(1) SISO Modeling

$$y(k+2)+5y(k+1)+6y(k)=u(k+2)+2u(k+1)+u(k)$$

num_1=[1 2 1];
den_1=[1 5 6];
sys_1=tf(num_1,den_1,-1)

(2) MIMO Modeling

$$G(s)=\left[\begin{array}{cc}\frac{s^2+2s+1}{s^2+5s+6}& \frac{s+5}{s+2}\\ \frac{2s+3}{s^3+6s^2+11s+6}& \frac{6}{2s+7}\end{array}\right]$$

num={[1 2 1] [1 5]; [2 3] [6]};
den={[1 5 6] [1 2]; [1 6 11 6] [2 7]};
sys_1=tf(num,den) 

(3) 状态空间模型

$$\begin{array}{l}\dot{x}=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]x+\left[\begin{array}{c}0\\ 1\end{array}\right]u\\ y=\left[\begin{array}{cc}1& 0\end{array}\right]x\end{array}$$

A=[0 1; -2 -3];
B=[0; 1];  C=[1 0];  
D=0;
sys=ss(A,B,C,D)

(4) 状态空间模型->传递函数

$$\begin{array}{l}\dot{x}=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]x+\left[\begin{array}{c}0\\ 1\end{array}\right]u\\ y=\left[\begin{array}{cc}1& 0\end{array}\right]x\end{array}$$

sys_tf=tf(sys)

(5) 传递函数->状态空间模型

• 转换函数ss（）

sys_1_ss=ss(sys_1)

• 规范形转换函数canon()

sys_can_ss=canon(sys_1,'modal')

• 常微分方程(传递函数)转换为状态空间模型函数dif2ss()
  $$y^{{}^{\prime \prime \prime }}+5y^{{}^{\prime \prime }}+8y^{{}^{\prime }}+4y=2u^{{}^{\prime \prime }}+14u^{{}^{\prime }}+24u$$

num_1=[2 14 24];  den_1=[1 5 8 4];
sys_1=tf(num_1,den_1);% 建立传递函数模型
sys_comp=dif2ss(sys_1,'companion')% 求传递函数的友矩阵形状态空间模型
sys_jord=dif2ss(num_1,den_1,'jordan')% 求传递函数的约旦规范形状态空间模型

(6) 状态空间模型变换

$$\{\begin{array}{l}\dot{x}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -6& -11& -6\end{array}\right]x+\left[\begin{array}{c}0\\ 0\\ 6\end{array}\right]u\\ y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]x\end{array}$$
和
$$P=\left[\begin{array}{ccc}1& 1& 1\\ -1& -2& -3\\ 1& 4& 9\end{array}\right]$$

A=[0 1 0; 0 0 1; -6 -11 -6];
B=[0; 0; 6];  C=[1 0 0];  D=0;
P=[1 1 1; -1 -2 -3; 1 4 9];
sys_in=ss(A,B,C,D);
sys_out=ss2ss(sys_in,inv(P)) 

(7) 特征值和特征向量

d = eig(A)%只计算所有特征值
[V,D] = eig(A)%同时得到所有特征向量和特征值

(8) 广义特征向量

J = jordan(A)%只计算A矩阵对应的约旦矩阵J
[V,J] = jordan(A)%所有广义特征向量和约旦矩阵J

(9) 状态空间模型->约旦型

$$\dot{x}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -4& -8& -5\end{array}\right]x+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u$$
$$y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]x$$

A=[0 1 0; 0 0 1; -4 -8 -5];
B=[0; 0; 1];  C=[1 0 0];  D=0;
sys_in=ss(A,B,C,D);
[P,J]=jordan(A);
sys_out=ss2ss(sys_in,inv(P))