Defining the state of a movement from the state of $t_0$ to the time of $t$ is a transfer process. Then the mathematical form describing such a process is written as: $$X(t)=\phi(t-t_0)X(t_0)$$ $\phi$ is called the state transition matrix.
the product term$C\phi(t-t_0)X(t_0)$ is called as zero input response,the convolution term$\lmoustache_{t_0}^tC\phi(t-\tau)BU(\tau)d\tau$ is called as zero state response,the remaining item $DU(t)$ is called as Input and Output Feed Forward。
$\phi(t-t_0)=e^{A(t-t_0)}X_0=I+A(t-t_0)+\frac{A^2}{2!}(t-t_0)^2+\cdots$
if,$U(\tau)=\delta(\tau)$(unit pulse function),and $X(t_0)=0$,then $Y(t)=h(t)$(unit impulse response),$h(t)=L^{-1}(C(sI-A)^{-1}B+D)$
expm(A*t)
syms x t
f1='sin(x)^2+cos(x)^3-3';
A=sym('[f1 0; 0 0]');
expm(A*t)
syms x t
f1='sin(x)^2+cos(x)^3-3'
A = sym('A',[2 2])
det(A*t)
expm(A*t);
f1 = 'sin(x)^2+cos(x)^3-3' A = [ A1_1, A1_2] [ A2_1, A2_2] ans = A1_1*A2_2*t^2 - A1_2*A2_1*t^2
For a state differential equation:$\dot X(t)=A(t)X(t)+B(t)U(t)$,it exists status response:$$X(t)=\Phi(t,t_0)X(t_0)+\lmoustache_{t_0}^t\Phi(t,\tau)BU(\tau)d\tau$$
the product term $\Phi(t,t_0)X(t_0)$ is called as zero input response,the convolution term $\lmoustache_{t_0}^t\Phi(t,\tau)BU(\tau)d\tau$ is called as zero state response
Obviously, the output response:$$Y(t)=C(t)X(t)+D(t)U(t)=C(t)\Phi(t,t_0)X(t_0)+C(t)\lmoustache_{t_0}^t\Phi(t,\tau)BU(\tau)d\tau+D(t)U(t)$$
$\Phi(t,t_0)=e^{\lmoustache_{t_0}^tA(\tau)d\tau}$
Therefore, the linear continuous system state transition equation can be uniformly represented as:$$X(t)=\Phi X(t_0)+\lmoustache_{t_0}^t\Phi BU(\tau)d\tau$$ where,
Then there is,$\begin{cases} G(kT)=\phi((k+1)T,kT) \\ H(T)=\lmoustache_{kT}^{(k+1)T}\Phi((k+1)T,\tau)B(\tau)d\tau \end{cases}$ preserve first-order precision ('zoh'), there is:$\begin{cases} G(T)=e^{\lmoustache_{kT}^{(k+1)T}A(t)dt} \\ H(T)=\lmoustache_{kT}^{(k+1)T}e^{\lmoustache_{kT}^{(k+1)T}A(t)dt}B(\tau)d\tau \end{cases}$
sys_dis=c2d(sys,Ts,'method') %①'zoh' :Zero-order hold (default),②'foh' :modified first-order hold
sys_con=d2c(sys_dis,'method')
A=[1 2;3 5];B=[2;2];C=[1 0];
sys=ss(A,B,C,[]);
sys_dis=c2d(sys,1,'zoh');
sys_con=d2c(sys_dis);
G=sys_dis.a
A=sys_con.a
G = 87.8953 149.7836 224.6754 387.4625 A = 1.0000 2.0000 3.0000 5.0000
impulse(sys)