基于事件的未知输入和状态融合估计算法

Event-Based Unknown Input and State Fusion Estimation Algorithm

摘要: 针对具有未知输入的离散时变不确定系统，提出了一种基于动态事件触发的多传感器融合估计算法。为了降低数据传输过程中的能源消耗，采用一种动态事件触发协议来决定当前测量值是否传输到滤波器。在此基础上，给定相应约束条件以解耦未知输入，进而构造一个递归滤波器来同时估计局部状态和未知输入，然后在每个采样时刻获得局部状态和未知输入的误差协方差的上界。利用完全平方法和拉格朗日乘子法，获得能够最小化所得上界的滤波器增益。对于局部状态估计，利用协方差交叉(CI)融合估计方法得到新的状态融合估计，并给出基于CI的融合估计算法的一致性。最后，通过一个数值仿真，验证了所提融合估计算法的有效性。

Abstract: A multi-sensor fusion estimation algorithm based on dynamic event triggering is proposed for discrete-time time-varying uncertain systems with unknown inputs. In order to reduce the energy consumption during data transmission, a dynamic event-triggered protocol is used to determine whether the current measured value is transmitted to the filter. On this basis, given the corresponding constraints to decouple the unknown input, a recursive filter is constructed to estimate the local state and the unknown input at the same time, and then the upper bound of the error covariance of the local state and the unknown input at each sampling time is obtained. By using the complete square method and the Lagrangian multiplier method, the filter gains that can minimize the upper bound are obtained. For local state estimation, the covariance intersection (CI) fusion estimation method is used to obtain new state fusion estimation and the consistency of CI-based fusion estimation algorithm is given. Finally, a numerical simulation is carried out to verify the effectiveness of the proposed fusion estimation algorithm.

1. 引言

随着信息科学和高新技术的发展，为了满足系统状态更高精度的要求，多传感器信息融合技术在军事、交通运输、生物医学、环境监测和人工智能等众多领域的广泛应用引起了科研人员的研究兴趣 [1] [2] [3] [4] 。如果将每个传感器收集的信息单独处理，不仅会增加计算工作量，还会切断传感器之间的内在联系，导致系统性能降低。传统的融合滤波算法主要分为集中式和分布式。集中式融合算法可直接将测量信息送至融合中心，得到全局最优估计，这往往需要较高的计算能力 [5] ；分布式融合算法可以结合局部或加权局部状态估计器得到全局最优或局部次优状态估计器，因其更好的鲁棒性和灵活性，目前已有大量的研究 [6] [7] 。在实际应用中，通常不容易得到交叉方差，文献 [8] 提出了协方差交叉(covariance intersection, CI)融合算法，并通过优化非线性性能指数得到CI融合器。因此，充分利用多个传感器的信息互补性，采取合理的信息融合方法，是现在研究的目的之一。

状态和未知输入联合估计问题在实际工程中，发挥着越来越重要的作用 [9] [10] [11] [12] 。所谓未知输入，就是未知的干扰或者未建模的动力学。例如，在机床系统中，刀具的刀削力无法直接测量，但可以基于其他可以测量的信息来估计这个未知输入 [13] 。文献 [14] 在没有未知输入先验信息的假设下，研究了一种最优递归状态滤波算法。文献 [15] 在此基础上，讨论了其稳定性和收敛条件。分析发现，未知输入估计问题在不同的情况下的滤波设计算法存在明显差异，例如文献 [14] 系统中的未知输入没有直接输入到输出方程，文献 [16] 讨论了直接馈通系统的未知输入和状态估计问题。另外，真实的工业系统可能经常遭受参数的不确定性，系统参数的不确定性也会影响系统估计误差 [17] 。因此，针对系统参数的不确定性和未知的干扰，设计一个合理的状态和未知输入联合滤波器具有重要的工程意义。

以固定周期传输数据的时间触发方案已被广泛应用于控制工程中。然而，基于时间触发的采样数据必然携带波动极小的数据包，频繁地传输会导致通信资源的浪费和大量的计算成本。因此，事件触发方案引起了众多研究人员的兴趣，并应用于滤波器 [11] 、网络控制系统 [18] 等等。传统的事件触发机制是基于一个静态规则，只有满足某一预设阈值时，数据才会被传输。但是，在含有未知输入和参数不确定性的系统中，很难直接给定一个合理的固定阈值。文献 [19] 引进一个内部动态变量，由系统的事件误差动态地调整触发条件。在有限的通信资源中，既能解决冗余传输问题，又能保证系统的性能，这也是本文的研究动机之一。

根据上述分析，本文研究了一类基于动态事件触发的不确定系统的未知输入和状态融合滤波问题。各传感器子系统先独自处理目标测量值，利用动态事件触发机制进行未知输入和状态估计，利用多传感器CI融合算法对目标状态进行融合，得到系统状态的融合估计，并给出CI融合估计算法的一致性。最后，通过数值仿真验证所提算法的有效性。

2. 问题描述

考虑以下离散时变不确定性系统：

$\{\begin{array}{l}x\left(k+1\right)=\left(A\left(k\right)+\Delta A\left(k\right)\right)x\left(k\right)+B\left(k\right)d\left(k\right)+\omega \left(k\right)\\ y_i\left(k\right)=\left(C_i\left(k\right)+\Delta C_i\left(k\right)\right)x\left(k\right)+D_i\left(k\right)d\left(k\right)+\nu \left(k\right)\end{array}$ (1)

式中， $x\left(k\right)\in R^{n_x}$ 为系统状态， $d\left(k\right)\in R^{n_d}$ 为未知输入， $y_i\left(k\right)\in R^{n_y}\left(i=1,2,\cdots ,M\right)$ 为第i个传感器的测量输出， $\omega \left(k\right)\in R^{n_x}$ 和 $\nu \left(k\right)\in R^{n_y}$ 分别表示过程噪声和测量噪声， $A\left(k\right)$ 、 $B\left(k\right)$ 、 $C_i\left(k\right)$ 和 $D_i\left(k\right)$ 为适当维数的已知矩阵，秩 $D_i\left(k\right)=n_d$ ， $\Delta A\left(k\right)$ 和 $\Delta C_i\left(k\right)$ 分别表示状态和测量中的不确定矩阵。

假设1：对于所有的 $k\in \left[0,N\right]$ ，不确定参数矩阵满足

$\begin{array}{l}\Delta A\left(k\right)=H\left(k\right)F\left(k\right)E\left(k\right)\\ \Delta C_i\left(k\right)=U_i\left(k\right)N_i\left(k\right)M_i(\; k\; )\end{array}$

且有 $F\left(k\right)F^{\text{T}}\left(k\right)\le I$ 、 $N_i\left(k\right)N_i^{\text{T}}\left(k\right)\le I$ 。

假设2： $\omega \left(k\right)$ 和 $\nu \left(k\right)$ 为互不相干的零均值高斯白噪声，协方差矩阵分别为：

$\begin{array}{l}W\left(k\right)=\mathbb{E}\left\{\omega \left(k\right){\omega }^{\text{T}}\left(k\right)\right\}>0\\ V\left(k\right)=\mathbb{E}\left\{\nu \left(k\right){\nu }^{\text{T}}\left(k\right)\right\}>0.\end{array}$

假设3：初始值 $x\left(0\right)$ 是一个随机向量，有均值 $\bar{x}\left(0\right)$ 和协方差 $P\left(0|0\right)$ ，并且与 $\omega \left(k\right)$ ， $\nu \left(k\right)$ 是互不相关的。

作为一种有效的节能方法，引入了动态事件触发机制来调节从传感器到局部滤波器的信号传输，这意味着只有在满足规定的条件时，测量信号才被传输到滤波器。对于传感器节点i，用 $0={\tau }_{i,0}<{\tau }_{i,1}<\cdots <{\tau }_{i,s}<\cdots$ 来表示触发序列，其中 ${\tau }_{i,s}$ 由以下迭代关系确定：

${\tau }_{i,s+1}=\min \left\{k\in N|k>{\tau }_{i,l},\frac{1}{{\eta }_i}{\zeta }_i\left(k\right)+{\delta }_i\le \Vert {\phi }_i\left(k\right)\Vert \right\}$ (2)

式中， ${\delta }_i$ 和 ${\eta }_i$ 是正标量，事件误差 ${\phi }_i\left(k\right)$ 被定义为 ${\phi }_i\left(k\right)=y_i\left(k\right)-y_i\left({\tau }_{i,s}\right)$ ， $y_i\left({\tau }_{i,s}\right)$ 为最新传输的测量值， ${\zeta }_i\left(k\right)$ 为内部动态变量，且满足：

${\zeta }_i\left(k+1\right)={\rho }_i{\zeta }_i\left(k\right)+{\delta }_i-\Vert {\phi }_i\left(k\right)\Vert ,{\zeta }_i\left(0\right)={\zeta }_{i,0}$ (3)

其中， ${\zeta }_{i,0}\ge 0$ 是给定的初始值， ${\rho }_i$ 是正标量，假设参数 ${\eta }_i$ 和 ${\rho }_i$ 满足 ${\rho }_i{\eta }_i\ge 1$ ，变量 ${\zeta }_i\left(k\right)$ 对所有的 $k\in \left[0,N\right]$ 满足 ${\zeta }_i\left(k\right)\ge 0$ 。

基于以上动态事件触发机制，对于节点i构造以下局部滤波器：

$\{\begin{array}{l}{\hat{x}}_i\left(k+1|k\right)=A\left(k\right){\hat{x}}_i\left(k|k\right)+B\left(k\right){\hat{d}}_i\left(k\right)\\ {\hat{d}}_i\left(k+1\right)=L_i\left(k+1\right)\left(y_i\left({\tau }_{i,s}\right)-C_i\left(k+1\right){\hat{x}}_i\left(k+1|k\right)\right)\\ {\hat{x}}_i\left(k+1|k+1\right)={\hat{x}}_i\left(k+1|k\right)+K_i\left(k+1\right)\left(y_i\left({\tau }_{i,s}\right)-C_i\left(k+1\right){\hat{x}}_i\left(k+1|k\right)-D_i\left(k+1\right){\hat{d}}_i\left(k+1\right)\right)\end{array}$ (4)

其中， ${\hat{x}}_i\left(k+1|k\right)$ 是k时刻的一步预测， ${\hat{d}}_i\left(k+1\right)$ 是k + 1时刻的未知输入 $d_i\left(k+1\right)$ 的估计， ${\hat{x}}_i\left(k+1|k+1\right)$ 是k + 1时刻的状态 $x_i\left(k+1|k+1\right)$ 的估计， $L_i\left(k+1\right)$ 和 $K_i\left(k+1\right)$ 为待确定的估计器增益。这里 $k+1\in \left[{\tau }_{i,s},{\tau }_{i,s+1}\right)\left(s\ge 0\right)$ 。

随后，状态预测误差、未知输入估计误差和状态估计误差分别定义为：

${\tilde{x}}_i\left(k+1|k\right)\triangleq x\left(k+1\right)-{\hat{x}}_i\left(k+1|k\right)$ (5)

${\tilde{d}}_i\left(k+1\right)\triangleq d\left(k+1\right)-{\hat{d}}_i\left(k+1\right)$ (6)

${\tilde{x}}_i\left(k+1|k+1\right)\triangleq x\left(k+1\right)-{\hat{x}}_i\left(k+1|k+1\right).$ (7)

于是，我们得到误差系统：

${\tilde{x}}_i\left(k+1|k\right)=\left(A\left(k\right)+\Delta A\left(k\right)\right){\tilde{x}}_i\left(k|k\right)+\Delta A\left(k\right){\hat{x}}_i\left(k|k\right)+B\left(k\right){\tilde{d}}_i\left(k\right)+\omega \left(k\right)$ (8)

$\begin{array}{c}{\tilde{d}}_i\left(k+1\right)=\left(I-L_i\left(k+1\right)D_i\left(k+1\right)\right)d\left(k\right)-L_i\left(k+1\right)\left(C_i\left(k+1\right)+\Delta C_i\left(k+1\right)\right){\tilde{x}}_i\left(k+1|k\right)\\ -L_i\left(k+1\right)\Delta C_i\left(k+1\right){\hat{x}}_i\left(k+1|k\right)-L_i\left(k+1\right)\nu \left(k+1\right)+L_i\left(k+1\right){\phi }_i\left(k+1\right)\end{array}$ (9)

$\begin{array}{c}{\tilde{x}}_i\left(k+1|k+1\right)=\left(I-K_i\left(k+1\right)\left(C_i\left(k+1\right)+\Delta C_i\left(k+1\right)\right)\right){\tilde{x}}_i\left(k+1|k\right)\\ -K_i\left(k+1\right)\left(D_i\left(k+1\right){\tilde{d}}_i\left(k+1\right)+\nu \left(k+1\right)-{\phi }_i\left(k+1\right)+\Delta C_i\left(k+1\right){\hat{x}}_i\left(k+1|k\right)\right).\end{array}$ (10)

为了消除未知输入在估计中的影响，我们给出以下约束：

$L_i\left(k+1\right)D_i\left(k+1\right)=I.$ (11)

注1：这里限定矩阵 $D_i\left(k+1\right)$ 的秩为 $n_d$ ，目的是为了保证约束(11)存在解 $L_i\left(k+1\right)$ ，进而把没有先验信息的未知输入解耦出来，分别估计状态和未知输入。这种解耦的方法在众多文献中都得到了采用 [11] [12] 。

于是，未知输入的估计误差(9)可重写为：

$\begin{array}{c}{\tilde{d}}_i\left(k+1\right)=-L_i\left(k+1\right)\left(C_i\left(k+1\right)+\Delta C_i\left(k+1\right)\right){\tilde{x}}_i\left(k+1|k\right)-L_i\left(k+1\right)\Delta C_i\left(k+1\right){\hat{x}}_i\left(k+1|k\right)\\ -L_i\left(k+1\right)\nu \left(k+1\right)+L_i\left(k+1\right){\phi }_i\left(k+1\right).\end{array}$ (12)

我们假设未知输入和状态的滤波误差协方差分别为：

$P_i^d\left(k+1\right)=E\left\{{\tilde{d}}_i\left(k+1\right){\tilde{d}}_i^{\text{T}}\left(k+1\right)\right\},$ (13)

$P_i^x\left(k+1|k+1\right)=E\left\{{\tilde{x}}_i\left(k+1|k+1\right){\tilde{x}}_i^{\text{T}}\left(k+1|k+1\right)\right\}.$ (14)

在本文中，我们的目标是找到局部滤波器的未知输入和状态估计误差协方差的上界，并找到使滤波误差协方差上界最小的增益矩阵 $K_i\left(k+1\right)$ 和 $L_i\left(k+1\right)$ 。另外，我们的另一个目标是开发一个合适的融合估计算法，利用所有的局部滤波器来推导出状态融合估计。

3. 主要结果

本节中，我们首先推导出局部滤波器的未知输入和状态估计误差协方差的上界 ${\Xi }_i^d\left(k+1\right)$ 和 ${\Xi }_i^x\left(k+1|k+1\right)$ 。然后，找出合适的估计器增益 $L_i\left(k+1\right)$ 和 $K_i\left(k+1\right)$ 最小化这些上界。最后，根据导出的局部状态估计，提出一种基于CI融合方法的状态融合估计方案。

3.1. 局部估计器设计

为方便后续推导，我们给出以下引理。

引理1：设未知矩阵F满足 $F{F}^{\text{T}}\le I$ ，对于具有适当维数的矩阵A、H、F、E满足 ${\gamma }^{-1}I-EX{E}^{\text{T}}>0$ 的任意正常数 $\gamma >0$ ，有以下不等式：

$\left(A+HFE\right)X{\left(A+HFE\right)}^{\text{T}}\le A{\left(X^{-1}-\gamma E{E}^{\text{T}}\right)}^{-1}{A}^{\text{T}}+{\gamma }^{-1}H{H}^{\text{T}}$ (15)

成立 [20] 。

引理 2：给定正标量 ${\gamma }_{i,j}\left(j=1,2\right)$ ，如果存在一组具有初始值 ${\bar{\Xi }}_i^{\zeta }\left(0\right)={\zeta }_{i,0}^2$ 的递归矩阵满足

${\bar{\Xi }}_i^{\zeta }\left(k+1\right)=\left(\left(1+{\gamma }_{i,1}\right)\left(1+{\gamma }_{i,2}\right){\rho }_i^2+\left(1+{\eta }_i\right)\left(1+{\gamma }_{i,1}^{-1}\right){\eta }_i^{-2}\right){\bar{\Xi }}_i^{\zeta }\left(k\right)+\left(\left(1+{\gamma }_{i,1}\right)\left(1+{\gamma }_{i,2}^{-1}\right)+\left(1+{\gamma }_{i,1}^{-1}\right)\left(1+{\eta }_i^{-1}\right)\right){\delta }_i^2,$ (16)

那么， ${\bar{\Xi }}_i^{\zeta }\left(k\right)$ 为 ${\Xi }_i^{\zeta }\left(k\right)=E\left\{{\zeta }_i\left(k\right){\zeta }_i^{\text{T}}\left(k\right)\right\}$ 的上界 [21] 。

证明：由事件触发的定义和初等不等式 $X{Y}^{\text{T}}+Y{X}^{\text{T}}\le aX{X}^{\text{T}}+a^{-1}Y{Y}^{\text{T}},\forall a>0$ ，我们可以得到：

${\phi }_i^{\text{T}}\left(k+1\right){\phi }_i\left(k+1\right)\le {\left(\frac{1}{{\eta }_i}{\zeta }_i\left(k+1\right)+{\delta }_i\right)}^2\le \frac{1+{\eta }_i}{{\eta }_i}{\zeta }_i^2\left(k+1\right)+\left(1+{\eta }_i^{-1}\right){\delta }_i^2.$ (17)

结合式(17)，可进一步得到

$\begin{array}{c}{\Xi }_i^{\zeta }\left(k+1\right)=E\left\{{\zeta }_i^2\left(k+1\right)\right\}=E\left\{{\left({\rho }_i{\zeta }_i\left(k\right)+{\delta }_i-\Vert {\phi }_i\left(k\right)\Vert \right)}^2\right\}\\ \le E\left\{\left(1+{\gamma }_{i,1}\right){\left({\rho }_i{\zeta }_i\left(k\right)+{\delta }_i\right)}^2+\left(1+{\gamma }_{i,1}^{-1}\right){\phi }_i\left(k\right){\phi }_i^{\text{T}}\left(k\right)\right\}\\ \le \left(1+{\gamma }_{i,1}\right)\left(1+{\gamma }_{i,2}\right){\rho }_i^{2}E\left\{{\zeta }_i^2\left(k\right)\right\}+\left(1+{\gamma }_{i,1}\right)\left(1+{\gamma }_{i,2}^{-1}\right){\delta }_i^2+\left(1+{\gamma }_{i,1}^{-1}\right){\phi }_i\left(k\right){\phi }_i^{\text{T}}\left(k\right)\\ \le \left(\left(1+{\gamma }_{i,1}\right)\left(1+{\gamma }_{i,2}\right){\rho }_i^2+\left(1+{\eta }_i\right)\left(1+{\gamma }_{i,1}^{-1}\right){\eta }_i^{-2}\right){\bar{\Xi }}_i^{\zeta }\left(k\right)+\left(\left(1+{\gamma }_{i,1}\right)\left(1+{\gamma }_{i,2}^{-1}\right)+\left(1+{\gamma }_{i,1}^{-1}\right)\left(1+{\eta }_i^{-1}\right)\right){\delta }_i^2\\ ={\bar{\Xi }}_i^{\zeta }\left(k+1\right).\end{array}$ (18)

定理1：对于目标系统(1)，构造一个基于动态触发条件(2)~(3)的局部滤波器(4)。给定正标量 ${\alpha }_{i,j}\left(j=1,2\right)$ ， ${\epsilon }_{i,j}\left(j=1,2,3\right)$ ， ${\lambda }_{i,j}\left(j=1,2,3,4\right)$ 和 ${\mu }_{i,j}\left(j=1,2,\cdots ,8\right)$ ，存在两个矩阵序列 ${\Xi }_i^d\left(k+1\right)$ 和 ${\Xi }_i^x\left(k+1|k+1\right)$ 满足以下方程：

${\Xi }_i^d\left(0\right)=P_i^d\left(0\right),{\Xi }_i^x\left(0|0\right)=P_i^x\left(0|0\right),$

${\alpha }_{i,1}^{-1}I-E\left(k\right){\Xi }_i^x\left(k|k\right)E^{\text{T}}\left(k\right)>0,{\alpha }_{i,2}^{-1}I-N_i\left(k\right){\Xi }_i^x\left(k+1|k\right)N_i^{\text{T}}\left(k\right)>0,$

$\begin{array}{c}{\Xi }_i^d\left(k+1\right)={\kappa }_{i,4}{L}_i\left(k+1\right){\left(C_i\left(k+1\right){\left({\Xi }_i^x\left(k+1|k\right)\right)}^{-1}-{\alpha }_{i,2}{M}_i^{\text{T}}\left(k+1\right)M_i\left(k+1\right)\right)}^{-1}{C}_i^{\text{T}}\left(k+1\right)L_i^{\text{T}}\left(k+1\right)\\ +{\kappa }_{i,5}{L}_i\left(k+1\right)U_i\left(k+1\right)U_i^{\text{T}}\left(k+1\right)L_i^{\text{T}}\left(k+1\right)+\left(1+{\lambda }_{i,4}\right)L_i\left(k+1\right)V\left(k+1\right)L_i^{\text{T}}\left(k+1\right)\\ +{\kappa }_{i,6}{L}_i\left(k+1\right)\left(\frac{1+{\eta }_i}{{\eta }_i}{\bar{\Xi }}_i^{\zeta }\left(k+1\right)+\left(1+{\eta }_i^{-1}\right){\delta }_i^2\right)L_i^{\text{T}}\left(k+1\right)\end{array}$ (19)

$\begin{array}{c}{\Xi }_i^x\left(k+1|k+1\right)={\kappa }_{i,7}\left(I-K_i\left(k+1\right)C_i\left(k+1\right)\right){\Xi }_i^x\left(k+1|k\right){\left(I-K_i\left(k+1\right)C_i\left(k+1\right)\right)}^{\text{T}}\\ +{\kappa }_{i,8}{K}_i\left(k+1\right)U_i\left(k+1\right)U_i^{\text{T}}\left(k+1\right)K_i^{\text{T}}\left(k+1\right)\\ +{\kappa }_{i,9}{K}_i\left(k+1\right)D_i\left(k+1\right){\Xi }_i^d\left(k+1\right)D_i^{\text{T}}\left(k+1\right)K_i^{\text{T}}\left(k+1\right)\\ +{\kappa }_{i,10}{K}_i\left(k+1\right)V_i\left(k+1\right)K_i^{\text{T}}\left(k+1\right)\\ +{\kappa }_{i,11}{K}_i\left(k+1\right)\left(\frac{1+{\eta }_i}{{\eta }_i}{\bar{\Xi }}_i^{\zeta }\left(k+1\right)+\left(1+{\eta }_i^{-1}\right){\delta }_i^2\right)K_i^{\text{T}}\left(k+1\right)\end{array}$ (20)

其中，

${\Xi }_i^x\left(k+1|k\right)={\kappa }_{i,1}A\left(k\right){\left({\left({\Xi }_i^x\left(k|k\right)\right)}^{-1}-{\alpha }_{i,1}{E}^{\text{T}}\left(k\right)E\left(k\right)\right)}^{-1}{A}^{\text{T}}\left(k\right)+{\kappa }_{i,2}H\left(k\right)H^{\text{T}}\left(k\right)+{\kappa }_{i,3}B\left(k\right){\Xi }_i^d\left(k\right)B^{\text{T}}\left(k\right)+W\left(k\right),$

${\kappa }_{i,1}=1+{\epsilon }_{i,1}+{\epsilon }_{i,2},{\kappa }_{i,2}=\left(1+{\epsilon }_{i,1}+{\epsilon }_{i,2}\right){\alpha }_{i,1}^{-1}+\left(1+{\epsilon }_{i,1}^{-1}+{\epsilon }_{i,3}\right){\beta }_{i,1},{\kappa }_{i,3}=1+{\epsilon }_{i,2}^{-1}+{\epsilon }_{i,3}^{-1},{\kappa }_{i,4}=1+{\lambda }_{i,1}+{\lambda }_{i,2},$

${\kappa }_{i,5}=\left(1+{\lambda }_{i,1}+{\lambda }_{i,2}\right){\alpha }_{i,2}^{-1}+\left(1+{\lambda }_{i,1}^{-1}+{\lambda }_{i,3}\right){\beta }_{i,2},{\kappa }_{i,6}=1+{\lambda }_{i,2}^{-1}+{\lambda }_{i,3}^{-1}+{\lambda }_{i,4}^{-1},{\kappa }_{i,7}=\left(1+{\mu }_{i,1}+{\mu }_{i,2}+{\mu }_{i,3}\right)\left(1+{\mu }_{i,9}\right),$

${\kappa }_{i,8}=\left(1+{\mu }_{i,1}+{\mu }_{i,2}+{\mu }_{i,3}\right)\left(1+{\mu }_{i,9}^{-1}\right){\beta }_{i,3}+\left(1+{\mu }_{i,3}^{-1}+{\mu }_{i,6}^{-1}+{\mu }_{i,8}^{-1}\right){\beta }_{i,2},{\kappa }_{i,9}=1+{\mu }_{i,1}^{-1}+{\mu }_{i,4}+{\mu }_{i,5}+{\mu }_{i,6},$

${\kappa }_{i,10}=1+{\mu }_{i,4}^{-1}+{\mu }_{i,7},{\kappa }_{i,11}=1+{\mu }_{i,2}^{-1}+{\mu }_{i,5}^{-1}+{\mu }_{i,7}^{-1}+{\mu }_{i,8},{\beta }_{i,1}={\lambda }_{\max }\left(E\left(k\right){\hat{x}}_i\left(k|k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)E^{\text{T}}\left(k\right)\right),$

${\beta }_{i,2}={\lambda }_{\max }\left(M_i\left(k+1\right){\hat{x}}_i\left(k+1|k\right){\hat{x}}_i^{\text{T}}\left(k+1|k\right)M_i^{\text{T}}\left(k+1\right)\right),{\beta }_{i,3}={\lambda }_{\max }\left(M_i\left(k+1\right){\Xi }_i^x\left(k+1|k\right)M_i^{\text{T}}\left(k+1\right)\right).$

此外， ${\Xi }_i^d\left(k+1\right)$ 和 ${\Xi }_i^x\left(k+1|k+1\right)$ 分别为第i个节点的未知输入和状态的估计误差协方差的上界，即

$P_i^d\left(k+1\right)\le {\Xi }_i^d\left(k+1\right),P_i^x\left(k+1|k+1\right)\le {\Xi }_i^x\left(k+1|k+1\right).$ (21)

证明：采用数学归纳法进行证明。首先，由初始条件可得 $P_x\left(0|0\right)\le {\Xi }_x\left(0|0\right)$ ， $P_d\left(0\right)\le {\Xi }_d\left(0\right)$ 。然后，假设在k时刻有 $P_x\left(k|k\right)\le {\Xi }_x\left(k|k\right)$ 和 $P_d\left(k\right)\le {\Xi }_d\left(k\right)$ 。接着，我们只需证明k + 1时刻有 $P_x\left(k+1|k+1\right)\le {\Xi }_x\left(k+1|k+1\right)$ 和 $P_d\left(k+1\right)\le {\Xi }_d\left(k+1\right)$ 。

由式(8)可得状态预测误差协方差：

$\begin{array}{l}{P}_i{}^x\left(k+1|k\right)\\ =E\{\left(\left(A\left(k\right)+\Delta A\left(k\right)\right){\tilde{x}}_i\left(k|k\right)+\Delta A\left(k\right){\hat{x}}_i\left(k|k\right)+B\left(k\right){\tilde{d}}_i\left(k\right)+\omega \left(k\right)\right)(\left(A\left(k\right)+\Delta A\left(k\right)\right){\tilde{x}}_i\left(k|k\right)\\ +{\Delta A\left(k\right){\hat{x}}_i\left(k|k\right)+B\left(k\right){\tilde{d}}_i\left(k\right)+\omega \left(k\right))}^{\text{T}}\}\end{array}$

$\begin{array}{l}=\left(A\left(k\right)+\Delta A\left(k\right)\right)p_i^x\left(k|k\right){\left(A\left(k\right)+\Delta A\left(k\right)\right)}^{\text{T}}+\Delta A\left(k\right){\hat{x}}_i\left(k|k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)\Delta A^{\text{T}}\left(k\right)\\ +B\left(k\right)p_i^d\left(k\right)B^{\text{T}}\left(k\right)+W\left(k\right)+\left(A\left(k\right)+\Delta A\left(k\right)\right)E\left\{{\tilde{x}}_i\left(k|k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)\right\}\Delta A^{\text{T}}\left(k\right)\\ +\Delta A\left(k\right)E\left\{{\hat{x}}_i\left(k|k\right){\tilde{x}}_i^{\text{T}}\left(k|k\right)\right\}{\left(A\left(k\right)+\Delta A\left(k\right)\right)}^{\text{T}}+\left(A\left(k\right)+\Delta A\left(k\right)\right)E\left\{{\tilde{x}}_i\left(k|k\right){\tilde{d}}_i^{\text{T}}\left(k\right)\right\}{B}^{\text{T}}\left(k\right)\\ +B\left(k\right)E\left\{{\tilde{d}}_i\left(k\right){\tilde{x}}_i^{\text{T}}\left(k|k\right)\right\}{\left(A\left(k\right)+\Delta A\left(k\right)\right)}^{\text{T}}+\Delta A\left(k\right)E\left\{{\hat{x}}_i\left(k|k\right){\tilde{d}}_i^{\text{T}}\left(k\right)\right\}{B}^{\text{T}}\left(k\right)\\ +B\left(k\right)E\left\{{\tilde{d}}_i\left(k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)\right\}\Delta A^{\text{T}}\left(k\right).\end{array}$ (22)

利用引理1可得

$\begin{array}{l}\left(A\left(k\right)+\Delta A\left(k\right)\right){\Xi }_i^x\left(k|k\right){\left(A\left(k\right)+\Delta A\left(k\right)\right)}^{\text{T}}\\ \le A\left(k\right){\left({\left({\Xi }_i^x\left(k|k\right)\right)}^{-1}-{\alpha }_{i,1}{E}^{\text{T}}\left(k\right)E\left(k\right)\right)}^{-1}{A}^{\text{T}}\left(k\right)+{\alpha }_{i,1}^{-1}H\left(k\right)H^{\text{T}}\left(k\right)\end{array}$ (23)

其中， ${\alpha }_{i,1}^{-1}I-E\left(k\right){\Xi }_i^x\left(k|k\right)E^{\text{T}}\left(k\right)>0$ 。

另一方面，注意到

$\begin{array}{l}\Delta A\left(k\right){\hat{x}}_i\left(k|k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)\Delta A{\left(k\right)}^{\text{T}}\\ =H\left(k\right)F\left(k\right)E\left(k\right){\hat{x}}_i\left(k|k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)E^{\text{T}}\left(k\right)F^{\text{T}}\left(k\right)H^{\text{T}}\left(k\right)\\ \le {\lambda }_{\max }\left(E\left(k\right){\hat{x}}_i\left(k|k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)E^{\text{T}}\left(k\right)\right)H\left(k\right)H^{\text{T}}\left(k\right)\\ ={\beta }_{i,1}H\left(k\right)H^{\text{T}}\left(k\right).\end{array}$ (24)

再次使用初等不等式，可得

$\begin{array}{l}\left(A\left(k\right)+\Delta A\left(k\right)\right)E\left\{{\tilde{x}}_i\left(k|k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)\right\}\Delta A^{\text{T}}\left(k\right)+\Delta A\left(k\right)E\left\{{\hat{x}}_i\left(k|k\right){\tilde{x}}_i^{\text{T}}\left(k|k\right)\right\}{\left(A\left(k\right)+\Delta A\left(k\right)\right)}^{\text{T}}\\ \le {\epsilon }_{i,1}\left(A\left(k\right)+\Delta A\left(k\right)\right)p_i^x\left(k|k\right){\left(A\left(k\right)+\Delta A\left(k\right)\right)}^{\text{T}}+{\epsilon }_{i,1}^{-1}\Delta A\left(k\right){\hat{x}}_i\left(k|k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)\Delta A^{\text{T}}\left(k\right)\end{array}$ (25)

$\begin{array}{l}\left(A\left(k\right)+\Delta A\left(k\right)\right)E\left\{{\tilde{x}}_i\left(k|k\right){\tilde{d}}_i^{\text{T}}\left(k\right)\right\}{B}^{\text{T}}\left(k\right)+B\left(k\right)E\left\{{\tilde{d}}_i\left(k\right){\tilde{x}}_i^{\text{T}}\left(k|k\right)\right\}{\left(A\left(k\right)+\Delta A\left(k\right)\right)}^{\text{T}}\\ \le {\epsilon }_{i,2}\left(A\left(k\right)+\Delta A\left(k\right)\right)p_i^x\left(k|k\right){\left(A\left(k\right)+\Delta A\left(k\right)\right)}^{\text{T}}+{\epsilon }_{i,2}^{-1}B\left(k\right)p_i^d\left(k\right)B^{\text{T}}\left(k\right)\end{array}$ (26)

$\begin{array}{l}\Delta A\left(k\right)E\left\{{\hat{x}}_i\left(k|k\right){\tilde{d}}_i^{\text{T}}\left(k\right)\right\}{B}^{\text{T}}\left(k\right)+B\left(k\right)E\left\{{\tilde{d}}_i\left(k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)\right\}\Delta A^{\text{T}}\left(k\right)\\ \le {\epsilon }_{i,3}\Delta A\left(k\right){\hat{x}}_i\left(k|k\right){\hat{x}}_i^{\text{T}}\left(k|k\right)\Delta A^{\text{T}}\left(k\right)+{\epsilon }_{i,3}^{-1}B\left(k\right)p_i^d\left(k\right)B^{\text{T}}\left(k\right)\end{array}$ (27)

由式(22)~(27)可得

$\begin{array}{c}{P}_i^x\left(k+1|k\right)\le \left(1+{\epsilon }_{i,1}+{\epsilon }_{i,2}\right)A\left(k\right){\left({\left({\Xi }_i^x\left(k|k\right)\right)}^{-1}-{\alpha }_{i,1}{E}^{\text{T}}\left(k\right)E\left(k\right)\right)}^{-1}{A}^{\text{T}}\left(k\right)+(\left(1+{\epsilon }_{i,1}+{\epsilon }_{i,2}\right){\alpha }_{i,1}^{-1}\\ +\left(1+{\epsilon }_{i,1}^{-1}+{\epsilon }_{i,3}\right){\beta }_{i,1})H\left(k\right)H^{\text{T}}\left(k\right)+\left(1+{\epsilon }_{i,2}^{-1}+{\epsilon }_{i,3}^{-1}\right)B\left(k\right){\Xi }_i^d\left(k\right)B^{\text{T}}\left(k\right)+W\left(k\right)\\ ={\kappa }_{i,1}A\left(k\right){\left({\left({\Xi }_i^x\left(k|k\right)\right)}^{-1}-{\alpha }_{i,1}{E}^{\text{T}}\left(k\right)E\left(k\right)\right)}^{-1}{A}^{\text{T}}\left(k\right)\\ +{\kappa }_{i,2}H\left(k\right)H^{\text{T}}\left(k\right)+{\kappa }_{i,3}B\left(k\right){\Xi }_i^d\left(k\right)B^{\text{T}}\left(k\right)+W\left(k\right)\\ \triangleq {\Xi }_i^x\left(k+1|k\right).\end{array}$ (28)

利用引理2，可得

$E\left\{{\phi }_i\left(k+1\right){\phi }_i^{\text{T}}\left(k+1\right)\right\}\le E\left\{{\phi }_i^{\text{T}}\left(k+1\right){\phi }_i\left(k+1\right)I\right\}\le \frac{1+{\eta }_i}{{\eta }_i}{\bar{\Xi }}_i^{\zeta }\left(k+1\right)+\left(1+{\eta }_i{}^{-1}\right){\delta }_i^2.$ (29)

采用(22)~(28)同样的方法，我们可以进一步得到：

$P_i^d\left(k+1\right)\le {\Xi }_i^d\left(k+1\right),P_i^x\left(k+1|k+1\right)\le {\Xi }_i^x\left(k+1|k+1\right).$

定理得证。

在下面的定理中设计适当的估计器增益，使得定理1中的协方差上界最小。

定理2：在约束条件(11)下，如果估计器增益满足：

$L_i\left(k+1\right)={\left(D_i^{\text{T}}\left(k+1\right){\Omega }_i^{-1}\left(k+1\right)D_i\left(k+1\right)\right)}^{-1}{D}_i^{\text{T}}\left(k+1\right){\Omega }_i^{-1}\left(k+1\right)$ (30)

$K_i\left(k+1\right)={\Xi }_i^x\left(k+1|k\right)C_i^{\text{T}}\left(k+1\right){\Phi }_i^{-1}\left(k+1\right)$ (31)

其中，

$\begin{array}{c}{\Omega }_i\left(k+1\right)={\kappa }_{i,4}(C_i\left(k+1\right){\left({\left({\Xi }_i^x\left(k+1|k\right)\right)}^{-1}-{\alpha }_{i,2}{M}_i^{\text{T}}\left(k+1\right)M_i\left(k+1\right)\right)}^{-1}{C}_i^{\text{T}}\left(k+1\right)\\ +{\kappa }_{i,5}{U}_i\left(k+1\right)U_i^{\text{T}}\left(k+1\right)+\left(1+{\lambda }_{i,4}\right)V\left(k+1\right)+{\kappa }_{i,6}\left(\frac{1+{\eta }_i}{{\eta }_i}{\bar{\Xi }}_i^{\zeta }\left(k+1\right)+\left(1+{\eta }_i^{-1}\right){\delta }_i^2\right)I\end{array}$

$\begin{array}{c}{\Phi }_i\left(k+1\right)={\kappa }_{i,7}{C}_i\left(k+1\right){\Xi }_i^x\left(k+1|k\right)C_i^{\text{T}}\left(k+1\right)+{\kappa }_{i,8}{U}_i\left(k+1\right)U_i^{\text{T}}\left(k+1\right)+{\kappa }_{i,10}{V}_i\left(k+1\right)\\ +{\kappa }_{i,9}{D}_i\left(k+1\right){\Xi }_i^d\left(k+1\right)D_i^{\text{T}}\left(k+1\right)+{\kappa }_{i,11}\left(\frac{1+{\eta }_i}{{\eta }_i}{\bar{\Xi }}_i^{\zeta }\left(k+1\right)+\left(1+{\eta }_i^{-1}\right){\delta }_i^2\right)I.\end{array}$

那么，协方差上界 ${\Xi }_i^d\left(k+1\right)$ 和 ${\Xi }_i^x\left(k+1|k+1\right)$ 得到了最小值。

证明：未知输入的估计误差协方差上界可表示为

${\Xi }_i^d\left(k+1\right)=L_i\left(k+1\right){\Omega }_i\left(k+1\right)L_i^{\text{T}}\left(k+1\right)$ (32)

利用拉格朗日乘子法求得增益矩阵 $L_i\left(k+1\right)$ ，使得 ${\Xi }_i^d\left(k+1\right)$ 最小。首先引入具有适当维数的对称矩阵 ${\Lambda }_i\left(k+1\right)$ ，并构造以下函数：

$\begin{array}{c}{\mathcal{H}}_i\left(L_i\left(k+1\right),{\Lambda }_i\left(k+1\right)\right)=L_i\left(k+1\right){\Omega }_i\left(k+1\right)L_i^{\text{T}}\left(k+1\right)+{\Lambda }_i\left(k+1\right){\left(I-L_i\left(k+1\right)D_i\left(k+1\right)\right)}^{\text{T}}\\ +\left(I-L_i\left(k+1\right)D_i\left(k+1\right)\right){\Lambda }_i^{\text{T}}\left(k+1\right).\end{array}$ (33)

使用完全平方法，我们可以进一步得到

$\begin{array}{l}{\mathcal{H}}_i\left(L_i\left(k+1\right),{\Lambda }_i\left(k+1\right)\right)\\ =L_i\left(k+1\right){\Omega }_i\left(k+1\right)L_i^{\text{T}}\left(k+1\right)+{\Lambda }_i\left(k+1\right)+{\Lambda }_i^{\text{T}}\left(k+1\right)-{\Lambda }_i\left(k+1\right)D_i^{\text{T}}\left(k+1\right)L_i^{\text{T}}\left(k+1\right)\\ -L_i\left(k+1\right)D_i\left(k+1\right){\Lambda }_i^{\text{T}}\left(k+1\right)+{\Lambda }_i\left(k+1\right)D_i^{\text{T}}\left(k+1\right){\Omega }_i^{-1}\left(k+1\right)D_i\left(k+1\right){\Lambda }_i^{\text{T}}\left(k+1\right)\\ -{\Lambda }_i\left(k+1\right)D_i^{\text{T}}\left(k+1\right){\Omega }_i^{-1}\left(k+1\right)D_i\left(k+1\right){\Lambda }_i^{\text{T}}\left(k+1\right)\\ =\left(L_i\left(k+1\right)-{\Lambda }_i\left(k+1\right)D_i^{\text{T}}\left(k+1\right){\Omega }_i^{-1}\left(k+1\right)\right){\Omega }_i\left(k+1\right){\left(L_i\left(k+1\right)-{\Lambda }_i\left(k+1\right)D_i^{\text{T}}\left(k+1\right){\Omega }_i^{-1}\left(k+1\right)\right)}^{\text{T}}\\ -{\Lambda }_i\left(k+1\right)D_i^{\text{T}}\left(k+1\right){\Omega }_i^{-1}\left(k+1\right)D_i\left(k+1\right){\Lambda }_i^{\text{T}}\left(k+1\right)+{\Lambda }_i\left(k+1\right)+{\Lambda }_i^{\text{T}}\left(k+1\right).\end{array}$ (34)

容易看出，当

$L_i\left(k+1\right)={\Lambda }_i\left(k+1\right)D_i^{\text{T}}\left(k+1\right){\Omega }_i^{-1}\left(k+1\right)$ (35)

时， ${\mathcal{H}}_i\left(L_i\left(k+1\right),{\Lambda }_i\left(k+1\right)\right)$ 取得最小值。

将式(35)代入到约束条件(11)中，可得

${\Lambda }_i\left(k+1\right)={\left(D_i^{\text{T}}\left(k+1\right){\Omega }_i^{-1}\left(k+1\right)D_i\left(k+1\right)\right)}^{-1}.$ (36)

因此，当增益 $L_i\left(k+1\right)$ 满足式(30)时，误差协方差上界 ${\Xi }_i^d\left(k+1\right)$ 最小。

同样地，状态估计误差协方差上界可表示为

$\begin{array}{l}{\Xi }_i^x\left(k+1|k+1\right)\\ ={\kappa }_{i,7}{\Xi }_i^x\left(k+1|k\right)-{\kappa }_{i,7}{K}_i\left(k+1\right)C_i\left(k+1\right){\Xi }_i^x\left(k+1|k\right)-{\kappa }_{i,7}{\Xi }_i^x\left(k+1|k\right)C_i^{\text{T}}\left(k+1\right)K_i^{\text{T}}\left(k+1\right)\\ +K_i\left(k+1\right){\Phi }_i\left(k+1\right)K_i^{\text{T}}\left(k+1\right)\\ ={\kappa }_{i,7}\left(K_i\left(k+1\right)-{\Xi }_i^x\left(k+1|k\right)C_i^{\text{T}}\left(k+1\right){\Phi }_i^{-1}\left(k+1\right)\right){\Phi }_i\left(k+1\right){\left(K_i\left(k+1\right)-{\Xi }_i^x\left(k+1|k\right)C_i^{\text{T}}\left(k+1\right){\Phi }_i^{-1}\left(k+1\right)\right)}^{\text{T}}\\ -{\kappa }_{i,7}{\Xi }_i^x\left(k+1|k\right)C_i^{\text{T}}\left(k+1\right){\Phi }_i^{-1}\left(k+1\right)C_i\left(k+1\right){\Xi }_i^x\left(k+1|k\right)+{\kappa }_{i,7}{\Xi }_i^x\left(k+1|k\right).\end{array}$ (37)

显然，当增益矩阵 $K_i\left(k+1\right)$ 满足式(31)时， ${\Xi }_i^x\left(k+1|k+1\right)$ 取得最小值。证毕。

3.2. 融合估计方案

针对本文所考虑的多传感器不确定系统，为避免互协方差的计算，保证系统的稳健性，我们采用CI融合估计方案。

分别用 ${\hat{x}}_f\left(k+1|k+1\right)$ 和 ${\Xi }_f^x\left(k+1|k+1\right)$ 表示状态融合估计和状态融合协方差。融合方案如下：

${\Xi }_f^x\left(k+1|k+1\right)={\left({\displaystyle \underset{i=1}{\overset{M}{\sum }}{w}_i}{\left({\Xi }_i^x\left(k+1|k+1\right)\right)}^{-1}\right)}^{-1}$ (38)

${\hat{x}}_f\left(k+1|k+1\right)={\Xi }_f^x\left(k+1|k+1\right){\displaystyle \underset{i=1}{\overset{M}{\sum }}{w}_i}{\left({\Xi }_i^x\left(k+1|k+1\right)\right)}^{-1}{\hat{x}}_i\left(k+1|k+1\right)$ (39)

其中 $w_i\ge 0$ ，且满足 ${\displaystyle \underset{i=1}{\overset{M}{\sum }}{w}_i}=1$ 。

为提高CI融合器的鲁棒精度，参数 $w_i$ 满足以下非线性约束最优化问题：

$\begin{array}{l}\underset{w_i}{\min }\left\{\text{tr}\left({\Xi }_f^x\left(k+1|k+1\right)\right)\right\}\\ \text{s}\text{.t}.{\displaystyle \underset{i=1}{\overset{M}{\sum }}{w}_i}=1,w_i\ge 0.\end{array}$ (40)

定理3：对于局部滤波器(4)，基于CI融合估计(38)~(40)是一致的，

$\begin{array}{l}{\bar{P}}_f\left(k+1|k+1\right)\\ =E\left\{\left(x\left(k+1|k+1\right)-{\hat{x}}_f\left(k+1|k+1\right)\right){\left(x\left(k+1|k+1\right)-{\hat{x}}_f\left(k+1|k+1\right)\right)}^{\text{T}}\right\}\\ \le {\Xi }_f^x\left(k+1|k+1\right).\end{array}$ (41)

证明：注意到 $P_x\left(k+1|k+1\right)\le {\Xi }_x\left(k+1|k+1\right)$ ，定理3的证明可直接由文献 [8] [21] 的证明得到。

4. 仿真实验

本节借助MATLAB数学工具，通过一个数值实例来验证所提算法的有效性。在有限时域N = 100内，系统参数设置如下：

$\begin{array}{l}A\left(k\right)=\left[\begin{array}{cc}0.26+0.02\sin \left(k\right)& 0.3\\ 0.2-0.02\cos \left(2k\right)& 0\end{array}\right],B\left(k\right)=\left[\begin{array}{cc}0.10& 0.12\\ 0.15& 0.13\end{array}\right]\\ C_1\left(k\right)=\left[\begin{array}{cc}0.31& 0.21\\ 0& -0.2\end{array}\right],C_2\left(k\right)=\left[\begin{array}{cc}0.30& 0.22\\ 0& -0.22\end{array}\right]\\ D_1\left(k\right)=\left[\begin{array}{cc}0.21& 0.14\\ 0.12& 0.15\end{array}\right],D_2\left(k\right)=\left[\begin{array}{cc}0.22& 0.12\\ 0.12& 0.13\end{array}\right]\\ W=0.00001I,V=0.00001I.\end{array}$

需要估计的未知输入为

$d\left(k\right)=\{\begin{array}{l}{\left[0.2\cos \left(k\right),0.2\cos \left(k\right)\right]}^{\text{T}},0\le k<50;\\ {\left[-0.2\sin \left(k\right),-0.2\cos \left(k\right)\right]}^{\text{T}},50\le k\le 100.\end{array}$

不确定参数为：

$\Delta A\left(k\right)=\left[\begin{array}{cc}0.01& 0.01\\ 0.06& -0.02\end{array}\right]\left[\begin{array}{cc}0.05\cos \left(k\right)& 0\\ 0& 0.03\sin \left(k\right)\end{array}\right]\left[\begin{array}{cc}0.01& -0.003\\ 0.01& 0.02\end{array}\right]$

$\Delta C_i\left(k\right)=\left[\begin{array}{cc}0.01& 0.01\\ 0.06& -0.02\end{array}\right]\left[\begin{array}{cc}0.02\cos \left(k\right)& 0\\ 0& 0.01\sin \left(k\right)\end{array}\right]\left[\begin{array}{cc}0.01& -0.003\\ 0.01& 0.02\end{array}\right],\left(i=1,2\right).$

对于动态事件触发协议(2)~(3)，我们给定参数 ${\eta }_i=4\left(i=1,2\right)$ ， ${\delta }_1=0.005$ ， ${\delta }_2=0.004$ ， ${\rho }_1=0.3$ ， ${\rho }_2=0.4$ ，内部动态变量初值 ${\zeta }_{i,0}=0.8$ 。假设系统状态和未知输入的初始值及其估计值分别为 $x\left(0\right)={\left[\begin{array}{cc}0.01& 0.01\end{array}\right]}^{\text{T}}$ ， ${\hat{x}}_1\left(0\right)={\hat{x}}_2\left(0\right)={\left[\begin{array}{cc}0& 0\end{array}\right]}^{\text{T}}$ ， $d\left(0\right)={\left[\begin{array}{cc}0.2& 0.2\end{array}\right]}^{\text{T}}$ ， ${\hat{d}}_1\left(0\right)={\hat{d}}_2\left(0\right)={\left[\begin{array}{cc}0& 0\end{array}\right]}^{\text{T}}$ 。令 ${\Xi }_i^x\left(0|0\right)=0.3I$ 。其他参数为 ${\alpha }_{i,j}=0.1\left(j=1,2\right)$ ， ${\epsilon }_{i,j}=0.1\left(j=1,2,3\right)$ ， ${\lambda }_{i,j}=0.1\left(j=1,2,3,4\right)$ 和 ${\mu }_{i,j}=0.1\left(j=1,2,\cdots ,8\right)$ 。

为了量化估计精度，我们使用均方误差(MSE)来反映估计值与真实值之间的差异程度，定义如下：

${\text{MSE}}_x^{}=\mathbb{E}\left\{{\left(x\left(k\right)-{\hat{x}}_i\left(k|k\right)\right)}^2\right\}$

${\text{MSE}}_d^{}=\mathbb{E}\left\{{\left(d_i\left(k\right)-\hat{d}\left(k\right)\right)}^2\right\}.$

基于以上参数，得到的仿真结果如图1~7所示。图1分别显示了状态的第1个分量和第2个分量的真实值、两个局部估计以及融合估计。图2绘制了未知输入的两个分量的真实值和估计值。可以看出，该基于动态事件触发的估计器可以准确估计出系统状态和未知输入。图3，图4分别绘制了状态的MSE及其上界，其中，图3和图4的上面两个子图表示状态第1个分量和第2个分量的局部估计情况，第三个表示相应的状态融合估计情况。图5，图6分别绘制了未知输入两个分量的log(MSE)及其上界。可以看出，所有log(MSE)都保持在相应上界的下方，符合我们的滤波设计期望。图7描述了每个传感器节点的触发时刻，触发次数明显减少，由此可以验证此滤波器具有节约能源的目的。

Figure 1. Actual value, local estimates and fusion estimate of the state

图1. 状态的实际值、局部估计和融合估计

Figure 2. Actual value and local estimates of the unknown input

图2. 未知输入的实际值和局部估计

Figure 3. Log(MSE) of the first component of state and its upper bound

图3. 状态的第1个分量的MSE及其上界

Figure 4. Log(MSE) of the second component of state and its upper bound

图4. 状态的第2个分量的MSE及其上界

Figure 5. Log(MSE) of the first component of unknown input and its upper bound

图5. 未知输入的第1个分量的MSE及其上界

Figure 6. Log(MSE) of the second component of unknown input and its upper bound

图6. 未知输入的第2个分量的MSE及其上界

Figure 7. Triggering instants of each sensor node

图7. 每个传感器节点的触发时刻

5. 结论

针对包含未知输入和不确定参数的多传感器系统，提出了一种基于动态事件触发的未知输入与状态局部递归滤波和状态融合估计算法。对于各个子系统，引入一个动态事件触发通信协议，减少不必要的冗余传输。利用数学归纳法得到局部的未知输入和状态估计的误差协方差矩阵的上界，然后利用拉格朗日乘子法和完全平方法选择合适的增益矩阵，使得协方差上界最小。随后，采用CI融合方法对所有的局部估计器进行融合，得到状态融合估计，并保证了所提状态融合方法的一致性。最后，通过一个数值仿真验证了所提局部联合滤波器和状态融合滤波器的有效性。

NOTES

^*通讯作者。