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 Edit
e
j
ω
=
cos
(
ω
)
+
j
sin
(
ω
)
{\displaystyle e^{j\omega }=\cos(\omega )+j\sin(\omega )}
(
a
∗
b
)
(
t
)
=
∫
−
∞
∞
a
(
τ
)
b
(
t
−
τ
)
d
τ
{\displaystyle (a*b)(t)=\int _{-\infty }^{\infty }a(\tau )b(t-\tau )d\tau }
L
[
f
(
t
)
∗
g
(
t
)
]
=
F
(
s
)
G
(
s
)
{\displaystyle {\mathcal {L}}[f(t)*g(t)]=F(s)G(s)}
L
[
f
(
t
)
g
(
t
)
]
=
F
(
s
)
∗
G
(
s
)
{\displaystyle {\mathcal {L}}[f(t)g(t)]=F(s)*G(s)}
[Characteristic Equation]
|
A
−
λ
I
|
=
0
{\displaystyle |A-\lambda I|=0}
A
v
=
λ
v
{\displaystyle Av=\lambda v}
w
A
=
λ
w
{\displaystyle wA=\lambda w}
d
B
=
20
log
(
C
)
{\displaystyle dB=20\log(C)}
Basic Inputs Edit
u
(
t
)
=
{
0
,
t
<
0
1
,
t
≥
0
{\displaystyle u(t)=\left\{{\begin{matrix}0,&t<0\\1,&t\geq 0\end{matrix}}\right.}
r
(
t
)
=
t
u
(
t
)
{\displaystyle r(t)=tu(t)}
[Unit Parabolic Function]
p
(
t
)
=
1
2
t
2
u
(
t
)
{\displaystyle p(t)={\frac {1}{2}}t^{2}u(t)}
Error Constants Edit
[Position Error Constant]
K
p
=
lim
s
→
0
G
(
s
)
{\displaystyle K_{p}=\lim _{s\to 0}G(s)}
K
p
=
lim
z
→
1
G
(
z
)
{\displaystyle K_{p}=\lim _{z\to 1}G(z)}
[Velocity Error Constant]
K
v
=
lim
s
→
0
s
G
(
s
)
{\displaystyle K_{v}=\lim _{s\to 0}sG(s)}
K
v
=
lim
z
→
1
(
z
−
1
)
G
(
z
)
{\displaystyle K_{v}=\lim _{z\to 1}(z-1)G(z)}
[Acceleration Error Constant]
K
a
=
lim
s
→
0
s
2
G
(
s
)
{\displaystyle K_{a}=\lim _{s\to 0}s^{2}G(s)}
K
a
=
lim
z
→
1
(
z
−
1
)
2
G
(
z
)
{\displaystyle K_{a}=\lim _{z\to 1}(z-1)^{2}G(z)}
System Descriptions Edit
[General System Description]
y
(
t
)
=
∫
−
∞
∞
g
(
t
,
r
)
x
(
r
)
d
r
{\displaystyle y(t)=\int _{-\infty }^{\infty }g(t,r)x(r)dr}
[Convolution Description]
y
(
t
)
=
x
(
t
)
∗
h
(
t
)
=
∫
−
∞
∞
x
(
τ
)
h
(
t
−
τ
)
d
τ
{\displaystyle y(t)=x(t)*h(t)=\int _{-\infty }^{\infty }x(\tau )h(t-\tau )d\tau }
[Transfer Function Description]
Y
(
s
)
=
H
(
s
)
X
(
s
)
{\displaystyle Y(s)=H(s)X(s)}
Y
(
z
)
=
H
(
z
)
X
(
z
)
{\displaystyle Y(z)=H(z)X(z)}
x
′
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
{\displaystyle x'(t)=Ax(t)+Bu(t)}
y
(
t
)
=
C
x
(
t
)
+
D
u
(
t
)
{\displaystyle y(t)=Cx(t)+Du(t)}
C
[
s
I
−
A
]
−
1
B
+
D
=
H
(
s
)
{\displaystyle C[sI-A]^{-1}B+D=\mathbf {H} (s)}
C
[
z
I
−
A
]
−
1
B
+
D
=
H
(
z
)
{\displaystyle C[zI-A]^{-1}B+D=\mathbf {H} (z)}
[Transfer Matrix Description]
Y
(
s
)
=
H
(
s
)
U
(
s
)
{\displaystyle \mathbf {Y} (s)=\mathbf {H} (s)\mathbf {U} (s)}
Y
(
z
)
=
H
(
z
)
U
(
z
)
{\displaystyle \mathbf {Y} (z)=\mathbf {H} (z)\mathbf {U} (z)}
M
=
y
o
u
t
y
i
n
=
∑
k
=
1
N
M
k
Δ
k
Δ
{\displaystyle M={\frac {y_{out}}{y_{in}}}=\sum _{k=1}^{N}{\frac {M_{k}\Delta \ _{k}}{\Delta \ }}}
Feedback Loops Edit
[Closed-Loop Transfer Function]
H
c
l
(
s
)
=
K
G
p
(
s
)
1
+
K
G
p
(
s
)
G
b
(
s
)
{\displaystyle H_{cl}(s)={\frac {KGp(s)}{1+KGp(s)Gb(s)}}}
[Open-Loop Transfer Function]
H
o
l
(
s
)
=
K
G
p
(
s
)
G
b
(
s
)
{\displaystyle H_{ol}(s)=KGp(s)Gb(s)}
[Characteristic Equation]
F
(
s
)
=
1
+
H
o
l
{\displaystyle F(s)=1+H_{ol}}
Transforms Edit
F
(
s
)
=
L
[
f
(
t
)
]
=
∫
0
∞
f
(
t
)
e
−
s
t
d
t
{\displaystyle F(s)={\mathcal {L}}[f(t)]=\int _{0}^{\infty }f(t)e^{-st}dt}
[Inverse Laplace Transform]
f
(
t
)
=
L
−
1
{
F
(
s
)
}
=
1
2
π
∫
c
−
i
∞
c
+
i
∞
e
s
t
F
(
s
)
d
s
{\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}
F
(
j
ω
)
=
F
[
f
(
t
)
]
=
∫
0
∞
f
(
t
)
e
−
j
ω
t
d
t
{\displaystyle F(j\omega )={\mathcal {F}}[f(t)]=\int _{0}^{\infty }f(t)e^{-j\omega t}dt}
[Inverse Fourier Transform]
f
(
t
)
=
F
−
1
{
F
(
j
ω
)
}
=
1
2
π
∫
−
∞
∞
F
(
j
ω
)
e
−
j
ω
t
d
ω
{\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 }
F
∗
(
s
)
=
L
∗
[
f
(
t
)
]
=
∑
i
=
0
∞
f
(
i
T
)
e
−
s
i
T
{\displaystyle F^{*}(s)={\mathcal {L}}^{*}[f(t)]=\sum _{i=0}^{\infty }f(iT)e^{-siT}}
X
(
z
)
=
Z
{
x
[
n
]
}
=
∑
i
=
−
∞
∞
x
[
n
]
z
−
n
{\displaystyle X(z)={\mathcal {Z}}\left\{x[n]\right\}=\sum _{i=-\infty }^{\infty }x[n]z^{-n}}
x
[
n
]
=
Z
−
1
{
X
(
z
)
}
=
1
2
π
j
∮
C
X
(
z
)
z
n
−
1
d
z
{\displaystyle x[n]=Z^{-1}\{X(z)\}={\frac {1}{2\pi j}}\oint _{C}X(z)z^{n-1}dz\ }
X
(
z
,
m
)
=
Z
(
x
[
n
]
,
m
)
=
∑
n
=
−
∞
∞
x
[
n
+
m
−
1
]
z
−
n
{\displaystyle X(z,m)={\mathcal {Z}}(x[n],m)=\sum _{n=-\infty }^{\infty }x[n+m-1]z^{-n}}
Transform Theorems Edit
x
(
∞
)
=
lim
s
→
0
s
X
(
s
)
{\displaystyle x(\infty )=\lim _{s\to 0}sX(s)}
x
[
∞
]
=
lim
z
→
1
(
z
−
1
)
X
(
z
)
{\displaystyle x[\infty ]=\lim _{z\to 1}(z-1)X(z)}
x
(
0
)
=
lim
s
→
∞
s
X
(
s
)
{\displaystyle x(0)=\lim _{s\to \infty }sX(s)}
State-Space Methods Edit
[General State Equation Solution]
x
(
t
)
=
e
A
t
−
t
0
x
(
t
0
)
+
∫
t
0
t
e
A
(
t
−
τ
)
B
u
(
τ
)
d
τ
{\displaystyle x(t)=e^{At-t_{0}}x(t_{0})+\int _{t_{0}}^{t}e^{A(t-\tau )}Bu(\tau )d\tau }
x
[
n
]
=
A
n
x
[
0
]
+
∑
m
=
0
n
−
1
A
n
−
1
−
m
B
u
[
n
]
{\displaystyle x[n]=A^{n}x[0]+\sum _{m=0}^{n-1}A^{n-1-m}Bu[n]}
[General Output Equation Solution]
y
(
t
)
=
C
e
A
t
−
t
0
x
(
t
0
)
+
C
∫
t
0
t
e
A
(
t
−
τ
)
B
u
(
τ
)
d
τ
+
D
u
(
t
)
{\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)}
y
[
n
]
=
C
A
n
x
[
0
]
+
∑
m
=
0
n
−
1
C
A
n
−
1
−
m
B
u
[
n
]
+
D
u
[
n
]
{\displaystyle y[n]=CA^{n}x[0]+\sum _{m=0}^{n-1}CA^{n-1-m}Bu[n]+Du[n]}
[Time-Variant General Solution]
x
(
t
)
=
ϕ
(
t
,
t
0
)
x
(
t
0
)
+
∫
t
0
t
ϕ
(
τ
,
t
0
)
B
(
τ
)
u
(
τ
)
d
τ
{\displaystyle x(t)=\phi (t,t_{0})x(t_{0})+\int _{t_{0}}^{t}\phi (\tau ,t_{0})B(\tau )u(\tau )d\tau }
x
[
n
]
=
ϕ
[
n
,
n
0
]
x
[
t
0
]
+
∑
m
=
n
0
n
ϕ
[
n
,
m
+
1
]
B
[
m
]
u
[
m
]
{\displaystyle x[n]=\phi [n,n_{0}]x[t_{0}]+\sum _{m=n_{0}}^{n}\phi [n,m+1]B[m]u[m]}
[Impulse Response Matrix]
G
(
t
,
τ
)
=
{
C
(
τ
)
ϕ
(
t
,
τ
)
B
(
τ
)
if
t
≥
τ
0
if
t
<
τ
{\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.}
G
[
n
]
=
{
C
A
k
−
1
N
if
k
>
0
0
if
k
≤
0
{\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 Edit
1
+
K
G
(
s
)
H
(
s
)
=
0
{\displaystyle 1+KG(s)H(s)=0}
1
+
K
G
H
¯
(
z
)
=
0
{\displaystyle 1+K{\overline {GH}}(z)=0}
∠
K
G
(
s
)
H
(
s
)
=
180
∘
{\displaystyle \angle KG(s)H(s)=180^{\circ }}
∠
K
G
H
¯
(
z
)
=
180
∘
{\displaystyle \angle K{\overline {GH}}(z)=180^{\circ }}
N
a
=
P
−
Z
{\displaystyle N_{a}=P-Z}
ϕ
k
=
(
2
k
+
1
)
π
P
−
Z
{\displaystyle \phi _{k}=(2k+1){\frac {\pi }{P-Z}}}
σ
0
=
∑
P
−
∑
Z
P
−
Z
{\displaystyle \sigma _{0}={\frac {\sum _{P}-\sum _{Z}}{P-Z}}}
[Breakaway Point Locations]
G
(
s
)
H
(
s
)
d
s
=
0
{\displaystyle {\frac {G(s)H(s)}{ds}}=0}
or
G
H
¯
(
z
)
d
z
=
0
{\displaystyle {\frac {{\overline {GH}}(z)}{dz}}=0}
Lyapunov Stability Edit
M
A
+
A
T
M
=
−
N
{\displaystyle MA+A^{T}M=-N}
Controllers and Compensators Edit
D
(
s
)
=
K
p
+
K
i
s
+
K
d
s
{\displaystyle D(s)=K_{p}+{K_{i} \over s}+K_{d}s}
D
(
z
)
=
K
p
+
K
i
T
2
[
z
+
1
z
−
1
]
+
K
d
[
z
−
1
T
z
]
{\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]}