| Continuous |
\(k\dfrac{s+Z}{s+p}\) |
\(K_{P}+\dfrac{K_{I}}{s}\) |
\(K_{P}+\dfrac{K_{I}}{s}+K_{D}s\) |
| Discrete |
\(k\dfrac{b_{0}+b_{1}z^{-1}}{a_{0}+a_{1}z^{-1}}\) |
\(\dfrac{b_{0}+b_{1}z^{-1}}{a_{0}+a_{1}z^{-1}}\) |
\(\dfrac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}}{a_{0}+a_{1}z^{-1}+a_{2}z^{-2}}\) |
| Differential equation |
\(\dfrac{dy}{dt}+py=k\left(\dfrac{dx}{dt}+Zx\right)\) |
\(y=K_{P}x+K_{I}\displaystyle\int_{0}^{t}x\,dt\) |
\(y=K_{P}x+K_{I}\displaystyle\int_{0}^{t}x\,dt+K_{D}\dfrac{dx}{dt}\) |
| Difference equation |
\(\begin{aligned}y(n)&=-\dfrac{a_{1}}{a_{0}}y(n-1)\\&+\dfrac{b_{0}}{a_{0}}x(n)\\&+\dfrac{b_{1}}{a_{0}}x(n-1)\end{aligned}\) |
\(\begin{aligned}y(n)&=-\dfrac{a_{1}}{a_{0}}y(n-1)\\&+\dfrac{b_{0}}{a_{0}}x(n)\\&+\dfrac{b_{1}}{a_{0}}x(n-1)\end{aligned}\) |
\(\begin{aligned}y(n)&=-\dfrac{a_{1}}{a_{0}}y(n-1)\\&-\dfrac{a_{2}}{a_{0}}y(n-2)\\&+\dfrac{b_{0}}{a_{0}}x(n)\\&+\dfrac{b_{1}}{a_{0}}x(n-1)\\&+\dfrac{b_{2}}{a_{0}}x(n-2)\end{aligned}\) |
| \(a_0\) |
\(1\) |
\(1\) |
\(1\) |
| \(a_1\) |
\((pT-2)/(pT+2)\) |
\(-1\) |
\(0\) |
| \(a_2\) |
|
|
\(-1\) |
| \(b_0\) |
\(k(ZT+2)/(pT+2)\) |
\(K_P+K_I T/2\) |
\(K_P+K_I T/2+2K_D/T\) |
| \(b_1\) |
\(k(ZT-2)/(pT+2)\) |
\(-K_P+K_I T/2\) |
\(K_I T-4K_D/T\) |
| \(b_2\) |
|
|
\(-K_P+K_I T/2+2K_D/T\) |