 54, No.9, (pp. 2100–2113), 2009. [6] P. Seiler, A. Pant and J. K. Hedrick “ Disturbance Propagation in Vehicle Strings”, IEEE Trans. Aut. Control, Vol.49, No.10, 2004. (pp. 1835–1841) DRAFT
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