এখানে পাঠ্যাংশ থেকে সংগৃহীত গুরুত্বপূর্ণ সমীকরণগুলো বিষয়ভিত্তিকভাবে উপস্থাপন করা হয়েছে। প্রতিটি সমীকরণের ভেরিয়েবল ও প্রতীকের অর্থ, ফাংশনগুলোর ব্যবহার এবং সমীকরণগুলোর বিশ্লেষণ সম্পর্কে বিস্তারিত জানতে মূল পাঠ্যাংশের সংশ্লিষ্ট পাতাগুলি দেখুন।
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)}
|
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)}
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)}
p
(
t
)
=
1
2
t
2
u
(
t
)
{\displaystyle p(t)={\frac {1}{2}}t^{2}u(t)}
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)}
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)}
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)}
y
(
t
)
=
∫
−
∞
∞
g
(
t
,
r
)
x
(
r
)
d
r
{\displaystyle y(t)=\int _{-\infty }^{\infty }g(t,r)x(r)dr}
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 }
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)}
[ট্রান্সফার ম্যাট্রিক্স ভিত্তিক বিবরণ]
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 \ }}}
[বন্ধ-লুপ ট্রান্সফার ফাংশন]
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)}}}
[খোলা-লুপ ট্রান্সফার ফাংশন]
H
o
l
(
s
)
=
K
G
p
(
s
)
G
b
(
s
)
{\displaystyle H_{ol}(s)=KGp(s)Gb(s)}
F
(
s
)
=
1
+
H
o
l
{\displaystyle F(s)=1+H_{ol}}
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}
[বিপরীত লাপ্লেস রূপান্তর]
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\}={\frac {1}{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}
[বিপরীত ফোরিয়ার রূপান্তর]
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}}
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)}
[সাধারণ স্টেট সমীকরণের সমাধান]
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^{A(t-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
[
m
]
{\displaystyle x[n]=A^{n}x[0]+\sum _{m=0}^{n-1}A^{n-1-m}Bu[m]}
[সাধারণ আউটপুট সমীকরণের সমাধান]
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^{A(t-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
[
m
]
+
D
u
[
n
]
{\displaystyle y[n]=CA^{n}x[0]+\sum _{m=0}^{n-1}CA^{n-1-m}Bu[m]+Du[n]}
[সময় পরিবর্তনশীল সাধারণ সমাধান]
x
(
t
)
=
ϕ
(
t
,
t
0
)
x
(
t
0
)
+
∫
t
0
t
ϕ
(
t
,
τ
)
B
(
τ
)
u
(
τ
)
d
τ
{\displaystyle x(t)=\phi (t,t_{0})x(t_{0})+\int _{t_{0}}^{t}\phi (t,\tau )B(\tau )u(\tau )d\tau }
x
[
n
]
=
ϕ
[
n
,
n
0
]
x
[
n
0
]
+
∑
m
=
n
0
n
ϕ
[
n
,
m
+
1
]
B
[
m
]
u
[
m
]
{\displaystyle x[n]=\phi [n,n_{0}]x[n_{0}]+\sum _{m=n_{0}}^{n}\phi [n,m+1]B[m]u[m]}
[ইমপালস রেসপন্স ম্যাট্রিক্স]
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.}
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}}}
[ব্রেকঅ্যাওয়ে পয়েন্টের অবস্থান]
d
d
s
[
G
(
s
)
H
(
s
)
]
=
0
{\displaystyle {\frac {d}{ds}}[G(s)H(s)]=0}
d
d
z
[
G
H
¯
(
z
)
]
=
0
{\displaystyle {\frac {d}{dz}}[{\overline {GH}}(z)]=0}
M
A
+
A
T
M
=
−
N
{\displaystyle MA+A^{T}M=-N}
D
(
s
)
=
K
p
+
K
i
s
+
K
d
s
{\displaystyle D(s)=K_{p}+{\frac {K_{i}}{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)}
![{\displaystyle {\mathcal {L}}[f(t)*g(t)]=F(s)G(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1da7a021eb1cc9ce933aee9c48544308fd9aea4b)
![{\displaystyle {\mathcal {L}}[f(t)g(t)]=F(s)*G(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a5783272e7260e8336adb5853f8a66ffc23d02)
![{\displaystyle C[sI-A]^{-1}B+D=\mathbf {H} (s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d37ee2188135370b2d768fcf965161d688cd12f6)
![{\displaystyle C[zI-A]^{-1}B+D=\mathbf {H} (z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/151e79dc41cb1fb26cf6964a0f9621bd4b69c7a1)
![{\displaystyle F(s)={\mathcal {L}}[f(t)]=\int _{0}^{\infty }f(t)e^{-st}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c38c98774feaf400f8af0dab5f8aed8aed5d06f)
![{\displaystyle F(j\omega )={\mathcal {F}}[f(t)]=\int _{0}^{\infty }f(t)e^{-j\omega t}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccad40df03382f6ead04ba196a52314f1b7a4291)
![{\displaystyle F^{*}(s)={\mathcal {L}}^{*}[f(t)]=\sum _{i=0}^{\infty }f(iT)e^{-siT}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30b1617367f1700032ffd3f5b8720159699648ed)
![{\displaystyle X(z)={\mathcal {Z}}\left\{x[n]\right\}=\sum _{i=-\infty }^{\infty }x[n]z^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af2109d64c2b69a9a5104d600e08fc9b4267360b)
![{\displaystyle x[n]=Z^{-1}\{X(z)\}={\frac {1}{2\pi j}}\oint _{C}X(z)z^{n-1}dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c47fa1780db635f78bf97c38f712f16f779bd4d)
![{\displaystyle X(z,m)={\mathcal {Z}}(x[n],m)=\sum _{n=-\infty }^{\infty }x[n+m-1]z^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85540c08a77b2c40043f7f7e84c0ed56437bb4a9)
![{\displaystyle x[\infty ]=\lim _{z\to 1}(z-1)X(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30f34e1769d16e186e49651f66f4401a8c589b3e)
![{\displaystyle x[n]=A^{n}x[0]+\sum _{m=0}^{n-1}A^{n-1-m}Bu[m]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eca2bee71a93c76ae995febf2ddc4d8cdf8fb75c)
![{\displaystyle y[n]=CA^{n}x[0]+\sum _{m=0}^{n-1}CA^{n-1-m}Bu[m]+Du[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7860669f2231604abcddf87a42e932f258c43e82)
![{\displaystyle x[n]=\phi [n,n_{0}]x[n_{0}]+\sum _{m=n_{0}}^{n}\phi [n,m+1]B[m]u[m]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/845ec15de688172421d4632584cf3b5de5597457)
![{\displaystyle G[n]=\left\{{\begin{matrix}CA^{k-1}N&{\mbox{ if }}k>0\\0&{\mbox{ if }}k\leq 0\end{matrix}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90177e0e9300b0985968117dde9fa1f3a5aec597)
![{\displaystyle {\frac {d}{ds}}[G(s)H(s)]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22b61d87f6f3e0bab60d72dfac427b3143e4f938)
![{\displaystyle {\frac {d}{dz}}[{\overline {GH}}(z)]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/488e7548eedc10f5e72f67818a7e261ec17e6fb4)
