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Design and analysis of quantizer for multi-agent systems with a limited rate of communication data
Commun Nonlinear Sci Numer Simulat 18 (2013) 282–290
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Design and analysis of quantizer for multi-agent systems with a limited rate of communication data Runsha Dong, Zhiyong Geng ⇑ The State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Article history: Received 19 October 2011 Received in revised form 5 June 2012 Accepted 24 June 2012 Available online 14 July 2012 Keywords: Multi-agent systems Consensus Quantizer design Quantized message passing
a b s t r a c t This paper concerns the problem of consensus for multi-agent systems with a limited rate of communication data. The effect of the quantization error on consensus is analyzed for the case that the nodes’ dynamics is described by the first order integrator with a quantized message input. In order to describe how exactly the consensus is achieved, the notion of consensus level is introduced. Based on these, a suboptimal method of resource allocation to reduce such effect of quantization error is proposed, and a dynamic quantizer is designed to meet the requirement of the consensus level, under the constraints that the total quantization rate of overall system is limited. At last, a numerical example is included for demonstration, and the outcome of the simulations suggests the validity of the presented theoretic results. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction A lot of interest in consensus control for multi-agent systems has been taken by the control community over the last decade [1–5]. The consensus problem is known as agreement on certain quantities of interest for groups of agents. Early efforts focuses on the real-valued data exchanging among agents with high-precision, which is hard to be guaranteed in the real digital networks when considering that communication channel has a limited capacity, and energy used for transmission is generally restrained. In light of the practical constraints on communication, the basic setup (real-valued data) has then been extended on quantized data. The consensus problem with quantized data passing is first proposed by Kashay et al. [6], in which each node with quantized integer value reached the approximate average of the overall initial value. Later in other works like in [7], quantized consensus is referred to consensus when the measurements were digital rather than analog. Quantized values of the state represent a limited information flow [8] which occurs over digital channels with a limited data rate [9,10]. A simple quantizer generally observes a single number and selects the nearest approximating value from a predetermined finite set of allowed numerical values [11]. Refs. [12–15,10] study how quantized information used for consensus control influences the final consensus value. And they point out that quantization error indeed has an impact on the process of reaching consensus and the final consensus value. In [16], the author propose a design of quantizer based on the convergence performance of the closed-loop contraction system. This paper considers the quantized consensus problem for multi-agent systems, and points out that under given quantized consensus model, all 3 factors of sampling period, network topology characteristic and the upper bound of quantization error can affect the final consensus value of the multi-agent systems. Motivated by Cui and Lau [16] which declares that ⇑ Corresponding author. Address: Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China. Tel.: +86 10 62761043. E-mail address: [email protected] (Z. Geng). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.06.024
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quantization error is determined by the design of quantizer, we develop and analyze the suboptimal quantizer based on resource allocation to reduce the effect of quantization error on consensus for multi-agent systems. In order to measure the total disagreement among agents when they have not reached consensus, the notion of consensus level is introduced and analyzed. To meet the requirement of consensus level, a dynamic quantizer is designed under a limited rate of communication data. The remainder of this paper is organized as follows. Section 2 introduces some basic concepts of graph theory, quantizer structure, and also some notations used throughout this paper. Section 3 presents a quantized consensus model, and the relation between practical consensus and design of quantizer. In Section 4, we demonstrate how to design a dynamic quantizer to meet the requirement of consensus level under limited rate of communication data. Section 5 shows a numerical simulation to verify the validity of our theoretical results; and Section 6 presents the conclusion of this paper. 2. Preliminaries 2.1. Notations Let R denote the set of real numbers, and Zþ denote the positive integer set. The largest (resp. smallest) singular value of ðAÞ (resp. rðAÞ). Denote matrix transposition by superscript T. Let k k2 be the usual 2-norm of a matrix A is denoted by r vector, and jj be the absolute value of a real number. Given bc and de mean, respectively, the largest integer less than the independent variable or equal to it, and the smallest integer greater than or equal to it. 2.2. Graph theory Graph theory plays an important role in modeling the communication topology of the multi-agent systems. In the following, some basic concepts in graph theory that will be used in this article are described. The communication topology of the multi-agent systems is modeled as an undirected graph G ¼ fV; E; Ag, where V ¼ f1; 2; . . . ; i; . . . ; ng is a set of n vertices with i representing the i-th agent, and E 2 V V is an edge set in which each edge is presented by a pair of vertices ði; jÞ, which means a communication channel between these two agents. Then, the set of neighbors of agent i is denoted by N i ¼ fj 2 V : ði; jÞ 2 Eg. A ¼ ½aij 2 Rnn is the weighted adjacency matrix of G, and there exist aii ¼ 0; aij > 0 if ði; jÞ 2 V but 0 otherwise. The Laplacian of G is defined as L ¼ D A, where D ¼ diagðdeg 1 ; . . . ; deg n Þ P and deg i ¼ nj¼1 aij . Assign a particular direction to each edge in G so that one end of each edge to be the head and the other to be the tail, and e The laplacian of G e ¼ fV; E e ; Ag. e is denoted by e e is the f0; 1git is obtained a directed graph G L. The incidence matrix B of G e respectively. For example, the i; f -entry of G e is matrix with rows and columns indexed by the vertices and the edges of G, e contains n vertices and m edges, then equal to 1 if the vertex i is the head of edge f, 1 if it is the tail, and 0 otherwise. If G e has order n m. Although there are many different ways to orient a given graph G, many of the results about the oriBð GÞ e are independent of the choice of the orientation such as Bð GÞB e T ð GÞ e ¼ L. Note that L is the laplacian of G [17]. ented graphs G 2.3. Uniform quantizer Define an M-point scalar quantizer q as a mapping q : R ! S, where S fy1 ; y2 ; y3 ; . . . ; yM g R is the prescribed output set, and yi is the output level. In practical application, M is generally finite. Hence, each element in S can be specified by a fixed number of binary digits (bits). The quantization rate r of the quantizer q, known as r log2 M, is the number of bits transmitted per sample. Kinds of deterministic quantizer structure include uniform quantization, logarithmic quantization and sign quantization. Stochastic quantizer structure include random rounding, probabilistic quantization, etc. [13]. Since the uniform quantzier q will be used in this paper, we define q : R ! S as follows:
8 z > ðK þ 1=2ÞD > < K; z 6 ðK þ 1=2ÞD qðzÞ ¼ K; > : z 1 þ ; ðK þ 1=2ÞD < z 6 ðK þ 1=2ÞD 2 2
ð1Þ
where z 2 R; K 2 Zþ is the saturation value of q. The output set is S ¼ f0; n j n ¼ 1; 2; . . . ; Kg. Note that here q is a symmetric quantizer since qðzÞ ¼ qðzÞ. D is called quantization step size and it could be a constant or varying with time. The number of output levels for q is 2K þ 1. Quantizer error is denoted by eq ¼ z qðzÞ; z 2 R and jeq j 6 D2 [11]. In this paper, a system quantizer Q of the multi-agent systems contains m elements with each a scalar quantizer q. Though, the quantization rate r (or code rate, for quantizer (1), r log 2 ð2K þ 1Þ) is inherently a property of the quantizer and the communication rate (transmission rate) is a property of the communication channel, we assume they are numerically equal. Let r i be defined as the quantization rate of i-th scalar quantizer contained in Q. For the constraint of a system P þ quantization rate m i¼1 r i ¼ R; r i 2 Z ; 1 6 i 6 m, if both r i and m is finite, we call this system with a limited rate of communication data.
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3. Problem formulation 3.1. Quantized consensus model Consider a network of n agents, in which the dynamics of i-th agent is described by the single integrator, that is
x_ i ðtÞ ¼ ui ðtÞ;
i ¼ 1; . . . ; n
ð2Þ
where xi ðtÞ 2 R and ui ðtÞ 2 R are, respectively, the state and the control input of agent i. A control law for the system (2) that guarantees xi ðtÞ ! xj ðtÞ as t ! 1 is called a consensus algorithm or consensus protocol. A common neighbor-based consensus protocol is, like in [1,2],
ui0 ðtÞ ¼
X
aij ðxi ðtÞ xj ðtÞÞ;
i ¼ 1; 2; . . . ; n
ð3Þ
j2N i
_ The compact form of (3) is xðtÞ ¼ Lx, where x ¼ ½ x1 xn T and L is the laplacian of the undirected communication graph G ¼ fV; E; Ag. This paper assumes each agent i makes use of quantized measurements qðxi xj Þ where j 2 N i and q is a quantizer. This model is suitable for the case that the sensors of agents are far away from their controller/actuator. By means of quantization, more information among agents exchange as soon as possible. Then, the control input ui ðtÞ with quantized message passing can be written in terms of
ui ðtÞ ¼
X
aij qðxi xj Þ
ð4Þ
j2N i
with (4), it is easy to show that [15]
x_ ¼ BT BQ ðxÞ
ð5Þ
~ ¼ 1; 2; . . . ; m, where xðtÞ ¼ ½ x1 ðtÞ xm ðtÞ T 2 Rm ; xk~ ¼ xi xj ; ði; jÞ 2 E; k and the system quantizer is Q ð xÞ ¼ ½ q1 ð x1 Þq2 ð x2 Þ qm ð xm Þ . Note that m is the number of edges in graph G and each element of Q is a scalar quantizer. 3.2. Measurement of disagreement In order to measure the total disagreement among agents when they have not reached consensus, the concept of consensus level is introduced. We define, at time t, the disagreement of the multi-agent systems as follows [13],
uðtÞ,
X aij ðxi ðtÞ xj ðtÞÞ2 ¼ xðtÞT LxðtÞ
ð6Þ
i;j
When agents approach consensus, there is uðtÞ ! 0 as t ! 1. If uðtÞ90 as t ! 1, but uðtÞ has met the requirement of consensus for multi-agent systems, then, this system reaches a practical consensus. Control with quantized message passing introduces quantization error to the closed-loop system. Hence, agents can not reach the strictly true consensus and the concept of e-level practical consensus is then introduced to describe the change in disagreement among multi-agents. Definition 1 (e-level practical consensus). Define
P
i;j ðxi ðtÞ
xj ðtÞÞ2
i;j ðxi ð0Þ
xj ð0ÞÞ2
/ðtÞ ¼ P
as the relative disagreement at time t. A consensus protocol achieves e-level practical consensus if for a given constant e > 0, there exists /ð1Þ ¼ limt!1 /ðtÞ 6 e Note that in the quantized consensus model, quantizer structure affects quantization error and the practical consensus value. In order to reduce the effect of quantization error on consensus, it is necessary to design a optimal quantizer under given constraints such as e-level practical consensus and a limited rate of communication data. 3.3. Analysis of quantization error Quantization process has a natural relation with discrete time. Sampling rate is one key factor in determining the final consensus result in our quantized consensus model. With uniform period sampling and zero-order holding, the control law (4) could be rewritted as,
ui ðtÞ ¼
X
aij qðxj ðkTÞ xi ðkTÞÞ;
t 2 ½kT; kT þ TÞ
ð7Þ
j2N i
where k ¼ 0; 1; 2; . . . ; i ¼ 1; . . . ; n. Then, (5) can be represented as follows:
xðkT þ TÞ xðkTÞ ¼ TBT B½xðkTÞ eðkTÞ
ð8Þ
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where eðkTÞ ¼ xðkTÞ Q ð xðkTÞÞ. It is easy to see that
xðkT þ TÞ ¼ ðI TBT BÞxðkTÞ þ TBT BeðkTÞ
ð9Þ
Since the property of matrix B and the sampling period T affect the asymptotic convergence of (9), before moving on, one assumption is presented. Assumption 1. The communication topology graph G is a tree. Remark 1. If G is a tree, the corresponding matrix B is positive definite[17]. So is BT B. This property is necessary for later analysis of the practical consensus. Though the tree-shaped graph is a very special case in the directed graph, it yet provides us a way of dealing with the problem of quantized consensus with the given control algorithm. And the directed graph which contains a spanning tree will be our work in the next. The relation between uðtÞ in (6) and xðkTÞ in (9) is described as follows:
uðkTÞ ¼ xT Lx ¼ xT BBT x ¼ ðBT xÞT ðBT xÞ ¼ xT x ¼ kxðkTÞk22
ð10Þ 2
2
It illustrates that, given the requirement of e-level practical consensus, / ¼ uð1Þ=uð0Þ ¼ jj xð1Þjj =jj xð0Þjj 6 e. To study the ultimate boundedness of the solution to difference equation (9), we introduce three lemmas. Lemma 1. If A; B are matrixes with compatible dimension, then
rðA þ BÞ P rðAÞ þ rðBÞ.
Lemma 2 ([18] Theorem 3.9). Assume Q is an element of Banach algebra. If kQ k < 1, then there exists ðI Q Þ1 and
ðI QÞ1 ¼
1 X Qk k¼0
Lemma 3. Assume Q and ðI Q Þ are both non-singular, ðQ Þ < 1, then r ðI Q Þ 6 1 rðQ Þ 1. if r ðQ Þ < 1 if and only if r ðI Q Þ 6 1 2. r The solution to (9) is
xððk þ 1ÞTÞ ¼ ðI TBT BÞkþ1 xð0Þ þ
k X ðI TBT BÞkr TBT BeðrTÞ r¼0
k X ¼ ðI TBT BÞkþ1 xð0Þ þ ðI TBT BÞr TBT BeðkT rTÞ
ð11Þ
r¼0
The relation between eðkTÞ and k xðkTÞk2 as k ! 1 is now presented. e; 8k. Then Theorem 4. Let BT B > 0; T < r ðB1T BÞ and keðkTÞk2 6 ^
limk!1 kxðkTÞk2 6 cd TBT B ^e
ðTBT BÞ=rðTBT BÞ is the conditioning number of matrix ðTBT BÞ. where ^e is a constant and cdðTBT BÞ ¼ r e. Note that from (11), it follows that Proof. Given that keðkTÞk2 6 ^e; 8k, so, keðkT rTÞk2 6 ^
! k X kþ1 I TBT B r ðI TBT BÞr r TBT B ^e kxððk þ 1ÞTÞk2 6 r kxð0Þk2 þ r¼0 T
T
kþ1
ðTBT BÞ < 1. By Lemma 3, there exjj xð0Þjj2 ! 0 as k ! 1. When T < r B1T B , it follow r ð Þ ðI TBT BÞ. By the above inequalities, one has ðI TBT BÞ 6 1 rðTBT BÞ < 1 and rðTBT BÞ 6 1 r ists r
ðI TB BÞ < 1, then r ðI TB BÞ If r
1 1 6 ðI TBT BÞ r TBT B 1r ðI TBT BÞ 6 1 rðTBT BÞ < 1. It can be verified that and r 1 X
r ðI TBT BÞr ¼
r¼0
1 1 6 ðI TBT BÞ rðTBT BÞ 1r
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ðTBT BÞ=rðTBT BÞ. Finally, it has Denote the conditional number of matrix ðTBT BÞ as cdðTBT BÞ ¼ r
lim jjxðkT Þjj2 6 cdðTBT BÞ^e:
k!1
Remark 2. By formula (10) and Theorem 4, the upper bound of quantization error, denoted by ^e, affects the disagreement ðTBT BÞ=rðTBT BÞ is fixed. Thereof multi-agent systems. For a given communication network, it is clear that cdðTBT BÞ ¼ r ^ fore, to reduce e is to reduce the final disagreement among agents and to meet the requirement of e-level practical consensus. P 2 12 ^ Since jjeðkTÞjj2 ¼ ð m i¼1 ðxi ðkTÞ qi ðxi ðkTÞÞÞ Þ ; e is relevant to xi ; k and qi , therefor, this paper design a dynamic quantizer based on a suboptimal method of allocation of communication resources, to meet the requirement of e-level practical consensus. In the next section, the main result is obtained.
4. Main results For later use, the discrete value set of xðkTÞ is defined as follows,
pi ,fxi ðkTÞ 2 Rxi ðkT þ TÞ ¼ fi ðxi ðkTÞ; xj ðkTÞÞ; xi ð0Þ ¼ xi;0 2 R; xj ð0Þ ¼ xj;0 2 Rg where fi is the state transfer function, xi;0 and xj;0 are, respectively, the initial value of xi and xj . Assume pi is a compact set. xi ðkTÞ 2 pi ; i ¼ 1; 2; . . . ; m. Define p ¼ p1 pm . For x0 2 p and the system quantizer Q, there exists Note that for 8k; xðkTÞ Q ð xðkTÞÞ 2 Rm ; k ¼ 0; 1; . . .. Let jjeð x0 ; Q Þjj1 ¼ supk jjeð x0 ; Q ÞðkTÞjj2 . By keðkT rTÞk2 6 ^e; 8k, and for eð x0 ; Q ÞðkTÞ ¼ 8x0 2 p and Q, we have jjeðx0 ; Q Þjj1 < 1. Therefor, eðx0 ; Q Þ 2 lm . In addition, x0 ; Q ÞðkTÞk2 6 maxx0 2p keð x0 ; Q Þk1 ¼ keð 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pm Pm 1 Let ^eðQ Þ ¼ maxxi;0 2pi maxk ð i¼1 jei ð xðkTÞ Q ð xðkTÞÞk2 ¼ maxxi;0 2pi maxk xi;0 ; qi ÞðkTÞj2 Þ2 . maxx0 2p maxk k i¼1 ei ðxi;0 ; qi ÞðkTÞ . The goal of quantizer design is to choose a optimal quantizer Q to minimize ^ eðQ Þ. Then, we give the following lemma. m Lemma 5. For eð x0 ; Q Þ 2 l1 , where x0 2 p, the inf Q ^ eðQ Þ exists when the system quantizer has a structure as follows:Q ð xÞ ¼ ½ ½q1 ð x1 Þ qm ð xm Þ T , where qi : xi # qi ð xi Þ is the i-th component of Q and a scalar uniform quantizer.
Proof. For a input signal xi 2 R with any probability density and a prescribed output set, the quantizer structure qi ðÞ which minimize ^ ei ðqi Þ is a scalar uniform quantizer[11]. Therefor, the structure of the system quantizer Q ð xÞ ¼ ½ ½q1 ð x1 Þ qm ð xm Þ T which minimize ^ eðQ Þ is the one with each element qi : xi # qi ð xi Þ; i ¼ 1; 2; . . . ; m, a scalar uniform quantizer. h
Now fix the quantization step size of the uniform quantizer qi such that DðkTÞ ¼ D0 ; k 2 ½0; k Þ, where k depends on the requirement of e-level practical consensus. In order to emphasize the relation between Q and r i , Q and D, denote Q as Q ðri ; D0 Þ and ^ e is expressed by ^eðQ ððr i ; D0 ÞÞÞ. To reach consensus, it is assumed each communication channel will be used for information flow. So, it has r i > 0. Define the quantization range of qi as jpi j ¼ maxxi 2pi xi minxi 2pi xi . pffiffiffiffiffi Lemma 6. If xi ðkTÞ is within the quantization range of quantizer qi , then jpi j j xi ð0Þ cdðTBT BÞD=ð2 mÞj. When D is small enough, there exists jpi j j xi ð0Þj. Proof. By Theorem 4 and ei ðrTÞ 6 D2 (i ¼ 1; 2; . . . ; m), it follows that
limk!1 jxi ðkT þ TÞj 6
DcdðTBT BÞ pffiffiffiffiffi ; i ¼ 1; 2; . . . ; m 2 m
When the quantization step size D is small enough, it can be verified that limk!1 j xi ðkT þ TÞj ! 0. So, the smallest jpi j is pffiffiffiffiffi j xi ð0Þ cdðTBT BÞD=ð2 mÞj. If D is small enough, then jpi j j xi ð0Þj. h Since the proof of Lemma 6 ignores the impact of coupling among agents, the result in Lemma 6 is conservative. Then, a convex optimization problem considered in this paper is described as follows. In the consensus problem of multi-agent systems, if the quantization range jpi j is defined by the quantization rate r i , then, P 12 P 2 12 m m jpi j ij 2 ¼ for the uniform quantizer qi ; D ¼ 2jrpi 1 and ei ¼ D2 ¼ j2priij. It follows that ^ e ¼ maxx2p kek2 ¼ maxxi 2pi . i¼1 ðei Þ i¼1 2ri Pm þ Under the constraint i¼1 ri ¼ R; ri 2 Z ; 1 6 i 6 m, the problem of finding a uniform quantizer to make agents reach practical consensus and to obtain the value of minQ ^e, is expressed by Problem 1,
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8 min^e > > < Q m X Problem1; > r i ¼ R; ri 2 Rþ ; 1 6 i 6 m > : s:t: i¼1
Since 0 < r i < R is a convex set and
Pm
i¼1 ðj
pi j2 22ri Þ is a convex function of ri , the equivalent form of the Problem 1 is [19]
8 mins > > fri g;s > > > m > X > < s:t: ðjpi j2 22ri Þ 6 s Problem2; i¼1 > > > m > > > Xr ¼ R; > ri P 0 ð1 6 i 6 mÞ : i i¼1
Then, we have the following theorem. P þ Theorem 7. Under the constraint of communication rate, that is, m i¼1 r i ¼ R; r i 2 Z ; 1 6 i 6 m, the scalar uniform quantizer qi (contained in the system quantizer Q) which makes agents reach the practical consensus and minimize ^e, has the following suboptimal quantization rate
r i ¼ where
1 jxi ð0Þj2 log2 2 s
! i ¼ 1; 2; . . . ; m Pm
s is a constant which satisfy 8 m X > < dr e; i 6 R br i c i ~r i ¼ i¼1 > : bri c; otherwise
i¼1 r i
¼ R. Since r i 2 Zþ , the suboptimal quantization rate will be:
Proof. Using the lagrange multiplier method to solve the transformed optimization problem (Problem 2), it follows P P Pm 2 2r i Lðs; ri ; k; t; lÞ ¼ s þ kð m sÞ m i¼1 jpi j 2 i¼1 ti r i þ lð i¼1 r i RÞ. The Karush–Kuhn–Tucker ðKKTÞ condition of the optimization problem 2 are [19]: Pm 1. jpi j2 22ri 6 s Pi¼1 m 2. i¼1 r i ¼ R; r i P 0; i ¼ 1; 2; . . . ; m 3. ti P 0ði ¼ 1; 2; . . . ; mÞ; l > 0; k P 0 4. kðjpi j2 22ri sÞ ¼ 0; ti ri ¼ 0; i ¼ 1; 2; . . . ; m @L 5. @r ¼ 2k lnð2Þjpi j2 22ri ti þ l ¼ 0; i ¼ 1; 2; . . . ; m i From the consensus condition r i > 0 and the KKT condition (4) ti ri ¼ 0, it is clear that ti ¼ 0. Then, by the KKT condition (5) and l > 0; k > 0 is obtained. It follows the solution to the KKT condition (4) is s ¼ jpi j2 22ri . Substitute it into the KKT condition 2
2 2r i l jpi j (5), then, one has k ¼ 2 lnð2Þ ; 8i, and 22ri ¼ 2k lnð2Þ ; l > 0. Since ri > 0, it can be verified s. Following this, l ¼ 2k lnð2Þjpi j 2 l 2 2 P m 2r i 2k lnð2Þjpi j jpi j 1 > 1. From r i ¼ 2 log2 s and i¼1 ri ¼ R; r i and s can be solved. For jpi j which is estimated as j that 2 ¼ xi ð0Þj, we l
know
that
ri
is
a
suboptimal
solution
minQ ^ e.
of 2
þ
2br i c
r i ¼ ðN 1Þ þ #; 0 < # < 1; N 2 Z , it is clear that jpi j 2 follows:
Note 2
2r i
> jpi j 2
that 2
ri 2 2dr i e
> jp i j 2
Zþ .
For
s ¼ jpi j2 22ri ,
assume
. Therefore, r should be chosen as
8 m X > < dr e; i 6 R br i c i ri ¼ i¼1 > : bri c; otherwise When jpi j jpj j; 1 6 i; j 6 m, the suboptimal quantization rate ~r i should be selected as:
8 m X > < d R e; i 6 R bmR c m ~r i ¼ i¼1 > : R bmc; otherwise
Remark 3. Assume the initial quantization step size D0 is small enough. The relation between allocation of communication resources (communication rate) and j xi ð0Þj illustrates that, the larger j xi ð0Þj is, the more ri is needed for qi .
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By theorem 7, an estimated method of allocating R to minimize ^ e is given. From the requirement of consensus level which pffi ekxð0Þk2 is defined by /ð1Þ ¼ k xð1Þk22 =k xð0Þk22 6 e, where k xð1Þk2 6 cd TBT B ^ e; ^e should satisfy ^ e 6 cdðTB to meet the requirement. T BÞ pffi xð0Þk2 e k However, if minQ ðri ;D0 Þ ^e > cdðTBT BÞ , then it is necessary to adjust the quantization step size D0 . T 1 ~ ffi BÞ and oð~ e ¼ minfri g; s s2 , q ¼ ep0 cdðTB e0 Þ is a infinitesimal of ~ e0 such as Define ~ eðQ ðr i ; DðtÞÞÞt¼0 ¼ ~e0 ðQ ðr i ; D0 ÞÞ ¼ minQ ðri ;D0 Þ ^ ekxð0Þk 2
oð~e0 Þ ¼ 0:01~ e0 . Note that Q contains m elements. The above propositiones lead to the following result. Lemma 8. For each quantizer qi ; i ¼ 1; 2 . . . ; m, with identical quantization step size at time kT; DðkTÞ, when 2 xðk TÞ 2 > ek xð0Þk22 , the multi-agent systems will reach the e -level practical consensus if each qi adjusts its D to DðkTÞ ¼ D0 =q; k P k þ 1 at k T, where k is selected when xðk TÞ 2 ¼ kxðkTÞk2 þ oðkxðkTÞk2 Þ. pffiffiffi pffiffiffi Proof. The requirement of e-level consensus means ^eð1Þ 6 ek xð0Þk2 =cdðTBT BÞ. Since ~ xð0Þk2 =cdðTBT BÞ, it needs e0 =q ¼ ek e0 þ oð~ e0 ÞÞ=q. That is, zooming out the initial to prove that when DðkTÞ ¼ D0 =q; k ¼ k þ 1; k þ 2 . . ., there exists ^eð1Þ 6 ð~ e0 Þ will zoom out the same time, that is, ^ eð1Þ 6 ð~e0 þ oð~e0 ÞÞ=q
quantization step size q times, then, ~e0 þ oð~ P 12 P 12 pffiffiffi m D0 2 m D0 2 ekxð0Þk2 =cdðTBT BÞ. Fix r i , it follows that eðkTÞ 6 ; k < k þ 1 and eðkTÞ 6 ; k P k þ 1. obviously, i¼1 ð 2 Þ i¼1 ð2q Þ one has ^ eð1Þ 6 ~ e0 =q ð~e0 þ oð~e0 ÞÞ=q. h xi decrease. However, since the relative states of agents are conRemark 4. Fix r i and zoom out D0 will make the range of verging as time goes on, it is reasonable to adjust D0 . The time consumed for finding a proper k is still acceptable. 5. Illustrative example The following is an illustrative example. Given a multi-agent system consists of 4 agents, the communication network is modeled as a undirected graph G ¼ fV; E; Ag. The corresponding adjacency matrix and incidence matix are, respectively,
0 61 6 A¼6 40 0 and
2
1
6 1 6 B¼6 4 0 0
1 0 1 0
0 1 0 1 0 1 1 0
3 0 07 7 7 15 0 0
3
0 7 7 7 1 5 1 5000 Any other Suboptimal
4500 4000 3500 3000 φ
2
2500 2000 1500 1000 500 0
0
5
10
15
20
25
30
k (T) Fig. 1. Disagreement comparison (two method of allocation).
35
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R. Dong, Z. Geng / Commun Nonlinear Sci Numer Simulat 18 (2013) 282–290
50 40 30
The relative state
20 10 0 −10 −20 −30 −40 −50
0
5
10
15
20
25
30
35
k (T) Fig. 2. State trajectories of x.
50 before adjusting Δ at k=35 after adjusting Δ at k=35
45 40 35
φ
30 25 20 15 10 5 0 10
15
20
25
30 k (T)
35
40
45
50
Fig. 3. Disagreement comparison (adjust D at k = 35).
P Let the initial state of x be x0 ¼ ½ 20:22 20:22 20:22 20:22 T and a communication rate constraint be m i¼1 r i ¼ 60. The initial T quantization step size of each scalar uniform quantizer is D0 ¼ 1. Take e ¼ 0:0001. By computation of B B , it is verified that T < 0.2929 s. Here, choose T = 0.1 s. Since T < 1=r xi ; i ¼ 1; 2; 3 converge around the origin, it is reasonable to estimate that jpi j jpj j; 1 6 i; j 6 3. So, a suboptimal quantization rate r i ¼ r j ¼ 20. In contrast, take any other way of allocation that r1 ¼ 36; r 2 ¼ 20; r3 ¼ 4. Then, the measurements of disagreement among agents under two methods as time goes on are given in Fig. 1. It should be noted this suboptimal allocation method is effective in reducing the disagreement and the corresponding quantization error. The state trajectories of the relative states among agents are depicted in Fig. 2. From x0 and e, it can be observed that u should satisfy uð1Þ 6 0:49. We note that uð1Þ is larger than 0.517 in Fig. 1. So adjust Dfrom 1 to 0.5, then, it is shown in Fig. 3, the final disagreement is less than 0.49.
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6. Conclusions This paper considers the problem of quantizer design under a limited rate of communication data for the multi-agent systems. It constructs a control law with quantized message passing to meet the requirement of consensus level. From discussion above, this paper obtains the following conclusions. To reduce the final disagreement of the multi-agent systems where each agent using quantized message for control, the system forces the allocation of communication resources to depend on the initial relative state. If physical conditions are allowed, adjusting the quantization step size dynamically will make the final practical consensus meet the requirement of consensus level. Further interesting research works include a discussion of the practical consensus for agents with complex linear dynamics. Acknowledgments This work is supported by the National Natural Science Foundation of China under grants 10832006 and 11072002. References [1] Ren W, Beard RW. Consensus seeking in multi-agent systems under dynamically changing interaction topologies. IEEE Trans Autom Control 2005;50:655–61. [2] Olfati-Saber R, Fax J, Murray R. Consensus and cooperation in networked multi-agent systems. Proc IEEE 2007;95:215–33. [3] Meng J, Egerstedt M. Distributed coordination control of multiagent systems while preserving connectedness. IEEE Trans Robot 2007;23:693–703. [4] Li Z, Duan Z, Chen G. Dynamic consensus of linear multi-agent systems. IET Control Theory Appl 2011;5:19–2 8. [5] Jiang H, Bi Q, Zheng S. Impulsive consensus in directed networks of identical nonlinear oscillators with switching topologies. Commun Nonlinear Sci Numer Simul 2012;17:378–87. [6] Kashay, Basar T, Srikant R. Quantized consensus. Automatica 2007;43:1192–203. [7] Ren W, Cao Y. Distributed coordination of multi-agent networks: emergent problems, models, and issues. Springer; 2011. [8] Bullo F, Liberzon D. Quantized cobtrol via locational optimization. IEEE Trans Autom Control 2006;51:2–13. [9] Nair GN, Evans RJ. Exponential stabilisability of finite-dimensional linear systems with limited data rates. Automatica 2003;39:585–93. [10] Li T, Fu M, Xie L, Zhang JF. Distributed consensus with limited communication data rate. IEEE Trans Autom Control 2011:56. [11] Gersho A, Gray R. Vector quantization and signal compression. Kluwer Academic; 1991. [12] Carli R, Fagnani F, Frasca P, Zampieri S. A probabilistic analysis of the average consensus algorithm with quantized communication. In: Proc 17th IFAC world congress; 2008. [13] Censi, Murray R. Real-valued average consensus over noisy quantized channels. In: American control conference (ACC ’09); 2009. p. 4361–6. [14] Fang J, Li H. Distributed consensus with quantized data via sequence averaging. IEEE Trans Signal Process 2010;58:944–8. [15] Dimarogonas DV, Johansson KH. Stability analysis for multi-agent systems using the incidence matrix: quantized communication and formation control. Automatica 2010;46:695–700. [16] Cui Y, Lau VKN. Convergence-optimal quantizer design of distributed contraction-based iterative algorithms with quantized message passing. IEEE Trans Signal Process 2010;58:5196–205. [17] Godsil C, Royle G. Algebraic graph theory. Springer-Verlag; 2001. [18] Dullerud GE, Paganini F. A course in robust control theory. Springer-Verlag; 2000. [19] Boyd S, Vandenberghe L. Convex optimization. Cambridge Univ. Press; 2004.
Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Design and analysis of quantizer for multi-agent systems with a limited rate of communication data Runsha Dong, Zhiyong Geng ⇑ The State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Article history: Received 19 October 2011 Received in revised form 5 June 2012 Accepted 24 June 2012 Available online 14 July 2012 Keywords: Multi-agent systems Consensus Quantizer design Quantized message passing
a b s t r a c t This paper concerns the problem of consensus for multi-agent systems with a limited rate of communication data. The effect of the quantization error on consensus is analyzed for the case that the nodes’ dynamics is described by the first order integrator with a quantized message input. In order to describe how exactly the consensus is achieved, the notion of consensus level is introduced. Based on these, a suboptimal method of resource allocation to reduce such effect of quantization error is proposed, and a dynamic quantizer is designed to meet the requirement of the consensus level, under the constraints that the total quantization rate of overall system is limited. At last, a numerical example is included for demonstration, and the outcome of the simulations suggests the validity of the presented theoretic results. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction A lot of interest in consensus control for multi-agent systems has been taken by the control community over the last decade [1–5]. The consensus problem is known as agreement on certain quantities of interest for groups of agents. Early efforts focuses on the real-valued data exchanging among agents with high-precision, which is hard to be guaranteed in the real digital networks when considering that communication channel has a limited capacity, and energy used for transmission is generally restrained. In light of the practical constraints on communication, the basic setup (real-valued data) has then been extended on quantized data. The consensus problem with quantized data passing is first proposed by Kashay et al. [6], in which each node with quantized integer value reached the approximate average of the overall initial value. Later in other works like in [7], quantized consensus is referred to consensus when the measurements were digital rather than analog. Quantized values of the state represent a limited information flow [8] which occurs over digital channels with a limited data rate [9,10]. A simple quantizer generally observes a single number and selects the nearest approximating value from a predetermined finite set of allowed numerical values [11]. Refs. [12–15,10] study how quantized information used for consensus control influences the final consensus value. And they point out that quantization error indeed has an impact on the process of reaching consensus and the final consensus value. In [16], the author propose a design of quantizer based on the convergence performance of the closed-loop contraction system. This paper considers the quantized consensus problem for multi-agent systems, and points out that under given quantized consensus model, all 3 factors of sampling period, network topology characteristic and the upper bound of quantization error can affect the final consensus value of the multi-agent systems. Motivated by Cui and Lau [16] which declares that ⇑ Corresponding author. Address: Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China. Tel.: +86 10 62761043. E-mail address: [email protected] (Z. Geng). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.06.024
R. Dong, Z. Geng / Commun Nonlinear Sci Numer Simulat 18 (2013) 282–290
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quantization error is determined by the design of quantizer, we develop and analyze the suboptimal quantizer based on resource allocation to reduce the effect of quantization error on consensus for multi-agent systems. In order to measure the total disagreement among agents when they have not reached consensus, the notion of consensus level is introduced and analyzed. To meet the requirement of consensus level, a dynamic quantizer is designed under a limited rate of communication data. The remainder of this paper is organized as follows. Section 2 introduces some basic concepts of graph theory, quantizer structure, and also some notations used throughout this paper. Section 3 presents a quantized consensus model, and the relation between practical consensus and design of quantizer. In Section 4, we demonstrate how to design a dynamic quantizer to meet the requirement of consensus level under limited rate of communication data. Section 5 shows a numerical simulation to verify the validity of our theoretical results; and Section 6 presents the conclusion of this paper. 2. Preliminaries 2.1. Notations Let R denote the set of real numbers, and Zþ denote the positive integer set. The largest (resp. smallest) singular value of ðAÞ (resp. rðAÞ). Denote matrix transposition by superscript T. Let k k2 be the usual 2-norm of a matrix A is denoted by r vector, and jj be the absolute value of a real number. Given bc and de mean, respectively, the largest integer less than the independent variable or equal to it, and the smallest integer greater than or equal to it. 2.2. Graph theory Graph theory plays an important role in modeling the communication topology of the multi-agent systems. In the following, some basic concepts in graph theory that will be used in this article are described. The communication topology of the multi-agent systems is modeled as an undirected graph G ¼ fV; E; Ag, where V ¼ f1; 2; . . . ; i; . . . ; ng is a set of n vertices with i representing the i-th agent, and E 2 V V is an edge set in which each edge is presented by a pair of vertices ði; jÞ, which means a communication channel between these two agents. Then, the set of neighbors of agent i is denoted by N i ¼ fj 2 V : ði; jÞ 2 Eg. A ¼ ½aij 2 Rnn is the weighted adjacency matrix of G, and there exist aii ¼ 0; aij > 0 if ði; jÞ 2 V but 0 otherwise. The Laplacian of G is defined as L ¼ D A, where D ¼ diagðdeg 1 ; . . . ; deg n Þ P and deg i ¼ nj¼1 aij . Assign a particular direction to each edge in G so that one end of each edge to be the head and the other to be the tail, and e The laplacian of G e ¼ fV; E e ; Ag. e is denoted by e e is the f0; 1git is obtained a directed graph G L. The incidence matrix B of G e respectively. For example, the i; f -entry of G e is matrix with rows and columns indexed by the vertices and the edges of G, e contains n vertices and m edges, then equal to 1 if the vertex i is the head of edge f, 1 if it is the tail, and 0 otherwise. If G e has order n m. Although there are many different ways to orient a given graph G, many of the results about the oriBð GÞ e are independent of the choice of the orientation such as Bð GÞB e T ð GÞ e ¼ L. Note that L is the laplacian of G [17]. ented graphs G 2.3. Uniform quantizer Define an M-point scalar quantizer q as a mapping q : R ! S, where S fy1 ; y2 ; y3 ; . . . ; yM g R is the prescribed output set, and yi is the output level. In practical application, M is generally finite. Hence, each element in S can be specified by a fixed number of binary digits (bits). The quantization rate r of the quantizer q, known as r log2 M, is the number of bits transmitted per sample. Kinds of deterministic quantizer structure include uniform quantization, logarithmic quantization and sign quantization. Stochastic quantizer structure include random rounding, probabilistic quantization, etc. [13]. Since the uniform quantzier q will be used in this paper, we define q : R ! S as follows:
8 z > ðK þ 1=2ÞD > < K; z 6 ðK þ 1=2ÞD qðzÞ ¼ K; > : z 1 þ ; ðK þ 1=2ÞD < z 6 ðK þ 1=2ÞD 2 2
ð1Þ
where z 2 R; K 2 Zþ is the saturation value of q. The output set is S ¼ f0; n j n ¼ 1; 2; . . . ; Kg. Note that here q is a symmetric quantizer since qðzÞ ¼ qðzÞ. D is called quantization step size and it could be a constant or varying with time. The number of output levels for q is 2K þ 1. Quantizer error is denoted by eq ¼ z qðzÞ; z 2 R and jeq j 6 D2 [11]. In this paper, a system quantizer Q of the multi-agent systems contains m elements with each a scalar quantizer q. Though, the quantization rate r (or code rate, for quantizer (1), r log 2 ð2K þ 1Þ) is inherently a property of the quantizer and the communication rate (transmission rate) is a property of the communication channel, we assume they are numerically equal. Let r i be defined as the quantization rate of i-th scalar quantizer contained in Q. For the constraint of a system P þ quantization rate m i¼1 r i ¼ R; r i 2 Z ; 1 6 i 6 m, if both r i and m is finite, we call this system with a limited rate of communication data.
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3. Problem formulation 3.1. Quantized consensus model Consider a network of n agents, in which the dynamics of i-th agent is described by the single integrator, that is
x_ i ðtÞ ¼ ui ðtÞ;
i ¼ 1; . . . ; n
ð2Þ
where xi ðtÞ 2 R and ui ðtÞ 2 R are, respectively, the state and the control input of agent i. A control law for the system (2) that guarantees xi ðtÞ ! xj ðtÞ as t ! 1 is called a consensus algorithm or consensus protocol. A common neighbor-based consensus protocol is, like in [1,2],
ui0 ðtÞ ¼
X
aij ðxi ðtÞ xj ðtÞÞ;
i ¼ 1; 2; . . . ; n
ð3Þ
j2N i
_ The compact form of (3) is xðtÞ ¼ Lx, where x ¼ ½ x1 xn T and L is the laplacian of the undirected communication graph G ¼ fV; E; Ag. This paper assumes each agent i makes use of quantized measurements qðxi xj Þ where j 2 N i and q is a quantizer. This model is suitable for the case that the sensors of agents are far away from their controller/actuator. By means of quantization, more information among agents exchange as soon as possible. Then, the control input ui ðtÞ with quantized message passing can be written in terms of
ui ðtÞ ¼
X
aij qðxi xj Þ
ð4Þ
j2N i
with (4), it is easy to show that [15]
x_ ¼ BT BQ ðxÞ
ð5Þ
~ ¼ 1; 2; . . . ; m, where xðtÞ ¼ ½ x1 ðtÞ xm ðtÞ T 2 Rm ; xk~ ¼ xi xj ; ði; jÞ 2 E; k and the system quantizer is Q ð xÞ ¼ ½ q1 ð x1 Þq2 ð x2 Þ qm ð xm Þ . Note that m is the number of edges in graph G and each element of Q is a scalar quantizer. 3.2. Measurement of disagreement In order to measure the total disagreement among agents when they have not reached consensus, the concept of consensus level is introduced. We define, at time t, the disagreement of the multi-agent systems as follows [13],
uðtÞ,
X aij ðxi ðtÞ xj ðtÞÞ2 ¼ xðtÞT LxðtÞ
ð6Þ
i;j
When agents approach consensus, there is uðtÞ ! 0 as t ! 1. If uðtÞ90 as t ! 1, but uðtÞ has met the requirement of consensus for multi-agent systems, then, this system reaches a practical consensus. Control with quantized message passing introduces quantization error to the closed-loop system. Hence, agents can not reach the strictly true consensus and the concept of e-level practical consensus is then introduced to describe the change in disagreement among multi-agents. Definition 1 (e-level practical consensus). Define
P
i;j ðxi ðtÞ
xj ðtÞÞ2
i;j ðxi ð0Þ
xj ð0ÞÞ2
/ðtÞ ¼ P
as the relative disagreement at time t. A consensus protocol achieves e-level practical consensus if for a given constant e > 0, there exists /ð1Þ ¼ limt!1 /ðtÞ 6 e Note that in the quantized consensus model, quantizer structure affects quantization error and the practical consensus value. In order to reduce the effect of quantization error on consensus, it is necessary to design a optimal quantizer under given constraints such as e-level practical consensus and a limited rate of communication data. 3.3. Analysis of quantization error Quantization process has a natural relation with discrete time. Sampling rate is one key factor in determining the final consensus result in our quantized consensus model. With uniform period sampling and zero-order holding, the control law (4) could be rewritted as,
ui ðtÞ ¼
X
aij qðxj ðkTÞ xi ðkTÞÞ;
t 2 ½kT; kT þ TÞ
ð7Þ
j2N i
where k ¼ 0; 1; 2; . . . ; i ¼ 1; . . . ; n. Then, (5) can be represented as follows:
xðkT þ TÞ xðkTÞ ¼ TBT B½xðkTÞ eðkTÞ
ð8Þ
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where eðkTÞ ¼ xðkTÞ Q ð xðkTÞÞ. It is easy to see that
xðkT þ TÞ ¼ ðI TBT BÞxðkTÞ þ TBT BeðkTÞ
ð9Þ
Since the property of matrix B and the sampling period T affect the asymptotic convergence of (9), before moving on, one assumption is presented. Assumption 1. The communication topology graph G is a tree. Remark 1. If G is a tree, the corresponding matrix B is positive definite[17]. So is BT B. This property is necessary for later analysis of the practical consensus. Though the tree-shaped graph is a very special case in the directed graph, it yet provides us a way of dealing with the problem of quantized consensus with the given control algorithm. And the directed graph which contains a spanning tree will be our work in the next. The relation between uðtÞ in (6) and xðkTÞ in (9) is described as follows:
uðkTÞ ¼ xT Lx ¼ xT BBT x ¼ ðBT xÞT ðBT xÞ ¼ xT x ¼ kxðkTÞk22
ð10Þ 2
2
It illustrates that, given the requirement of e-level practical consensus, / ¼ uð1Þ=uð0Þ ¼ jj xð1Þjj =jj xð0Þjj 6 e. To study the ultimate boundedness of the solution to difference equation (9), we introduce three lemmas. Lemma 1. If A; B are matrixes with compatible dimension, then
rðA þ BÞ P rðAÞ þ rðBÞ.
Lemma 2 ([18] Theorem 3.9). Assume Q is an element of Banach algebra. If kQ k < 1, then there exists ðI Q Þ1 and
ðI QÞ1 ¼
1 X Qk k¼0
Lemma 3. Assume Q and ðI Q Þ are both non-singular, ðQ Þ < 1, then r ðI Q Þ 6 1 rðQ Þ 1. if r ðQ Þ < 1 if and only if r ðI Q Þ 6 1 2. r The solution to (9) is
xððk þ 1ÞTÞ ¼ ðI TBT BÞkþ1 xð0Þ þ
k X ðI TBT BÞkr TBT BeðrTÞ r¼0
k X ¼ ðI TBT BÞkþ1 xð0Þ þ ðI TBT BÞr TBT BeðkT rTÞ
ð11Þ
r¼0
The relation between eðkTÞ and k xðkTÞk2 as k ! 1 is now presented. e; 8k. Then Theorem 4. Let BT B > 0; T < r ðB1T BÞ and keðkTÞk2 6 ^
limk!1 kxðkTÞk2 6 cd TBT B ^e
ðTBT BÞ=rðTBT BÞ is the conditioning number of matrix ðTBT BÞ. where ^e is a constant and cdðTBT BÞ ¼ r e. Note that from (11), it follows that Proof. Given that keðkTÞk2 6 ^e; 8k, so, keðkT rTÞk2 6 ^
! k X kþ1 I TBT B r ðI TBT BÞr r TBT B ^e kxððk þ 1ÞTÞk2 6 r kxð0Þk2 þ r¼0 T
T
kþ1
ðTBT BÞ < 1. By Lemma 3, there exjj xð0Þjj2 ! 0 as k ! 1. When T < r B1T B , it follow r ð Þ ðI TBT BÞ. By the above inequalities, one has ðI TBT BÞ 6 1 rðTBT BÞ < 1 and rðTBT BÞ 6 1 r ists r
ðI TB BÞ < 1, then r ðI TB BÞ If r
1 1 6 ðI TBT BÞ r TBT B 1r ðI TBT BÞ 6 1 rðTBT BÞ < 1. It can be verified that and r 1 X
r ðI TBT BÞr ¼
r¼0
1 1 6 ðI TBT BÞ rðTBT BÞ 1r
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ðTBT BÞ=rðTBT BÞ. Finally, it has Denote the conditional number of matrix ðTBT BÞ as cdðTBT BÞ ¼ r
lim jjxðkT Þjj2 6 cdðTBT BÞ^e:
k!1
Remark 2. By formula (10) and Theorem 4, the upper bound of quantization error, denoted by ^e, affects the disagreement ðTBT BÞ=rðTBT BÞ is fixed. Thereof multi-agent systems. For a given communication network, it is clear that cdðTBT BÞ ¼ r ^ fore, to reduce e is to reduce the final disagreement among agents and to meet the requirement of e-level practical consensus. P 2 12 ^ Since jjeðkTÞjj2 ¼ ð m i¼1 ðxi ðkTÞ qi ðxi ðkTÞÞÞ Þ ; e is relevant to xi ; k and qi , therefor, this paper design a dynamic quantizer based on a suboptimal method of allocation of communication resources, to meet the requirement of e-level practical consensus. In the next section, the main result is obtained.
4. Main results For later use, the discrete value set of xðkTÞ is defined as follows,
pi ,fxi ðkTÞ 2 Rxi ðkT þ TÞ ¼ fi ðxi ðkTÞ; xj ðkTÞÞ; xi ð0Þ ¼ xi;0 2 R; xj ð0Þ ¼ xj;0 2 Rg where fi is the state transfer function, xi;0 and xj;0 are, respectively, the initial value of xi and xj . Assume pi is a compact set. xi ðkTÞ 2 pi ; i ¼ 1; 2; . . . ; m. Define p ¼ p1 pm . For x0 2 p and the system quantizer Q, there exists Note that for 8k; xðkTÞ Q ð xðkTÞÞ 2 Rm ; k ¼ 0; 1; . . .. Let jjeð x0 ; Q Þjj1 ¼ supk jjeð x0 ; Q ÞðkTÞjj2 . By keðkT rTÞk2 6 ^e; 8k, and for eð x0 ; Q ÞðkTÞ ¼ 8x0 2 p and Q, we have jjeðx0 ; Q Þjj1 < 1. Therefor, eðx0 ; Q Þ 2 lm . In addition, x0 ; Q ÞðkTÞk2 6 maxx0 2p keð x0 ; Q Þk1 ¼ keð 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pm Pm 1 Let ^eðQ Þ ¼ maxxi;0 2pi maxk ð i¼1 jei ð xðkTÞ Q ð xðkTÞÞk2 ¼ maxxi;0 2pi maxk xi;0 ; qi ÞðkTÞj2 Þ2 . maxx0 2p maxk k i¼1 ei ðxi;0 ; qi ÞðkTÞ . The goal of quantizer design is to choose a optimal quantizer Q to minimize ^ eðQ Þ. Then, we give the following lemma. m Lemma 5. For eð x0 ; Q Þ 2 l1 , where x0 2 p, the inf Q ^ eðQ Þ exists when the system quantizer has a structure as follows:Q ð xÞ ¼ ½ ½q1 ð x1 Þ qm ð xm Þ T , where qi : xi # qi ð xi Þ is the i-th component of Q and a scalar uniform quantizer.
Proof. For a input signal xi 2 R with any probability density and a prescribed output set, the quantizer structure qi ðÞ which minimize ^ ei ðqi Þ is a scalar uniform quantizer[11]. Therefor, the structure of the system quantizer Q ð xÞ ¼ ½ ½q1 ð x1 Þ qm ð xm Þ T which minimize ^ eðQ Þ is the one with each element qi : xi # qi ð xi Þ; i ¼ 1; 2; . . . ; m, a scalar uniform quantizer. h
Now fix the quantization step size of the uniform quantizer qi such that DðkTÞ ¼ D0 ; k 2 ½0; k Þ, where k depends on the requirement of e-level practical consensus. In order to emphasize the relation between Q and r i , Q and D, denote Q as Q ðri ; D0 Þ and ^ e is expressed by ^eðQ ððr i ; D0 ÞÞÞ. To reach consensus, it is assumed each communication channel will be used for information flow. So, it has r i > 0. Define the quantization range of qi as jpi j ¼ maxxi 2pi xi minxi 2pi xi . pffiffiffiffiffi Lemma 6. If xi ðkTÞ is within the quantization range of quantizer qi , then jpi j j xi ð0Þ cdðTBT BÞD=ð2 mÞj. When D is small enough, there exists jpi j j xi ð0Þj. Proof. By Theorem 4 and ei ðrTÞ 6 D2 (i ¼ 1; 2; . . . ; m), it follows that
limk!1 jxi ðkT þ TÞj 6
DcdðTBT BÞ pffiffiffiffiffi ; i ¼ 1; 2; . . . ; m 2 m
When the quantization step size D is small enough, it can be verified that limk!1 j xi ðkT þ TÞj ! 0. So, the smallest jpi j is pffiffiffiffiffi j xi ð0Þ cdðTBT BÞD=ð2 mÞj. If D is small enough, then jpi j j xi ð0Þj. h Since the proof of Lemma 6 ignores the impact of coupling among agents, the result in Lemma 6 is conservative. Then, a convex optimization problem considered in this paper is described as follows. In the consensus problem of multi-agent systems, if the quantization range jpi j is defined by the quantization rate r i , then, P 12 P 2 12 m m jpi j ij 2 ¼ for the uniform quantizer qi ; D ¼ 2jrpi 1 and ei ¼ D2 ¼ j2priij. It follows that ^ e ¼ maxx2p kek2 ¼ maxxi 2pi . i¼1 ðei Þ i¼1 2ri Pm þ Under the constraint i¼1 ri ¼ R; ri 2 Z ; 1 6 i 6 m, the problem of finding a uniform quantizer to make agents reach practical consensus and to obtain the value of minQ ^e, is expressed by Problem 1,
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8 min^e > > < Q m X Problem1; > r i ¼ R; ri 2 Rþ ; 1 6 i 6 m > : s:t: i¼1
Since 0 < r i < R is a convex set and
Pm
i¼1 ðj
pi j2 22ri Þ is a convex function of ri , the equivalent form of the Problem 1 is [19]
8 mins > > fri g;s > > > m > X > < s:t: ðjpi j2 22ri Þ 6 s Problem2; i¼1 > > > m > > > Xr ¼ R; > ri P 0 ð1 6 i 6 mÞ : i i¼1
Then, we have the following theorem. P þ Theorem 7. Under the constraint of communication rate, that is, m i¼1 r i ¼ R; r i 2 Z ; 1 6 i 6 m, the scalar uniform quantizer qi (contained in the system quantizer Q) which makes agents reach the practical consensus and minimize ^e, has the following suboptimal quantization rate
r i ¼ where
1 jxi ð0Þj2 log2 2 s
! i ¼ 1; 2; . . . ; m Pm
s is a constant which satisfy 8 m X > < dr e; i 6 R br i c i ~r i ¼ i¼1 > : bri c; otherwise
i¼1 r i
¼ R. Since r i 2 Zþ , the suboptimal quantization rate will be:
Proof. Using the lagrange multiplier method to solve the transformed optimization problem (Problem 2), it follows P P Pm 2 2r i Lðs; ri ; k; t; lÞ ¼ s þ kð m sÞ m i¼1 jpi j 2 i¼1 ti r i þ lð i¼1 r i RÞ. The Karush–Kuhn–Tucker ðKKTÞ condition of the optimization problem 2 are [19]: Pm 1. jpi j2 22ri 6 s Pi¼1 m 2. i¼1 r i ¼ R; r i P 0; i ¼ 1; 2; . . . ; m 3. ti P 0ði ¼ 1; 2; . . . ; mÞ; l > 0; k P 0 4. kðjpi j2 22ri sÞ ¼ 0; ti ri ¼ 0; i ¼ 1; 2; . . . ; m @L 5. @r ¼ 2k lnð2Þjpi j2 22ri ti þ l ¼ 0; i ¼ 1; 2; . . . ; m i From the consensus condition r i > 0 and the KKT condition (4) ti ri ¼ 0, it is clear that ti ¼ 0. Then, by the KKT condition (5) and l > 0; k > 0 is obtained. It follows the solution to the KKT condition (4) is s ¼ jpi j2 22ri . Substitute it into the KKT condition 2
2 2r i l jpi j (5), then, one has k ¼ 2 lnð2Þ ; 8i, and 22ri ¼ 2k lnð2Þ ; l > 0. Since ri > 0, it can be verified s. Following this, l ¼ 2k lnð2Þjpi j 2 l 2 2 P m 2r i 2k lnð2Þjpi j jpi j 1 > 1. From r i ¼ 2 log2 s and i¼1 ri ¼ R; r i and s can be solved. For jpi j which is estimated as j that 2 ¼ xi ð0Þj, we l
know
that
ri
is
a
suboptimal
solution
minQ ^ e.
of 2
þ
2br i c
r i ¼ ðN 1Þ þ #; 0 < # < 1; N 2 Z , it is clear that jpi j 2 follows:
Note 2
2r i
> jpi j 2
that 2
ri 2 2dr i e
> jp i j 2
Zþ .
For
s ¼ jpi j2 22ri ,
assume
. Therefore, r should be chosen as
8 m X > < dr e; i 6 R br i c i ri ¼ i¼1 > : bri c; otherwise When jpi j jpj j; 1 6 i; j 6 m, the suboptimal quantization rate ~r i should be selected as:
8 m X > < d R e; i 6 R bmR c m ~r i ¼ i¼1 > : R bmc; otherwise
Remark 3. Assume the initial quantization step size D0 is small enough. The relation between allocation of communication resources (communication rate) and j xi ð0Þj illustrates that, the larger j xi ð0Þj is, the more ri is needed for qi .
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By theorem 7, an estimated method of allocating R to minimize ^ e is given. From the requirement of consensus level which pffi ekxð0Þk2 is defined by /ð1Þ ¼ k xð1Þk22 =k xð0Þk22 6 e, where k xð1Þk2 6 cd TBT B ^ e; ^e should satisfy ^ e 6 cdðTB to meet the requirement. T BÞ pffi xð0Þk2 e k However, if minQ ðri ;D0 Þ ^e > cdðTBT BÞ , then it is necessary to adjust the quantization step size D0 . T 1 ~ ffi BÞ and oð~ e ¼ minfri g; s s2 , q ¼ ep0 cdðTB e0 Þ is a infinitesimal of ~ e0 such as Define ~ eðQ ðr i ; DðtÞÞÞt¼0 ¼ ~e0 ðQ ðr i ; D0 ÞÞ ¼ minQ ðri ;D0 Þ ^ ekxð0Þk 2
oð~e0 Þ ¼ 0:01~ e0 . Note that Q contains m elements. The above propositiones lead to the following result. Lemma 8. For each quantizer qi ; i ¼ 1; 2 . . . ; m, with identical quantization step size at time kT; DðkTÞ, when 2 xðk TÞ 2 > ek xð0Þk22 , the multi-agent systems will reach the e -level practical consensus if each qi adjusts its D to DðkTÞ ¼ D0 =q; k P k þ 1 at k T, where k is selected when xðk TÞ 2 ¼ kxðkTÞk2 þ oðkxðkTÞk2 Þ. pffiffiffi pffiffiffi Proof. The requirement of e-level consensus means ^eð1Þ 6 ek xð0Þk2 =cdðTBT BÞ. Since ~ xð0Þk2 =cdðTBT BÞ, it needs e0 =q ¼ ek e0 þ oð~ e0 ÞÞ=q. That is, zooming out the initial to prove that when DðkTÞ ¼ D0 =q; k ¼ k þ 1; k þ 2 . . ., there exists ^eð1Þ 6 ð~ e0 Þ will zoom out the same time, that is, ^ eð1Þ 6 ð~e0 þ oð~e0 ÞÞ=q
quantization step size q times, then, ~e0 þ oð~ P 12 P 12 pffiffiffi m D0 2 m D0 2 ekxð0Þk2 =cdðTBT BÞ. Fix r i , it follows that eðkTÞ 6 ; k < k þ 1 and eðkTÞ 6 ; k P k þ 1. obviously, i¼1 ð 2 Þ i¼1 ð2q Þ one has ^ eð1Þ 6 ~ e0 =q ð~e0 þ oð~e0 ÞÞ=q. h xi decrease. However, since the relative states of agents are conRemark 4. Fix r i and zoom out D0 will make the range of verging as time goes on, it is reasonable to adjust D0 . The time consumed for finding a proper k is still acceptable. 5. Illustrative example The following is an illustrative example. Given a multi-agent system consists of 4 agents, the communication network is modeled as a undirected graph G ¼ fV; E; Ag. The corresponding adjacency matrix and incidence matix are, respectively,
0 61 6 A¼6 40 0 and
2
1
6 1 6 B¼6 4 0 0
1 0 1 0
0 1 0 1 0 1 1 0
3 0 07 7 7 15 0 0
3
0 7 7 7 1 5 1 5000 Any other Suboptimal
4500 4000 3500 3000 φ
2
2500 2000 1500 1000 500 0
0
5
10
15
20
25
30
k (T) Fig. 1. Disagreement comparison (two method of allocation).
35
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50 40 30
The relative state
20 10 0 −10 −20 −30 −40 −50
0
5
10
15
20
25
30
35
k (T) Fig. 2. State trajectories of x.
50 before adjusting Δ at k=35 after adjusting Δ at k=35
45 40 35
φ
30 25 20 15 10 5 0 10
15
20
25
30 k (T)
35
40
45
50
Fig. 3. Disagreement comparison (adjust D at k = 35).
P Let the initial state of x be x0 ¼ ½ 20:22 20:22 20:22 20:22 T and a communication rate constraint be m i¼1 r i ¼ 60. The initial T quantization step size of each scalar uniform quantizer is D0 ¼ 1. Take e ¼ 0:0001. By computation of B B , it is verified that T < 0.2929 s. Here, choose T = 0.1 s. Since T < 1=r xi ; i ¼ 1; 2; 3 converge around the origin, it is reasonable to estimate that jpi j jpj j; 1 6 i; j 6 3. So, a suboptimal quantization rate r i ¼ r j ¼ 20. In contrast, take any other way of allocation that r1 ¼ 36; r 2 ¼ 20; r3 ¼ 4. Then, the measurements of disagreement among agents under two methods as time goes on are given in Fig. 1. It should be noted this suboptimal allocation method is effective in reducing the disagreement and the corresponding quantization error. The state trajectories of the relative states among agents are depicted in Fig. 2. From x0 and e, it can be observed that u should satisfy uð1Þ 6 0:49. We note that uð1Þ is larger than 0.517 in Fig. 1. So adjust Dfrom 1 to 0.5, then, it is shown in Fig. 3, the final disagreement is less than 0.49.
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6. Conclusions This paper considers the problem of quantizer design under a limited rate of communication data for the multi-agent systems. It constructs a control law with quantized message passing to meet the requirement of consensus level. From discussion above, this paper obtains the following conclusions. To reduce the final disagreement of the multi-agent systems where each agent using quantized message for control, the system forces the allocation of communication resources to depend on the initial relative state. If physical conditions are allowed, adjusting the quantization step size dynamically will make the final practical consensus meet the requirement of consensus level. Further interesting research works include a discussion of the practical consensus for agents with complex linear dynamics. Acknowledgments This work is supported by the National Natural Science Foundation of China under grants 10832006 and 11072002. References [1] Ren W, Beard RW. Consensus seeking in multi-agent systems under dynamically changing interaction topologies. IEEE Trans Autom Control 2005;50:655–61. [2] Olfati-Saber R, Fax J, Murray R. Consensus and cooperation in networked multi-agent systems. Proc IEEE 2007;95:215–33. [3] Meng J, Egerstedt M. Distributed coordination control of multiagent systems while preserving connectedness. IEEE Trans Robot 2007;23:693–703. [4] Li Z, Duan Z, Chen G. Dynamic consensus of linear multi-agent systems. IET Control Theory Appl 2011;5:19–2 8. [5] Jiang H, Bi Q, Zheng S. Impulsive consensus in directed networks of identical nonlinear oscillators with switching topologies. Commun Nonlinear Sci Numer Simul 2012;17:378–87. [6] Kashay, Basar T, Srikant R. Quantized consensus. Automatica 2007;43:1192–203. [7] Ren W, Cao Y. Distributed coordination of multi-agent networks: emergent problems, models, and issues. Springer; 2011. [8] Bullo F, Liberzon D. Quantized cobtrol via locational optimization. IEEE Trans Autom Control 2006;51:2–13. [9] Nair GN, Evans RJ. Exponential stabilisability of finite-dimensional linear systems with limited data rates. Automatica 2003;39:585–93. [10] Li T, Fu M, Xie L, Zhang JF. Distributed consensus with limited communication data rate. IEEE Trans Autom Control 2011:56. [11] Gersho A, Gray R. Vector quantization and signal compression. Kluwer Academic; 1991. [12] Carli R, Fagnani F, Frasca P, Zampieri S. A probabilistic analysis of the average consensus algorithm with quantized communication. In: Proc 17th IFAC world congress; 2008. [13] Censi, Murray R. Real-valued average consensus over noisy quantized channels. In: American control conference (ACC ’09); 2009. p. 4361–6. [14] Fang J, Li H. Distributed consensus with quantized data via sequence averaging. IEEE Trans Signal Process 2010;58:944–8. [15] Dimarogonas DV, Johansson KH. Stability analysis for multi-agent systems using the incidence matrix: quantized communication and formation control. Automatica 2010;46:695–700. [16] Cui Y, Lau VKN. Convergence-optimal quantizer design of distributed contraction-based iterative algorithms with quantized message passing. IEEE Trans Signal Process 2010;58:5196–205. [17] Godsil C, Royle G. Algebraic graph theory. Springer-Verlag; 2001. [18] Dullerud GE, Paganini F. A course in robust control theory. Springer-Verlag; 2000. [19] Boyd S, Vandenberghe L. Convex optimization. Cambridge Univ. Press; 2004.