基于凸优化的非线性滤波算法研究
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摘要
随着科学技术的飞速发展，我们所面对的系统的复杂程度越来越高。由于系统中非线性和不确定性等因素的存在，以及对系统性能要求的不断提高，对系统的分析和综合手段提出了更高的要求。其中，非线性滤波方法近年来已成为国内外研究的热点。
在凸优化问题里，任何局部极值点都是全局极值点，并且当凸优化问题为严格凸优化时，它的全局极小值点是唯一的。因此，如果待求解的问题能表示成为凸优化问题，则能够很好地避免算法初始化、搜索步长选择和陷入局部极值等问题，并且可以获得该问题的全局最优解。正是因为凸性具有很多的优良特性，当采用凸优化方法解决非线性滤波问题时，可以从滤波组合和非线性回归的角度进行分析，把系统线性与非线性特性转化为凸与非凸问题，从而能够使问题的求解得到简化。因而开展基于凸优化的滤波理论研究，改进凸优化滤波方法具有重要的现实意义。
本文结合组合优化算法、统计学习理论的方法和多模型系统，针对惯性导航系统中非线性滤波算法及其工程应用中存在的问题，从以下五个部分进行深入研究：
针对非线性系统状态滤波问题，在凸优化的基本理论框架下，研究了Lagrange对偶和最优性条件。在此基础上，把非线性滤波问题转化为凸二次优化问题，并对凸线性组合在动态系统滤波中的应用进行了理论推导，得出组合之后的滤波效果优于未经组合的单一滤波方法的结论。这也为本文的后续研究奠定了基础。
研究了采用支持向量机进行非线性滤波的方法，该方法本质上是求解凸二次优化问题。为了提高求解的实时性及降低求解的复杂程度，本文提出一种最小二乘支持向量机算法，该算法用解线性等式来代替求解标准的支持向量机凸二次优化问题。
针对惯性导航系统的初始对准问题，本文研究了凸组合方法在非线性滤波中的应用，提出了利用多组支持向量机以凸线性方法组合起来并行滤波，再通过第二层支持向量机进行线性回归的方法求得组合系数，得到有两层结构形式的凸线性组合支持向量机。
针对解决非线性滤波的实时性与稳定性问题，本文提出了一种新的基于凸优化的自适应联合滤波算法，解决了由于系统环境复杂多变，噪声的分布特性具有不确定性而产生的系统精度降低甚至容易发散的问题。通过对SINS/CNS/GNSS组合导航系统进行自适应融合处理，实现了动态系统的实时滤波。仿真验证了算法的可行性和有效性。
研究了组合导航H2/H混合滤波。针对系统的不确定性和噪声的非高斯性，提出了基于凸优化的自适应H2/H混合滤波算法。该算法基于凸优化的方法来实时地调整滤波增益矩阵，实现H2和H滤波的优势互补，具有更好的鲁棒性。该算法是基于多模型滤波算法的一个特例，是一种新的自适应滤波方法。
With the rapid development of science and technology, the system is becoming more and more complicated. Due to the nonlinear effects, uncertainty and the higher requirements for the system performance, a more advanced method for system analysis and synthesis is needed. Among all the methods, a special non-linear filtering method is being widely studied at home and abroad.
In convex optimization problems, any local extreme point is a global extreme point. Moreover, when the convex optimization problem is a strict convex optimization, the global minimum point is unique. If the problem can be expressed as a convex optimization problem. The global optimal solution of the problem can be acquired without worrying about the algorithm initialization, the step length selection and trap of local minima problems. The convexity can be applied into the nonlinear filtering problem to simplify it due to this great feature. The problem of linear/nonlinear turns into one of convex/non-convex through the view of filter integration and non-linear regression filtering. Therefore, there is great practical significance to study on the convex optimization based filtering method.
The optimization algorithms, statistical learning theory and multi-model systems are combined to solve the problem in nonlinear filtering algorithm and its application in inertial systems. The study is conducted in the following aspects.
Firstly, to solve the state filtering problem for nonlinear systems, the Lagrange Duality and optimality conditions have been studied under the basic theory of convex optimization. Based on this, nonlinear filtering problem is transferred into convex quadratic optimization problem. The application of convex linear combination in dynamic filtering systems is also derived. The conclusion is drawn that combined filtering is better than single filtering method. The conclusion lays foundation for the following research.
Secondly, nonlinear filtering method using support vector machine is studied, which is actually a convex quadratic optimization problem. In order to improve the real-time performance and reduce complexity, a least squares support vector machine is proposed. It solves linear equations instead of convex quadratic optimization problems in the standard support vector machine.
Thirdly, to solve problems in the initial alignment of inertial navigation system, the application of convex combination method in nonlinear filtering is discussed. A new method is proposed. The SVM of two layers in convex linear combination can be acquired by using a convex linear combination of support vector machine to filter and then calculating the combination coefficients by the second layers of SVM through linear regression method.
In order to improve the real-time performance and stability of nonlinear filtering, a new adaptive federated filter algorithm based on convex optimization is proposed. It solves the problem of precision reduction or even divergence, which results from the change in system environment and the uncertainty of the noise distribution. Real-time filtering of the dynamic system is realized through the adaptive fusion processing of SINS/CNS/GNSS integrated navigation system. Simulations verify the feasibility and effectiveness of the algorithm.
Finally, the hybrid filter of H2/H combined navigation is studied. Due to the system uncertainty and non-Gaussian noise, the adaptive H2/H filtering algorithm based on convex optimization is proposed. Based on convex optimization, the algorithm adjusts the filter gain matrix, takes advantages of the H2and H filers, and has better robustness. It is a special case of multi-model filter algorithm and a new adaptive filtering method.
在凸优化问题里，任何局部极值点都是全局极值点，并且当凸优化问题为严格凸优化时，它的全局极小值点是唯一的。因此，如果待求解的问题能表示成为凸优化问题，则能够很好地避免算法初始化、搜索步长选择和陷入局部极值等问题，并且可以获得该问题的全局最优解。正是因为凸性具有很多的优良特性，当采用凸优化方法解决非线性滤波问题时，可以从滤波组合和非线性回归的角度进行分析，把系统线性与非线性特性转化为凸与非凸问题，从而能够使问题的求解得到简化。因而开展基于凸优化的滤波理论研究，改进凸优化滤波方法具有重要的现实意义。
本文结合组合优化算法、统计学习理论的方法和多模型系统，针对惯性导航系统中非线性滤波算法及其工程应用中存在的问题，从以下五个部分进行深入研究：
针对非线性系统状态滤波问题，在凸优化的基本理论框架下，研究了Lagrange对偶和最优性条件。在此基础上，把非线性滤波问题转化为凸二次优化问题，并对凸线性组合在动态系统滤波中的应用进行了理论推导，得出组合之后的滤波效果优于未经组合的单一滤波方法的结论。这也为本文的后续研究奠定了基础。
研究了采用支持向量机进行非线性滤波的方法，该方法本质上是求解凸二次优化问题。为了提高求解的实时性及降低求解的复杂程度，本文提出一种最小二乘支持向量机算法，该算法用解线性等式来代替求解标准的支持向量机凸二次优化问题。
针对惯性导航系统的初始对准问题，本文研究了凸组合方法在非线性滤波中的应用，提出了利用多组支持向量机以凸线性方法组合起来并行滤波，再通过第二层支持向量机进行线性回归的方法求得组合系数，得到有两层结构形式的凸线性组合支持向量机。
针对解决非线性滤波的实时性与稳定性问题，本文提出了一种新的基于凸优化的自适应联合滤波算法，解决了由于系统环境复杂多变，噪声的分布特性具有不确定性而产生的系统精度降低甚至容易发散的问题。通过对SINS/CNS/GNSS组合导航系统进行自适应融合处理，实现了动态系统的实时滤波。仿真验证了算法的可行性和有效性。
研究了组合导航H2/H混合滤波。针对系统的不确定性和噪声的非高斯性，提出了基于凸优化的自适应H2/H混合滤波算法。该算法基于凸优化的方法来实时地调整滤波增益矩阵，实现H2和H滤波的优势互补，具有更好的鲁棒性。该算法是基于多模型滤波算法的一个特例，是一种新的自适应滤波方法。
With the rapid development of science and technology, the system is becoming more and more complicated. Due to the nonlinear effects, uncertainty and the higher requirements for the system performance, a more advanced method for system analysis and synthesis is needed. Among all the methods, a special non-linear filtering method is being widely studied at home and abroad.
In convex optimization problems, any local extreme point is a global extreme point. Moreover, when the convex optimization problem is a strict convex optimization, the global minimum point is unique. If the problem can be expressed as a convex optimization problem. The global optimal solution of the problem can be acquired without worrying about the algorithm initialization, the step length selection and trap of local minima problems. The convexity can be applied into the nonlinear filtering problem to simplify it due to this great feature. The problem of linear/nonlinear turns into one of convex/non-convex through the view of filter integration and non-linear regression filtering. Therefore, there is great practical significance to study on the convex optimization based filtering method.
The optimization algorithms, statistical learning theory and multi-model systems are combined to solve the problem in nonlinear filtering algorithm and its application in inertial systems. The study is conducted in the following aspects.
Firstly, to solve the state filtering problem for nonlinear systems, the Lagrange Duality and optimality conditions have been studied under the basic theory of convex optimization. Based on this, nonlinear filtering problem is transferred into convex quadratic optimization problem. The application of convex linear combination in dynamic filtering systems is also derived. The conclusion is drawn that combined filtering is better than single filtering method. The conclusion lays foundation for the following research.
Secondly, nonlinear filtering method using support vector machine is studied, which is actually a convex quadratic optimization problem. In order to improve the real-time performance and reduce complexity, a least squares support vector machine is proposed. It solves linear equations instead of convex quadratic optimization problems in the standard support vector machine.
Thirdly, to solve problems in the initial alignment of inertial navigation system, the application of convex combination method in nonlinear filtering is discussed. A new method is proposed. The SVM of two layers in convex linear combination can be acquired by using a convex linear combination of support vector machine to filter and then calculating the combination coefficients by the second layers of SVM through linear regression method.
In order to improve the real-time performance and stability of nonlinear filtering, a new adaptive federated filter algorithm based on convex optimization is proposed. It solves the problem of precision reduction or even divergence, which results from the change in system environment and the uncertainty of the noise distribution. Real-time filtering of the dynamic system is realized through the adaptive fusion processing of SINS/CNS/GNSS integrated navigation system. Simulations verify the feasibility and effectiveness of the algorithm.
Finally, the hybrid filter of H2/H combined navigation is studied. Due to the system uncertainty and non-Gaussian noise, the adaptive H2/H filtering algorithm based on convex optimization is proposed. Based on convex optimization, the algorithm adjusts the filter gain matrix, takes advantages of the H2and H filers, and has better robustness. It is a special case of multi-model filter algorithm and a new adaptive filtering method.
引文
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