# 0.22 I/o and i/s/o relationships in time and frequency

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## I/o and i/s/o representation of siso linear systems

I/O I/S/O
variables: $\left(u,y\right)$ variables: $\left(u,x,y\right)$
$\frac{d q}{d t}}y(t)=\frac{d p}{d t}}u(t),n=\mathrm{deg}(q)\ge \mathrm{deg}(p)$ $\frac{d x(t)}{d t}}=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t)$
$u(t),y(t)\in \mathbb{R}$ $x(t)\in \mathbb{R}^{n},\begin{pmatrix}A & B\\ C & D\\ \end{pmatrix}\in \mathbb{R}^{(n+1\times n+1)}$
Impulse Response
$\frac{d q}{d t}}h(t)=\frac{d p}{d t}}\delta (t)$ $h(t)=D\delta (t)+Ce^{At}B,t\ge 0$
$H(s)=ℒ(h(t))=\frac{p(s)}{q(s)}$ $H(s)=D+CsI-A^{(-1)}B$
Poles - characteristic roots - eigenfrequencies
${\lambda }_{i},q({\lambda }_{i})=0,I=1,\dots ,n$ $\det ({\lambda }_{i}I-A)=0$
Zeros
$H({z}_{i})=0⇔p({z}_{i}),1,\dots ,n$ $\begin{vmatrix}{z}_{i}I-A & \mathrm{-B}\\ \mathrm{-C} & \mathrm{-D}\end{vmatrix}=0$
Matrix exponential
$(e^{At}=\sum_{k=0} )\implies$ t k k A k t A t A A t A t A
$ℒ(e^{At})=sI-A^{(-1)}$
BIBO stability
$y=(h, u)$ , requirement
$\exists \forall u\colon ({\mathrm{Norm}(u)}_{})\implies$ u
$⇔{(h)}_{1}=\int_{0} \,d t$ h t
$⇔\Re ({\lambda }_{i})< 0⇔\mathrm{poles}\in \mathrm{LHP}$
Solution in the time domain
$y(t)={y}_{\mathrm{zi}}(t)+{y}_{\mathrm{zs}}(t)$ $x(t)={x}_{\mathrm{zi}}(t)+{x}_{\mathrm{zs}}(t)$
$y(t)=\sum_{I=1}^{n} {c}_{i}e^{{\lambda }_{i}t}+\int_{{0}^{-}}^{t} h(t-\tau )u(\tau )\,d \tau$ $x(t)=e^{At}x({0}^{-})+\int_{{0}^{-}}^{t} e^{A(t-\tau )}Bu(\tau )\,d \tau$
$y(t)=Ce^{At}x({0}^{-})+\int_{{0}^{-}}^{t} (D\delta (t-\tau )+Ce^{A(t-\tau )}B)u(\tau )\,d \tau ,h(·)=D\delta (t-\tau )+Ce^{A(t-\tau )}B$
$y(t)=Ce^{At}x({0}^{-})+\int_{{0}^{-}}^{t} h(t-\tau )u(\tau )\,d \tau$
Laplace Transform: Solution in the frequency domain
$Y(s)=\frac{r(s)}{q(s)}+H(s)U(s)$ $X(s)=sI-A^{(-1)}x({0}^{-})+sI-A^{(-1)}BU(s)$
$Y(s)=CsI-A^{(-1)}x({0}^{-})+(D+CsI-A^{(-1)}B)U(s),H(s)=D+CsI-A^{(-1)}B$

## Definition of state from i/o description

Let $H(s)=D+\frac{\langle p(s)\rangle }{q(s)}$ , $\mathrm{deg}(\langle p\rangle )< \mathrm{deg}(q)$ . Define $w$ so that $\frac{d q}{d t}}w(t)=u(t)$ , $(y(t)=\frac{d \langle p\rangle }{d t}}w+Du(t))\implies ({x}^{T}=\begin{pmatrix}w & w^{1} & \dots & w^{(n-1)}\\ \end{pmatrix}\in \mathbb{R}^{n})$ , $n$ : degree of $q(s)$ .

## Various responses

Zero-input or free response
response due to initial conditions alone.
Zero-state or forced response
response due to input (forcing function) alone (zero initial condition).
Homogeneous solution
general form of free-response (arbitrary initial conditions).
Particular solution
forced response.
response obtained for large balues of time $T\to$ .
Transient response
full response minus steady minus state response.

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why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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absolutely yes
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it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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what is the actual application of fullerenes nowadays?
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That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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is Bucky paper clear?
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
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What is lattice structure?
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or in general
Ebrahim
in general
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Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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