systems with homogeneous linear time-invariant dynamics in the presence of matched or .... ARIA Resort & Casino. December 12-14, 2016, Las Vegas, USA.
2016 IEEE 55th Conference on Decision and Control (CDC) ARIA Resort & Casino December 12-14, 2016, Las Vegas, USA
Distributed Decoupling of Linear Multiagent Systems with Interconnected Nonlinear Uncertainties Vahid Rezaei? and Margareta Stefanovic† Abstract— The distributed controllers that have been designed based on the decoupled nominal linear models of agents do not guarantee stabilization of the entire multiagent system in the presence of unknown nonlinearities or agents’ interconnections. In this article, we consider a class of multiagent systems with homogeneous linear time-invariant dynamics in the presence of matched or unmatched heterogeneous interconnected unknown nonlinearity. We only measure the relativestate information and design distributed decoupling algorithms enabling agents to independently operate at their desired operating conditions. This goal is achieved by reformulating this distributed decoupling task as an equivalent leader-follower consensus problem. Finally, we investigate feasibility of the proposed ideas through simulation studies.
I. I NTRODUCTION Graph-theoretic control of multiagent systems (MASs) has received significant attention during the past two decades. While the initial research was around the consensus in networks of single-integrator [1], double-integrator [2], highorder integrator [3], and completely known linear timeinvariant (LTI) [4] agent models; recently, MASs with uncertain or interconnected dynamics have also gone under investigation. Reference [5] studied the consensus of LTI MASs in the presence of linear state-dependent uncertainties and [6] introduced a consensus strategy for LTI MASs where system matrices were polynomials of an unknown parameter. Reference [7] proposed a consensus algorithm for highorder integrator agents subject to a scalar nonlinearity that was parameterized in terms of some basis functions, [8] developed consensus algorithms for linear MASs subject to the Lipschitz nonlinearity, and [9] discussed the consensus of MASs under the norm-bounded matched unknown nonlinearities. Nevertheless, in these studies, each agent’s uncertainty was a function of its own variables. In parallel, the control of interconnected MASs has been discussed in the literature. Reference [10] proposed the concept of state-dependent graphs caused by the relative-state information exchange in the distributed algorithm. Reference [11] further studied coupled-state, -input, and -output MASs. Considering the state-coupled MASs, [12] proposed a codesign algorithm to synthesize the graph and the distributed controller in order to minimize the effect of disturbance on the agreement value. However, the effect of interconnection was completely known in these references. ?† V. Rezaei and M. Stefanovic are with the department of Electrical and Computer Engineering, University of Denver, Denver, CO 80210, USA (vrezaei(at)du.edu and a.v.rezaei(at)gmail.com) and (margareta.stefanovic(at)du.edu).
978-1-5090-1837-6/16/$31.00 ©2016 IEEE
In addition to the stability issues, the consensus performance has received attention in the literature. For a group of agents with completely known LTI dynamics, [13] proposed a linear quadratic regulator-based (LQR-based) consensus strategy and [14] designed a LQR framework for optimal consensus of MASs in the presence of unknown persistent disturbances. Moreover, [6] developed LQR-based optimal consensus algorithms in the presence of matched and unmatched uncertainties in LTI MASs, without any nonlinearities or any explicit interconnections. In this paper, we consider a MAS model where agents are described by homogeneous LTI dynamics subject to the unknown heterogeneous nonlinearities that are functions of lumped relative-output measurements. Each matched or unmatched nonlinearity is unknown and may result in instability of the interconnected MAS. The objective is to design distributed algorithms in order to cancel the effect of these interconnected unknown nonlinearities on agents’ nominal dynamics and enable agents to operate at their desired operating-points using only local feedback controllers. By reformulating this task as a leader-follower consensus problem and using only the relative-state measurements, we develop two LQR-based optimal distributed strategies and decouple agents’ nominal dynamics from each other. In each scenario, we design control signals for all leader and follower agents. Reference [15] used a LMI-based strategy and addressed the gain-scheduling leader-follower tracking problem of a linearly interconnected MAS without designing the leader’s control signal. The paper is organized as follows: Section II provides some preliminary concepts that are required in this paper. Section III introduces the MAS model with matched and unmatched interconnected nonlinear uncertainties, and reformulate the decoupling task as a leader-follower consensus problem. Section IV explains the main contribution of this paper. Section V presents the simulation verification results. Finally, Section VI gives a summary of the paper. II. N OTATION AND P RELIMINARIES We follow the standard notation in this paper. Briefly, 0 denotes a matrix of all zeros, 1N indicates a N × 1 vector of all ones, A B (A < B) represents a positive (semi-) definite matrix A − B, the symbol R+ stands for the set of positive scalar real numbers, R0+ stands for the set of non-negative scalar real numbers, diag{[.]} represents a diagonal matrix with [.] vector as its diagonal term, and Diagb {[.]} indicates a block diagonal matrix with the matrices in [.] as its diagonal blocks.
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The follower agents communicate over the undirected graph G with the graph Laplacian matrix L . The leader is connected to only some of the followers through a set of directed edges over the graph Gl f . We define B = diag{[b1 , b2 , ..., bN ]} where bi = 1 when the ith follower receives information from the leader and bi = 0 otherwise. We ignore two trivial cases where all bi = 1 and all bi = 0. Then, H = L + B denotes the generalized Laplacian matrix for Gl f . The generalized graph Laplacian is a positive definite matrix with eigenvalues 0 < µ1 ≤ µ2 ≤ ... ≤ µN if the Gl f is connected (see [16], Lemma 4). Fact 1: ([17]) The scalar λiA λ jB gives an eigenvalue of A ⊗ B where λiA denotes the ith eigenvalue of A ∈ RN×N , λ jB represents the jth eigenvalue of B ∈ RM×M , i ∈ {1, 2, ..., N}, and j ∈ {1, 2, .., M}. Definition 1: For the vector x = [x1 , x2 , ..., xn ]T ∈ Rn , the entry-wise absolute-value is defined to be |x| = [|x1 |, |x2 |, .., |xn |]T where |xi | ∈ R0+ indicates the absolutevalue of xi ∈ R for i ∈ {1, 2, ..., n}. Definition 2: For the vectors x and y ∈ Rn , we define the inequality |x| ≤ |y| ⇔ |xi | ≤ |yi | ∀ i ∈ {1, 2, ..., n}
where Bψ ∈ Rnx ×nψ and Cψ ∈ Rnψ ×nx ; and yi ∈ Rnψ and ψi (yi ,t) ∈ Rnψ . Assumption 2: The nonlinear functions ψi (yi ;t) : Rnψ × 0+ R → Rnψ ∀ i ∈ {1, ..., N} satisfy similar conditions to the Assumption 1 substituting φi by ψi , nu by nψ , and Γui by Γψi . C. Decoupling of Agents in a MAS with Uncertain Nonlinear Interconnection: Leader-Follower Consensus Formulation In this paper, we want to design a decoupling controller in order to asymptotically mitigate the effect of interconnected unknown nonlinearities in (1) or (2) such that the MAS behaves as a group of N decoupled agents (3): x˙i = Axi + Bu ui
This objective should be achieved using only relative-state measurements for i ∈ {1, 2, ..., N}. Because the variables xi and ui are defined as state and control input deviations from the operating-point values of the ith agent (see (1)), this decoupling task is achieved if the following condition is satisfied for MASs (1) or (2) under any initial conditions and over the fixed-graph G :
III. M ODELING OF MAS S WITH U NKNOWN N ONLINEAR I NTERCONNECTIONS : P ROBLEM S TATEMENT
lim xi (t) = 0
t→∞
A. MAS with Matched Nonlinear Interconnection We consider a MAS with homogeneous linear dynamics and matched heterogeneous nonlinear interconnections over G (we refer to (1) as the ith follower’s dynamics): ( x˙i = Axi + Bu (ui + φi (zi ;t)) (1) zi = Cu ∑ j∈Ni (xi − x j ) where xi ∈ Rnx stands for the state deviation from the operating-point and ui ∈ Rnu indicates the control input deviation from the operating-point; and A ∈ Rnx ×nx and Bu ∈ Rnx ×nu . Also, zi ∈ Rnu denotes the input of ith follower’s nonlinearity φi (zi ;t) ∈ Rnu and Cu ∈ Rnu ×nx indicates the mapping matrix from the lumped relative-state measurement ∑ j∈Ni (xi − x j ) to zi . Assumption 1: The nonlinear functions φi (zi ;t) : Rnu × 0+ R → Rnu ∀ i ∈ {1, ..., N} satisfy the followings: 1) Each function φi (zi ;t) has separate nonlinearities φi (zi ;t) , [φi1 (zi1 ;t), φi2 (zi2 ;t), ..., φinu (zinu ;t)]T such that φim (zim ;t) : R × R+ → R for m = {1, 2, ..., nu } and zi = [zi1 , zi2 , ..., zinu ]T (see [18]), 2) Each separate nonlinearity φim (zi ;t) satisfies the sector condition −γim |zim | ≤ φim (zim ;t) ≤ γim |zim | where γim ∈ R0+ (e.g., see [19], p. 105) such that −Γui |zi | ≤ φi (zi ;t) ≤ Γui |zi | where Γui = diag{[γui1 , γui2 , ..., γuinu ]},
(4)
In order to achieve this distributed decoupling, we introduce the leader agents (5) for the matched case and (6) for the unmatched scenario: ( x˙0 = Ax0 + Bu (u0 + φ0 (z0 ;t)) (5) z0 = Cu x0 ( x˙0 = Ax0 + Bu u0 + Bψ ψ0 (y0 ;t) y0 = Cψ x0
(6)
where x0 ∈ Rnx and u0 ∈ Rnu are defined similar to the variables in (1). The functions φ0 (z0 ;t) ∈ Rnu and ψ0 (y0 ;t) ∈ Rnψ satisfy the Assumption 1 and Assumption 2 for i = 0, respectively. We further adopt the follower models (1) and (2) as (7) and (8), respectively: ( x˙i = Axi + Bu (ui + φi (zi ;t)) (7) zi = Cu (∑ j∈Ni (xi − x j ) + bi (xi − x0 )) ( x˙i = Axi + Bu ui + Bψ ψi (yi ,t) yi = Cψ (∑ j∈Ni (xi − x j ) + bi (xi − x0 ))
(8)
Now, the distributed decoupling task (4) for (1) or (2) is achieved when the leader-follower consensus problem (9) is solved for (5) and (7), or (6) and (8): lim (xi (t) − x0 (t)) = 0
B. MAS with Unmatched Nonlinear Interconnection
t→∞
In this scenario, we introduce the (follower) agents’ dynamics with unmatched heterogeneous nonlinear interconnections: ( x˙i = Axi + Bu ui + Bψ ψi (yi ,t) (2) yi = Cψ ∑ j∈Ni (xi − x j )
(3)
(9)
by setting the new control objective to be finding the control signals u0 and ui that simultaneously stabilize the uncertain leader dynamics (5) (or (6)) and drive the followers’ states xi in (7) (or (8)) to the leader state x0 , under any initial state conditions and over the fixed graph Gl f . The following assumptions are satisfied in this paper:
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Assumption 3: The matrix A is Hurwitz, (A, B) characterizes a stabilizable model, there exists a direct path from the leader to each follower over Gl f , and x0 is known but u0 is unknown to the followers connected to the leader. Assumption 4: Nonlinear functions φi (zi ;t) and ψi (yi ;t) are unknown, Γui in Assumption 1 and Γψi in Assumption 2 are known matrices, there exists a local (agent-level) lookuptable scheduling system or feedback tracking controller such that each agent’s nominal model (3) operates at the desired operating-point (xiopt , uopt i ), and the distributed controller does not have access to (xiopt , uopt i ). Remark 1: By applying the change of variables xi = xiact − opt xiopt and ui = uact i − ui , we can transform the system (3) to its actual coordinate with the state variable xiact and control input uact i , and design local tracking controllers in a decentralized manner (independent of the agent’s neighbors). Remark 2: An alternative approach is to only consider the virtual reference model x˙0 = Ax0 without control input and nonlinearity. However, we assume that the leader is a new physical agent such that u0 and φ0 (or ψ0 ) appear in (5) (or (6)). In other words, we propose a class of leaderfollower consensus problem under matched (or unmatched) uncertainties in both leader and follower agents. The case where the leader has a bounded control input, with no uncertainty, has been addressed in the literature via adaptive control techniques (e.g., see [20]) whereas our solutions provide fixed-gain optimal distributed consensus algorithms. IV. D ISTRIBUTED D ECOUPLING OF U NCERTAIN I NTERCONNECTED MAS S In this section, we solve (9) based on the matched scenario (7) and unmatched scenario (8) (or solve (4) for (1) and (2)).
˜ , K˜ = where ξ = [x0T , ε T ]T , τ = [uT0 , ν T ]T = Kξ ¯ Diagb {[K0 , K]}, η = [φ0T , ΦTµ ], and Φµ = µ11 Φ. Also, ¯ A˜ = Diagb {[A, A]}, A¯ = IN ⊗ A, B˜ u = Diagb {[Bu , B¯ u ]}, ¯ B¯ u = IN ⊗ µ1 Bu , E˜ = Diagb {[0, E]}, and E¯ = E¯ T = (( µ11 H − IN ) ⊗ Inu ) < 0. We further define Γu , diag{[γu1 , γu2 , ..., γunu ]} where γum , maxi {γuim } for m ∈ {1, ..., nu } and i ∈ {1, 2, ..., N}. We find Φµ (u0 , z0 , z;t) ≤ µ11 ΦM (u0 , z0 , z;t) where the upper bound function ΦM (u0 , z0 , z;t) is given by (12): ΦM , (IN ⊗ Γu )|z| + (1N ⊗ Γu0 )|z0 | + (1N ⊗ Inu )|u0 | (12) Moreover, using the Rayleigh-Ritz inequality and Fact 1, the quadratic upper bound on the unknown nonlinearity η is given by (13): T ˜ ˜ ≤ ε T R¯ ε ε + x0T Rx0 x0 + uT0 Ru0 u0 =: ηM η T Rη RηM (13)
where R˜ = R˜ T = Diagb {[Rl , R¯ f ]}, Rl = RTl 0, R¯ f = IN ⊗ R f , µ2 R f = RTf = r f Inu 0, R¯ ε = IN ⊗Rε , Rε = Rε = 2r f N2 CuT Γ2uCu , f
x
1
0 and Rl = RTl 0 are two design matrices. Z ∞
J0 (x0 (0)) = min u0
In order to achieve the consensus in the MAS (5) and (7), we propose the following control signals u0 and ui : ui = K(
(10)
∑ (xi − x j ) + bi (xi − x0 )) j∈Ni
where K0 ∈ Rnu ×nx represents the leader’s control gain and K ∈ Rnu ×nx stands for the followers’ control gain. By introducing the leader-follower tracking error εi , xi − x0 , we find the leader-follower tracking error dynamics ε˙i = Aεi + Bu ui + Bu Φi (u0 , z0 , zi ;t) where Φi (u0 , z0 , zi ;t) , φi (zi ;t) − φ0 (z0 ;t) − u0 and ui = K(∑ j∈Ni (εi − ε j ) + bi εi ). Also, let ε , [ε1T , ε2T , ..., εNT ]T be the aggregated tracking error vector, u , [uT1 , uT2 , ..., uTN ]T be the aggregated control-input, Φ , [ΦT1 , ΦT2 , ..., ΦTN ]T = φ (z;t)−(1N ⊗Inu )φ0 (z0 ;t)−(1N ⊗Inu )u0 be the aggregated matched unknown nonlinearity, φ (z;t) , [φ1T (z1 ;t), φ2T (z2 ;t), ..., φNT (zN ;t)]T , and z = (H ⊗Cu )ε. Over Gl f , we decompose u = (H ⊗ K)ε as u = (H ⊗ Inu )ν and ¯ = (IN ⊗ K)ε, and find the following augmented ν = Kε leader-follower MAS dynamics: ˜ + B˜ u τ ξ˙ = Aξ | {z }
Networked nominal dynamics
˜ + B˜ u η(u0 , z0 , z;t) (11) + B˜ u Eτ {z } | Modeling uncetainty
1
u
Ru0 = R f 0 = 4Nr f µ12 Inu . Theorem 1 provides sufficient con1 ditions to achieve the leader-follower consensus (9) using (11) and, consequently, to solve (4) for a MAS of (1). T Theorem 1: Let u0 = K0 x0 = −R−1 1l Bu P1l x0 be the optimal control signal that minimizes the cost function (14) subject to the dynamics (15) where the condition (16) is satisfied. The matrix P1l denotes solution of the algebraic Riccati equation x x (ARE) (17), Q1l = Ql + Rx0 , Rx0 = Rl 0 + R f0 , R1l = Rl + Ru0 , 1 T u x u T 2 R 0 0 = Cu Γu0 R 0 Γu0Cu = 4Nr f µ 2 Cu Γu0Cu , and Ql = QTl
A. MAS with Matched Nonlinear Interconnection
u0 = K0 x0
µ1
x
Rx0 = Rl 0 + R f0 = CuT Γu0 Rl Γu0Cu + 4Nr f µ12 CuT Γ2u0Cu , and
0
(x0T Q1l x0 + uT0 R1l u0 )dt
(14)
x˙0 = Ax0 + Bu u0
(15)
Ql − Ru0 x0 − K0T Ru0 K0 0
(16)
T AT P1l + P1l A + Q1l − P1l Bu R−1 1l Bu P1l = 0
(17)
T Also, let νi = Kεi = −µ1 R−1 1 f Bu P1 f εi ∀i ∈ {1, 2, ..., N} be th the i follower’s optimal control signal that minimizes the cost function (18) subject to the dynamics (19) where P1 f represents solution of the ARE (20), Q1 f = Q f + Rε , R1 f = R f , and Q f = QTf 0 and R f = RTf = r f Inu 0 for r f ∈ R+ are two design matrices.
Z ∞
Ji (εi (0)) = min νi
0
(εiT Q1 f εi + νiT R1 f νi )dt
(18)
ε˙i = Aεi + µ1 Bu νi
(19)
T AT P1 f + P1 f A + Q1 f − µ12 P1 f Bu R−1 1 f Bu P1 f = 0
(20)
Then, the closed-loop system (11) is asymptotically stable and the distributed decoupling problem (4) is solved in the presence of heterogeneous matched interconnected nonlinear uncertainties.
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Proof: Using (14) and (18) for i ∈ {1, 2, ..., N}, we find R ˜ + τ T Rτ ˜ + the augmented cost function J(ξ (0)) = 0∞ (ξ T Qξ T ˜ ¯ ¯ ˜ ηM RηM )dt where Q = Diagb {[Ql , Q f ]} and Q f = IN ⊗ Q f . Using (15) and (19) for i ∈ {1, 2, ..., N}, we find the networked nominal dynamics in (11). Hence, the control signals ¯ minimize the augmented leaderu0 = K0 x0 and ν = Kε follower cost function J(ξ (0)) subject to the networked ˜ nominal dynamics in (11). We need to show that τ = Kξ stabilizes the entire uncertain dynamics (11). We propose the following Lyapunov candidate function for t ≥ 0: Z ∞
V (ξ (t)) = min τ
t
to Γu in Subsection IV-A. Over Gl f , the augmented leaderfollower dynamics are as follow: ˜ + B˜ u σ + B˜ u Eσ ˜ + B˜ u0 u0 + B˜ ψ Ψt (y0 , y;t) (22) ζ˙ = Aζ {z } | | {z }
Networked nominal dyn.
B˜ ψ = Diagb {[Bψ , B¯ ψ ]}, and B¯ ψ = IN ⊗ Bψ . We find Ψ ≤ (IN ⊗ Γψ )|y| + (1N ⊗ Γψ0 )|y0 |. We define the following auxiliary leader-follower MAS dynamics:
T ˜ ˜ + τ T Rτ ˜ + ηM (ξ T Qξ RηM )dt 0 0
that, since V (ξ (0)) = J(ξ (0)), satisfies the Hamilton˜ + τ T Rτ ˜ + Jaccobi-Bellman (HJB) equation minτ {ξ T Qξ T T ˜ ˜ M + V (Aξ + B˜ u τ)} = 0 (e.g., see Sec. 3.11 in [21]). ηM Rη ξ ˜ satisfies Particularly, the optimal solution τ = τ ? = Kξ T Rη ˜ + τ T Rτ ˜ + B˜ u τ) = 0 ˜ + ηM ˜ M +V T (Aξ the equalities ξ T Qξ ξ and 2τ T (R˜ + R˜ τ ) + VξT B˜ u = 0 where R˜ τ = Diagb {[Ru0 , 0]}. Differentiation of V (ξ (t)) along the uncertain dynamics (11) results in:
˜ + B˜ u σ + B˜ u θ + B˜ ψ β ζ˙ = Aζ 0 = [θ1T , ..., θNT ]T
≤ −x0T Ql x0 + x0T Ru0 x0 x0 + uT0 Ru0 u0 ≤ −x0T (Ql − Ru0 x0 − K0T Ru0 K0 )x0 ≺ 0 where we have used E¯ = E¯ T < 0 such that −2τ T (R˜ + T Rη ˜ M+ ˜ = −2r f ν T Eν ¯ ≤ 0 and η T (R˜ + R˜ τ )η ≤ ηM R˜ τ )Eτ T (R ˜ + R˜ τ )ηM . Now, based on the Lyapunov x0T Ru0 x0 x0 =: ηM theorem (e.g., see [19], p. 47), the leader-follower MAS dynamics (11) are asymptotically stable for all initial state values and over the fixed-graph Gl f . Based on the reformulation in Subsection III-C, the distributed decoupling of agents is achieved. We proposed a similar Lyapunov candidate function in [6] for a different consensus problem in MASs (see Section I). Also, this has been used in [22] (Sec. 5.4) for a single dynamical system.
¯ T0 = uT0 (NS)u0 uT0 Su
(24)
T ˜ ΨtT W˜ Ψt ≤ ε T W¯ ε ε + x0T W x0 x0 =: ΨtM W ΨtM
(25)
Z ∞
J0 (x0 (0)) = min
u0 ,β0 0
In this subsection, we consider the unmatched nonlinear uncertainty scenario. Wo only introduce new variables and the rest can be found in Subsection IV-A. We propose the control signals u0 and ui :
j∈Ni
∑ j∈Ni
where G0 ∈ Rnu ×nx denotes the leader’s control gain and G ∈ Rnu ×nx indicates the followers’ control gain. The leader-follower tracking error dynamics are given by ε˙i = Aεi + Bu ui − Bu u0 + Bψ Ψi (y0 , yi ;t) where the unknown nonlinear functions Ψi (y0 , y;t) = ψi (yi ;t) − ψ0 (y0 ;t) satisfy Ψi (y0 , z;t) ≤ Γψ |yi | + Γψ0 |y0 | where Γψ is defined similar
(23)
= [β0T , β1T , ..., βNT ]T
where S¯ = IN ⊗ S, S = ST 0, W˜ = Diagb {[Wl , W¯ f ]}, Wl = WlT 0, W¯ f = IN ⊗ W f , W f = W fT 0, W¯ ε = W¯ fε = IN ⊗ x x W fε , W fε = 2µN2 CψT Γψ W f Γψ Cψ , and W x0 = Wl 0 + W f 0 = T T Cψ Γψ0 Wl Γψ0 Cψ + 2NCψ Γψ0 W f Γψ0 Cψ . The next theorem characterizes sufficient conditions for stabilization of (22) resulting in the leader-follower consensus (9) for (6) and (8) or, equivalently, solving (2). (4) for BTu P2l −R−1 u0 G0 2l Theorem 2: Let = x = x be β0 L0 0 −Wl−1 BTψ P2l 0 the control signal that minimizes the cost function (26) subject to the auxiliary system (27) where the condition (28) or (29) is satisfied. The matrix P2l denotes solution of the ARE (30); Q2l = Ql + W x0 , R2l = Rl + NS; Ql = QTl 0, Rl = RTl 0, and Wl = WlT = 0 are three design matrices, BUl = [Bu , Bψ ], and RUl = Diagb {[R2l ,Wl ]}.
B. MAS with Unmatched Nonlinear Interconnection
(21) (εi − ε j ) + bi εi )
∈ RNnu
where θ and β ∈ R(N+1)nψ are two auxiliary control inputs corresponding to two unmatched uncertainties u0 (unknown to followers) and Ψt (y0 , y;t), respectively. The quadratic (upper) bounds on these uncertainties are given by (24) and (25), respectively:
V˙ = VξT ξ˙
u0 = G0 x0 ui = G( ∑ (xi − x j ) + bi (xi − x0 )) = G(
Modeling uncertainty
˜ = where ζ = [x0T , ε T ]T , σ = [uT0 , ν T ]T = Gζ ¯ , G¯ = IN ⊗ G, u0 = −(1N ⊗ Inu )u0 ∈ RNnu , Diagb {G0 , G}ζ Ψt = [ψ0T , ΨT ]T , Ψ(y0 , y;t) = [ΨT1 , ΨT2 , ..., ΨTN ]T = ψ(y;t) − (1N ⊗ Inu )ψ0 (y0 ;t), ψ(y;t) = [ψ1T , ψ2T , ..., ψNT ]T , and y = (H ⊗Cψ )ε. Also, B˜ u0 = [0T , B¯ Tu0 ]T , B¯ u0 = IN ⊗ Bu ,
(x0T Q2l x0 + uT0 R2l u0 + β0T Wl β0 )dt (26)
x˙0 = Ax0 + Bu u0 + Bψ β0
(27)
Ql − 2L0T Wl L0 0
(28)
Ql + GT0 Rl G0 − 2L0T Wl L0 0
(29)
−1 T AT P2l + P2l A + Q2l − P2l BUl RUl BUl P2l = 0 (30) T −µ1 R−1 νi G 2 f Bu P2 f −1 Also, let βi = L εi = −W f BTψ P2 f εi be the θi H −S−1 BTu P2 f control signal that minimizes the cost function (31) subject to the auxiliary system (32) where the condition (33) or (34) is satisfied. The matrix P2 f stands for the solution of the ARE
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(35), Q2 f = Q f + W ε , and R2 f = R f . Also, Q f = QTf 0, R f = RTf = r f Inu 0 for r f ∈ R+ , W f = W fT 0, and S = ST 0 are four design matrices. Moreover, BU f = [µ1 Bu , Bψ , Bu ] and RU f = Diagb {[R2 f ,W f , S]}. Z ∞
Ji (εi (0)) = min
νi ,βi ,θi 0
( εiT Q2 f εi + νiT R2 f νi + βiT W f βi + θiT Sθ )dt
(32)
Q f − 2LT W f L − 2H T SH 0
(33)
Q f + GT R f G − 2LT W f L − 2H T SH 0
(34)
AT P2 f + P2 f A + Q2 f − P2 f BU f RU−1f BUT f P2 f = 0
(35)
σ ,θ ,β t
In this section, we investigate the feasibility of our ideas through simulation studies over the graph topology that is depicted in Fig. 1. Decentralized nominal systems are stable dynamics subject to the initial conditions, given by A = 0 1 and Bu = [0, 1]T . Furthermore, x0 (0) = [15, 15]T , −5 −3 x1 (0) = [−10, 20]T , x2 (0) = [15, −15]T , x3 (0) = [10, 15]T , x4 (0) = [−30, 20]T , and x5 (0) = [20, −30]T . A. MAS with Matched Nonlinear Interconnection
Then, the closed-loop system (22) is asymptotically stable and the distributed decoupling problem (4) is solved. Proof: We find (23) using (27) and (32). Also, based on (26) and (31), we find the augmented leader-follower cost R ¯ + ˜ + σ T Rσ ˜ + θ T Sθ function J(ζ (0)) = minσ ,θ ,β 0∞ {ζ T Qζ T T T ¯ ˜ ˜ β W β + u0 Su0 + ΨtMW ΨtM }dt. The optimal control signals σ , θ , and β minimize the cost function J(ζ (0)) subject to (23). We need to show that the uncertain closed-loop system (22) will be stabilized using σ (i.e., there is no need for θ and β for the implementation purpose). We propose the following Lyapunov candidate function for t ≥ 0: Z ∞
V. S IMULATION V ERIFICATION
(31)
ε˙i = Aεi + µ1 Bu νi + Bψ βi + Bu θi
V (ζ (t)) = min
Remark 3: The condition (29) is essentially a less conservative version of (28) with an added positive term GT0 Rl G0 (similarly, see (34) and (33)). The choice of the MAS-level Lyapunov candidate function is inspired by [6]. In [22] (Sec. 6.3), this function is used for a single dynamical system.
˜ + σ T Rσ ¯ + β T W˜ β ˜ + θ T Sθ {ζ T Qζ
In the matched scenario, Cu = [0, 1.5] and the unknown nonlinear functions are φ0 (z0 ) = 0.1sin(z0 ), φ1 (z1 ) = 0.7z1 , φ2 (z2 ) = −0.2sin(z2 ), φ3 (z3 ) = 0.5tanh(z3 ), φ4 (z4 ) = −0.5tanh(z4 ), and φ5 (z5 ) = −0.3sin(z5 ) where tanh(.) refers to the hyperbolic tangent. We first simulate the open-loop interconnected MAS, without the distributed decoupling controller of Subsection IV-A, over the leaderless graph G . The unstable behavior of the interconnected MAS in Fig. 2 indicates the need for a (distributed) decoupling controller. Figure 3 represents the stable closed-loop MAS behavior, using the controllers of Theorem 1, in terms of the state variables’ deviations from the operating-point.
T ˜ ¯ 0 + ΨtM +uT0 Su W ΨtM }dt 0 0
where V (ζ (0)) = J(ζ (0)) is the optimal cost function subject to the augmented auxiliary system (23). Hence, the HJB ¯ 0 +β T W¯ β + ˜ +σ T Rσ ¯ +uT Su ˜ +θ Sθ equation minσ ,θ ,β {ζ T Qζ 0 T T ˜ ˜ ˜ ˜ ˜ ΨtMW ΨtM + Vζ (Aζ + Bu σ + Bu0 θ + Bψ β )} = 0 is satisfied (e.g., see Sec. 3.11 in [21]). Consequently, the optimal control signals σ = σ ? , θ = θ ? , and β = β ? satisfy the ¯ + β T W˜ β + ˜ + σ T Rσ ˜ + θ T Sθ following four equations: ζ T Qζ T T T T ˜ ¯ ˜ ˜ ˜ u0 Su0 + ΨtMW ΨtM + Vζ (Aζ + Bu σ + Bu0 θ + B˜ ψ β ) = 0, 2σ T (R˜ + R˜ σ ) +VζT B˜ u = 0, 2θ T S¯ +VζT B˜ u0 = 0, and 2β T W˜ + ) and R˜ σ = Diagb {[NS, 0]}. The VζT B˜ ψ = 0 where Vζ = ∂V∂ (ζ ζ differentiation along the uncertain trajectory (22) results in:
v2
v3
v0
v4 v1
v5
3
−1
0
−1
−1 L = 0 −1
3
−1
0
−1
2
−1
0
−1
3
−1
−1
0
−1
−1
−1 0
−1 3
B = diag{[1, 1, 0, 0, 0]}
Fig. 1: The communication topology (left), L corresponding to G (remove the node v0 and the directed edges originating from that) (right top), and B to find the generalized graph Laplacian H for Gl f (right bottom).
V˙ (ζ ) = VζT ζ˙ ˜ − σ T Rσ ¯ + 2β T W˜ β ˜ + 2θ T Sθ ≤ −ζ T Qζ ˜ + 2θ T Sθ ¯ + 2β T W˜ β ≤ −ζ T Qζ that can be rewritten as V˙ (ζ ) ≤ −ζ T (Q˜ + G˜ T R˜ G˜ − 2L˜ T W˜ L˜ − ˜ and V˙ (ζ ) ≤ −ζ T (Q˜ − 2L˜ T W˜ L˜ − 2H˜ T S˜H)ζ ˜ where 2H˜ T S˜H)ζ L˜ = Diagb {[L0 , IN ⊗ L]}, H˜ = Diagb {[0, IN ⊗ H]}, and S˜ = Diagb {[0, IN ⊗ S]}. Based on the conditions (28)-(29) and (33)-(34), we find V˙ ≺ 0. Hence, using the Lyapunov theorem (e.g., see [19], p. 47), the closed-loop MAS (22) is asymptotically stable and decoupling of uncertain nonlinearly interconnected agents is achieved.
Fig. 2: Matched scenario: State deviation variables of all agents without the distributed decoupling controller (see the definition in (1)) where xi1 and xi2 denote the first and second state deviation variables of the ith agent, respectively.
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B. MAS with Unmatched Nonlinear Interconnection In this subsection, we implement the controllers of Theorem 2 for a MAS with Cψ = [0.5, 1.5] and Bψ = [0.2, 1]T . Substituting zi variables by yi , the nonlinear functions are the same as Subsection V-A. The unstable open-loop behavior of the MAS over G is given by Fig. 4. The closed-loop MAS simulation result, Fig. 5, shows that agents can independently operate at their desired operating-points.
the effect of these unknown nonlinear interconnections and, independent of the neighbors, enable agents to operate at the desired operating-points using their local control systems. We reformulate this task as the leader-follower consensus with uncertain leader and followers. We prove that this robust consensus problem can be solved by appropriate LQR formulations under some sufficient conditions. The simulation results verify feasibility of the proposed ideas. R EFERENCES
VI. S UMMARY In this paper, we consider a group of homogeneous LTI systems in the presence of the heterogeneous interconnected unknown nonlinearities. These nonlinear mismatches depend on the agents’ relative-output information and are represented as the matched and unmatched uncertainties. We design distributed decoupling algorithms in order to completely cancel
Fig. 3: Matched scenario: State deviation variables of all agents with distributed decoupling controller, respectively.
Fig. 4: Unmatched scenario: State deviation variables of all agents without distributed decoupling controller.
Fig. 5: Unmatched scenario: The first actual state variables act of all agents with distributed decoupling controller (see xi1 act is not shown in this paper.) the definition in Remark 1). (xi2
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