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]}