

 State space systems

(Blank Abstract)
I/o and i/s/o representation of siso linear systems
I/O 
I/S/O 
variables:
$\left(u,y\right)$ 
variables:
$(u,x,y)$ 
$\frac{d q}{d t}y(t)=\frac{d p}{d t}u(t),n=\mathrm{deg}(q)\ge \mathrm{deg}(p)$ 
$\frac{d x(t)}{d t}=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t)$ 
$u(t),y(t)\in \mathbb{R}$ 
$x(t)\in \mathbb{R}^{n},\begin{pmatrix}A & B\\ C & D\\ \end{pmatrix}\in \mathbb{R}^{(n+1\times n+1)}$ 
Impulse Response 
$\frac{d q}{d t}h(t)=\frac{d p}{d t}\delta (t)$ 
$h(t)=D\delta (t)+Ce^{At}B,t\ge 0$ 
$H(s)=\mathcal{L}(h(t))=\frac{p(s)}{q(s)}$ 
$H(s)=D+CsIA^{(1)}B$ 
Poles 
characteristic roots  eigenfrequencies 
${\lambda}_{i},q({\lambda}_{i})=0,I=1,\dots ,n$ 
$\det ({\lambda}_{i}IA)=0$ 
Zeros 
$H({z}_{i})=0\iff p({z}_{i}),1,\dots ,n$ 
$\begin{vmatrix}{z}_{i}IA & \mathrm{B}\\ \mathrm{C} & \mathrm{D}\end{vmatrix}=0$ 
Matrix exponential 
$(e^{At}=\sum_{k=0} )\implies $∞
t
k
k
A
k
t
A
t
A
A
t
A
t
A 
$\mathcal{L}(e^{At})=sIA^{(1)}$ 
BIBO stability 
$y=(h, u)$ , requirement 
$\exists \forall u\colon ({\mathrm{Norm}(u)}_{})\implies $∞ ∞
u ∞ ∞ 
$\iff {(h)}_{1}=\int_{0} \,d t$∞
h
t ∞ 
$\iff \Re ({\lambda}_{i})< 0\iff \mathrm{poles}\in \mathrm{LHP}$ 
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 $ 

$y(t)=Ce^{At}x({0}^{})+\int_{{0}^{}}^{t} (D\delta (t\tau )+Ce^{A(t\tau )}B)u(\tau )\,d \tau ,h(\xb7)=D\delta (t\tau )+Ce^{A(t\tau )}B$ 

$y(t)=Ce^{At}x({0}^{})+\int_{{0}^{}}^{t} h(t\tau )u(\tau )\,d \tau $ 
Laplace
Transform: Solution in the frequency domain 
$Y(s)=\frac{r(s)}{q(s)}+H(s)U(s)$ 
$X(s)=sIA^{(1)}x({0}^{})+sIA^{(1)}BU(s)$ 

$Y(s)=CsIA^{(1)}x({0}^{})+(D+CsIA^{(1)}B)U(s),H(s)=D+CsIA^{(1)}B$ 
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
$\frac{d q}{d t}w(t)=u(t)$ ,
$(y(t)=\frac{d \langle p\rangle}{d t}w+Du(t))\implies ({x}^{T}=\begin{pmatrix}w & w^{1} & \dots & w^{(n1)}\\ \end{pmatrix}\in \mathbb{R}^{n})$ ,
$n$ : degree of
$q(s)$ .
Various responses
Zeroinput or free response
 response due to initial conditions alone.
Zerostate or forced response
 response due to input (forcing function) alone (zero
initial condition).
Homogeneous solution
 general form of
freeresponse (arbitrary initial conditions).
Particular solution
 forced response.
Steadystate response
 response obtained
for large balues of time
$T\to $∞ .
Transient response
 full response minus steady
minus state response.
Questions & Answers
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Shanjida
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what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is system testing?
AMJAD
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field .
1Electronicsmanufacturad IC ,RAM,MRAM,solar panel etc
2Helth and MedicalNanomedicine,Drug Dilivery for cancer treatment etc
3 Atomobile MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:
OpenStax, State space systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10143/1.3
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