线性系统理论课程笔记
对应教材内容第二章 线性系统的状态空间描述的内容
状态和状态空间
状态是一个列向量，由各个状态变量构成。 \[X(t) = \left[ {\begin{array}{*{20}{c}} { {x_1}(t)} \\ \vdots \\ { {x_n}(t)} \end{array}} \right]t \geqslant {t_0}\] 状态空间是状态向量的集合，状态空间的维数等于状态的维数。
线性系统的状态空间描述
图1 动力学系统结构示意图
输入变量组：\({u_1},{u_2}, \cdots
,{u_p}\)（环境对系统的作用）
输出变量组：\({y_1},{y_2}, \cdots
,{y_q}\)（系统对环境的作用）
状态变量组：\({x_1},{x_2}, \cdots
,{x_n}\)（能完全表征其时间域行为的一个最小内部变量组，刻画系统在每个时刻所处状况的变量，体现了系统的行为）
系统的外部描述：又称作输入—输出描述。外部描述是不完全的描述，不表征系统的内部结构和内部变量，只反映外部变量间的因果关系，即输出和输入间的因果关系。
线性定常、单输入—单输出系统，外部描述为线性常系数微分方程。 \[{y^{(n)}} + {a_{n - 1}}{y^{(n - 1)}} + \cdots +
{a_1}{y^{(1)}} + {a_0}y = {b_{n - 1}}{u^{(n - 1)}} + {b_{n - 2}}{u^{(n -
2)}} + \cdots + {b_1}{u^{(1)}} + {b_0}u\]
假定初始条件为零，取拉氏变换。得到其复频率域描述，即传递函数。 \[G(s) = \frac{ { {b_{n - 1}}{s^{n - 1}}
+ \cdots + {b_1}s + {b_0}}}{ { {s^n} + {a_{n - 1}}{s^{n - 1}}
+ \cdots + {a_1}s + {a_0}}}\]
系统的内部描述：这里指状态空间描述。内部描述是完全的描述，能够完全反映系统的所有动力学特性。内部描述需要由两个数学方程表征（状态方程和输出方程，统称为状态空间表达式）。
状态方程：微分方程或差分方程（状态变量组和输入变量组间的因果关系）。
一般的情况下，为一阶非线性时变微分方程组。 \[\left\{ {\begin{array}{*{20}{c}} { { {\dot x}_1}
= {f_1}({x_1}, \cdots ,{x_n};{u_1}, \cdots ,{u_p},t)} \\ \cdots \\ { {
{\dot x}_n} = {f_n}({x_1}, \cdots ,{x_n};{u_1}, \cdots ,{u_p},t)}
\end{array}} \right.t \geqslant {t_0}\] 向量方程形式为\(\dot X = f(x,u,t),t \geqslant {t_0}\)
输出方程：代数方程（状态变量组、输入变量组和输出变量组间的转换关系）。
一般的情况下，输出方程为 \[\left\{
{\begin{array}{*{20}{c}}{ {y_1} = {g_1}({x_1}, \cdots ,{x_n};{u_1},
\cdots ,{u_p},t)} \\ \cdots \\ { {y_q} = {g_q}({x_1}, \cdots
,{x_n};{u_1}, \cdots ,{u_p},t)}
\end{array}} \right.t \geqslant {t_0}\] 向量方程形式为\(Y = g(x,u,t),t \geqslant {t_0}\)
图2 连续时间线性系统结构图
图3 离散时间线性系统结构图
图4
RLC电路网络结构图
选取电感电流\(i_L\)与电容电压\(u_C\)作为状态变量，列写回路方程： \[\left\{ {\begin{array}{*{20}{c}}
{i_L=(u-L\frac{di_L}{dt})\frac{1}{R_1}}+C\frac{du_C}{dt} \\
{L\frac{di_L}{dt}+u_C+C\frac{du_C}{dt}R_2=u}
\end{array}}\right.\] 整理得 \[\left\{
{\begin{array}{*{20}{c}}
{\frac{di_L}{dt}=\frac{u}{L}-\frac{i_L}{L}(\frac{R_1R_2}{R_1+R_2})-\frac{u_C}{L}(\frac{R_1}{R_1+R_2})}
\\ {\frac{du_C}{dt}=\frac{R_1}{C(R_1+R_2)}i_L-\frac{1}{C(R_1+R_2)}u_C}
\end{array}}\right.\] 取状态变量\(x_1=i_L\)、\(x_2=u_C\)，进一步整理得到状态方程 \[\left\{ {\begin{array}{*{20}{c}} {\dot
x_1=-\frac{1}{L}(\frac{R_1R_2}{R_1+R_2})x_1-\frac{1}{L}(\frac{R_1}{R_1+R_2})x_2+\frac{1}{L}u}
\\ {\dot x_2=\frac{R_1}{C(R_1+R_2)}x_1-\frac{1}{C(R_1+R_2)}x_2}
\end{array}}\right.\] 该状态方程可简化为\(\dot x = A(t)x + B(t)u\)，其中\(A = \left[ {\begin{array}{*{20}{c}}
-\frac{1}{L}(\frac{R_1R_2}{R_1+R_2})&-\frac{1}{L}(\frac{R_1}{R_1+R_2})
\\ \frac{R_1}{C(R_1+R_2)}&-\frac{1}{C(R_1+R_2)} \end{array}}
\right]\)，\(B=\left[
{\begin{array}{*{20}{c}} \frac{1}{L} \\ 0 \end{array}}
\right]\)
选取\(u_{R2}\)作为输出变量\(y\)，则有： \[u_{R2}=R_2 i_C=R_2 C\frac{dU_c}{dt}=\frac{R_1
R_2}{R_1+R_2}i_L-\frac{R_2}{R_1+R_2}u_C\] \[y=\frac{R_1
R_2}{R_1+R_2}x_1-\frac{R_2}{R_1+R_2}x_2\] 该输出方程可简化为\(y = C(t)x + D(t)u\)，其中\(C=\left[ {\begin{array}{*{20}{c}} \frac{R_1
R_2}{R_1+R_2} & -\frac{R_2}{R_1+R_2} \end{array}}
\right]\)，\(D=[0]\)
系统按其状态空间描述的分类
- 线性系统和非线性系统
- 时变系统和时不变系统
- 连续时间系统和离散时间系统
- 确定性系统和不确定性系统
化输入—输出描述为状态空间描述
结论1——由输入输出描述导出状态空间描述
给定单输入，单输出线性时不变系统的输入输出描述： \[{y^{(n)}} + {a_{n - 1}}{y^{(n - 1)}} + \cdots +
{a_1}{y^{(1)}} + {a_0}y = {b_m}{u^{(m)}} + {b_{m - 1}}{u^{(m - 1)}}
+ \cdots + {b_1}{u^{(1)}} + {b_0}u\] \[G(s) = \frac{Y(s)}{U(s)} = \frac{ { {b_m}{s^m} +
{b_{m - 1}}{s^{m - 1}} + \cdots + {b_1}{s^1} + {b_0}}}{ { {s^n} +
{a_{n - 1}}{s^{n - 1}} + \cdots + {a_1}s + {a_0}}}\] \(m < n\)时，有 \[\begin{gathered}\dot x = \left[
{\begin{array}{*{20}{c}}0&1&0& \cdots &0 \\ \vdots
&{}& \ddots &{}& \vdots \\ 0&{}&{}& \ddots
&0 \\ 0&0&0& \cdots &1 \\ { - {a_0}}&{ -
{a_1}}&{ - {a_2}}& \cdots &{ - {a_{n - 1}}} \end{array}}
\right]x + \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ \vdots \\ 0 \\ 1
\end{array}} \right]u \hfill \\ \hfill \\ y = \left[
{\begin{array}{*{20}{c}}{b_0}&{b_1}& \cdots
&{b_m}&0& \cdots &0 \end{array}} \right]x \hfill \\
\end{gathered}\] 【例1】：\({y^{(3)}} +
16{y^{(2)}} + 194{y^{(1)}} + 640y = 160{u^{(1)}} + 720u\) \[\begin{gathered}\dot x = \left[
{\begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \\ { -
640}&{ - 194}&{ - 16} \end{array}} \right] x + \left[
{\begin{array}{*{20}{c}} 0 \\ 0 \\ 1 \end{array}} \right]u \hfill \\
\hfill \\ y = \left[ {\begin{array}{*{20}{c}} {720}&{160}&0
\end{array}} \right] x \hfill \\ \end{gathered}\] \(m = n\)时，有 \[\begin{gathered}\dot x = \left[
{\begin{array}{*{20}{c}}0&1&0& \cdots &0 \\ \vdots
&{}& \ddots &{}& \vdots \\ 0&{}&{}& \ddots
&0 \\ 0&0&0& \cdots &1 \\ { - {a_0}}&{ -
{a_1}}&{ - {a_2}}& \cdots &{ - {a_{n - 1}}} \end{array}}
\right]x + \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ \vdots \\ 0 \\ 1
\end{array}} \right]u \hfill \\ \hfill \\ y = \left[
{\begin{array}{*{20}{c}} {({b_0} - {b_n}{a_0}),}&{({b_1} -
{b_n}{a_1}),}&{ \cdots ,}&{({b_{n - 1}} - {b_n}{a_{n- 1}})}
\end{array}} \right]x + {b_n}u \hfill \\ \end{gathered}\]
【例2】：\({y^{(3)}} + 16{y^{(2)}} +
194{y^{(1)}} + 640y = 4{u^{(3)}} + 160{u^{(1)}} + 720u\)
\(b_0 - b_3 a_0 = 720 - 4 \times 640 =
-1840\)
\(b_1 - b_3 a_1 = 160 - 4 \times 194 =
-616\)
\(b_0 - b_3 a_0 = 0 - 4 \times 16 =
-64\) \[\begin{gathered} \dot x =
\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \\ { -
640}&{ - 194}&{ - 16} \end{array}} \right]x + \left[
{\begin{array}{*{20}{c}} 0 \\ 0 \\ 1 \end{array}} \right]u \hfill \\
\hfill \\ y = \left[
{\begin{array}{*{20}{c}}{-1840}&{-616}&{-64} \end{array}}
\right]x + 4u \hfill \\ \end{gathered}\]
结论2——由输入输出描述导出状态空间描述
给定单输入，单输出线性时不变系统的输入输出描述：
\(m = 0\)时，输入输出描述为： \[{y^{(n)}} + {a_{n - 1}}{y^{(n - 1)}} + \cdots +
{a_1}{y^{(1)}} + {a_0}y = {b_0}u\] \[G(s) = \frac{b_0}{ { {s^n} + {a_{n - 1}}{s^{n -
1}} + \cdots + {a_1}s + {a_0}}}\] 其对应的状态空间描述为：
\[\begin{gathered}\dot x = \left[
{\begin{array}{*{20}{c}} 0&1&0& \cdots &0 \\ \vdots
&{}& \ddots &{}& \vdots \\ 0&{}&{}& \ddots
&0 \\ 0&0&0& \cdots &1 \\ { - {a_0}}&{ -
{a_1}}&{ - {a_2}}& \cdots &{ - {a_{n - 1}}} \end{array}}
\right]x + \left[ {\begin{array}{*{20}{c}}0 \\ 0 \\ \vdots \\ 0 \\ {b_0}
\end{array}} \right]u \hfill \\ y = \left[
{\begin{array}{*{20}{c}}{1,}&{0,}&{ \cdots ,}&0 \end{array}}
\right]x \hfill \\ \end{gathered}\] \(m
\ne 0\)时，输入输出描述为： \[{y^{(n)}} + {a_{n - 1}}{y^{(n - 1)}} + \cdots +
{a_1}{y^{(1)}} + {a_0}y = {b_n}{u^{(n)}} + {b_{n - 1}}{u^{(n - 1)}}
+ \cdots + {b_1}{u^{(1)}} + {b_0}u\] \[G(s) = \frac{ { {b_n}{s^n} + {b_{n - 1}}{s^{n -
1}} + \cdots + {b_1}{s^1} + {b_0}}}{ { {s^n} + {a_{n - 1}}{s^{n - 1}}
+ \cdots + {a_1}s + {a_0}}}\] 其对应的状态空间描述为： \[\begin{gathered}\dot x = \left[
{\begin{array}{*{20}{c}}0&1&0& \cdots &0 \\ \vdots
&{}& \ddots &{}& \vdots \\ 0&{}&{}& \ddots
&0 \\ 0&0&0& \cdots &1 \\ { - {a_0}}&{ -
{a_1}}&{ - {a_2}}& \cdots &{ - {a_{n - 1}}}\end{array}}
\right]x + \left[ {\begin{array}{*{20}{c}}{\beta _1} \\ {\beta _2} \\
\vdots \\ {\beta _{n - 1}} \\ {\beta _n} \end{array}} \right]u \hfill
\\ \hfill \\ \begin{array}{*{20}{c}} {}&{} \end{array}y = \left[
{\begin{array}{*{20}{c}}{1,}&{0,}&{ \cdots ,}&0 \end{array}}
\right]x + b_n u \hfill \\ \end{gathered}\] 其中\({\beta _0} = {b_n}\)
\({\beta _1} = {b_{n - 1}} - {a_{n - 1}}{\beta
_0}\)
\({\beta _2} = {b_{n - 2}} - {a_{n - 1}}{\beta
_1} - {a_{n - 2}}{\beta _0}\)
\(\vdots\)
\({\beta _n} = {b_0} - {a_{n - 1}}{\beta _{n -
1}} - {a_{n - 2}}{\beta _{n - 2}} - \cdots - {a_1}{\beta _1} -
{a_0}{\beta _0}\)
【例3】：\({y^{(3)}} + 16{y^{(2)}} +
194{y^{(1)}} + 640y = 160{u^{(1)}} + 720u\)
\({\beta _0} = {b_3} = 0\)
\({\beta _1} = {b_2} - {a_2}{\beta _0} = 0-16
\times 0=0\)
\({\beta _2} = {b_1} - {a_2}{\beta _1} -
{a_1}{\beta _0}=160-16 \times 0-194 \times 0=160\)
\({\beta _3} = {b_0} - {a_2}{\beta _2} -
{a_1}{\beta _1} - {a_0}{\beta _0}=720-16 \times 160-194 \times 0-640
\times 0=-1840\)
\[\begin{gathered}\dot x = \left[
{\begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \\ { -
640}&{ - 194}&{ - 16} \end{array}} \right] x + \left[
{\begin{array}{*{20}{c}} 0 \\ 160 \\ -1840 \end{array}} \right]u \hfill
\\ \hfill \\ y = \left[ {\begin{array}{*{20}{c}} 1&0&0
\end{array}} \right] x \hfill \\ \end{gathered}\]
注：与例1相比，相同的输入输出描述得到的状态空间描述不同，说明同一个输入输出描述可以对应多个状态空间描述。
结论3——由传递函数描述导出状态空间描述
给定单输入、单输出线性时不变系统的传递函数描述为： \[G(s) = \frac{ { {b_m}{s^m} + {b_{m - 1}}{s^{m -
1}} + \cdots + {b_1}{s^1} + {b_0}}}{ { {s^n} + {a_{n - 1}}{s^{n - 1}}
+ \cdots + {a_1}s + {a_0}}}\] 当分母方程的根\({\lambda_1},{\lambda_2},\cdots,{\lambda_n}\)两两相异
\(m < n\)时，有
\(\begin{gathered}G(s) = \frac{k_1}{s -
{\lambda_1}} + \frac{k_2}{s - {\lambda_2}} + \cdots + \frac{k_n}{s -
{\lambda_n}} \hfill \\{k_i} = \mathop {\lim }\limits_{s \to \infty }
G(s)(s - {\lambda_i}),\begin{array}{*{20}{c}}{}&{} \end{array}i =
1,2, \cdots ,n \hfill \\ \end{gathered}\)
对应的状态空间描述为： \[\begin{gathered}\dot
x = \left[ {\begin{array}{*{20}{c}}{\lambda _1}&{}&{}&{} \\
{}&{\lambda _2}&{}&{} \\ {}&{}& \ddots &{} \\
{}&{}&{}&{\lambda _n} \end{array}} \right]x + \left[
{\begin{array}{*{20}{c}}{k_1} \\ {k_2} \\ \vdots \\ {k_n} \end{array}}
\right]u \hfill \\y = \left[ {\begin{array}{*{20}{c}}{1,}&{1,}&{
\cdots ,}&1 \end{array}} \right]x \hfill \\ \end{gathered}\]
\(m = n\)时，有
\(\begin{gathered}G(s) = \frac{ { {b_n}{s^n} +
{b_{n - 1}}{s^{n - 1}} + \cdots + {b_1}s + {b_0}}}{ { {s^n} + {a_{n -
1}}{s^{n - 1}} + \cdots + {a_1}s + {a_0}}} = {b_n} + \bar G(s) \hfill \\
\bar G(s) = \frac{ {\left( { {b_{n - 1}} - {b_n}{a_{n - 1}}}
\right){s^{n - 1}} + \cdots + \left( { {b_0} - {b_n}{a_0}} \right)}}{ {
{s^n} + {a_{n - 1}}{s^{n - 1}} + \cdots + {a_1}s + {a_0}}} \hfill \\{
{\bar k}_i} = \mathop {\lim }\limits_{s \to {\lambda _i}} \bar G(s)(s -
{\lambda_i}),\begin{array}{*{20}{c}}{}&{} \end{array}i = 1,2, \cdots
,n \hfill \\ \end{gathered}\)
对应的状态空间描述为： \[\begin{gathered}\dot
x = \left[ {\begin{array}{*{20}{c}}{\lambda_1}&{}&{}&{} \\
{}&{\lambda_2}&{}&{} \\ {}&{}& \ddots &{} \\
{}&{}&{}&{\lambda_n} \end{array}} \right]x + \left[
{\begin{array}{*{20}{c}}{ {\bar k}_1} \\ { {\bar k}_2} \\ \vdots \\ {
{\bar k}_n} \end{array}} \right]u \hfill \\y = \left[
{\begin{array}{*{20}{c}}{1,}&{1,}&{\cdots ,}&1 \end{array}}
\right]x + {b_n}u \hfill \\\end{gathered}\] 【例4】：\(G(s)=\frac{7s^2+2s+1}{s^3+6s^2+11s+6}\)
\(D(s)=s^3+6s^2+11s+6=(s+1)(s+2)(s+3)\)
\(k_1=\mathop {\lim }\limits_{s \to -1 }
\frac{7s^2+2s+1}{(s+2)(s+3)}\)，同理\(k_2=-25\)、\(k_3=29\)。 对应的状态空间描述为： \[\begin{gathered}\dot x = \left[
{\begin{array}{*{20}{c}} -1&0&0 \\ 0&-2&0 \\
0&0&-3 \end{array}} \right] x + \left[ {\begin{array}{*{20}{c}}
3 \\ -25 \\ 29 \end{array}} \right]u \hfill \\ \hfill \\ y = \left[
{\begin{array}{*{20}{c}} 1&1&1 \end{array}} \right] x \hfill \\
\end{gathered}\]
由方块图描述导出状态空间描述
【例5】
图5
系统方块图
图6
等效方块图
可以得到变量间的关系：
\(\left\{ \begin{gathered}x_1 =
\frac{5}{s+4}(u-x_3) \hfill \\x_2 = \frac{2}{s+1}(u-x_3) \hfill \\x_3 =
\frac{1}{s+2}(x_1+x_2) \hfill \\y = x_1 + x_2 \hfill \\ \end{gathered}
\right. \Rightarrow \left\{ \begin{gathered}{ {\dot x}_1} = - 4{x_1} +
5(u - {x_3}) \hfill \\{ {\dot x}_2} = - {x_2} + 2(u - {x_3}) \hfill \\{
{\dot x}_3} = - 2{x_3} + y \hfill \\y = {x_1} + {x_2} \hfill \\
\end{gathered} \right.\)
得到状态空间描述为： \[\begin{gathered}{\dot
x} = \left[ {\begin{array}{*{20}{c}}{ - 4}&0&{ - 5} \\ 0&{ -
1}&{ - 2} \\ 1&1&{ - 2} \end{array}} \right]x + \left[
{\begin{array}{*{20}{c}}5 \\ 2 \\ 0 \end{array}} \right]u \hfill \\ y =
\left[ {\begin{array}{*{20}{c}} 1&1&0 \end{array}} \right]x
\hfill \\ \end{gathered}\] 注：这里有\(sx_1={\dot x_1}\)、\(sx_2={\dot x_2}\)、\(sx_3={\dot x_3}\)
线性时不变系统的特征结构
对于连续线性时不变系统\(\dot x = Ax +
Bu\)，其特征矩阵定义为\((sI-A)\)。
特征多项式\(\alpha (s) = \det (sI - A) = {s^n}
+ {\alpha _{n - 1}}{s^{n - 1}} + \cdots + {\alpha _1}s + {\alpha
_0}\)
Caley-Hamilton定理：\(\alpha (A) = {A^n} +
{\alpha _{n - 1}}{A^{n - 1}} + \cdots + {\alpha _1}A + {\alpha _0}I =
0\)
矩阵的迹：\(trH=(h_{11}+h_{22}+\cdots+h_{nn})=\sum\limits_{i=1}^n{h_{ii}}\)
基于迹计算的特征多项式的迭代算法： \[\begin{array}{*{20}{c}}
{ {R_{n - 1}} = I}&{ {\alpha _{n - 1}} = - \frac{ {tr{R_{n -
1}}A}}{1}} \\
{ {R_{n - 2}} = {R_{n - 1}}A+{\alpha _{n - 1}}I}&{ {\alpha _{n -
2}} = - \frac{ {tr{R_{n - 2}}A}}{2}} \\
{ {R_{n - 3}} = {R_{n - 2}}A+{\alpha _{n - 2}}I}&{ {\alpha _{n -
3}} = - \frac{ {tr{R_{n - 3}}A}}{3}} \\
{\vdots}&{\vdots} \\
{ {R_1} = {R_2}A+{\alpha _2}I}&{ {\alpha _1} = - \frac{
{tr{R_1}A}}{n-1}} \\
{ {R_0} = {R_1}A+{\alpha _1}I}&{ {\alpha _0} = - \frac{
{tr{R_0}A}}{n}}
\end{array}\]
特征值：特征方程\(\alpha (s) = \det (sI -
A) = {s^n} + {\alpha_{n - 1}}{s^{n - 1}} + \cdots + {\alpha_1}s +
{\alpha_0} = 0\)的根。
特征向量：设\(\lambda_i\)为系统矩阵A的特征值，则右特征向量为满足\({\lambda_i}{v_i} = A{v_i}\)的\(n\times 1\)非零向量\(v_i\)，则左特征向量为满足\(\bar v_i^T{\lambda_i} = \bar
v_i^TA\)的\(1\times
n\)非零向量\(\bar
v_i^T\)。
【例6】\(A=\left[ {\begin{array}{*{20}{c}}
-2&0&1&1 \\ 1&-1&1&2 \\ 1&2&-1&2 \\
1&1&1&2 \end{array}} \right]\) \[\begin{array}{*{20}{c}}
{R_3=I=\left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\
0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1
\end{array}} \right]}&{R_3 A=R_3=\left[ {\begin{array}{*{20}{c}}
-2&0&1&1 \\ 1&-1&1&2 \\ 1&2&-1&2 \\
1&1&1&2 \end{array}} \right]}&{\alpha_3=-\frac{trR_3
A}{1}}=2 \\
{R_2=R_3 A+\alpha_3 I=\left[ {\begin{array}{*{20}{c}}
0&0&1&1 \\ 1&1&1&2 \\ 1&2&1&1 \\
1&1&1&4 \end{array}} \right]}&{R_2 A=\left[
{\begin{array}{*{20}{c}} 2&3&0&3 \\ 2&3&3&8 \\
2&1&3&8 \\ 4&5&5&12 \end{array}}
\right]}&{\alpha_2=-\frac{trR_2 A}{2}=-10} \\
{R_1=R_2 A+\alpha_2 I=\left[ {\begin{array}{*{20}{c}}
-8&3&0&3 \\ 2&-7&3&8 \\ 2&1&-7&8 \\
4&5&5&2 \end{array}} \right]}&{R_1 A=\left[
{\begin{array}{*{20}{c}} 22&0&-2&4 \\ 0&21&0&7
\\ -2&-7&18&13 \\ 4&7&6&23 \end{array}}
\right]}&{\alpha_1=-\frac{trR_1 A}{3}=-28} \\
{R_0=R_1 A+\alpha_1 I=\left[ {\begin{array}{*{20}{c}}
-6&0&-2&4 \\ 0&-7&0&7 \\
-2&-7&-10&13 \\ 4&7&6&-5 \end{array}}
\right]}&{R_0 A=\left[ {\begin{array}{*{20}{c}} 14&0&0&0
\\ 0&14&0&0 \\ 0&0&14&0 \\ 0&0&0&14
\end{array}} \right]}&{\alpha_0=-\frac{trR_0 A}{4}=-14}
\end{array}\] \(\alpha (s)=
s^4+2s^3-10s^2-28s-14\)
状态方程的对角线规范形和约当规范形
对角线规范性
对于连续线性时不变系统\(\dot x = Ax +
Bu\)，其特征值\({\lambda_1},{\lambda_2},\cdots,{\lambda_n}\)两两相异，构造变换阵\(P=[v_1,v_2,\cdots,v_n]\)，则状态方程可通过线性非奇异变换\(\bar X = {P^{ - 1}}X\)化为对角线规范性。
\[\dot {\bar X} = \left[
{\begin{array}{*{20}{c}} {\lambda _1}&{}&{}&{} \\
{}&{\lambda _2}&{}&{} \\ {}&{}& \ddots &{} \\
{}&{}&{}&{\lambda _n} \end{array}} \right]\bar X + \bar
Bu\] 其中\(\bar X = {P^{ -
1}}X\)，\(\bar B = {P^{ -
1}}B\)。
【例7】\(\dot X = \left[
{\begin{array}{*{20}{c}}2&{-1}&{-1} \\ 0&{-1}&0 \\
0&2&1 \end{array}} \right]X + \left[ {\begin{array}{*{20}{c}} 7
\\ 2 \\ 3 \end{array}} \right]u\)
由\(|\lambda I-A|=0\)⇒\((\lambda-2)(\lambda+1)(\lambda-1)=0\)，特征值为\(\lambda_1 = 2,\lambda _2 = -1,\lambda _3 =
1\)。
由等式\(\lambda_i
v_i=Av_i\)，解方程组\(|\lambda_i
I-A|v_i=0\)，得到特征向量
\[{\begin{array}{*{20}{c}} {v_1=\left[
{\begin{array}{*{20}{c}} 1\\0\\0 \end{array}}\right]}&{v_2=\left[
{\begin{array}{*{20}{c}} 1\\0\\1 \end{array}}\right]}&{v_3=\left[
{\begin{array}{*{20}{c}} 0\\1\\-1 \end{array}}\right]}
\end{array}}\] 得到\(P =[v_1,v_2,v_3] =
\left[ {\begin{array}{*{20}{c}}1&1&0 \\ 0&0&1 \\
0&1&-1\end{array}} \right]\)，求其逆矩阵\(P^{-1} = \left[
{\begin{array}{*{20}{c}}1&-1&-1 \\ 0&1&1 \\
0&1&0 \end{array}} \right]\)
\(\bar A = P^{-1}AP = \left[
{\begin{array}{*{20}{c}}2&0&0 \\ 0&-1&0 \\ 0&0&1
\end{array}} \right]\)，\(\bar
B=P^{-1}B = \left[ {\begin{array}{*{20}{c}}2\\5\\2\end{array}}
\right]\)
\(\dot{\bar X} = \left[
{\begin{array}{*{20}{c}}2&0&0 \\ 0&-1&0 \\ 0&0&1
\end{array}} \right]\bar X + \left[ {\begin{array}{*{20}{c}}2\\5\\2
\end{array}} \right]u\)
约当规范形
系统的特征值为并非两两相异时，状态方程可变换为约当规范形。
设系统特征值\({\lambda_1},{\lambda_2},\cdots,{\lambda_l}\)分别为\({\sigma_1},{\sigma_2},\cdots,{\sigma_l}\)重特征值，则状态方程可通过变换\(\bar X = {Q^{ - 1}}X\)（\(Q\)为可逆矩阵）化为约当规范型： \[\dot {\bar X}=\left[
{\begin{array}{*{20}{c}}{J_1}&{}&{} \\ {}& \ddots &{} \\
{}&{}&{J_l} \end{array}} \right]\bar X + \bar Bu\]
其中\(\bar X = {Q^{ - 1}}X\)，\(\bar B = {Q^{ - 1}}B\)，\(J_1,\cdots,J_l\)为对应特征值\({\lambda_1},\cdots,{\lambda_l}\)的约当块，其形式为：
\[{J_{i}} = \left[
{\begin{array}{*{20}{c}}{\lambda_i}&1&{}&{} \\ {}&
\ddots & \ddots &{} \\ {}&{}& \ddots &1 \\
{}&{}&{}& \end{array}} \right]\]
约当块的行数（列数）与所对应特征值的重数相等。在约当规范形中，每一个状态变量的方程和下一序号的状态变量构成耦合。
【例8】求\(A=\left[{\begin{array}{*{20}{c}}0&1&0\\0&0&1\\2&3&0\end{array}}\right]\)
的约当阵。
由\(|\lambda I-A|=0\)⇒\(\lambda^3-3\lambda-2=0\)，特征值为\(\lambda_1 = \lambda_2 = -1,\lambda_3 =
2\)。
对于\(\lambda_1=-1\)，有\(\lambda_1 Q_1 - AQ_1 = 0\)，解得\(Q_1=\left[{\begin{array}{*{20}{c}}1\\-1\\1\end{array}}\right]\)
对于\(\lambda_2=-1\)，有\(\lambda_2 Q_2 - AQ_2 =
-Q_1\)，解得\(Q_2=\left[{\begin{array}{*{20}{c}}1\\0\\1\end{array}}\right]\)
对于\(\lambda_3=2\)，有\(\lambda_3 Q_3 - AQ_3 = 0\)，解得\(Q_3=\left[{\begin{array}{*{20}{c}}1\\2\\4\end{array}}\right]\)
得到\(Q =[Q_1,Q_2,Q_3] = \left[
{\begin{array}{*{20}{c}}1&1&1 \\ -1&0&2 \\
1&1&4\end{array}} \right]\)，求其逆矩阵\(P^{-1} = \frac{1}{3}\left[
{\begin{array}{*{20}{c}}-2&-3&2 \\ 6&3&-3 \\
-1&0&1 \end{array}} \right]\)
得到约当阵\(\bar A = P^{-1}AP = \left[
{\begin{array}{*{20}{c}}-1&1&0 \\ 0&-1&0 \\
0&0&2 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
J_1&0 \\ 0&J_2 \end{array}} \right]\)。
其中\(J_1=\left[ {\begin{array}{*{20}{c}}
-1&1 \\ 0&-1 \end{array}} \right]\)，\(J_2=[2]\)。
由状态空间描述导出传递函数矩阵
对于连续时间线性时不变系统\(\dot x = Ax +
Bu\)，\(y = Cx + Du\)
其传递函数矩阵\(G(s) = C{(sI - A)^{ - 1}}B +
D\)
【例9】 \[\begin{gathered}\dot x = \left[
{\begin{array}{*{20}{c}} -1&0 \\ 0&1 \end{array}} \right] x +
\left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right]u \hfill \\
\hfill \\ y = \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}}
\right] x \hfill \\ \end{gathered}\] \(sI-A=\left[ {\begin{array}{*{20}{c}} s+1&0 \\
0&s-1 \end{array}} \right]\)
\((sI-A)^{-1}=\left[ {\begin{array}{*{20}{c}}
\frac{1}{s+1}&0 \\ 0&\frac{1}{s-1} \end{array}}
\right]\)
\(G(s)=\frac{Y(s)}{R(s)}=C{(sI-A)^{-1}}B+D=\left[
{\begin{array}{*{20}{c}} 1&0 \end{array}} \right] \left[
{\begin{array}{*{20}{c}} \frac{1}{s+1}&0 \\ 0&\frac{1}{s-1}
\end{array}} \right] \left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}}
\right]=\frac{1}{s+1}\)
对于多输入多输出系统，计算特征多项式\(\alpha (s) \equiv \det (sI - A) = {s^n} + {\alpha
_{n - 1}}{s^{n - 1}} + \cdots + {\alpha _1}s + {\alpha
_0}\)
\[\left\{ {\begin{array}{*{20}{l}}
{E_{n - 1} = CB} \\
{E_{n - 2} = CAB + {\alpha _{n - 1}}CB} \\
{ \cdots \cdots } \\
{E_1} = C{A^{n - 2}B + {\alpha _{n - 1}}C{A^{n - 3}}B + \cdots +
{\alpha _2}CB} \\
{E_0} = C{A^{n - 1}B + {\alpha _{n - 1}}C{A^{n - 2}}B + \cdots +
{\alpha _1}CB}
\end{array}} \right.\] 则有\(G(s) =
\frac{1}{\alpha (s)}[{E_{n - 1}}{s^{n - 1}} + {E_{n - 2}}{s^{n - 2}} +
\cdots + {E_1}s + {E_0}] + D\)
【例10】 \[\begin{gathered}\dot X = \left[
{\begin{array}{*{20}{c}}2&0&0 \\ 0&2&0 \\ 0&3&1
\end{array}} \right]X + \left[ {\begin{array}{*{20}{c}}1&2 \\
1&0 \\ 3&1 \end{array}} \right]u \hfill \\y = \left[
{\begin{array}{*{20}{c}}1&1&2 \end{array}} \right]X \hfill \\
\end{gathered}\] 特征多项式\(\alpha (s)
= \det (sI - A) = {(s - 2)^2}(s - 1) = {s^3} - 5{s^2} + 8s -
4\)
\(\alpha_2=-5,\alpha_1=8\)
\({E_2} = CB = \left[ {\begin{array}{*{20}{c}}
1&1&2 \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
1&2 \\ 1&0 \\ 3&1 \end{array}} \right] = \left[
{\begin{array}{*{20}{c}}8&4 \end{array}} \right]\)
\({E_1} = CAB + {\alpha _2}CB = \left[
{\begin{array}{*{20}{c}}1&1&2 \end{array}} \right]\left[
{\begin{array}{*{20}{c}}2&0&0 \\ 0&2&0 \\ 0&3&1
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1&2 \\ 1&0 \\
3&1 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{ -
40}&{ - 20} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{
- 24}&{ - 14} \end{array}} \right]\)
\({E_0} = C{A^2}B + {\alpha _2}CAB + {\alpha
_1}CB = \left[ {\begin{array}{*{20}{c}}1&1&2 \end{array}}
\right]\left[ {\begin{array}{*{20}{c}}2&0&0 \\ 0&2&0 \\
0&3&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}}2&4
\\ 2&0 \\ 6&1 \end{array}} \right]\)
\(+\left[ {\begin{array}{*{20}{c}}{ -
80}&{ - 30} \end{array}} \right] + \left[
{\begin{array}{*{20}{c}}{64}&{32} \end{array}} \right] = \left[
{\begin{array}{*{20}{c}}{16}&{12} \end{array}}
\right]\)
\(G(s) = \frac{1}{\alpha (s)}[{E_2}{s^2} +
{E_1}s + {E_0}]=\left[ {\begin{array}{*{20}{c}}{\frac{8{s^2} - 24s +
16}{s^3 - 5s^2 + 8s-4}}&{\frac{4{s^2} - 14s + 12}{s^3 - 5s^2 + 8s -
4}} \end{array}}\right]\)
线性系统的坐标变化
注：自学内容，仅供参考。
设状态在基底\(\{e_1,e_2,\cdots,e_n\}\)上的表征为\(X=[x_1,x_2,\cdots,x_n]^T\)
而在另一个基底\(\{ {\bar e_1},{\bar
e_2},\cdots,{\bar e_n}\}\)上的表征为\(X=[{\bar x_1},{\bar x_2},\cdots,{\bar
x_n}]^T\)
\(n\)维状态空间中有且仅有\(n\)个线性无关的向量，而\(\{e_1,e_2,\cdots,e_n\}\)必是线性无关的，因此\(\{e_1,e_2,\cdots,e_n\}\)可表示为\(\{ {\bar e_1},{\bar e_2},\cdots,{\bar
e_n}\}\)的线性组合，且表示法唯一。
结论1：给定线性定常系统的状态空间描述为\(\dot X = AX + Bu\)，\(y = CX + Du\)
引入变换\({\bar X}=PX\)(\(detP \ne 0\))得到\(\dot {\bar X} = {\bar A}{\bar X} + {\bar
B}u\)，\(y = {\bar C}{\bar X} + {\bar
D}u\)
则有\(\bar A = PA{P^{-1}},\bar B = PB,\bar C =
C{P^{-1}},\bar D = D\)
结论2：变换前后特征值不变。即\({\lambda
_i}(A)={\lambda _i}(\bar A),i=1,2,\cdots,n\)
两个状态空间描述存在变换关系，称为代数等价。
同一系统采用不同的状态变量组，所导出的两个状态空间描述，必然是代数等价的。
两个代数等价的状态空间描述，可化为相同的对角线规范形或约当规范形。
结论：线性定常系统的传递函数矩阵在坐标变换下保持不变。
物理含义：当系统的输入和输出变量确定后，不管如何选取状态变量组，系统的输出—输入特性将总是一样的。
组合系统的状态空间描述
子系统并联
图7
子系统并联
子系统并联的条件：\(dim(u_1)=dim(u_2),dim(y_1)=dim(y_2)\)
其中dim表示向量的维数。
并联后：\(u_1=u_2=u,y_1+y_2=y\)
\(\dot x_1=A_1 x_1+B_1 u_1\)
\(\dot x_2=A_2 x_2+B_2 u_2\)
\(y=y_1+y_2=C_1 x_1+C_2 x_2+D_1 u_1+D_2
u_2\) \[\left[
{\begin{array}{*{20}{c}}\dot x_1 \\ \dot x_2 \end{array}} \right] =
\left[ {\begin{array}{*{20}{c}} A_1&0 \\ 0&A_2 \end{array}}
\right]\left[ {\begin{array}{*{20}{c}} x_1 \\ x_2 \end{array}} \right] +
\left[ {\begin{array}{*{20}{c}} B_1 \\ B_2 \end{array}}
\right]u\] \[y = \left[
{\begin{array}{*{20}{c}}C_1&C_2 \end{array}} \right]\left[
{\begin{array}{*{20}{c}}x_1 \\ x_2 \end{array}} \right] +
(D_1+D_2)u\]
子系统串联
图8
子系统串联
子系统串联的条件：\(dim(y_1)=dim(u_2)\)
串联后：\(u = u_1, u_2 = y_1, y_2 =
u\)
\(\dot x_1=A_1 x_1+B_1 u\)①
\(\dot x_2=A_2 x_2+B_2 u_2\)②
\(y_1=C_1 x_1+D_1 u\)③
\(y_1=u_2\)④
\(y_2=C_2 x_2+D_2 u_2\)⑤
\(y_2=y\)⑥
由②③④知\(\dot x_2 = A_2 x_2 + B_2 C_1 x_1 +
B_2 D_1 u\)
由③④⑤⑥知\(y = C_2 x_2 + D_2 C_1 x_1 + D_2 D_1
u\) \[\left[
{\begin{array}{*{20}{c}}\dot x_1 \\ \dot x_2 \end{array}} \right] =
\left[ {\begin{array}{*{20}{c}}A_1&0 \\ B_2 C_1&A_2 \end{array}}
\right]\left[ {\begin{array}{*{20}{c}}x_1 \\ x_2 \end{array}} \right] +
\left[ {\begin{array}{*{20}{c}}B_1 \\ B_2 D_1 \end{array}}
\right]u\] \[y = \left[
{\begin{array}{*{20}{c}}D_2 C_1&C_2 \end{array}} \right]\left[
{\begin{array}{*{20}{c}}x_1 \\ x_2 \end{array}} \right] + (D_1
D_2)u\]
子系统的反馈
图9
子系统的反馈
子系统的反馈的条件：\(dim(u_1)=dim(y_2),dim(u_2)=dim(y_1)\)
这里取\(D_i=0\)的情况
反馈后：\(u_1=u-y_2,y_1=y=u_2\)①
\(\dot x_1=A_1 x_1+B_1 u_1\)②
\(y_1=C_1 x_1\)③
\(\dot x_2=A_2 x_2+B_2 u_2\)④
\(y_2=C_2 x_2\)⑤
由①②⑤知\(\dot x_1 = A_1 x_1 + B_1 u - B_1 C_2
x_2\)
由①③④知\(\dot x _2 = A_2 x_2 + B_2 C_1
x_1\) \[\left[
{\begin{array}{*{20}{c}}\dot x_1 \\ \dot x_2 \end{array}} \right] =
\left[ {\begin{array}{*{20}{c}}A_1& -B_1 C_2 \\ B_2 C_1&A_2
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}x_1 \\ x_2
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}B_1 \\ 0
\end{array}} \right]u\] \[y = \left[
{\begin{array}{*{20}{c}}C_1&0 \end{array}} \right]\left[
{\begin{array}{*{20}{c}}x_1 \\ x_2 \end{array}} \right]\]