A performance guaranteed sampled-data event-triggered consensus approach for linear multi-agent systems

doi:10.1016/j.ins.2019.01.057

Information Sciences

Volume 484, May 2019, Pages 338-349

https://doi.org/10.1016/j.ins.2019.01.057 Get rights and content

Abstract

The paper proposes a novel performance guaranteed sampled-data event-triggered consensus (PSEC) algorithm for linear multi-agent systems configured as directed networks. To reduce information exchanges and preserve communication resources, a sampled-data event detector is incorporated at each agent. Communication between agents is based on the fulfillment of distributed state-dependent event-triggering conditions. PSEC ensures a guaranteed exponential convergence rate and is resilient to norm-bounded uncertainties in control gains resulting from implementation distortions. The Lyapunov–Krasovskii theorem is used to incorporate the performance objectives. The design parameters, namely, the heterogeneous control gains and a transmission threshold, are simultaneously computed using a constrained convex optimization framework with linear matrix inequalities. Numerical simulations based on an experimental spacecraft formation flying multi-agent system quantify the effectiveness of the proposed PSEC approach.

Introduction

Multi-agent systems (MASs) are usually employed to fulfill cooperative tasks by allowing a group of agents to operate in collaboration with each other. These collaborative tasks include formation control, containment control, rendezvous, and consensus. Among such cooperative behaviours, consensus has attracted overwhelming attention due to its applications in many areas such as average consensus [31], Unmanned Aerial Vehicles (UAV) [27], Unmanned Surface Vehicles (USV) [24], and Autonomous Underwater Vehicles (AUV) [10] to name a few. Early studies in this field address the unrestricted consensus problem in which agents are allowed to constantly transmit their information to their neighbours. There has been a recent surge of interest in event-triggered schemes due to their ability in reducing the number of transmissions and cope with bandwidth constraints [13]. Motivated by early results in event-triggered control schemes, Reference [11] proposes a method to investigate the event-triggered consensus (ETC) in first-order integrator agents. More recently, the event-triggered strategy has been extended to a wide variety of consensus problems and its potential has been investigated from a different aspects [8], [16], [18], [19], [20], [29], [34], [38]. Despite introducing several advantageous properties, event-triggered strategies face at least one of the following three limitations.

Strong Assumptions: Dealing with event-triggered strategies imposes analytical difficulties and strong assumptions are often considered to simplify the ETC problem. For instance, many researchers limit their study to MASs with first-order [19] or second-order dynamics [20]. From a network point of view, bidirectional communication between the neighbouring agents is another requirement in some implementations [8]. As a result, these schemes are not generalizable to directed network topologies. Moreover, it is a common assumption in the context of the ETC problem that the control gains are precisely implementable with their computed nominal values. It is widely known that the control parameters are often subject to uncertainties due to physical restrictions from software andor hardware aspects. The uncertainties occurring in the realization of the control gains can cause considerable performance deterioration, especially in event-triggered mechanisms where a limited number of transmissions is used. In this regard, resilient (non-fragile) control design techniques, which take into account the uncertainties of the controller realization, are of great interest [4]. To the best of our knowledge, no studies on non-fragility of control parameters in event-based consensus for general linear MASs have been completed. Another strong assumption often considered in the literature is that the event-triggering condition is constantly monitored during the consensus process [16]. Along with the implementation difficulties of such approaches, constant measurement and monitoring of event-triggering conditions waste valuable energy resources allocated to agents. To relax this limitation, a sampled-data approach allows the event-triggering condition to be monitored only at periodic samples of the systems [9], [14], [17], [21], [33], [34]. Most existing sampled-data event-triggered approaches are limited to first-order [34] or second-order MASs [14] with undirected topologies [17].

Convergence Rate: An important criteria in distributed networked control systems is the convergence rate of the proposed implementation [12]. In this regard, a satisfactory fast or a finite-time convergence scheme is of paramount importance. Ensuring only asymptotic rate of convergence may result in conservative performance in practical scenarios, i.e., the system may not converge to the desired objective in the required amount of time. Considering constant-time transmission among the agents, reference [15] proposes a finite-time algorithm to reach consensus. To maintain a satisfactory rate of convergence in an event-triggered fashion, an exponentially fast approach is developed in [39] which requires continuous-time measurement and detection of the event-triggering conditions. An event-triggered method is also proposed in [37] that guarantees a finite-time consensus within first-order agents. In more practical scenarios where agents may be of higher order dynamics and only sampled measurements are available, maintaining a minimum rate of consensus convergence in an event-triggered fashion is a challenging and surprisingly a disregarded task.

Unified Design: The implementation of an ETC scheme often involves designing a control gain (used by the distributed control law) and transmission thresholds (used by the distributed event detectors). A third drawback observed in existing strategies lies in independent design of control gain and event-triggering parameters. More precisely, control gains are either assumed as a priori information [20] or designed as the first step based on different interpretations of the Hurwitz stability of the closed-loop system. Then, as the second step, sufficient conditions are derived to compute transmission thresholds [38]. In such approaches, the design of transmission thresholds is treated as a secondary stage which totally depends on the first design step (control gain design). Hence, the upper bounds derived for the thresholds are conditioned on the previously designed control gains and may not be the best choices. In an impartial approach, however, control parameters and event-triggering thresholds should be co-designed. A method is suggested in [18] to iteratively compute and adjust the control and transmission parameters based on a set of initial guesses. To the best of our knowledge, there is no efficient unified framework to simultaneously design these parameters through a non-iterative single optimization step.

Motivated by the aforementioned discussion, the paper develops a Performance guaranteed Sampled-data Event-triggered Consensus algorithm (PSEC) for general linear multi-agent systems configured in directed networks. In summary, the distinguishing features of the PSEC algorithm are as follows:

•
Unlike previous implementations that guarantee asymptotic convergence for the sampled-data event-triggered consensus, the PSEC algorithm guarantees a bound for the exponential consensus convergence rate for general linear MASs configured as a directed network. Our results are more practical and flexible.
•
The computed consensus control gains within the PSEC framework are resilient to additive uncertainty and some level of variations from their nominal designed values.
•
As opposed to Guo et al. [18], all consensus parameters, i.e., the control gains and transmission threshold, are computed simultaneously from a non-iterative single optimization step. The design stage in the PSEC algorithm is less sensitive to the choice of initialized values used in the optimization.

As an efficient design method, linear matrix inequality (LMI) optimization guarantees system stability for desired design objectives through semidefinite programming (SDP). With the emergence of the interior-point methods that can solve the convex optimization problems in an efficient fashion, the LMI-based algorithms offer the state-of-the-art solution in many control applications [5], [30], [32]. The proposed PSEC algorithm is formulated using existing SDP techniques to benefit from the advantages of convex optimization. Developing a convex framework to simultaneously compute the consensus parameters with the guaranteed convergence rate and control gain resilience is unique and non-trivial.

The rest of the paper is organized as follows: In Section 2, we introduce necessary notation and assumptions. Section 3 formulates the sampled-data event-triggered consensus problem and derives a theorem to compute consensus parameters, namely, the control gains and a transmission threshold that collectively satisfy the desired objectives. To validate our theoretical results, simulation examples, including application to spacecraft formation flying control, are included in Section 4. Finally, Section 5 summarizes the paper.

Section snippets

Notation and problem statement

Throughout the paper, we use bold alphabets to denote matrices or vectors, and normal letters to specify scalars. Let $A_{m \times n} = {a_{i, j}}$ denote a (m × n) dimensional real-valued matrix. ‖A‖ denotes the Frobenius norm; A^† represents the pseudo inverse of A; Notation A > 0 implies that A is symmetric positive definite; $A^{T}$ stands for the matrix transpose. Row vector a_(i,•) is the i-th row of matrix A, i.e., $a_{(i, •)} = [a_{i, 1}, \dots, a_{i, n}]$ . Matrices I and 0, respectively, stand for the identity matrix and zero

Problem formulation

Let $e_{i} (t) = {\hat{x}}_{i} (t) - x_{i} (t),$ $x (t) = {[x_{1}^{T} (t), \dots, x_{N}^{T} (t)]}^{T},$ $\hat{x} (t) = {[{\hat{x}}_{1}^{T} (t), \dots, {\hat{x}}_{N}^{T} (t)]}^{T},$ and $e (t) = {[e_{1}^{T} (t), \dots, e_{N}^{T} (t)]}^{T} .$ Then, it holds that $e (t) = \hat{x} (t) - x (t)$ . Incorporating (2) in (1) leads to the following closed-loop system $\begin{matrix} \dot{x} (t) = A_{[N]} x (t) + B (K + Δ_{K}) L_{[n]} (x (n T) + e (n T)), \end{matrix}$ where $K = diag (K_{1}, \dots, K_{N}),$ $B = diag (B_{1}, \dots, B_{N}),$ $Δ_{K} = diag ({Δ_{K}}_{1}, \dots, {Δ_{K}}_{N}),$ $L_{[n]} = L \otimes I_{n},$ and $A_{[N]} = I_{N} \otimes A$ . From Assumption 1, we obtain ||Δ_K||  ≤  δ, where $δ ≜ \sqrt{δ_{1}^{2} + \dots + δ_{N}^{2}}$ . Expression (3) is a differential equation with a discontinuous term in the right hand side. It is

Simulations

In this section, we report results from numerical examples taken from the spacecraft formation flying. To solve required convex optimizations, we use the YALMIP parser [25] and MOSEK solver [3] with an optimization accuracy³ of $10^{- 6}$ . Numerical results are computed on a system with Intel(R) Core(TM) i7-4790 CPU @3.6 GHz

Conclusion

Using the theory of linear matrix inequalities (LMI), the paper proposes the PSEC algorithm as new approach for sampled-data event-triggered distributed consensus in linear multi-agent systems. We transform the consensus problem to an equivalent stability problem and introduce objectives such as an exponential convergence rate and degree of resilience in our design procedure. Within the LMI optimization framework, the stability condition is incorporated with performance constraints. The design

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A performance guaranteed sampled-data event-triggered consensus approach for linear multi-agent systems

Abstract

Introduction

Section snippets

Notation and problem statement

Problem formulation

Simulations

Conclusion

Mech. Syst. Sig. Process.

Syst. Control Lett.

Inf. Sci.

Automatica

Automatica

Inf. Sci.

Inf. Sci.

Chaos Soliton. Fract.

Inf. Sci.

Inf. Sci.

Linear Algebra Appl.

Neurocomputing

Inf. Sci.

Inf. Sci.

Automatica

Syst. Control Lett.

Inf. Sci.

Appl. Math. Lett.

Interior point methods in semidefinite programming with applications to combinatorial optimization

SIAM J. Optim.