[发明专利]一种泊松白噪声激励下非线性系统的追踪控制方法

权利要求书

1.一种泊松白噪声激励下非线性系统的追踪控制方法，其特征在于，包括如下步骤：

1)将高维系统动力学方程

$\stackrel{\·}{X}=\mathrm{\ϵ}F(X,\mathrm{\ξ}(t\left)\right)=\mathrm{\ϵ}F(X,t)$

对状态矢量的增量作如下展开

$X\left(t\right)-x_0=H(x_0,t,t_0)\≡H(x_0,t)={\{\underset{i=1}{\overset{\mathrm{\∞}}{\Σ}}{\mathrm{\ϵ}}^i{H}_{1i}(x_0,t),...,\underset{i=1}{\overset{\mathrm{\∞}}{\Σ}}{\mathrm{\ϵ}}^i{H}_{ni}(x_0,t)\}}^T$

2)将上述方程的右端项在x₀处作Taylor级数展开

$\begin{array}{c}\mathrm{\ϵ}{\stackrel{\·}{H}}_{i1}+{\mathrm{\ϵ}}^2{\stackrel{\·}{H}}_{i2}+{\mathrm{\ϵ}}^3{\stackrel{\·}{H}}_{i3}+{\mathrm{\ϵ}}^4{\stackrel{\·}{H}}_{i4}+...\\ ={\mathrm{\ϵF}}_i+\underset{j=1}{\overset{n}{\Σ}}{\mathrm{\ϵH}}_{j1}+{\mathrm{\ϵ}}^2{H}_{j2}+{\mathrm{\ϵ}}^3{H}_{j3}+...\·\mathrm{\ϵ}\frac{\∂}{\∂x_j}{F}_i\\ +\frac{1}{2!}\underset{j=1}{\overset{n}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}{\mathrm{\ϵH}}_{j1}+{\mathrm{\ϵ}}^2{H}_{j2}+...\·{\mathrm{\ϵH}}_{k1}+{\mathrm{\ϵ}}^2{H}_{k2}+...\·\mathrm{\ϵ}\frac{\∂}{\∂x_j}\frac{\∂}{\∂x_k}{F}_i\\ +\frac{1}{3!}\underset{j=1}{\overset{n}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}\underset{l=1}{\overset{n}{\Σ}}{\mathrm{\ϵH}}_{j1}+...\·{\mathrm{\ϵH}}_{k1}+...\·{\mathrm{\ϵH}}_{l1}...\·\mathrm{\ϵ}\frac{\∂}{\∂x_j}\frac{\∂}{\∂x_k}\frac{\∂}{\∂x_l}{F}_i+...\\ i=1,...,n\end{array}$

3)令方程中左右两端不同ε幂次项分别相等，进而可以得到用F(x₀,t)表达的H(x₀,t)的各阶摄动展开分量

$H_{i1}={\∫}_{t_0}^t{F}_i(x_0,t_1){\mathrm{dt}}_1$

$H_{i2}=\underset{j=1}{\overset{n}{\Σ}}{\∫}_{t_0}^t{\mathrm{dt}}_2\frac{\∂F_i(x_0,t_2)}{\∂x_j}{\∫}_{t_0}^{t_2}{F}_j(x_0,t_1){\mathrm{dt}}_1$

$\begin{array}{c}{H}_{i3}=\underset{j=1}{\overset{n}{\Σ}}{\∫}_{t_0}^t{\mathrm{dt}}_3\frac{\∂F_i(x_0,t_3)}{\∂x_j}\\ \left(\underset{k=1}{\overset{n}{\Σ}}{\∫}_{t_0}^{t_3}{\mathrm{dt}}_2\frac{\∂F_j(x_0,t_2)}{\∂x_k}{\∫}_{t_0}^{t_2}{F}_k\right(x_0,t_1\left){\mathrm{dt}}_1\right)\\ +\frac{1}{2!}\underset{j=1}{\overset{n}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}{\∫}_{t_0}^t{\mathrm{dt}}_3\frac{{\∂}^2{F}_i(x_0,t_3)}{\∂x_j\∂x_k}{\∫}_{t_0}^{t_3}{F}_j(x_0,t_1){\mathrm{dt}}_1{\∫}_{t_0}^{t_3}{F}_k(x_0,t_2){\mathrm{dt}}_2\\ =\underset{j=1}{\overset{n}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}\left\{{\∫}_{t_0}^t{\mathrm{dt}}_3\frac{\∂F_i(x_0,t_3)}{\∂x_j}\left({\∫}_{t_0}^{t_3}{\mathrm{dt}}_2\frac{\∂F_j(x_0,t_2)}{\∂x_k}{\∫}_{t_0}^{t_2}{F}_k\right(x_0,t_1){\mathrm{dt}}_1\right)\\ +\frac{1}{2!}{\∫}_{t_0}^t{\mathrm{dt}}_3\frac{{\∂}^2{F}_i(x_0,t_3)}{\∂x_j\∂x_k}{\∫}_{t_0}^{t_3}{F}_j(x_0,t_1){\mathrm{dt}}_1{\∫}_{t_0}^{t_3}{F}_k(x_0,t_2){\mathrm{dt}}_2\},...\end{array}$

4)求得关于p＝p(x|x₀)的平均广义FPK方程：

$\begin{array}{c}\frac{\∂p}{\∂t}=\underset{i=1}{\overset{n}{\Σ}}\left(\frac{\∂}{\∂x_j}\right)\[A_{i}p\]+\underset{i=1}{\overset{n}{\Σ}}\underset{j=1}{\overset{n}{\Σ}}\left(\frac{\∂}{\∂x_i}\right)\left(\frac{\∂}{\∂x_j}\right)\[B_{ij}p\]\\ =\underset{i=1}{\overset{n}{\Σ}}\underset{j=1}{\overset{n}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}\left(\frac{\∂}{\∂x_i}\right)\left(\frac{\∂}{\∂x_j}\right)\left(\frac{\∂}{\∂x_k}\right)\[C_{ijk}p\]\\ =\underset{i=1}{\overset{n}{\Σ}}\underset{j=1}{\overset{n}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}\underset{l=1}{\overset{n}{\Σ}}\left(\frac{\∂}{\∂x_i}\right)\left(\frac{\∂}{\∂x_j}\right)\left(\frac{\∂}{\∂x_k}\right)\left(\frac{\∂}{\∂x_l}\right)\[D_{ijkl}p\]+O\left({\mathrm{\ϵ}}^5\right)\end{array}$

5)根据给定的目标概率密度，由上述平均广义FPK方程求得追踪控制律：

$\begin{array}{c}{u}_i=\{-{\mathrm{\Φ}}_i/\mathrm{\ρ}-\frac{{\mathrm{\ϵ}}^2}{2!}\underset{j=1}{\overset{n}{\Σ}}\underset{s=1}{\overset{r}{\Σ}}{g}_{js}\frac{\∂g_{is}}{\∂x_j}{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}K\[{\mathrm{\ξ}}_s,{\mathrm{\ξ}}_s^{\mathrm{\τ}}\]d\mathrm{\τ}\\ -\frac{{\mathrm{\ϵ}}^3}{3!}\underset{j=1}{\overset{n}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}\underset{s=1}{\overset{r}{\Σ}}{g}_{ks}\frac{\∂}{\∂x_k}\left(g_{js}\frac{\∂g_{is}}{\∂x_j}\right)\\ {\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}K\[{\mathrm{\ξ}}_s,{\mathrm{\ξ}}_s^{\mathrm{\τ}},{\mathrm{\ξ}}_s^{\mathrm{\σ}}\]d\mathrm{\τ}d\mathrm{\σ}-\frac{{\mathrm{\ϵ}}^4}{4!}\underset{j=1}{\overset{n}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}\underset{l=1}{\overset{n}{\Σ}}\underset{s=1}{\overset{r}{\Σ}}\\ g_{ls}\frac{\∂}{\∂x_l}\[g_{ks}\frac{\∂}{\∂x_k}\left(g_{js}\frac{\∂g_{is}}{\∂x_j}\right)\]{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}\\ \×K\[{\mathrm{\ξ}}_s,{\mathrm{\ξ}}_s^{\mathrm{\τ}},{\mathrm{\ξ}}_s^{\mathrm{\σ}},{\mathrm{\ξ}}_s^{\mathrm{\γ}}\]d\mathrm{\τ}d\mathrm{\σ}d\mathrm{\γ}\}/{\mathrm{\ϵ}}^2-{\stackrel{\‾}{f}}_i\end{array}$