[200722] PID without a PhD

Learn and practice on “PID without a PhD” by Tim Wescott[3]

Math

time domain

$$u(t)=K_p[e(t)+\frac{1}{T_i}\int e(t)dt+T_d\ast d\frac{e(t)}{dt}]$$

transfer function

$$G(s)=K_p(1+\frac{T_s/T_i}{s}+\frac{T_d}{T_s}\ast s)$$

This can be simplified as:
$$G(s)=K_p+\frac{K_i}{s}+K_d\ast s$$
wherein,
$$K_i=K_p\ast \frac{T_s}{T_i}$$
$$K_d=K_p\ast \frac{T_d}{T_s}$$
$T_s$ is the sampling period.

discretization

absolute class

$$u(k)=K_p\ast e(k)+K_i\ast \sum _{i=0}^{k}e(i)+K_d\ast [e(k)-e(k-1)]$$
wherein,
$$K_i=K_p\ast \frac{T_s}{T_i}$$
$$K_d=K_p\ast \frac{T_d}{T_s}$$
$T_s$ is the sampling period.

位置式PID是一种非递推式算法，可直接控制执行机构（如平衡小车），u(k)的值和执行机构的实际位置（如小车当前角度）是一一对应的，因此在执行机构不带积分部件的对象中可以很好应用等优点，但每次输出均与过去的状态有关，计算时要对e(k)进行累加，运算工作量大等缺点。[1]

incremental class

根据以上位置式PID算法公式推算增量式PID算法，可以求得如下形式：[1]

$$u(k)=u(k-1)+\Delta u(k)$$
wherein,
$$\Delta u(k)=K_p\ast [e(k)-e(k-1)]+K_i\ast e(k)+K_d[e(k)-2e(k-1)+e(k-2)]$$

增量式PID算法是一种递推式算法，误动作时影响小，必要时可用逻辑判断的方法去掉出错数据，手动/自动切换时冲击小，便于实现无扰动切换，当计算机故障时，仍能保持原值，算式中不需要累加，控制增量Δu(k)的确定仅与最近3次的采样值有关等优点，但积分截断效应大，有稳态误差，溢出的影响大，有的被控对象用增量式则不太好等缺点。

Implementation code for PI regulator

absolute class

typedef struct
{
double dState; // Last position input
double iState; // Integrator state
double iMax, iMin;
// Maximum and minimum allowable integrator state
double iGain, // integral gain
pGain, // proportional gain
dGain; // derivative gain
} SPid;
double UpdatePID(SPid * pid, double error, double
position)
{
double pTerm,
dTerm, iTerm;
pTerm = pid->pGain * error;
// calculate the proportional term
// calculate the integral state with appropriate limiting
pid->iState += error;
pid->iState = MAX(pid->iState, pid->iMin);
pid->iState = MIN(pid->iState, pid->iMax);

iTerm = pid->iGain * iState; // calculate the integral term

return pTerm + iTerm;
}

incremental class

To be added.

reference

[1] https://www.cnblogs.com/AChenWeiqiangA/p/12718473.html
[2] https://blog.csdn.net/kilotwo/article/details/79829669
[3] PID without a PhD