Compensators & PID formulas

Applies to unity feedback with a stable closed loop, for inputs $Au(t)$, $Atu(t)$ and $\frac{A}{2}{t}^{2}u(t)$. A type-$N$ loop ($N$ poles at the origin) has zero error for inputs of lower order than $t^N$, a finite error for $t^N$, and an infinite error for higher orders.

Exact for the unity-feedback loop $\frac{{\omega }_n^2}{s(s+2\zeta {\omega }_n)}$, whose closed loop is a pure second-order system. The straight-line rule $\text{PM}\approx 100\zeta$ degrees stays within about $3.5^{\circ }$ of the exact value for $0<\zeta \le 0.6$ and drifts off above that ($\zeta =0.7$ gives $65.2^{\circ }$, not $70^{\circ }$). For a higher-order loop it is only an estimate, valid when one complex pole pair dominates. Lead and lag design from the Bode plot start by turning a $\zeta$ or overshoot specification into a required PM this way.

Has one pole at the origin and two zeros, the roots of $K_d{s}^2+K_{p}s+K_i=0$. The zeros are real when $K_p^2\ge 4K_i{K}_d$ (equivalently $T_i\ge 4T_d$) and a complex pair otherwise. With $K_i\ne 0$ it raises the system type by 1.

Assumes $K_p>0$ and $T_i>0$. Pole at $s=0$ and zero at $s=-1/T_i=-K_i/K_p$. Phase $=-{\tan }^{-1}\frac{1}{\omega T_i}$, which lies between $-90^{\circ }$ and $0^{\circ }$. Raises the type by 1.

Assumes $K_p>0$ and $T_d>0$. Zero at $s=-1/T_d=-K_p/K_d$ and no pole. Phase $=+{\tan }^{-1}(\omega T_d)$, which lies between $0^{\circ }$ and $+90^{\circ }$. The system type does not change.

The closed-loop characteristic equation becomes $s^2+(2\zeta {\omega }_n+{\omega }_n^2{T}_d)s+{\omega }_n^2=0$. So ${\omega }_n$ and $K_v={\omega }_n/(2\zeta )$ are unchanged and the damping increases. The closed loop also gains a zero at $s=-1/T_d$. For an underdamped ${\zeta }^{\prime }$ this zero makes the response rise faster and overshoot more than a pure second-order system with the same ${\zeta }^{\prime }$, and the effect is stronger the closer the zero lies to the origin compared with ${\zeta }^{\prime }{\omega }_n$.

The inner loop reduces to $\frac{{\omega }_n^2}{s(s+2\zeta {\omega }_n+K_t{\omega }_n^2)}$, so the characteristic equation is the same as with PD and $T_d=K_t$, and there is no closed-loop zero. But $K_v$ falls below ${\omega }_n/(2\zeta )$, so the ramp error grows. PD in the forward path keeps $K_v$ unchanged.

Ideal op-amp, $\frac{V_o}{V_i}=-\frac{Z_2}{Z_1}$ with $Z_1$ the input impedance and $Z_2$ the feedback impedance. PI: $K_p=R_2/R_1$ and $T_i=R_2{C}_2$. PD: $K_p=R_2/R_1$ and $T_d=R_1{C}_1$. A second inverting stage of gain $-R_4/R_3$ removes the minus sign.

Corner frequencies are $\frac{1}{aT}$ (zero) and $\frac{1}{T}$ (pole). The gain rises from 1 (0 dB) at low frequency to $a$ ($20{\log }_{10}a$ dB) at high frequency, which is a high-pass shape.

Valid for the time-constant form with $a>1$. ${\varphi }_m$ depends only on the ratio $a$ of the two corner frequencies, not on $T$. For any finite $a$, ${\varphi }_m<90^{\circ }$.

The geometric mean of the two corner frequencies, which is their midpoint on a log scale.

Holds for the unity-DC-gain form. In lead design, ${\omega }_m$ is placed at the new gain crossover, where the uncompensated magnitude is $-10{\log }_{10}a$ dB.

$\alpha =1/a$. The pole-zero form has DC gain $z/p<1$ and high-frequency gain 1. Its phase is identical to the time-constant form with the same corners.

Series arm: $R_1$ in parallel with $C$. Shunt arm: $R_2$ across the output. Assumes an ideal source and an unloaded output. $a=\frac{R_1+R_2}{R_2}$, $aT=R_{1}C$, $T=\frac{R_1{R}_{2}C}{R_1+R_2}$. The network attenuates DC by $1/a$, so an amplifier of gain $a$ is needed to keep the error constant.

High-frequency gain is $1/b$, an attenuation of $20{\log }_{10}b$ dB above the zero corner.

Same algebra as the lead, with the sign reversed. ${\omega }_m$ is the geometric mean of the corners $\frac{1}{bT}$ and $\frac{1}{T}$.

Holds at the same loop gain $K$, for a stable compensated loop. DC gain is $z/p$ and high-frequency gain is 1. When $z$ and $p$ both sit close to the origin (a dipole), the angle they add at the dominant poles is small. The transient response then barely changes while the error constant is multiplied by $z/p$.

Series arm: $R_1$. Shunt arm: $R_2$ in series with $C$, with the output taken across this shunt arm. Assumes an unloaded output. DC gain is 1, so no amplifier is needed to hold the low-frequency gain.

Ideal op-amps. The first inverting stage gives $-\frac{R_2}{R_1}\cdot \frac{1+R_1{C}_{1}s}{1+R_2{C}_{2}s}$ and the second stage cancels the sign. Zero at $-1/(R_1{C}_1)$, pole at $-1/(R_2{C}_2)$: a lead if $R_1{C}_1>R_2{C}_2$ and a lag if $R_1{C}_1<R_2{C}_2$. Unlike the passive networks, the DC gain $\frac{R_2{R}_4}{R_1{R}_3}$ can be set independently of the pole-zero ratio.

The first factor is the lag section and the second is the lead section. Both lag corners lie well below both lead corners.

Realised by one RC network: series arm $R_1$ in parallel with $C_1$, shunt arm $R_2$ in series with $C_2$, output across the shunt arm, unloaded. Then $T_1=R_1{C}_1$ and $T_2=R_2{C}_2$, and the denominator $1+(R_1{C}_1+R_2{C}_2+R_1{C}_2)s+R_1{C}_1{R}_2{C}_2{s}^2$ factors as $\left(1+\frac{T_1}{\beta }s\right)(1+\beta T_{2}s)$, so the lead and lag sections are forced to share the same ratio $\beta$. The phase is a lag below ${\omega }_0$ and a lead above it. DC and high-frequency gains are both 1, with a dip in between.

Both PID rows set $T_i=4T_d$, which is exactly the condition $K_p^2=4K_i{K}_d$ for a double zero. So both give a pole at the origin and a double zero on the negative real axis, at $-1/L$ and at $-4/P_u$ respectively. The leading coefficient is $K_d$, which makes a quick check. For a process gain $K\ne 1$, divide the step-response result by $K$.