Combining least-squares and gradient-based algorithms for the identification of a co-current flow heat exchanger

Because of the high-dimensional nature of partial differential equations (PDEs), identifying accurate models of processes, the behaviour of which is governed by PDEs, is a challenging problem which still deserves a lot of attention. We address the problem of identifying a grey-box model of a heat exchanger by combining equation-error and output-error-based algorithms. First, in order to estimate rough but reliable values of the sought physical parameters characterising the heat exchanger behaviour, we use the interesting properties of the reinitialised partial moments (RPMs) developed initially for ordinary differential equations to deal with the problem of inaccessible partial derivatives of the PDE. Such an adaptation of the RPM features to PDEs leads to a direct continuous-time system identification problem for which convex least-squares solutions can be found. Second, thanks to a description of the heat exchanger dynamics with a 2D linear time-invariant Roesser model, the aforementioned rough estimates are used as reliable initial guesses for the nonlinear optimisation of a standard non-convex cost function introduced to estimate the state-space matrices of the Roesser model we want to identify. The efficiency of this two-step approach in terms of physical parameter estimation is validated through the simulation of a co-current flow heat exchanger.

由于偏微分方程（partial differential equations，PDEs）具有高维特性，针对动态特性受其支配的过程构建精确模型仍是一个极具挑战性且值得广泛关注的研究问题。本文针对换热器的灰箱模型（grey-box model）辨识问题，结合基于方程误差（equation-error）与输出误差（output-error）的两类算法展开研究。首先，为了估算表征换热器动态特性的待求物理参数的粗糙但可靠的初始值，我们利用最初针对常微分方程（ordinary differential equations）提出的重置偏矩（reinitialised partial moments，RPMs）的优良特性，解决了PDE中无法获取偏导数的难题；将RPM特性适配至PDE场景后，可得到一个可求解凸最小二乘解的直接连续时间系统辨识问题。其次，通过采用二维线性时不变Roesser模型描述换热器动力学，我们将此前得到的粗糙参数估计值作为可靠初始猜测，用于对为辨识目标Roesser模型的状态空间矩阵（state-space matrices）而引入的标准非凸代价函数（non-convex cost function）进行非线性优化（nonlinear optimisation）。最终通过并流换热器（co-current flow heat exchanger）的仿真实验，验证了该两步法在物理参数估计方面的有效性。