非定常气动力建模除了要准确描述气动力的非定常特性，还要反映其非线性特性。Volterra级数因为对系统非线性具有很强的描述能力正日益受到重视。一阶Volterra核只能表达线性特性，要建立反映非线性特性的非定常气动力模型，需要引入二阶核甚至更高阶核的影响。高阶Volterra核辨识的主要困难在于待辨识参数的数量随着核阶次的增加而呈指数增加导致计算难度急剧加大，即出现所谓维数灾难问题。以一种分段二次多小波为基函数将Volterra核展开，求解一个高维病态方程组来计算展开系数，利用小波的多分辨分析在时间和频率两个维度的分解特性将方程降维，最终将问题转化为求解一个低维方程组得到稳定解。通过NACA0012翼型在马赫数0.8下作沉浮运动时，升力系数、阻力系数和俯仰力矩系数的二阶核和三阶核的辨识构建非定常气动力模型，然后由此计算不同减缩频率下的气动力并与CFD结果进行比较，验证了Volterra级数对非线性非定常气动力的描述能力和多小波处理方法的有效性。
Unsteady aerodynamics modeling must accurately describe the nonlinear aerodynamic characteristics in addition to unsteady aerodynamic characteristics. The Volterra series has got more and more attention as a powerful tool for nonlinear system modeling. The first order Volterra kernel can only describe the linear characteristics of the system. It is thus essential to incorporate the influence of the second order kernel or higher order kernels to build a nonlinear unsteady aerodynamics model. The main difficulty of higher order kernels identification is the number of parameters needed to be identified increases exponentially with the order of the kernel. This results in a dramatic increase of computational difficulty, and the so-called dimensional disaster arises. This paper expands the Volterra kernels using the piecewise-quadratic multiwavelet as the basis function. In the process of solving high dimensional and ill-posed equations, the paper utilizes the decomposition of multiwavelet multiresolution analysis in time and frequency to reduce the dimension of equations, and finally turns the problem into solving low dimensional equations and gets a stable solution. By identifying the second order kernel and the third order kernel of the lift coefficient, drag coefficient and pitching moment coefficient of the NACA0012 airfoil in plunging motion at Mach number 0.8, a nonlinear unsteady aerodynamics model is constructed. The aerodynamics in different reduced frequency is computed, and is compared with the CFD results to verify the ability of the Volterra series to describe nonlinear and unsteady aerodynamics and the effectiveness of the multiwavelet processing method.
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