Coordination networks under noisy measurements and sensor biases
Shi, Mingming
DOI:
10.33612/diss.99968844
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date: 2019
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Shi, M. (2019). Coordination networks under noisy measurements and sensor biases. Rijksuniversiteit Groningen. https://doi.org/10.33612/diss.99968844
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum.
4
Benefits of saturating
information in coordination
networks with noise
abstract
In this chapter, we propose a new control algorithm to mitigate the state drift phenomenon of consensus dynamics under communication noise by assigning each node a so-called favourite interval that characterizes the system desired convergence region. The algorithm is based on the self-triggered framework given in the previous chapter. If the nodes do not share a global clock, the net-work can operate in a fully asynchronous mode. By this algorithm, we show that the state evolution is confined around the favourite interval and the node disagreement is bounded by a simple linear function of the noise magnitude, without requiring any priori information on the noise. We also show that if the nodes share some global information, then the algorithm can be adjusted to make the nodes evolve into the favourite interval, improve on the disagree-ment bound and achieve asymptotic consensus in the noiseless case.
Published as:
M. Shi, C. D. Persis, and P. Tesi, “On the benefits of saturating information in coordin-ation networks with noise,” In Preparcoordin-ation.
4.1
introduction
Although the proposed coordination method in the previous chapter has been shown to be robust to the communication noise, the node disagreement de-pends on the norm of the initial state, which may cause a large consensus error. Moreover, the bound on the state depends on the noise magnitude and is un-predictable under the assumption that the magnitude of the noise is unknown. On the other hand, in real scenarios network systems are usually expected to evolve in a certain region. Hence in this chapter we propose a new method which, while still retaining the same properties of the algorithm in the previ-ous chapter, also ensures that the bound on the disagreement is independent of the norm of the initial conditions and the state bound are unrelated to the noise magnitude, thus guaranteeing a convergence result that is uniform over the system initial conditions.
The new algorithm assigns the nodes a “favourite” interval where the state aims at evolving and lets the nodes saturate the received information when the received state lies outside the interval. The favourite interval embodies the idea of a preferred operating range where the nodes are supposed to take values on. The favourite interval leads to a saturated control, which as shown in Franceschelli et al. (2017) and Senejohnny et al. (2019) is effective in con-trasting perturbations from outliers and uncooperative agents. We can show that the system achieves consensus with a tolerance that can be made arbit-rarily small and has a linear dependence on the noise magnitude in the noisy case. Moreover the state is confined around the given operating region for all time. The idea of using saturated version of the state to limit the system ex-cursion is inspired by the so-called interval consensus of Fontan et al. (2017). In this chapter, we demonstrate that saturation is also useful in countering the spreading of noise over the network.
Same as the algorithm in the previous chapter, the proposed method also ad-opts a self-triggered control scheme, which was first proposed in De Persis and Frasca (2013). At each update time, each node collects the state information from the neighbours, compute its next update instant and determines the con-trol value over the next sampling interval, resulting in a dynamical network with fully asynchronous information transmission and no global clock. Mo-tivated by wireless communication, event/self triggered methods have been prevailing in the network control systems literature recently, since they en-joy the advantage of packet-based data exchange among the agents Seyboth et al. (2013)-Kadowaki and Ishii (2014). Compared with the event-triggered
method, the self-triggered one does not require the agent to continuously mon-itor its state or listen to the communication channel, hence it can reduce the energy cost of sensing Kartakis et al. (2018). Moreover, it has been shown to be robust to various kind of malfunctions in the network, such as data loss Senejohnny et al. (2018) and misbehaving nodes Senejohnny et al. (2019).
4.2 problem formulation – self-triggered interval
consensus
We consider an undirected connected network with n nodes and its corres-ponding graph G ={I, E}, with with I = {1, 2, ..., n} the set of nodes and E the set of edges. For each node i∈ I, we assume its dynamics is described by
˙xi = ui (4.1)
where xi ∈ R is the state and ui ∈ R is the control input. The nodes aim at
evolving towards a common value, which is not agreed upon a priori. To fulfil this task, each node needs to obtain the state information from other nodes using communication or measurements. The information transmission may be affected by noise, so we assume that node i ∈ Nj receives the following
noisy state of node j
xwij(t) = xj(t) + wij(t) (4.2)
where wij(t)∈ R represents the communication noise when node j transmits
information to node i at time t. Same as in chapter 3, the noise is assumed to be bounded in magnitude, namely|wij(t)| ≤ |w|∞for all{i, j} ∈ E and all t ∈ R≥0.
Even though|w|∞may be estimated by empirical tests in some cases, this may not be always convenient, and we assume its value is not known.
We assume that all the nodes have a so-called favourite interval [p, q] ⊂ R in which the node would like its state to evolve. This interval is shared among all the nodes. Based on this favourite interval, for each neighbour j∈ Ni, node i
takes the saturated version of the received state of j, namely
where sat(z) = p ifz < p z ifp≤ z ≤ q q ifq > z (4.4)
Node i then computes the saturated noisy average
zw i(t) =
∑
j∈Ni
(yij(t)− xi(t)) (4.5)
When the communication network is ideal, i.e., it is noise-free we drop the superscript w. We also denote the actual average of each node i∈ I by
avei(t) =
∑
j∈Ni
(xj(t)− xi(t)) (4.6)
For each node i ∈ I, let {tik}k∈Z≥0, with t
i
0 = 0, be the sequence of triggering
times at which node i access the communication network. At these times, the node collects the state information from it neighbours, updates its control ac-tion and determines the next triggering time.
The control signals take values in the set U := {−1, 0, +1}, and the specific quantizer of choice is signα: R→ U, α > 0, which is given by
signα(z) :=
{
sign(z) if|z| ≥ α
0 otherwise (4.7)
The control action is given by
ui(t) = signϵ ( zwi(tik) ) (4.8) for t∈ [ti
k,tik+1[, where ϵ > 0 is a design threshold determining the consensus
The triggering times are given by ti k+1=tik+Δ i k, where Δik:= |zw i(tik)| 4di if |z w i(tik)| ≥ ϵ ϵ 4di otherwise (4.9)
Notice that the algorithm is Zeno free since the inter-sampling times are always bounded from below by a positive value by construction.
Remark 4.1. The state favourite interval should be chosen according to the
physical constraints and task requirements. In the distributed estimation prob-lem, all the sensors may seek to agree on the value of some physical variable. In this case, the interval [p, q] can be pre-set according to the operating range of the sensors. If the task is cooperative surveillance of a specific region, the multi-agent system may need to create a formation and stay within the target region. In this scenario, the interval can be set as the boundary of the region.
4.3 main result
We present the main result of the proposed self-triggered consensus method in this section.
4.3.1 state convergence interval
Denote dmin = mini∈Vdiand dmax = maxi∈Vdi, and let x(t) = maxixi(t) and
x(t) = minixi(t). We start by showing that the system state is always bounded
during the evolution and converges to an interval which depends on the end values of the favourite interval, the threshold ϵ and the noise magnitude.
Theorem 4.1. Consider a network of n dynamical systems as in (4.1), which are
in-terconnected over an undirected connected graph G = (I, E). Let each local control input be generated in accordance with (4.3)-(4.9). Then for all t∈ R≥0
x(t)≤ max{x(0), q + ϵ dmin} (4.10) x(t)≥ min{x(0), p − ϵ dmin} (4.11)
Moreover, there exists a finite time T such that xi(t)∈ [ p− ϵ dmin , q + ϵ dmin ] (4.12)
for all t≥ T and all i ∈ I.
Proof. We only prove the upper bounds, since the conclusion for the lower
bound can be derived in a similar way. Let γ = q + dminϵ . We first show two facts that will be used later.
Fact 1. For any node i∈ I and any t′≥ 0, if xi(t′)≤ γ, then it will never exceed
γ for all time t≥ t′.
This fact can be checked as follows. Let ti
m = max{tik∈ R≥0,tki ≤ t′}. If ui(t′)≤
0, then ui(t)≤ 0 for all t ∈ [tim,tim+1[, hence xi(t)≤ xi(t′)≤ γ for all t ∈ [t′,tim+1[.
While if ui(t′) =1, according to (4.8), the noisy saturated average at timshould
satisfy zwi(tim) = ∑ j∈Ni (yij(tim)− xi(tim))≥ ϵ (4.13) which implies xi(tim) ≤ 1 di (∑ j∈Ni yij(tim)− ϵ) = 1 di (∑ j∈Ni (sat(xj(tim) +wij(tim)))− ϵ) ≤ q − ϵ di (4.14)
where the last inequality comes from the definition of the saturation function (4.4). Consider the evolution of xi(t) for t∈ [tim,tim+1[, we have
xi(t) = xi(tim) +ui(tim)(t− tim) ≤ xi(tim) +Δ i m = xi(tim) + 1 4di|z w i (tim)|
= xi(tim) + 1 4di ∑ j∈Ni (yij(tim)− xi(tim)) ≤ xi(tim) + 1 4(q− xi(t i m)) = 3 4xi(t i m) + 1 4q ≤ q − 3ϵ 4di <γ (4.15)
where we used (4.13) in the third equality, (4.14) in the third inequality. By the induction argument, we conclude that xi(t)≤ γ for all t ≥ t′. Note that
by the analogous reasoning as above, this fact also holds if we replace γ by q.
Fact 2. For any node i∈ I and any t′ ≥ 0, if xi(t′) > γ, it decreases at time t′,
namely ui(t′) <0.
This fact can be checked as follows. From Fact 1, we have xi(tim)should be
larger than γ, otherwise xi(t′)≤ γ. Consider the saturated average
zwi (tim) = ∑ j∈Ni (yij(tim)− xi(tim)) < di(q− γ) = − di dmin ϵ≤ −ϵ (4.16)
which implies that ui(t′) =ui(tim) =−1. Hence xidecreases at t = t′.
We now finalize the proof. If x(0)≤ γ, by Fact 1, we have that xi(t)≤ γ for all
t≥ 0. If x(0) > γ, for the nodes with xi(0)≤ γ, xi(t)≤ γ ≤ x(0) holds for all
t≥ 0 by Fact 1. While for the nodes whose initial condition is greater than γ,
according to Fact 2, their states decrease with the same constant rate until the state is no greater than γ. This implies that x(t)≤ x(0) for all t > 0, which ends the proof of the first claim. The second claim follows again from the previous
analysis. ■
Remark 4.2. There is a discrepancy between the actual state convergence
in-terval and [p, q]. The actual inin-terval to which the state converges enlarges the interval [p, q] on both directions by dminϵ . However, according to the last com-ment in the proof of Fact 1, if the initial state is already within [p, q]n, one can
show that the state will never evolve outside [p, q]n. To make the state
use q− ϵ and p + ϵ as the upper and lower switching points of the saturation function, respectively. This yields
sat(z) = p + ϵ if z < p + ϵ z ifp + ϵ≤ z ≤ q − ϵ q− ϵ if z > q − ϵ (4.17)
Based on Theorem 4.1, by this strategy, the state converges in finite time to the interval [p + ϵ−dminϵ ,q− ϵ +dminϵ ]⊂ [p, q].
4.3.2
practical consensus
We then consider the consensus property of the proposed method. As shown in the following result, the node disagreement will be bounded by a term which scales linearly with the noise magnitude.
Theorem 4.2. Consider a network of n dynamical systems as in (4.1), which are
interconnected over an undirected connected graph G = (I, E). Let each local control input be generated in accordance with (4.3)-(4.9). Then there exists a finite time T′ such that for all i∈ I
|avei(t)| ≤ 5 4ϵ + max { dmax dmin ϵ, dmax|w|∞ } (4.18) for all t≥ T′.
Remark 4.3. Theorem 4.2 shows a convergence result for general unknown but
bounded noise. If [x(0), x(0)] ⊆ [p, q] and the noise is sufficiently small such that w≤ ϵ
2dmax, then by a similar proof as the one of (Shi et al., 2017, Theorem
3), we can show that xi(t) for all i ∈ I and all t ≥ 0 will remain in [x(0), x(0)]
and converge in a finite time to a point at which | avei| ≤ ϵ + dmaxw for all
i ∈ I. Thus, in a special yet interesting case, the state converges to a point in
[x(0), x(0)].
To prove this theorem, we need some intermediate results. We first introduce two sets for each node i as follows
Si1 := { tik:| avei(tik)| ≥ Li } (4.19) Si2 := { tik:| avei(tik)| < Li } (4.20)
where
Li:= 5
4ϵ + diw
′ (4.21)
with w′ = max{ϵ/dmin,|w|∞}. Clearly, tik∈ Si1∪ Si2for every k∈ Z≥0.
The following result shows that if at certain time node i enters Si2, it will remain
in this set.
Lemma 4.1 (Invariant set). For the network of n dynamical systems as in (4.1), which
are interconnected over an undirected connected graph G = (I, E), let each local control input be generated in accordance with (4.3)-(4.9). Consider the system evolution for t≥ T with T as in Theorem 4.1. If ti
k∈ Si2, then tiM∈ S/ i1for all integers M≥ k + 1.
Moreover,| avei(t)| < Lifor all t≥ tik.
Proof. We first show that the following inequality
|yij(t)− xi(t)| ≤ w′ (4.22)
holds for all t≥ T and all i ∈ I.
By Theorem 4.1, for all t≥ T, each node i’s state satisfies
p− ϵ/dmin≤ xi(t)≤ q + ϵ/dmin (4.23)
we consider the following three cases for t≥ T.
Case 1, p≤ xw
ij(t)≤ q. By (4.3), yij(t) = xwi(t). Hence
|yij(t)− xi(t)| = |xwij(t)− xi(t)| ≤ |w|∞≤ w′ (4.24)
where the second inequality comes from (4.2).
Case 2, xw
ij(t)≥ q. Then by (4.3), yij(t) = q. Since xi(t) = xwi (t)− wij(t), we have
q− |w|∞≤ xi(t)≤ q + ϵ/dmin, which implies that
|yij(t)− xi(t)| ≤ max{ϵ/dmin,|w|∞} = w′ (4.25)
Case 3, xw
ij(t)≤ p. Then by (4.3) and (4.2), we have yij(t) = p and p− ϵ/dmin≤
xi(t)≤ p + |w|∞, which implies that
For each node i, we then let φi(t) =∑ j∈Ni (yij(t)− xj(t)) (4.27) By (4.6) and (4.22), we have zwi (t) = avei(t) + φi(t) (4.28) |φi(t)| ≤ diw′ (4.29) for all t≥ T.
Now consider the following two sub-cases,
Sub-case a.|zw
i(tik)| ≥ ϵ. Without loss of generality, we assume zwi (tik)≥ ϵ, which
implies that ui(t) = 1 for all t∈ [tik,tik+1[and
ϵ− φi(tik)≤ avei(tik)≤ Li (4.30)
For t∈ [ti
k,tik+1[, the average is given by
avei(t) = avei(tik) + ∫ t ti k ∑ j∈Ni (uj(τ)− 1)dτ (4.31)
since|uj(t)| ≤ 1, we have avei(t)≤ avei(tik)≤ Liand
avei(t) ≥ avei(tki)− 2di(t− tik) ≥ avei(tik)− 2diΔk = avei(tik)− 1 2|z w i(tik)| = avei(tik)− 1 2(avei(t i k) +φi(tik)) = 1 2(avei(t i k)− φi(t i k)) ≥ 1 2(ϵ− 2φi(t i k)) ≥ 1 2ϵ− diw ′ >−L i (4.32)
where the third inequality comes from (4.30), the fourth from (4.29) and the last from the definition of Li.
Sub-case b.|zw i(tik)| < ϵ. Then Δ i k= 4dϵi, ui(t) = 0 for all t∈ [t i k,tik+1[and | avei(tik)| = |ziw(tik)− φi(t i k)| ≤ |zw i(tik)| + |φi(tik)| < ϵ + diw′ = Li−ϵ 4 (4.33) Hence for t∈ [ti
k,tik+1[, the average satisfies
| avei(t)| = | avei(tik) + ∫ t ti k ∑ j∈Ni uj(τ)dτ| ≤ | avei(tik)| + | ∫ t ti k ∑ j∈Ni uj(τ)dτ| ≤ | avei(tik)| + |diΔik| < Li− ϵ 4+ ϵ 4 =Li (4.34)
By induction as above for all the integers M≥ k + 1 and all the corresponding time intervals [ti
M−1,tiM[, we prove the result. ■
The next result shows that the average preserves the sign as long as its absolute value remains large enough.
Lemma 4.2. For the network of n dynamical systems as in (4.1), which are
intercon-nected over an undirected conintercon-nected graph G = (I, E), let each local control input be generated in accordance with (4.3)-(4.9). Consider the system evolution for t ≥ T, with T as in Theorem 4.1. For any i∈ I and any positive integer M, if
| avei(tik+m)| ≥ Li,m = 0, 1,· · · , M (4.35)
then
sign(avei(tik+m)) = sign(avei(tik)),
Proof. Notice that, since t > T, the inequality (4.29) always holds in the
follow-ing analysis. Suppose w.l.o.g that avei(tik)≥ Li>0, we know
zwi(tik) = avei(tik) +φi(t i k) ≥ Li− diw′ = ϵ +ϵ 4 >ϵ (4.37)
This implies that ui(t) = 1 for all t∈ [tik,tik+1[. Hence same as (4.32), we have
avei(tik+1) ≥ 1 2(avei(t i k)− φi(tik)) ≥ 1 2(Li− diw ′) ≥ 5 8ϵ (4.38)
which has the same sign as avei(tik). ■
Proof of Theorem 4.2. Notice that Li = 5 4ϵ + diw ′ = 5 4ϵ + dimax { ϵ dmin ,w } ≤ 5 4ϵ + max { dmax dmin ϵ, dmax|w|∞ } (4.39)
Accordingly, it is sufficient to show that there exists a time t′≥ T with T given as in Theorem 4.1, such that| avei(t)| < Lifor all t ≥ t′. We claim that there
exist a finite sampling time tis≥ T such that | avei(tis)| < Li. Suppose this is not
true, assume that | avei(tik)| ≥ Li holds for every tik ≥ T. By Lemma 4.2, the
average preserves its sign. Assume without loss of generality that avei(tik)≥ Li
for all ti
k ≥ T. Let tir = min{tik : tik ≥ T}, then by Lemma 4.2, ui(t) = 1 for
all t ≥ ti
r and sign(avei(tik)) = sign(avei(tir)) > 0 for all tik ≥ tir. However, this
implies that xi(t) will increase to infinity, which contradicts Theorem 4.1 and
proves the existence of finite ti s.
Now consider the system evolution for the time t ≥ ti
s. Since| avei(tis)| < Li,
by the invariant property shown in Lemma 4.1, we have| avei(t)| < Lifor all
t≥ ti
s. This ends the proof. ■
4.4 asymptotic consensus
In the noiseless case, the algorithm in Section 4.2 only achieves practical con-sensus. In this section, we show that the algorithm can be modified to achieve
asymptotic consensus with positive lower bounded inter-sampling times by
re-quiring some global information. As shown later, with this modification, all nodes employ a common time-varying threshold ϵ(t) and control magnitude
α(t). Compared with the algorithm in Section 4.2, this in fact implies that the
nodes have access to a common clock. Nonetheless, we emphasize that the nodes still need not to synchronously communicate states.
In detail, the system dynamics (4.1) becomes
˙xi(t) = α(t)ui(t) (4.40)
where the control input between two successive sampling times ti
kand tik+1[is given by ui(t) = signϵ(ti k)(z w i(tik)) (4.41)
and the inter-sampling time satisfies
Δik:= |zw i(tik)| 4α(ti k)di , if |zw i(tik)| ≥ ϵ(tik) ϵ(t) 4α(tik)di , otherwise (4.42)
The signals ϵ(t) : R≥0 → R≥0and α(t) : R≥0 → R≥0are positive monotone
decreasing functions and satisfy the following conditions
lim t→∞ϵ(t) = tlim→∞α(t) = 0 ∫ ∞ t α(τ)dτ = +∞, ϵ(t) α(t) ≥ c (4.43)
for all t≥ 0, with c being a positive value. The first two equalities guarantee that the system will achieve consensus and the inequality is used to rule out the Zeno behaviour.
We then can provide the following result on the state boundedness.
Theorem 4.3. Consider a network of n dynamical systems as in (4.40), which are
interconnected over an undirected connected graph G = (I, E). Let each local control input be generated in accordance with (4.41) and (4.42). If ϵ(t) and α(t) satisfy (4.43), then for all t∈ R≥0,
x(t)≤ max { x(0), q + ϵ(0) dmin } (4.44) x(t)≥ min { x(0), p−ϵ(0) dmin } (4.45)
Moreover, lim sup
t→+∞ xi(t)≤ q and lim inft→+∞xi(t)≥ p for all i ∈ I.
The consensus property can be described as follows.
Theorem 4.4. Consider a network of n dynamical systems as in (4.40), which are
interconnected over an undirected connected graph G = (I, E). Let each local control input be generated in accordance with (4.41) and (4.42). If ϵ(t) and α(t) satisfy (4.43), then for all i∈ I, it holds
lim sup
t→∞ |avei(t)| ≤ dmax|w|∞ (4.46)
The proofs of Theorems 4.3 and 4.4 are provided in Appendix B. Here we stress that by the time-varying threshold and control magnitude in (4.43), when
w ≡ 0, the algorithm achieves exact consensus while making the state
con-verge to the desired favourite interval. Moreover, in the noisy case, the system state is still confined within the desired favourite interval. Finally, if the noise converges to zero, one can verify that the consensus error will approach zero.
4.5
numerical examples
In this section, we perform simulations to verify the results. We consider a 10-node network with the communication graph illustrated in Fig. 4.1. From the figure, we know dmin=2 and dmax=6. For all the simulations we assume
-20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 1 1 1 2 3 3 3 3 4 4 5 5 6 6 6 6 7 7 8 1 2 3 4 5 6 7 8 9 10
Figure 4.1: Graph of the consensus network considered in Section 4.5
4.5.1 constant threshold
In this subsection we assume the consensus threshold ϵ = 0.05. We first con-sider the noiseless situation and let the initial state of each node be generated as a random value within the interval [−2, 4]. The simulation result is given in Fig. 4.2. Fig. 4.2(a) shows the state trajectory, where the green dashed lines represent the boundary of [p, q] and the red dashed lines represent the real state bounds (γ = 1.025 and γ =−1.025) from Theorem 4.3. Fig. 4.2(b) shows the absolute values of the averages, where the red dashed line represents the bound on the average from Theorem 4.2 (0.2125). Fig. 4.2(c) shows the con-trol inputs for the nodes with odd indices. From the figure, we can find that the nodes which are outside [γ, γ] move toward it with constant velocity until they converge inside the interval. All the nodes enter the set [γ, γ] within 3 seconds. There are some nodes whose states are larger than q, in conformity with Theorem 4.1. From Fig. 4.2(b), all the absolute averages are less than the bound given in Theorem 4.2 within 3 seconds and the absolute averages are much less than the theoretical bound. Moreover, from these simulations it appears that all the control inputs eventually become zero even though we do not provide yet a proof of this.
In the second example, we consider the noisy case where wij(t),{i, j} ∈ E is
a positive random noise with|w|∞ = 0.1. The initial state of each node is generated randomly within [−1, 1]. The results are presented in Fig. 4.3. From
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Time x 3.986 3.988 3.99 3.992 0.98 1 1.02 (a) State 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 7 8 9 10 Times |ave| 4.4 4.6 4.8 0.05 0.1 0.15 0.2 0.25
(b) Absolute value of the noiseless averages
−1 0 1 u1 −1 0 1 u3 −1 0 1 u5 −1 0 1 u7 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −1 0 1 u9 Time
(c) Control inputs at the odd nodes
0 0.5 1 1.5 2 2.5 3 3.5 4 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time x (a) State 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 Times |ave| 3 3.5 4 4.5 0 0.02 0.04
(b) Absolute value of the noiseless averages
−1 0 1 u1 −1 0 1 u3 −1 0 1 u5 −1 0 1 u7 0 0.5 1 1.5 2 2.5 3 3.5 4 −1 0 1 u9 Time
(c) Control inputs at the odd nodes
Figure 4.3: Network behaviour for constant threshold and positive random noise with|w|∞=0.1.
these figures, we can see that the states are driven by the noise to increase, however, they never exceed the upper bound of the favourite interval. The absolute values of all the averages become smaller than the theoretical bound (0.6625) within 2 seconds. The bound on the absolute averages value is 0.035, much less than the theoretical bound. The control inputs become zero after 2 seconds. This is reasonable since when all the states are within [q− ϵ/dmax,q],
with positive noise, all yij(t) are also within [q−ϵ/dmax,q] and the control inputs
for all nodes will be zero by (4.8).
In the third example, we consider the noisy case where wij(t),{i, j} ∈ E is a
ran-dom noise with|w|∞=0.1, with initial state of each node generated randomly
within [−1, 1]. The results are presented in Fig. 4.4. From these figures, we can see that the bound on the absolute averages value is 0.17, less than the theor-etical bound (0.6625). Moreover, the control inputs are non-zeros for most of the time.
4.5.2
time-varying threshold
In this subsection, we consider the threshold and control magnitude function from De Persis and Frasca (2013), given in the following form
α(t) = 1
1 + t
ϵ(t) = cα(t)
where c = 0.05 is the constant mentioned in Section 4.4, by which we know the initial magnitude of the control input is unitary and ϵ(0) = 0.05.
We only present the simulation for the noiseless case since the numerical result for the noisy case in this subsection is similar to that in the last subsection. In the simulation, the initial state is the same as that of the first example in the constant threshold case. The simulation results are given in Fig. 4.5. In Fig. 4.5(b), the red dashed line represents the bound calculated in (4.18) with
ϵ = 0.05. From these figures, we can see that the absolute value of the average
decreases asymptotically and all the states approach [p, q].
4.6
conclusion
In this chapter, we presented a new self-triggered coordination method in the presence of communication noise. The nodes adopt a common favourite
inter-0 0.5 1 1.5 2 2.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time x (a) State 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 Times |ave| 1.8 2 2.2 2.4 0 0.05 0.1 0.15
(b) Absolute value of the noiseless averages
−1 0 1 u1 −1 0 1 u3 −1 0 1 u5 −1 0 1 u7 0 0.5 1 1.5 2 2.5 −1 0 1 u9 Time
(c) Control inputs at the odd nodes
Figure 4.4: Network behaviour for constant threshold and random noise with
0 5 10 15 20 25 30 35 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Time x 37.2 37.4 37.6 37.8 0.9995 1 1.0005 (a) State 0 5 10 15 20 25 30 35 0 1 2 3 4 5 6 7 8 9 10 Times |ave| 25 30 35 0 2 4 x 10−3
(b) Absolute value of the noiseless averages
−1 0 1 u2 −1 0 1 u5 0 2 4 6 8 10 12 14 −1 0 1 u9 Time
(c) Control inputs at the odd nodes
Figure 4.5: Network behaviour with time-varying threshold ϵ(t) = 20(t+1)1 and
val of evolution and saturate the received states from the neighbours. For con-stant thresholds, the method can achieve approximate consensus and make the system state converge within a set around the favourite interval in finite time. Compared with the result in the previous chapter, the node disagreement in this chapter is independent of the initial condition of the system. Moreover, with time-varying threshold and control magnitude, the algorithm can achieve asymptotic consensus in the noiseless case, while preserving state bounded-ness in the noisy case.
4.7 appendix b: proofs of the results of the
asymptotic consensus
4.7.1 proof of theorem 4.3
Notice that under the algorithm (4.40)-(4.42), the two facts in Theorem 4.1 also hold for γ defined with ϵ = ϵ(0), hence the first claim can be derived by the same analysis in Theorem 4.1.
For the second claim, we need some intermediate supporting results. For any
η ∈ (0, ϵ(0)], let γ(η) = q + dminη and γ(η) = p− dminη . We then prove the following two lemmas for the upper bound. Similar results can be established for the lower bound, we omit the proofs due to space limitations.
Lemma 4.3. For any node i∈ I and any t′≥ 0, if the state xi(t′)≤ γ(η), then it will
never exceed γ(η) for all time t≥ t′. Proof. Let ti
m = max{tik ∈ R≥0,tik ≤ t′}. If ui(t′) ≤ 0, then ui(t) ≤ 0 for all
t∈ [ti
m,tim+1[, hence xi(t)≤ xi(t′)≤ γ for all t ∈ [t′,tim+1[. If instead, ui(t′) =1,
zwi(tim)≥ ϵ(tim)according to (4.41), which implies
xi(tim) ≤ 1 di (∑ j∈Ni (sat(xw ij(tim))− ϵ(tim)) ≤ q − ϵ(tim) di (4.47)
(4.4). Consider the evolution of xi(t) for t∈ [tim,tim+1[, we have xi(t) = xi(tim) + ∫ t ti m α(τ)ui(τ)dτ ≤ xi(tim) +α(tmi )(tim+1− tim) = xi(tim) + 1 4di|z w i (tim)| = xi(tim) + 1 4di ∑ j∈Ni (yij(tim)− xi(tim)) ≤ xi(tim) + 1 4(q− xi(t i m)) = 3 4xi(t i m) + 1 4q ≤ q −3ϵ(tim) 4di <γ(η) (4.48)
where the first inequality follows from that α(t) is a time decreasing function, the second equality from (4.42) and the third inequality from (4.47).
Considering both cases, we obtain that xi(t)≤ γ(η) for all t ≥ t′by induction.
■
Lemma 4.4. For any η ∈ (0, ϵ(0)], let t′(η) be a time satisfying ϵ(t′(η)) ≤ η. If xi(t′(η)) > γ(η), then there exists a time t′′>t′(η) such that xi(t′′)≤ γ(η).
Proof. First, since ϵ(t) is a positive time decreasing function with limit value
being zero, t′(η) always exists. Let ti
m = max{tik ∈ R≥0,tik ≤ t′(η)}. From
Lemma 4.3, we have xi(tim)must be larger than γ(η), otherwise xi(t′(η))≤ γ(η).
We consider the saturated average at ti m, zwi(tim) = ∑ j∈Ni (sat(xwij(tim))− xi(tim)) < di(q− γ(η)) = − diη dmin ≤ 0 (4.49)
Assume ui(t) = −1 for all t ∈ [tim,tim+1[. Suppose there exists no t′′ > t′(η)
such that xi(t′′)≤ γ(η), then for all tik≥ tim+1, xi(tik) > γ(η) and the saturated
average should satisfy
zwi (tik) < di(q− γ(η))
= − diη
dmin ≤ −η
≤ −ϵ(t′(η)) <−ϵ(ti
k) (4.50)
where the second inequality comes from dmin≤ difor all i∈ I, the third from
the assumption that ϵ(t′(η)) ≤ η and the last from the property that ϵ(t) is
time-decreasing and ti
k ≥ tim+1 > t′(η). This, along with the property that
ui(t) = −1 for t ∈ [tim,tm+1i [, shows that ui(t) = −1 for all t ≥ tim. However,
by∫t+∞α(s)ds = +∞, this would imply that x(t) = xi(tim) +
∫t ti mα(s)u(s)ds = xi(tim)− ∫t ti
mα(s)ds diverges to−∞ as t → +∞. This contradicts the assumption
that there is no t′′ > t′(η) such that xi(t′′) ≤ γ(η). Hence there must exist a
finite time t′′>t′(η) such that xi(t′′)≤ γ(η).
Next assume ui(t) = 0 for all t ∈ [tim,tim+1[. We have xi(tim+1) = xi(tim) >
γ(η). Again suppose the instant t′′ > t′(η) such that xi(t′′) ≤ γ(η) does not
exist, then for all the sampling time ti
k≥ tm+1, zwi(tik)should also satisfy (4.50).
This implies that ui(t) =−1 for all t ≥ tim+1. By the same argument as in the
case ui(t) = −1 for t ∈ [tim,tim+1[, we have limt→+∞x(t) = −∞. This leads to a
contradiction and proves the existence of t′′. ■
As a final step, we prove that lim sup
t→+∞ xi(t) = q and lim inft→+∞xi(t) = p for all i∈ I.
For each η∈ (0, ϵ(0)], let t′(η) = min{t ≥ 0, ϵ(t) ≤ η}. For each node i ∈ I, if
xi(t′(η)) ≤ γ(η), then xi(t) ≤ γ(η) for all t ≥ t′(η) by Lemma 4.3. If instead
xi(t′(η)) > γ(η), by Lemma 4.4, there should exist a finite time t′′>t′(η) such
that xi(t′′)≤ γ(η). By Lemma 4.3, we further have xi(t)≤ γ(η) for all t ≥ t′′.
This shows that for any η ∈ (0, ϵ(0)], there exists a time T1(η) ≥ t′(η) such
that xi(t) ≤ γ(η) for all i ∈ I and all t ≥ T1(η). For the lower bound, by the
same analysis, we have that there exists a finite time T2(η) ≥ t′(η) such that
xi(t)≥ γ(η) for all i ∈ I and all t ≥ T2(η). Let T(η) = max{T1(η), T2(η)}, then
xi(t)∈ [γ(η), γ(η)] for all i ∈ I and all t ≥ T(η). As t → +∞, γ(η) → γ(0) and
4.7.2
appendix c: proof of theorem 4.4
To prove this theorem, we need some intermediate results. For each η∈ (0, ϵ(0)], we introduce two sets for node i as
Si1(η) := { tik:| avei(tik)| ≥ Li(η) } (4.51) Si2(η) := { tik:| avei(tik)| < Li(η) } (4.52)
where Li(η) := 54η + diw′(η), with w′(η) = max{η/dmin,|w|∞}. Clearly, tik ∈
Si1(η)∪ Si2(η) for every k∈ Z≥0.
The following result shows that if at certain time node i enters Si2(η), it will
indefinitely remain in this set.
Lemma 4.5 (Invariant set). For the network of n dynamical systems as in (4.40),
which are interconnected over an undirected connected graph G = (I, E), let each local control input be generated in accordance with (4.41) and (4.42) with ϵ(t) and α(t) satisfying (4.43). For each η ∈ (0, ϵ(0)], define T(η) as in the proof of Theorem 4.3. If ti
k ≥ T(η) and belongs to Si2(η), then tiM ∈ S/ i1(η) for all integers M ≥ k + 1.
Moreover,| avei(t)| < Li(η) for all t≥ tik.
Proof. By the proof of Theorem 4.3, for each η∈ (0, ϵ(0)], T(η) exists and
satis-fies ϵ(T(η))≤ η. Moreover, for all i ∈ I and all t ≥ T(η), p − η/dmin=γ(η)≤
xi(t)≤ γ(η) = q + η/dmin. Then by the same proof as for inequality (4.22), for
all t≥ T(η), we have
|yij(t)− xi(t)| ≤ w′(η) (4.53)
Hence for all t≥ T(η), it holds
|φi(t)| ≤ diw′(η) (4.54)
with φi(t) given in (4.27).
We consider the following two cases,
Case 1. |zwi(tik)| ≥ ϵ(t i
k). Without loss of generality, we assume z w
i(tik) ≥ ϵ(t i k),
which implies that ui(t) = 1 for all t∈ [tik,tik+1[and
ϵ(tik)− φi(t i
For t∈ [ti
k,tik+1[, the average satisfies
avei(t) = avei(tik) + ∫ t ti k α(τ)∑ j∈Ni (uj(τ)− 1)dτ (4.56)
Since|uj(t)| ≤ 1, we have avei(t)≤ avei(tik) <Li(η) by (4.55) and
avei(t) ≥ avei(tik)− 2di ∫ t ti k α(τ)dτ > avei(tik)− 2diα(tik)(tik+1− tik) = avei(tik)− |zw i (tik)| 2 = avei(tik)− 1 2(avei(t i k) +φi(tik)) = 1 2(avei(t i k)− φi(t i k)) ≥ 1 2(ϵ(t i k)− 2φi(t i k)) ≥ 1 2ϵ(t i k)− diw′(η) > −Li(η) (4.57)
where the second inequality comes from the assumption that α(t) is a time decreasing function, the first equality from (4.42), the third inequality from (4.55), the fourth inequality from the bound on |φi(t)| and the last from the definition of Li(η). Case 2. |zw i (tik)| < ϵ(tik). We have Δ i k = ϵ(ti k)
4di and ui(t) = 0 for all t ∈ [t
i k,tik+1[.
Since ϵ(t) is time decreasing and ti
k≥ T(η) with ϵ(T(η)) ≤ η, we have ϵ(tik)≤ η and | avei(tik)| = |ziw(tik)− φi(t i k)| ≤ |zw i(tik)| + |φi(t i k)| < ϵ(tik) +diw′(η) = 5 4ϵ(t i k) +diw′(η)− ϵ(ti k) 4
≤ Li(η)−
ϵ(ti k)
4 (4.58)
where the first equality follows from (4.28). Hence for t∈ [ti
k,tik+1[, the average satisfies | avei(t)| = | avei(tik) + ∫ t ti k α(τ)∑ j∈Ni uj(τ)dτ| ≤ | avei(tik)| + | ∫ t ti k α(τ)∑ j∈Ni uj(τ)dτ| < | avei(tik)| + diα(tik)Δ i k < Li(η)− ϵ(ti k) 4 + ϵ(ti k) 4 =Li(η) (4.59) where the last inequality comes from (4.58). As before, by induction for all the integers M≥ k + 1 and all the corresponding time intervals [ti
M−1,tiM[, we
prove the result. ■
The next result shows that the average preserves the sign as long as its absolute value remains large enough compared with Li(η).
Lemma 4.6. For the network of n dynamical systems as in (4.40), which are
intercon-nected over an undirected conintercon-nected graph G = (I, E), let each local control input be generated in accordance with (4.41) and (4.42) with ϵ(t) and α(t) satisfying (4.43). For each η∈ (0, ϵ(0)], define T(η) as in the proof of Theorem 4.3. For any i ∈ I and any positive integer M, if ti
k≥ T(η) and
| avei(tik+m)| ≥ Li(η), m = 0, 1,· · · , M (4.60)
then sign(avei(tik+m)) = sign(avei(tik))for m = 1, 2,· · · , M + 1.
Proof. Since ti
k>T(η),|φi(t)| ≤ diw′(η) always holds for t≥ tik. Suppose w.l.o.g
that avei(tik)≥ Li(η) > 0, we know
zwi(tik) = avei(tik) +φi(t i k) ≥ Li(η)− diw′(η) = 5η 4 >ϵ(t i k) (4.61)
where the last inequality descends from ϵ(t) being a time decreasing signal and ti
k≥ T(η), with ϵ(T(η)) ≤ η.
This implies that ui(t) = 1 for all t∈ [tik,tik+1[. Hence
avei(tik+1) > avei(tik)− 2diα(tik)Δ i k = 1 2(avei(t i k)− φi(t i k)) ≥ 1 2(Li(η)− diw ′(η))≥ 5 8η (4.62)
where the first equality comes from (4.57). This shows that avei(tik+1)has the
same sign as avei(tik). ■
We then finalize the proof of Theorem 4.4. First notice that
Li(η) = 5 4η + diw ′(η) ≤ 5 4η + max { dmax dmin η, dmax|w|∞ } (4.63)
and, as η approaches zero, Li(η) approaches dmax|w|∞. Then we show that
for any η ∈ (0, ϵ(0)], there exists a finite time t′i(η) ≥ T(η) with T(η) given
in the Proof of Theorem 4.3, such that| avei(t)| < Li(η) for all t ≥ t′i(η). To
see this, we first claim that there exists a finite sampling time tis(η) ≥ T(η),
such that| avei(tis(η))| < Li(η). Suppose this is not true W.l.o.g. we assume
avei(tik)≥ Li(η) for all tki ≥ T(η). Let tir = min{tik:tik≥ T(η)}, then by Lemma
4.6 and the definition of Li(η), sign(avei(tik)) = sign(avei(tir)) > 0 for all tik ≥
T(η) and ui(t) = 1 for all t ≥ tir. However, this implies that xi(t) = xi(tir) +
∫∞
ti
r α(τ)dτ will increase to infinity, which contradicts Theorem 4.3 and proves
the existence of a finite tis(η) ≥ T(η) with the property | avei(tis(η))| < Li(η).
Since| avei(tis(η))| < Li(η), tis(η)∈ Li2(η). This along with tis(η)≥ T(η) shows
that| avei(t)| < Li(η) for all t≥ tis(η) ≥ T(η) by Lemma 4.5. The existence of
t′i(η) follows by letting t′i(η) = ti s(η).
