数字 PID 控制器

\begin{array}{r} u (t) = K_{p} [e (t) + \frac{1}{T_{i}} \int_{0}^{t} e (τ) d τ + T_{d} \frac{d e (t)}{d t}] \Leftrightarrow D (s) = \frac{U (s)}{E (s)} = K_{p} (1 + \frac{1}{T_{i} s} + T_{d} s) \end{array}

\begin{array}{r} {\begin{cases} u (t) \approx u (k) \\ e (t) \approx e (k) \\ \int_{0}^{t} e (τ) d τ \approx \sum_{j = 0}^{k} e (j) T \\ \frac{d e (t)}{d t} \approx \frac{e (k) - e (k - 1)}{T} \end{cases} \end{array}

\begin{array}{r} u (k) = K_{p} e (k) + K_{i} \sum_{j = 0}^{k} e (j) + K_{d} [e (k) - e (k - 1)] \end{array}

稳态精度高
直观易于理解
计算量大
积分饱和

\begin{aligned} Δ u & = u (k) - u (k - 1) \\ = K_{p} [e (k) - e (k - 1)] + K_{i} e (k) + K_{d} [e (k) - 2 e (k - 1) + e (k - 2)] \\ = (K_{p} + K_{i} + K_{d}) e (k) - (K_{p} + 2 K_{d}) e (k - 1) + K_{d} e (k - 2) \end{aligned}

无积分环节积累，可能存在一定误差

两种 PID 形式的优缺点

一般常用模拟控制器离散化#二、差分变换法，用后向差分，或双线性差分
并写为控制器的形式

在系统误差较大时，取消积分作用；当误差减小到一定值时，再重新积分。 $e_{0}$ 为积分分离阈值
$| e (k) | \leq e_{0}$ 采用 PID 控制，保证稳态误差为 0
$| e (k) | > e_{0}$ 采用 PD 控制，大幅度减小超调量

\begin{array}{r} u (k) = K_{p} [e (k) + \frac{T}{T_{i}} \sum_{i = 0}^{k} e (i) + \frac{T_{d}}{T} (e (k) - e (k - 1))] \end{array}

\sum_{i = 0}^{k} e (i) = {\begin{cases} \sum_{i = 0}^{k} e (i) | e (k) | \leq e_{0} \\ \sum_{i = 0}^{k - 1} e (i) | e (k) | > e_{0} \end{cases}

输出限幅，输出超限时不积分

\sum_{i = 0}^{k} e (i) = {\begin{cases} \sum_{i = 0}^{k} e (i) u_{m i n} \leq u (k) \leq u_{m a x} \\ \sum_{i = 0}^{k - 1} e (i) u_{m i n} > u (k) or u (k) > u_{m a x} \end{cases}

\sum_{i = 0}^{k} e (i) = {\begin{cases} \sum_{i = 0}^{k} e (i) {\begin{cases} | u (k) | \leq u_{0} \\ u (k) > u_{0} and e (k) < 0 \\ u (k) < u_{0} and e (k) > 0 \end{cases} \\ \sum_{i = 0}^{k - 1} e (i) {\begin{cases} u (k) > u_{0} and e (k) > 0 \\ u (k) < u_{0} and e (k) < 0 \end{cases} \end{cases}

纯微分环节对噪声很敏感

可以串接一个惯性环节来抑制高频影响

\begin{array}{r} G_{f} = \frac{1}{T_{f} s + 1} = \frac{U (s)}{U_{1} (s)} \Rightarrow T_{f} \frac{d u (t)}{d t} + u (t) = u_{1} (t) \Rightarrow T_{f} \frac{u (k) - u (k - 1)}{T} + u (k) = u_{1} (k) \end{array}

\begin{array}{r} u (k) = \frac{T_{f}}{T_{f} + T} u (k - 1) + \frac{T}{T_{f} + T} u_{1} (k) \end{array}

输出量微分：只对输出量进行微分，避免因给定值变化给控制系统带来超调量过大、调节阀动作剧烈的冲击
偏差量微分：对给定值和输出量均微分

控制器的正反作用实现：
反作用： $e = r - y$
正作用： $e = y - r$

e (k) = {\begin{cases} e (k) & | e (k) | > e_{0} \\ 0 & | e (k) | \leq e_{0} \end{cases}