极点配置与系统镇定

控制系统的性能主要取决于系统极点在根平面上的分布，给定一组极点或根据时域指标转换为一组等价的期望极点，作为综合系统性能指标的一种形式。

Description

极点实际上只取决于系统矩阵，而经过反馈，可以改变系统矩阵
核心就是设计反馈矩阵，使得反馈后的特征多项式等于期望极点系统对应的多项式。

采用状态反馈对系统任意配置极点的充分必要条件为系统完全能控

首先要进行能控性判别
将系统进行线性变换 $T_{c_{1}}$ 转化为能控标准型
设出线性变换后的状态反馈增益矩阵 $\bar{K} = T_{c_{1}} K = (\begin{matrix} {\bar{k}}_{0} {\bar{k}}_{1} \dots {\bar{k}}_{n - 1} \end{matrix})$

\begin{array}{r} u = K x + v \dot{x} = A x + B (v + K x) = (A + B K) x + B v \end{array}

原系统：

\begin{array}{r} det [λ I - A] = f (λ) = \prod_{i = 1}^{n} (λ - λ_{i}) = λ^{n} + a_{n - 1} λ^{n - 1} + \dots + a_{1} λ + a_{0} \end{array}

期望极点系统：

\begin{array}{r} f^{*} (λ) = \prod_{i = 1}^{n} (λ - λ_{i}^{*}) = λ^{n} + a_{n - 1}^{*} λ^{n - 1} + \dots + a_{1}^{*} λ + a_{0}^{*} \end{array}

设计线性变换后的反馈矩阵 $\bar{K}$ 系统：

\begin{array}{r} det [λ I - (A + B K)] = \prod_{i = 1}^{n} (λ - λ_{i}^{*}) = λ^{n} + (a_{n - 1} - {\bar{k}}_{n - 1}) λ^{n - 1} + \dots + (a_{1} - {\bar{k}}_{1}) λ + (a_{0} - {\bar{k}}_{0}) \end{array}

\begin{array}{r} T_{c_{1}} = (\begin{array}{c} A^{n - 1} b & A^{n - 2} b & \dots & A b & b \end{array}) (\begin{array}{c} 1 \\ a_{n - 1} & 1 \\ ⋮ & ⋮ & ⋱ \\ a_{2} & a_{3} & \dots & 1 \\ a_{1} & a_{2} & \dots & a_{n - 1} & 1 \end{array}) \end{array}

\begin{aligned} \bar{A} & = T_{c_{1}}^{- 1} A T_{c_{1}} = (\begin{array}{c} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ - a_{0} & - a_{1} & - a_{2} & \dots & - a_{n - 1} \end{array}) \\ \bar{b} & = T_{c_{1}}^{- 1} b = (\begin{array}{c} 0 \\ 0 \\ ⋮ \\ 1 \end{array}) \bar{c} = c T_{c_{1}} = (\begin{array}{c} β_{0}, β_{1}, \dots, β_{n - 1} \end{array}) \end{aligned}

\begin{array}{r} \bar{A} + \bar{b} \bar{K} = (\begin{array}{c} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ - a_{0} + {\bar{k}}_{0} & - a_{1} + {\bar{k}}_{1} & - a_{2} + {\bar{k}}_{2} & \dots & - a_{n - 1} + {\bar{k}}_{n - 1} \end{array}) \end{array}

\begin{array}{r} \bar{K} = (\begin{array}{c} a_{0} - a_{0}^{*} a_{1} - a_{1}^{*} \dots a_{n - 1} - a_{n - 1}^{*} \end{array}) K = \bar{K} T_{c_{1}}^{- 1} \end{array}

注意

如果维度较低，可以直接利用设计后系统的特征多项式和期望极点系统的特征多项式进行比较

采用输出到状态变量导数的反馈对系统任意配置极点的充分必要条件为系统完全能观

\begin{array}{r} \dot{x} = A x + B u + G y = A x + B u + G C x \dot{x} = (A + G C) x + B u \end{array}

原系统：

\begin{array}{r} det [λ I - A] = f (λ) = \prod_{i = 1}^{n} (λ - λ_{i}) = λ^{n} + a_{n - 1} λ^{n - 1} + \dots + a_{1} λ + a_{0} \end{array}

期望极点系统：

\begin{array}{r} det [λ I - (A + G C)] = f^{*} (λ) = \prod_{i = 1}^{n} (λ - λ_{i}^{*}) = λ^{n} + a_{n - 1}^{*} λ^{n - 1} + \dots + a_{1}^{*} λ + a_{0}^{*} \end{array}

设计线性变换后的反馈矩阵 $\bar{G}$ 系统：

\begin{array}{r} det [λ I - (A + G C)] = f^{*} (λ) = \prod_{i = 1}^{n} (λ - λ_{i}^{*}) = λ^{n} + (a_{n - 1} - {\bar{g}}_{n - 1}) λ^{n - 1} + \dots + (a_{1} - {\bar{g}}_{1}) λ + (a_{0} - {\bar{g}}_{0}) \end{array}

\begin{aligned} \tilde{A} & = T_{o_{2}}^{- 1} A T_{o_{2}} = (\begin{array}{c} 0 & 0 & 0 & \dots & 0 & - a_{0} \\ 1 & 0 & 0 & \dots & 0 & - a_{1} \\ 0 & 1 & 0 & \dots & 0 & - a_{2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 1 & - a_{n - 1} \end{array}) \\ \tilde{b} & = T_{o_{2}}^{- 1} b = (\begin{array}{c} β_{0} \\ β_{1} \\ ⋮ \\ β_{n - 1} \end{array}) \tilde{c} = c T_{o_{2}} = (\begin{array}{c} 0, 0, \dots, 1 \end{array}) \end{aligned}

\begin{aligned} \tilde{A} + \bar{G} \bar{C} & = (\begin{array}{c} 0 & 0 & 0 & \dots & 0 & - a_{0} + {\bar{g}}_{0} \\ 1 & 0 & 0 & \dots & 0 & - a_{1} + {\bar{g}}_{1} \\ 0 & 1 & 0 & \dots & 0 & - a_{2} + {\bar{g}}_{2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 1 & - a_{n - 1} + {\bar{g}}_{n - 1} \end{array}) \end{aligned}

\begin{array}{r} \bar{G} = (\begin{array}{c} a_{0} - a_{0}^{*} a_{1} - a_{1}^{*} \dots a_{n - 1} - a_{n - 1}^{*} \end{array}) G = T_{o_{2}} \bar{G} \end{array}

对于完全能控的单输入单输出系统，

系统镇定：对受控系统通过反馈使得极点均具有负实部，保证系统渐近稳定。（实际上是极点配置的特殊情况）

已知传递函数如下，设计状态反馈使得系统极点为 $- 2, - 1 \pm j$

\begin{array}{r} W (s) = \frac{10}{s (s + 1) (s + 2)} \end{array}

原系统特征多项式： $f (λ) = λ^{3} + 3 λ^{2} + 2 λ$
直接写为能控标准型：

\begin{array}{r} \dot{x} = (\begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & - 2 & - 3 \end{array}) x + (\begin{array}{c} 0 \\ 0 \\ 1 \end{array}) u y = (\begin{array}{c} 10 & 0 & 0 \end{array}) u \end{array}

设状态反馈矩阵 $K = (\begin{matrix} k_{0} k_{1} k_{2} \end{matrix})$

引入状态反馈后特征多项式：

\begin{array}{r} f (λ) = | λ I - (A + b K) | = λ^{3} + (3 - k_{2}) λ^{2} + (2 - k_{1}) λ + (- k_{0}) \end{array}

期望特征多项式：

\begin{array}{r} f^{*} (λ) = (λ + 2) (λ + 1 - j) (λ + 1 + j) = λ^{3} + 4 λ^{2} + 6 λ + 4 \end{array}

对比系数得到： $K = (\begin{matrix} - 4 - 4 - 1 \end{matrix})$