Minimax Robust Kalman Filtering under Multistep Random Measurement Delays and Packet Dropouts
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摘要: 该文研究了多步随机观测滞后和丢包系统的极大极小鲁棒Kalman滤波问题。系统噪声方差不确定但有已知保守上界，传感器到估值器的多步随机观测滞后和丢包通过一组概率已知的伯努利分布随机变量描述。利用哈达玛乘积改进模型转换方法，设计了极大极小鲁棒时变Kalman估值器。利用矩阵初等变换、盖尔圆盘定理和哈达玛乘积定理证明了广义李雅普诺夫方程解的半正定性，进而应用矩阵分解和李雅普诺夫方程方法证明了所设计估值器的鲁棒性，即对所有容许的不确定性，确保实际估计误差方差有最小上界。给出时变广义李雅普诺夫方程存在稳态唯一半正定解的条件，进而设计了鲁棒稳态估值器。证明了时变和稳态估值器的按实现收敛性。仿真实例验证了其有效性。Abstract:
Objective Networked Control Systems (NCSs) provide advantages such as flexible installation, convenient maintenance, and reduced cost, but they also present challenges arising from random measurement delays and packet dropouts caused by communication network unreliability and limited bandwidth. Moreover, system noise variance may fluctuate significantly under strong electromagnetic interference. In NCSs, time delays are random and uncertain. When a set of Bernoulli-distributed random variables is used to describe multistep random measurement delays and packet dropouts, the fictitious noise method in existing studies introduces autocorrelation among different components, which complicates the computation of fictitious noise variances and makes it difficult to establish robustness. This study presents a solution for minimax robust Kalman filtering in systems characterized by uncertain noise variance, multistep random measurement delays, and packet dropouts. Methods The main challenges lie in model transformation and robustness verification. When a set of Bernoulli-distributed random variables is employed to represent multistep random measurement delays and packet dropouts, a series of strategies are applied to address the minimax robust Kalman filtering problem. First, a new model transformation method is proposed based on the flexibility of the Hadamard product in multidimensional data processing, after which a robust time-varying Kalman estimator is designed in a unified framework following the minimax robust filtering principle. Second, the robustness proof is established using matrix elementary transformation, strictly diagonally dominant matrices, the Gerŝgorin circle theorem, and the Hadamard product theorem within the framework of the generalized Lyapunov equation method. Additionally, by converting the Hadamard product into a matrix product through matrix factorization, a sufficient condition for the existence of a steady-state estimator is derived, and the robust steady-state Kalman estimator is subsequently designed. Results and Discussions The proposed minimax robust Kalman filter extends the robust Kalman filtering framework and provides new theoretical support for addressing the robust fusion filtering problem in complex NCSs. The curves (Fig. 5) present the actual accuracy ${\text{tr}}{{\mathbf{\bar P}}^l}(N)$, $l = a,b,c,d$ as a function of $ 0.1 \le {\alpha _0},{\alpha _1},{\alpha _2} \le 1 $. It is observed that situation (a) achieves the highest robust accuracy, followed by situations (b) and (c), whereas situation (d) exhibits poorer accuracy. This difference arises because the estimators in situation (a) receive measurements with one-step random delay, whereas situation (d) experiences a higher packet loss rate. The curves (Fig. 5) confirm the validity and effectiveness of the proposed method. Another simulation is conducted for a mass–spring–damper system. The comparison between the proposed approach and the optimal robust filtering method (Table 2, Fig. 7) indicates that although the proposed method ensures that the actual prediction error variance attains the minimum upper bound, its actual accuracy is slightly lower than the optimal prediction accuracy. Conclusions The minimax robust Kalman filtering problem is investigated for systems characterized by uncertain noise variance, multistep random measurement delays, and packet dropouts. The system noise variance is uncertain but bounded by known conservative upper limits, and a set of Bernoulli-distributed random variables with known probabilities is used to represent the multistep random measurement delays and packet dropouts between the sensor and the estimator. The Hadamard product is used to enhance the model transformation method, followed by the design of a minimax robust time-varying Kalman estimator. Robustness is demonstrated through matrix elementary transformation, the Gerŝgorin circle theorem, the Hadamard product theorem, matrix factorization, and the Lyapunov equation method. A sufficient condition is established for the time-varying generalized Lyapunov equation to possess a unique steady-state positive semidefinite solution, based on which a robust steady-state estimator is constructed. The convergence between the time-varying and steady-state estimators is also proven. Two simulation examples verify the effectiveness of the proposed approach. The presented methods overcome the limitations of existing techniques and provide theoretical support for solving the robust fusion filtering problem in complex NCSs. -
图 6 实际的一步平滑器精度$ {\text{tr}}{\boldsymbol{\bar P}}(1) $随$ 0.1 \le \delta ,\alpha \le 0.9 $变化
表 1 鲁棒稳态Kalman估值器的实际和鲁棒精度
${\text{tr}}{\boldsymbol{P}}( - 1)$ ${\text{tr}}{\boldsymbol{P}}(0)$ ${\text{tr}}{\boldsymbol{P}}(1)$ ${\text{tr}}{\boldsymbol{P}}(2)$ 1.5742 1.3780 1.1550 1.0702 ${\text{tr}}{\boldsymbol{\bar P}}( - 1)$ ${\text{tr}}{\boldsymbol{\bar P}}(0)$ ${\text{tr}}{\boldsymbol{\bar P}}(1)$ ${\text{tr}}{\boldsymbol{\bar P}}(2)$ 1.2292 1.0779 0.9043 0.8381 
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