An adaptive observer for a class of parabolic PDEs based on a convex optimization approach for backstepping PDE design
An adaptive observer for a class of parabolic PDEs based on a convex optimization approach for backstepping PDE design
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基于反步偏微分方程设计凸优化方法的一类抛物线偏微分方程的自适应观测器
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发表时间：
2016
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通讯作者：
T. Parisini
中科院分区：
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作者：
Pedro Ascencio;A. Astolfi;T. Parisini
本文解决了一类一维线性抛物线偏微分方程的同时状态和参数估计的观测器设计问题。该设计基于反推偏微分方程 (backstepping PDE) 方法，包括修正积分变换以补偿参数不确定性。所得核偏微分方程的解通过平方和公式重新转换为凸优化问题，并通过多项式优化技术（半定规划）求解。这允许以快速和直接的方式计算每个采样时间的状态和参数观测器增益。除了基于 Volterra 变换的观测器之外，还提出了一种通过 Fredholm 型变换进行两个边界测量的观测器设计方法。对于连续函数空间中的多项式核，证明了该变换的唯一性和可逆性。通过数值模拟说明了该方法的有效性。
This article addresses the observer design problem for simultaneous state and parameter estimation for a class of one-dimensional linear parabolic PDEs. The design is based on the backstepping PDE methodology, including a modified integral transformation to compensate for the parameter uncertainty. The solution of the resulting Kernel-PDE is recast as a convex optimization problem via a Sum-of-Squares formulation which is solved by polynomial optimization techniques (semidefinite programming). This allows computing - in a fast and direct way - the state and parameter observer gains at every sampling time. In addition to an observer based on the Volterra transformation, an observer design method with two boundary measures via a Fredholm-type transformation is presented. The uniqueness and invertibility of this transformation are proved for polynomial kernels in the space of continuous functions. The effectiveness of this approach is illustrated by numerical simulations.