Questions tagged [differential-equation]
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37
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Not getting the same step response from Laplace transform and it's respective difference equation
I am trying to simulate a plant on a microcontroller. The transfer function of the plant is
$$ G_{p} \left( s \right) = \frac{2}{\left( s + 3 \right) \left( s - 1 \right)} \tag{1} \label{1}$$
The step ...
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0
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Block Diagram 2nd Butterworth Low-pass
I would like to represent a second-order butterworth low-pass filter in a block diagram, how can I get started?
As per the attached image
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1
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114
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Impulse response of an LTI system
I want to find the impulse response of the linear time invariant system given by
$y''(t)+4y'(t)+3y(t)=x'(t)+2x(t)$
using time domain analysis.
As solved in the textbook "Continuous and Discrete ...
1
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1
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74
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Show that decomposition does not hold for non-linear system
The solution to an inhomogeneous differential equation can be split up into homogeneous solution and a particular solution (forced response).
Another way to split up the solution to an inhomogeneous ...
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0
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41
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Fit a harmonic oscilator via step response
Suppose we are studying a dynamic system believed to be well modeled via
$$
x'' + \frac bm x' + \frac cm x = \frac Fm$$
$x''$ - acceleration
$b$ - damping factor
$c$ - spring rate
$F$ - external force....
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Solving for the Kernel of a system using impulse balancing with diracs delta's derivative
I wanted to practice Impulse balancing and tried using it to solve this IAR system in one go as opposed to using its LTI attributes. The exercise wants me to find the impulse response of the system:
$$...
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1
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70
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When $x(t)$ and the output $y(t)$ are related by linear differential equations, why is the system unstable when $M>N$?
This is two linear differential systems equating each other.
\begin{equation}
\frac{d^Ny(t)}{dt^N}+a_1\frac{d^{N-1}y(t)}{dt^{N-1}}+\cdots+a_{N-1}\frac{dy(t)}{dt}+a_Ny(t)=\\
b_{N-M}\frac{d^Mx(t)}{dt^M}+...
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1
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Why Is PDE Based Image Processing Not as Active as It Used to Be?
I can see from the literature that PDE-based image processing (especially inpainting) was a "hot topic" at one point. Why does it not seem to be an active area of research anymore, yet it ...
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How to solve this degree $2N$ polynomial equation of filter's cut-off frequency?
In a paper I am interested in, the following equation is given to determine the cut-off frequency $\omega$ of a filter:
$$t^{2N}+\left[-2\left(1-\frac{1}{2Q^2}\right)\right]t^N +\left[-\left(\frac{|H(\...
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Resolution of linear constant coefficient difference equation (homogeneus + particular)
I'm studying how to solve linear constant coefficients difference equations but I have some troubles with a step of the procedure.
Consider the system
\begin{equation}
\begin{cases}
y[n]+y[n-1]-6 y[n-...
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3
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241
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Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions?
Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions $y(t)$??
With output I am meaning that $y(t)$ is such ...
- 183
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41
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Why is this minimum cost function returning an insane cost of registration value?
I have a sample program that uses the minimum cost function to align image B to image A.
The method takes two arrays of points and does the summation of all the points in the images for both Matrix A ...
- 101
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938
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Difference equation from a transfer function of a low pass filter
I need to get the difference equation from this transfer function:
$H(z) = g \frac{1+a_1}{1+a_1z^-1}$
My math skills are too many years old, but I remember I need to get the Y(output) on one side and ...
- 103
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0
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Which conditions must fulfill a time-limited signal (so of unlimited bandwidth) $f(t)$ to have a bounded maximum slew rate?
Which conditions must fulfill a time-limited signal (so of unlimited bandwidth) $f(t)$ to have a bounded maximum slew rate?
I want to know about which conditions must fulfill a real-valued time-...
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2
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339
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Why does Simulink generate this code for a PID controller?
For the Simulink PID Controller model
The Simulink generated code (rewrite for better understanding) is:
...
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1
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603
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Why is particular solution zero for an impulse excitation signal?
We were being taught the impulse response for a series RC Circuit- consisting of simply one resistance, one capacitor, and an impulse excitation all in series.
I get that the homogeneous part of the ...
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Square pulse test of Upwind Finite Differences
I`m analyzing the numerical methods for the 1D convection equation for stability, consistency, and accuracy. I want to see How does this square pulse move in domain and time?
Here is my code
% ...
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1
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3k
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RC high pass filter differential equation
I have the question to derive:
$$RC \frac{dy(t)}{dt} + y(t) = RC \frac{dx(t)}{dt}$$
which describes a high pass filter or $CR$ circuit with $x(t)$ the input and $y(t)$ the output.
I am really ...
3
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Causal and Non-memoryless LTI sytems described by LCCDE
I was wondering about the nature of stable systems (in the BIBO sense) that are causal with memory for which we wish to represent them by LCCDE (if they may exist). How frequent do LCCDE exist such ...
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2
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3k
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From transfer function to differential equation
I have the below detailed solution (boxed in blue) that I don't understand completely:
I can reconstitute the differential equation from:
$$ (1+Ts) X(s) = K_v U(s) $$
$$ x(t) + T\dot x(t) = K_v u(t) $...
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863
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Particular Solution to Difference Equation
I have a system given by
$$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$
I want to solve for $y[n]$ for $x[n]=(\frac{1}{2})^nu[n]$.
The complementary solution evaluates to $[k_1(\frac{1}{2})...
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160
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Augment State Space Model with Derivative of State
Problem
How do you augment a state space model with the derivative of a state? I know how to augment a state space model with the integral of a state by doing the following. Given a linear system
$$ \...
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26
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Parameter tracking using Augmented state vector approach and unscented Kalman filter
I'm trying to reproduce and extend figure 9 results in
Nonlinear dynamical system identification from uncertain and indirect measurements"HU Voss, J Timmer, J Kurths - International Journal of ...
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1
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463
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Stability of a system in time-domain
Consider a system for which the input $x(t)$ and output $y(t)$ are related by the differential equation $$\frac{d^2y(t)}{dt^2} + \frac{3}{2}\frac{dy(t)}{dt} - y(t) = x(t) \tag{1}$$Determine $h(t)$ for ...
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158
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How does the intuitive notion of causality fit in with control systems?
Edit: By causality, in this question, I do not mean the traditional mathematical definition in the theory o signals and systems; I mean causality as in an intuitive 'what's moving/pushing what notion'....
1
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1
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659
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Confusion in initial condition of differential equation using Laplace transform transform
I'm confused in solving linear constant coefficients differential equations (LCCDEs) by Laplace transform if initial conditions are given at time
just before $t=0$
just after $t=0$
exactly at $t=0$
...
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Impulse response if initial conditions are given
In most books, impulse response of LTI systems are calculated by assuming initial rest condition, but how to calculate response of an impulse input if there are some initial conditions present ?
My ...
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Finding the system output by convolution
pretty new to this concept, so do bear with me.
A linear dynamic system is described by the following differential equation:
Transfer function H(s) is calculated to be =
I've already found the ...
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0
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106
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Output Noise Power for ODE
My system is described by the following ODE:
$$
\frac{\mathrm{d}y(t)}{\mathrm{d}t} = a-y(t)x(t)
$$
Where $a$ is a constant and x(t) is a poisson process so that:
$$
E[x(t)x(s)] = qI\delta(t-s)
$$
...
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106
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Is a LTI system with dead-time equivalent with a LTI system with no dead-time but delayed input signal?
I wonder if the system parameters would remain the same after all.
According to the definitions, the gain is the change of the output induced by the change of the input if we assume that the gain is ...
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0
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Is non-causal, non-LTI solution to difference equation correct?
Find a solution to the difference equation $y[n]-\frac{5}{6}y[n-1]+\frac{1}{6}y[n-2]=\frac{1}{3}x[n-1]$ that is neither casual nor LTI, where $y[0]=y[1]=1$ and $x[n]=\delta[n]$
The homogenous ...
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Proof for the solution of homogenous difference equation
Suppose the output $y_h[n]$ of a linear, time-invariant system is described by by the following equation for input $x[n]=0$,
$$\sum_{k=0}^{N}a_k y_h[n-k] = 0$$
My book states the $y_h[n]$ is in fact ...
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110
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Difference equation with variable coefficients in MATLAB
Given the following difference equation
$$y[n]-\left(\dfrac{n}{n+1}\right)y[n-1] = n\cdot x[n]$$
How can we use MATLAB to solve it?
I know if the coefficients are constant we can simply use ...
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1
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Determine the system function H(s) of a system and find out the differential equation
I have created the following system for practice purposes. From this system I want to determine the system function H(s).
In the picture I have worked with auxiliary (dummy) variables, which should ...
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2
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LCCDE in simple words?
What is LCCDE?I only know its abbreviation/full form :linear constant-coefficient difference equation
I know that in s domain we have differential equations and in z domain we have difference ...
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1
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167
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Damped spring mass system - parameter estimation
I've tried to calculate the parameters of a damped spring mass system of the form
$m~ y''(t)+d~y'(t)+c~y(t)=F(t)$
but I have some problems determining the mass m of the system.
The damped ...
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483
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Transform a Continuous Time Differential Equation of a Low Pass Filter to Discrete Time
Assuming I have a linear differential equation of first order low pass filter:
$$RC*y(t)'+y(t) = x(t) $$
And Assuming i know now that the input x(t) is being sampled at a rate of $T_s$
how can now ...
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