最近在做一些 LQR 的研究，發現涉及到了大量的二次型求導以及矩陣求導問題，故簡單整理如下。

因為是 LQR 的問題，故從一個哈密頓 Hamilton 函數開始吧。

有哈密頓函數
$$H=\frac{1}{2}{x}^{T}Qx+\frac{1}{2}{u}^{T}Ru+{\lambda }^{T}Ax+{\lambda }^{T}Bu$$

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求駐值點時會用到

$$\frac{?}{?u}\frac{1}{2}{u}^{T}Ru=Ru$$

$$\begin{array}{rl}& \frac{?}{?u}\frac{1}{2}\left[\begin{array}{cc}{u}_1& u_2\end{array}\right]\left[\begin{array}{cc}{r}_{11}& r_{12}\\ r_{12}& r_{22}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]\\ =& \frac{?}{?u}\frac{1}{2}\left[\begin{array}{cc}{u}_1{r}_{11}+u_2{r}_{12}& u_1{r}_{12}+u_2{r}_{22}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]\\ =& \frac{?}{?u}\frac{1}{2}\left[\begin{array}{c}{u}_1^2{r}_{11}+u_1{u}_2{r}_{12}+u_1{u}_2{r}_{12}+u_2^2{r}_{22}\end{array}\right]\\ =& \left[\begin{array}{c}{u}_1{r}_{11}+u_2{r}_{12}\\ u_1{r}_{12}+u_2{r}_{22}\end{array}\right]\\ =& \left[\begin{array}{cc}{r}_{11}& r_{12}\\ r_{12}& r_{22}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]\\ =& Ru\end{array}$$

$$\frac{?}{?u}{\lambda }^{T}Bu=B^T\lambda$$

$$\begin{array}{rl}& \frac{?}{?u}\left[\begin{array}{cc}{\lambda }_1& {\lambda }_2\end{array}\right]\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]u\\ =& \frac{?}{?u}({\lambda }_1{b}_1+{\lambda }_2{b}_2)u\\ =& {\lambda }_1{b}_1+{\lambda }_2{b}_2\\ =& B^T\lambda \end{array}$$

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求正則方程時有

$$\frac{?}{?\lambda }{\lambda }^{T}Ax=Ax$$

$$\frac{?}{?\lambda }{\lambda }^{T}Bu=Bu$$

$$\begin{array}{rl}& \frac{?}{?\lambda }\left[\begin{array}{cc}{\lambda }_1& {\lambda }_2\end{array}\right]\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]u\\ =& \frac{?}{?\lambda }({\lambda }_1{b}_{1}u+{\lambda }_2{b}_{2}u)\\ =& \left[\begin{array}{c}{b}_{1}u\\ b_{2}u\end{array}\right]\\ =& Bu\end{array}$$

$$\frac{?}{?x}\frac{1}{2}{x}^{T}Qx=Qx$$

$$\frac{?}{?x}{\lambda }^{T}Ax=A^T\lambda$$

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總結

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