Output Synchronization of Linear Parameter-varying Systems Via ...
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Output Synchronization of Linear Parameter-varying Systems via Dynamic Couplings Georg S. Seyboth∗ , Gerd S. Schmidt, Frank Allg¨ower Abstract—We address the output synchronization problem for groups of linear parameter-varying (LPV) agents. The agents are assumed to have identical LPV structure depending on different local parameters which make the group heterogeneous. First, we show that homogeneous groups of LPV systems with static diffusive couplings synchronize over uniformly connected graphs. Then, based thereon, we propose a constructive and scalable solution to the output synchronization problem for groups of LPV systems with heterogeneous parameters. The results are illustrated in a numerical example.
I. I NTRODUCTION In recent years, distributed and cooperative control became one of the major research areas in control theory. Specially consensus and synchronization problems in multiagent systems receive considerable attention due to the wide application range, e.g., motion coordination for autonomous vehicles [1]. Initially, consensus problems in single-integrator networks were investigated [2], and it was shown that consensus can be guaranteed under very mild assumptions on the network connectivity [3]. One development in the consensus and synchronization literature was the successive shift to agent dynamics with higher complexity. Networks of identical linear time-invariant (LTI) systems are addressed in [4]– [6]. Recently synchronization problems in heterogeneous groups of LTI agents are investigated [7]–[11]. It has been discovered in [7], [8] that there is a close connection between the synchronization problem in dynamical networks and the classical output regulation problem [12]. In this paper we consider heterogeneous multi-agent systems which consist of linear parameter-varying (LPV) systems. This system class has hardly been studied in the context of cooperative control [13], [14] even though it is suitable to describe a variety of physical systems such as airplanes, satellites, wind turbines and many more. Linear systems with constant uncertain parameters are addressed in [9]. In [14], a synchronization problem for parameter-varying systems is addressed and solved under noisy measurements via dissipativity techniques. In contrast to the present work, all parameters are globally available to the controllers of all subsystems. In the present work the parameters are assumed to be timevarying and available as real-time measurements. The LPV ∗ Corresponding author. The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart and within the Priority Programme 1305 “Control Theory of Digitally Networked Dynamical Systems”. All authors are with the Institute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany. {georg.seyboth, gerd.schmidt, allgower}@ist.uni-stuttgart.de.
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
systems may depend on parameters which influence all agents (global parameters) and on parameters which influence only individual agents (local parameters). The local parameters can be different for each agent, which causes the heterogeneity in the multi-agent system. Such multi-agent systems may for example describe a formation of UAVs in which the global parameter is air pressure and local parameters are quantities such as fuel mass or angle of attack. The main contribution of this paper is a constructive solution of the output synchronization problem of heterogeneous LPV systems. We propose distributed dynamic LPV controllers which solve the synchronization problem. The design procedure is applicable to multi-agent systems with arbitrarily many agents. Furthermore discuss the connection of the present problem to output regulation of LPV systems [15]. The remainder of this paper is organized as follows. Section II contains some preliminaries and the formulation of the output synchronization problem. Section III presents our main results including the constructive solution to the synchronization problem. The theoretical results are illustrated by an example in Section IV and Section V concludes the paper. II. P RELIMINARIES AND P ROBLEM F ORMULATION A. Communication graphs The communication topology of the multi-agent system is modeled by a directed, weighted, and time-varying graph G (t) = (V , E (t), A(t)). We use the same notation as in [5], [8] which is briefly summarized here for completeness. The graph G (t) consists of the vertex set V = {v1 , ..., vN }, N ∈ N, the edge set E (t) ⊂ V × V and the time-varying adjacency matrix A(t). Vertex vk corresponds to agent k in the network, while the edges in E (t) model information flow, i.e., (v j , vk ) ∈ E (t) if and only if agent k receives information from j at time t. The adjacency matrix A(t) encodes the graph structure and edge weights, i.e., element ak j (t) ≥ α for some α > 0 if and only if (v j , vk ) ∈ E (t) and ak j (t) = 0 otherwise. In the following A(t) is assumed to be piecewise continuous and bounded. The Laplacian matrix of G (t) is defined as L(t) = diag(A(t)1N ) − A(t). We adopt the following definitions for graph connectivity. The graph G (t) is connected at time t if there exists a vertex vk such that there is a path consisting of consecutive edges in E (t) from vk to every other node. The graph G (t) is uniformly connected if there exists a time horizon T > 0 such that the union graph G ([t,t + T ]) = (V , E ([t,t + T ]), A([t,t + T ])) is connected for all t, where E ([t,t +T ]) = {(v j , vk ) ∈ V ×V : a jk ([t,t +T ]) ≥ R α} and A([t,t + T ]) = 1/T tt+T A(τ)dτ.
5128
agent k (1)
This assumption guarantees convergence to consensus in single-integrator networks [3].
x˙k = A(γ(t), λk (t))xk + B(γ(t), λk (t))uk
controller k (2) vk
ζk network G (t) (3)
(1)
yk = Cxk
Fig. 1.
Rg
yk
uk
B. Problem Formulation We consider multi-agent systems consisting of LPV agents. All agents have an identical structure but may depend on nonidentical, local parameters as well as on global parameters. Each agent k = 1, ..., N is described by
Dynamically coupled systems.
Rl
where γ(t) ∈ are global parameters and λk (t) ∈ are local parameters. The parameters are piecewise continuous functions of time which take values in compact sets, i.e., γ(t) ∈ Pγ ⊂ Rg , λk (t) ∈ Pλ ⊂ Rl for all t and k = 1, ..., N. Throughout the paper, we assume continuous parameterdependence of the system matrices. Furthermore xk (t) ∈ Rn is the state, yk (t) ∈ R p is the output and uk (t) ∈ Rq the input of agent k. The parameters are assumed to be available as realtime measurements. In particular, each agent k has access to γ(t) and its own local parameter λk (t). In order to simplify the notation, the argument of γ and λk is suppressed in the following. Note that the output matrix C is assumed to be constant and identical for all agents. It is always possible to obtain a constant output matrix C from parameter-dependent output matrices by appropriate filtering of the outputs yk , cf. [16]. Our goal is to synchronize the outputs yk of all agents in the network via local interaction rules. We consider general dynamic LPV controllers of the form ξ˙k = D(γ, λk )ξk + E(γ, λk )yk + F(γ, λk )vk uk = G(γ, λk )ξk
(2)
ζk = Hξk , i.e., the controller matrices may depend on the locally available parameters γ and λk . The controller state is ξk (t) ∈ Rnc and the controller output to the network is ζk (t) ∈ Rr . We assume that the controllers can exchange relative information with their neighbors over the network, i.e., they are diffusively coupled according to N
vk =
∑ ak j (t)(ζ j − ζk ),
(3)
j=1
where ak j (t) are the elements of the adjacency matrix A(t) corresponding to the graph G (t). The controller of agent k has access to the output yk and communicates over the network as depicted in Fig. 1. Here we consider an output synchronization problem as follows: Output Synchronization Problem (OSP). For a given set of N agents (1), find dynamic LPV controllers (2), (3) such that the outputs yk and ζk synchronize, i.e., for all k, j = 1, ..., N, (yk − y j ) → 0
and
(ζk − ζ j ) → 0
exponentially as t → ∞. Furthermore, the state z(t) ∈ RN(n+nc ) of the closed-loop system has to be bounded for all times t.
We say the agents reach non-trivial output synchronization if the outputs yk (t) synchronize and do not converge to zero for all initial conditions. Trivial synchronization, i.e., yk (t) → 0 as t → ∞, k = 1, ..., N, is of minor interest since it does not require coordination among the subsystems. If the OSP is solved, then the state of the closed-loop system converges exponentially to the synchronous subspace S = {z ∈ RN(n+nc ) : y1 = ... = yN , ζ1 = ... = ζN },
(4)
where z is the stack vector z = [x1T , ξ1T , ..., xNT , ξNT ]T of all agent and controller states. Hence, the subspace S has to be attractive. Note that S is in general not invariant with respect to the closed-loop dynamics. III. M AIN R ESULTS This section presents a constructive solution for the OSP. The OSP is addressed in two steps. First, we show that homogeneous groups of LPV agents with static diffusive couplings synchronize exponentially over uniformly connected graphs. Second, we present dynamic diffusive couplings which solve the OSP for heterogeneous groups of agents. Last, we discuss the necessity of a condition for the existence of such a solution. A. Synchronization in Homogeneous Groups In order to solve the OSP in heterogeneous groups of agents, we adapt a procedure from LTI multi-agent systems [8]. We first solve a synchronization problem in a homogeneous group of LPV agents in this section and then, based thereon, we solve the OSP for heterogeneous groups in the next section. The following result is an extension of [5, Lemma 1] to LPV systems and goes back to the analysis in [3]. Theorem 1. Consider a group of N identical LPV systems with static diffusive couplings, given by N
ζ˙k = S(γ)ζk + ∑ ak j (t)(ζ j − ζk )
(5)
j=1
for k = 1, ..., N, where ζk (t) ∈ Rm and γ(t) ∈ Pγ for all t. Suppose that the underlying graph G (t) is uniformly connected and the state transition matrix ΦS (t, 0) of w˙ = S(γ)w is bounded for all t. Then, (ζk − ζ j ) → 0 exponentially as t → ∞ for all k, j = 1, ..., N, i.e., the systems (5) synchronize.
5129
Proof: We perform a state transformation zk = ΦS (0,t)ζk , where ΦS (t, 0) is the transition matrix of w˙ = S(γ)w, to be understood after fixing the parameter trajectory γ(t) ∈ Pγ . Since for every trajectory γ(t) there exists a constant c > 0 such that kΦS (t, 0)k < c for all t by assumption, this is a Lyapunov transformation, i.e., stability is preserved under the state transformation [17]. From (5) we obtain
c2 c
and the proof is complete with x˜0 = x0 + θ , c2 = 0µ 1 . Since the systems (5) synchronize exponentially by Theorem 1, they are asymptotically unforced. Therefore, by Lemma 1, there exists a w0 ∈ Rm such that for k = 1, ..., N, lim kζk (t) − ΦS (t, 0)w0 k = 0,
t→∞
(7)
! i.e., the synchronous trajectories are generated by the unforced LPV system w˙ = S(γ)w, γ(t) ∈ Pγ . z˙k = −ΦS (0,t)S(γ)ζk + ΦS (0,t) S(γ)ζk + ∑ ak j (t)(ζ j − ζk ) N
j=1
B. Synchronization in Heterogeneous Groups
N
= ΦS (0,t) ∑ ak j (t)(ΦS (t, 0)z j − ΦS (t, 0)zk )
In this section we propose parameter-dependent dynamic controllers of the form (2), which solve the OSP. These controllers can be constructed under the following assumptions.
j=1
N
=
∑ ak j (t)(z j − zk )
j=1
for k = 1, ..., N. We know from [3] that these systems converge to consensus exponentially under uniformly connected topologies G (t). Hence, (ζk −ζ j ) → 0 exponentially as t → ∞ for all k, j = 1, ..., N. The following lemma is useful in order to describe the synchronous trajectories of (5). It characterizes the solution of asymptotically unforced linear time-varying (LTV) systems. A system (6) is called asymptotically unforced if limt→∞ u(t) = 0. Lemma 1. Consider an LTV system of the form x˙ = A(t)x + u
(6)
with x(0) = x0 and A(t) piecewise continuous. Assume that the state transition matrix Φ(t, 0) of x˙ = A(t)x is bounded by kΦ(t, 0)k ≤ c0 for some c0 > 0 and all t, and ku(t)k ≤ c1 e−µt for some c1 , µ > 0. Then, there exists c2 > 0 and for all x0 exists an x˜0 such that kx(t) − Φ(t, 0)x˜0 k ≤ c2 e
−µt
.
In words, the solution x(t) of (6) converges exponentially to a solution of the unforced system x˙ = A(t)x. Proof: The solution x(t) of (6) is given by Z t x(t) = Φ(t, 0) x0 + Φ(0, τ)u(τ)dτ . 0
With the given bounds, we obtain c0 c1 e−µτ as norm bound for the integrand. Hence there exists a vector θ such that
Assumption 1 (stabilizability). There exists a continuously parameter-dependent matrix K(γ, λ ) ∈ Rq×n such that x˙ = [A(γ, λ )+B(γ, λ )K(γ, λ )]x is uniformly asymptotically stable for all parameter trajectories γ(t) ∈ Pγ , λ (t) ∈ Pλ . Assumption 2 (detectability). There exists a continuously parameter-dependent matrix L(γ, λ ) ∈ Rn×p such that x˙ = [A(γ, λ ) + L(γ, λ )C]x is uniformly asymptotically stable for all parameter trajectories γ(t) ∈ Pγ , λ (t) ∈ Pλ . Note that these assumptions are weaker than quadratic stabilizability and detectability of LPV systems as used in [16]. Assumption 3. There exists an integer m, a matrix Π ∈ Rn×m and continuously parameter-dependent matrices S(γ) ∈ Rm×m and Γ(γ, λ ) ∈ Rq×m such that A(γ, λ )Π + B(γ, λ )Γ(γ, λ ) = ΠS(γ)
for all γ ∈ Pγ , λ ∈ Pλ . Furthermore, the state transition matrix ΦS (t, 0) corresponding to w˙ = S(γ)w is bounded for all t. We propose the following dynamic LPV controllers in order to solve the OSP. Theorem 2. Consider a group of N agents (1) with uniformly connected graph G (t). Suppose that Assumptions 1, 2, 3 are satisfied. Then, the dynamic controllers N
ζ˙k = S(γ)ζk + ∑ ak j (t)(ζ j − ζk ),
Z ∞
Φ(0, τ)u(τ)dτ = θ .
j=1
0
Consequently,
Z t
Z
Φ(0, τ)u(τ)dτ − θ =
0
≤
t
Z ∞
c0 c1
e−µτ dτ =
t
x˙ˆk = A(γ, λk )xˆk + B(γ, λk )uk + L(γ, λk )(yˆk − yk ), ∞
Φ(0, τ)u(τ)dτ
(9)
uk = K(γ, λk )(xˆk − Πζk ) + Γ(γ, λk )ζk , solve the OSP. Furthermore, there exists a w0 ∈ Rm such that
c0 c1 −µt e . µ
Finally, we obtain
(8)
lim kyk (t) −CΠΦS (t, 0)w0 k = 0
t→∞
for k = 1, ..., N. Proof: From Theorem 1 we know that the systems
kx(t) − Φ(t, 0)(x0 + θ )k
Z t
c20 c1 −µt
= Φ(t, 0) Φ(0, τ)u(τ)dτ − θ
≤ µ e 0
N
ζ˙k = S(γ)ζk + ∑ ak j (t)(ζ j − ζk ) j=1
5130
(10)
synchronize, i.e., (ζk − ζ j ) → 0 exponentially as t → ∞ for all k, j = 1, ..., N. We consider new state variables εk = xk − Πζk and ek = xk − xˆk . From (1) and (9) we obtain e˙k = [A(γ, λk ) + L(γ, λk )C] ek .
(11)
By Assumption 2, (11) is uniformly asymptotically stable and thus ek → 0 as t → ∞. Furthermore, we have ε˙k = A(γ, λk )xk + B(γ, λk ) [K(γ, λk )(xˆk − Πζk ) + Γ(γ, λk )ζk ] " # N
− Π S(γ)ζk + ∑ ak j (t)(ζ j − ζk ) j=1
= [A(γ, λk ) + B(γ, λk )K(γ, λk )] εk "
#
N
− B(γ, λk )K(γ, λk )ek + Π ∑ ak j (t)(ζ j − ζk ) j=1
+ [A(γ, λk )Π + B(γ, λk )Γ(γ, λk ) − ΠS(γ)] ζk , and with (8), we get ε˙k = [A(γ, λk ) + B(γ, λk )K(γ, λk )] εk − vk ,
(12)
where vk = B(γ, λk )K(γ, λk )ek + Π ∑Nj=1 ak j (t)(ζ j − ζk ). The inputs vk vanish, i.e., vk → 0 as t → ∞, due to the exponential stability of (ζ j − ζk ) and ek . Since ε˙k = [A(γ, λk ) + B(γ, λk )K(γ, λk )] εk is uniformly asymptotically stable by Assumption 1, εk → 0 as t → ∞. Hence, (xk − Πζk ) → 0 as t → ∞. Since uniform asymptotic stability and exponential stability are equivalent for LTV systems [18] the OSP is solved. In fact, the agents reach state synchronization, i.e., (xk − x j ) → 0 as t → ∞, which implies synchronization of the outputs. Furthermore, with (7), we obtain (10). The structure of the dynamic controllers is shown in Fig. 2. The proposed controllers (9) have the form (2), where xk yk
uk xˆk ζk ζk vk
ζk G (t)
Fig. 2. Structure of the dynamically coupled network with dynamic LPV controllers (9) in the dashed box.
ξk = [xˆkT , ζkT ]T is the controller state. Each controller contains a dynamical subsystem which is identical for all controllers in the network. These homogeneous subsystems with states ζk are synchronized over the network via static diffusive couplings. The remainder of the dynamic controller performs output regulation of the agent output yk with respect to ζk .
This idea is adapted from [8], where output synchronization in heterogeneous networks of LTI systems is studied. We would like to discuss the main differences between [8] and the present work. First, the design of controllers (9) is scalable. Due to the identical LPV structure of all agents, the controller matrices K(γ, λ ), L(γ, λ ), Γ(γ, λ ), S(γ), and Π have to be designed only once and can then be applied for each agent. The synthesis problem is thus independent of the number of agents N in the network, whereas in [8] the controllers have to be designed for each agent separately. The design can be performed with standard LPV control methods, cf. [16], [19]. Second, the assumptions in Theorem 2 are only sufficient for the existence of a solution to the OSP, whereas [8] provides a necessary and sufficient condition in terms of an internal model principle. This aspect is discussed in more detail in the following section. Third, LPV systems provide more flexibility in modeling heterogeneous multi-agent systems than LTI systems. In particular, the multi-agent system (1) contains the case of heterogeneous LTI agents by setting the parameter λk to a different constant value for each k = 1, ..., N. The system w˙ = S(γ)w, which is contained in the dynamics of each subsystem in the network, is an LPV system itself, which depends on the global parameter γ. The class of LPV systems with kΦS (t, 0)k ≤ c includes the case of skew symmetric matrices S(γ), which may generate non-stationary sinusoidal trajectories [15]. The case of single-integrator exosystems is included as well with S(γ) = 0. C. Necessary condition for a solution of the OSP In this section, we discuss the necessity of Assumption 3 for the existence of a solution for the OSP. Suppose that the agents (1) are stabilizable and detectable (Assumptions 1 and 2). Then, according to Theorem 2, the existence of a solution to the parameter-dependent regulator equation (8) (Assumption 3) is sufficient for the existence of a solution for the OSP. Equation (8) is a parameter-dependent Sylvester equation. In the context of LTI systems, regulator equations of this form are well known and appear in the classical output regulation problem [12]. It is well known that in output regulation problems for LTV systems [20] and LPV systems [15], the regulator equation becomes a Sylvester-type matrix differential equation. This complicates the constructive solution of the output regulation problem. In [7], [8] it is shown that an internal model principle, expressed in terms of a regulator equation, is necessary and sufficient for output synchronization of LTI systems. However, in the present case the existence of a non-trivial solution for the parameter-dependent regulator equation (8) is not guaranteed even if the OSP is solved. In the following, we show that there exists a solution for (8) if an additional invariance condition is satisfied. Equation (8) admits the trivial solution Π = 0, Γ = 0. However, this solution is of minor interest since it corresponds to trivial synchronization according to (10).
5131
The closed-loop system consisting of agents (1) and dynamic controllers (2) can be written compactly as z˙ = Acl (γ, λ1 , ..., λN )z + Bcl (γ, λ1 , ..., λN )v y = Ccl z ζ = Hcl z v = [L(t) ⊗ Ir ] ζ . The matrices Acl , Bcl , Ccl , Hcl are block diagonal, i.e., Acl = diag(A1 , ..., AN ), Bcl = diag(B1 , ..., BN ), Ccl = diag(C1 , ...,CN ), Hcl = diag(H1 , ..., HN ), with parameterdependent blocks A(γ, λk ) B(γ, λk )G(γ, λk ) Ak (γ, λk ) = , E(γ, λk )C D(γ, λk ) Bk (γ, λk ) = [0T F(γ, λk )T ]T , Ck = [C 0], and Hk = [0 H], respectively. Suppose that the OSP is solved and there exists an invariant subspace S ∗ ⊆ S with m = dim(S ∗ ). Suppose the state z(t) of the closed-loop system converges exponentially to S ∗ as t → ∞. For a definition of invariant subspaces with respect to LPV systems, the reader is referred to [21]. Then, there exists a matrix Ψ ∈ RN(n+nc )×m and a continuously parameter-dependent matrix S(γ, λ1 , ..., λN ) ∈ Rm×m such that Acl (γ, λ1 , ..., λN )Ψ = ΨS(γ, λ1 , ..., λN ),
(13)
where S ∗ = im(Ψ), cf. [22] for the LTI case. Ψ can T T T T be written as Ψ = [ΠT according to the 1 , Σ1 , ..., ΠN , ΣN ] partition of z into agent and controller states, i.e., Πk ∈ Rn×m and Σk ∈ Rnc ×m for k = 1, ..., N. Due to the block-diagonal structure of Acl (γ, λ1 , ..., λN ), we obtain from (13) the set of equations A(γ, λk ) B(γ, λk )G(γ, λk ) Πk Πk = S(γ, λ1 , ..., λN ) E(γ, λk )C D(γ, λk ) Σk Σk for k = 1, ..., N. Every pair Πk , Σk solves the equation for all points in Pλ . Since all local parameters take values in the same set Pλ , we solve in fact the same equation N times, i.e., there exist Π and Σ such that A(γ, λk ) B(γ, λk )G(γ, λk ) Π Π = S(γ, λ1 , ..., λN ) E(γ, λk )C D(γ, λk ) Σ Σ (14) for all γ ∈ Pγ , λk ∈ Pλ , k = 1, ..., N. Assume now that γ and λk are fixed for some k ∈ {1, ..., N} and let λ j , j 6= k, vary in Pλ . Then the left-hand side of (14) is constant. Note that [ΠT , ΣT ]T has full column rank. Since (14) has to be satisfied for all parameter combinations, the right-hand side has to be independent of λ j , j 6= k. Otherwise the right-hand side would not be constant for varying λ j and (14) would be violated. By fixing a different k and the same argument, we conclude that the matrix S(γ, λ1 , ..., λN ) has to be independent of all local parameters λk , k = 1, ..., N. Finally, by choosing Γ(γ, λ ) = G(γ, λ )Σ, we conclude that there exist matrices Π, Γ(γ, λ ), and S(γ), such that A(γ, λ )Π + B(γ, λ )Γ(γ, λ ) = ΠS(γ)
for all γ ∈ Pγ , λ ∈ Pλ . Since the trajectories in S ∗ stay bounded by assumption, it follows that the transition matrix ΦS (t, 0) of the dynamics restricted to S ∗ , i.e., w˙ = S(γ)w, has to be bounded for all t. This allows the following conclusion. Remark 1. Suppose the OSP is solved. Suppose there exists an invariant subspace S ∗ ⊆ S and the state z(t) of the closed-loop system converges exponentially to S ∗ as t → ∞. Then, Assumption 3 is satisfied. However, as already mentioned, Assumption 3 may not be satisfied in general. A formulation of the general necessary condition for output synchronization of LPV systems along the lines of [15], [20] is beyond the scope of the present paper. IV. E XAMPLE In this section, the main results are illustrated by a numerical example. We consider a network of 5 agents with dynamics 0 2.5(1 + γ) 1 0 0 −2.5(1 + γ) 0 0 0 0 x + u , x˙k = 0 0 1 1 k 0 k 3 − 0.2λk 0 −λk 3λk 1 1 0 0 0 yk = x, 0 1 0 0 k for k = 1, ..., 5, where the time-varying parameters are scalar and satisfy γ(t) ∈ [−0.5, 0.5] and λk (t) ∈ [−2, 2] for all t. The graph G (t) has 5 nodes and switches between the edge sets E1 = {(1, 2), (2, 3), (3, 1)}, E2 = {(1, 3), (3, 5), (5, 1)}, E3 = {(3, 4), (4, 5), (5, 3)} periodically every 2.5 time units. G (t) is uniformly connected over the time horizon T = 7.5. The system matrix A(γ, λk ) is affine in the scalar parameters γ and λk and can thus be written as A(γ, λk ) = A0 + γA1 + λk A2 for some constant matrices A0 , A1 , A2 . Hence, (8) becomes (A0 + γA1 + λ A2 )Π + BΓ(γ, λ ) = ΠS(γ). With an affine ansatz Γ(γ, λ ) = Γ0 + γΓ1 + λ Γ2 and S(γ) = S0 + γS1 , and since 0 ∈ Pγ , 0 ∈ Pλ , we obtain the equations A0 Π + BΓ0 = ΠS0 , A1 Π + BΓ1 = ΠS1 , A2 Π + BΓ2 = 0. (15) Thus, the problem of finding a solution of (8) in the present example means to find a solution of the equations (15) with a common Π. Here, a possible choice for S0 , S1 is 0 2.5 S0 = S1 = . −2.5 0 The system w˙ = (S0 + γS1 )w generates non-stationary sinusoidal trajectories. The following solution of the equations above is found: 1 0 Γ0 = −3 0 , 0 1 Γ1 = 0 0 , Π= 0 0 , Γ2 = 0.2 0 . 0 0
5132
Since the agents are polytopic LPV systems [16], the design of suitable K(γ, λ ) and L(γ, λ ) can be formulated as SDP and solved efficiently. We obtain dynamic, parameter-dependent controllers (2). The global and non-identical local parameters vary periodically between their maximal and minimal values. The simulation results in Fig. 3 and 4 show the evolution of the controller and agent outputs ζk and yk , k = 1, ..., 5, respectively. The controller outputs ζk converge as expected from Theorem 1 to a non-stationary sinusoidal signal with frequency ω(t) = 2.5(1 + γ(t)) generated by w˙ = S(γ)w for some w0 ∈ R2 . The agent outputs yk are regulated to ζk , i.e., they synchronize as well as expected from Theorem 2.
(1)
ζk (t)
1 0 −1 0
5
10
15
20
25
30
0
5
10
15
20
25
30
(2)
ζk (t)
1 0 −1
Fig. 3. Evolution of the first component (top) and second component (bottom) of the controller outputs ζk (t), k = 1, ..., N.
(1)
yk (t)
1 0 −1 0
5
10
15
20
25
30
0
5
10
15
20
25
30
(2)
yk (t)
1 0 −1
Fig. 4. Evolution of the first component (top) and second component (bottom) of the agent outputs yk (t), k = 1, ..., N.
V. C ONCLUSION We have formulated an output synchronization problem (OSP) for a group of LPV agents with heterogeneous time-varying parameters. The agents are interconnected by dynamic diffusive couplings over an uniformly connected graph. We have shown that homogeneous groups of LPV systems with static diffusive couplings and uniformly connected graph synchronize exponentially. Based on this result,
we have proposed dynamic LPV controllers which solve the considered OSP. The design procedure is scalable to arbitrarily large networks, since the controller has to be designed for only one agent and can then be applied to each agent in the network. We have shown that a parameterdependent regulator equation is a sufficient condition for the existence of a solution for the OSP. Under additional assumptions, this condition is also necessary. R EFERENCES [1] W. Ren and R. W. Beard, Distributed consensus in multi-vehicle cooperative control: Theory and applications. Springer, 2008. [2] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. Automat. Control, vol. 49, no. 9, pp. 1520–1533, 2004. [3] L. Moreau, “Stability of continuous-time distributed consensus algorithms,” 2004, pp. 3998–4003. [4] S. E. Tuna, “Conditions for synchronizability in arrays of coupled linear systems,” IEEE Trans. Automat. Control, vol. 54, no. 10, pp. 2416–2420, 2009. [5] L. Scardovi and R. Sepulchre, “Synchronization in networks of identical linear systems,” Automatica, vol. 45, no. 11, pp. 2557–2562, 2009. [6] C. Ma and J. Zhang, “Necessary and sufficient conditions for consensusability of linear multi-agent systems,” IEEE Trans. Automat. Control, vol. 55, no. 5, pp. 1263–1268, 2010. [7] P. Wieland, “An internal model principle for consensus in heterogeneous linear multi-agent systems,” in Proc. 1st IFAC Workshop on Estimation and Control of Networked Systems, 2009, pp. 7–12. [8] P. Wieland, R. Sepulchre, and F. Allg¨ower, “An internal model principle is necessary and sufficient for linear output synchronization,” Automatica, vol. 47, no. 5, pp. 1068–1074, 2011. [9] H. Kim, H. Shim, and J. H. Seo, “Output consensus of heterogeneous uncertain linear multi-agent systems,” IEEE Trans. Automat. Control, vol. 56, no. 1, pp. 200–206, 2011. [10] K. D. Listmann, A. Wahrburg, J. Strubel, J. Adamy, and U. Konigorski, “Partial-state synchronization of linear heterogeneous multi-agent systems,” in Proc. 50th IEEE Conf. Decision and Control (CDC), 2011, pp. 3440–3445. [11] J. Lunze, “An internal-model principle for the synchronisation of autonomous agents with individual dynamics,” in Proc. 50th IEEE Conf. Decision and Control (CDC), 2011, pp. 2106–2111. [12] B. A. Francis and W. Wonham, “The internal model principle of control theory,” Automatica, vol. 12, no. 5, pp. 457–465, 1976. [13] G. S. Seyboth, G. S. Schmidt, and F. Allg¨ower, “Cooperative control of linear parameter-varying systems,” in Proc. American Control Conference (ACC), 2012, pp. 2407–2412. [14] V. Ugrinovskii, “Gain-scheduled synchronization of uncertain parameter varying systems via relative H∞ consensus,” in Proc. 50th IEEE Conf. Decision and Control (CDC), 2011, pp. 4251–4256. [15] H. K¨oroglu and C. W. Scherer, “Scheduled control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies,” Automatica, vol. 47, no. 3, pp. 504–514, 2011. [16] P. Apkarian, “Self-scheduled H∞ control of linear parameter-varying systems: A design example,” Automatica, vol. 31, no. 9, pp. 1251– 1261, 1995. [17] W. J. Rugh, Linear system theory, 2nd ed. Prentice Hall, 1996. [18] F. M. Callier and C. A. Desoer, Linear system theory. Springer, 1991. [19] F. Wu, X. H. Yang, A. Packard, and G. S. Becker, “Induced L2 norm control for LPV systems with bounded parameter variation rates,” International Journal of Robust and Nonlinear Control, vol. 6, no. 910, pp. 983–998, 1996. [20] A. Ichikawa and H. Katayama, “Output regulation of time-varying systems,” Systems & Control Letters, vol. 55, no. 12, pp. 999–1005, 2006. [21] G. Balas, J. Bokor, and Z. Szabo, “Invariant subspaces for LPV systems and their applications,” IEEE Trans. Automat. Control, vol. 48, no. 11, pp. 2065–2069, 2003. [22] H. W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie. Springer, 1985.
5133
Output Synchronization of Linear Parameter-varying Systems via Dynamic Couplings Georg S. Seyboth∗ , Gerd S. Schmidt, Frank Allg¨ower Abstract—We address the output synchronization problem for groups of linear parameter-varying (LPV) agents. The agents are assumed to have identical LPV structure depending on different local parameters which make the group heterogeneous. First, we show that homogeneous groups of LPV systems with static diffusive couplings synchronize over uniformly connected graphs. Then, based thereon, we propose a constructive and scalable solution to the output synchronization problem for groups of LPV systems with heterogeneous parameters. The results are illustrated in a numerical example.
I. I NTRODUCTION In recent years, distributed and cooperative control became one of the major research areas in control theory. Specially consensus and synchronization problems in multiagent systems receive considerable attention due to the wide application range, e.g., motion coordination for autonomous vehicles [1]. Initially, consensus problems in single-integrator networks were investigated [2], and it was shown that consensus can be guaranteed under very mild assumptions on the network connectivity [3]. One development in the consensus and synchronization literature was the successive shift to agent dynamics with higher complexity. Networks of identical linear time-invariant (LTI) systems are addressed in [4]– [6]. Recently synchronization problems in heterogeneous groups of LTI agents are investigated [7]–[11]. It has been discovered in [7], [8] that there is a close connection between the synchronization problem in dynamical networks and the classical output regulation problem [12]. In this paper we consider heterogeneous multi-agent systems which consist of linear parameter-varying (LPV) systems. This system class has hardly been studied in the context of cooperative control [13], [14] even though it is suitable to describe a variety of physical systems such as airplanes, satellites, wind turbines and many more. Linear systems with constant uncertain parameters are addressed in [9]. In [14], a synchronization problem for parameter-varying systems is addressed and solved under noisy measurements via dissipativity techniques. In contrast to the present work, all parameters are globally available to the controllers of all subsystems. In the present work the parameters are assumed to be timevarying and available as real-time measurements. The LPV ∗ Corresponding author. The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart and within the Priority Programme 1305 “Control Theory of Digitally Networked Dynamical Systems”. All authors are with the Institute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany. {georg.seyboth, gerd.schmidt, allgower}@ist.uni-stuttgart.de.
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
systems may depend on parameters which influence all agents (global parameters) and on parameters which influence only individual agents (local parameters). The local parameters can be different for each agent, which causes the heterogeneity in the multi-agent system. Such multi-agent systems may for example describe a formation of UAVs in which the global parameter is air pressure and local parameters are quantities such as fuel mass or angle of attack. The main contribution of this paper is a constructive solution of the output synchronization problem of heterogeneous LPV systems. We propose distributed dynamic LPV controllers which solve the synchronization problem. The design procedure is applicable to multi-agent systems with arbitrarily many agents. Furthermore discuss the connection of the present problem to output regulation of LPV systems [15]. The remainder of this paper is organized as follows. Section II contains some preliminaries and the formulation of the output synchronization problem. Section III presents our main results including the constructive solution to the synchronization problem. The theoretical results are illustrated by an example in Section IV and Section V concludes the paper. II. P RELIMINARIES AND P ROBLEM F ORMULATION A. Communication graphs The communication topology of the multi-agent system is modeled by a directed, weighted, and time-varying graph G (t) = (V , E (t), A(t)). We use the same notation as in [5], [8] which is briefly summarized here for completeness. The graph G (t) consists of the vertex set V = {v1 , ..., vN }, N ∈ N, the edge set E (t) ⊂ V × V and the time-varying adjacency matrix A(t). Vertex vk corresponds to agent k in the network, while the edges in E (t) model information flow, i.e., (v j , vk ) ∈ E (t) if and only if agent k receives information from j at time t. The adjacency matrix A(t) encodes the graph structure and edge weights, i.e., element ak j (t) ≥ α for some α > 0 if and only if (v j , vk ) ∈ E (t) and ak j (t) = 0 otherwise. In the following A(t) is assumed to be piecewise continuous and bounded. The Laplacian matrix of G (t) is defined as L(t) = diag(A(t)1N ) − A(t). We adopt the following definitions for graph connectivity. The graph G (t) is connected at time t if there exists a vertex vk such that there is a path consisting of consecutive edges in E (t) from vk to every other node. The graph G (t) is uniformly connected if there exists a time horizon T > 0 such that the union graph G ([t,t + T ]) = (V , E ([t,t + T ]), A([t,t + T ])) is connected for all t, where E ([t,t +T ]) = {(v j , vk ) ∈ V ×V : a jk ([t,t +T ]) ≥ R α} and A([t,t + T ]) = 1/T tt+T A(τ)dτ.
5128
agent k (1)
This assumption guarantees convergence to consensus in single-integrator networks [3].
x˙k = A(γ(t), λk (t))xk + B(γ(t), λk (t))uk
controller k (2) vk
ζk network G (t) (3)
(1)
yk = Cxk
Fig. 1.
Rg
yk
uk
B. Problem Formulation We consider multi-agent systems consisting of LPV agents. All agents have an identical structure but may depend on nonidentical, local parameters as well as on global parameters. Each agent k = 1, ..., N is described by
Dynamically coupled systems.
Rl
where γ(t) ∈ are global parameters and λk (t) ∈ are local parameters. The parameters are piecewise continuous functions of time which take values in compact sets, i.e., γ(t) ∈ Pγ ⊂ Rg , λk (t) ∈ Pλ ⊂ Rl for all t and k = 1, ..., N. Throughout the paper, we assume continuous parameterdependence of the system matrices. Furthermore xk (t) ∈ Rn is the state, yk (t) ∈ R p is the output and uk (t) ∈ Rq the input of agent k. The parameters are assumed to be available as realtime measurements. In particular, each agent k has access to γ(t) and its own local parameter λk (t). In order to simplify the notation, the argument of γ and λk is suppressed in the following. Note that the output matrix C is assumed to be constant and identical for all agents. It is always possible to obtain a constant output matrix C from parameter-dependent output matrices by appropriate filtering of the outputs yk , cf. [16]. Our goal is to synchronize the outputs yk of all agents in the network via local interaction rules. We consider general dynamic LPV controllers of the form ξ˙k = D(γ, λk )ξk + E(γ, λk )yk + F(γ, λk )vk uk = G(γ, λk )ξk
(2)
ζk = Hξk , i.e., the controller matrices may depend on the locally available parameters γ and λk . The controller state is ξk (t) ∈ Rnc and the controller output to the network is ζk (t) ∈ Rr . We assume that the controllers can exchange relative information with their neighbors over the network, i.e., they are diffusively coupled according to N
vk =
∑ ak j (t)(ζ j − ζk ),
(3)
j=1
where ak j (t) are the elements of the adjacency matrix A(t) corresponding to the graph G (t). The controller of agent k has access to the output yk and communicates over the network as depicted in Fig. 1. Here we consider an output synchronization problem as follows: Output Synchronization Problem (OSP). For a given set of N agents (1), find dynamic LPV controllers (2), (3) such that the outputs yk and ζk synchronize, i.e., for all k, j = 1, ..., N, (yk − y j ) → 0
and
(ζk − ζ j ) → 0
exponentially as t → ∞. Furthermore, the state z(t) ∈ RN(n+nc ) of the closed-loop system has to be bounded for all times t.
We say the agents reach non-trivial output synchronization if the outputs yk (t) synchronize and do not converge to zero for all initial conditions. Trivial synchronization, i.e., yk (t) → 0 as t → ∞, k = 1, ..., N, is of minor interest since it does not require coordination among the subsystems. If the OSP is solved, then the state of the closed-loop system converges exponentially to the synchronous subspace S = {z ∈ RN(n+nc ) : y1 = ... = yN , ζ1 = ... = ζN },
(4)
where z is the stack vector z = [x1T , ξ1T , ..., xNT , ξNT ]T of all agent and controller states. Hence, the subspace S has to be attractive. Note that S is in general not invariant with respect to the closed-loop dynamics. III. M AIN R ESULTS This section presents a constructive solution for the OSP. The OSP is addressed in two steps. First, we show that homogeneous groups of LPV agents with static diffusive couplings synchronize exponentially over uniformly connected graphs. Second, we present dynamic diffusive couplings which solve the OSP for heterogeneous groups of agents. Last, we discuss the necessity of a condition for the existence of such a solution. A. Synchronization in Homogeneous Groups In order to solve the OSP in heterogeneous groups of agents, we adapt a procedure from LTI multi-agent systems [8]. We first solve a synchronization problem in a homogeneous group of LPV agents in this section and then, based thereon, we solve the OSP for heterogeneous groups in the next section. The following result is an extension of [5, Lemma 1] to LPV systems and goes back to the analysis in [3]. Theorem 1. Consider a group of N identical LPV systems with static diffusive couplings, given by N
ζ˙k = S(γ)ζk + ∑ ak j (t)(ζ j − ζk )
(5)
j=1
for k = 1, ..., N, where ζk (t) ∈ Rm and γ(t) ∈ Pγ for all t. Suppose that the underlying graph G (t) is uniformly connected and the state transition matrix ΦS (t, 0) of w˙ = S(γ)w is bounded for all t. Then, (ζk − ζ j ) → 0 exponentially as t → ∞ for all k, j = 1, ..., N, i.e., the systems (5) synchronize.
5129
Proof: We perform a state transformation zk = ΦS (0,t)ζk , where ΦS (t, 0) is the transition matrix of w˙ = S(γ)w, to be understood after fixing the parameter trajectory γ(t) ∈ Pγ . Since for every trajectory γ(t) there exists a constant c > 0 such that kΦS (t, 0)k < c for all t by assumption, this is a Lyapunov transformation, i.e., stability is preserved under the state transformation [17]. From (5) we obtain
c2 c
and the proof is complete with x˜0 = x0 + θ , c2 = 0µ 1 . Since the systems (5) synchronize exponentially by Theorem 1, they are asymptotically unforced. Therefore, by Lemma 1, there exists a w0 ∈ Rm such that for k = 1, ..., N, lim kζk (t) − ΦS (t, 0)w0 k = 0,
t→∞
(7)
! i.e., the synchronous trajectories are generated by the unforced LPV system w˙ = S(γ)w, γ(t) ∈ Pγ . z˙k = −ΦS (0,t)S(γ)ζk + ΦS (0,t) S(γ)ζk + ∑ ak j (t)(ζ j − ζk ) N
j=1
B. Synchronization in Heterogeneous Groups
N
= ΦS (0,t) ∑ ak j (t)(ΦS (t, 0)z j − ΦS (t, 0)zk )
In this section we propose parameter-dependent dynamic controllers of the form (2), which solve the OSP. These controllers can be constructed under the following assumptions.
j=1
N
=
∑ ak j (t)(z j − zk )
j=1
for k = 1, ..., N. We know from [3] that these systems converge to consensus exponentially under uniformly connected topologies G (t). Hence, (ζk −ζ j ) → 0 exponentially as t → ∞ for all k, j = 1, ..., N. The following lemma is useful in order to describe the synchronous trajectories of (5). It characterizes the solution of asymptotically unforced linear time-varying (LTV) systems. A system (6) is called asymptotically unforced if limt→∞ u(t) = 0. Lemma 1. Consider an LTV system of the form x˙ = A(t)x + u
(6)
with x(0) = x0 and A(t) piecewise continuous. Assume that the state transition matrix Φ(t, 0) of x˙ = A(t)x is bounded by kΦ(t, 0)k ≤ c0 for some c0 > 0 and all t, and ku(t)k ≤ c1 e−µt for some c1 , µ > 0. Then, there exists c2 > 0 and for all x0 exists an x˜0 such that kx(t) − Φ(t, 0)x˜0 k ≤ c2 e
−µt
.
In words, the solution x(t) of (6) converges exponentially to a solution of the unforced system x˙ = A(t)x. Proof: The solution x(t) of (6) is given by Z t x(t) = Φ(t, 0) x0 + Φ(0, τ)u(τ)dτ . 0
With the given bounds, we obtain c0 c1 e−µτ as norm bound for the integrand. Hence there exists a vector θ such that
Assumption 1 (stabilizability). There exists a continuously parameter-dependent matrix K(γ, λ ) ∈ Rq×n such that x˙ = [A(γ, λ )+B(γ, λ )K(γ, λ )]x is uniformly asymptotically stable for all parameter trajectories γ(t) ∈ Pγ , λ (t) ∈ Pλ . Assumption 2 (detectability). There exists a continuously parameter-dependent matrix L(γ, λ ) ∈ Rn×p such that x˙ = [A(γ, λ ) + L(γ, λ )C]x is uniformly asymptotically stable for all parameter trajectories γ(t) ∈ Pγ , λ (t) ∈ Pλ . Note that these assumptions are weaker than quadratic stabilizability and detectability of LPV systems as used in [16]. Assumption 3. There exists an integer m, a matrix Π ∈ Rn×m and continuously parameter-dependent matrices S(γ) ∈ Rm×m and Γ(γ, λ ) ∈ Rq×m such that A(γ, λ )Π + B(γ, λ )Γ(γ, λ ) = ΠS(γ)
for all γ ∈ Pγ , λ ∈ Pλ . Furthermore, the state transition matrix ΦS (t, 0) corresponding to w˙ = S(γ)w is bounded for all t. We propose the following dynamic LPV controllers in order to solve the OSP. Theorem 2. Consider a group of N agents (1) with uniformly connected graph G (t). Suppose that Assumptions 1, 2, 3 are satisfied. Then, the dynamic controllers N
ζ˙k = S(γ)ζk + ∑ ak j (t)(ζ j − ζk ),
Z ∞
Φ(0, τ)u(τ)dτ = θ .
j=1
0
Consequently,
Z t
Z
Φ(0, τ)u(τ)dτ − θ =
0
≤
t
Z ∞
c0 c1
e−µτ dτ =
t
x˙ˆk = A(γ, λk )xˆk + B(γ, λk )uk + L(γ, λk )(yˆk − yk ), ∞
Φ(0, τ)u(τ)dτ
(9)
uk = K(γ, λk )(xˆk − Πζk ) + Γ(γ, λk )ζk , solve the OSP. Furthermore, there exists a w0 ∈ Rm such that
c0 c1 −µt e . µ
Finally, we obtain
(8)
lim kyk (t) −CΠΦS (t, 0)w0 k = 0
t→∞
for k = 1, ..., N. Proof: From Theorem 1 we know that the systems
kx(t) − Φ(t, 0)(x0 + θ )k
Z t
c20 c1 −µt
= Φ(t, 0) Φ(0, τ)u(τ)dτ − θ
≤ µ e 0
N
ζ˙k = S(γ)ζk + ∑ ak j (t)(ζ j − ζk ) j=1
5130
(10)
synchronize, i.e., (ζk − ζ j ) → 0 exponentially as t → ∞ for all k, j = 1, ..., N. We consider new state variables εk = xk − Πζk and ek = xk − xˆk . From (1) and (9) we obtain e˙k = [A(γ, λk ) + L(γ, λk )C] ek .
(11)
By Assumption 2, (11) is uniformly asymptotically stable and thus ek → 0 as t → ∞. Furthermore, we have ε˙k = A(γ, λk )xk + B(γ, λk ) [K(γ, λk )(xˆk − Πζk ) + Γ(γ, λk )ζk ] " # N
− Π S(γ)ζk + ∑ ak j (t)(ζ j − ζk ) j=1
= [A(γ, λk ) + B(γ, λk )K(γ, λk )] εk "
#
N
− B(γ, λk )K(γ, λk )ek + Π ∑ ak j (t)(ζ j − ζk ) j=1
+ [A(γ, λk )Π + B(γ, λk )Γ(γ, λk ) − ΠS(γ)] ζk , and with (8), we get ε˙k = [A(γ, λk ) + B(γ, λk )K(γ, λk )] εk − vk ,
(12)
where vk = B(γ, λk )K(γ, λk )ek + Π ∑Nj=1 ak j (t)(ζ j − ζk ). The inputs vk vanish, i.e., vk → 0 as t → ∞, due to the exponential stability of (ζ j − ζk ) and ek . Since ε˙k = [A(γ, λk ) + B(γ, λk )K(γ, λk )] εk is uniformly asymptotically stable by Assumption 1, εk → 0 as t → ∞. Hence, (xk − Πζk ) → 0 as t → ∞. Since uniform asymptotic stability and exponential stability are equivalent for LTV systems [18] the OSP is solved. In fact, the agents reach state synchronization, i.e., (xk − x j ) → 0 as t → ∞, which implies synchronization of the outputs. Furthermore, with (7), we obtain (10). The structure of the dynamic controllers is shown in Fig. 2. The proposed controllers (9) have the form (2), where xk yk
uk xˆk ζk ζk vk
ζk G (t)
Fig. 2. Structure of the dynamically coupled network with dynamic LPV controllers (9) in the dashed box.
ξk = [xˆkT , ζkT ]T is the controller state. Each controller contains a dynamical subsystem which is identical for all controllers in the network. These homogeneous subsystems with states ζk are synchronized over the network via static diffusive couplings. The remainder of the dynamic controller performs output regulation of the agent output yk with respect to ζk .
This idea is adapted from [8], where output synchronization in heterogeneous networks of LTI systems is studied. We would like to discuss the main differences between [8] and the present work. First, the design of controllers (9) is scalable. Due to the identical LPV structure of all agents, the controller matrices K(γ, λ ), L(γ, λ ), Γ(γ, λ ), S(γ), and Π have to be designed only once and can then be applied for each agent. The synthesis problem is thus independent of the number of agents N in the network, whereas in [8] the controllers have to be designed for each agent separately. The design can be performed with standard LPV control methods, cf. [16], [19]. Second, the assumptions in Theorem 2 are only sufficient for the existence of a solution to the OSP, whereas [8] provides a necessary and sufficient condition in terms of an internal model principle. This aspect is discussed in more detail in the following section. Third, LPV systems provide more flexibility in modeling heterogeneous multi-agent systems than LTI systems. In particular, the multi-agent system (1) contains the case of heterogeneous LTI agents by setting the parameter λk to a different constant value for each k = 1, ..., N. The system w˙ = S(γ)w, which is contained in the dynamics of each subsystem in the network, is an LPV system itself, which depends on the global parameter γ. The class of LPV systems with kΦS (t, 0)k ≤ c includes the case of skew symmetric matrices S(γ), which may generate non-stationary sinusoidal trajectories [15]. The case of single-integrator exosystems is included as well with S(γ) = 0. C. Necessary condition for a solution of the OSP In this section, we discuss the necessity of Assumption 3 for the existence of a solution for the OSP. Suppose that the agents (1) are stabilizable and detectable (Assumptions 1 and 2). Then, according to Theorem 2, the existence of a solution to the parameter-dependent regulator equation (8) (Assumption 3) is sufficient for the existence of a solution for the OSP. Equation (8) is a parameter-dependent Sylvester equation. In the context of LTI systems, regulator equations of this form are well known and appear in the classical output regulation problem [12]. It is well known that in output regulation problems for LTV systems [20] and LPV systems [15], the regulator equation becomes a Sylvester-type matrix differential equation. This complicates the constructive solution of the output regulation problem. In [7], [8] it is shown that an internal model principle, expressed in terms of a regulator equation, is necessary and sufficient for output synchronization of LTI systems. However, in the present case the existence of a non-trivial solution for the parameter-dependent regulator equation (8) is not guaranteed even if the OSP is solved. In the following, we show that there exists a solution for (8) if an additional invariance condition is satisfied. Equation (8) admits the trivial solution Π = 0, Γ = 0. However, this solution is of minor interest since it corresponds to trivial synchronization according to (10).
5131
The closed-loop system consisting of agents (1) and dynamic controllers (2) can be written compactly as z˙ = Acl (γ, λ1 , ..., λN )z + Bcl (γ, λ1 , ..., λN )v y = Ccl z ζ = Hcl z v = [L(t) ⊗ Ir ] ζ . The matrices Acl , Bcl , Ccl , Hcl are block diagonal, i.e., Acl = diag(A1 , ..., AN ), Bcl = diag(B1 , ..., BN ), Ccl = diag(C1 , ...,CN ), Hcl = diag(H1 , ..., HN ), with parameterdependent blocks A(γ, λk ) B(γ, λk )G(γ, λk ) Ak (γ, λk ) = , E(γ, λk )C D(γ, λk ) Bk (γ, λk ) = [0T F(γ, λk )T ]T , Ck = [C 0], and Hk = [0 H], respectively. Suppose that the OSP is solved and there exists an invariant subspace S ∗ ⊆ S with m = dim(S ∗ ). Suppose the state z(t) of the closed-loop system converges exponentially to S ∗ as t → ∞. For a definition of invariant subspaces with respect to LPV systems, the reader is referred to [21]. Then, there exists a matrix Ψ ∈ RN(n+nc )×m and a continuously parameter-dependent matrix S(γ, λ1 , ..., λN ) ∈ Rm×m such that Acl (γ, λ1 , ..., λN )Ψ = ΨS(γ, λ1 , ..., λN ),
(13)
where S ∗ = im(Ψ), cf. [22] for the LTI case. Ψ can T T T T be written as Ψ = [ΠT according to the 1 , Σ1 , ..., ΠN , ΣN ] partition of z into agent and controller states, i.e., Πk ∈ Rn×m and Σk ∈ Rnc ×m for k = 1, ..., N. Due to the block-diagonal structure of Acl (γ, λ1 , ..., λN ), we obtain from (13) the set of equations A(γ, λk ) B(γ, λk )G(γ, λk ) Πk Πk = S(γ, λ1 , ..., λN ) E(γ, λk )C D(γ, λk ) Σk Σk for k = 1, ..., N. Every pair Πk , Σk solves the equation for all points in Pλ . Since all local parameters take values in the same set Pλ , we solve in fact the same equation N times, i.e., there exist Π and Σ such that A(γ, λk ) B(γ, λk )G(γ, λk ) Π Π = S(γ, λ1 , ..., λN ) E(γ, λk )C D(γ, λk ) Σ Σ (14) for all γ ∈ Pγ , λk ∈ Pλ , k = 1, ..., N. Assume now that γ and λk are fixed for some k ∈ {1, ..., N} and let λ j , j 6= k, vary in Pλ . Then the left-hand side of (14) is constant. Note that [ΠT , ΣT ]T has full column rank. Since (14) has to be satisfied for all parameter combinations, the right-hand side has to be independent of λ j , j 6= k. Otherwise the right-hand side would not be constant for varying λ j and (14) would be violated. By fixing a different k and the same argument, we conclude that the matrix S(γ, λ1 , ..., λN ) has to be independent of all local parameters λk , k = 1, ..., N. Finally, by choosing Γ(γ, λ ) = G(γ, λ )Σ, we conclude that there exist matrices Π, Γ(γ, λ ), and S(γ), such that A(γ, λ )Π + B(γ, λ )Γ(γ, λ ) = ΠS(γ)
for all γ ∈ Pγ , λ ∈ Pλ . Since the trajectories in S ∗ stay bounded by assumption, it follows that the transition matrix ΦS (t, 0) of the dynamics restricted to S ∗ , i.e., w˙ = S(γ)w, has to be bounded for all t. This allows the following conclusion. Remark 1. Suppose the OSP is solved. Suppose there exists an invariant subspace S ∗ ⊆ S and the state z(t) of the closed-loop system converges exponentially to S ∗ as t → ∞. Then, Assumption 3 is satisfied. However, as already mentioned, Assumption 3 may not be satisfied in general. A formulation of the general necessary condition for output synchronization of LPV systems along the lines of [15], [20] is beyond the scope of the present paper. IV. E XAMPLE In this section, the main results are illustrated by a numerical example. We consider a network of 5 agents with dynamics 0 2.5(1 + γ) 1 0 0 −2.5(1 + γ) 0 0 0 0 x + u , x˙k = 0 0 1 1 k 0 k 3 − 0.2λk 0 −λk 3λk 1 1 0 0 0 yk = x, 0 1 0 0 k for k = 1, ..., 5, where the time-varying parameters are scalar and satisfy γ(t) ∈ [−0.5, 0.5] and λk (t) ∈ [−2, 2] for all t. The graph G (t) has 5 nodes and switches between the edge sets E1 = {(1, 2), (2, 3), (3, 1)}, E2 = {(1, 3), (3, 5), (5, 1)}, E3 = {(3, 4), (4, 5), (5, 3)} periodically every 2.5 time units. G (t) is uniformly connected over the time horizon T = 7.5. The system matrix A(γ, λk ) is affine in the scalar parameters γ and λk and can thus be written as A(γ, λk ) = A0 + γA1 + λk A2 for some constant matrices A0 , A1 , A2 . Hence, (8) becomes (A0 + γA1 + λ A2 )Π + BΓ(γ, λ ) = ΠS(γ). With an affine ansatz Γ(γ, λ ) = Γ0 + γΓ1 + λ Γ2 and S(γ) = S0 + γS1 , and since 0 ∈ Pγ , 0 ∈ Pλ , we obtain the equations A0 Π + BΓ0 = ΠS0 , A1 Π + BΓ1 = ΠS1 , A2 Π + BΓ2 = 0. (15) Thus, the problem of finding a solution of (8) in the present example means to find a solution of the equations (15) with a common Π. Here, a possible choice for S0 , S1 is 0 2.5 S0 = S1 = . −2.5 0 The system w˙ = (S0 + γS1 )w generates non-stationary sinusoidal trajectories. The following solution of the equations above is found: 1 0 Γ0 = −3 0 , 0 1 Γ1 = 0 0 , Π= 0 0 , Γ2 = 0.2 0 . 0 0
5132
Since the agents are polytopic LPV systems [16], the design of suitable K(γ, λ ) and L(γ, λ ) can be formulated as SDP and solved efficiently. We obtain dynamic, parameter-dependent controllers (2). The global and non-identical local parameters vary periodically between their maximal and minimal values. The simulation results in Fig. 3 and 4 show the evolution of the controller and agent outputs ζk and yk , k = 1, ..., 5, respectively. The controller outputs ζk converge as expected from Theorem 1 to a non-stationary sinusoidal signal with frequency ω(t) = 2.5(1 + γ(t)) generated by w˙ = S(γ)w for some w0 ∈ R2 . The agent outputs yk are regulated to ζk , i.e., they synchronize as well as expected from Theorem 2.
(1)
ζk (t)
1 0 −1 0
5
10
15
20
25
30
0
5
10
15
20
25
30
(2)
ζk (t)
1 0 −1
Fig. 3. Evolution of the first component (top) and second component (bottom) of the controller outputs ζk (t), k = 1, ..., N.
(1)
yk (t)
1 0 −1 0
5
10
15
20
25
30
0
5
10
15
20
25
30
(2)
yk (t)
1 0 −1
Fig. 4. Evolution of the first component (top) and second component (bottom) of the agent outputs yk (t), k = 1, ..., N.
V. C ONCLUSION We have formulated an output synchronization problem (OSP) for a group of LPV agents with heterogeneous time-varying parameters. The agents are interconnected by dynamic diffusive couplings over an uniformly connected graph. We have shown that homogeneous groups of LPV systems with static diffusive couplings and uniformly connected graph synchronize exponentially. Based on this result,
we have proposed dynamic LPV controllers which solve the considered OSP. The design procedure is scalable to arbitrarily large networks, since the controller has to be designed for only one agent and can then be applied to each agent in the network. We have shown that a parameterdependent regulator equation is a sufficient condition for the existence of a solution for the OSP. Under additional assumptions, this condition is also necessary. R EFERENCES [1] W. Ren and R. W. Beard, Distributed consensus in multi-vehicle cooperative control: Theory and applications. Springer, 2008. [2] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. Automat. Control, vol. 49, no. 9, pp. 1520–1533, 2004. [3] L. Moreau, “Stability of continuous-time distributed consensus algorithms,” 2004, pp. 3998–4003. [4] S. E. Tuna, “Conditions for synchronizability in arrays of coupled linear systems,” IEEE Trans. Automat. Control, vol. 54, no. 10, pp. 2416–2420, 2009. [5] L. Scardovi and R. Sepulchre, “Synchronization in networks of identical linear systems,” Automatica, vol. 45, no. 11, pp. 2557–2562, 2009. [6] C. Ma and J. Zhang, “Necessary and sufficient conditions for consensusability of linear multi-agent systems,” IEEE Trans. Automat. Control, vol. 55, no. 5, pp. 1263–1268, 2010. [7] P. Wieland, “An internal model principle for consensus in heterogeneous linear multi-agent systems,” in Proc. 1st IFAC Workshop on Estimation and Control of Networked Systems, 2009, pp. 7–12. [8] P. Wieland, R. Sepulchre, and F. Allg¨ower, “An internal model principle is necessary and sufficient for linear output synchronization,” Automatica, vol. 47, no. 5, pp. 1068–1074, 2011. [9] H. Kim, H. Shim, and J. H. Seo, “Output consensus of heterogeneous uncertain linear multi-agent systems,” IEEE Trans. Automat. Control, vol. 56, no. 1, pp. 200–206, 2011. [10] K. D. Listmann, A. Wahrburg, J. Strubel, J. Adamy, and U. Konigorski, “Partial-state synchronization of linear heterogeneous multi-agent systems,” in Proc. 50th IEEE Conf. Decision and Control (CDC), 2011, pp. 3440–3445. [11] J. Lunze, “An internal-model principle for the synchronisation of autonomous agents with individual dynamics,” in Proc. 50th IEEE Conf. Decision and Control (CDC), 2011, pp. 2106–2111. [12] B. A. Francis and W. Wonham, “The internal model principle of control theory,” Automatica, vol. 12, no. 5, pp. 457–465, 1976. [13] G. S. Seyboth, G. S. Schmidt, and F. Allg¨ower, “Cooperative control of linear parameter-varying systems,” in Proc. American Control Conference (ACC), 2012, pp. 2407–2412. [14] V. Ugrinovskii, “Gain-scheduled synchronization of uncertain parameter varying systems via relative H∞ consensus,” in Proc. 50th IEEE Conf. Decision and Control (CDC), 2011, pp. 4251–4256. [15] H. K¨oroglu and C. W. Scherer, “Scheduled control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies,” Automatica, vol. 47, no. 3, pp. 504–514, 2011. [16] P. Apkarian, “Self-scheduled H∞ control of linear parameter-varying systems: A design example,” Automatica, vol. 31, no. 9, pp. 1251– 1261, 1995. [17] W. J. Rugh, Linear system theory, 2nd ed. Prentice Hall, 1996. [18] F. M. Callier and C. A. Desoer, Linear system theory. Springer, 1991. [19] F. Wu, X. H. Yang, A. Packard, and G. S. Becker, “Induced L2 norm control for LPV systems with bounded parameter variation rates,” International Journal of Robust and Nonlinear Control, vol. 6, no. 910, pp. 983–998, 1996. [20] A. Ichikawa and H. Katayama, “Output regulation of time-varying systems,” Systems & Control Letters, vol. 55, no. 12, pp. 999–1005, 2006. [21] G. Balas, J. Bokor, and Z. Szabo, “Invariant subspaces for LPV systems and their applications,” IEEE Trans. Automat. Control, vol. 48, no. 11, pp. 2065–2069, 2003. [22] H. W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie. Springer, 1985.
5133
