Robustness Analysis of Asynchronous Sampled-Data Multi-Agent Networks With Time-Varying Delays††thanks: This work was supported by the National Natural Science Foundation of China under Grant 61422302, the Program for New Century Excellent Talents in University under Grant NCET-13-0178, and the 111 Project under Grant B17048.
In this paper, we study the simultaneous stability problem of a finite number of locally inter-connected linear subsystems under practical constraints, including asynchronous and aperiodic sampling, time-varying delays, and measurement errors. We establish a new Lyapunov-based stability result for such a decentralized system. This system has a particular simple structure of inter-connections, but it captures some key characteristics of a large class of intermediate models derived from the consensus analysis of multi-agent systems. The stability result is applicable to the estimation of the maximum allowable inter-sampling periods and time delays based on individual dynamics and coupling structures in the scenarios of consensus control via asynchronous sampling of relative states and asynchronous broadcasting of self-sampled states respectively. The asynchrony of aperiodic sampling and the existence of measurement errors allow the utilization of some kinds of quantizing devices, such as Logarithmic quantizers, in the process of data sampling, and allow the introduction of a period of dwell time after each update of state measurement to eliminate the Zeno behavior of events in event-based control. The extension in the case with input saturations and input delays is also discussed.
Asynchronous multi-agent systems, simultaneous stability, consensus, aperiodic sampling, time-varying delays, event-triggered control.
Due to the attractive advantages in signal processing and transmission, digital devices have found their wide applications in modern control systems. An analog-to-digital converter is responsible for converting a continuous-time signal into a digital signal by sampling and quantization. A digital controller gathers all input signals and compute an output to achieve the desired control purpose. Sampled-data control deals with such hybrid continuous-time and discrete-time systems and usually assumes that these digital devices share the same clocks and process their data periodically and synchronously. This ideal assumption can formulate the considered digital systems as standard LTI discrete-time systems by plant discretization and servers as the basis of many fundamental results, for example, in stability and optimal control[3, 4]. When these digital components, such as A/D samplers and D/A zero-order holds, work at different frequencies or they are spatially scattered in a large area without a central clock, the sampled-data systems become inherently asynchronous. Although synchronous sampled-data models may be used to approximate the asynchronous ones in some of such cases, this treatment could sacrifice some dynamic details in plants.
Literature Review: Asynchronous sampling has been brought to the attention of researchers in the control community for more than three decades. It has obtained enormous results, and is now developing rapidly with the growth of multi-agent theory as one of the main research areas. Generally, asynchronous sampling can be found in the following three types of systems:
(1) Multi-rate systems. Multi-rate sampling is one of the earliest motivating examples for studying asynchronous systems, where sampling and hold elements work periodically at different rates with irrational ratios[6, 5, 9, 8, 10]. Examples include dual-rate linear systems with a single sampler and a single hold [5, 8] and multi-rate linear systems with multiple asynchronous samplers and zero-order holds[6, 9, 10]. The corresponding control problems include the optimal LQG control[5, 8] and stability problems[6, 9, 10].
(2) Networked control systems (NCSs). Theoretically, some NCSs can be modeled as multi-rate systems. Due to the long-distance transmission of information, communication networks in NCSs may suffer from packet loss and time delays[11, 14, 15, 13, 12] and often have multiple independent information channels[14, 16, 13, 12]. To ensure effective information transmission with reduced costs, asynchronous event-triggered sampling also has been considered in NCSs[14, 15]
(3) Multi-agent systems. The asynchronous property in signal processing becomes more prominent in multi-agent networks, in which, finite numbers of subsystems, equipped with independent signal-sensing and data-processing devices, are inter-connected to perform some cooperative tasks[17, 18]. For example, in the formation control of multi-robot systems, each robot should detect the positions and velocities of other adjacent robots for route planning. However, the installed sensors, such as ultrasonic sensors or laser sensors, usually cannot monitor all objects in -degree coverage at a time. Furthermore, they could also be affected by environmental interference. So in such cases, it is practically preferable that each robot collects the data of its neighbors in some order (not necessarily periodically) according to its own schedule and within its sensing/processing capacities. These kinds of realistic scenarios raise the problem of asynchronous and aperiodic sampling. The sensor scheduling can be time-driven[19, 20, 21], event-driven, or mixture (of time and event)-driven[23, 24].
With asynchronous sampling, even if the involved digital devices operate periodically, sampled-data systems become non-periodic and time-varying . Asynchronous sampling can destroy the stability of systems which are stable in synchronous environments , and induce additional time-varying delays in system analysis. So the analysis of asynchronous systems is more difficult and challenging than their synchronous counterparts, and the asynchronous mechanism also sets a strict requirement on the robustness of designed controllers or algorithms with respect to aperiodic sensor scheduling, sensing errors and processing/communcation delays. In asynchronous multi-agent systems, most results have been developed based on the simple individual models of single-integrators and double-integrators[19, 28, 27], but relatively few studies have been done on the general linear models, with the exception of the event-triggered control[31, 29, 30].
Contributions: This paper will establish a Lyapunov-based stability result for an asynchronous multi-agent system with the consideration of the above-mentioned practical issues and then apply it to solving several representative asynchronous consensus problems in one framework. The contributions are summarized as follows:
First, we set up a basic stability model for asynchronous coordination of multi-agent systems and solve the problem of how to estimate the maximum length of sampling intervals by subsystem matrices and inter-connection structures. In the model, all subsystems, represented by the general linear state-space models, share the same system matrix and feedback matrix. They communicate with each other via discrete-time signals produced by samplers and zero-order holds. Examples of such a model without sampling can be found in the coordination analysis of a large range of multi-agent systems[32, 33]. We impose relaxed assumptions on the A/D and D/A devices and information channels, including asynchronous and aperiodic sampling, measurement errors, and time-varying delays. The proposed result well describes the robustness of the asynchronous sampled-data system based on the stability of the continuous-time system without sampling. In , by a Lyapunov-Krasovskii method and LMIs, the authors studied the protocol design problem of a similar model. But it is different from ours in the following two aspects: (1) in our model, given any inter-connection structure, the subsystems are inherently coupled by discrete-time signals; in , the subsystems are coupled by continuous-time signals and the feedback matrices and scheduling protocols for discrete-time signals need to be designed accordingly; (2) In our model, the information is aperiodically sampled with measurement errors and transmitted with time delays; in , the information is sampled and transmitted according to the proposed Round-Robin (RR) or Try-Once-Discard (TOD) protocol. Note that the sampling with time-varying periods is also referred to as asynchronous sampling in [34, 35, 36] but it is a different definition from the “asynchronous sampling” in this paper.
Second, we solve the sampled-data consensus problems of the following asynchronous multi-agent systems in one framework: (1) networks of general linear agents with asynchronous sampling of relative states; (2) networks of single-integrators and marginally stable systems with asynchronous broadcasting of self-sampled states. Feedbacks with relative states and broadcast communication have been widely used in the consensus and formation control of multi-agent systems; particularly, broadcasting is an important way of keeping the state average unchanged; see [37, 41, 42, 38, 39, 44, 43, 28, 45] and references therein. Our work on asynchronous broadcasting is partly inspired by this observation and the protocol design method presented for the event-based control of single-integrator and double-integrator agents in . Due to the challenging difficulties of asynchronous consensus analysis, the existing results on sampled-data consensus mainly focus on synchronous sampling; that is, all data should be sampled at the same time with constant or variant frequencies. The involved individual models are usually single-integrators, double-integrators[48, 47, 49], or second-order oscillators. There are relatively rare reports on asynchronous sampled-data consensus of general linear agents and they mostly deal with event-triggered consensus [31, 29, 30], which is intrinsically different from the time-driven style of asynchrony in this paper. Our work is also different from the previous results based on dynamic outputs of controllers [19, 27].
Finally, taking advantages of asynchronous sampling and allowable measurement errors, we can extend the obtained results further to deal with some of the general cases with quantization in discretizing continuous-time signals, and we rigorously prove the effectiveness of the method for eliminating the Zeno behavior by introducing a dwell time after each measurement in event-based control. To the best of our knowledge, the related problems on general linear multi-agent systems have not been studied in the existing literature. This extension is illustrated by the implementation of Logarithmic quantizers  and event-triggering conditions based on state errors [39, 23]. We also show the application of the results in the case with input saturations and input delays.
This paper is organized as follows. In Section II, we set up the asynchronous sampled-data model and give sufficient conditions for stability by a Lyapunov-based approach. In Section III, the stability result is applied to solving some asynchronous consensus problems. In Section IV, further extensions of the stability result are discussed. The paper is concluded in Section V.
Notations: denotes the identity matrix in ; denotes the Kronecker product; .
Ii Lyapunov-based asynchronous stability
In this paper, we are interested in the protocol design and robustness/effectiveness analysis in coordinating agents’ states in a general scenario of aperiodic asynchronous sampling and time-varying transmission delays.
Assume that there are subsystems in a multi-agent system with zero-order holds and their states are represented by , , respectively, in a common state space . The dynamics of subsystem is given as follows:
where is the common system matrix of subsystems, represents the coupling weight, can be viewed as the common state feedback gain, and is the sampled state of subsystem .
Let , , be the sampling instants of , , , be the corresponding time delays in sampling, transmission, or computation, and , , be the measurement errors. The sampled state , incorporating time delays, is given by
which is a piece-wise constant function of time . Here, the sequences of , , and 111Notations , , and stand for the sequences , , , , , , and , , , respectively. are completely independent with respect to different , and satisfy the following assumption:
where is the maximum sampling period, is the maximum time delay, and .
The simultaneous stability model, with each individual represented by system (1), originates from the consensus coordination of multi-agent systems. Several examples will be given in Section III. Furthermore, this model is also of interest in the networked situation where some individual systems cannot stabilize themselves by their sampled self-states especially when they cannot access their own states due to device failure. Particularly, when , system (1) can be viewed a networked control system with aperiodic discrete-time signals and time-varying delays, which is also of its own significance .
Assumption 1 (4) says that the measurement error is of multiplicative type and proportional to the measurement. The additive errors can be modeled as uncertainties, which will be considered in Section III.
It can be shown that Assumption 1 (4) can be ensured by
Note that the sequences of , , and are only indexed by the th subsystem. So it can be understood that the information is transmitted via broadcasting. Another interpretation will be given in Section III.
Denote and . The overall system is given by
Let denote the largest eigenvalue of and let , , and denote the largest singular values of , , and , respectively. We have the following result:
In (3), if , the stability analysis becomes trivial. So we only consider the case with positive .
The left side of (4) is a continuous function with respect to variables and . If , then the left side converges to as parameters and approach to . Therefore, when , from (4), it can be seen that we can always find maximum allowable sampling period and the corresponding maximum time delay to ensure the stability of the system. However, if , then the decrease of cannot be decided by (3) and thus the stability of system (2) cannot be decided either.
Iii Asynchronous consensus
Consider a group of linear autonomous agents interacting with each other through local information flow. Label these agents with to and suppose that the -th agent takes the following dynamics:
where is the state, is an input signal, called protocol in multi-agent coordination and designed based on the information received from locally linked agents, and system matrix and input matrix are with compatible dimensions.
Iii-a Asynchronous sampling of relative states
Assume that the interaction topology is modeled by an undirected simple graph with vertices , which represent the agents respectively. Assume that there are total edges, labeled with , , , . The existence of an edge represents that agents and have the capacity of knowing the relative state at the same time, and in such a case, suppose that agents and sample the relative state at discrete times , and they receive the data with measurement error at time , . Here, , and . For different pair of adjacent agents, the initial sampling of relative states may start at different times, and their subsequent sampling instants are also independent. This is referred to as asynchrony, which is an intrinsic characteristic in distributed networks without any global clocks to synchronize agents’ actions. Our assumption about time delay is also quite general. Time delay can be changing with respect to and independent of those on other channels. Let denote the set of all s, such that exists, and let , , , . In what follows, if is indexed by (the -th edge), , , and are also used instead of , , and .
With the utilization of zero-order holds, the protocol takes the following form:
We will design the feedback matrix and give sufficient conditions in terms of sampling period and time delays to ensure that , which implies that the system solves the average consensus problem.
In the protocol, we require that , and , which ensure the symmetry of information sharing and are the least requirement for distributed average consensus. To get this symmetric information, each pair of adjacent agents should communicate beforehand to synchronize each mutual sampling and ensure the sampled data, like , used in both controllers, is updated at the same time. The synchronization of data sampling and controller update for any pair of adjacent agents is technically possible because it is only based on local communication. In many engineering applications, such as formation control and attitude alignment, relative states (position, attitude, etc), as a whole, are easily obtainable, which is one of the reasons why we assume the same time delays on both parts of each relative state.
Iii-A1 Edge dynamics
Assign each edge an arbitrary direction in interaction topology and define the incidence matrix by (see )
Denote and , which is the vector obtained by stacking the relative states corresponding to edges , , , in sequence. Let with , . For the th edge , if it is oriented from to in the definition of , denote , , and , . Denote . The system (5) with protocol (6) has the following equivalent representation:
Matrix is called the graph Laplacian of , which is independent of the selection of . Let the eigenvalues of be , , , in the increasing order. Then . is called the algebraic connectivity of , which is positive when is connected . It can be observed that matrices and share the same non-zero eigenvalues with the same algebraic multiplicities. So also serves the purpose of algebraic characterization of and is called edge Laplacian .
Iii-A2 Average consensus
The feedback matrix is designed with the requirement that there exist a positive definite matrix and a positive number satisfying the following Lyapunov inequalities:
Let denote the largest eigenvalue of , denote the largest singular value of matrix , denote the largest singular value of matrix , and
In the proof of Theorem 2, we will see that if there exists such that and , then the derivative of the employed Lyapunov candidate is always negative (in the case with nonzero ; see (13)), and thus the system solves the average consensus problem for any sampling periods, which is usually not possible for the systems with zero-order holds. So Theorem 2 only considers the case that .
Since is stabilizable, we can always find proper matrices , , and positive number such that inequalities (8) holds. One feasible solution can be obtained by solving the following Riccati equation in the LQR problem with :
Note that the above Riccati equation has been used in the design of consensus protocols in the literature [33, 29]. Furthermore, can be estimated only by the number of agents without the knowledge of the algebraic connectivity of .
Proof of Theorem 2: Consider the following Lyapunov candidate:
Then by (7),
Let , where and is the diagonal matrix with eigenvalues of in the diagonal positions. In equation (11), we have that
Thus, for any ,
Iii-B Asynchronous broadcasting of self-sampled data
Consider the multi-agent system (5) with an undirected interaction topology . Different from the sampling of relative states in the previous subsection, we assume that each agent samples its own state with measurement errors and then broadcasts them to its neighbors with time delays. When all neighbors get the information, the agent and the neighbors all update their controllers. Denote the sampling instants of agent , the associated measurement errors and time delays by , , and , , respectively. Denote for , . Under Assumption 1 (1-3), we give the following protocol:
The matrix will be designed later.
Let be defined by (14) with , and let . For , let and , , . Denote and . Then we have
By the definition of incidence matrix , if the system solves a consensus problem, then the final consensus trajectory should be . However, it does not satisfy the differential equation (16) with asynchronous sampling and unstable system matrix ; that is, in most cases. Thus in such cases, the asynchronous consensus cannot be achieved.
In the case of multiplicative type of measurement errors, even if the consensus is reached, the final consensus state is usually not zero and thus may never converge to zero as . Therefore, the multiplicative type of measurement errors can destroy the consensus of the considered system (16).
Iii-B1 Asynchronous consensus of single-integrators
In the case of multiple single-integrators with , , and , is time-invariant and so is . Therefore, choose and by (17),
Consider the following Lyapunov candidate
With the same arguments as in proving (13), we have
where is the largest singular value of matrix , and with .
By Theorem 1, we have the following result:
In system (5), suppose that each agent takes the single-integrator dynamics without measurement errors, and suppose that the sampling instants and time delays satisfy Assumption 1 (1-3), . If the interaction topology is connected and
then protocol (15) with solves the average consensus problem.
In , it has been shown that the necessary and sufficient condition for the synchronous sampled-data consensus of single-integrators with sampling period is . Compared with this condition, the condition (19) may be a bit conservative, which is understandable since this condition is valid in the general asynchronous sampling case and obtained based on the general model of linear subsystems.
Iii-B2 Asynchronous consensus of marginally stable systems
Suppose that matrix is marginally stable. Then is bounded and by Lemma 1, we have222In case of a -valued serving as a denominator, the value of the expression is set to the limit when .
Denote the right side of the above equation by . Clearly, .
For any , denote and , , . Denote and . Then and
If , then
Choose the feedback matrix with a positive definite matrix and a positive number by (10) and consider the following Lyapunov function candidate
Then we have
where is the largest eigenvalue of , is the largest singular value of , is the largest singular value of , and and are any positive numbers with .
It can be seen that equations (17) and (21) do not exactly match the model (2), (3). However, when the largest sampling period and measurement errors are small, the system represented by (20) and (21) can be seen as the model (2), (3) with uncertainties. Let denote the largest singular value of ,
where denotes the maximum row sum matrix norm. By the same arguments as in the proof of Theorem 1, we can obtain the following result:
In system (5), assume that is stabilizable, is marginally stable, Assumption 1 (1-3) holds, , is lower bounded by a positive number independent of , , and the interaction topology is connected. Under the proposed protocol (15) with given by (10), if , then for any time and any , there exists some , , such that
Proof The proof is omitted here. ∎
Iv-a Quantized sampling
This subsection gives a simple example to show the effectiveness of Theorem 1 in the scenario of quantization. Further discussions will continue in the next subsection.