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Leader–follower H∞ consensus of linear multi-agent systems with aperiodic sampling and switching connected topologies
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Research article
Leader–follower H1 consensus of linear multi-agent systems with aperiodic sampling and switching connected topologies$ Dan Zhang a,b,n, Zhenhua Xu a, Qing-Guo Wang c, Yun-Bo Zhao a a
Department of Automation, Zhejiang University of Technology, Hangzhou 310023, PR China School of Automation, Hangzhou Dianzi University, Hangzhou 310018, PR China c Institute for Intelligent Systems, The University of Johannesburg, Johannesburg, South Africa b
art ic l e i nf o
a b s t r a c t
Article history: Received 20 June 2016 Received in revised form 1 January 2017 Accepted 1 January 2017
This paper is concerned with the distributed H∞ consensus of leader–follower multi-agent systems with aperiodic sampling interval and switching topologies. Under the assumption that the sampling period takes values from a given set, a new discrete-time model is proposed for the tracking error system. For the multi-agent systems with time-varying sampling period, switching topologies and external disturbance, the considered tracking problem is converted to a robust H∞ control problem. With help of the Lyapunov stability theory, a sufficient condition for the existence of mode-dependent controller is established and it guarantees the exponential stability of tracking error system and a prescribed H∞ disturbance attenuation level. The influence of sampling period on the overall control performance is also discussed. Two simulation examples are given to show the effectiveness of the proposed control algorithm. & 2017 Published by Elsevier Ltd. on behalf of ISA.
Keywords: Multi-agent systems Leader–follower consensus Aperiodic sampling Switching topology Linear matrix inequality (LMI)
1. Introduction In recent years, consensus of multi-agent systems has received increasing research attention due to its wide applications in various fields such as vehicle formation [1], cooperative target tracking in sensor networks [2,3], and power dispatch [4]. Generally, the consensus can be classified into two categories, i.e., leaderless consensus [5] and leader–follower consensus [6]. The main difference is whether or not there exist leaders in such coupled dynamical systems. So far, many interesting results have been reported in the literature for the consensus issue. For example, the distributed H∞ consensus of high order linear dynamics and switching topologies was addressed in [7]. By using tools of graph theory and the switched system approach, the consensus is achieved if the coupling strength among agents is larger than a desired positive value. In the presence of the actuator saturation, the leader–follower consensus problem for linear multi-agent system was investigated in [8], and the domain of consensus attraction of origin was estimated under the magnitude bounded ☆ This work was partially supported by the National Natural Science Foundation of China (61403341, 61673350), and the Open Foundation of the First Level Zhejiang Key in Key Discipline of Control Science and Engineering. n Corresponding author at: Department of Automation, Zhejiang University of Technology, Hangzhou 310023, PR China. E-mail addresses: [email protected] (D. Zhang), [email protected] (Q.-G. Wang).
disturbance. In [9], the internal model principle was applied onto the output synchronization control of non-identical linear multiagent systems. Very recently, the leader–follower consensus problem for a class of nonlinear agents was addressed in [10] and the pinning control method was utilized to achieve the consensus. To handle the nonlinearity and disturbance, an adaptive control approach [11] was also introduced, the details of which are given in [12]. More recent advances on the consensus of multi-agent system can be found in [13–17] and the references therein. It is noted that most works reported in the consensus literature are in the framework of continuous time, which requires the continuous communication among agents. However, many practical multi-agent systems are communicating by wireless network and such a continuous-time communication may consume a lot of transmission power. In order to save power of wireless nodes, it is more desirable that the agent only uses the sampled-data to update its dynamics [18], which may lead to periodic sampling or even aperiodic sampling. There have been some discussions on the consensus of multi-agent systems with sampled-data information. The consensus problem for a class of general linear multi-agent system with sampled-data was studied in [19] and an allowable upper bound of sampling period was given to achieve the consensus. By converting the sampled-data system to a discrete-time system with delays, the consensus of second-order multi-agent systems with sampled-data and switching topologies was investigated in [20]. It is shown that if the union graph of all direct
http://dx.doi.org/10.1016/j.isatra.2017.01.001 0019-0578/& 2017 Published by Elsevier Ltd. on behalf of ISA.
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2
graph has a spanning tree, then the consensus can be reached. The consensus of multi-agent system governed by nonlinear dynamics with sampled-data was also studied in [21], where the sampleddata system was transformed into an equivalent nonlinear system with a time-varying input delay, the upper bound of the maximal allowable sampling period was obtained by solving LMIs. Very recently, the similar approach was used in [22] for the leader– follower consensus of multi-agent systems with sampled-data and switching topologies. Based on the free weighting matrix method, a sufficient condition was obtained to achieve the consensus and the upper bound of the sampling period was also calculated. It should be noted that for the multi-agent systems with sampleddata, the input-delay approach is commonly used, and some useful delay information is usually missing in the derivation of their main t results. For example, the term − ∫ eṪ (s )(In ⊗ R )e(̇ s )ds was ent−h
larged to be − ∫
t
t − τ (t )
eṪ (s )(In ⊗ R )e(̇ s )ds in [22], where h and τ(t ) are
the upper bound of the sampling period and the time-delay, respectively, In is an Identity matrix with n dimension, R is a positive-definite matrix and ⊗ is the Kronecker product. Moreover, the designed controllers in the above literatures are all independent on the variation of the sampling period. It is well known that the mode-dependent controller can achieve a better control performance than the mode-independent one. Hence, consensus of linear multi-agent systems with time-varying sampled-data and switching topologies needs further improvement, which motivates the present study. In this paper, we study the leader–follower consensus problem of general linear multi-agent systems with switching topologies and external disturbance. The agent dynamics are homogeneous and only the sampled-data information is utilized for the controller design. The topology is allowed to be switching but keeps connected. By assuming that the sampling period is varying in a given set, a new discrete-time approach is proposed for the analysis of the tracking error system. Based on the Lyapunov stability theory, a sufficient condition is obtained such that the tracking error system is exponentially stable and achieves a desired H∞ disturbance attenuation level. An optimization problem is proposed to calculate the controller gains such that the H∞ performance index is minimized. Finally, two simulation examples are introduced to illustrate the effectiveness of the proposed control algorithm. Notations: Rn denotes the n dimensional Euclidean space. The superscript “T ” is used to stand for the matrix transposition. L2[0, ∞) stands for the space of square-integrable vector function over [0, ∞), • refers to the induced matrix two-norm. The symbol “n” is used to describe the symmetry of a matrix, and “⊗” is the Kronecker product. We use diag{⋯} to describe the block-diagonal matrix. χmin and χmax are the minimal and maximal eigenvalues, respectively. I and 0 represent the identity matrix and zero matrix with appropriate dimensions, respectively.
where x i(t ) ∈ Rn is the state of the i-th follower, i = 1, 2, … , N . ui(t ) ∈ Rm is the control input, and wi(t ) ∈ Rq is the disturbance, belonging into L2[0, ∞). and are some constant matrices with appropriate dimensions. Suppose all followers and the leader are clock synchronized and during each sampling period, the followers use the available sampled-data to update its state. Let the sampling instant be tk, and the sampling period is defined as hk = tk + 1 − tk , which is a time-varying value. We assume that the sampling period hk only takes a value from Φ = T1, T2, …, TN1 , where Tj = δjT0, j = 1, 2, … , N1,
{
}
δj is a positive scalar, T0 is a given sampling period, and N1 is the
total number of sampling periods. For the sampling period hk and by using a zero-order hold device, then for t ∈ [tk, tk + 1), the dynamics of the leader–follower can be described as
⎧ xi (tk + 1) = kxi (tk ) + kui (tk ) + kwi(tk ) ⎨ ⎩ x 0(tk + 1) = kx 0(tk ) where k = e hk , k =
(3)
h
h
∫0 k e τ dτ , and k = ∫0 k e τ dτ . Since the
Φ, then k = ( 0) k , l
sampling period can only take values from l −1
t
k = ∑tk= 0 ( 0) 0 , and T0 τ e dτ , 0
0 = ∫
t
l −1
k = ∑tk= 0 ( 0) 0 , where
0 = e T0 ,
T0 τ e dτ , 0
N1
0 = ∫
and lk =
{ δ , δ , …, δ }. It can 1
2
be seen that the values of k , k and k can change from one to another when the sampling period switches. We now introduce a switching signal ρ( tk ) ∈ Φ1 = { 1, 2, … , N1}, and the following discrete-time system model is obtained:
⎧ ⎪ x i (tk + 1) = ρ(t )x i (tk ) + ρ(t )ui (tk ) + ρ(t )wi(tk ) k k k ⎨ ⎪ ⎩ x 0(tk + 1) = ρ(tk)x 0(tk ) where δρ(t ) ∑t = 1k
ρ(t ) = ( 0) k
δρ(t ) k
,
ρ(t ) = k
δρ(t ) ∑t = 1k
(4) t−1
( 0)
0,
and
t−1
( 0) 0 . Since we have N1 numbers of sampling periods, each one may vary from one to another, thus a nature problem is how such a variation affects the control performance? To answer this problem, we define the variation numbers of these sampling periods over the time interval ( k0, k ) by Nρ , then the variation frequency can be
ρ(t ) = k
calculated by ϑ =
Nρ k − k0
. We will show in the next section that this
parameter contributes much to the control performance. Due to the fact that the followers are usually moving and the topology of the communication network may be dynamically changing [23,24]. In this paper, we assume that there are N2 possible topologies. As is well known, the interaction among followers can be described by a graph. Then, we denote the set of topologies of follower's interaction by G(t ) = ⎡⎣ gij(t )⎤⎦ , the eleN×N ment gij(t ) = 1, i ≠ j , if the i-th agent can receive information from the j-th element, otherwise gij(t ) = 0. Moreover, gii(t ) = 0. In this paper, we assume that the topology switching is finite, and it takes values from Θ2 = G1, G2, … , G N2 . Denote the corresponding La-
{
2. Problem formulation Consider a multi-agent system with one leader and N agents. The dynamics of the leader is represented by the following continuous-time linear time-invariant (LTI) model:
ẋ0(t ) = x 0(t )
(1) n
where x0(t ) ∈ R is the state vector of the leader, and the system matrix is constant, which may not be Hurwitz stable. The dynamics of the i-th follower is represented by the following continuous-time LTI model:
xi̇ (t ) = xi (t ) + ui (t ) + wi(t )
(2)
}
placian matrices by Li, i = 1, 2, … , N2 . The connection matrix between the leader and followers are represented by , where the element mi(t ) = 1 if the i-th follower M (t ) = ⎡⎣ mi(t )⎤⎦ N×N can receive information from the leader, otherwise it is zero. The interaction number of the leader–follower network is also assumed to be N2. Based on the above discussion, we see that L(t ) + M (t ) ∈ { L i + Mi, i = 1, 2, … , N2}. For consensus purpose, we assume that the topology Gi is connected and at each time instant, there is at least one follower has the information of leader node. To achieve the tracking performance, the following consensusbased control algorithm is applied:
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⎛ N ui = − ρ(tk)⎜⎜ ∑ gij (tk ) xi (tk ) − xj (tk ) ⎝ j=1
(
condition, ∑k = 0 eT (tk )e(tk ) ≤ ∑k = 0 γ 2wT (tk )w (tk ) is true. +∞
⎞ ⎟ ⎟ ⎠
)
− ρ(tk)mi (tk )( xi (tk ) − x 0(tk ))
(5)
where ρ(tk ) is the controller gains to be determined. Define the tracking error signal as ei(tk ) = x i(tk ) − x0(tk ), and based on the above analysis, we have the following closed-loop system:
¯ ρ(t )e(tk ) + ⎡⎣ IN ⊗ ρ(t )⎤⎦w (tk ) e(tk + 1) = k k
)
T e( tk ) = ⎡⎣ e1T ( tk ) e2T ( tk ) ⋯ eNT ( tk )⎤⎦ , T w ( tk ) = ⎡⎣ w1T ( tk ) w2T ( tk ) ⋯ wNT ( tk )⎤⎦ ,
(tk ) = L(tk ) + M (tk ). Remark 1. Compared with the existing results in [18–22], a new discrete-time closed-loop model has been obtained in (6), in which no delay is involved. This new model enables us to design the mode-dependent controller that fully uses the sampling information. It should be noted that we construct the controller on a linear multi-agent system. But the main results can be applied to the nonlinear case if a nonlinear multi-agent system can be linearized to be a linear dynamic system, e.g., using the fuzzy blending [25] or first order Taylor approximation. Let the eigenvalues of i be χij , i = 1, 2, … , N2, j = 1, 2, … , N . Since the topology Gi is undirected and connected and at least one follower detects the leader at each time instant, then it is easy to
{ }
{ }
verify that χij > 0. Denote χmin = min χij , χmax = max χij , and χmin + χmax , 2
definite matrix
Δi =
i σ
δ=
χmax − χmin , χmin + χmax
the symmetric and positive
i can be written by
− IN . The eigenvalues of
2χij − χmax − χmin , χmin + χmax
which
satisfy
3. Main results In this section, a sufficient condition is first presented, which guarantees the existence of controller (5) such that the above control performance is achieved. Theorem 1. For the given scalars 0 < λ i < 1, μ ≥ 1, and τ > 0, the tracking error system (7) is exponentially stable with a prescribed H∞ disturbance attenuation level γ * = τ 1 − λa , if the variation frequency 1 − λλb ln λ of the sampling period is bounded by ϑ ≤ − ln , and there exist poμ sitive-definite matrices Pi > 0 such that the following inequalities hold for all s = 1, − 1, and i, j ∈ Φ1:
¯ ρ(t ) = ⎡⎣ IN ⊗ ρ(t ) − (tk ) ⊗ ρ(t )ρ(t )⎤⎦, k k k k
define σ =
+∞
(6)
where IN is an N dimensional identity matrix, and
(
3
i = σ ( IN + Δi ), where
⎤ ⎡ ^T ⎢ −λ iPi 0 i Pi I ⎥ ⎥ ⎢ 2 T ⎢ ⁎ −τ I Di Pi 0 ⎥ < 0 ⎢ ⁎ ⁎ −Pi 0 ⎥ ⎥ ⎢ ⁎ ⁎ −I ⎦ ⎣ ⁎
(8)
Pi ≤ μPj
(9)
^ = − σ (1 + sδ ) , λ = min{λ }, λ = max{λ }, λ > λ . where a i i i i i b i b Proof. To derive the stability and H∞ disturbance attenuation performance of system (7), we choose the following Lyapunov function:
(
)
Vρ(tk)(tk ) = eT (tk ) IN ⊗ Pρ(tk) e(tk ) Then, for each ρ(tk ) = i ∈ Φ1, we have
˜ iw (tk )⎤⎦T IN ⊗ Pρ(t ) ˜ ie(tk ) + Vi (tk + 1) − λ iVi (tk ) + Γ (tk ) = ⎡⎣ k
(
χmin + χmax
≤ δIN .
(
T
where
˜ ρ(t )e(tk ) + ⎡⎣ IN ⊗ ρ(t )⎤⎦w (tk ) e(tk + 1) = k k
T η(k ) = ⎡⎣ eT (k ) wT (k )⎤⎦ , then
(
)
Definition 1. System (7) is called robustly exponentially stable, if there exist some scalars π > 0 and 0 < ζ < 1, such that the solution e of system (7) satisfies e(tk ) < πζ (k − k0) e(t0) , ∀ tk ≥ t0 . Definition 2. For a given scalar γ > 0, system (7) is said to be robustly exponentially stable and achieves a prescribed H∞ performance γ. If it is exponentially stable and under zero initial
2
T
+ e (tk )e(tk ) − τ w (tk )w (tk )
(tk ) = σ ( IN + Δi (tk )), where Δi (tk ) ≤ δIN , the closed-loop system (6) can now be written as
˜ ρ(t ) = IN ⊗ ρ(t ) − σ ( IN + Δ(tk )) ⊗ ρ(t )ρ(t ) . where k k k k The considered consensus tracking problem is transformed as follows: design the controller in the form of (5) such that the closed-loop tracking error system (7) is exponentially stable with a prescribed H∞ disturbance attenuation level γ. In the next section, attention is focused on how to design the distributed controller (5) that achieves a desired control performance. Moreover, the impact of sampling period variation on the control performance will also need to be addressed. To help readers understand the control purpose more clearly, the following definitions are first introduced.
)
− λ ieT (tk ) IN ⊗ Pρ(tk) e(tk )
Then,
(7)
)
˜ iw (tk )⎤⎦ ˜ ie(tk ) + × ⎡⎣
Δi can be determined by 2χij − χmax − χmin
(10)
T
2
T
Γ (tk ) = e (tk )e(tk ) − τ w (tk )w (tk ),
D˜ i = IN ⊗ i .
Vi (tk + 1) − λ iVi (tk ) + Γ (tk ) = ηT (tk ) ⎡⎣ Ω + Ω1T P˜iΩ1 + Ω2T Ω2⎤⎦ η(tk )
(11) Define
(12)
where
P˜i = IN ⊗ Pi, ⎡ −λ P˜ 0 ⎤ i i ⎥, Ω=⎢ ⎢⎣ ⁎ −τ 2I ⎥⎦ ˜ i IN ⊗ i⎤⎦, Ω1 = ⎡⎣ Ω2 = ⎡⎣ I 0⎤⎦. By using the Schur complement, Ω + Ω1T (IN ⊗ Pi )Ω1 + Ω2T Ω2 < 0 is equivalent to
⎡ Ω Ω T (I ⊗ P ) Ω T ⎤ 1 N i 2 ⎥ ⎢ ⎢ ⁎ −(IN ⊗ Pi ) 0 ⎥<0 ⎢ ⎥ ⁎ −I ⎦ ⎣⁎ that is
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(13)
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⎡ ˜ ˜ iT ⊗ Pi ⎢ −λ iPi 0 ⎢ 2 T ⎢ ⁎ −τ I IN ⊗ i Pi ⎢ ⁎ −P˜i ⎢ ⁎ ⎢⎣ ⁎ ⁎ ⁎
(
⎤ I ⎥ ⎥ 0⎥ <0 ⎥ 0⎥ −I ⎥⎦
)
from condition ϑ ≤ −
lnλ lnμ
and λ > λb . According to Definition 1, the
tracking error system (7) is exponentially stable when w (k ) = 0. For H∞ performance level, we consider w (k ) ≠ 0. Under the zero initial condition, it follows from (16) that
(14)
Obviously, the left side of the above inequality can be written as
k−1
k−1
μ Nρ(s, k − 1) λ ak − s − 1eT (s )e(s ) ≤ τ 2
∑ s= k0
∑
μ Nρ(s, k − 1) λbk − s − 1w T (s )w (s )
s= k0
(18)
Ξ = IN ⊗ Ξ1 + Δ(tk ) ⊗ Ξ2.
where Nρ(s, k − 1) is the switching numbers over time interval
with
(s, k − 1). It can be seen that
⎡ ⎤ T T ⎢ −λ iPi 0 i Pi − σ ( i i) Pi I ⎥ ⎢ ⎥ 2 0 ⎥, iT Pi Ξ1 = ⎢ ⁎ −τ I ⎢ ⁎ 0⎥ ⁎ −Pi ⎢ ⎥ ⁎ ⁎ −I ⎦ ⎣ ⁎ ⎡ ⎢0 ⎢ Ξ2 = ⎢ ⁎ ⎢⁎ ⎢⎣ ⁎
Nρ(s, k − 1)
ln μ < lnλ readily seen that
∑
(15)
s = kl
It follows from (9) and (15) that k−1
∑ λρk(−kls)− 1Γ (s)
s = kl
∑ λρk(−kls)− 1Γ (s)
s = kl
⋮ ≤ μ Nϖ (k 0, k) λρk(−k k) lλρk(lk− kl −) 1⋯λρk(1k− k) 0 Vρ(k 0)(k 0) − Θ(Γ ) l
l−1
0
(16)
where l− 1
Θ(Γ ) =
μ Nρ λρk(−k k) l l
∏ s= 1
λρk(sk+)1− ks s
l− 1
k
l
j
s= 2
k1− 1
∑ s= k0
− kj
+ μ Nρ λρk(−k k) l ∏ λρ(jk+)1
λ
w T (s )w (s ) (19)
λρk(1k− 1) − sΓ (s ) 0
k2 − 1
∑ s = k1
+∞
λ ak − s − 1 < τ 2
∑ k = s+ 1
+∞
∑
+∞
w T (s )w (s )
s= k0
Since ∑k = s + 1 λak − s − 1 = +∞
∑
1 1 − λa
k − s− 1
∑ ( λb λ ) k = s+ 1
+∞
k−s−1
and ∑k = s + 1 ( λb λ)
=
1 , 1 − ( λb λ)
(20) we have
+∞
eT (s )e(s ) < γ 2
s= k0
where γ = τ
∑
w T (s )w (s )
s= k0 1 − λa 1 − λb λ
(21)
. It is noted that λb < λ , which ensures γ > 0. Let
k0 = 0, the tracking error system (7) is exponentially stable in the mean-square sense and achieves a prescribed H∞ performance □ level γ. This completes the proof. Remark 2. In our results, the exponential decay rate ζ = λbμ ϑ is a monotonic increasing function of the variation frequency ϑ. It means that a poor tracking performance is obtained if the sampling period is varying very frequently. Based on Theorem 1, we are on the stage to determine the controller gains.
k−1 l
. Then, it can be
k − s− 1 k − s− 1
∑ ( λb λ )
+∞
eT (s )e(s )
s= k0
∑ λlk − s − 1Γ (s)
≤ λρk(−k k) lμVρ(kl − 1)(kl ) −
<λ
μ > 1, we obtain
s= k0
+∞
∑
k−1
l
lnλ . Since lnμ −(k − s − 1)
Summing the above inequality from k = k0 + 1 to k = ∞ and changing the order of summation yields
Due to the fact that Δ(tk ) ≤ δIN , Ξ < 0 holds for all the uncertainty Δ(tk ) if and only if Ξ < 0 holds when Δ(tk ) = − δIN and Δ(tk ) = δIN . It follows from (8) that Ξ < 0 is true indeed. Thus Vi (tk + 1) − λ iVi (tk ) + Γ (tk ) < 0. For simplicity of sequential analysis, we replace tk by k. Define the switching time instants k0 < k1 < ⋯ < kl < ⋯ < kt < k , l = 1, 2, … , t , one has
Vρ(kl)(k ) ≤ λρk(−k k) lVρ(kl)(kl ) −
k−1
λ ak − s − 1eT (s )e(s ) < τ 2
s= k0
<−
k−s−1 Nρ(s, k − 1)
, and 1 < μ
k−1
⎤ T 0 −σ ( i i) Pi 0⎥ 0 0 0 ⎥. ⎥ ⁎ 0 0⎥ ⁎ ⁎ 0⎥⎦
Vl (k ) ≤ λlk − kl Vl (kl ) −
−(k − s − 1)
Nρ(s, k − 1)
λρk(2k−) 1 − sΥ (s ) 1
Theorem 2. For the given scalars 0 < λ i < 1, μ ≥ 1 and τ > 0, the considered tracking problem is solvable if the variation frequency of ln λ , and there exist positivethe sampling period is bounded by ϑ ≤ − ln μ definite matrices Q i > 0 such that the following inequalities
⎤ ⎡ T ˜T ⎢ − λ iQ i 0 Q i i + i Q i ⎥ ⎢ ⁎ −τ 2I 0⎥<0 iT ⎥ ⎢ ⎢ ⁎ ⁎ −Q i 0⎥ ⎥ ⎢ ⁎ ⁎ −I ⎦ ⎣ ⁎
(22)
Q j ≤ μQ i
(23)
k−1
+ ⋯ + μ0
∏ λρk(−kl1) − sΓ (s).
s = kl
Now, we consider the exponential stability of system (7) with w (k ) = 0. One has
Proof. Multiplying the left and right sides of (8) by diag{Pi−1, I , Pi−1, I} and its transpose, respectively, (8) is equivalent to
Vρ(kl)(k ) ≤ μ Nρ λρk(−k k) lλρk(lk− kl −) 1⋯λρk(1k− k) 0Vρ(k 0)(k 0) l
l−1
0
≤ μ Nρ λbk − k 0Vρ(k 0)(k 0)
(
≤ μ ϑ λb
k − k0
)
which yields e(k )
Vρ(k 0)(k 0) = ζ 2(k − k 0)Vρ(k 0)(k 0)
2
≤
φ2 2(k − k ) 0 ζ φ1
hold for all s = 1, − 1, and i, j ∈ Φ1. Moreover, the controller gain is ¯ iQ i−1, where ˜ Ti = − σ (1 + sδ ) Ti BiT . determined by i =
(17)
2
e(k0) , where φ1 = mini ∈ Φ1 χmin (Pi ),
φ2 = max i ∈ Φ1 χmax (Pi ), ζ = λbμ ϑ , χmin (Pi ) and χmax (Pi ) are the minimal and maximal eigenvalues of Pi. Therefore, one can readily obtain ζ < 1
⎡ ^ T P −1⎤ −1 0 Pi−1 ⎢ −λ iPi i i ⎥ ⎥ ⎢ T 2 ⁎ − τ I 0 ⎥<0 ⎢ i ⎥ ⎢ ⁎ −Pi−1 0 ⎥ ⎢ ⁎ ⎢⎣ ⁎ ⁎ ⁎ −I ⎥⎦
Please cite this article as: Zhang D, et al. (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.001i
(24)
D. Zhang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ T Let Q i = Pi−1 and ¯ i = Pi−1 Ti , (24) is equivalent to (22). In addition,
replacing Pi in (9) by Q i−1, then (9) can be written as Q i−1 ≤ μQ −j 1, which is (23). The proof is completed. □ When the follower's dynamics is not affected by disturbance, □ i.e., wi(t ) = 0, we have the following corollary. Corollary 1. For the given scalars 0 < λ i < 1, μ ≥ 1 and τ > 0, the modified tracking error system is exponentially stable if the varlnλ , iation frequency of the sampling period is bounded by ϑ ≤ − ln μ and there exist positive definite matrices Q i > 0, such that the inequality (23) and the following inequality
⎡ T ¯ T T⎤ ⎢ −λ iQ i Q i i − σ (1 + sδ ) i i ⎥ < 0 ⎥⎦ ⎢⎣ ⁎ −Q i
(25)
holds for all s = 1, − 1, and i, j ∈ Φ1. Moreover, the stabilizing ¯ iQ i−1. controller gain is determined by i = Remark 3. To obtain the minimal H∞ performance level γ, we can transform the conditions in Theorem 2 into the following LMIbased optimization problem:
min τ s. t.
(22), (23)
(26)
When the above optimization problem has a solution τ*, the optimal H∞ disturbance attenuation level is determined as
γ* = τ*
1 − λa 1 − λλb
. The corresponding controller gain can also be
obtained. Remark 4. To solve the optimization problem (26), the parameters λi and μ have to be chosen. One can see that LMI (23) is easy to be satisfied for a larger μ, but the bound of variation frequency of sampling period ϑ becomes smaller. Moreover, a larger μ can lead to a smaller H∞ disturbance attenuation level. Hence, the parameter μ should be chosen by considering the trade-off between the variation frequency of sampling period and the H∞ disturbance attenuation level. While, the choice of parameters λi is similar to the case of μ. This statement is demonstrated in simulation Example 2.
4. Simulation examples In this section, two simulation examples are introduced to demonstrate the effectiveness of the proposed control algorithm. Example 1 ([22]). Consider a leader–follower multi-agent networks with one leader and four followers. The state matrices are
⎡0 0 0 ⎤ ⎢ ⎥ = ⎢ 1 0 −1⎥, ⎣0 1 1 ⎦
= I,
= 0. (27)
⎡0 ⎢ 0 G1 = ⎢ ⎢0 ⎢⎣ 1
0 0 1 1
0 1 0 0
1⎤ ⎥ 1⎥ , 0⎥ ⎥ 0⎦
⎡1 ⎢ 0 M1 = ⎢ ⎢0 ⎢⎣ 0
0 0 0 0
0 0 1 0
1 0 0 0
1 0 0 1
0⎤ ⎥ 0⎥ , 1⎥ 0⎥⎦
⎡0 ⎢ 0 M2 = ⎢ ⎢0 ⎢⎣ 0
0 1 0 0
0 0 0 0
0⎤ ⎥ 0⎥ 0⎥ 1⎥⎦
(29)
When the above network is working under a time-varying sampling period, the upper bound of sampling period is obtained as 0.17 s in [22]. In our results, we assume that the sampling period is switching between { 0.2 s, 0.4 s}, i.e., T1 = 0.2 s, T2 = 0.4 s, and choose λ1 = 0.96, λ2 = 0.98, μ = 1.01, a feasible controller is obtained as
⎡ 1.8576 0.0031 0.0013 ⎤ ⎥ ⎢ 1 = ⎢ 0.4732 2.1033 −0.2233⎥, ⎣ −0.2477 0.2212 2.3327 ⎦ ⎡ 0.8826 0.0120 0.0109 ⎤ ⎢ ⎥ 2 = ⎢ 0.6130 1.2174 −0.2774 ⎥. ⎣ −0.3287 0.2780 1.5064 ⎦ One can see that the maximal allowable sampling period based on Corollary 1 is much larger than Theorem 7 in [22], which shows that our results are better than the ones in [22]. It follows from the above parameters λ1, λ2 and μ that we have λa = 0.96 , λb = 0.98, let λ = 0.99, then ϑ = 1.01 > 1, which means that the follower can track the leader with an exponential decay rate no matter how the sampling period is varying. In this simulation setup, we assume that the variation of the two sampling period is periodical, and the first one period is given as T1, T1, T1, T1, T2, T2, T2. To verify the control performance, the following initial conditions are taken:
⎧ ⎡ ⎪ x0 = ⎣ 4 ⎪ ⎪ x1 = ⎡⎣ 1 ⎪ ⎨ x2 = ⎡⎣ 3 ⎪ ⎪ x = ⎡⎣ 2 ⎪ 3 ⎪ ⎩ x4 = ⎣⎡ 5
T
8 10⎤⎦ , T 2 3⎤⎦ , T
6 9⎤⎦ , T
4 8⎤⎦ , T
9 7⎤⎦ .
(30)
The trajectories of tracking errors are depicted in Figs. 1–3. One can see that the tracking error system is stable. Now we compare our results with the ones in [22] and we focus on the control performance. We take the sampling period as T1 = 0.05 s, T2 = 0.1 s. By choosing the same parameters for λi and μ, a feasible controller is obtained as
⎡ 6.3270 ⎢ 1 = ⎢ 0.1572 ⎣ 0.0015 ⎡ 3.1645 ⎢ 2 = ⎢ 0.1609 ⎣ 0.0027
−0.0010 0.0004 ⎤ ⎥ 6.3263 −0.1602⎥, 0.1589 6.4827 ⎦ 0.0026 0.0002 ⎤ ⎥ 3.1621 −0.1616⎥. 0.1607 3.3274 ⎦
In order to perform the comparison study fairly, we choose the same initial conditions, the same sampling time instant and the same topology switching sequence. Figs. 4–6 show the simulation results. One can see our control algorithm can provide a better tracking performance than the one in [22]. Example 2. Consider a multi-agent network with one leader and four followers, and the dynamics is governed by the following double-integrator model:
The topology is switching between G1 and G2 with
0⎤ ⎥ 0⎥ 0⎥ 0⎥⎦
⎡0 ⎢ 1 G2 = ⎢ ⎢1 ⎢⎣ 0
5
(28)
⎧ ri = vi ⎨ ⎩ vi = ui + wi,
i = 1, 2, 3, 4
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(31)
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Time(k) Fig. 2. Trajectories of second element of ei.
Fig. 3. Trajectories of third element of ei.
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where r i, υ i, u i and wi are the position, velocity, control input and disturbance of the i-th agent, respectively. The above system (27) can be modeled by a linear state-space model with 0 1 0 0 = ⎡⎣ 0 0 ⎤⎦, = ⎡⎣ 1 ⎤⎦ and = ⎡⎣ 1 ⎤⎦. The above second-order multi-agent system has been applied to model the outer-loop dynamics of unmanned aerial vehicle (UAV), see [26] for more details. Due to the fact that we are lack of instruments and UAVs in our lab, we only perform a simulation study here as in [15–22]. In reality, one shall note that the topology of a multiagent network is usually switching due to the moving of the agent, we assume that the topology is varying between G1 and G2 with
⎡0 ⎢ 0 G1 = ⎢ ⎢1 ⎢⎣ 1
0 0 1 1
1 1 0 0
1⎤ ⎥ 1⎥ , 0⎥ ⎥ 0⎦
⎡1 ⎢ 0 M1 = ⎢ ⎢0 ⎢⎣ 0
0 1 0 0
0 0 1 0
0⎤ ⎥ 0⎥ 0⎥ 0⎥⎦
(32)
⎡0 ⎢ 1 G2 = ⎢ ⎢1 ⎢⎣ 0
1 0 0 0
1 0 0 1
0⎤ ⎥ 0⎥ , 1⎥ ⎥ 0⎦
⎡0 ⎢ 0 M2 = ⎢ ⎢0 ⎢⎣ 0
0 1 0 0
0 0 1 0
0⎤ ⎥ 0⎥ 0⎥ 1⎥⎦
(33)
The purpose is to design a distributed control algorithm such that all following agents track the leader by using the sampled-data.
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Time(k) Fig. 7. Trajectories of position error.
Suppose the agent is working under two sampling periods T ∈ { 0.1 s, 0.2 s}, i.e., T1 ¼ 0.1, and T2 = 0.2 s, we have
⎧ ⎪ 1 = ⎪ ⎨ ⎪ ⎪ 1 = ⎩
⎡ 1 0.1⎤ ⎢⎣ ⎥, 0 1⎦ ⎡ 0.005⎤ ⎢⎣ ⎥ 0.1 ⎦
⎧ ⎡ 1 0.2⎤ ⎪ 2 = ⎢ , ⎣ 0 1 ⎥⎦ ⎪ ⎨ ⎪ = ⎡ 0.02⎤ ⎪ 2 ⎢⎣ 0.2 ⎥⎦ ⎩
⎡ 0.005⎤ 1 = ⎢ , ⎣ 0.1 ⎥⎦ (34)
⎡ 0.02⎤ 2 = ⎢ , ⎣ 0.2 ⎥⎦
(35)
By choosing λ1 = 0.97, λ2 = 0.98, μ = 1.02, and solving the optimization problem (26), the controller gains are determined to be
1 = ⎡⎣ 2.4937 3.7890⎤⎦, 2 = ⎡⎣ 1.2540 1.9576⎤⎦.
(36)
Based on these parameters, we have λa = 0.97, λb = 0.98. Then, we take λ = 0.985, the upper bound of sampling period variation is
calculated to be ϑ* = −
lnλ lnμ
= 0.7632, and the H∞ disturbance at-
tenuation level is γ * = 2.0031. For the simulation purpose, the sampling period is periodically switching and the first period is given as: T1, T1, T1, T1, T2, T2, T2. We run simulation for 50 s. It follows from this sequence that the variation frequency is much smaller than ϑ*. Take the initial conT dition of the leader x0 = ⎡⎣ 1 10⎤⎦ , and the initial conditions of the followers as zero. The disturbance signal is assumed to be wi(k ) = sin(0.1k )⁎exp( − 0.1k ). Figs. 7 and 8 show the tracking errors of position and velocity. It is seen that the tracking error approaches to zero. We also depict the tracking trajectories in Figs. 9 and 10, when the initial conditions of the leader and followers are randomly generated from [0, 10]. This simulation example shows that a desired tracking performance can be obtained when the agent dynamics is affected by disturbance. We now discuss how the parameters λ i and μ affect the system performance. For the fixed λ1 = 0.97 and λ2 = 0.98, we still take λ = 0.985. The relation between μ and the H∞ performance level γ is listed in Table 1. One can see that a larger μ can help obtain a better H∞ performance level. Table 2 shows the relation between μ and variation bound of sampling period ϑ, and it indicates that the sampling period is not allowed to be varying frequently when a larger μ is chosen. For a given μ = 1.02, and λ2 = 0.98, we verify the
4 2 0 e12
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Time(k) Fig. 8. Trajectories of velocity error.
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Table 1 Relation between μ and γ *. μ
γ*
1.01 2.0129
1.02 2.0031
1.03 1.9936
1.04 1.9843
1.05 1.9753
1.03 0.5113
1.04 0.3853
1.05 0.3098
0.94 3.2129
0.95 2.8012
0.96 2.4027
Table 2 Relation between μ and ϑ. μ ϑ
1.01 1.5189
5. Conclusions 1.02 0.7632
Table 3 Relation between λ1 and γ *. λ1 γ*
0.92 4.1241
sampling period variation ϑ thus becomes smaller. For example, if we take λ = 0.992, then ϑ = 0.4056, which is smaller than ϑ = 0.7632. Hence, the parameters λi and μ have certain impacts on the system performance. As applying of the main results to real systems, one may consider the trade-off between the control performance and application range.
0.93 3.6500
relation between λ1 and system performance. The relation between λ1 and H∞ performance level γ is depicted in Table 3, and it shows that a larger λ1 can achieve a better H∞ performance level. Moreover, when we take λ1 = 0.99, which is larger than λ2, then λ must be chosen to be larger than 0.99, and the upper bound of
We have investigated the consensus tracking problem for a class of linear dynamic systems subject to time-varying sampling period, switching topologies and external disturbances. The sampling period is assumed to vary in a given set, and the topology is switching but keeps connected. A new discrete-time switched system model has been introduced for the analysis of closed-loop tracking error system. By applying the piece-wise Lyapunov functional method, we showed that the consensus is achieved if the variation frequency of sampling period is upper bounded. An LMI based optimization problem is given for the calculation of controller gains. Future discussions on the effect of sampling period variation are addressed as well. In our future, the finitetime consensus tracking problem for a class of linear or nonlinear agents is of our interest. The recent works on the finite-time
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10
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control in [27] may provide some insights to our future work. In addition, the so-called event-triggered transmission protocol [3,28–30] has recently attracted much attention, and it has shown that such a novel transmission can also reduce the transmission rate. Then how to further incorporate the event-based transmission protocol into the aperiodic sampling scheme is an interesting topic to be investigated in the future.
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[14] Zhao Y, Guo G, Ding L. Guaranteed cost control of mobile sensor networks with Markov switching topologies. ISA Trans 2015;58(September):206–13. [15] Xia YQ, Na XT, Sun ZQ, Chen J. Formation control and collision avoidance for multi-agent systems based on position estimation. ISA Trans 2016;61 (March):287–96. [16] Liang HJ, Zhang HG, Wang ZS. Distributed-observer-based cooperative control for synchronization of linear discrete-time multi-agent systems. ISA Trans 2015;58(September):206–13. [17] Zhang H, Yang RH, Yan HC, Yang FW. H∞ consensus of event-based multi-agent systems with switching topology. Inf Sci 2016;370–371(November):623–35. [18] Wang YW, Bian T, Xiao JW, Wen CY. Global synchronization of complex dynamical networks through digital communication with limited data rate. IEEE Trans Neural Netw Learn Syst 2015;26(October (10)):2487–99. [19] Zhang Y, Tian Y. Allowable sampling period for consensus control of multiple general linear dynamical agents in random networks. Int J Control 2010;83:2368–77. [20] Gao YP, Wang L. Sampled-data based consensus of continuous-time multiagent systems with time-varying topology. IEEE Trans Autom Control 2011;56:1226–31. [21] Wen GH, Duan ZS, Yu WW, Chen GR. Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach. Int J Robust Nonlinear Control 23;2013:602–19. [22] Zhang Y, Tian YP. Consensus tracking in sensor networks with periodic sensing and switching connected topologies. Syst Control Lett 2015;84(October):44–51. [23] Wu L, Yang R, Shi P, Su X. Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings. Automatica 2015;59(January (1)):206–15. [24] Zhang D, Shi P, Wang QG. Energy-efficient distributed control of large-scale systems: a switched system approach. Int J Robust Nonlinear Control 2016;26 (14):3101–17. [25] Zhou Q, Li H, Wu C, Wang L, Ahn CK. Adaptive fuzzy control of nonstrictfeedback nonlinear systems with unmodeled dynamics and input saturation using small-gain approach. IEEE Trans Syst Man Cybern: Syst, http://dx.doi. org/10.1109/TSMC.2016.2586108. [26] Dong XW, Zhou Y, Ren Z, Zhong YS. Time-varying formation control for unmanned aerial vehicles with switching interaction topologies. Control Eng Pract 2016;46(January):26–36. [27] Zhang HL, Cheng J, Wang HL, Chen YP, Xiang HL. Robust finite-time eventtriggered H∞ boundedness for network-based Markovian jump nonlinear systems. ISA Trans 2016;63:32–8. [28] Li H, Chen Z, Wu L, Lam HK. Event-triggered control for nonlinear systems under unreliable communication links. IEEE Trans Fuzzy Syst, http://dx.doi. org/10.1109/TFUZZ.2016.2578346. [29] Zhang H, Feng G, Yan HC, Chen QJ. Observer-based output feedback eventtriggered control for consensus of multi-agent systems. IEEE Trans Ind Electron 2014;61(9):4885–94. [30] Zhang D, Shi P, Wang QG, Yu L. Analysis and synthesis of networked control systems: a survey of recent advances and challenges. ISA Trans, http://dx.doi. org/10.1016/j.isatra.2016.09.026.
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Research article
Leader–follower H1 consensus of linear multi-agent systems with aperiodic sampling and switching connected topologies$ Dan Zhang a,b,n, Zhenhua Xu a, Qing-Guo Wang c, Yun-Bo Zhao a a
Department of Automation, Zhejiang University of Technology, Hangzhou 310023, PR China School of Automation, Hangzhou Dianzi University, Hangzhou 310018, PR China c Institute for Intelligent Systems, The University of Johannesburg, Johannesburg, South Africa b
art ic l e i nf o
a b s t r a c t
Article history: Received 20 June 2016 Received in revised form 1 January 2017 Accepted 1 January 2017
This paper is concerned with the distributed H∞ consensus of leader–follower multi-agent systems with aperiodic sampling interval and switching topologies. Under the assumption that the sampling period takes values from a given set, a new discrete-time model is proposed for the tracking error system. For the multi-agent systems with time-varying sampling period, switching topologies and external disturbance, the considered tracking problem is converted to a robust H∞ control problem. With help of the Lyapunov stability theory, a sufficient condition for the existence of mode-dependent controller is established and it guarantees the exponential stability of tracking error system and a prescribed H∞ disturbance attenuation level. The influence of sampling period on the overall control performance is also discussed. Two simulation examples are given to show the effectiveness of the proposed control algorithm. & 2017 Published by Elsevier Ltd. on behalf of ISA.
Keywords: Multi-agent systems Leader–follower consensus Aperiodic sampling Switching topology Linear matrix inequality (LMI)
1. Introduction In recent years, consensus of multi-agent systems has received increasing research attention due to its wide applications in various fields such as vehicle formation [1], cooperative target tracking in sensor networks [2,3], and power dispatch [4]. Generally, the consensus can be classified into two categories, i.e., leaderless consensus [5] and leader–follower consensus [6]. The main difference is whether or not there exist leaders in such coupled dynamical systems. So far, many interesting results have been reported in the literature for the consensus issue. For example, the distributed H∞ consensus of high order linear dynamics and switching topologies was addressed in [7]. By using tools of graph theory and the switched system approach, the consensus is achieved if the coupling strength among agents is larger than a desired positive value. In the presence of the actuator saturation, the leader–follower consensus problem for linear multi-agent system was investigated in [8], and the domain of consensus attraction of origin was estimated under the magnitude bounded ☆ This work was partially supported by the National Natural Science Foundation of China (61403341, 61673350), and the Open Foundation of the First Level Zhejiang Key in Key Discipline of Control Science and Engineering. n Corresponding author at: Department of Automation, Zhejiang University of Technology, Hangzhou 310023, PR China. E-mail addresses: [email protected] (D. Zhang), [email protected] (Q.-G. Wang).
disturbance. In [9], the internal model principle was applied onto the output synchronization control of non-identical linear multiagent systems. Very recently, the leader–follower consensus problem for a class of nonlinear agents was addressed in [10] and the pinning control method was utilized to achieve the consensus. To handle the nonlinearity and disturbance, an adaptive control approach [11] was also introduced, the details of which are given in [12]. More recent advances on the consensus of multi-agent system can be found in [13–17] and the references therein. It is noted that most works reported in the consensus literature are in the framework of continuous time, which requires the continuous communication among agents. However, many practical multi-agent systems are communicating by wireless network and such a continuous-time communication may consume a lot of transmission power. In order to save power of wireless nodes, it is more desirable that the agent only uses the sampled-data to update its dynamics [18], which may lead to periodic sampling or even aperiodic sampling. There have been some discussions on the consensus of multi-agent systems with sampled-data information. The consensus problem for a class of general linear multi-agent system with sampled-data was studied in [19] and an allowable upper bound of sampling period was given to achieve the consensus. By converting the sampled-data system to a discrete-time system with delays, the consensus of second-order multi-agent systems with sampled-data and switching topologies was investigated in [20]. It is shown that if the union graph of all direct
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2
graph has a spanning tree, then the consensus can be reached. The consensus of multi-agent system governed by nonlinear dynamics with sampled-data was also studied in [21], where the sampleddata system was transformed into an equivalent nonlinear system with a time-varying input delay, the upper bound of the maximal allowable sampling period was obtained by solving LMIs. Very recently, the similar approach was used in [22] for the leader– follower consensus of multi-agent systems with sampled-data and switching topologies. Based on the free weighting matrix method, a sufficient condition was obtained to achieve the consensus and the upper bound of the sampling period was also calculated. It should be noted that for the multi-agent systems with sampleddata, the input-delay approach is commonly used, and some useful delay information is usually missing in the derivation of their main t results. For example, the term − ∫ eṪ (s )(In ⊗ R )e(̇ s )ds was ent−h
larged to be − ∫
t
t − τ (t )
eṪ (s )(In ⊗ R )e(̇ s )ds in [22], where h and τ(t ) are
the upper bound of the sampling period and the time-delay, respectively, In is an Identity matrix with n dimension, R is a positive-definite matrix and ⊗ is the Kronecker product. Moreover, the designed controllers in the above literatures are all independent on the variation of the sampling period. It is well known that the mode-dependent controller can achieve a better control performance than the mode-independent one. Hence, consensus of linear multi-agent systems with time-varying sampled-data and switching topologies needs further improvement, which motivates the present study. In this paper, we study the leader–follower consensus problem of general linear multi-agent systems with switching topologies and external disturbance. The agent dynamics are homogeneous and only the sampled-data information is utilized for the controller design. The topology is allowed to be switching but keeps connected. By assuming that the sampling period is varying in a given set, a new discrete-time approach is proposed for the analysis of the tracking error system. Based on the Lyapunov stability theory, a sufficient condition is obtained such that the tracking error system is exponentially stable and achieves a desired H∞ disturbance attenuation level. An optimization problem is proposed to calculate the controller gains such that the H∞ performance index is minimized. Finally, two simulation examples are introduced to illustrate the effectiveness of the proposed control algorithm. Notations: Rn denotes the n dimensional Euclidean space. The superscript “T ” is used to stand for the matrix transposition. L2[0, ∞) stands for the space of square-integrable vector function over [0, ∞), • refers to the induced matrix two-norm. The symbol “n” is used to describe the symmetry of a matrix, and “⊗” is the Kronecker product. We use diag{⋯} to describe the block-diagonal matrix. χmin and χmax are the minimal and maximal eigenvalues, respectively. I and 0 represent the identity matrix and zero matrix with appropriate dimensions, respectively.
where x i(t ) ∈ Rn is the state of the i-th follower, i = 1, 2, … , N . ui(t ) ∈ Rm is the control input, and wi(t ) ∈ Rq is the disturbance, belonging into L2[0, ∞). and are some constant matrices with appropriate dimensions. Suppose all followers and the leader are clock synchronized and during each sampling period, the followers use the available sampled-data to update its state. Let the sampling instant be tk, and the sampling period is defined as hk = tk + 1 − tk , which is a time-varying value. We assume that the sampling period hk only takes a value from Φ = T1, T2, …, TN1 , where Tj = δjT0, j = 1, 2, … , N1,
{
}
δj is a positive scalar, T0 is a given sampling period, and N1 is the
total number of sampling periods. For the sampling period hk and by using a zero-order hold device, then for t ∈ [tk, tk + 1), the dynamics of the leader–follower can be described as
⎧ xi (tk + 1) = kxi (tk ) + kui (tk ) + kwi(tk ) ⎨ ⎩ x 0(tk + 1) = kx 0(tk ) where k = e hk , k =
(3)
h
h
∫0 k e τ dτ , and k = ∫0 k e τ dτ . Since the
Φ, then k = ( 0) k , l
sampling period can only take values from l −1
t
k = ∑tk= 0 ( 0) 0 , and T0 τ e dτ , 0
0 = ∫
t
l −1
k = ∑tk= 0 ( 0) 0 , where
0 = e T0 ,
T0 τ e dτ , 0
N1
0 = ∫
and lk =
{ δ , δ , …, δ }. It can 1
2
be seen that the values of k , k and k can change from one to another when the sampling period switches. We now introduce a switching signal ρ( tk ) ∈ Φ1 = { 1, 2, … , N1}, and the following discrete-time system model is obtained:
⎧ ⎪ x i (tk + 1) = ρ(t )x i (tk ) + ρ(t )ui (tk ) + ρ(t )wi(tk ) k k k ⎨ ⎪ ⎩ x 0(tk + 1) = ρ(tk)x 0(tk ) where δρ(t ) ∑t = 1k
ρ(t ) = ( 0) k
δρ(t ) k
,
ρ(t ) = k
δρ(t ) ∑t = 1k
(4) t−1
( 0)
0,
and
t−1
( 0) 0 . Since we have N1 numbers of sampling periods, each one may vary from one to another, thus a nature problem is how such a variation affects the control performance? To answer this problem, we define the variation numbers of these sampling periods over the time interval ( k0, k ) by Nρ , then the variation frequency can be
ρ(t ) = k
calculated by ϑ =
Nρ k − k0
. We will show in the next section that this
parameter contributes much to the control performance. Due to the fact that the followers are usually moving and the topology of the communication network may be dynamically changing [23,24]. In this paper, we assume that there are N2 possible topologies. As is well known, the interaction among followers can be described by a graph. Then, we denote the set of topologies of follower's interaction by G(t ) = ⎡⎣ gij(t )⎤⎦ , the eleN×N ment gij(t ) = 1, i ≠ j , if the i-th agent can receive information from the j-th element, otherwise gij(t ) = 0. Moreover, gii(t ) = 0. In this paper, we assume that the topology switching is finite, and it takes values from Θ2 = G1, G2, … , G N2 . Denote the corresponding La-
{
2. Problem formulation Consider a multi-agent system with one leader and N agents. The dynamics of the leader is represented by the following continuous-time linear time-invariant (LTI) model:
ẋ0(t ) = x 0(t )
(1) n
where x0(t ) ∈ R is the state vector of the leader, and the system matrix is constant, which may not be Hurwitz stable. The dynamics of the i-th follower is represented by the following continuous-time LTI model:
xi̇ (t ) = xi (t ) + ui (t ) + wi(t )
(2)
}
placian matrices by Li, i = 1, 2, … , N2 . The connection matrix between the leader and followers are represented by , where the element mi(t ) = 1 if the i-th follower M (t ) = ⎡⎣ mi(t )⎤⎦ N×N can receive information from the leader, otherwise it is zero. The interaction number of the leader–follower network is also assumed to be N2. Based on the above discussion, we see that L(t ) + M (t ) ∈ { L i + Mi, i = 1, 2, … , N2}. For consensus purpose, we assume that the topology Gi is connected and at each time instant, there is at least one follower has the information of leader node. To achieve the tracking performance, the following consensusbased control algorithm is applied:
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D. Zhang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
⎛ N ui = − ρ(tk)⎜⎜ ∑ gij (tk ) xi (tk ) − xj (tk ) ⎝ j=1
(
condition, ∑k = 0 eT (tk )e(tk ) ≤ ∑k = 0 γ 2wT (tk )w (tk ) is true. +∞
⎞ ⎟ ⎟ ⎠
)
− ρ(tk)mi (tk )( xi (tk ) − x 0(tk ))
(5)
where ρ(tk ) is the controller gains to be determined. Define the tracking error signal as ei(tk ) = x i(tk ) − x0(tk ), and based on the above analysis, we have the following closed-loop system:
¯ ρ(t )e(tk ) + ⎡⎣ IN ⊗ ρ(t )⎤⎦w (tk ) e(tk + 1) = k k
)
T e( tk ) = ⎡⎣ e1T ( tk ) e2T ( tk ) ⋯ eNT ( tk )⎤⎦ , T w ( tk ) = ⎡⎣ w1T ( tk ) w2T ( tk ) ⋯ wNT ( tk )⎤⎦ ,
(tk ) = L(tk ) + M (tk ). Remark 1. Compared with the existing results in [18–22], a new discrete-time closed-loop model has been obtained in (6), in which no delay is involved. This new model enables us to design the mode-dependent controller that fully uses the sampling information. It should be noted that we construct the controller on a linear multi-agent system. But the main results can be applied to the nonlinear case if a nonlinear multi-agent system can be linearized to be a linear dynamic system, e.g., using the fuzzy blending [25] or first order Taylor approximation. Let the eigenvalues of i be χij , i = 1, 2, … , N2, j = 1, 2, … , N . Since the topology Gi is undirected and connected and at least one follower detects the leader at each time instant, then it is easy to
{ }
{ }
verify that χij > 0. Denote χmin = min χij , χmax = max χij , and χmin + χmax , 2
definite matrix
Δi =
i σ
δ=
χmax − χmin , χmin + χmax
the symmetric and positive
i can be written by
− IN . The eigenvalues of
2χij − χmax − χmin , χmin + χmax
which
satisfy
3. Main results In this section, a sufficient condition is first presented, which guarantees the existence of controller (5) such that the above control performance is achieved. Theorem 1. For the given scalars 0 < λ i < 1, μ ≥ 1, and τ > 0, the tracking error system (7) is exponentially stable with a prescribed H∞ disturbance attenuation level γ * = τ 1 − λa , if the variation frequency 1 − λλb ln λ of the sampling period is bounded by ϑ ≤ − ln , and there exist poμ sitive-definite matrices Pi > 0 such that the following inequalities hold for all s = 1, − 1, and i, j ∈ Φ1:
¯ ρ(t ) = ⎡⎣ IN ⊗ ρ(t ) − (tk ) ⊗ ρ(t )ρ(t )⎤⎦, k k k k
define σ =
+∞
(6)
where IN is an N dimensional identity matrix, and
(
3
i = σ ( IN + Δi ), where
⎤ ⎡ ^T ⎢ −λ iPi 0 i Pi I ⎥ ⎥ ⎢ 2 T ⎢ ⁎ −τ I Di Pi 0 ⎥ < 0 ⎢ ⁎ ⁎ −Pi 0 ⎥ ⎥ ⎢ ⁎ ⁎ −I ⎦ ⎣ ⁎
(8)
Pi ≤ μPj
(9)
^ = − σ (1 + sδ ) , λ = min{λ }, λ = max{λ }, λ > λ . where a i i i i i b i b Proof. To derive the stability and H∞ disturbance attenuation performance of system (7), we choose the following Lyapunov function:
(
)
Vρ(tk)(tk ) = eT (tk ) IN ⊗ Pρ(tk) e(tk ) Then, for each ρ(tk ) = i ∈ Φ1, we have
˜ iw (tk )⎤⎦T IN ⊗ Pρ(t ) ˜ ie(tk ) + Vi (tk + 1) − λ iVi (tk ) + Γ (tk ) = ⎡⎣ k
(
χmin + χmax
≤ δIN .
(
T
where
˜ ρ(t )e(tk ) + ⎡⎣ IN ⊗ ρ(t )⎤⎦w (tk ) e(tk + 1) = k k
T η(k ) = ⎡⎣ eT (k ) wT (k )⎤⎦ , then
(
)
Definition 1. System (7) is called robustly exponentially stable, if there exist some scalars π > 0 and 0 < ζ < 1, such that the solution e of system (7) satisfies e(tk ) < πζ (k − k0) e(t0) , ∀ tk ≥ t0 . Definition 2. For a given scalar γ > 0, system (7) is said to be robustly exponentially stable and achieves a prescribed H∞ performance γ. If it is exponentially stable and under zero initial
2
T
+ e (tk )e(tk ) − τ w (tk )w (tk )
(tk ) = σ ( IN + Δi (tk )), where Δi (tk ) ≤ δIN , the closed-loop system (6) can now be written as
˜ ρ(t ) = IN ⊗ ρ(t ) − σ ( IN + Δ(tk )) ⊗ ρ(t )ρ(t ) . where k k k k The considered consensus tracking problem is transformed as follows: design the controller in the form of (5) such that the closed-loop tracking error system (7) is exponentially stable with a prescribed H∞ disturbance attenuation level γ. In the next section, attention is focused on how to design the distributed controller (5) that achieves a desired control performance. Moreover, the impact of sampling period variation on the control performance will also need to be addressed. To help readers understand the control purpose more clearly, the following definitions are first introduced.
)
− λ ieT (tk ) IN ⊗ Pρ(tk) e(tk )
Then,
(7)
)
˜ iw (tk )⎤⎦ ˜ ie(tk ) + × ⎡⎣
Δi can be determined by 2χij − χmax − χmin
(10)
T
2
T
Γ (tk ) = e (tk )e(tk ) − τ w (tk )w (tk ),
D˜ i = IN ⊗ i .
Vi (tk + 1) − λ iVi (tk ) + Γ (tk ) = ηT (tk ) ⎡⎣ Ω + Ω1T P˜iΩ1 + Ω2T Ω2⎤⎦ η(tk )
(11) Define
(12)
where
P˜i = IN ⊗ Pi, ⎡ −λ P˜ 0 ⎤ i i ⎥, Ω=⎢ ⎢⎣ ⁎ −τ 2I ⎥⎦ ˜ i IN ⊗ i⎤⎦, Ω1 = ⎡⎣ Ω2 = ⎡⎣ I 0⎤⎦. By using the Schur complement, Ω + Ω1T (IN ⊗ Pi )Ω1 + Ω2T Ω2 < 0 is equivalent to
⎡ Ω Ω T (I ⊗ P ) Ω T ⎤ 1 N i 2 ⎥ ⎢ ⎢ ⁎ −(IN ⊗ Pi ) 0 ⎥<0 ⎢ ⎥ ⁎ −I ⎦ ⎣⁎ that is
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(13)
D. Zhang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
⎡ ˜ ˜ iT ⊗ Pi ⎢ −λ iPi 0 ⎢ 2 T ⎢ ⁎ −τ I IN ⊗ i Pi ⎢ ⁎ −P˜i ⎢ ⁎ ⎢⎣ ⁎ ⁎ ⁎
(
⎤ I ⎥ ⎥ 0⎥ <0 ⎥ 0⎥ −I ⎥⎦
)
from condition ϑ ≤ −
lnλ lnμ
and λ > λb . According to Definition 1, the
tracking error system (7) is exponentially stable when w (k ) = 0. For H∞ performance level, we consider w (k ) ≠ 0. Under the zero initial condition, it follows from (16) that
(14)
Obviously, the left side of the above inequality can be written as
k−1
k−1
μ Nρ(s, k − 1) λ ak − s − 1eT (s )e(s ) ≤ τ 2
∑ s= k0
∑
μ Nρ(s, k − 1) λbk − s − 1w T (s )w (s )
s= k0
(18)
Ξ = IN ⊗ Ξ1 + Δ(tk ) ⊗ Ξ2.
where Nρ(s, k − 1) is the switching numbers over time interval
with
(s, k − 1). It can be seen that
⎡ ⎤ T T ⎢ −λ iPi 0 i Pi − σ ( i i) Pi I ⎥ ⎢ ⎥ 2 0 ⎥, iT Pi Ξ1 = ⎢ ⁎ −τ I ⎢ ⁎ 0⎥ ⁎ −Pi ⎢ ⎥ ⁎ ⁎ −I ⎦ ⎣ ⁎ ⎡ ⎢0 ⎢ Ξ2 = ⎢ ⁎ ⎢⁎ ⎢⎣ ⁎
Nρ(s, k − 1)
ln μ < lnλ readily seen that
∑
(15)
s = kl
It follows from (9) and (15) that k−1
∑ λρk(−kls)− 1Γ (s)
s = kl
∑ λρk(−kls)− 1Γ (s)
s = kl
⋮ ≤ μ Nϖ (k 0, k) λρk(−k k) lλρk(lk− kl −) 1⋯λρk(1k− k) 0 Vρ(k 0)(k 0) − Θ(Γ ) l
l−1
0
(16)
where l− 1
Θ(Γ ) =
μ Nρ λρk(−k k) l l
∏ s= 1
λρk(sk+)1− ks s
l− 1
k
l
j
s= 2
k1− 1
∑ s= k0
− kj
+ μ Nρ λρk(−k k) l ∏ λρ(jk+)1
λ
w T (s )w (s ) (19)
λρk(1k− 1) − sΓ (s ) 0
k2 − 1
∑ s = k1
+∞
λ ak − s − 1 < τ 2
∑ k = s+ 1
+∞
∑
+∞
w T (s )w (s )
s= k0
Since ∑k = s + 1 λak − s − 1 = +∞
∑
1 1 − λa
k − s− 1
∑ ( λb λ ) k = s+ 1
+∞
k−s−1
and ∑k = s + 1 ( λb λ)
=
1 , 1 − ( λb λ)
(20) we have
+∞
eT (s )e(s ) < γ 2
s= k0
where γ = τ
∑
w T (s )w (s )
s= k0 1 − λa 1 − λb λ
(21)
. It is noted that λb < λ , which ensures γ > 0. Let
k0 = 0, the tracking error system (7) is exponentially stable in the mean-square sense and achieves a prescribed H∞ performance □ level γ. This completes the proof. Remark 2. In our results, the exponential decay rate ζ = λbμ ϑ is a monotonic increasing function of the variation frequency ϑ. It means that a poor tracking performance is obtained if the sampling period is varying very frequently. Based on Theorem 1, we are on the stage to determine the controller gains.
k−1 l
. Then, it can be
k − s− 1 k − s− 1
∑ ( λb λ )
+∞
eT (s )e(s )
s= k0
∑ λlk − s − 1Γ (s)
≤ λρk(−k k) lμVρ(kl − 1)(kl ) −
<λ
μ > 1, we obtain
s= k0
+∞
∑
k−1
l
lnλ . Since lnμ −(k − s − 1)
Summing the above inequality from k = k0 + 1 to k = ∞ and changing the order of summation yields
Due to the fact that Δ(tk ) ≤ δIN , Ξ < 0 holds for all the uncertainty Δ(tk ) if and only if Ξ < 0 holds when Δ(tk ) = − δIN and Δ(tk ) = δIN . It follows from (8) that Ξ < 0 is true indeed. Thus Vi (tk + 1) − λ iVi (tk ) + Γ (tk ) < 0. For simplicity of sequential analysis, we replace tk by k. Define the switching time instants k0 < k1 < ⋯ < kl < ⋯ < kt < k , l = 1, 2, … , t , one has
Vρ(kl)(k ) ≤ λρk(−k k) lVρ(kl)(kl ) −
k−1
λ ak − s − 1eT (s )e(s ) < τ 2
s= k0
<−
k−s−1 Nρ(s, k − 1)
, and 1 < μ
k−1
⎤ T 0 −σ ( i i) Pi 0⎥ 0 0 0 ⎥. ⎥ ⁎ 0 0⎥ ⁎ ⁎ 0⎥⎦
Vl (k ) ≤ λlk − kl Vl (kl ) −
−(k − s − 1)
Nρ(s, k − 1)
λρk(2k−) 1 − sΥ (s ) 1
Theorem 2. For the given scalars 0 < λ i < 1, μ ≥ 1 and τ > 0, the considered tracking problem is solvable if the variation frequency of ln λ , and there exist positivethe sampling period is bounded by ϑ ≤ − ln μ definite matrices Q i > 0 such that the following inequalities
⎤ ⎡ T ˜T ⎢ − λ iQ i 0 Q i i + i Q i ⎥ ⎢ ⁎ −τ 2I 0⎥<0 iT ⎥ ⎢ ⎢ ⁎ ⁎ −Q i 0⎥ ⎥ ⎢ ⁎ ⁎ −I ⎦ ⎣ ⁎
(22)
Q j ≤ μQ i
(23)
k−1
+ ⋯ + μ0
∏ λρk(−kl1) − sΓ (s).
s = kl
Now, we consider the exponential stability of system (7) with w (k ) = 0. One has
Proof. Multiplying the left and right sides of (8) by diag{Pi−1, I , Pi−1, I} and its transpose, respectively, (8) is equivalent to
Vρ(kl)(k ) ≤ μ Nρ λρk(−k k) lλρk(lk− kl −) 1⋯λρk(1k− k) 0Vρ(k 0)(k 0) l
l−1
0
≤ μ Nρ λbk − k 0Vρ(k 0)(k 0)
(
≤ μ ϑ λb
k − k0
)
which yields e(k )
Vρ(k 0)(k 0) = ζ 2(k − k 0)Vρ(k 0)(k 0)
2
≤
φ2 2(k − k ) 0 ζ φ1
hold for all s = 1, − 1, and i, j ∈ Φ1. Moreover, the controller gain is ¯ iQ i−1, where ˜ Ti = − σ (1 + sδ ) Ti BiT . determined by i =
(17)
2
e(k0) , where φ1 = mini ∈ Φ1 χmin (Pi ),
φ2 = max i ∈ Φ1 χmax (Pi ), ζ = λbμ ϑ , χmin (Pi ) and χmax (Pi ) are the minimal and maximal eigenvalues of Pi. Therefore, one can readily obtain ζ < 1
⎡ ^ T P −1⎤ −1 0 Pi−1 ⎢ −λ iPi i i ⎥ ⎥ ⎢ T 2 ⁎ − τ I 0 ⎥<0 ⎢ i ⎥ ⎢ ⁎ −Pi−1 0 ⎥ ⎢ ⁎ ⎢⎣ ⁎ ⁎ ⁎ −I ⎥⎦
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(24)
D. Zhang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ T Let Q i = Pi−1 and ¯ i = Pi−1 Ti , (24) is equivalent to (22). In addition,
replacing Pi in (9) by Q i−1, then (9) can be written as Q i−1 ≤ μQ −j 1, which is (23). The proof is completed. □ When the follower's dynamics is not affected by disturbance, □ i.e., wi(t ) = 0, we have the following corollary. Corollary 1. For the given scalars 0 < λ i < 1, μ ≥ 1 and τ > 0, the modified tracking error system is exponentially stable if the varlnλ , iation frequency of the sampling period is bounded by ϑ ≤ − ln μ and there exist positive definite matrices Q i > 0, such that the inequality (23) and the following inequality
⎡ T ¯ T T⎤ ⎢ −λ iQ i Q i i − σ (1 + sδ ) i i ⎥ < 0 ⎥⎦ ⎢⎣ ⁎ −Q i
(25)
holds for all s = 1, − 1, and i, j ∈ Φ1. Moreover, the stabilizing ¯ iQ i−1. controller gain is determined by i = Remark 3. To obtain the minimal H∞ performance level γ, we can transform the conditions in Theorem 2 into the following LMIbased optimization problem:
min τ s. t.
(22), (23)
(26)
When the above optimization problem has a solution τ*, the optimal H∞ disturbance attenuation level is determined as
γ* = τ*
1 − λa 1 − λλb
. The corresponding controller gain can also be
obtained. Remark 4. To solve the optimization problem (26), the parameters λi and μ have to be chosen. One can see that LMI (23) is easy to be satisfied for a larger μ, but the bound of variation frequency of sampling period ϑ becomes smaller. Moreover, a larger μ can lead to a smaller H∞ disturbance attenuation level. Hence, the parameter μ should be chosen by considering the trade-off between the variation frequency of sampling period and the H∞ disturbance attenuation level. While, the choice of parameters λi is similar to the case of μ. This statement is demonstrated in simulation Example 2.
4. Simulation examples In this section, two simulation examples are introduced to demonstrate the effectiveness of the proposed control algorithm. Example 1 ([22]). Consider a leader–follower multi-agent networks with one leader and four followers. The state matrices are
⎡0 0 0 ⎤ ⎢ ⎥ = ⎢ 1 0 −1⎥, ⎣0 1 1 ⎦
= I,
= 0. (27)
⎡0 ⎢ 0 G1 = ⎢ ⎢0 ⎢⎣ 1
0 0 1 1
0 1 0 0
1⎤ ⎥ 1⎥ , 0⎥ ⎥ 0⎦
⎡1 ⎢ 0 M1 = ⎢ ⎢0 ⎢⎣ 0
0 0 0 0
0 0 1 0
1 0 0 0
1 0 0 1
0⎤ ⎥ 0⎥ , 1⎥ 0⎥⎦
⎡0 ⎢ 0 M2 = ⎢ ⎢0 ⎢⎣ 0
0 1 0 0
0 0 0 0
0⎤ ⎥ 0⎥ 0⎥ 1⎥⎦
(29)
When the above network is working under a time-varying sampling period, the upper bound of sampling period is obtained as 0.17 s in [22]. In our results, we assume that the sampling period is switching between { 0.2 s, 0.4 s}, i.e., T1 = 0.2 s, T2 = 0.4 s, and choose λ1 = 0.96, λ2 = 0.98, μ = 1.01, a feasible controller is obtained as
⎡ 1.8576 0.0031 0.0013 ⎤ ⎥ ⎢ 1 = ⎢ 0.4732 2.1033 −0.2233⎥, ⎣ −0.2477 0.2212 2.3327 ⎦ ⎡ 0.8826 0.0120 0.0109 ⎤ ⎢ ⎥ 2 = ⎢ 0.6130 1.2174 −0.2774 ⎥. ⎣ −0.3287 0.2780 1.5064 ⎦ One can see that the maximal allowable sampling period based on Corollary 1 is much larger than Theorem 7 in [22], which shows that our results are better than the ones in [22]. It follows from the above parameters λ1, λ2 and μ that we have λa = 0.96 , λb = 0.98, let λ = 0.99, then ϑ = 1.01 > 1, which means that the follower can track the leader with an exponential decay rate no matter how the sampling period is varying. In this simulation setup, we assume that the variation of the two sampling period is periodical, and the first one period is given as T1, T1, T1, T1, T2, T2, T2. To verify the control performance, the following initial conditions are taken:
⎧ ⎡ ⎪ x0 = ⎣ 4 ⎪ ⎪ x1 = ⎡⎣ 1 ⎪ ⎨ x2 = ⎡⎣ 3 ⎪ ⎪ x = ⎡⎣ 2 ⎪ 3 ⎪ ⎩ x4 = ⎣⎡ 5
T
8 10⎤⎦ , T 2 3⎤⎦ , T
6 9⎤⎦ , T
4 8⎤⎦ , T
9 7⎤⎦ .
(30)
The trajectories of tracking errors are depicted in Figs. 1–3. One can see that the tracking error system is stable. Now we compare our results with the ones in [22] and we focus on the control performance. We take the sampling period as T1 = 0.05 s, T2 = 0.1 s. By choosing the same parameters for λi and μ, a feasible controller is obtained as
⎡ 6.3270 ⎢ 1 = ⎢ 0.1572 ⎣ 0.0015 ⎡ 3.1645 ⎢ 2 = ⎢ 0.1609 ⎣ 0.0027
−0.0010 0.0004 ⎤ ⎥ 6.3263 −0.1602⎥, 0.1589 6.4827 ⎦ 0.0026 0.0002 ⎤ ⎥ 3.1621 −0.1616⎥. 0.1607 3.3274 ⎦
In order to perform the comparison study fairly, we choose the same initial conditions, the same sampling time instant and the same topology switching sequence. Figs. 4–6 show the simulation results. One can see our control algorithm can provide a better tracking performance than the one in [22]. Example 2. Consider a multi-agent network with one leader and four followers, and the dynamics is governed by the following double-integrator model:
The topology is switching between G1 and G2 with
0⎤ ⎥ 0⎥ 0⎥ 0⎥⎦
⎡0 ⎢ 1 G2 = ⎢ ⎢1 ⎢⎣ 0
5
(28)
⎧ ri = vi ⎨ ⎩ vi = ui + wi,
i = 1, 2, 3, 4
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(31)
D. Zhang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −3
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Time(k) Fig. 1. Trajectories of first element of ei.
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Time(k) Fig. 2. Trajectories of second element of ei.
Fig. 3. Trajectories of third element of ei.
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D. Zhang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1
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our results
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Fig. 4. Comparative results on trajectories of first element of ei.
2
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our results
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Fig. 5. Comparative results on trajectories of second element of ei.
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our results
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[22]
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Fig. 6. Comparative results on trajectories of third element of ei.
where r i, υ i, u i and wi are the position, velocity, control input and disturbance of the i-th agent, respectively. The above system (27) can be modeled by a linear state-space model with 0 1 0 0 = ⎡⎣ 0 0 ⎤⎦, = ⎡⎣ 1 ⎤⎦ and = ⎡⎣ 1 ⎤⎦. The above second-order multi-agent system has been applied to model the outer-loop dynamics of unmanned aerial vehicle (UAV), see [26] for more details. Due to the fact that we are lack of instruments and UAVs in our lab, we only perform a simulation study here as in [15–22]. In reality, one shall note that the topology of a multiagent network is usually switching due to the moving of the agent, we assume that the topology is varying between G1 and G2 with
⎡0 ⎢ 0 G1 = ⎢ ⎢1 ⎢⎣ 1
0 0 1 1
1 1 0 0
1⎤ ⎥ 1⎥ , 0⎥ ⎥ 0⎦
⎡1 ⎢ 0 M1 = ⎢ ⎢0 ⎢⎣ 0
0 1 0 0
0 0 1 0
0⎤ ⎥ 0⎥ 0⎥ 0⎥⎦
(32)
⎡0 ⎢ 1 G2 = ⎢ ⎢1 ⎢⎣ 0
1 0 0 0
1 0 0 1
0⎤ ⎥ 0⎥ , 1⎥ ⎥ 0⎦
⎡0 ⎢ 0 M2 = ⎢ ⎢0 ⎢⎣ 0
0 1 0 0
0 0 1 0
0⎤ ⎥ 0⎥ 0⎥ 1⎥⎦
(33)
The purpose is to design a distributed control algorithm such that all following agents track the leader by using the sampled-data.
Please cite this article as: Zhang D, et al. (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.001i
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8
0.5 0 −0.5
e11 e2
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e31
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Time(k) Fig. 7. Trajectories of position error.
Suppose the agent is working under two sampling periods T ∈ { 0.1 s, 0.2 s}, i.e., T1 ¼ 0.1, and T2 = 0.2 s, we have
⎧ ⎪ 1 = ⎪ ⎨ ⎪ ⎪ 1 = ⎩
⎡ 1 0.1⎤ ⎢⎣ ⎥, 0 1⎦ ⎡ 0.005⎤ ⎢⎣ ⎥ 0.1 ⎦
⎧ ⎡ 1 0.2⎤ ⎪ 2 = ⎢ , ⎣ 0 1 ⎥⎦ ⎪ ⎨ ⎪ = ⎡ 0.02⎤ ⎪ 2 ⎢⎣ 0.2 ⎥⎦ ⎩
⎡ 0.005⎤ 1 = ⎢ , ⎣ 0.1 ⎥⎦ (34)
⎡ 0.02⎤ 2 = ⎢ , ⎣ 0.2 ⎥⎦
(35)
By choosing λ1 = 0.97, λ2 = 0.98, μ = 1.02, and solving the optimization problem (26), the controller gains are determined to be
1 = ⎡⎣ 2.4937 3.7890⎤⎦, 2 = ⎡⎣ 1.2540 1.9576⎤⎦.
(36)
Based on these parameters, we have λa = 0.97, λb = 0.98. Then, we take λ = 0.985, the upper bound of sampling period variation is
calculated to be ϑ* = −
lnλ lnμ
= 0.7632, and the H∞ disturbance at-
tenuation level is γ * = 2.0031. For the simulation purpose, the sampling period is periodically switching and the first period is given as: T1, T1, T1, T1, T2, T2, T2. We run simulation for 50 s. It follows from this sequence that the variation frequency is much smaller than ϑ*. Take the initial conT dition of the leader x0 = ⎡⎣ 1 10⎤⎦ , and the initial conditions of the followers as zero. The disturbance signal is assumed to be wi(k ) = sin(0.1k )⁎exp( − 0.1k ). Figs. 7 and 8 show the tracking errors of position and velocity. It is seen that the tracking error approaches to zero. We also depict the tracking trajectories in Figs. 9 and 10, when the initial conditions of the leader and followers are randomly generated from [0, 10]. This simulation example shows that a desired tracking performance can be obtained when the agent dynamics is affected by disturbance. We now discuss how the parameters λ i and μ affect the system performance. For the fixed λ1 = 0.97 and λ2 = 0.98, we still take λ = 0.985. The relation between μ and the H∞ performance level γ is listed in Table 1. One can see that a larger μ can help obtain a better H∞ performance level. Table 2 shows the relation between μ and variation bound of sampling period ϑ, and it indicates that the sampling period is not allowed to be varying frequently when a larger μ is chosen. For a given μ = 1.02, and λ2 = 0.98, we verify the
4 2 0 e12
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e4
2
−6 −8 −10
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Time(k) Fig. 8. Trajectories of velocity error.
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180
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9
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Time(k) Fig. 9. Trajectories of position error.
4 2 0 e12
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Time(k) Fig. 10. Trajectories of velocity error.
Table 1 Relation between μ and γ *. μ
γ*
1.01 2.0129
1.02 2.0031
1.03 1.9936
1.04 1.9843
1.05 1.9753
1.03 0.5113
1.04 0.3853
1.05 0.3098
0.94 3.2129
0.95 2.8012
0.96 2.4027
Table 2 Relation between μ and ϑ. μ ϑ
1.01 1.5189
5. Conclusions 1.02 0.7632
Table 3 Relation between λ1 and γ *. λ1 γ*
0.92 4.1241
sampling period variation ϑ thus becomes smaller. For example, if we take λ = 0.992, then ϑ = 0.4056, which is smaller than ϑ = 0.7632. Hence, the parameters λi and μ have certain impacts on the system performance. As applying of the main results to real systems, one may consider the trade-off between the control performance and application range.
0.93 3.6500
relation between λ1 and system performance. The relation between λ1 and H∞ performance level γ is depicted in Table 3, and it shows that a larger λ1 can achieve a better H∞ performance level. Moreover, when we take λ1 = 0.99, which is larger than λ2, then λ must be chosen to be larger than 0.99, and the upper bound of
We have investigated the consensus tracking problem for a class of linear dynamic systems subject to time-varying sampling period, switching topologies and external disturbances. The sampling period is assumed to vary in a given set, and the topology is switching but keeps connected. A new discrete-time switched system model has been introduced for the analysis of closed-loop tracking error system. By applying the piece-wise Lyapunov functional method, we showed that the consensus is achieved if the variation frequency of sampling period is upper bounded. An LMI based optimization problem is given for the calculation of controller gains. Future discussions on the effect of sampling period variation are addressed as well. In our future, the finitetime consensus tracking problem for a class of linear or nonlinear agents is of our interest. The recent works on the finite-time
Please cite this article as: Zhang D, et al. (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.001i
10
D. Zhang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
control in [27] may provide some insights to our future work. In addition, the so-called event-triggered transmission protocol [3,28–30] has recently attracted much attention, and it has shown that such a novel transmission can also reduce the transmission rate. Then how to further incorporate the event-based transmission protocol into the aperiodic sampling scheme is an interesting topic to be investigated in the future.
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Please cite this article as: Zhang D, et al. (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.001i