Steady-state Analysis of Discrete Systems
一、稳态误差计算
误差传递函数：误差信号与输入信号之比 为开环传递函数， 为误差传递函数：
\varPhi_{k}(z)=D(z)G(z)\quad \varPhi_{e}(z)= \dfrac{E(z)}{R(z)} = \dfrac{1}{1+\varPhi_{k}(z)}\; {\color{red}\Rightarrow} \; E(z)=\varPhi_{e}(z)R(z)=\dfrac{R(z)}{1+\varPhi_{k}(z)} \end{align}$$ 根据[[z 变换#4. 极限定理\|终值定理]]，稳态误差表示为： $$\begin{align} e(\infty)= \lim\limits_{ z \to 1 } (z-1) R(z) \varPhi_{e}(z) \end{align}$$ ### 二、误差系数计算 位置误差系数： $$\begin{align} e(\infty)= \lim\limits_{ z \to 1 } (z-1) \dfrac{1}{1+\varPhi_{k}(z)} \dfrac{1}{1-z^{-1}}= \dfrac{1}{1+\varPhi_{k}(z)} = \dfrac{1}{1+K_{p}}\; {\color{red}\Rightarrow} \; K_{p}= \varPhi_{k}(1) \end{align}$$ 速度误差系数： $$\begin{align} e(\infty)= \lim\limits_{ z \to 1 } (z-1) \dfrac{1}{1+\varPhi_{k}(z)} \dfrac{Tz}{(z-1)^{2}}= \dfrac{T}{\varPhi_{k}(z)(z-1)} =\dfrac{1}{K_{v}} \; {\color{red}\Rightarrow} \; K_{v}= \dfrac{(z-1)\varPhi_{k}(z)}{T} \end{align}$$ 加速度误差系数： $$\begin{align} e(\infty)= \lim\limits_{ z \to 1 } (z-1) \dfrac{1}{1+\varPhi_{k}(z)} \dfrac{T^{2}z(z+1)}{2(z-1)^{3}}= \dfrac{1}{K_{a}}\; {\color{red}\Rightarrow} \; K_{a}= \dfrac{(z-1)^{2}\varPhi_{k}(z)}{T^{2}} \end{align}$$