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Distributed State Estimation by a Network of Observers under Communication and Measurement Delays
7th IFAC Workshop on Distributed Estimation and 7th IFAC on Distributed 7th IFACinWorkshop Workshop onSystems Distributed Estimation Estimation and and Control Networked 7th IFACinWorkshop onSystems Distributed Estimation Availableand online at www.sciencedirect.com Control Networked Control inWorkshop Networked Systems 7th IFAC on Distributed Estimation and Chicago, IL, USA, September 16-17, 2019 Control in Networked Systems16-17, 2019 Chicago, USA, Chicago,inIL, IL, USA, September September 16-17, 2019 Control Networked Systems Chicago, IL, USA, September 16-17, 2019 Chicago, IL, USA, September 16-17, 2019
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IFAC PapersOnLine 52-20 (2019) 13–18
Distributed Distributed State State Estimation Estimation by by a a Network Network Distributed State Estimation by a Network Distributed State Estimation by a Network of Observers under Communication and of Observers under Communication and of Observers under Communication and of Observers under Communication and Measurement Delays Measurement Delays Measurement Delays Measurement Delays ∗∗ ∗ ∗ ∗∗
∗∗ Himadri Basu ∗∗ Se Young Yoon Himadri Himadri Basu Basu Se Se Young Young Yoon Yoon ∗∗ Himadri Basu ∗∗ Se Young Yoon ∗∗ Himadri Basu Se Young Yoon ∗∗ ∗ ∗ ∗ Department of Electrical ∗ Department of Electrical and and Computer Computer Engineering, Engineering, University University of of ∗ Department of Electrical and Computer Engineering, University of New Hampshire, NH 03824, USA (e-mail: of ElectricalDurham, and Computer Engineering, University of ∗ Department New Hampshire, Durham, NH 03824, USA (e-mail: New Hampshire, Department of Electrical and Computer Engineering, University of Durham, [email protected]) New Hampshire, Durham, NH NH 03824, 03824, USA USA (e-mail: (e-mail: [email protected]) [email protected]) New Hampshire, Durham, NH 03824, USA (e-mail: ∗∗ ∗∗ ∗∗ Department Department of of Electrical Electrical and Computer Computer Engineering, University University of [email protected]) ∗∗ and of Electrical and Computer Engineering, Engineering, University of of [email protected]) ∗∗ Department New Hampshire, Durham, NHEngineering, 03824, USA USAUniversity of of Electrical andDurham, Computer ∗∗ Department New Hampshire, NH 03824, Hampshire, Durham, NHEngineering, 03824, USAUniversity of DepartmentNew of Electrical and Computer New (e-mail:[email protected]) Hampshire, Durham, NH 03824, USA (e-mail:[email protected]) New (e-mail:[email protected]) Hampshire, Durham, NH 03824, USA (e-mail:[email protected]) (e-mail:[email protected]) Abstract: In this this paper we we study the the distributed state state estimation of of autonomous dynamic dynamic Abstract: Abstract: In In this paper paper we study study the distributed distributed state estimation estimation of autonomous autonomous dynamic systems, under time-invariant communication and measurement delays. In the studied problem, Abstract: In this paper we study the distributed state estimation of autonomous dynamic systems, under time-invariant communication and measurement measurement delays.ofIn Inautonomous the studied studied problem, problem, systems, under time-invariant and delays. the Abstract: Inthat this paper we communication study the distributed state estimation dynamic we consider consider the deployed network of observer observer agents cannot determine all the the states systems, under time-invariant communication and measurement delays.determine In the studied problem, we that the deployed network of agents cannot all states we consider that the deployed network of observer agents cannot determine all the states systems, under time-invariant communication and measurement delays. In the studied problem, we consider that the deployed network of observer agents cannot determine all the states of the the observed observed plant plant from from their their own own collected collected measurements. measurements. To To reconstruct reconstruct the the correct correct of of the observed from their own collected measurements. To determine reconstructallthe we consider that plant the deployed network of observer agents cannot thecorrect states state estimation, they must collaborate with neighboring agents to agree on the coordinated coordinated of theestimation, observed plant from collaborate their own with collected measurements. Toagree reconstruct the correct state they neighboring agents on state estimation, they must must collaborate with neighboring agents to to agree on the the coordinated of the observed plant from their own collected measurements. To reconstruct the correct estimation data. Our Our current work considers considers the presence of ofagents time-invariant communication and state estimation, theycurrent must collaborate withthe neighboring to agree communication on the coordinated estimation data. work presence time-invariant and estimation data. Our work considers the presence ofagents time-invariant communication and state estimation, theycurrent collaborate withand neighboring to agreeapproach on the coordinated measurement latency ofmust the observer observer agents, then following a low-gain low-gain we propose propose estimation data. Our current work considers the then presence of time-invariant communication and measurement latency of the agents, and following a approach we measurement latency of agents, and following aa low-gain approach we estimation data. Our current work considers the then presence of time-invariant communication measurement latency of the the observer observer agents, and then following low-gainfor approach we propose propose a framework for constructing distributed observers. Sufficient conditions for the stability of and the a constructing distributed observers. Sufficient conditions the of a framework framework for for constructing distributed observers. Sufficient conditions for the stability stability of the the measurement latency of the observer agents, and then following a low-gain approach we propose corresponding observation error dynamics are derived, including an upper bound for the lowa framework for constructing distributed observers. Sufficient conditions for the stability of the corresponding error dynamics are including an for corresponding observation error dynamicsobservers. are derived, derived, including an upper upper bound for the theoflowlowa framework forobservation distributed Sufficient conditions for bound the stability the gain parameter ofconstructing the observer equations. An illustrative example is also presented to verify the corresponding observation errorequations. dynamicsAn areillustrative derived, including analso upper bound to forverify the lowgain of example presented the gain parameter parameterobservation of the the observer observer equations. An illustrative example is is also presented to verify the corresponding error dynamics are derived, including an upper bound for the loweffectiveness of the the theoretical derivations in the the paper. example is also presented to verify the gain parameter of the observer derivations equations. An illustrative effectiveness of theoretical in paper. effectiveness of theoretical in paper. gain parameter of the observer derivations equations. An illustrative effectiveness of the the theoretical derivations in the the paper. example is also presented to verify the effectiveness of the theoretical derivations in the Copyright © 2019. The Authors. Published by Elsevier Ltd. paper. All rights reserved. Keywords: distributed state state estimation, communication communication delay, low-gain low-gain feedback, Keywords: Keywords: distributed distributed state estimation, estimation, communication delay, delay, low-gain feedback, feedback, leader-follower synchronization Keywords: distributed state estimation, communication delay, low-gain feedback, leader-follower synchronization leader-follower synchronization Keywords: distributed state estimation, communication delay, low-gain feedback, leader-follower synchronization leader-follower synchronization 1. INTRODUCTION cannot independently reconstruct the leader’s trajectory. 1. cannot 1. INTRODUCTION INTRODUCTION cannot independently independently reconstruct reconstruct the the leader’s leader’s trajectory. trajectory. Such distributed state estimation problems were studied 1. INTRODUCTION cannot independently reconstruct the leader’s trajectory. Such distributed state estimation problems were studied 1. INTRODUCTION cannot independently reconstruct the leader’s trajectory. Such distributed state estimation problems were The objective objective of of the the distributed distributed state state estimation estimation problem problem Such The in the works of Basu Yoon (2018). distributed stateand estimation problems were studied studied The objective of the distributed state estimation problem in the works of Basu and Yoon (2018). distributed state estimation problems were studied in the The of theis statethe estimation problem for aa objective dynamic plant plant isdistributed to reconstruct plant state vector Such for dynamic to plant vector the works works of of Basu Basu and and Yoon Yoon (2018). (2018). for a objective dynamic plant to reconstruct reconstruct the plant state state vector in The ofobservers theisdistributed statethe estimation problem In the works of Park and Martins (2017), the design in the works of Basu and Yoon (2018). In the works of Park and Martins (2017), the the design design of of by a network of using limited decentralized plant for a dynamic plant is to reconstruct the plant state vector by aaa network network observers using limited decentralized plant In the works of Park and Martins (2017), for dynamicofplant is to reconstruct theshared plant between state vector by observers using decentralized plant distributed discrete LTI observers aa scalability the works of Park and Martinssubject (2017),to the design of of measurements and the communication the In by a network of ofand observers using limited limited shared decentralized plant distributed discrete LTI observers subject to scalability measurements the communication between the the works of studied, Park andwhile Martins (2017), the design of distributed discrete LTI subject toMorse aa scalability measurements and the communication shared between the In by a network observers using limited decentralized plant constraint was Wang and (2017) discrete LTI observers observers subject scalability observers. Toofand estimate target stateshared evolving from measurements the communication between theaa distributed constraint was was studied, while Wang Wang and to Morse (2017) observers. To estimate aa target target state evolving from distributed discrete LTI observers subject to a scalability constraint studied, while and Morse (2017) observers. To estimate a state evolving from a measurements and the communication shared between the investigated the same problem for the continuous counwhile Wang Morse (2017) observers. To estimate a target state evolving from by a constraint investigatedwas the studied, same problem problem for the theand continuous coundynamic process, process, distributed observers were studied studied by dynamic distributed observers were was studied, while Wang and Morse (2017) investigated the same for continuous counobservers. To asestimate a target state evolving from a constraint dynamic process, distributed observers were studied by terpart with a preassigned observer spectrum. In case of investigated the same problem for the continuous counauthors such Park and Martins (2017); Wang et al. dynamic process, distributed observers were studied by terpart with a preassigned observer spectrum. In case of authors such as Park Park and Martins (2017); et the same problem for the continuous counterpart with aa preassigned observer spectrum. In case of authors such as and Martins (2017); Wang et al. al. dynamic process, distributed observers wereWang studied by investigated any abrupt of the network over withchanges preassigned observertopology spectrum. In time case or of (2017). such authors as Park and Martins (2017); Wang et al. terpart any abrupt changes of the network topology over time or (2017). withchanges a preassigned observertopology spectrum. In case of any abrupt of over time or (2017). such as Park and Martins (2017); Wang et al. terpart authors packet dropouts, aa hybrid observer comprising of a any abrupt changes of the the network network topology over time or (2017). packet dropouts, hybrid observer comprising of aa local local packet dropouts, a hybrid observer comprising of local any abrupt changes of the network topology over time or In contrast to the decentralized estimation scheme as (2017). In contrast contrast to to the the decentralized decentralized estimation estimation scheme scheme as as packet a hybrid observer comprising of a local observer and local parameter estimator was by observerdropouts, and aaa local local parameter estimator was designed designed by In a hybrid observer comprising of a local observer and parameter estimator was by noted in the thetoworks works of Su Su and and Huang Huang (2012),scheme where as at packet In contrast the decentralized estimation noted in of (2012), where at observer and a local parameter estimator was designed designed by Wang et etdropouts, al. (2017). (2017). noted in thetoworks of Su and Huang (2012), where at In contrast the decentralized estimation scheme as Wang al. and(2017). a local parameter estimator was designed by Wang et least one one observer independently estimates the entire entire plant noted in observer the works of Su and estimates Huang (2012), where at observer least independently the plant et al. al. (2017). least one independently the entire plant noted in observer the and works ofestimation Su and estimates Huang (2012), where at Wang In all the above mentioned papers, the inter-agent comWang et al. (2017). In all the above mentioned papers, papers, the the inter-agent inter-agent comcomstate vector the is then propagated to least one observer independently estimates the entire plant state one vector and independently the estimation estimationestimates is then then the propagated to In all the above mentioned state vector and the is propagated to least observer entire plant In all the above mentioned papers, the inter-agent communication or the leader-follower communication was imthe remaining observers through a consensus protocol, state vector and the estimation is then propagated to munication or the leader-follower communication was imthe remaining remaining observers through isaa then consensus protocol, all the above mentioned papers, the inter-agent communication or leader-follower communication was imthe observers through consensus protocol, state vector and the estimation to In plicitly assumed to be instantaneous. However, due to unmunication or the the leader-follower communication was imin the the current problem we consider that propagated each protocol, observer the remaining observers through a consensus plicitly assumed to be instantaneous. However, due to unin current problem we consider that each observer munication or the leader-follower communication was implicitly assumed to instantaneous. However, due to in the current problem we consider that each observer the remaining through a consensus avoidable time-delay the communication between assumed to be bein instantaneous. However, due agents to ununin the the current observers problem we consider that receives each protocol, observer the distributed observation framework receives only aa plicitly avoidable time-delay in the between agents in distributed observation framework only avoidable time-delay ininstantaneous. the communication communication between agents assumed to be However, due to unthe distributed observation framework receives only a plicitly in the current problem we consider that each observer in multi-agent system, the effects of the time-delay time-delay in the communication between agents in aaa multi-agent multi-agent system, the effects effects of of the the time-delay portion of the the plant output measurement needed for for the in the distributed observation framework receives onlythe a avoidable portion of output needed time-delay in the communication between agents in system, the time-delay portion of the plant plant output measurement measurement needed for in the distributed observation framework receives onlythea avoidable in a multi-agent system, the effects of the time-delay should be explicitly taken into consideration. The coninto consideration. The conestimation of the system states. The limited plant output portion of the plant output measurement needed for the should be explicitly taken estimation of the system states. The limited plant output in a multi-agent system, the effects of the time-delay should be explicitly taken into consideration. The conestimation of the system states. The limited plant output portion of the plant output measurement needed for the sensus problem for first-order agents with a time-varying should be explicitly taken into consideration. The coninformationofis isthe notsystem enoughstates. for independent independent reconstruction estimation The limitedreconstruction plant output should sensus problem problem for first-order agents with a time-varying information not enough for be explicitly taken into consideration. The confor agents with information not enough for independent reconstruction estimation ofis limited plantInstead, output sensus communication was studied by Zhenzua et al. (2016), sensus problem delay for first-order first-order agents with aa time-varying time-varying information isthe notsystem enough for reconstruction communication delay was studied by Zhenzua et of the the entire entire state vectorstates. by independent a The single observer. of state vector by a single observer. Instead, communication delay was studied by Zhenzua et al. al. (2016), (2016), sensus problem for first-order agents with a time-varying information is not enough for independent reconstruction of the entire state vector by a single observer. Instead, while for higher-order the consensus delay wassystems, studied by et al.problem (2016), while for for higher-order higher-order systems, theZhenzua consensus problem observers disseminate theirby local information overInstead, com- communication of the entire state vector a single observer. observers disseminate their local information over aaa comdelay was studied by Zhenzua et input al.problem (2016), while systems, the consensus observers disseminate their local information over com- communication of the entire state vector by a single observer. Instead, while for higher-order systems, the consensus problem was studied in Zhou and Lin (2013) for fixed and munication network, and collaborate to jointly synthesize observers disseminate their local information over a comwas studied in Zhou and Lin (2013) for fixed input and munication network, and collaborate to jointly synthesize while for higher-order systems, the consensus problem was studied in Zhou and Lin (2013) for fixed input and munication network, and collaborate to jointly synthesize observers disseminate their local information over a comcommunication delays. was studied in Zhou and Lin (2013) for fixed input and an estimation estimation of the the plant plant dynamics. to jointly synthesize was munication network, and collaborate communication delays. an of dynamics. studied in Zhou and Lin (2013) for fixed input and communication delays. an of dynamics. munication network, and collaborate delays. an estimation estimation of the the plant plant dynamics. to jointly synthesize communication In this work we study the distributed state estimation communication delays. In the framework of the distributed observation problem, an estimation of the plant dynamics. In this work we study the distributed distributed state state estimation estimation In the the framework framework of of the the distributed distributed observation this work we study the In observation problem, problem, In problem by a network of observers under distinct comIn this work we study the distributed state estimation each observer can of bethe viewed as aa follower follower agent,problem, and the the In In theobserver framework distributed observation problem by a network of observers under distinct comeach can be viewed as agent, and this work study of theobservers distributed state estimation problem by aa we network under distinct comeach can be viewed as aa follower agent, and the In theobserver framework of the distributed observation problem, munication and measurement delays. We construct a disproblem by network of observers under distinct comeach observer can be viewed as follower agent, and the munication and measurement delays. We construct a displant to be observed as the leader agent. The design of plant to be observed as the leader agent. The design of problem by a network of observers under distinct communication and measurement delays. We construct a displant observer to be observed as the as leader agent. agent, The design of munication each can be can viewed aregarded follower andcase the tributed solution following the low-gain approach and and measurement delays. We construct a distributed solution following the low-gain approach and distributed observers then be as a special plant to be observed as the leader agent. The design of distributed observers can then be regarded regarded asThe special case and measurement delays. We construct a and distributed solution following the low-gain approach distributed can be as aa special case plant to beobservers observed as then the problem, leader agent. design of munication Lyapunov equation. Sufficient stability of following theconditions low-gain for approach of aa leader-follower leader-follower consensus where each follower distributed observers can thenproblem, be regarded as aeach special case tributed Lyapunovsolution equation. Sufficient conditions for stabilityand of of consensus where follower solution following theconditions low-gain for approach and Lyapunov equation. Sufficient stability of of a leader-follower consensus where follower distributed observers can thenproblem, be regarded as aeach special case tributed Lyapunov equation. Sufficient conditions for stability of of a leader-follower consensus problem, where each follower Lyapunov equation. Sufficient conditions for stability of of a leader-follower consensus problem, wherebyeach follower 2405-8963 Copyright © 2019. The Authors. Published Elsevier Ltd. All rights reserved.
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the observation error dynamics, including an upper bound for the low-gain parameter, are found for such arbitrary large delays. Compared to the parametric Riccati equation based solution in Liu et al. (2018), the results in our current work offers a more general Hurwitz condition for the convergence of the state estimation. Such generalization is achieved by offering an alternative formulation and proof of the main results.
Let the connections between the plant (1b) and the N distributed observers be described by the graph G = (V, E), V = {0, 1, 2, · · · , N }. Under the multi-agent system representation, the leader agent (1b) is the zeroth node of V, while the follower agents are the remaining N nodes. The distributed observer equation in the presence of known inter-agent communication delays τ1 , and delayed measurement yi in (1a), are given as aij eτ1 S (ηj (t − τ1 ) − ηi (t − τ1 )) η˙ i = Sηi + µ j∈Ni (2) τ2 S + µai0 e Gi Ci (w(t−τ2 )−ηi (t−τ2 )) ,
The remainder of the paper is organized in the following way. The problem formulation and algebraic graph theoretic properties are briefly revisited in Section 2. Next, the stability conditions for the distributed observer dynamics coupled with communication and measurement delays are derived in Section 3. An illustrative example to verify the effectiveness of the proposed approach is presented in Section 4. The paper ends with a conclusion in Section 5.
where ηi is the state estimation by the ith observer, i = 1, 2, · · · , N , aij is the weighting factor for the communication between follower agents i and j in the communication digraph G, Gi and µ are respectively the observer gain and the low-gain scalar to be determined later.
Notations. We now briefly introduce some notations and symbols to be used throughout the text. The Kronecker product of matrices is denoted by ⊗. A vector 1N is a column vector in RN of all ones. Z+ is the set of all positive integers. Iq and 0q respectively denote the identity matrix and zero matrix of dimension q × q. Unless mentioned otherwise, for matrices Ai , i = 1, 2, · · · , N , T T T A¯ = col(A1 , A2 , · · · , AN ) = AT and A = 1 , A2 , · · · , AN blk diag(A1 , A2 , · · · , AN ) represents a block diagonal matrix with the ith block being Ai . For a non-zero vector x, and matrix X, x , and X respectively stand for the L2 norm for vectors and the spectral norm for matrices. For a square matrix X , let the minimum, maximum and sum of eigenvalues be respectively denoted by λmin (X ), λmax (X ) and Tr(X ). For the positive scalar τ , let C([−τ, 0], Rm ) denote the Banach space of all continuous functions mapping the interval [−τ, 0] into Rm endowed with the supremum norm.
In case when the time-delay is present only in the measurements, the distributed observer dynamics in (2) can be reformulated in a way such that the stability result becomes independent of the delay and thus the problem does not offer a significant challenge. However, due to the presence of the communication delay in (2), the distributed observers cannot be designed without explicitly considering the effects of the delays into account. For τ = mini=1,2 τi , τ¯ = maxi=1,2 τi and θ ∈ [−¯ τ , 0], let the ¯initial conditions w(θ) = w(θ) and ηi (θ) = η i (θ), where w, η i ∈ C([−¯ τ , 0], Rq ). Denote the estimation error ˜ i (θ), with η ˜i ∈ η˜i = ηi − w with η˜i (θ) = ηi (θ) − w(θ) = η C([−¯ τ , 0], Rq ). The estimation error dynamics obtained from (1b) and (2) is given as η˜˙ i = S η˜i + µ aij eτ1 S (˜ ηj (t − τ1 ) − η˜i (t − τ1 )) j∈Ni (3) τ2 S − µai0 e Gi Ci η˜i (t − τ2 ).
2. PROBLEM FORMULATION Consider a network of N observer agents under a communication network, with each agent i sensing a timedelayed measurement yi of an autonomous system with states w ∈ Rq , yi (t) = Ci w(t − τ2 ), i = 1, 2, · · · , N, (1a) w(t) ˙ = Sw(t), (1b) for known measurement delay τ2 ≥ 0. The objective of the group of N distributed observers is to provide an asymptotic estimation of the plant state vector w(t). When the pair (S, Ci ) is detectable for some agent i, the distributed estimation problem becomes a well studied decentralized state observation problem under measurement delay. On the other hand, when the previous detectability condition is not satisfied, a cooperative effort is required by the observers over the communication network to jointly provide an estimation for w(t). The estimation of the states w(t) is more challenging when we include the latency in the communication between the distributed observers to the formulation of the cooperative observation problem.
We will now introduce the following assumptions to guarantee the solvability of the distributed estimation problem. Assumption 1. S has all eigenvalues on the imaginary axis. Remark 1. As noted in Basu and Yoon (2018), Assumption 1 is common in the literature of cooperative output regulation problem (CORP), and the leader-follower synchronization problem to be later considered in this work. Stable eigenvalues of S are ignored without loss of generality, as their corresponding modes will asymptotically approach zero and the solution to the distributed estimation problem becomes trivial. On the other hand, estimating exponentially unstable state trajectories is also rare in practical applications, especially when these states serve as leader tracking trajectories in the leader-follower synchronization problem. ¯ is detectable, where C¯ = Assumption 2. The pair (S, C) col (C1 , C2 , · · · , CN ). Remark 2. The condition in Assumption 2 redefines the classical detectability property to the multi-agent framework, and it is referred to as the “combined detectability” property in Basu and Yoon (2018). An equivalent observability condition appears in Park and Martins (2017); Wang and Morse (2017); Wang et al. (2017); Liu et al. (2018).
2.1 Problem Statement The observers along with the plant model in (1b) can be viewed as a multi-agent system with the plant being the leader agent and the observers being the followers. 14
2019 IFAC NecSys Chicago, IL, USA, September 16-17, 2019 Himadri Basu et al. / IFAC PapersOnLine 52-20 (2019) 13–18
Assumption 3. All N follower agents form a strongly connected partition of the digraph G. Remark 3. The strongly connected partition of G in Assumption 3 can be seen as a way for an observer agent i to collect information on modes that may not be detectable by the pair (S, Ci ). In such case, Assumption 2 guarantees that information on such mode is indeed collected by some observer agents in the system, and Assumption 3 provides a path for the information to travel to the i agent.
considered to be strictly positive definite. Here we extend the result to semi-definite matrices. For a real symmetric matrix M0 , it can be orthogonally decomposed as M0 = JDM0 J−1 where JT = J−1 , and DM0 is a real diagonal matrix with all diagonal elements being the eigenvalues of M0 . Then the integral on the right hand side of (9) yields the following form γ2 ω T (β)M0 ω(β)dβ (γ 2 − γ 1 ) γ1 γ2 = (γ 2 − γ 1 ) ω T (β)JDM0 JT ω(β)dβ, γ1 γ2 = (γ 2 − γ 1 ) ω T (β)DM0 ω(β)dβ, (10)
Now we are ready to define the problem statement as follows. Definition 1. Distributed state estimation problem: Design observer gains Gi , i = 1, 2, · · · , N , and feedback gain µ such that the estimation error dynamics (3) is ˜ i (θ) ∈ exponentially stable, i.e., for given τ1 , τ2 > 0 and η C([−¯ τ , 0], Rq ), limt→∞ η˜i (t) = 0 for i = 1, 2, · · · , N .
γ1
where ω(β) = JT ω(β), DM0 = blk diag(λ1 , λ2 , · · · , λn ). Suppose M0 has r non-zero eigenvalues, i.e., with no loss of generality let λi = 0, i = r + 1, r + 2, · · · , n and ω(β) = col(ω 1 (β), ω 2 (β), · · · , ω n (β)). Then by virtue of the result in Gu (2000), the integral expression in (10) reduces to the form γ2 ω T (β)M0 ω(β)dβ (γ 2 − γ 1 )
Before we present the main contributions of this work, we first establish the following results, which will be used in the next section. Lemma 1. Given the plant dynamics (1b) with the state matrix S satisfying Assumption 1, it holds that
γ1
∗
T
e−S t e−St ≥ e−ωγ t Iq , (4) ∗ for a positive scalar γ = max(∈, γ min ), γ min = min{γ ≥ 0 : Q = S T + S + γI > 0}, an arbitrary small positive scalar ∈> 0, and ω = q − 1.
= (γ 2 − γ 1 ) =
Proof. If S T + S is negative (semi)-definite or indefinite, the matrix Q can be made positive definite by selecting any γ ≥ γ min . Then by using Cholesky decomposition Q = W W T we obtain S T + S − W W T = −γ min Iq , (5) and thus Lemma 1 of Zhou et al. (2012) yields that ∗
T
eS t eSt ≤ eωγ min t Iq = eωγ t Iq , ω = q − 1. ∗
≥ =
e−St e−S
T
t
∗
≥ e−ωγ t Iq .
(6) = (7) =
On the other hand, if S T + S is positive definite, then γ min = 0. Furthermore, for an arbitrarily small positive scalar ∈, S T + S+ ∈ I > 0 and γ ∗ =∈. Therefore in a similar manner as above, we obtain from Lemma 1 in Zhou et al. (2012) e−St e−S
T
t
∗
≥ e−ω∈t Iq = e−ωγ t Iq .
=
2
i=1 r i=1 n
γ2
γ1
r i=1
λi (γ 2 − γ 1 ) λi λi γ2
T
γ2 γ1 γ2 γ1
λi ω T i (β)ω i (β) dβ, γ2
γ1
ωT i (β)dβ ωT i (β)dβ
ω (β)dβ DM0 γ1
γ2
T
ωT i (β)ω i (β)dβ
ω (β)dβ JDM0 J
γ1 γ2 γ1
ω T (β)dβ M0
γ2
,
ω i (β)dβ , γ1 γ2
ω i (β)dβ , γ1
γ2
ω(β)dβ , γ1
T
γ2
ω(β)dβ , γ1
γ2
ω(β)dβ . γ1
(11)
This concludes the proof for this lemma.
(8)
3. MAIN RESULT
By multiplying the left hand side of (7) by eSt and its right hand side by e−St we obtain (4). This concludes the proof. Lemma 2. For any positive semi-definite matrix M0 ≥ 0, two scalars γ 1 and γ 2 with γ 2 ≥ γ 1 , and a vector valued function ω : [γ 1 , γ 2 ] → Rn , the inequality γ2 γ2 T ω (β) dβ M0 ω(β) dβ γ1 γ1 (9) γ ≤ (γ 2 − γ 1 )
r
i=1
T
Since eωγ t Iq − eS t eSt ≥ 0, then
15
In this section, we will present the stability results for the estimated error dynamics in (3), which will eventually lead to the design of observer gains Gi and low-gain µ. The current work also provides an upper bound for µ to ensure stability of the error dynamics. Since Ci = 0 when ai0 = 0, the composite estimation error dynamics of ˜ = col(˜ ˜ 2, · · · , η ˜ N ), and η˜ = col(˜ η1 , η˜2 , · · · , η˜N ) with η η1 , η η˜i = col(˜ ηi1 , η˜i2 , · · · , η˜iq ) is then obtained as η − µ e(IN ⊗τ2 S) GC η˜(t − τ2 ) η˜˙ = (IN ⊗ S)˜ (12) − µ L ⊗ eτ1 S η˜(t − τ1 ), t > 0,
ω T (β) M0 ω(β) dβ
γ1
holds if the integrals are well defined.
˜ (θ), η ˜ ∈ C([−¯ η˜(θ) = η τ , 0], RN q ), ∀θ ∈ [−¯ τ , 0], N ×N where L ∈ R is the Laplacian matrix corresponding to the strongly connected partition of the network of
Proof. A version of this lemma appears in Yoon and Lin (2013); Zhou et al. (2010, 2012); Gu (2000), where M0 was 15
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N observer agents, G = blk diag(G1 , G2 , · · · , GN ), and C = blk diag(C1 , C2 , · · · , CN ). Let us now introduce some notations which will be used throughout the text. Denote σG = GC 2 , σL = L 2 , G = [G 1 G 2 · · · G N ] , G i = ζi Gi where ζi is the ith entry of the left eigenvector T ζ = [ζ1 , ζ2 , · · · , ζN ] of L corresponding to zero eigenvalue.
Next, by substituting (15) and (16) in (12), we obtain η˜˙ (t) = IN ⊗ eτ2 S M IN ⊗ e−τ2 S η˜(t) (17) − µ2 (π 1 + π 2 + π 3 + π 4 ) , ∀t ≥ τ¯, where M = (IN ⊗ S) − µ (L⊗Iq ) − µGC, π i = L ⊗ eτ1 S π ∗i , i = 1, 2, π j = IN ⊗ eτ2 S GCπ ∗j , j = 3, 4. Let us define the initial state of the redefined estimation error dynamics (17) as ˜ (¯ η τ + θ), ∀θ ∈ [−2¯ τ , −¯ τ] ¯ ˜ (θ) = η (18) η˜(¯ τ + θ), ∀θ ∈ [−¯ τ , 0],
First, we will check the boundedness of the estimation error dynamics (12), ∀t ∈ [0, τ¯] and then verify the asymptotic stability of (12) for t ≥ τ¯. For the first part, with no loss of generality, we assume that τ¯ = mτ + ∈ ¯ will where m ∈ Z+ , ∈< τ . In a step-by-step manner we ¯ evaluate the bounds of η˜(t) across each such sub-intervals.
where η˜(t), ∀t ∈ [0, τ¯] is shown to be bounded. As noted in Zhou et al. (2010), the solution to the estimation error ˜ (θ), θ ∈ dynamics (12), for t ≥ 0 with initial condition η [−¯ τ , 0] coincides with the solution to (17) with initial ¯˜ (θ). conditions η
To do this, from (12) we obtain t ˜ (0)−µ IN ⊗eS(t−s+τ2 ) GC η ˜ (s−τ2 )ds η˜(t) = IN ⊗eSt η 0 t η (s−τ1 )ds, ∀t < τ − µ IN ⊗eS(t−s+τ1 ) (L⊗Iq )˜ ¯ 0 Sθ S(τ2 +θ) √ ˜ η ≤ max e + µτ e σG ¯ θ∈[0,τ ] ¯ √ η C , + eS(τ1 +θ) σL ˜ (13)
To analyze the stability of (17), we consider a Lyapunov function of the form T η, V(˜ η ) = η˜T (Σ ⊗ e−τ2 S P e−τ2 S )˜ where P is positive definite matrix Σ = diag(ζ) is a entry being ζi . Since diagonal matrix with the ith diagonal T the matrix Σ ⊗e−τ2 S P e−τ2 S is symmetric and have all positive eigenvalues, V(˜ η ) > 0, ∀˜ η = 0. By differentiating V(˜ η (t)) along the trajectories of (17), we obtain ˙ η ) = η˜T M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d V(˜
where ˜ η C = maxθ∈[−¯τ ,0] ˜ η (θ) . It follows that η˜(t), ∀t ∈ ˜ [0, τ ] is bounded as η ∈ C([−τ, 0], RN q ). ¯ For the subsequent intervals with t > τ , we obtain from ¯ (12), t η˜(t) = IN ⊗eS¯τ η˜(t − τ ) − µ (π1 (s)+π2 (s))ds, (14) ¯ t−τ
d
T
−τ2 S T P e−τ2 S )˜ η − 2µ2 (π T 1 + π 2 )(Σ ⊗ e T −τ2 S 2 T T −τ2 S Pe )˜ η, − 2µ π 3 + π 4 (Σ ⊗ e
¯
where π1 = IN ⊗ eS(t−s+τ2 ) GC η˜(s − τ2 )ds, π2 = IN ⊗ η (s − τ1 )ds. Since η˜(t), t ∈ [0, τ ] and eS(t−s+τ1 ) (L ⊗ Iq )˜ ¯ that ˜ (θ), ∀θ ∈ [−τ, 0] are bounded, it follows from (14) η ¯ η˜(t), ∀t ∈ [τ, 2τ ] is bounded. ¯ ¯ In a similar manner, we can evaluate ˜ η (t) , ∀t ∈ [j τ, (j + ¯ can 1)τ ], j ≥ 2 and by using the method of induction we ¯ show η˜(t) is bounded for t ∈ [0, mτ ]. Again for t ≥ mτ , ¯ ¯ from (12) it yields t η˜(t) = IN ⊗eSm¯τ η˜(t−mτ )−µ (π1 (s) + π2 (s))ds, ¯ t−µτ
˙ 0 + 4µ2 V + µ2 ≤V + µ2
i=1
¯
where π ∗1 = π ∗2 = π ∗3
=
π ∗4 =
η˜(t − τ2 ) = e
t
−(IN ⊗Sτ2 )
e
t−τ1 t
+
π4∗ ),
(15)
t−τ2 t t−τ2
i=3
T LT ΣL ⊗ e(τ1 −τ2 )S P e(τ1 −τ2 )S π ∗i , (19)
t→∞
(16)
¯ with if the low-gain parameter µ satisfies µ < µ, α ¯= , µ λmax (P )ζmax (4εd +c1 c2 )
e(IN ⊗S)(t−s) (L ⊗ Iq )˜ η (s − τ1 ) ds,
t−τ1 t (IN ⊗S)(t−s)
η˜(t) +
µ(π3∗
π ∗i
T
T
π ∗i C T GT (Σ ⊗ P ) GCπ ∗i
t where V 0 = 0 η˜dT (s) M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d (s) ds, η˜d (t) = IN ⊗ e−τ2 S η˜(t). Next, we consider the following Theorem to evaluate (19). Theorem 3. Consider the distributed observers (2) satisfying Assumptions 1-3, and gains Gi selected such that the matrix R = S − µGC¯ is a Hurwitz matrix for any µ ∈ (0, 1). Let then P ∈ Rq×q > 0 be a solution to the inequality RT P + P R < 0, (20) T such that the matrix P (S − µGi Ci ) + (S − µGi Ci ) P has non-positive eigenvalues for all follower agents i. Then the observer states ηi (t) converges to w(t) asymptotically, i.e., lim (ηi (t) − w(t)) = 0, i = 1, 2, · · · , N,
which also implies that η˜(t), ∀t ∈ [0, τ¯] is bounded as ˜ (θ), ∀θ ∈ [−¯ η τ , 0] and η˜(t), ∀t ∈ [0, mτ ] are bounded. ¯ Since we obtained that η˜(t) is bounded for t ∈ [0, τ¯], we now need to show the stability of the closed-loop system (12) for t ≥ τ¯. In this regard, from (12) we evaluate η˜(t−τ1 ) and η˜(t − τ2 ) as η˜(t − τ1 ) = e−(IN ⊗Sτ1 ) η˜(t) + µ(π ∗1 + π ∗2 ),
2
4
(21)
∗
with M T (Σ⊗P )+(Σ⊗P )M < −αI, εd = eωγ τ2 e−τ2 S 2 , ∗ ∗ ∗ c1 = σG eωγ τ2 + σL eωγ τ1 , and c2 = τ12 σL eωγ τ1 + ∗ τ22 σG eωγ τ2 .
IN ⊗e(τ2 −τ1 )S GC η˜(s−τ2 ) ds,
e(IN ⊗S)(t−s) (L ⊗ e(τ1 −τ2 )S )˜ η (s − τ1 ) ds,
Proof. Let G satisfy Assumption 3 with at least one agent being the child node of the leader. Now, we evaluate the first term on the right-hand side of the inequality in (19). In this regard, we have
e(IN ⊗S)(t−s) GC η˜(s − τ2 ) ds. 16
2019 IFAC NecSys Chicago, IL, USA, September 16-17, 2019 Himadri Basu et al. / IFAC PapersOnLine 52-20 (2019) 13–18
17
˙ 0 = η˜T Σ ⊗ (S T P + P S) − µC T GT (Σ ⊗ P ) V d −µ(Σ ⊗ P )GC − µ(Lˆ ⊗ Iq ) η˜d (22) ≤ η˜dT Σ ⊗ (S T P + P S) − µC T GT (Σ ⊗ P ) −µ(Σ ⊗ P )GC] η˜d ≤ 0,
since the matrix (Lˆ ⊗ Iq ) is positive semi-definite and by assumption P (S − µGi Ci ) + (S − µGi Ci )T P is negative semi-definite. We will now investigate the largest invariant ˙ 0 = 0. We note here that the term V set of η˜d on which T ˆ −µ˜ ηd L ⊗ Iq η˜d in (22) becomes zero non-trivially only ˙ 0 in (22) thus when η˜d = 1N ⊗ η˜f , for any η˜f ∈ Rq , and V reduces to ˙ 0 = η˜f RT P + P R η˜f < 0, (23) V
Fig. 1. Directed communication network G with strongly connected partititon for agents 1, 2, 3 ¯ with µ ¯ in (21). Therefore from Lyapunovfor µ < µ Krasovskii stability theorem, limt→∞ η˜i (t) = 0, i = 1, 2, · · · , N . This concludes the proof. Remark 4. R can be rewritten as ¯ R = S − µGC. (29)
˙ by assumption. Therefore, for any non-zero η˜d , V0 < 0, ˙0 = 0 i.e. η˜dT M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d < 0, and V only when η˜d = 0. Furthermore, η˜dT M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d is continuous and finite for all µ ∈ (0, 1). Itthen implies that the term η˜dT M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d is bounded for all values of µ ∈ (0, 1) and thus there exists a positive scalar α > 0 such that ηd 2 η˜dT M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d < −α˜ (24) ∗ < −αe−ωγ τ2 ˜ η 2 ,
By Assumption 2, there exists a matrix G so that R in (29) is Hurwitz. While Lemma 3 allows for any G such that R is Hurwitz, the equivalent observer gain in Liu et al. (2018) is tied to the solution of a parametric Riccati equation. It is also true that any solution to the parametric Riccati equation in Liu et al. (2018) and the corresponding observer gain G results in R being Hurwitz, but the opposite may not necessarily hold. ∗
In case when τ1 = τ2 = τ , then c1 = eωγ τ (σG + σL ), ¯ reduces to c2 = τ 2 c1 , and µ α . µ ¯= 2 λmax (P ) ζmax 4εd + τ 2 (σL + σG ) e2ωγ ∗ τ
by virtue of Lemma 1.
Now we evaluate the rest of the terms on the right-hand side of the inequality (19) as follows 2 T T π ∗i LT ΣL ⊗ e(τ1 −τ2 )S P e(τ1 −τ2 )S π ∗i µ2
On the other hand, if S has all semi-simple eigenvalues, then it is unitarily diagonalizable. As a result, the bound on the matrix exponentials in (6) and (7) becomes T T eSt eS t = Iq = e−St e−S t which further yields α . µ ¯= λmax (P )ζmax (4 + (σG + σL )(τ12 σL + τ22 σG )) As τ1 and τ2 become larger, from (21), we observe that the low-gain bound µ ¯ decreases accordingly. Furthermore, when the leader dynamics is controlled by an input u to follow a certain trajectory, i.e. w˙ = Sw(t) + Q0 u(t), the results of the distributed estimation problem will remain unaffected by the selected observation protocol (2).
i=1
2 ωγ ∗ (2∗τ1 −τ2 ) ˙1 , ≤ λmax (P )µ2 τ1 ζmax σL e τ1 ˜ η 2−V ∗ ˙2 , +σL σG eωγ τ1 τ1 ˜ η 2−V (25)
µ2
4 i=3
T
π ∗i C T GT (Σ ⊗ P ) GCπ ∗i
∗ ˙3 ≤ λmax (P )µ2 τ2 ζmax σG σL eωγ τ1 τ2 ˜ η 2 − V 2 ωγ ∗ τ2 ˙4 , + σG e η 2 − V τ2 ˜ (26) τi +τ1 t where V1 = τi η˜T (σ)˜ η (σ)dσ ds, i = 1, 2, Vj = t−s τ2 +τj−2 t T η˜ (σ)˜ η (σ)dσ ds, j = 3, 4. We define a new τj−2 t−s Lyapunov function Vφ as 2 ωγ ∗ (2τ1 −τ2 ) e V1 Vφ = V + λmax (P )µ2 ζmax τ1 σL ∗ 2 ωγ ∗ τ2 + σL σG eωγ τ1 (τ1 V2 + τ2 V3 ) + τ2 σG e V4 , (27)
4. ILLUSTRATIVE EXAMPLE In this section we consider an illustrative example to evaluate the effectiveness of our proposed algorithm. Let T the plant (1b) have state vector w = [w1 w2 w3 w4 ] and state matrix 0 −1 0 −2 , S2 = . (30) S = blk diag(S1 , S2 ), S1 = 1 0 2 0 The measured output matrices Ci , i = 1, 2, 3, are given as follows: C1 = [I2 02 ] , C2 = [02 I2 ] , C3 = 02×4 , from which we verify that none of the pairs (S, Ci ), i = 1, 2, 3, are detectable. Instead, the combined detectability property in Assumption 2 is satisfied for the given Ci .
η 2 . Since V > 0, where V ≤ λmax (P ) ζmax e−τ2 S 2 ˜ the Lyapunov function Vφ is also positive. Then by substituting the results from (25), (26) into (19), we obtain ˙ φ ≤ −αe−ωγ ∗ τ2 + 4λmax (P )µ2 ζmax e−τ2 S 2 V 2 2ωγ ∗ (2τ1 −τ2 ) 2 ωγ ∗ τ2 + λmax (P )µ2 ζmax τ12 σL e + τ22 σG e ∗ η 2 < 0, +(τ12 + τ22 )σL σG eωγ τ1 ˜ (28)
The communication network between the observers is represented by the digraph G satisfying Assumption 3, as 17
2019 IFAC NecSys 18 Chicago, IL, USA, September 16-17, 2019 Himadri Basu et al. / IFAC PapersOnLine 52-20 (2019) 13–18
By the combined detectability condition and a “strongly connected” network property of the observer agents, we guarantee that the local state estimation of all agents converges to the states of the observed plant, including states which are not detectable from local measurements. To guarantee the convergence of estimation errors to zero in the presence of time-delays, we designed a low-gain parameter and observers based on generalized sufficient conditions which are obtained without requiring to solve extensive matrix inequalities. For any time-delay in the network, a corresponding low-gain bound can always be obtained to ensure the consensus on the estimated states. An extension of this presented result for time-varying measurement and communication latency is our future research endeavor. REFERENCES
1.4
δ1 δ2 δ3
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1
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0 0
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500
600
700
800
900
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1000
Time(seconds)
Fig. 2. Observation error of three observers for τ1 = 0.8 and τ2 = 1.2 illustrated in Figure 1,with the Laplacian matrix L given as follows 1 −1 0 L = 0 1 −1 . −1 0 1 The strongly connected partition of G allows the exchange of estimated states between agents, and complements for the incomplete measurements that each agent receives from the plant. Let τ1 = 0.8 and τ2 = 1.2. From the Laplacian matrix L of digraph G, we find ζmin = ζmax = 1/3. Next, we select the observer gains as T
T
G1 = [I2 02 ] , G2 = [02 I2 ] , G3 = 04×2 (31) such that the matrix −µ/3 −1 −µ/3 −2 R = blk diag , 1 −µ/3 2 −µ/3
in (29) is Hurwitz with the eigenvalues located at −µ/3 ± j, −µ/3 ± 2j. Then, from the Lyapunov equation RT P + P R = −µI, we find the solution P as ∞ ∞ T T eR s eRs ds = eR +Rs ds = 1.5I. P = 0
0
Next, from the inequality (28) we obtain the upper bound ¯ as µ 0.1 ¯= µ = 0.029. (32) 1 + 1.5τ12 + τ22 ¯ The simulated response of the We select µ = 0.022 < µ. distributed observes are presented in Figure 2 through the 4 sum of the observation errors δi (t) = q=1 ˜ ηiq (t), where T
ηi1 η˜i2 η˜i3 η˜i4 ] and i = 1, 2, 3. η˜i = [˜
The simulated response in Fig. 2 shows that δi (t) and η˜i converges asymptotically to zero, and thus the objectives of the distributed observation problem are achieved. 5. CONCLUSION In this work we studied the distributed state estimation problem for a marginally stable autonomous dynamic system under communication and measurement delays. 18
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Distributed Distributed State State Estimation Estimation by by a a Network Network Distributed State Estimation by a Network Distributed State Estimation by a Network of Observers under Communication and of Observers under Communication and of Observers under Communication and of Observers under Communication and Measurement Delays Measurement Delays Measurement Delays Measurement Delays ∗∗ ∗ ∗ ∗∗
∗∗ Himadri Basu ∗∗ Se Young Yoon Himadri Himadri Basu Basu Se Se Young Young Yoon Yoon ∗∗ Himadri Basu ∗∗ Se Young Yoon ∗∗ Himadri Basu Se Young Yoon ∗∗ ∗ ∗ ∗ Department of Electrical ∗ Department of Electrical and and Computer Computer Engineering, Engineering, University University of of ∗ Department of Electrical and Computer Engineering, University of New Hampshire, NH 03824, USA (e-mail: of ElectricalDurham, and Computer Engineering, University of ∗ Department New Hampshire, Durham, NH 03824, USA (e-mail: New Hampshire, Department of Electrical and Computer Engineering, University of Durham, [email protected]) New Hampshire, Durham, NH NH 03824, 03824, USA USA (e-mail: (e-mail: [email protected]) [email protected]) New Hampshire, Durham, NH 03824, USA (e-mail: ∗∗ ∗∗ ∗∗ Department Department of of Electrical Electrical and Computer Computer Engineering, University University of [email protected]) ∗∗ and of Electrical and Computer Engineering, Engineering, University of of [email protected]) ∗∗ Department New Hampshire, Durham, NHEngineering, 03824, USA USAUniversity of of Electrical andDurham, Computer ∗∗ Department New Hampshire, NH 03824, Hampshire, Durham, NHEngineering, 03824, USAUniversity of DepartmentNew of Electrical and Computer New (e-mail:[email protected]) Hampshire, Durham, NH 03824, USA (e-mail:[email protected]) New (e-mail:[email protected]) Hampshire, Durham, NH 03824, USA (e-mail:[email protected]) (e-mail:[email protected]) Abstract: In this this paper we we study the the distributed state state estimation of of autonomous dynamic dynamic Abstract: Abstract: In In this paper paper we study study the distributed distributed state estimation estimation of autonomous autonomous dynamic systems, under time-invariant communication and measurement delays. In the studied problem, Abstract: In this paper we study the distributed state estimation of autonomous dynamic systems, under time-invariant communication and measurement measurement delays.ofIn Inautonomous the studied studied problem, problem, systems, under time-invariant and delays. the Abstract: Inthat this paper we communication study the distributed state estimation dynamic we consider consider the deployed network of observer observer agents cannot determine all the the states systems, under time-invariant communication and measurement delays.determine In the studied problem, we that the deployed network of agents cannot all states we consider that the deployed network of observer agents cannot determine all the states systems, under time-invariant communication and measurement delays. In the studied problem, we consider that the deployed network of observer agents cannot determine all the states of the the observed observed plant plant from from their their own own collected collected measurements. measurements. To To reconstruct reconstruct the the correct correct of of the observed from their own collected measurements. To determine reconstructallthe we consider that plant the deployed network of observer agents cannot thecorrect states state estimation, they must collaborate with neighboring agents to agree on the coordinated coordinated of theestimation, observed plant from collaborate their own with collected measurements. Toagree reconstruct the correct state they neighboring agents on state estimation, they must must collaborate with neighboring agents to to agree on the the coordinated of the observed plant from their own collected measurements. To reconstruct the correct estimation data. Our Our current work considers considers the presence of ofagents time-invariant communication and state estimation, theycurrent must collaborate withthe neighboring to agree communication on the coordinated estimation data. work presence time-invariant and estimation data. Our work considers the presence ofagents time-invariant communication and state estimation, theycurrent collaborate withand neighboring to agreeapproach on the coordinated measurement latency ofmust the observer observer agents, then following a low-gain low-gain we propose propose estimation data. Our current work considers the then presence of time-invariant communication and measurement latency of the agents, and following a approach we measurement latency of agents, and following aa low-gain approach we estimation data. Our current work considers the then presence of time-invariant communication measurement latency of the the observer observer agents, and then following low-gainfor approach we propose propose a framework for constructing distributed observers. Sufficient conditions for the stability of and the a constructing distributed observers. Sufficient conditions the of a framework framework for for constructing distributed observers. Sufficient conditions for the stability stability of the the measurement latency of the observer agents, and then following a low-gain approach we propose corresponding observation error dynamics are derived, including an upper bound for the lowa framework for constructing distributed observers. Sufficient conditions for the stability of the corresponding error dynamics are including an for corresponding observation error dynamicsobservers. are derived, derived, including an upper upper bound for the theoflowlowa framework forobservation distributed Sufficient conditions for bound the stability the gain parameter ofconstructing the observer equations. An illustrative example is also presented to verify the corresponding observation errorequations. dynamicsAn areillustrative derived, including analso upper bound to forverify the lowgain of example presented the gain parameter parameterobservation of the the observer observer equations. An illustrative example is is also presented to verify the corresponding error dynamics are derived, including an upper bound for the loweffectiveness of the the theoretical derivations in the the paper. example is also presented to verify the gain parameter of the observer derivations equations. An illustrative effectiveness of theoretical in paper. effectiveness of theoretical in paper. gain parameter of the observer derivations equations. An illustrative effectiveness of the the theoretical derivations in the the paper. example is also presented to verify the effectiveness of the theoretical derivations in the Copyright © 2019. The Authors. Published by Elsevier Ltd. paper. All rights reserved. Keywords: distributed state state estimation, communication communication delay, low-gain low-gain feedback, Keywords: Keywords: distributed distributed state estimation, estimation, communication delay, delay, low-gain feedback, feedback, leader-follower synchronization Keywords: distributed state estimation, communication delay, low-gain feedback, leader-follower synchronization leader-follower synchronization Keywords: distributed state estimation, communication delay, low-gain feedback, leader-follower synchronization leader-follower synchronization 1. INTRODUCTION cannot independently reconstruct the leader’s trajectory. 1. cannot 1. INTRODUCTION INTRODUCTION cannot independently independently reconstruct reconstruct the the leader’s leader’s trajectory. trajectory. Such distributed state estimation problems were studied 1. INTRODUCTION cannot independently reconstruct the leader’s trajectory. Such distributed state estimation problems were studied 1. INTRODUCTION cannot independently reconstruct the leader’s trajectory. Such distributed state estimation problems were The objective objective of of the the distributed distributed state state estimation estimation problem problem Such The in the works of Basu Yoon (2018). distributed stateand estimation problems were studied studied The objective of the distributed state estimation problem in the works of Basu and Yoon (2018). distributed state estimation problems were studied in the The of theis statethe estimation problem for aa objective dynamic plant plant isdistributed to reconstruct plant state vector Such for dynamic to plant vector the works works of of Basu Basu and and Yoon Yoon (2018). (2018). for a objective dynamic plant to reconstruct reconstruct the plant state state vector in The ofobservers theisdistributed statethe estimation problem In the works of Park and Martins (2017), the design in the works of Basu and Yoon (2018). In the works of Park and Martins (2017), the the design design of of by a network of using limited decentralized plant for a dynamic plant is to reconstruct the plant state vector by aaa network network observers using limited decentralized plant In the works of Park and Martins (2017), for dynamicofplant is to reconstruct theshared plant between state vector by observers using decentralized plant distributed discrete LTI observers aa scalability the works of Park and Martinssubject (2017),to the design of of measurements and the communication the In by a network of ofand observers using limited limited shared decentralized plant distributed discrete LTI observers subject to scalability measurements the communication between the the works of studied, Park andwhile Martins (2017), the design of distributed discrete LTI subject toMorse aa scalability measurements and the communication shared between the In by a network observers using limited decentralized plant constraint was Wang and (2017) discrete LTI observers observers subject scalability observers. Toofand estimate target stateshared evolving from measurements the communication between theaa distributed constraint was was studied, while Wang Wang and to Morse (2017) observers. To estimate aa target target state evolving from distributed discrete LTI observers subject to a scalability constraint studied, while and Morse (2017) observers. To estimate a state evolving from a measurements and the communication shared between the investigated the same problem for the continuous counwhile Wang Morse (2017) observers. To estimate a target state evolving from by a constraint investigatedwas the studied, same problem problem for the theand continuous coundynamic process, process, distributed observers were studied studied by dynamic distributed observers were was studied, while Wang and Morse (2017) investigated the same for continuous counobservers. To asestimate a target state evolving from a constraint dynamic process, distributed observers were studied by terpart with a preassigned observer spectrum. In case of investigated the same problem for the continuous counauthors such Park and Martins (2017); Wang et al. dynamic process, distributed observers were studied by terpart with a preassigned observer spectrum. In case of authors such as Park Park and Martins (2017); et the same problem for the continuous counterpart with aa preassigned observer spectrum. In case of authors such as and Martins (2017); Wang et al. al. dynamic process, distributed observers wereWang studied by investigated any abrupt of the network over withchanges preassigned observertopology spectrum. In time case or of (2017). such authors as Park and Martins (2017); Wang et al. terpart any abrupt changes of the network topology over time or (2017). withchanges a preassigned observertopology spectrum. In case of any abrupt of over time or (2017). such as Park and Martins (2017); Wang et al. terpart authors packet dropouts, aa hybrid observer comprising of a any abrupt changes of the the network network topology over time or (2017). packet dropouts, hybrid observer comprising of aa local local packet dropouts, a hybrid observer comprising of local any abrupt changes of the network topology over time or In contrast to the decentralized estimation scheme as (2017). In contrast contrast to to the the decentralized decentralized estimation estimation scheme scheme as as packet a hybrid observer comprising of a local observer and local parameter estimator was by observerdropouts, and aaa local local parameter estimator was designed designed by In a hybrid observer comprising of a local observer and parameter estimator was by noted in the thetoworks works of Su Su and and Huang Huang (2012),scheme where as at packet In contrast the decentralized estimation noted in of (2012), where at observer and a local parameter estimator was designed designed by Wang et etdropouts, al. (2017). (2017). noted in thetoworks of Su and Huang (2012), where at In contrast the decentralized estimation scheme as Wang al. and(2017). a local parameter estimator was designed by Wang et least one one observer independently estimates the entire entire plant noted in observer the works of Su and estimates Huang (2012), where at observer least independently the plant et al. al. (2017). least one independently the entire plant noted in observer the and works ofestimation Su and estimates Huang (2012), where at Wang In all the above mentioned papers, the inter-agent comWang et al. (2017). In all the above mentioned papers, papers, the the inter-agent inter-agent comcomstate vector the is then propagated to least one observer independently estimates the entire plant state one vector and independently the estimation estimationestimates is then then the propagated to In all the above mentioned state vector and the is propagated to least observer entire plant In all the above mentioned papers, the inter-agent communication or the leader-follower communication was imthe remaining observers through a consensus protocol, state vector and the estimation is then propagated to munication or the leader-follower communication was imthe remaining remaining observers through isaa then consensus protocol, all the above mentioned papers, the inter-agent communication or leader-follower communication was imthe observers through consensus protocol, state vector and the estimation to In plicitly assumed to be instantaneous. However, due to unmunication or the the leader-follower communication was imin the the current problem we consider that propagated each protocol, observer the remaining observers through a consensus plicitly assumed to be instantaneous. However, due to unin current problem we consider that each observer munication or the leader-follower communication was implicitly assumed to instantaneous. However, due to in the current problem we consider that each observer the remaining through a consensus avoidable time-delay the communication between assumed to be bein instantaneous. However, due agents to ununin the the current observers problem we consider that receives each protocol, observer the distributed observation framework receives only aa plicitly avoidable time-delay in the between agents in distributed observation framework only avoidable time-delay ininstantaneous. the communication communication between agents assumed to be However, due to unthe distributed observation framework receives only a plicitly in the current problem we consider that each observer in multi-agent system, the effects of the time-delay time-delay in the communication between agents in aaa multi-agent multi-agent system, the effects effects of of the the time-delay portion of the the plant output measurement needed for for the in the distributed observation framework receives onlythe a avoidable portion of output needed time-delay in the communication between agents in system, the time-delay portion of the plant plant output measurement measurement needed for in the distributed observation framework receives onlythea avoidable in a multi-agent system, the effects of the time-delay should be explicitly taken into consideration. The coninto consideration. The conestimation of the system states. The limited plant output portion of the plant output measurement needed for the should be explicitly taken estimation of the system states. The limited plant output in a multi-agent system, the effects of the time-delay should be explicitly taken into consideration. The conestimation of the system states. The limited plant output portion of the plant output measurement needed for the sensus problem for first-order agents with a time-varying should be explicitly taken into consideration. The coninformationofis isthe notsystem enoughstates. for independent independent reconstruction estimation The limitedreconstruction plant output should sensus problem problem for first-order agents with a time-varying information not enough for be explicitly taken into consideration. The confor agents with information not enough for independent reconstruction estimation ofis limited plantInstead, output sensus communication was studied by Zhenzua et al. (2016), sensus problem delay for first-order first-order agents with aa time-varying time-varying information isthe notsystem enough for reconstruction communication delay was studied by Zhenzua et of the the entire entire state vectorstates. by independent a The single observer. of state vector by a single observer. Instead, communication delay was studied by Zhenzua et al. al. (2016), (2016), sensus problem for first-order agents with a time-varying information is not enough for independent reconstruction of the entire state vector by a single observer. Instead, while for higher-order the consensus delay wassystems, studied by et al.problem (2016), while for for higher-order higher-order systems, theZhenzua consensus problem observers disseminate theirby local information overInstead, com- communication of the entire state vector a single observer. observers disseminate their local information over aaa comdelay was studied by Zhenzua et input al.problem (2016), while systems, the consensus observers disseminate their local information over com- communication of the entire state vector by a single observer. Instead, while for higher-order systems, the consensus problem was studied in Zhou and Lin (2013) for fixed and munication network, and collaborate to jointly synthesize observers disseminate their local information over a comwas studied in Zhou and Lin (2013) for fixed input and munication network, and collaborate to jointly synthesize while for higher-order systems, the consensus problem was studied in Zhou and Lin (2013) for fixed input and munication network, and collaborate to jointly synthesize observers disseminate their local information over a comcommunication delays. was studied in Zhou and Lin (2013) for fixed input and an estimation estimation of the the plant plant dynamics. to jointly synthesize was munication network, and collaborate communication delays. an of dynamics. studied in Zhou and Lin (2013) for fixed input and communication delays. an of dynamics. munication network, and collaborate delays. an estimation estimation of the the plant plant dynamics. to jointly synthesize communication In this work we study the distributed state estimation communication delays. In the framework of the distributed observation problem, an estimation of the plant dynamics. In this work we study the distributed distributed state state estimation estimation In the the framework framework of of the the distributed distributed observation this work we study the In observation problem, problem, In problem by a network of observers under distinct comIn this work we study the distributed state estimation each observer can of bethe viewed as aa follower follower agent,problem, and the the In In theobserver framework distributed observation problem by a network of observers under distinct comeach can be viewed as agent, and this work study of theobservers distributed state estimation problem by aa we network under distinct comeach can be viewed as aa follower agent, and the In theobserver framework of the distributed observation problem, munication and measurement delays. We construct a disproblem by network of observers under distinct comeach observer can be viewed as follower agent, and the munication and measurement delays. We construct a displant to be observed as the leader agent. The design of plant to be observed as the leader agent. The design of problem by a network of observers under distinct communication and measurement delays. We construct a displant observer to be observed as the as leader agent. agent, The design of munication each can be can viewed aregarded follower andcase the tributed solution following the low-gain approach and and measurement delays. We construct a distributed solution following the low-gain approach and distributed observers then be as a special plant to be observed as the leader agent. The design of distributed observers can then be regarded regarded asThe special case and measurement delays. We construct a and distributed solution following the low-gain approach distributed can be as aa special case plant to beobservers observed as then the problem, leader agent. design of munication Lyapunov equation. Sufficient stability of following theconditions low-gain for approach of aa leader-follower leader-follower consensus where each follower distributed observers can thenproblem, be regarded as aeach special case tributed Lyapunovsolution equation. Sufficient conditions for stabilityand of of consensus where follower solution following theconditions low-gain for approach and Lyapunov equation. Sufficient stability of of a leader-follower consensus where follower distributed observers can thenproblem, be regarded as aeach special case tributed Lyapunov equation. Sufficient conditions for stability of of a leader-follower consensus problem, where each follower Lyapunov equation. Sufficient conditions for stability of of a leader-follower consensus problem, wherebyeach follower 2405-8963 Copyright © 2019. The Authors. Published Elsevier Ltd. All rights reserved.
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the observation error dynamics, including an upper bound for the low-gain parameter, are found for such arbitrary large delays. Compared to the parametric Riccati equation based solution in Liu et al. (2018), the results in our current work offers a more general Hurwitz condition for the convergence of the state estimation. Such generalization is achieved by offering an alternative formulation and proof of the main results.
Let the connections between the plant (1b) and the N distributed observers be described by the graph G = (V, E), V = {0, 1, 2, · · · , N }. Under the multi-agent system representation, the leader agent (1b) is the zeroth node of V, while the follower agents are the remaining N nodes. The distributed observer equation in the presence of known inter-agent communication delays τ1 , and delayed measurement yi in (1a), are given as aij eτ1 S (ηj (t − τ1 ) − ηi (t − τ1 )) η˙ i = Sηi + µ j∈Ni (2) τ2 S + µai0 e Gi Ci (w(t−τ2 )−ηi (t−τ2 )) ,
The remainder of the paper is organized in the following way. The problem formulation and algebraic graph theoretic properties are briefly revisited in Section 2. Next, the stability conditions for the distributed observer dynamics coupled with communication and measurement delays are derived in Section 3. An illustrative example to verify the effectiveness of the proposed approach is presented in Section 4. The paper ends with a conclusion in Section 5.
where ηi is the state estimation by the ith observer, i = 1, 2, · · · , N , aij is the weighting factor for the communication between follower agents i and j in the communication digraph G, Gi and µ are respectively the observer gain and the low-gain scalar to be determined later.
Notations. We now briefly introduce some notations and symbols to be used throughout the text. The Kronecker product of matrices is denoted by ⊗. A vector 1N is a column vector in RN of all ones. Z+ is the set of all positive integers. Iq and 0q respectively denote the identity matrix and zero matrix of dimension q × q. Unless mentioned otherwise, for matrices Ai , i = 1, 2, · · · , N , T T T A¯ = col(A1 , A2 , · · · , AN ) = AT and A = 1 , A2 , · · · , AN blk diag(A1 , A2 , · · · , AN ) represents a block diagonal matrix with the ith block being Ai . For a non-zero vector x, and matrix X, x , and X respectively stand for the L2 norm for vectors and the spectral norm for matrices. For a square matrix X , let the minimum, maximum and sum of eigenvalues be respectively denoted by λmin (X ), λmax (X ) and Tr(X ). For the positive scalar τ , let C([−τ, 0], Rm ) denote the Banach space of all continuous functions mapping the interval [−τ, 0] into Rm endowed with the supremum norm.
In case when the time-delay is present only in the measurements, the distributed observer dynamics in (2) can be reformulated in a way such that the stability result becomes independent of the delay and thus the problem does not offer a significant challenge. However, due to the presence of the communication delay in (2), the distributed observers cannot be designed without explicitly considering the effects of the delays into account. For τ = mini=1,2 τi , τ¯ = maxi=1,2 τi and θ ∈ [−¯ τ , 0], let the ¯initial conditions w(θ) = w(θ) and ηi (θ) = η i (θ), where w, η i ∈ C([−¯ τ , 0], Rq ). Denote the estimation error ˜ i (θ), with η ˜i ∈ η˜i = ηi − w with η˜i (θ) = ηi (θ) − w(θ) = η C([−¯ τ , 0], Rq ). The estimation error dynamics obtained from (1b) and (2) is given as η˜˙ i = S η˜i + µ aij eτ1 S (˜ ηj (t − τ1 ) − η˜i (t − τ1 )) j∈Ni (3) τ2 S − µai0 e Gi Ci η˜i (t − τ2 ).
2. PROBLEM FORMULATION Consider a network of N observer agents under a communication network, with each agent i sensing a timedelayed measurement yi of an autonomous system with states w ∈ Rq , yi (t) = Ci w(t − τ2 ), i = 1, 2, · · · , N, (1a) w(t) ˙ = Sw(t), (1b) for known measurement delay τ2 ≥ 0. The objective of the group of N distributed observers is to provide an asymptotic estimation of the plant state vector w(t). When the pair (S, Ci ) is detectable for some agent i, the distributed estimation problem becomes a well studied decentralized state observation problem under measurement delay. On the other hand, when the previous detectability condition is not satisfied, a cooperative effort is required by the observers over the communication network to jointly provide an estimation for w(t). The estimation of the states w(t) is more challenging when we include the latency in the communication between the distributed observers to the formulation of the cooperative observation problem.
We will now introduce the following assumptions to guarantee the solvability of the distributed estimation problem. Assumption 1. S has all eigenvalues on the imaginary axis. Remark 1. As noted in Basu and Yoon (2018), Assumption 1 is common in the literature of cooperative output regulation problem (CORP), and the leader-follower synchronization problem to be later considered in this work. Stable eigenvalues of S are ignored without loss of generality, as their corresponding modes will asymptotically approach zero and the solution to the distributed estimation problem becomes trivial. On the other hand, estimating exponentially unstable state trajectories is also rare in practical applications, especially when these states serve as leader tracking trajectories in the leader-follower synchronization problem. ¯ is detectable, where C¯ = Assumption 2. The pair (S, C) col (C1 , C2 , · · · , CN ). Remark 2. The condition in Assumption 2 redefines the classical detectability property to the multi-agent framework, and it is referred to as the “combined detectability” property in Basu and Yoon (2018). An equivalent observability condition appears in Park and Martins (2017); Wang and Morse (2017); Wang et al. (2017); Liu et al. (2018).
2.1 Problem Statement The observers along with the plant model in (1b) can be viewed as a multi-agent system with the plant being the leader agent and the observers being the followers. 14
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Assumption 3. All N follower agents form a strongly connected partition of the digraph G. Remark 3. The strongly connected partition of G in Assumption 3 can be seen as a way for an observer agent i to collect information on modes that may not be detectable by the pair (S, Ci ). In such case, Assumption 2 guarantees that information on such mode is indeed collected by some observer agents in the system, and Assumption 3 provides a path for the information to travel to the i agent.
considered to be strictly positive definite. Here we extend the result to semi-definite matrices. For a real symmetric matrix M0 , it can be orthogonally decomposed as M0 = JDM0 J−1 where JT = J−1 , and DM0 is a real diagonal matrix with all diagonal elements being the eigenvalues of M0 . Then the integral on the right hand side of (9) yields the following form γ2 ω T (β)M0 ω(β)dβ (γ 2 − γ 1 ) γ1 γ2 = (γ 2 − γ 1 ) ω T (β)JDM0 JT ω(β)dβ, γ1 γ2 = (γ 2 − γ 1 ) ω T (β)DM0 ω(β)dβ, (10)
Now we are ready to define the problem statement as follows. Definition 1. Distributed state estimation problem: Design observer gains Gi , i = 1, 2, · · · , N , and feedback gain µ such that the estimation error dynamics (3) is ˜ i (θ) ∈ exponentially stable, i.e., for given τ1 , τ2 > 0 and η C([−¯ τ , 0], Rq ), limt→∞ η˜i (t) = 0 for i = 1, 2, · · · , N .
γ1
where ω(β) = JT ω(β), DM0 = blk diag(λ1 , λ2 , · · · , λn ). Suppose M0 has r non-zero eigenvalues, i.e., with no loss of generality let λi = 0, i = r + 1, r + 2, · · · , n and ω(β) = col(ω 1 (β), ω 2 (β), · · · , ω n (β)). Then by virtue of the result in Gu (2000), the integral expression in (10) reduces to the form γ2 ω T (β)M0 ω(β)dβ (γ 2 − γ 1 )
Before we present the main contributions of this work, we first establish the following results, which will be used in the next section. Lemma 1. Given the plant dynamics (1b) with the state matrix S satisfying Assumption 1, it holds that
γ1
∗
T
e−S t e−St ≥ e−ωγ t Iq , (4) ∗ for a positive scalar γ = max(∈, γ min ), γ min = min{γ ≥ 0 : Q = S T + S + γI > 0}, an arbitrary small positive scalar ∈> 0, and ω = q − 1.
= (γ 2 − γ 1 ) =
Proof. If S T + S is negative (semi)-definite or indefinite, the matrix Q can be made positive definite by selecting any γ ≥ γ min . Then by using Cholesky decomposition Q = W W T we obtain S T + S − W W T = −γ min Iq , (5) and thus Lemma 1 of Zhou et al. (2012) yields that ∗
T
eS t eSt ≤ eωγ min t Iq = eωγ t Iq , ω = q − 1. ∗
≥ =
e−St e−S
T
t
∗
≥ e−ωγ t Iq .
(6) = (7) =
On the other hand, if S T + S is positive definite, then γ min = 0. Furthermore, for an arbitrarily small positive scalar ∈, S T + S+ ∈ I > 0 and γ ∗ =∈. Therefore in a similar manner as above, we obtain from Lemma 1 in Zhou et al. (2012) e−St e−S
T
t
∗
≥ e−ω∈t Iq = e−ωγ t Iq .
=
2
i=1 r i=1 n
γ2
γ1
r i=1
λi (γ 2 − γ 1 ) λi λi γ2
T
γ2 γ1 γ2 γ1
λi ω T i (β)ω i (β) dβ, γ2
γ1
ωT i (β)dβ ωT i (β)dβ
ω (β)dβ DM0 γ1
γ2
T
ωT i (β)ω i (β)dβ
ω (β)dβ JDM0 J
γ1 γ2 γ1
ω T (β)dβ M0
γ2
,
ω i (β)dβ , γ1 γ2
ω i (β)dβ , γ1
γ2
ω(β)dβ , γ1
T
γ2
ω(β)dβ , γ1
γ2
ω(β)dβ . γ1
(11)
This concludes the proof for this lemma.
(8)
3. MAIN RESULT
By multiplying the left hand side of (7) by eSt and its right hand side by e−St we obtain (4). This concludes the proof. Lemma 2. For any positive semi-definite matrix M0 ≥ 0, two scalars γ 1 and γ 2 with γ 2 ≥ γ 1 , and a vector valued function ω : [γ 1 , γ 2 ] → Rn , the inequality γ2 γ2 T ω (β) dβ M0 ω(β) dβ γ1 γ1 (9) γ ≤ (γ 2 − γ 1 )
r
i=1
T
Since eωγ t Iq − eS t eSt ≥ 0, then
15
In this section, we will present the stability results for the estimated error dynamics in (3), which will eventually lead to the design of observer gains Gi and low-gain µ. The current work also provides an upper bound for µ to ensure stability of the error dynamics. Since Ci = 0 when ai0 = 0, the composite estimation error dynamics of ˜ = col(˜ ˜ 2, · · · , η ˜ N ), and η˜ = col(˜ η1 , η˜2 , · · · , η˜N ) with η η1 , η η˜i = col(˜ ηi1 , η˜i2 , · · · , η˜iq ) is then obtained as η − µ e(IN ⊗τ2 S) GC η˜(t − τ2 ) η˜˙ = (IN ⊗ S)˜ (12) − µ L ⊗ eτ1 S η˜(t − τ1 ), t > 0,
ω T (β) M0 ω(β) dβ
γ1
holds if the integrals are well defined.
˜ (θ), η ˜ ∈ C([−¯ η˜(θ) = η τ , 0], RN q ), ∀θ ∈ [−¯ τ , 0], N ×N where L ∈ R is the Laplacian matrix corresponding to the strongly connected partition of the network of
Proof. A version of this lemma appears in Yoon and Lin (2013); Zhou et al. (2010, 2012); Gu (2000), where M0 was 15
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N observer agents, G = blk diag(G1 , G2 , · · · , GN ), and C = blk diag(C1 , C2 , · · · , CN ). Let us now introduce some notations which will be used throughout the text. Denote σG = GC 2 , σL = L 2 , G = [G 1 G 2 · · · G N ] , G i = ζi Gi where ζi is the ith entry of the left eigenvector T ζ = [ζ1 , ζ2 , · · · , ζN ] of L corresponding to zero eigenvalue.
Next, by substituting (15) and (16) in (12), we obtain η˜˙ (t) = IN ⊗ eτ2 S M IN ⊗ e−τ2 S η˜(t) (17) − µ2 (π 1 + π 2 + π 3 + π 4 ) , ∀t ≥ τ¯, where M = (IN ⊗ S) − µ (L⊗Iq ) − µGC, π i = L ⊗ eτ1 S π ∗i , i = 1, 2, π j = IN ⊗ eτ2 S GCπ ∗j , j = 3, 4. Let us define the initial state of the redefined estimation error dynamics (17) as ˜ (¯ η τ + θ), ∀θ ∈ [−2¯ τ , −¯ τ] ¯ ˜ (θ) = η (18) η˜(¯ τ + θ), ∀θ ∈ [−¯ τ , 0],
First, we will check the boundedness of the estimation error dynamics (12), ∀t ∈ [0, τ¯] and then verify the asymptotic stability of (12) for t ≥ τ¯. For the first part, with no loss of generality, we assume that τ¯ = mτ + ∈ ¯ will where m ∈ Z+ , ∈< τ . In a step-by-step manner we ¯ evaluate the bounds of η˜(t) across each such sub-intervals.
where η˜(t), ∀t ∈ [0, τ¯] is shown to be bounded. As noted in Zhou et al. (2010), the solution to the estimation error ˜ (θ), θ ∈ dynamics (12), for t ≥ 0 with initial condition η [−¯ τ , 0] coincides with the solution to (17) with initial ¯˜ (θ). conditions η
To do this, from (12) we obtain t ˜ (0)−µ IN ⊗eS(t−s+τ2 ) GC η ˜ (s−τ2 )ds η˜(t) = IN ⊗eSt η 0 t η (s−τ1 )ds, ∀t < τ − µ IN ⊗eS(t−s+τ1 ) (L⊗Iq )˜ ¯ 0 Sθ S(τ2 +θ) √ ˜ η ≤ max e + µτ e σG ¯ θ∈[0,τ ] ¯ √ η C , + eS(τ1 +θ) σL ˜ (13)
To analyze the stability of (17), we consider a Lyapunov function of the form T η, V(˜ η ) = η˜T (Σ ⊗ e−τ2 S P e−τ2 S )˜ where P is positive definite matrix Σ = diag(ζ) is a entry being ζi . Since diagonal matrix with the ith diagonal T the matrix Σ ⊗e−τ2 S P e−τ2 S is symmetric and have all positive eigenvalues, V(˜ η ) > 0, ∀˜ η = 0. By differentiating V(˜ η (t)) along the trajectories of (17), we obtain ˙ η ) = η˜T M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d V(˜
where ˜ η C = maxθ∈[−¯τ ,0] ˜ η (θ) . It follows that η˜(t), ∀t ∈ ˜ [0, τ ] is bounded as η ∈ C([−τ, 0], RN q ). ¯ For the subsequent intervals with t > τ , we obtain from ¯ (12), t η˜(t) = IN ⊗eS¯τ η˜(t − τ ) − µ (π1 (s)+π2 (s))ds, (14) ¯ t−τ
d
T
−τ2 S T P e−τ2 S )˜ η − 2µ2 (π T 1 + π 2 )(Σ ⊗ e T −τ2 S 2 T T −τ2 S Pe )˜ η, − 2µ π 3 + π 4 (Σ ⊗ e
¯
where π1 = IN ⊗ eS(t−s+τ2 ) GC η˜(s − τ2 )ds, π2 = IN ⊗ η (s − τ1 )ds. Since η˜(t), t ∈ [0, τ ] and eS(t−s+τ1 ) (L ⊗ Iq )˜ ¯ that ˜ (θ), ∀θ ∈ [−τ, 0] are bounded, it follows from (14) η ¯ η˜(t), ∀t ∈ [τ, 2τ ] is bounded. ¯ ¯ In a similar manner, we can evaluate ˜ η (t) , ∀t ∈ [j τ, (j + ¯ can 1)τ ], j ≥ 2 and by using the method of induction we ¯ show η˜(t) is bounded for t ∈ [0, mτ ]. Again for t ≥ mτ , ¯ ¯ from (12) it yields t η˜(t) = IN ⊗eSm¯τ η˜(t−mτ )−µ (π1 (s) + π2 (s))ds, ¯ t−µτ
˙ 0 + 4µ2 V + µ2 ≤V + µ2
i=1
¯
where π ∗1 = π ∗2 = π ∗3
=
π ∗4 =
η˜(t − τ2 ) = e
t
−(IN ⊗Sτ2 )
e
t−τ1 t
+
π4∗ ),
(15)
t−τ2 t t−τ2
i=3
T LT ΣL ⊗ e(τ1 −τ2 )S P e(τ1 −τ2 )S π ∗i , (19)
t→∞
(16)
¯ with if the low-gain parameter µ satisfies µ < µ, α ¯= , µ λmax (P )ζmax (4εd +c1 c2 )
e(IN ⊗S)(t−s) (L ⊗ Iq )˜ η (s − τ1 ) ds,
t−τ1 t (IN ⊗S)(t−s)
η˜(t) +
µ(π3∗
π ∗i
T
T
π ∗i C T GT (Σ ⊗ P ) GCπ ∗i
t where V 0 = 0 η˜dT (s) M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d (s) ds, η˜d (t) = IN ⊗ e−τ2 S η˜(t). Next, we consider the following Theorem to evaluate (19). Theorem 3. Consider the distributed observers (2) satisfying Assumptions 1-3, and gains Gi selected such that the matrix R = S − µGC¯ is a Hurwitz matrix for any µ ∈ (0, 1). Let then P ∈ Rq×q > 0 be a solution to the inequality RT P + P R < 0, (20) T such that the matrix P (S − µGi Ci ) + (S − µGi Ci ) P has non-positive eigenvalues for all follower agents i. Then the observer states ηi (t) converges to w(t) asymptotically, i.e., lim (ηi (t) − w(t)) = 0, i = 1, 2, · · · , N,
which also implies that η˜(t), ∀t ∈ [0, τ¯] is bounded as ˜ (θ), ∀θ ∈ [−¯ η τ , 0] and η˜(t), ∀t ∈ [0, mτ ] are bounded. ¯ Since we obtained that η˜(t) is bounded for t ∈ [0, τ¯], we now need to show the stability of the closed-loop system (12) for t ≥ τ¯. In this regard, from (12) we evaluate η˜(t−τ1 ) and η˜(t − τ2 ) as η˜(t − τ1 ) = e−(IN ⊗Sτ1 ) η˜(t) + µ(π ∗1 + π ∗2 ),
2
4
(21)
∗
with M T (Σ⊗P )+(Σ⊗P )M < −αI, εd = eωγ τ2 e−τ2 S 2 , ∗ ∗ ∗ c1 = σG eωγ τ2 + σL eωγ τ1 , and c2 = τ12 σL eωγ τ1 + ∗ τ22 σG eωγ τ2 .
IN ⊗e(τ2 −τ1 )S GC η˜(s−τ2 ) ds,
e(IN ⊗S)(t−s) (L ⊗ e(τ1 −τ2 )S )˜ η (s − τ1 ) ds,
Proof. Let G satisfy Assumption 3 with at least one agent being the child node of the leader. Now, we evaluate the first term on the right-hand side of the inequality in (19). In this regard, we have
e(IN ⊗S)(t−s) GC η˜(s − τ2 ) ds. 16
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17
˙ 0 = η˜T Σ ⊗ (S T P + P S) − µC T GT (Σ ⊗ P ) V d −µ(Σ ⊗ P )GC − µ(Lˆ ⊗ Iq ) η˜d (22) ≤ η˜dT Σ ⊗ (S T P + P S) − µC T GT (Σ ⊗ P ) −µ(Σ ⊗ P )GC] η˜d ≤ 0,
since the matrix (Lˆ ⊗ Iq ) is positive semi-definite and by assumption P (S − µGi Ci ) + (S − µGi Ci )T P is negative semi-definite. We will now investigate the largest invariant ˙ 0 = 0. We note here that the term V set of η˜d on which T ˆ −µ˜ ηd L ⊗ Iq η˜d in (22) becomes zero non-trivially only ˙ 0 in (22) thus when η˜d = 1N ⊗ η˜f , for any η˜f ∈ Rq , and V reduces to ˙ 0 = η˜f RT P + P R η˜f < 0, (23) V
Fig. 1. Directed communication network G with strongly connected partititon for agents 1, 2, 3 ¯ with µ ¯ in (21). Therefore from Lyapunovfor µ < µ Krasovskii stability theorem, limt→∞ η˜i (t) = 0, i = 1, 2, · · · , N . This concludes the proof. Remark 4. R can be rewritten as ¯ R = S − µGC. (29)
˙ by assumption. Therefore, for any non-zero η˜d , V0 < 0, ˙0 = 0 i.e. η˜dT M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d < 0, and V only when η˜d = 0. Furthermore, η˜dT M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d is continuous and finite for all µ ∈ (0, 1). Itthen implies that the term η˜dT M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d is bounded for all values of µ ∈ (0, 1) and thus there exists a positive scalar α > 0 such that ηd 2 η˜dT M T (Σ ⊗ P ) + (Σ ⊗ P )M η˜d < −α˜ (24) ∗ < −αe−ωγ τ2 ˜ η 2 ,
By Assumption 2, there exists a matrix G so that R in (29) is Hurwitz. While Lemma 3 allows for any G such that R is Hurwitz, the equivalent observer gain in Liu et al. (2018) is tied to the solution of a parametric Riccati equation. It is also true that any solution to the parametric Riccati equation in Liu et al. (2018) and the corresponding observer gain G results in R being Hurwitz, but the opposite may not necessarily hold. ∗
In case when τ1 = τ2 = τ , then c1 = eωγ τ (σG + σL ), ¯ reduces to c2 = τ 2 c1 , and µ α . µ ¯= 2 λmax (P ) ζmax 4εd + τ 2 (σL + σG ) e2ωγ ∗ τ
by virtue of Lemma 1.
Now we evaluate the rest of the terms on the right-hand side of the inequality (19) as follows 2 T T π ∗i LT ΣL ⊗ e(τ1 −τ2 )S P e(τ1 −τ2 )S π ∗i µ2
On the other hand, if S has all semi-simple eigenvalues, then it is unitarily diagonalizable. As a result, the bound on the matrix exponentials in (6) and (7) becomes T T eSt eS t = Iq = e−St e−S t which further yields α . µ ¯= λmax (P )ζmax (4 + (σG + σL )(τ12 σL + τ22 σG )) As τ1 and τ2 become larger, from (21), we observe that the low-gain bound µ ¯ decreases accordingly. Furthermore, when the leader dynamics is controlled by an input u to follow a certain trajectory, i.e. w˙ = Sw(t) + Q0 u(t), the results of the distributed estimation problem will remain unaffected by the selected observation protocol (2).
i=1
2 ωγ ∗ (2∗τ1 −τ2 ) ˙1 , ≤ λmax (P )µ2 τ1 ζmax σL e τ1 ˜ η 2−V ∗ ˙2 , +σL σG eωγ τ1 τ1 ˜ η 2−V (25)
µ2
4 i=3
T
π ∗i C T GT (Σ ⊗ P ) GCπ ∗i
∗ ˙3 ≤ λmax (P )µ2 τ2 ζmax σG σL eωγ τ1 τ2 ˜ η 2 − V 2 ωγ ∗ τ2 ˙4 , + σG e η 2 − V τ2 ˜ (26) τi +τ1 t where V1 = τi η˜T (σ)˜ η (σ)dσ ds, i = 1, 2, Vj = t−s τ2 +τj−2 t T η˜ (σ)˜ η (σ)dσ ds, j = 3, 4. We define a new τj−2 t−s Lyapunov function Vφ as 2 ωγ ∗ (2τ1 −τ2 ) e V1 Vφ = V + λmax (P )µ2 ζmax τ1 σL ∗ 2 ωγ ∗ τ2 + σL σG eωγ τ1 (τ1 V2 + τ2 V3 ) + τ2 σG e V4 , (27)
4. ILLUSTRATIVE EXAMPLE In this section we consider an illustrative example to evaluate the effectiveness of our proposed algorithm. Let T the plant (1b) have state vector w = [w1 w2 w3 w4 ] and state matrix 0 −1 0 −2 , S2 = . (30) S = blk diag(S1 , S2 ), S1 = 1 0 2 0 The measured output matrices Ci , i = 1, 2, 3, are given as follows: C1 = [I2 02 ] , C2 = [02 I2 ] , C3 = 02×4 , from which we verify that none of the pairs (S, Ci ), i = 1, 2, 3, are detectable. Instead, the combined detectability property in Assumption 2 is satisfied for the given Ci .
η 2 . Since V > 0, where V ≤ λmax (P ) ζmax e−τ2 S 2 ˜ the Lyapunov function Vφ is also positive. Then by substituting the results from (25), (26) into (19), we obtain ˙ φ ≤ −αe−ωγ ∗ τ2 + 4λmax (P )µ2 ζmax e−τ2 S 2 V 2 2ωγ ∗ (2τ1 −τ2 ) 2 ωγ ∗ τ2 + λmax (P )µ2 ζmax τ12 σL e + τ22 σG e ∗ η 2 < 0, +(τ12 + τ22 )σL σG eωγ τ1 ˜ (28)
The communication network between the observers is represented by the digraph G satisfying Assumption 3, as 17
2019 IFAC NecSys 18 Chicago, IL, USA, September 16-17, 2019 Himadri Basu et al. / IFAC PapersOnLine 52-20 (2019) 13–18
By the combined detectability condition and a “strongly connected” network property of the observer agents, we guarantee that the local state estimation of all agents converges to the states of the observed plant, including states which are not detectable from local measurements. To guarantee the convergence of estimation errors to zero in the presence of time-delays, we designed a low-gain parameter and observers based on generalized sufficient conditions which are obtained without requiring to solve extensive matrix inequalities. For any time-delay in the network, a corresponding low-gain bound can always be obtained to ensure the consensus on the estimated states. An extension of this presented result for time-varying measurement and communication latency is our future research endeavor. REFERENCES
1.4
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1000
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Fig. 2. Observation error of three observers for τ1 = 0.8 and τ2 = 1.2 illustrated in Figure 1,with the Laplacian matrix L given as follows 1 −1 0 L = 0 1 −1 . −1 0 1 The strongly connected partition of G allows the exchange of estimated states between agents, and complements for the incomplete measurements that each agent receives from the plant. Let τ1 = 0.8 and τ2 = 1.2. From the Laplacian matrix L of digraph G, we find ζmin = ζmax = 1/3. Next, we select the observer gains as T
T
G1 = [I2 02 ] , G2 = [02 I2 ] , G3 = 04×2 (31) such that the matrix −µ/3 −1 −µ/3 −2 R = blk diag , 1 −µ/3 2 −µ/3
in (29) is Hurwitz with the eigenvalues located at −µ/3 ± j, −µ/3 ± 2j. Then, from the Lyapunov equation RT P + P R = −µI, we find the solution P as ∞ ∞ T T eR s eRs ds = eR +Rs ds = 1.5I. P = 0
0
Next, from the inequality (28) we obtain the upper bound ¯ as µ 0.1 ¯= µ = 0.029. (32) 1 + 1.5τ12 + τ22 ¯ The simulated response of the We select µ = 0.022 < µ. distributed observes are presented in Figure 2 through the 4 sum of the observation errors δi (t) = q=1 ˜ ηiq (t), where T
ηi1 η˜i2 η˜i3 η˜i4 ] and i = 1, 2, 3. η˜i = [˜
The simulated response in Fig. 2 shows that δi (t) and η˜i converges asymptotically to zero, and thus the objectives of the distributed observation problem are achieved. 5. CONCLUSION In this work we studied the distributed state estimation problem for a marginally stable autonomous dynamic system under communication and measurement delays. 18