# Design of Adaptive Kalman Consensus Filters (a-KCF)
## 1 Introduction  
The problem of distributed estimation for linear dynamic systems via the cooperation and coordination of multiple fixed/mobile agents has been extensively explored by numerous researchers [1; 2; 3; 4; 5]. Distributed Kalman filtering has been broadly utilized as a powerful and efficient tool to solve the aforementioned scenario [1; 6; 7; 8; 9; 10]. An essential problem involving cooperation of multi-agent systems that has attracted much attention is the _consensus problem_. In order to reconstruct the states of the dynamic system, a network of sensing agents is adopted; these may form either a homogeneous or a heterogeneous network. Each agent transmits state estimates to its immediate neighbors based on the communication topology. Through the consensus strategy, all agents in use eventually agree with each other on a common estimation value regarding the states of the dynamic system. The early formal study of consensus problems [11] formed the basis of distributed computing [12], which has found wide and popular utilization in sensor network applications [13; 14]. The dynamic consensus problem appears frequently in the cooperation and coordination of multi-agent systems, including scenarios such as formation control [5; 9; 15], self-alignment, flocking [4; 16], and distributed filtering [1; 17]. The typical consensus protocol and its performance analysis were first introduced by [PERSON] and [PERSON] in the continuous-time model (see [3; 7]). In [7], the authors considered a cluster of first-order integrators working cooperatively under the average consensus control algorithm, in which each agent finally agrees on a common value that is the average of initial states of all agents as the individual ultimate state. In [1], a distributed Kalman filtering (DKF) algorithm was proposed in which data fusion was achieved through dynamic consensus protocols [18]. Later, in [17], the same author extended the results of [1] to use two identical consensus filters for sensor fusion with different observation matrices, then presented [1] an alternative distributed Kalman filtering algorithm which applies consensus to the state estimates. This idea forms the foundation of the present paper, in which we propose an adaptive Kalman consensus filtering algorithm. In [9], the authors presented a differentview towards designing consensus protocols based on the Kalman filter structure. By adjusting the time-varying consensus gains, [9] proved that consensus can be achieved asymptotically under a no-noise condition. In addition, graph theory [7, 19, 20] has been adopted to construct the communication topology among distributed agents. In this paper, we assume a fixed topology; however, it is not necessarily relaxed to the point of all-to-all connection. This means that each agent is not constrained in communication with others, a consideration that would be more practical in real-world applications. Recently, many extensions of consensus protocols have been explored to improve the convergence rate of dynamic systems among cooperative agents. This includes the study of communication topology design [21, 22], optimal consensus-based estimation algorithms [2, 23, 24], and adaptive consensus algorithms [25, 26, 27] in both continuous-time and discrete-time scenarios. Other extensions for system control purposes have been studied to realize finite time consensus among agents, methods of which involves event-triggered and sliding mode control [28, 29]. These research fruits have been considered and embedded in Kalman filtering algorithms [30, 31], although the complexity of the algorithms makes practical implementation challenging, particularly when compared to the adaptive weight parameter method proposed in this paper. The main contribution of this paper is to derive an adaptive Kalman consensus filtering (a-KCF) strategy in a continuous-time model and analyze its stability and convergence properties.