Output Synchronization of Systems in Chained Form - Semantic Scholar
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThA06.4
Output Synchronization of Systems in Chained Form Kim D. Listmann and Craig A. Woolsey
Abstract— In this article we extend previous results for backstepping and passivity-based design of cooperative control laws to a class of chained form systems that includes certain driftfree nonholonomic systems. We exploit the cascaded structure and feedback equivalence to passive systems to derive suitable control laws using a modified backstepping methodology. A virtual output is obtained and shared within the group in such a way that full state synchronization for the group is achieved. Given a static, strongly connected and balanced communication network we prove stability of the proposed controller. Further, an extension to higher order chained forms is shown, providing a methodology to coordinate general nonholonomic systems. The effectiveness of the method is demonstrated by synchronizing a collective of unicycles.
I. I NTRODUCTION Motivated by many applications in biology, physics and engineering, problems involving cooperation among members of a group of autonomous agents have enjoyed considerable attention recently. These problems generally involve a high-level goal that the group must agree on collectively. Coordination problems generally fall in two categories. In the first category of problems, one addresses the complexity of the information exchange among several agents, modeling the dynamics of each agent as simply as possible, and seeks conditions on the exchange of information which guarantee agreement. Such problems are referred to as consensus problems [20]. In the second category of problems, one incorporates more complicated agent dynamics, though with simplifying assumptions about the communication topology. Nair and Leonard, for example, consider unstable, underactuated mechanical systems that are amenable to the method of controlled Lagrangians [16]. In this second category, the coordination task may be referred to as synchronization. In this article, we will consider on the latter category of problems. The notion of passivity underlies a number of proposed methods for cooperative control design [1], [3], [18] and we build upon this theme of passivity-based design, extending previous results in this area [14]. Due to the structure of the systems that we consider here, however, we must explicitly address boundedness and convergence of the control laws that we design. In our previous work [14], these properties were not considered due to an appropriate assumption on the systems examined. This work was supported by the German Research Foundation (DFG) within the GRK 1362 “Cooperative, Adaptive and Responsive Monitoring of Mixed Mode Environments” (www.gkmm.tu-darmstadt.de) K.D. Listmann is with the Control Theory and Robotics Lab, Technische Universit¨at Darmstadt, Landgraf-Georg-Str. 4, 64283 Darmstadt, GERMANY. [email protected] C.A. Woolsey is with the Department of Aerospace and Oceanic Engineering, Virginia Polytechnic University, Blacksburg, VA 24061, USA [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
A variety of driftless nonholonomic systems, including the unicycle [10], are contained in the class of chained form systems considered here. Higher order extensions of the chained form include still more underactuated systems [15]. One way to achieve stabilization for these systems, is to exploit their cascaded structure and employ a recursive design technique [7], [17], [22], namely backstepping [11]. We adopt this methodology, as well, and propose a backstepping-based design that exploits the feedback equivalence of chained form systems to passive systems. By recognizing this feedback equivalence, it is possible to use prior results [1], [3] to derive suitable coordination laws that will achieve synchronization. Additionally, we present an extension of our methodology to achieve synchronization for general nonholonomic and underactuated systems given in higher order chained form. This results in a general solution of the synchronization problem for the stated class of chained form systems. The paper is organized as follows. In the next section, we introduce the necessary definitions and notation. We provide the design methodology and state our main result in Section III. The extension to higher order chained form systems is presented in Section IV. An illustrative example is given in Section V by demonstrating the methodology on the synchronization of a collective of unicycles. Finally, conclusions are presented in Section VI. II. P RELIMINARIES A. Passivity Passivity is a system property that can be understood in terms of energy dissipation in an input-output system. Informally, an input-output system is considered passive if it does not “generate energy,” where the term “energy” generalizes a familiar concept for mechanical systems. We provide a brief review below. For a thorough introduction, see [23]. Consider the system x˙ = f(x) + G(x)u, y = h(x),
(1)
where x(t) ∈ Rn , u(t) ∈ Rr and y(t) ∈ Rr . The drift vector fields f(·), the matrix G(·) of input vector fields, and the output vector field h(·) are assumed to be sufficiently smooth. We assume that f(0) = 0, h(0) = 0. The input history u(t) is taken to be piecewise continuous and locally square integrable. Definition 1: The nonlinear system (1) is said to be passive if there exists a scalar C1 storage function V (x) ≥ 0, V (0) = 0 and a function S(x) ≥ 0 such that for all t ≥ 0
3341
V˙ (x) = u⊤ (t)y(t) − S(x(t)).
ThA06.4 The system (1) is strictly passive if S(x) > 0 and lossless if S(x) = 0. A positive definite storage function V (x) would serve as a Lyapunov function to guarantee stability of the equilibrium at the origin for the system (1). However, the definition of passivity allows a positive semidefinite storage function V (x) so that, in general, one may have to check additional conditions to guarantee stability. In order to determine the necessary conditions for the system (1) to be passive, we present a well known result [2]. Theorem 1: Suppose x = 0 is a regular point for (1). Then, (1) is feedback equivalent to a passive system with a C2 storage function V , which is positive definite, if and only if (1) has a vector relative degree {1, . . . , 1} at x = 0 and is weakly minimum phase. Having verified that the systems of interest are feedback equivalent to passive systems, we seek synchronizing control laws [1], [3]. B. System Description In this paper, we consider systems given in chained form [10], that is ξ˙0 = u1 , ξ˙1 = u1 ξ2 , .. .
(2)
ξ˙k−1 = u1 ξk , ξ˙k = u2 , where z, ξi , and ui are scalar variables. Typically, driftless nonholonomic systems can be (locally) transformed to a chained form by input and state transformation. That is why this system description is widely used and should be exploited for control design [10]. One possibility is to use a recursive backstepping design [7], [11], [22] to asymptotically stabilize these systems. The backstepping procedure defines a virtual output such that cascaded systems, e.g. system (2), are feedback equivalent to passive systems [21]. In our approach we will make use of this and slightly alter the standard backstepping design to address group coordination. For the remainder of the paper, a system with dynamics (2) is denoted as an agent. C. Graph Theory and Group Communication Key to coordination algorithms is information exchange within the group, and hence, the underlying communication network. This network can be modeled using graphs. We refer to [6] for a thorough treatment of graphs and their properties, and give condensed definitions following [3]. Definition 2: A graph G is a finite set of elements V (G ) = {v1 , ..., vN }, the vertices of a graph, and a set E (G ) ⊂ V ×V called the edges of a graph. If for all (vi , v j ) ∈ E , (v j , vi ) ∈ E as well, the graph is said to be undirected, otherwise it is called directed or Digraph. The in-degree of a vertex vi ∈ V defines the number of edges incoming to this vertex, whereas the out-degree defines the number of edges
outgoing from this vertex. A graph is balanced, if for each vi ∈ V the in-degree equals the out-degree. Clearly every undirected graph is balanced. Definition 3: A path of length k in a Digraph is a sequence v0 , ..., vk of k + 1 distinct vertices such that for every i ∈ {0, ..., k − 1}, (vi , vi+1 ) ∈ E and a weak path is such that (vi , vi+1 ) ∈ E or (vi+1 , vi ) ∈ E . A Digraph is fully connected if, for all vi , v j ∈ V , (vi , v j ) ∈ E and (v j , vi ) ∈ E . It is strongly connected if any two vertices can be connected by a path and it is said to be weakly connected if any two vertices can be connected by a weak path. An undirected graph can only be connected or disconnected, since there is no distinction between paths and weak paths. D. Consensus Algorithms Consider the continuous dynamics N i
x˙ = − ∑ ai j j=1
x − jx ,
i
i = 1, . . . , N
(3)
known as consensus protocol for a group with N members [20]. We use ix to denote a state x of the i-th member, whenever distinction between the group members is needed. Rewriting this dynamics for the whole group results in x˙ = −Lx.
(4)
Here, x = [ 1x, . . . , Nx]⊤ and L denotes the Laplacian matrix associated to the graph, modeling the communication network of the group. It is defined as L = li j ∈ RN×N , where li j = −1 for i 6= j, if (vi , v j ) ∈ E (G ), zero otherwise and on the diagonal we have lii = di , where di denotes the in-degree of vertex i. We say that consensus is achieved if | ix − jx| → 0 for t → ∞. Thus, the properties of the Laplacian determine the behavior of (4). In this context, the rate of convergence of such consensus protocols was identified as the second smallest eigenvalue λ2 of L [9]. This fact will later be used to assure convergence of our control laws. Concerning the network topology of the group of agents, we need the following assumption. Assumption 1: The agents form a static, balanced and strongly connected communication network at all times. III. C ONTROLLER D ESIGN In this section we present a novel methodology designing cooperative control laws for the agents considered. Based on the backstepping procedure we show that system (2) is feedback equivalent to a passive system. Hence, it is possible to develop a coordination controller exploiting this property [1], [3]. Finally, we prove stabilization of the aggregate motion and show that synchronization is achieved. Definition 4: A group of N agents is said to be synchronized, if lim iξ p − jξ p = 0, with p = 0, . . . , k. t→∞
for every i, j ∈ {1, 2, . . . , N}, i 6= j. Note that our method is agent based, such that each of the next steps has to be done for all agents, accordingly.
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ThA06.4 Firstly, consider the first subsystem of system (2), that is i˙ ξ0
= iu1 ,
(5)
where the first input of the system already appears. Trivially, this system is passive with output ξ0 , input u1 and storage function W1 = 21 ξ02 . Following a standard consensus protocol [20], we set i u1 = −γη , γ > 0, (6) where
η=
ξ0 − jξ0 .
j∈Ni
This time, our approach differs from the standard backstepping procedure in that we take
α2 = (2γ k1 − 1)ξ1 − γ k1u1 y1 + k1
with Ni denoting the set of neighbors from whom agent i receives information. Note that the rate of convergence of ξ0 is specified by γ and, due to the consensus protocol, by the second smallest eigenvalue λ2 of the Laplacian of the graph associated to the communication network of the group. Next, we start employing our backstepping-based procedure by considering the second subsystem = iu1 iξ2
(7)
of (2). (In the following, we will omit the left superscript except in situations where it is necessary to distinguish individual agents.) Take ξ2 = α1 as a virtual control input to stabilize this subsystem such that there exists a function W2 (ξ1 ) ≥ 0, with W2 (0) = 0 and W˙ 2 ≤ 0. Then W2 is used as a candidate Lyapunov function to show stability of (7). Choosing 1 (8) W2 (ξ1 ) = ξ12 , 2 we have (9) W˙ 2 (ξ1 ) = ξ1 u1 ξ2 . Setting the virtual control to
α1 = ξ2 = k1
ξ1 , η
k1 > 0,
y1 α˙ 1 + η u1
(13)
to obtain
where
e˙ 3 − γ k1 y21 ≤ 0, ˙3 = W W
∀ (ξ1 , y1 ) 6= 0,
e˙ 3 = −γ k1 (ξ1 − u1y1 )2 . W
In this step α2 is a function of αu˙ 11 , that is, of ηξ12 . Using a similar argument as before, we require that k1 > 2λ2 . For all further steps, introduce the error coordinates ym = ξm+1 − αm , where m = 2, . . . , k − 2, and extend (11) giving
ξ˙1 = u1 (α1 + y1 ), y˙1 = u1 (α2 + y2 ) − α˙ 1 , .. . y˙m−1 = u1 (αm + ym ) − α˙ m−1 , y˙m = u1 ξm+2 − α˙ m .
(14)
Using the Lyapunov functions 1 Wm+2 (ξ1 , y1 , . . . , ym ) = Wm+1 (ξ1 , y1 , . . . , ym−1 ) + y2m , (15) 2 and choosing the virtual control inputs ξm+2 = αm+1 as
αm+1 = (2γ k1 − 1)ym−1 − γ k1 u1 ym + k1
(10)
ym α˙ m + η u1
(16)
results in
results in W˙ 2 (ξ1 ) = −γ k1 ξ12 = −2γ k1W2 ≤ 0,
exp (−γ k1t) = k1 exp (−γ [k1 − λ2 ]t). exp (−γλ2t)
In order to assure boundedness and convergence of α1 , we require that k1 > λ2 . Next, we introduce the error coordinate y1 = ξ2 − α1 and extend (7), applying this change of coordinates, to
ξ˙1 = u1 (α1 + y1 ), y˙1 = u1 ξ3 − α˙ 1 .
e˙ m+2 − γ k1 y2m ≤ 0, ˙ m+2 = W W
∀ ξ1 6= 0,
which ensures asymptotic stability of (7). Clearly, as W2 decays with 2γ k1 , ξ1 decays with γ k1 . We require that α1 remain bounded and that it converge to zero as t → ∞. As ξ0 , and thus η , decays with γλ2 , using (10) we have
α1 = k1
W˙ 3 = ξ1 ξ˙1 + y1y˙1 = −γ k1 ξ12 + y1 (u1 [ξ1 + ξ3] − α˙ 1 ) .
i
∑
i˙ ξ1
and W˙ 3 ≤ 0, ∀ (ξ1 , y1 ) 6= 0 using the candidate Lyapunov function 1 W3 (ξ1 , y1 ) = W2 (ξ1 ) + y21 . (12) 2 Computing the derivative with respect to time, we have
(11)
This time, take ξ3 = α2 as a virtual control input to stabilize this system such that there exists a W3 (ξ1 , y1 ) ≥ 0, W3 (0) = 0
∀ (ξ1 , y1 , . . . , ym ) 6= 0,
where
e˙ m+2 = −γ k1 (ξ1 − u1y1 )2 + W
m
∑ (y p−1 − u1y p)2
p=2
!
. (17)
In each of the preceding steps, the condition k1 > (m + 1)λ2 is imposed to assure boundedness and convergence of the virtual controls. Finally, in the last step, we introduce the error coordinate yk−1 = ξk − αk−1 and apply this change of coordinates to obtain ξ˙1 = u1 (α1 + y1 ), y˙1 = u1 (α2 + y2 ) − α˙ 1 , .. (18) . y˙k−2 = u1 (αk−1 + yk−1 ) − α˙ k−2 , y˙k−1 = u2 − α˙ k−1 .
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ThA06.4 At this point the second input u2 appears for the first time in the equations. Hence, by Theorem 1, system (18) is feedback equivalent to a passive system, as the relative degree restriction can be satisfied if yk−1 is taken as an output for the ξ1 , . . . , ξk subsystem of (2). Further, by construction, the zero dynamics of (18) are stable in the sense of Lyapunov. With the Lyapunov function V = W1 + Wk + 21 y2k−1 we get e˙ k − γ k1 y2k−2 + yk−1 (u1 yk−2 + u2 − α˙ k−1 ) . (19) V˙ = ξ0 u1 + W
If we then take
u2 = (2γ k1 − 1)u1yk−2 − γ k1u21 yk−1 + α˙ k−1 + τ , we obtain
(20)
V˙ = Ve˙ + ξ0u1 + yk−1 τ ,
where
(21)
e˙ = W e˙ k − γ k1(yk−2 − u1yk−1 )2 . V
(22)
Here, the condition k1 > (k − 1)λ2 from the final step above is sufficient to assure boundedness and convergence of u2 . Moreover, we see that system (2) is passive with input u = [u1 , τ ]⊤ , output y = [ξ0 , yk−1 ]⊤ and storage function V . Exploiting this property we set [1], [3] τ = −k2 ∑ iyk−1 − jyk−1 , k2 > 0, (23)
As U˙ is only negative semidefinite we need to employ LaSalle’s invariance principle [8] to show synchronization. Hence, let B ⊂ R × R × ···× R | {z } (k+1)N times
be a compact set that is positively invariant with respect to N copies of (18). Then, U ≥ 0 is a scalar C1 function such that U˙ ≤ 0 in B. We need to identify the largest invariant set contained in the set M = {[ξ 0 , . . . , ξ k ] ∈ B | U˙ ≡ 0}, where ξ p = [ 1ξ p , . . . , Nξ p ]⊤ for each p ∈ {0, . . . , k}. Recall that # " k−1 ˙ 2 2 e = −γ k1 (ξ1 − u1y1 ) + ∑ (ym−1 − u1ym ) , V m=2
for each agent, and hence, M is identified by iξ0 = jξ , iy j i i i i i i 0 k−1 = yk−1 , ξ1 = u1 y1 and ym−1 = u1 ym for m = 2, . . . , k − 2, i = 1, . . . , N and j ∈ Ni . Applying (10), (13), (16) and (6), and (20) together with (23) to each agent, (5) and (18) reduce to i˙ ξ0 = iu1 = −γ ∑ iξ0 − jξ0 ≡ 0, j∈Ni
i˙ ξ1 i y˙1
j∈Ni
i
= u 1 y 1 − γ k 1 ξ1 = − γ k 1 ξ1 ,
= iu1 iy2 − γ k1y1 + u1 [(2γ k1 )ξ1 − γ k1 u1 y1 ] = −γ k1 y1 , .. . i y˙m = iu1 iym+1 − γ k1 iym + iu1 (2γ k1 ) iym−1 − γ k1 iu1 iym
accordingly, leading to our main result. Theorem 2: Consider a homogeneous group of N agents, interconnected through a static, balanced, and strongly connected communication graph G . Then, synchronization in the sense of Definition 4 is achieved for the group with the controls i u1 = −γ ∑ iξ0 − jξ0 ,
= −γ k1 iym , .. . i yk−1 = u1 (2γ k1 ) iyk−1 − γ k1 iu1 iyk−1 − k2 ∑ iyk−1 − jyk−1
j∈Ni
u2 = (2γ k1 − 1)u1 yk−2 − γ k1 u21 yk−1 + α˙ k−1 − k2 ∑ iyk−1 − jyk−1 ,
i
i
j∈Ni
≡ 0.
j∈Ni
for each agent, with k1 > (k − 1)λ2 , γ > 0, k2 > 0 and where Ni = { j | (v j , vi ) ∈ E (G )}. Proof: Take U = 2 ∑Ni=1 iV as a Lyapunov function for the motion of the group. Its derivative with respect to time is N U˙ = 2 ∑ iVe˙ − γ ∑ iξ0 iξ0 − jξ0 i=1
j∈Ni
− k2
i
∑
yk−1
i
j
yk−1 − yk−1
j∈Ni
For balanced graphs we have [3] N
2K ∑
∑
N
i ⊤ i
x
i=1 j∈Ni
x=K∑
∑
N
i ⊤ i
x
i=1 j∈Ni
x+K ∑
∑
.
j ⊤ j
x
so that N
˙ ie
U˙ = ∑ 2 V − γ i=1
≤0
∑ j∈Ni
i
j
2
ξ0 − ξ0 − k 2
x,
i=1 j∈Ni
∑ j∈Ni
i
j
yk−1 − yk−1
! 2
Therefore, the largest invariant set contained in M is the set S = {[ξ 0 , . . . , ξ k ] ∈ B | iξ0 = jξ0 , iyk−1 = jyk−1 , ξ 1 = 0, y1 = · · · = yk−2 = 0} and synchronization in the sense of Definition 4 is achieved, completing the proof. Remark 1: The existence of a positively invariant set B is guaranteed by the construction of the Lyapunov function U. As U is radially unbounded, one may take B = {ξ ∈ R(k+1)N | U(ξ ) ≤ c} for any c > 0 [8]. Remark 2: One may assign different gains ki in each step of the procedure and assure convergence by setting ki > iλ2 for i = 1, . . . , k − 1. We just used one gain k1 for simplicity. Remark 3: The discontinuity in the control laws, imposed by a division by u1 , is not very restrictive as we may set u1 to a constant value when ∑ j∈Ni iξ0 − jξ0 = 0 [22]. In any case, the situation does not arise for generic initial states. IV. E XTENSION TO H IGHER O RDER C HAINED F ORMS Extending the results from Section III to higher order chained form systems is straightforward. Higher order
3344
ThA06.4 Lyapunov function candidate Wl+1,1 = Wl,κl + 12 y2l+1,1 . For this step assign
chained forms1 are given by the equations (κ 0 )
ξ0
(κ ) ξ1 1
(κ
= u1 ,
αl+1,κl +1 = α˙ l,κl + u1yl,κl −1 − 2yl+1,1
= u 1 ξ2 , .. .
(24)
to obtain e˙ l,κ − u1 yl,κ −1 − yl+1,1 2 − y2l+1,1. W˙ l+1,1 = W l l
)
ξk−1k−1 = u1 ξk , (κ k )
ξk
= u2 ,
(κ )
where ξi i denotes the κi -th derivative of the scalar ξi with respect to time. Well-known examples for such systems include the knife edge [10] and a class of underactuated mechanical systems with κi = 2 for all i, e.g. planar underactuated manipulators, the PVTOL without gravity, simplified underwater vehicle models etc. [15]. We note that stabilization of this system (24) can also be achieved using backstepping [17]. Considering synchronization for (24), we adapt our design (κ ) methodology as follows. First, transform ξ0 0 = u1 into a chain of integrators ξ˙0,1 = ξ0,2 , . . . , ξ˙0,κ0 = u1 . Since this is a simple strict feedback system, use every ξ0,m , m = 2, . . . , κ0 − 1 as virtual controls and employ the methodology introduced in [14] to define iu1 such that state components iξ = [ iξ , . . . , iξ ⊤ for all agents i ∈ {1, . . . , N} are 0,1 0,κ0 ] 0 synchronized. (κ ) Next, again transform every subsystem ξl l = u1 ξl+1 for l = 1, . . . , k − 1 into the chain of integrators ξ˙l,1 = ξl,2 , . . . , ξ˙l,κl = u1 ξl+1 and proceed as above, following [14], in the first κl − 1 steps. Next, in the κl -th step, the input u1 appears in the dynamics of the extended system and a combination of [14] and the approach presented in Section III must be made. To this end, introduce the error coordinate yl,κl −1 = ξl,κl − αl,κl −1 and consider Wl,κl = Wl,κl −1 + 21 y2l,κl −1 as a candidate Lyapunov function. Additionally, take ξl+1 = αl,κl as virtual control and set
αl,κl =
In the following, use [14] again to define the virtual control laws ξl+1,m for m = 2, . . . , κl+1 − 1, and repeat the procedure κ above for all ξq q with q = 1, . . . , k − 1. Finally, the same (κ ) approach as for u1 is used for the last subsystem ξk k = u2 of (24). This time u1 is related to a system of order κ0 , leading to the problem of assuring u1 6= 0 to avoid discontinuities in the subsequent virtual controls (25). This problem can be solved by the additional constraint on the parameters of (κ ) the controller for ξ0 0 = u1 , derived using [14], to produce a non-oscillating closed loop dynamics. For a second order linear, time-invariant system, e.g. ξ¨0 = u1 , this corresponds to the critically or over-damped case. V. E XAMPLE As an example illustrating the design methodology we choose the synchronization of unicycles [4], [5], [12], [13], [19]. Hence, the dynamics of each agent are given by x˙ = v cos θ , y˙ = v sin θ , θ˙ = ω ,
where
e˙ l,κ − u21y2l,κ −1 , ˙ l,κ = W W l l l
(26)
e˙ l,κ −1 − kl yl,κ −2 − yl,κ −1 2 . e˙ l,κ = W W l l l l (κ
)
For the next step, transform ξl+1l+1 = u1 ξl+2 into the chain of integrators ξ˙l+1,1 = ξl+1,2 , . . . , ξ˙l+1,κl+1 = u1 ξl+2 and observe that ξl+1 = ξl+1,1 . Hence, we introduce the error coordinate yl+1,1 = ξl+1,1 − αl,κl and proceed by using the 1 According
(28)
where x, y, θ denote the position and the heading of the unicycle with respect to an inertial frame. Using the state transformation ξ0 = θ , ξ1 = x sin θ − y cos θ , ξ2 = x cos θ + y sin θ and the input transformation u1 = ω , u2 = v − ξ2 ω [10], results in the chained form
ξ˙0 = u1 , ξ˙1 = u1 ξ2 ,
(2kl − 1)yl,κl −2 − kl yl,κl −1 + α˙ l,κl −2 − u1 yl,κl −1 (25) u1
to stabilize the cascade up to this point. Here, yl,κl −2 and α˙ l,κl −2 denote the previously (in the step κl − 1) defined error coordinate and derivative of the virtual control, respectively. Note that similar to Section III, kl is used to assure boundedness and convergence of the virtual control laws. For the derivative of the Lyapunov function, we have
(27)
(29)
ξ˙2 = u2 . For the control design, we start by assigning the standard consensus algorithm to the first input, that is i u1 = −γη = −γ ∑ iξ0 − jξ0 , γ > 0. (30) j∈Ni
In this example, we assume an all-to-all communication scheme, i.e. the underlying graph is fully connected. Hence, the algebraic connectivity λ2 = N, which defines the rate of convergence of the headings of the group. In the second step, consider (31) ξ˙1 = u1 ξ2 and take W1 = 12 ξ12 as a Lyapunov function candidate and ξ2 = α1 as a virtual control input to stabilize this subsystem. The derivative of W1 with respect to time is W˙ 1 = ξ1 u1 α1 = −γ k1 ξ12 ,
to [15] no unifying nomenclature for such systems exists.
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ThA06.4 where we set α1 = k1 ξη1 . To assure convergence, we must have k1 > N. Next, introduce the error coordinate y1 = ξ2 − α1 and rewrite (29) to obtain
ξ˙1 = u1 (α1 + y1 ), y˙1 = u2 − α˙ 1 .
(32)
In order to achieve synchronization, set u2 =(2γ k1 − 1)u1ξ1 − γ k1 u21 y1 + α˙ 1 −k2
i
y1 − jy1
∑ j∈Ni
for every agent in the group, where
α˙ 1 = k1
ξ˙1 η − ξ1 ∑ j∈Ni η2
iu
1−
ju
1
(33)
is needed to complete the derivation. Results for N = 4 agents with initial conditions x0 = [−10, 12, 10, 5]⊤, y0 = 1 [12, −12, 8, −15]⊤, θ 0 = [ π4 , − π2 , 23π , π ]⊤ and gains γ = N+1 , k1 = N + 1, k2 = 2 are given in Fig. 1. 25 position ix (m)
20 15 10 5
agent agent agent agent
0 −5 0 50
5
1 2 3 4
10
15
20
25
30
35
10
15
20
25
30
35
15 20 time (s)
25
30
35
position iy (m)
40 30 20 10 0 −10 0
5
heading iθ (rad)
3 2 1 0 −1 0
Fig. 1.
5
10
Original state of four unicycles with controls (30), (33)
VI. C ONCLUSION AND F UTURE W ORK We presented a methodology to design controllers synchronizing systems given in chained form. The approach exploits feedback equivalence to passive systems and extends
results on passivity-based coordination to a broader class of nonholonomic systems. The primary goal for the future is to relax the tight constraints on the network topology, such that time-varying and unbalanced communication networks are possible and to define the control law such that only relative information is needed to achieve synchronization. R EFERENCES [1] M. Arcak. Passivity as a design tool for group coordination. IEEE Transactions on Automatic Control, 52(8):1380–1390, 2007. [2] C.I. Byrnes, A. Isidori, and J.C. Willems. Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Transactions on Automatic Control, 36(11):1228–1240, 1991. [3] N. Chopra and M.W. Spong. Passivity-based control of multi-agent systems. In Advances in Robot Control: From Everyday Physics to Human-Like Movements, pages 107–134. Springer Verlag, 2006. [4] D.V. Dimarogonas and K.J. Kyriakopoulos. On the rendezvous problem for multiple nonholonomic agents. IEEE Transactions on Automatic Control, 52(5):916–922, 2007. [5] W. Dong and J. A. Farrell. Cooperative control of multiple nonholonomic mobile agents. IEEE Transactions on Automatic Control, 53(6):1434–1448, 2008. [6] C. Godsil and G. Royle. Algebraic Graph Theory. Number 207 in Graduate Texts in Mathematics. Springer Verlag, 2001. [7] Z.-P. Jiang and H. Nijmeijer. A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Transactions on Automatic Control, 44(2):265–279, 1999. [8] H. Khalil. Nonlinear Systems. Prentice-Hall, 3rd edition, 2001. [9] Y. Kim and M. Mesbahi. On maximizing the second smallest eigenvalue of a state-dependent graph laplacian. IEEE Transactions on Automatic Control, 51(1):116–120, 2006. [10] I. Kolmanovsky and N.H. McClamroch. Developments in nonholonomic control problems. IEEE Control Systems Magazine, 15(6):20– 36, 1995. [11] M. Kristi´c, I. Kanellakopoulos, and P. Kokotovi´c. Nonlinear and Adaptive Control Design. Wiley-Interscience, 1995. [12] Z. Lin, B. Francis, and M. Maggiore. Necessary and sufficient graphical conditions for formation control of unicycles. IEEE Transactions on Automatic Control, 50(1):121–127, 2005. [13] K.D. Listmann, M.V. Masalawala, and J. Adamy. Consensus for formation control of nonholonomic mobile robots. In 2009 IEEE International Conference on Robotics and Automation, pages 3886– 3891, 2009. [14] K.D. Listmann, C.A. Woolsey, and J. Adamy. Passivity-based coordination of multi-agent systems: A backstepping approach. In Proceedings of the European Control Conference 2009, pages 2450– 2455, 2009. [15] D.A. Lizarraga, N.P.I. Aneke, and H. Nijmeijer. Robust point stabilization of underactuated mechanical systems via the extended chained form. SIAM J. Control Optim., 42(6):2172–2199, 2004. [16] S. Nair and N.E. Leonard. Stable synchronization of mechanical system networks. SIAM J. Control Optim., 47(2):661–683, 2008. [17] T.-K. Nam, J. Jongusuk, and T. Mita. Control of high order nonholonomic systems by backstepping. In 40th IEEE Conference on Decision and Control, pages 1775–1780, 2001. [18] A. Y. Pogromsky. Passivity based design of synchronizing systems. International Journal of Bifurcation and Chaos, 8(2):295–319, 1998. [19] Z. Qu, J. Wang, and R.A. Hull. Cooperative control of dynamical systems with application to autonomous vehicles. IEEE Transactions on Automatic Control, 53(4):894–911, 2008. [20] W. Ren, R.W. Beard, and E.M. Atkins. Information consensus in multivehicle cooperative control. IEEE Control Systems Magazine, 27(2):71–82, 2007. [21] R. Sepulchre, M. Jankovi´c, and P. Kokotovi´c. Constructive Nonlinear Control. Springer Verlag, 1996. [22] A. Tayebi, M. Tadjine, and A. Rachid. Discontinuous control design for the stabilization of nonholonomic systems in chained form using the backstepping approach. In 36th IEEE Conference on Decision and Control, pages 3089–3090, 1997. [23] A. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear Control. Springer Verlag, 2nd edition, 2000.
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ThA06.4
Output Synchronization of Systems in Chained Form Kim D. Listmann and Craig A. Woolsey
Abstract— In this article we extend previous results for backstepping and passivity-based design of cooperative control laws to a class of chained form systems that includes certain driftfree nonholonomic systems. We exploit the cascaded structure and feedback equivalence to passive systems to derive suitable control laws using a modified backstepping methodology. A virtual output is obtained and shared within the group in such a way that full state synchronization for the group is achieved. Given a static, strongly connected and balanced communication network we prove stability of the proposed controller. Further, an extension to higher order chained forms is shown, providing a methodology to coordinate general nonholonomic systems. The effectiveness of the method is demonstrated by synchronizing a collective of unicycles.
I. I NTRODUCTION Motivated by many applications in biology, physics and engineering, problems involving cooperation among members of a group of autonomous agents have enjoyed considerable attention recently. These problems generally involve a high-level goal that the group must agree on collectively. Coordination problems generally fall in two categories. In the first category of problems, one addresses the complexity of the information exchange among several agents, modeling the dynamics of each agent as simply as possible, and seeks conditions on the exchange of information which guarantee agreement. Such problems are referred to as consensus problems [20]. In the second category of problems, one incorporates more complicated agent dynamics, though with simplifying assumptions about the communication topology. Nair and Leonard, for example, consider unstable, underactuated mechanical systems that are amenable to the method of controlled Lagrangians [16]. In this second category, the coordination task may be referred to as synchronization. In this article, we will consider on the latter category of problems. The notion of passivity underlies a number of proposed methods for cooperative control design [1], [3], [18] and we build upon this theme of passivity-based design, extending previous results in this area [14]. Due to the structure of the systems that we consider here, however, we must explicitly address boundedness and convergence of the control laws that we design. In our previous work [14], these properties were not considered due to an appropriate assumption on the systems examined. This work was supported by the German Research Foundation (DFG) within the GRK 1362 “Cooperative, Adaptive and Responsive Monitoring of Mixed Mode Environments” (www.gkmm.tu-darmstadt.de) K.D. Listmann is with the Control Theory and Robotics Lab, Technische Universit¨at Darmstadt, Landgraf-Georg-Str. 4, 64283 Darmstadt, GERMANY. [email protected] C.A. Woolsey is with the Department of Aerospace and Oceanic Engineering, Virginia Polytechnic University, Blacksburg, VA 24061, USA [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
A variety of driftless nonholonomic systems, including the unicycle [10], are contained in the class of chained form systems considered here. Higher order extensions of the chained form include still more underactuated systems [15]. One way to achieve stabilization for these systems, is to exploit their cascaded structure and employ a recursive design technique [7], [17], [22], namely backstepping [11]. We adopt this methodology, as well, and propose a backstepping-based design that exploits the feedback equivalence of chained form systems to passive systems. By recognizing this feedback equivalence, it is possible to use prior results [1], [3] to derive suitable coordination laws that will achieve synchronization. Additionally, we present an extension of our methodology to achieve synchronization for general nonholonomic and underactuated systems given in higher order chained form. This results in a general solution of the synchronization problem for the stated class of chained form systems. The paper is organized as follows. In the next section, we introduce the necessary definitions and notation. We provide the design methodology and state our main result in Section III. The extension to higher order chained form systems is presented in Section IV. An illustrative example is given in Section V by demonstrating the methodology on the synchronization of a collective of unicycles. Finally, conclusions are presented in Section VI. II. P RELIMINARIES A. Passivity Passivity is a system property that can be understood in terms of energy dissipation in an input-output system. Informally, an input-output system is considered passive if it does not “generate energy,” where the term “energy” generalizes a familiar concept for mechanical systems. We provide a brief review below. For a thorough introduction, see [23]. Consider the system x˙ = f(x) + G(x)u, y = h(x),
(1)
where x(t) ∈ Rn , u(t) ∈ Rr and y(t) ∈ Rr . The drift vector fields f(·), the matrix G(·) of input vector fields, and the output vector field h(·) are assumed to be sufficiently smooth. We assume that f(0) = 0, h(0) = 0. The input history u(t) is taken to be piecewise continuous and locally square integrable. Definition 1: The nonlinear system (1) is said to be passive if there exists a scalar C1 storage function V (x) ≥ 0, V (0) = 0 and a function S(x) ≥ 0 such that for all t ≥ 0
3341
V˙ (x) = u⊤ (t)y(t) − S(x(t)).
ThA06.4 The system (1) is strictly passive if S(x) > 0 and lossless if S(x) = 0. A positive definite storage function V (x) would serve as a Lyapunov function to guarantee stability of the equilibrium at the origin for the system (1). However, the definition of passivity allows a positive semidefinite storage function V (x) so that, in general, one may have to check additional conditions to guarantee stability. In order to determine the necessary conditions for the system (1) to be passive, we present a well known result [2]. Theorem 1: Suppose x = 0 is a regular point for (1). Then, (1) is feedback equivalent to a passive system with a C2 storage function V , which is positive definite, if and only if (1) has a vector relative degree {1, . . . , 1} at x = 0 and is weakly minimum phase. Having verified that the systems of interest are feedback equivalent to passive systems, we seek synchronizing control laws [1], [3]. B. System Description In this paper, we consider systems given in chained form [10], that is ξ˙0 = u1 , ξ˙1 = u1 ξ2 , .. .
(2)
ξ˙k−1 = u1 ξk , ξ˙k = u2 , where z, ξi , and ui are scalar variables. Typically, driftless nonholonomic systems can be (locally) transformed to a chained form by input and state transformation. That is why this system description is widely used and should be exploited for control design [10]. One possibility is to use a recursive backstepping design [7], [11], [22] to asymptotically stabilize these systems. The backstepping procedure defines a virtual output such that cascaded systems, e.g. system (2), are feedback equivalent to passive systems [21]. In our approach we will make use of this and slightly alter the standard backstepping design to address group coordination. For the remainder of the paper, a system with dynamics (2) is denoted as an agent. C. Graph Theory and Group Communication Key to coordination algorithms is information exchange within the group, and hence, the underlying communication network. This network can be modeled using graphs. We refer to [6] for a thorough treatment of graphs and their properties, and give condensed definitions following [3]. Definition 2: A graph G is a finite set of elements V (G ) = {v1 , ..., vN }, the vertices of a graph, and a set E (G ) ⊂ V ×V called the edges of a graph. If for all (vi , v j ) ∈ E , (v j , vi ) ∈ E as well, the graph is said to be undirected, otherwise it is called directed or Digraph. The in-degree of a vertex vi ∈ V defines the number of edges incoming to this vertex, whereas the out-degree defines the number of edges
outgoing from this vertex. A graph is balanced, if for each vi ∈ V the in-degree equals the out-degree. Clearly every undirected graph is balanced. Definition 3: A path of length k in a Digraph is a sequence v0 , ..., vk of k + 1 distinct vertices such that for every i ∈ {0, ..., k − 1}, (vi , vi+1 ) ∈ E and a weak path is such that (vi , vi+1 ) ∈ E or (vi+1 , vi ) ∈ E . A Digraph is fully connected if, for all vi , v j ∈ V , (vi , v j ) ∈ E and (v j , vi ) ∈ E . It is strongly connected if any two vertices can be connected by a path and it is said to be weakly connected if any two vertices can be connected by a weak path. An undirected graph can only be connected or disconnected, since there is no distinction between paths and weak paths. D. Consensus Algorithms Consider the continuous dynamics N i
x˙ = − ∑ ai j j=1
x − jx ,
i
i = 1, . . . , N
(3)
known as consensus protocol for a group with N members [20]. We use ix to denote a state x of the i-th member, whenever distinction between the group members is needed. Rewriting this dynamics for the whole group results in x˙ = −Lx.
(4)
Here, x = [ 1x, . . . , Nx]⊤ and L denotes the Laplacian matrix associated to the graph, modeling the communication network of the group. It is defined as L = li j ∈ RN×N , where li j = −1 for i 6= j, if (vi , v j ) ∈ E (G ), zero otherwise and on the diagonal we have lii = di , where di denotes the in-degree of vertex i. We say that consensus is achieved if | ix − jx| → 0 for t → ∞. Thus, the properties of the Laplacian determine the behavior of (4). In this context, the rate of convergence of such consensus protocols was identified as the second smallest eigenvalue λ2 of L [9]. This fact will later be used to assure convergence of our control laws. Concerning the network topology of the group of agents, we need the following assumption. Assumption 1: The agents form a static, balanced and strongly connected communication network at all times. III. C ONTROLLER D ESIGN In this section we present a novel methodology designing cooperative control laws for the agents considered. Based on the backstepping procedure we show that system (2) is feedback equivalent to a passive system. Hence, it is possible to develop a coordination controller exploiting this property [1], [3]. Finally, we prove stabilization of the aggregate motion and show that synchronization is achieved. Definition 4: A group of N agents is said to be synchronized, if lim iξ p − jξ p = 0, with p = 0, . . . , k. t→∞
for every i, j ∈ {1, 2, . . . , N}, i 6= j. Note that our method is agent based, such that each of the next steps has to be done for all agents, accordingly.
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ThA06.4 Firstly, consider the first subsystem of system (2), that is i˙ ξ0
= iu1 ,
(5)
where the first input of the system already appears. Trivially, this system is passive with output ξ0 , input u1 and storage function W1 = 21 ξ02 . Following a standard consensus protocol [20], we set i u1 = −γη , γ > 0, (6) where
η=
ξ0 − jξ0 .
j∈Ni
This time, our approach differs from the standard backstepping procedure in that we take
α2 = (2γ k1 − 1)ξ1 − γ k1u1 y1 + k1
with Ni denoting the set of neighbors from whom agent i receives information. Note that the rate of convergence of ξ0 is specified by γ and, due to the consensus protocol, by the second smallest eigenvalue λ2 of the Laplacian of the graph associated to the communication network of the group. Next, we start employing our backstepping-based procedure by considering the second subsystem = iu1 iξ2
(7)
of (2). (In the following, we will omit the left superscript except in situations where it is necessary to distinguish individual agents.) Take ξ2 = α1 as a virtual control input to stabilize this subsystem such that there exists a function W2 (ξ1 ) ≥ 0, with W2 (0) = 0 and W˙ 2 ≤ 0. Then W2 is used as a candidate Lyapunov function to show stability of (7). Choosing 1 (8) W2 (ξ1 ) = ξ12 , 2 we have (9) W˙ 2 (ξ1 ) = ξ1 u1 ξ2 . Setting the virtual control to
α1 = ξ2 = k1
ξ1 , η
k1 > 0,
y1 α˙ 1 + η u1
(13)
to obtain
where
e˙ 3 − γ k1 y21 ≤ 0, ˙3 = W W
∀ (ξ1 , y1 ) 6= 0,
e˙ 3 = −γ k1 (ξ1 − u1y1 )2 . W
In this step α2 is a function of αu˙ 11 , that is, of ηξ12 . Using a similar argument as before, we require that k1 > 2λ2 . For all further steps, introduce the error coordinates ym = ξm+1 − αm , where m = 2, . . . , k − 2, and extend (11) giving
ξ˙1 = u1 (α1 + y1 ), y˙1 = u1 (α2 + y2 ) − α˙ 1 , .. . y˙m−1 = u1 (αm + ym ) − α˙ m−1 , y˙m = u1 ξm+2 − α˙ m .
(14)
Using the Lyapunov functions 1 Wm+2 (ξ1 , y1 , . . . , ym ) = Wm+1 (ξ1 , y1 , . . . , ym−1 ) + y2m , (15) 2 and choosing the virtual control inputs ξm+2 = αm+1 as
αm+1 = (2γ k1 − 1)ym−1 − γ k1 u1 ym + k1
(10)
ym α˙ m + η u1
(16)
results in
results in W˙ 2 (ξ1 ) = −γ k1 ξ12 = −2γ k1W2 ≤ 0,
exp (−γ k1t) = k1 exp (−γ [k1 − λ2 ]t). exp (−γλ2t)
In order to assure boundedness and convergence of α1 , we require that k1 > λ2 . Next, we introduce the error coordinate y1 = ξ2 − α1 and extend (7), applying this change of coordinates, to
ξ˙1 = u1 (α1 + y1 ), y˙1 = u1 ξ3 − α˙ 1 .
e˙ m+2 − γ k1 y2m ≤ 0, ˙ m+2 = W W
∀ ξ1 6= 0,
which ensures asymptotic stability of (7). Clearly, as W2 decays with 2γ k1 , ξ1 decays with γ k1 . We require that α1 remain bounded and that it converge to zero as t → ∞. As ξ0 , and thus η , decays with γλ2 , using (10) we have
α1 = k1
W˙ 3 = ξ1 ξ˙1 + y1y˙1 = −γ k1 ξ12 + y1 (u1 [ξ1 + ξ3] − α˙ 1 ) .
i
∑
i˙ ξ1
and W˙ 3 ≤ 0, ∀ (ξ1 , y1 ) 6= 0 using the candidate Lyapunov function 1 W3 (ξ1 , y1 ) = W2 (ξ1 ) + y21 . (12) 2 Computing the derivative with respect to time, we have
(11)
This time, take ξ3 = α2 as a virtual control input to stabilize this system such that there exists a W3 (ξ1 , y1 ) ≥ 0, W3 (0) = 0
∀ (ξ1 , y1 , . . . , ym ) 6= 0,
where
e˙ m+2 = −γ k1 (ξ1 − u1y1 )2 + W
m
∑ (y p−1 − u1y p)2
p=2
!
. (17)
In each of the preceding steps, the condition k1 > (m + 1)λ2 is imposed to assure boundedness and convergence of the virtual controls. Finally, in the last step, we introduce the error coordinate yk−1 = ξk − αk−1 and apply this change of coordinates to obtain ξ˙1 = u1 (α1 + y1 ), y˙1 = u1 (α2 + y2 ) − α˙ 1 , .. (18) . y˙k−2 = u1 (αk−1 + yk−1 ) − α˙ k−2 , y˙k−1 = u2 − α˙ k−1 .
3343
ThA06.4 At this point the second input u2 appears for the first time in the equations. Hence, by Theorem 1, system (18) is feedback equivalent to a passive system, as the relative degree restriction can be satisfied if yk−1 is taken as an output for the ξ1 , . . . , ξk subsystem of (2). Further, by construction, the zero dynamics of (18) are stable in the sense of Lyapunov. With the Lyapunov function V = W1 + Wk + 21 y2k−1 we get e˙ k − γ k1 y2k−2 + yk−1 (u1 yk−2 + u2 − α˙ k−1 ) . (19) V˙ = ξ0 u1 + W
If we then take
u2 = (2γ k1 − 1)u1yk−2 − γ k1u21 yk−1 + α˙ k−1 + τ , we obtain
(20)
V˙ = Ve˙ + ξ0u1 + yk−1 τ ,
where
(21)
e˙ = W e˙ k − γ k1(yk−2 − u1yk−1 )2 . V
(22)
Here, the condition k1 > (k − 1)λ2 from the final step above is sufficient to assure boundedness and convergence of u2 . Moreover, we see that system (2) is passive with input u = [u1 , τ ]⊤ , output y = [ξ0 , yk−1 ]⊤ and storage function V . Exploiting this property we set [1], [3] τ = −k2 ∑ iyk−1 − jyk−1 , k2 > 0, (23)
As U˙ is only negative semidefinite we need to employ LaSalle’s invariance principle [8] to show synchronization. Hence, let B ⊂ R × R × ···× R | {z } (k+1)N times
be a compact set that is positively invariant with respect to N copies of (18). Then, U ≥ 0 is a scalar C1 function such that U˙ ≤ 0 in B. We need to identify the largest invariant set contained in the set M = {[ξ 0 , . . . , ξ k ] ∈ B | U˙ ≡ 0}, where ξ p = [ 1ξ p , . . . , Nξ p ]⊤ for each p ∈ {0, . . . , k}. Recall that # " k−1 ˙ 2 2 e = −γ k1 (ξ1 − u1y1 ) + ∑ (ym−1 − u1ym ) , V m=2
for each agent, and hence, M is identified by iξ0 = jξ , iy j i i i i i i 0 k−1 = yk−1 , ξ1 = u1 y1 and ym−1 = u1 ym for m = 2, . . . , k − 2, i = 1, . . . , N and j ∈ Ni . Applying (10), (13), (16) and (6), and (20) together with (23) to each agent, (5) and (18) reduce to i˙ ξ0 = iu1 = −γ ∑ iξ0 − jξ0 ≡ 0, j∈Ni
i˙ ξ1 i y˙1
j∈Ni
i
= u 1 y 1 − γ k 1 ξ1 = − γ k 1 ξ1 ,
= iu1 iy2 − γ k1y1 + u1 [(2γ k1 )ξ1 − γ k1 u1 y1 ] = −γ k1 y1 , .. . i y˙m = iu1 iym+1 − γ k1 iym + iu1 (2γ k1 ) iym−1 − γ k1 iu1 iym
accordingly, leading to our main result. Theorem 2: Consider a homogeneous group of N agents, interconnected through a static, balanced, and strongly connected communication graph G . Then, synchronization in the sense of Definition 4 is achieved for the group with the controls i u1 = −γ ∑ iξ0 − jξ0 ,
= −γ k1 iym , .. . i yk−1 = u1 (2γ k1 ) iyk−1 − γ k1 iu1 iyk−1 − k2 ∑ iyk−1 − jyk−1
j∈Ni
u2 = (2γ k1 − 1)u1 yk−2 − γ k1 u21 yk−1 + α˙ k−1 − k2 ∑ iyk−1 − jyk−1 ,
i
i
j∈Ni
≡ 0.
j∈Ni
for each agent, with k1 > (k − 1)λ2 , γ > 0, k2 > 0 and where Ni = { j | (v j , vi ) ∈ E (G )}. Proof: Take U = 2 ∑Ni=1 iV as a Lyapunov function for the motion of the group. Its derivative with respect to time is N U˙ = 2 ∑ iVe˙ − γ ∑ iξ0 iξ0 − jξ0 i=1
j∈Ni
− k2
i
∑
yk−1
i
j
yk−1 − yk−1
j∈Ni
For balanced graphs we have [3] N
2K ∑
∑
N
i ⊤ i
x
i=1 j∈Ni
x=K∑
∑
N
i ⊤ i
x
i=1 j∈Ni
x+K ∑
∑
.
j ⊤ j
x
so that N
˙ ie
U˙ = ∑ 2 V − γ i=1
≤0
∑ j∈Ni
i
j
2
ξ0 − ξ0 − k 2
x,
i=1 j∈Ni
∑ j∈Ni
i
j
yk−1 − yk−1
! 2
Therefore, the largest invariant set contained in M is the set S = {[ξ 0 , . . . , ξ k ] ∈ B | iξ0 = jξ0 , iyk−1 = jyk−1 , ξ 1 = 0, y1 = · · · = yk−2 = 0} and synchronization in the sense of Definition 4 is achieved, completing the proof. Remark 1: The existence of a positively invariant set B is guaranteed by the construction of the Lyapunov function U. As U is radially unbounded, one may take B = {ξ ∈ R(k+1)N | U(ξ ) ≤ c} for any c > 0 [8]. Remark 2: One may assign different gains ki in each step of the procedure and assure convergence by setting ki > iλ2 for i = 1, . . . , k − 1. We just used one gain k1 for simplicity. Remark 3: The discontinuity in the control laws, imposed by a division by u1 , is not very restrictive as we may set u1 to a constant value when ∑ j∈Ni iξ0 − jξ0 = 0 [22]. In any case, the situation does not arise for generic initial states. IV. E XTENSION TO H IGHER O RDER C HAINED F ORMS Extending the results from Section III to higher order chained form systems is straightforward. Higher order
3344
ThA06.4 Lyapunov function candidate Wl+1,1 = Wl,κl + 12 y2l+1,1 . For this step assign
chained forms1 are given by the equations (κ 0 )
ξ0
(κ ) ξ1 1
(κ
= u1 ,
αl+1,κl +1 = α˙ l,κl + u1yl,κl −1 − 2yl+1,1
= u 1 ξ2 , .. .
(24)
to obtain e˙ l,κ − u1 yl,κ −1 − yl+1,1 2 − y2l+1,1. W˙ l+1,1 = W l l
)
ξk−1k−1 = u1 ξk , (κ k )
ξk
= u2 ,
(κ )
where ξi i denotes the κi -th derivative of the scalar ξi with respect to time. Well-known examples for such systems include the knife edge [10] and a class of underactuated mechanical systems with κi = 2 for all i, e.g. planar underactuated manipulators, the PVTOL without gravity, simplified underwater vehicle models etc. [15]. We note that stabilization of this system (24) can also be achieved using backstepping [17]. Considering synchronization for (24), we adapt our design (κ ) methodology as follows. First, transform ξ0 0 = u1 into a chain of integrators ξ˙0,1 = ξ0,2 , . . . , ξ˙0,κ0 = u1 . Since this is a simple strict feedback system, use every ξ0,m , m = 2, . . . , κ0 − 1 as virtual controls and employ the methodology introduced in [14] to define iu1 such that state components iξ = [ iξ , . . . , iξ ⊤ for all agents i ∈ {1, . . . , N} are 0,1 0,κ0 ] 0 synchronized. (κ ) Next, again transform every subsystem ξl l = u1 ξl+1 for l = 1, . . . , k − 1 into the chain of integrators ξ˙l,1 = ξl,2 , . . . , ξ˙l,κl = u1 ξl+1 and proceed as above, following [14], in the first κl − 1 steps. Next, in the κl -th step, the input u1 appears in the dynamics of the extended system and a combination of [14] and the approach presented in Section III must be made. To this end, introduce the error coordinate yl,κl −1 = ξl,κl − αl,κl −1 and consider Wl,κl = Wl,κl −1 + 21 y2l,κl −1 as a candidate Lyapunov function. Additionally, take ξl+1 = αl,κl as virtual control and set
αl,κl =
In the following, use [14] again to define the virtual control laws ξl+1,m for m = 2, . . . , κl+1 − 1, and repeat the procedure κ above for all ξq q with q = 1, . . . , k − 1. Finally, the same (κ ) approach as for u1 is used for the last subsystem ξk k = u2 of (24). This time u1 is related to a system of order κ0 , leading to the problem of assuring u1 6= 0 to avoid discontinuities in the subsequent virtual controls (25). This problem can be solved by the additional constraint on the parameters of (κ ) the controller for ξ0 0 = u1 , derived using [14], to produce a non-oscillating closed loop dynamics. For a second order linear, time-invariant system, e.g. ξ¨0 = u1 , this corresponds to the critically or over-damped case. V. E XAMPLE As an example illustrating the design methodology we choose the synchronization of unicycles [4], [5], [12], [13], [19]. Hence, the dynamics of each agent are given by x˙ = v cos θ , y˙ = v sin θ , θ˙ = ω ,
where
e˙ l,κ − u21y2l,κ −1 , ˙ l,κ = W W l l l
(26)
e˙ l,κ −1 − kl yl,κ −2 − yl,κ −1 2 . e˙ l,κ = W W l l l l (κ
)
For the next step, transform ξl+1l+1 = u1 ξl+2 into the chain of integrators ξ˙l+1,1 = ξl+1,2 , . . . , ξ˙l+1,κl+1 = u1 ξl+2 and observe that ξl+1 = ξl+1,1 . Hence, we introduce the error coordinate yl+1,1 = ξl+1,1 − αl,κl and proceed by using the 1 According
(28)
where x, y, θ denote the position and the heading of the unicycle with respect to an inertial frame. Using the state transformation ξ0 = θ , ξ1 = x sin θ − y cos θ , ξ2 = x cos θ + y sin θ and the input transformation u1 = ω , u2 = v − ξ2 ω [10], results in the chained form
ξ˙0 = u1 , ξ˙1 = u1 ξ2 ,
(2kl − 1)yl,κl −2 − kl yl,κl −1 + α˙ l,κl −2 − u1 yl,κl −1 (25) u1
to stabilize the cascade up to this point. Here, yl,κl −2 and α˙ l,κl −2 denote the previously (in the step κl − 1) defined error coordinate and derivative of the virtual control, respectively. Note that similar to Section III, kl is used to assure boundedness and convergence of the virtual control laws. For the derivative of the Lyapunov function, we have
(27)
(29)
ξ˙2 = u2 . For the control design, we start by assigning the standard consensus algorithm to the first input, that is i u1 = −γη = −γ ∑ iξ0 − jξ0 , γ > 0. (30) j∈Ni
In this example, we assume an all-to-all communication scheme, i.e. the underlying graph is fully connected. Hence, the algebraic connectivity λ2 = N, which defines the rate of convergence of the headings of the group. In the second step, consider (31) ξ˙1 = u1 ξ2 and take W1 = 12 ξ12 as a Lyapunov function candidate and ξ2 = α1 as a virtual control input to stabilize this subsystem. The derivative of W1 with respect to time is W˙ 1 = ξ1 u1 α1 = −γ k1 ξ12 ,
to [15] no unifying nomenclature for such systems exists.
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ThA06.4 where we set α1 = k1 ξη1 . To assure convergence, we must have k1 > N. Next, introduce the error coordinate y1 = ξ2 − α1 and rewrite (29) to obtain
ξ˙1 = u1 (α1 + y1 ), y˙1 = u2 − α˙ 1 .
(32)
In order to achieve synchronization, set u2 =(2γ k1 − 1)u1ξ1 − γ k1 u21 y1 + α˙ 1 −k2
i
y1 − jy1
∑ j∈Ni
for every agent in the group, where
α˙ 1 = k1
ξ˙1 η − ξ1 ∑ j∈Ni η2
iu
1−
ju
1
(33)
is needed to complete the derivation. Results for N = 4 agents with initial conditions x0 = [−10, 12, 10, 5]⊤, y0 = 1 [12, −12, 8, −15]⊤, θ 0 = [ π4 , − π2 , 23π , π ]⊤ and gains γ = N+1 , k1 = N + 1, k2 = 2 are given in Fig. 1. 25 position ix (m)
20 15 10 5
agent agent agent agent
0 −5 0 50
5
1 2 3 4
10
15
20
25
30
35
10
15
20
25
30
35
15 20 time (s)
25
30
35
position iy (m)
40 30 20 10 0 −10 0
5
heading iθ (rad)
3 2 1 0 −1 0
Fig. 1.
5
10
Original state of four unicycles with controls (30), (33)
VI. C ONCLUSION AND F UTURE W ORK We presented a methodology to design controllers synchronizing systems given in chained form. The approach exploits feedback equivalence to passive systems and extends
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