ROBUST CONSENSUS CONTROL BY STATE ... - IEEE Xplore

ROBUST CONSENSUS CONTROL BY STATE-DEPENDENT DITHERS Lijian Xu∗ , Le Yi Wang† , George Yin‡

Wei Xing Zheng§

Wayne State University Detroit, MI 48202

University of Western Sydney Penrith NSW 2751, Australia

ABSTRACT This paper introduces a new method for enhancing robustness against communication uncertainties in consensus control by using a state and sampling-interval dependent dither in signal transmission. This method is based on the principle of Itˆo’s formula for stochastic differential equation in which the diffusion term introduces a quadratic term in stability analysis. It is revealed that this feature can be utilized to provide robustness against communication multiplicative uncertainties, much beyond the ability of traditional feedback robustness design. Algorithms are introduced and their convergence properties are established. It is shown that appropriate design of the dithers can create a highly robust consensus control. Simulation results are used to illustrate the benefits of this method. Keywords. Consensus control, networked systems, feedback robustness, communication channels, multiplicative uncertainty, dithers. 1. INTRODUCTION Recent advance in communication technologies and networked systems has generated intensified research efforts on networked control systems that involve both feedback systems and communication channels [2, 7, 5, 8, 11]. Research topics include minimal channel capacities required to stabilize an unstable plant for noise free channels [2, 6]; optimal channel coding with error correcting control for feedback systems [7]; output variance minimization of systems involving additive noisy channels in the feedback loop; control design with channel delays [8, 11]. One of the main methodologies in networked control is consensus control in which a group of subsystems use local information exchange to achieve a global mission [3]. Consensus control has drawn increased attention recently in a variety of application areas, including load balancing in parallel computing [12], sensor networks [10], decentralized filtering, estimation, mobile agents, etc. In ∗ Research of this author was supported in part by Wayne State University Graduate Research Assistantship. † Research of this author was supported in part by the National Science Foundation under DMS-0624849, and in part by the Air Force Office of Scientific Research under FA9550-10-1-0210. ‡ The research of this author was supported in part by the National Science Foundation under DMS-0907753, and in part by the Air Force Office of Scientific Research under FA9550-10-1-0210. § The research of this author was supported in part by a research grant from the Australian Research Council.

our recent work [14], a Markov model is used to treat a large class of noises, where the network graph is modulated by a discrete-time Markov chain. Our work in [14] also provides convergence and rates of convergence for the corresponding recursive algorithms. Some of the useful features of [14] have been extended to the constrained consensus methodology used in this paper. In addition, the technique of post-iterate averaging is employed to enhance the power flow control algorithm. Communication channels introduce unique challenges to networked control systems. Typically, communication uncertainties are modeled by additive noises, which do not affect directly feedback stability. On the other hand, communication uncertainties involve many types of multiplicative nature, such as signal power loss, gain uncertainties, latency (time delays), phase shift, etc. Such multiplicative uncertainties impact directly on feedback stability and are more difficult to handle. This paper is focused on channel multiplicative uncertainties, mostly transmission gains. When a signal is transmitted, it is broadcasted and propagates through multiple pathways, depending on terrain conditions, buildings, weather conditions, echoes, interferences, and correlations with other signals. They are then collected at the receiver, combined, and decoded. Such a scenario is better represented by variations on transmission gains whose values can vary over a large range and may change signs as well. Due to fundamental limitations of feedback systems, large gain uncertainties, especially sign changes, cannot be overcome by the traditional feedback mechanism alone. This paper introduces a new method to enhance robustness of consensus control in this unique aspect. The idea was initially investigated for first-order systems in [13]. Although the method was shown to be highly effective for first-order systems, its utility in general higher dimensional systems turns out to be very challenging. However, in this paper, we will show that the method is of substantial utility in networked consensus control. The main idea of a stochastic dither for feedback robustness is based on a fundamental property in Itˆo’s formula for stochastic differential equations (SDE) [1] in which a stochastic signal affects the stability of the limit SDE with a squared gain factor. This distinct feature implies that if a scaled dither is used in transmitting a signal, it will be immune to sign changes in transmission gains. This feature and inherent feedback robustness can potentially extend feedback robustness to a much expanded uncertainty set. The rest of the work is organized in the following sec-

978-1-4673-0818-2/12/$31.00 ©2012 IEEE

tions. Section 2 describes the basic ideas of scaled dithers and their potential impact of stability robustness for closedloop systems. The main results of the paper are first presented in Section 3 on consensus control. Consensus control may be viewed as a networked first-order system, which can potentially benefit from state-dependent dithers. Section 4 first illustrates convergence of the consensus control when no communication gain uncertainties are involved. Then, a case study shows that communication gain uncertainty may destabilize the consensus control. Section 5 establishes gain robustness of the dithered consensus control. It is shown that by appropriate design of the dithers, a large gain uncertainty set on network connections can be tolerated. Finally, Section 6 summarizes the main findings of this paper. 2. PRELIMINARIES Communication channels introduce uncertainties of various types. The basic noisy channel models often assume additive noises. This model captures transmission channel uncertainties that are independent of signals. In general, such additive noises will affect system performance, such as control accuracy and error bounds, but not robust stability, due to the fact that they do not alter feedback systems. For more fundamental feedback stability and robustness, more severe uncertainties are those dependent on signals, especially gain uncertainties, phase shifts, and time delays. In the case of linear systems, these are multiplicative uncertainties which are proportional to signals. Consequently, they directly change feedback systems, and hence feedback robustness. These communication uncertainties are typically associated with multi-path signal propagations due to echoes and signal interferences, channel fading, channels with memories, and communication latency. To illustrate the issues and main ideas, we use an example from [13] which deals with first-order systems. Consider an unstable open-loop system 𝑥˙ = 𝑎𝑥 + 𝑏𝑢

(1)

with 𝑎 > 0 and 𝑏 > 0. Under selected sampling intervals 𝜏𝑘 , which may change with time and irregular, its sampled system is 𝑥𝑘+1 = 𝑥𝑘 = 𝜏𝑘 (𝑎𝑥𝑘 + 𝑏𝑢𝑘 ), ∑𝑘 where starting from 𝑡0 = 0, 𝑡𝑘 = 𝑙=1 𝜏𝑙 , 𝑥𝑘 = 𝑥(𝑡𝑘 ). Suppose that the feedback is 𝑢𝑘 = 𝑔𝑥𝑘 + 𝑒𝑘 with (uncertain) gain 𝑔 and additive noise 𝑒𝑘 . 𝑢𝑘 is then mapped to 𝑢(𝑡) by the standard ZOH. For sufficiently small 𝜏𝑘 , the stability of the closed-loop system is determined by its continuous-time limit system 𝑥˙ = (𝑎 − 𝑏𝑔)𝑥. It is obvious that the additive uncertainty 𝑒 will not affect system stability. If the communication channels create an uncertain gain 𝑔 ∈ [𝑔𝑚𝑖𝑛 , 𝑔𝑚𝑎𝑥 ], the closed-loop system is not robustly stable if 𝑎 − 𝑏𝑔𝑚𝑖𝑛 > 0.

Instead of sending only 𝑦𝑘 , a scaled dither can be added to form a new signal 𝑧𝑘 to be sent through the communication channel 𝑧𝑘 = 𝑥𝑘 + 𝛼(𝜏𝑘 , 𝑥𝑘 )𝑑𝑘 where 𝑑𝑘 is an independent and identically distributed (i.i.d) Gaussian-distributed random dither such that 𝐸𝑑𝑘 = 0 and 𝐸𝑑2𝑘 = 1. The scaling factor 𝛼(𝜏𝑘 , 𝑦𝑘 ) is both signal dependent and sampling interval dependent 𝛾 𝛼(𝜏𝑘 , 𝑦𝑘 ) = √ 𝑦𝑘 . 𝜏𝑘

(2)

The communication channel introduces uncertainties and generates a received signal 𝑧ˆ𝑘 = 𝑔𝑧𝑘 + 𝑒𝑘 . The feedback becomes 𝛾 𝑧𝑘 = −𝑔𝑧𝑘 − 𝑒𝑘 == −𝑔(𝑥𝑘 + √ 𝑥𝑘 𝑑𝑘 ) − 𝑒𝑘 . 𝑢𝑘 = −ˆ 𝜏𝑘 (3) where the gain 𝑔 is an unknown constant. Consequently, the closed-loop system becomes √ 𝑥𝑘+1 = 𝑥𝑘 + 𝜏𝑘 (𝑎 − 𝑏𝑔)𝑥𝑘 − 𝜏𝑘 𝑏𝑔𝛾𝑑𝑘 − 𝜏𝑘 𝑏𝑒𝑘 . (4) The enhancement of stability robustness by scaled dithers is based on Itˆo’s formula in stochastic differential equations [1, 9]. In its applications to linear time invariant systems, suppose that 𝑥(𝑡) ∈ IR𝑛 is a real-valued stochastic process satisfying ∫ 𝑡 ∫ 𝑡 𝑥(𝑡) = 𝑥(𝑡0 ) + 𝑎𝑥(𝑟)𝑑𝑟 + 𝑔1 𝑥(𝑟)𝑑𝑤(𝑟), 𝑡0

𝑡0

where 𝑤 is a standard Brownain motion. In this case, 𝑥(𝑡) is said to satisfy the stochastic differential equation 𝑑𝑥 = 𝑚𝑥𝑑𝑡 + 𝑔1 𝑥𝑑𝑤.

(5)

Here the first term is the drift term and the second term is the multiplicative diffusion. In our approach, the multiplicative diffusion is created by the added scaled dither. By Itˆo’s Formula [1, 9], the solution to (5) is 1

2

𝑥(𝑡) = 𝑒(𝑚− 2 𝑔1 )𝑡+𝑔1 𝑤 𝑥(0). Consequently, the SDE (5) is stable if and only if 𝑚𝑐 = 𝑚 − 12 𝑔12 < 0. For the system (1), 𝑚 = 𝑎 − 𝑏𝑔 and 𝑔1 = 𝑏𝑔𝛾. Hence, the closed-loop system SDE has 𝑚𝑐 = 𝑎 − 𝑔𝑏 − 12 𝑔 2 𝛾 2 𝑏2 . If we define 𝑔0 = 𝑔𝑏, this becomes 1 𝑚𝑐 = 𝑎 − 𝑔0 − 𝑔02 𝛾 2 . 2 Suppose that the gain uncertainty on 𝑔0 is characterized by an uncertainty set Ω. Then, the robust stability requires that (6) sup 𝑚𝑐 < 0. 𝑔0 ∈Ω

Note that since 𝑎 > 0, representing an unstable openloop system, 𝑔0 ∕= 0 is necessary for stability. When the regular feedback is used without dither, robust stability

is determined by 𝑎 − 𝑏𝑔 < 0. So, the robustness range for the uncertain gain is (𝑎/𝑏, ∞). When a scaled dither is used, the robustness is enhanced against a much larger uncertainty set Ω. The main reason for this enhancement 𝑔2 𝛾 2 is that the term − 02 is always negative, regardless the sign and size of the uncertain gain 𝑔0 . However, using scaled dithers in higher-order systems demand caution. It can be shown that adding such a dither without careful analysis of stability may destabilize a stable system! This paper will focus on the important area of networked consensus control. 3. NETWORK CONSENSUS CONTROL WITH SCALED DITHERS A networked system consists of 𝑟 node states denoted by 𝑥𝑛 = [𝑥1𝑛 , . . . , 𝑥𝑟𝑛 ]′ . At the control step 𝑛, the value of 𝑥 will be updated 𝑥𝑛 to 𝑥𝑛+1 by the amount 𝑢𝑛 𝑥𝑛+1 = 𝑥𝑛 + 𝑢𝑛

(7)

with 𝑢𝑛 = [𝑢1𝑛 , . . . , 𝑢𝑟𝑛 ]′ . The node subsystems are linked by a network, represented by a directed graph 𝒢 whose element (𝑖, 𝑗) indicates a connection between node 𝑖 and node 𝑗, namely estimation of the state 𝑥𝑗𝑛 by node 𝑖 via a communication link. For node 𝑖, (𝑖, 𝑗) ∈ 𝒢 is a departing edge and (𝑙, 𝑖) ∈ 𝒢 is an entering edge. For symmetric networks, if (𝑖, 𝑗) ∈ 𝒢 then (𝑗, 𝑖) ∈ 𝒢. The total number of communication links in 𝒢 is 𝑙𝑠 . From its physical meaning, node 𝑖 can always observe its own state, which will not be considered as a link in 𝒢. Using power flow (or goods transportation, or computing job balancing) control problems as an example, a power transfer 𝑝𝑖𝑗 𝑛 from node (a bus in a power grid) 𝑖 to node 𝑗 is the independent decision variable. The bus power control 𝑢𝑖𝑛 is determined by the link control 𝑝𝑖𝑗 𝑛. The power control at node 𝑖 is ∑ ∑ 𝑢𝑖𝑛 = − 𝑝𝑖𝑗 𝑝𝑗𝑖 (8) 𝑛 + 𝑛. (𝑖,𝑗)∈𝒢

𝑖=1

𝑥𝑖𝑛 =

˜𝑛 = 𝐻2 𝑥𝑛 − 𝐻1 𝑥𝑛 − 𝑑𝑛 = 𝐻𝑥𝑛 − 𝑑𝑛 , 𝛿𝑛 = 𝐻 2 𝑥 𝑛 − 𝑥 (12) where 𝐻2 is an 𝑙𝑠 × 𝑟 matrix whose rows are elementary ˜ is 𝑥 vectors such that if the ℓth element of 𝜁(𝑘) ˆ𝑖𝑗 then the ℓth row in 𝐻2 is the row vector of all zeros except for a “1” at the 𝑖th position, and 𝐻 = 𝐻2 − 𝐻1 . Due to network constraints, the information 𝛿𝑛𝑖𝑗 can only be used by nodes 𝑖 and 𝑗. When the power control is linear, time invariant, and memoryless, we have 𝑖𝑗 𝑝𝑖𝑗 𝑛 = 𝜇𝑔𝑖𝑗 𝛿𝑛 where 𝑔𝑖𝑗 = 𝑔𝑗𝑖 is the link control gain and 𝜇 is a global scaling factor which will be used in state updating algorithms as the recursive step size. Let 𝐺 be the 𝑙𝑠 × 𝑙𝑠 diagonal matrix that has 𝑔𝑖𝑗 as its diagonal element. In this case, the bus power control becomes 𝑢𝑛 = −𝜇𝑛 𝐻 ′ 𝐺𝛿𝑛 . For convergence analysis, we note that 𝜇𝑛 is a global control variable and we may represent 𝑢𝑛 equivalently as 𝑢𝑛

= −𝜇𝑛 𝐻 ′ 𝐺(𝐻𝑥𝑛 − 𝑑𝑛 ) = −𝜇𝑛 (𝐻 ′ 𝐺𝐻𝑥𝑛 − 𝐻 ′ 𝐺𝑑𝑛 ) = 𝜇𝑛 (𝑀 𝑥𝑛 + 𝑊 𝑑𝑛 ),

with 𝑀 = −𝐻 ′ 𝐺𝐻 and 𝑊 = 𝐽 ′ 𝐺. It can be directly verified that 11′ 𝑀 = 0, 11′ 𝑊 = 0, 𝑀 11 = 0. For stability consideration, the following assumption is imposed on the network. (A0)

(1) All link gains are positive, 𝑔𝑖𝑗 > 0. (2) 𝒢 contains a complete tree.

We can show that under Assumption (A0), 𝑀 has rank 𝑟 − 1 and is negative semi-definite. As a result, 𝑀 is a generator of a continuous-time irreducible Markov chain.

(𝑗,𝑖)∈𝒢

The most relevant implication in this control scheme is that for all 𝑛, 𝑟 ∑

row in 𝐻1 is the row vector of all zeros except for a “1” at the 𝑗th position. Each link in 𝒢 provides information ˆ𝑖𝑗 𝛿𝑛𝑖𝑗 = 𝑥𝑖𝑛 − 𝑥 𝑛 , an estimated difference between weighted 𝑖 𝑗 𝑥𝑛 and 𝑥𝑛 . This information may be represented by a vector 𝛿𝑛 of size 𝑙𝑠 containing all 𝛿𝑛𝑖𝑗 in the same order as 𝑥 ˜𝑛 . 𝛿𝑛 can be written as

𝑟 ∑

𝑥𝑖0 = 1.

(9)

𝑖=1

Consensus control seeks control algorithms such that 𝑥𝑛 → 𝛽11 under the constraint (9). 𝑗 A link (𝑖, 𝑗) ∈ 𝒢 entails an estimate 𝑥 ˆ𝑖𝑗 𝑛 , of 𝑥𝑛 by node 𝑖𝑗 𝑖 with estimation error 𝑑𝑛 . That is, 𝑗 𝑖𝑗 𝑥 ˆ𝑖𝑗 𝑛 = 𝑥 𝑛 + 𝑑𝑛 .

(10)

Let 𝑥 ˜𝑛 and 𝑑𝑛 be the 𝑙𝑠 dimensional vectors that contain 𝑖𝑗 all 𝑥 ˆ𝑖𝑗 𝑛 and 𝑑𝑛 in a selected order, respectively. Then, (10) can be written as 𝑥 ˜𝑛 = 𝐻1 𝑥𝑛 + 𝑑𝑛

(11)

where 𝐻1 is an 𝑙𝑠 × 𝑟 matrix whose rows are elementary ˆ𝑖𝑗 then the ℓth vectors such that if the ℓth element of 𝜁˜𝑛 is 𝑥

3.1. Consensus Control Algorithms and Convergence We begin by considering the following state updating algorithm 𝑥𝑛+1 = 𝑥𝑛 + 𝜇𝑛 𝑀 𝑥𝑛 + 𝜇𝑛 𝑊 𝑑𝑛 ,

(13)

together with the constraint 11′ 𝑥𝑛 = 𝐿,

(14)

where {𝜇𝑛 } is a sequence of stepsizes, 𝑀 is a generator of a continuous-time Markov chain (hence 11′ 𝑀 = 0), {𝑑𝑛 } is a noise sequence. The stepsize satisfies the following conditions: 𝜇𝑛 ≥ 0, 𝜇𝑛 → 0 as 𝑛 → ∞, and ∑ 𝜇 𝑛 𝑛 = ∞. Some commonly used stepsize sequence includes 𝜇𝑛 = 𝑎/𝑛, 𝜇𝑛 = 𝑎/𝑛𝛼 for 0 < 𝛼 ≤ 1. Since algorithm (13) is a stochastic approximation procedure, we can use the general framework in Kushner and Yin [4] to analyze the asymptotic properties. Since 11′ 𝑀 = 0 and 11′ 𝑊 = 0, the constraint 11′ 𝑥𝑛 = 𝑟𝐿 is always satisfied by the algorithm structure.

(A1) The noise {𝑑𝑛 } is a sequence of i.i.d. random variables such that 𝐸𝑑𝑛 = 0, 𝐸∣𝑑𝑛 ∣2 < ∞.

jX23

P3 L3

To study the convergence of the algorithm, using the stochastic approximation methods developed in [4], instead of working with the discrete-time iterations, we examine sequences defined in an appropriate function space. This will enable us to get a limit ordinary differential equation (ODE). The significance of the ODE is that the stationary points are exactly the true parameter we wish to estimate.

L2

jX34

P5

and this constraint lead to the following system of equations { 𝑀𝑥 = 0 (16) 11′ 𝑥 = 𝐿. The irreducibility of 𝑀 then implies that (16) has a unique solution 𝑥∗ = 𝛽Ψ−1 11 = 𝛽𝛾. 4. COMMUNICATION GAIN UNCERTAINTIES AND CONSENSUS CONTROL The convergence of the constrained consensus control relies on the matrix 𝑀 in (15). It was proved in [14] that if all link gains are positive, 𝑀 has one eigenvalue at 0 (this is due to the constraint) and the rest eigenvalues are all stable (namely, they are in the left-half complex plane). However, if communication channels introduce gain uncertainties that change the sign of any link gains, convergence of the consensus control will be lost. We use the following example to illustrate this issue.

jX12

jX45

L4

Theorem 1 Under Assumption (A1), the iterates generated by the stochastic approximation algorithm (13) satisfies Ψ𝑥𝑛 → 𝛽11 w.p.1 as 𝑛 → ∞. Furthermore, the algorithm (13) together with 𝑥′𝑛 11 = 𝐿 leads to the desired consensus. The equilibria of the limit ODE 𝑥(𝑡) ˙ = 𝑀 𝑥(𝑡). (15)

P2

P4

L1

P1

Fig. 1. A grid of five buses

and ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ 𝐻1 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 1 0 0 0 0 0 0

1 0 0 1 0 0 0 0

0 0 1 0 0 1 0 0

0 0 0 0 1 0 0 1

0 0 0 0 0 0 1 0

⎡

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ; 𝐻2 = ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

1 0 0 0 0 0 0 0

0 1 1 0 0 0 0 0

0 0 0 1 1 0 0 0

0 0 0 0 0 1 1 0

0 0 0 0 0 0 0 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

It follows that 𝐻 =⎡ 𝐻2 − 𝐻1 1 −1 0 ⎢ −1 1 0 ⎢ ⎢ 0 1 −1 ⎢ ⎢ 0 −1 1 =⎢ ⎢ 0 0 1 ⎢ ⎢ 0 0 −1 ⎢ ⎣ 0 0 0 0 0 0

0 0 0 0 0 0 0 0 −1 0 1 0 1 −1 −1 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎦

Suppose the control gains are 𝐺 = diag[0.3, 0.3, 0.5, 0.7, 0.9, 0.9, 1, 1]. It follows that

Example 1 A five-bus grid has transmission lines between Buses 1 and 2, 2 and 3, 3 and 4, and 4 and 5, shown in Figure 1. The initial per-unit load distributions on the buses are not balanced with 𝑥0 = [0.1, 0.2, 0.3, 0.4, 0]′ . The grid network set of 𝒢 is {(1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4)}. The network state vector 𝑥 is [𝑃 1 , 𝑃 2 , 𝑃 3 , 𝑃 4 , 𝑃 5 ]′ . By choosing the order for the links as (1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4), we have 𝑥 ˜ = [ˆ 𝑥12 , 𝑥 ˆ21 , 𝑥 ˆ23 , 𝑥 ˆ32 , 𝑥 ˆ34 , 𝑥 ˆ43 , 𝑥 ˆ45 , 𝑥 ˆ54 ]′

𝑀 =⎡ −𝐻 ′ 𝐺𝐻 = −0.6 0.6 ⎢ 0.6 −1.8 ⎢ 1.2 =⎢ ⎢ 0 ⎣ 0 0 0 0

0 1.2 −3 1.8 0

0 0 1.8 −3.8 2

0 0 0 2 −2

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

In this case, the eigenvalues of 𝑀 are −6.0125, −3.2432, −1.5016, −0.4426, 0. Since all eigenvalues (except the single 0) are stable, the control achieves the weighted consensus, as shown in Figure 2. Example 2 Now, suppose that communication channel gain uncertainties cause the link gain matrix to change its values to 𝐺 = diag[0.3, 0.3, 0.5, −0.7, −0.9, 0.9, 1, 1]. Correspondingly, the 𝑀 matrix is changed to

where 𝑀 = −𝐻 ′ 𝐺𝐻, 𝐺1 = 𝛾𝐻 ′ 𝐺𝐻1 , and 𝑤 a standard Brownian motion.

Power Distribution Trajectoies 0.4

States

0.3

Assumption 1. (1) All link gains are non-zero 𝑔𝑖𝑗 ∕= 0. (2) The network topology 𝐺 contains a full tree.

0.2

0.1

0

0

50

100

150

200

250

Proposition 1 Under Assumption 1, 𝑀 is of rank 𝑟 − 1.

300

Consensus Error Trajectories 0.35

Lemma 1 Under Assumption 1, (1) 𝐻 ′ 𝐺𝐻1 has rank 𝑟 − 1. (2) 𝐻 ′ 𝐺𝐻1 𝐻1′ 𝐺𝐻 has exactly one zero eigenvalue and all other eigenvalues are positive.

0.2 0.15 0.1 0.05 0

0

50

100

150

200

250

Example 3 Consider the same system as in Example 2. By adding a dither with 𝜎 2 = 9, we obtain

300

Iteration Number

Fig. 2. Power flow control under positive link gains

⎡ ⎢ ⎢ 𝑀 = −𝐻 ′ 𝐺𝐻 = ⎢ ⎢ ⎣

−0.6 0.6 0 0 0

0.6 −0.4 −0.2 0 0

0 −0.2 0.2 0 0

0 0 0 0 0 0 −2 2 2 −2

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

whose eigenvalues are −4, −1.1211, 0.3211, 0, 0. The inclusion of an unstable eigenvalue indicates that the consensus control becomes unstable. This is shown in Figure 3.

′ 𝑀 = −𝐻 ⎡ 𝐺𝐻 = −1.41 0.465 ⎢ 0.465 −2.65 ⎢ 4.21 =⎢ ⎢ 0.945 ⎣ 0 −2.025 0 0

0 0 4.05 6.95 −11

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Power Distribution Trajectoies 0.8 0.6 0.4 0.2 0

−0.4

0.4

−0.6 0.3

0

5

10

15

20

25

30

35

40

45

50

40

45

50

Consensus Error Trajectories

0.2

1

0.1

0.8

0

5

10

15

20

25

30

35

40

45

Error Norm

States

0 −2.025 5.265 −10.19 6.95

−0.2

Power Distribution Trajectoies

0

0.945 4.21 −14.47 5.265 4.05

whose eigenvalues are −19.3659, −17.1275, −2.0336, −1.1931, 0. The restoration of stable eigenvalues recovers convergence of the consensus control under communication uncertainty. This is shown in Figure 4.

States

Error Norm

0.3 0.25

50

Consensus Error Trajectories

0.6 0.4 0.2

0.35

Error Norm

0 0.3

0

5

10

15

20

25

30

35

Iteration Number

0.25

Fig. 4. Power flow control under perturbed link gains but with a scaled dither added to each observation link

0.2

0

5

10

15

20

25

30

35

40

45

50

Iteration Number

Fig. 3. Power flow control under perturbed link gains 6. CONCLUSIONS

5. ROBUST STABILITY IN CONSENSUS CONTROL BY SCALED DITHERS We now explore the scaled dither to improve robust stability of the consensus control against communication uncertainties. Suppose that for each observation link, a scaled dither is added. The dither is a state and sampling interval scaled Gaussian i.i.d. variable of mean 0. If the node 𝑗 is √ observed by node 𝑖, then the added dither is 𝛾𝑥𝑗𝑘 𝑑𝑖𝑗 𝑘 / 𝜏𝑘 with 𝑑𝑖𝑗 𝑘 an i.i.d. standard Gaussian random variable. It can be verified that this will lead to a limit stochastic differential equation 𝑑𝑥 = 𝑀 𝑥𝑑𝑡 + 𝐺1 𝑥𝑑𝑤

(17)

This paper introduces the approach of scaled dithers to expand robustness capabilities of networked consensus control systems. The approach is effective in networked systems with communication channels in which transmission channel gain uncertainties are naturally present and often involve sign changes. Effectiveness of such dithers in more complicated systems and broader types of communication uncertainties are currently under investigation. 7. REFERENCES [1] L.C. Evans, “An Introduction to Stochastic Differential Equations,” Online Resource: http://math.berkeley.edu/ evans/SDE.course.pdf Version 1.2.

[2] J. S. Freudenberg and R. H. Middleton, Feedback control performance over a noisy communication channel. Proceedings of the 2008 Information Theory Workshop, Porto, Portugal, pp. 232-236, May 2008. [3] M. Huang and J.H. Manton. “Coordination and consensus of networked agents with noisy measurements: stochastic algorithms and asymptotic behavior,” SIAM J. Control Optim., vol. 48, no. 1, pp. 134161, 2009. [4] H.J. Kushner and G. Yin, “Stochastic Approximation Algorithms and Applications,” Springer-Verlag, New York, 2nd Ed., 2003. [5] R. Luck and A. Ray, “Experimental verification of a delay compensation algorithm for integrated communication and control system. ” International Journal of Control, vol. 59, pp. 1357-1372, 1994. [6] L. Moreau, “Stability of multiagent systems with time- dependent communication links, ” IEEE Trans. Autom. Control, vol. 50, no. 2, pp. 169-182, Feb. 2005. [7] G. N. Nair and R. J. Evans, “Exponential stabilisability of finite-dimensional linear systems with limited data rates,” Automatica, 39 (2003) 585-593. [8] R. Obstovasky, Y. Radbani, L. J. Schulman, “ErrorCorrecting Codes for Automatic Control, ” Information Theory, IEEE Transactions 2009 [9] B. Oksendal, “Stochastic Differential Equations: An Introduction with Applications,” 6th Edition, , 2007. [10] P. Ogren, E. Fiorelli, and N.E. Leonard, “Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment, ” IEEE Trans. Autom. Control, vol. 49, no. 8, pp. 1292-1302, Apr. 2005. [11] A. J. Rojas, J. H. Braslavsky, and R. H, Middleton, “Fundamental Limitations in Control over a Communication Channel ’ Automatica 44(2008), pp. 31473151. [12] J.N. Tsitsiklis, D.P. Bertsekas, and M. Athans, “Distributed asynchronous deterministic and stochastic gradient optimization algorithms, ” IEEE Trans. Automat. Control, 31, no. 9, pp. 803-812, 1986. [13] L.J. Xu, L.Y. Wang, and G. Yin, “Enhanced feedback robustness against communication channel gain uncertainties vis scaled dithers, ” Accepted ISSPA Montreal, Quebec, 2012. [14] G. Yin, Y. Sun, and L.Y. Wang, “Asymptotic properties of consensus-type algorithms for networked systems with regime-switching topologies, ” Automatica, 47 (2011) 1366–1378. [15] G. Yin, “On extensions of Polyak’s averaging approach to stochastic approximation, ” Stochastics Rep., 36 (1991), 245-264.

[16] G. Yin and Q. Zhang, “ Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, ” Springer-Verlag, New York, 1998.