The following is a list of the important equations from the text, arranged by subject. For more information about these equations, including the meaning of each variable and symbol, the uses of these functions, or the derivations of these equations, see the relevant pages in the main text.

Fundamental Equations

${\displaystyle e^{j\omega }=\cos(\omega )+j\sin(\omega )}$

${\displaystyle (a*b)(t)=\int _{-\infty }^{\infty }a(\tau )b(t-\tau )d\tau }$

${\displaystyle {\mathcal {L}}[f(t)*g(t)]=F(s)G(s)}$

${\displaystyle {\mathcal {L}}[f(t)g(t)]=F(s)*G(s)}$

${\displaystyle |A-\lambda I|=0}$

${\displaystyle Av=\lambda v}$

${\displaystyle wA=\lambda w}$

${\displaystyle dB=20\log(C)}$

Basic Inputs

${\displaystyle u(t)=\left\{{\begin{matrix}0,&t<0\\1,&t\geq 0\end{matrix}}\right.}$

${\displaystyle r(t)=tu(t)}$

${\displaystyle p(t)={\frac {1}{2}}t^{2}u(t)}$

Error Constants

${\displaystyle K_{p}=\lim _{s\to 0}G(s)}$

${\displaystyle K_{p}=\lim _{z\to 1}G(z)}$

${\displaystyle K_{v}=\lim _{s\to 0}sG(s)}$

${\displaystyle K_{v}=\lim _{z\to 1}(z-1)G(z)}$

${\displaystyle K_{a}=\lim _{s\to 0}s^{2}G(s)}$

${\displaystyle K_{a}=\lim _{z\to 1}(z-1)^{2}G(z)}$

System Descriptions

${\displaystyle y(t)=\int _{-\infty }^{\infty }g(t,r)x(r)dr}$

${\displaystyle y(t)=x(t)*h(t)=\int _{-\infty }^{\infty }x(\tau )h(t-\tau )d\tau }$

${\displaystyle Y(s)=H(s)X(s)}$

${\displaystyle Y(z)=H(z)X(z)}$

${\displaystyle x'(t)=Ax(t)+Bu(t)}$

${\displaystyle y(t)=Cx(t)+Du(t)}$

${\displaystyle C[sI-A]^{-1}B+D=\mathbf {H} (s)}$

${\displaystyle C[zI-A]^{-1}B+D=\mathbf {H} (z)}$

${\displaystyle \mathbf {Y} (s)=\mathbf {H} (s)\mathbf {U} (s)}$

${\displaystyle \mathbf {Y} (z)=\mathbf {H} (z)\mathbf {U} (z)}$

${\displaystyle M={\frac {y_{out}}{y_{in}}}=\sum _{k=1}^{N}{\frac {M_{k}\Delta \ _{k}}{\Delta \ }}}$

Feedback Loops

${\displaystyle H_{cl}(s)={\frac {KGp(s)}{1+KGp(s)Gb(s)}}}$

${\displaystyle H_{ol}(s)=KGp(s)Gb(s)}$

${\displaystyle F(s)=1+H_{ol}}$

Transforms

${\displaystyle F(s)={\mathcal {L}}[f(t)]=\int _{0}^{\infty }f(t)e^{-st}dt}$

${\displaystyle f(t)={\mathcal {L}}^{-1}\left\{F(s)\right\}={1 \over {2\pi }}\int _{c-i\infty }^{c+i\infty }e^{st}F(s)\,ds}$

${\displaystyle F(j\omega )={\mathcal {F}}[f(t)]=\int _{0}^{\infty }f(t)e^{-j\omega t}dt}$

${\displaystyle f(t)={\mathcal {F}}^{-1}\left\{F(j\omega )\right\}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(j\omega )e^{-j\omega t}d\omega }$

${\displaystyle F^{*}(s)={\mathcal {L}}^{*}[f(t)]=\sum _{i=0}^{\infty }f(iT)e^{-siT}}$

${\displaystyle X(z)={\mathcal {Z}}\left\{x[n]\right\}=\sum _{i=-\infty }^{\infty }x[n]z^{-n}}$

${\displaystyle x[n]=Z^{-1}\{X(z)\}={\frac {1}{2\pi j}}\oint _{C}X(z)z^{n-1}dz\ }$

${\displaystyle X(z,m)={\mathcal {Z}}(x[n],m)=\sum _{n=-\infty }^{\infty }x[n+m-1]z^{-n}}$

Transform Theorems

${\displaystyle x(\infty )=\lim _{s\to 0}sX(s)}$

${\displaystyle x[\infty ]=\lim _{z\to 1}(z-1)X(z)}$

${\displaystyle x(0)=\lim _{s\to \infty }sX(s)}$

State-Space Methods

${\displaystyle x(t)=e^{At-t_{0}}x(t_{0})+\int _{t_{0}}^{t}e^{A(t-\tau )}Bu(\tau )d\tau }$

${\displaystyle x[n]=A^{n}x[0]+\sum _{m=0}^{n-1}A^{n-1-m}Bu[n]}$

${\displaystyle y(t)=Ce^{At-t_{0}}x(t_{0})+C\int _{t_{0}}^{t}e^{A(t-\tau )}Bu(\tau )d\tau +Du(t)}$

${\displaystyle y[n]=CA^{n}x[0]+\sum _{m=0}^{n-1}CA^{n-1-m}Bu[n]+Du[n]}$

${\displaystyle x(t)=\phi (t,t_{0})x(t_{0})+\int _{t_{0}}^{t}\phi (\tau ,t_{0})B(\tau )u(\tau )d\tau }$

${\displaystyle x[n]=\phi [n,n_{0}]x[t_{0}]+\sum _{m=n_{0}}^{n}\phi [n,m+1]B[m]u[m]}$

${\displaystyle G(t,\tau )=\left\{{\begin{matrix}C(\tau )\phi (t,\tau )B(\tau )&{\mbox{ if }}t\geq \tau \\0&{\mbox{ if }}t<\tau \end{matrix}}\right.}$

${\displaystyle G[n]=\left\{{\begin{matrix}CA^{k-1}N&{\mbox{ if }}k>0\\0&{\mbox{ if }}k\leq 0\end{matrix}}\right.}$

Root Locus

${\displaystyle 1+KG(s)H(s)=0}$

${\displaystyle 1+K{\overline {GH}}(z)=0}$

${\displaystyle \angle KG(s)H(s)=180^{\circ }}$

${\displaystyle \angle K{\overline {GH}}(z)=180^{\circ }}$

${\displaystyle N_{a}=P-Z}$

${\displaystyle \phi _{k}=(2k+1){\frac {\pi }{P-Z}}}$

${\displaystyle \sigma _{0}={\frac {\sum _{P}-\sum _{Z}}{P-Z}}}$

${\displaystyle {\frac {G(s)H(s)}{ds}}=0}$ or ${\displaystyle {\frac {{\overline {GH}}(z)}{dz}}=0}$

Lyapunov Stability

${\displaystyle MA+A^{T}M=-N}$

Controllers and Compensators

${\displaystyle D(s)=K_{p}+{K_{i} \over s}+K_{d}s}$

${\displaystyle D(z)=K_{p}+K_{i}{\frac {T}{2}}\left[{\frac {z+1}{z-1}}\right]+K_{d}\left[{\frac {z-1}{Tz}}\right]}$