4 votes
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When $x(t)$ and the output $y(t)$ are related by linear differential equations, why is the system unstable when $M>N$?

Let's try the simple $N=0$ and $M=1$ case: $$a_0 y(t) = b_0 \frac{d x(t)}{ dt} + b_1 x(t) \tag{1}$$ Now, let's think about stability. A standard approach to stability is that for any bounded (finite-...
Peter K.'s user avatar
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4 votes
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Why is particular solution zero for an impulse excitation signal?

The correct form of the statement should be the particular solution is zero for $t>0$ This is simply the case because the input $\delta(t)$ is zero for $t>0$. So for $t>0$, the impulse ...
Matt L.'s user avatar
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3 votes
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LCCDE in simple words?

An $N$th-order linear constant-coefficient difference equation (LCCDE) is of the form $$y[n]=\sum_{k=0}^{M}b_kx[n-k]-\sum_{k=1}^{N}a_ky[n-k]\tag{1}$$ It is linear because the sequences $x[n]$ and $y[...
Matt L.'s user avatar
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3 votes
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Determine the system function H(s) of a system and find out the differential equation

Your transfer function looks correct. Note that you can rewrite $H(s)$ as $$H(s)=-\frac{3s}{5s+3}\tag{1}$$ or, equivalently, $$5sY(s)+3Y(s)=-3sX(s)\tag{2}$$ Since multiplication by $s$ corresponds ...
Matt L.'s user avatar
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2 votes

LCCDE in simple words?

A difference equation simultaneously characterises a system and enables the practical computation of its output $y[n]$ for a given input $x[n]$ and stated initial conditions. An LCCDE (MattL Eq(1)) is ...
Fat32's user avatar
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2 votes

Damped spring mass system - parameter estimation

I think I got the answer: I look at the step response of the system. $H(s) = \frac{1}{ms^2+ds+c}$ The limit of the step response equals my parameter $1/c$ $y(\infty)=\lim_{s\rightarrow 0} s\cdot \...
Phobos's user avatar
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2 votes
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Confusion in initial condition of differential equation using Laplace transform transform

Initial conditions are always given at $t=0^-$, because they define the state of the system before any input is applied, and - by definition - the input is applied at $t=0$. The state at $t=0^+$ is ...
Matt L.'s user avatar
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2 votes
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Stability of a system in time-domain

Solving the characteristic equation $$s^2+\frac32 s-1=0\tag{1}$$ gives the following homogeneous solution: $$y(t)=c_1e^{-2t}+c_2e^{t/2}\tag{2}$$ The requirements of causality and stability are taken ...
Matt L.'s user avatar
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2 votes
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Why Is PDE Based Image Processing Not as Active as It Used to Be?

PDE for image processing had its glory days when we didn't have a good model for images. Back in those days, PDE's were the best models as they were mathematically understandable and in many times ...
Royi's user avatar
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2 votes
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Show that decomposition does not hold for non-linear system

Assume that $y_0(t)$ is the zero-input response. Then $y_0(t)$ must satisfy $$\dot{y}_0(t)+y_0(t)=1\tag{1}$$ because $x(t)=0$. Now let $y_1(t)$ be the zero-state response to an input $x(t)$, ...
Matt L.'s user avatar
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2 votes
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Solving for the Kernel of a system using impulse balancing with diracs delta's derivative

The mistake lies in the fact that you assumed $h'(0^+)=0$, which is not the case because $h''(t)$ contains a Dirac delta impulse. The result of the first integration should be $$h'(0^+)+4h(0^+)=2\tag{...
Matt L.'s user avatar
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1 vote
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Impulse response of an LTI system

I don't see how the first method would result in $c_1=0$ and $c_2=1$. With $$h(t)=\big[c_1e^{-3t}+c_2e^{-t}\big]u(t)$$ you obtain $$h'(t)=\big[\ldots\big]u(t)+(c_1+c_2)\delta(t)$$ and $$h''(t)=\big[\...
Matt L.'s user avatar
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1 vote

How to solve this degree $2N$ polynomial equation of filter's cut-off frequency?

Since only $\omega$ is unknown, I can see that the equation has the form $x^{2N}a + x^{N}b + xc +d$. Exactly, you're looking for the roots (i.e., points where the value is 0) of a polynomial in $t$. ...
Marcus Müller's user avatar
1 vote
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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?

I have found the following paper named "Finite time differential equations" by V. T. Haimo (1985), where continuous time differential equations with finite-duration solutions are studied, an ...
Joako's user avatar
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1 vote

Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions?

You may be overthinking this. If the input has length $T_x$ and the impulse response of the LTI system has length $T_h$ than the output will have have length $T_x + T_h$. If either input or impulse ...
Hilmar's user avatar
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Why does Simulink generate this code for a PID controller?

I'm not sur exactly where you went wrong because your haven't fully explained your approach but for the derivate component of the PID, you should convert the C++ lines to this : $$ D_{out}[n] = N(err[...
Ben's user avatar
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1 vote

How does the intuitive notion of causality fit in with control systems?

As near as we can tell by experiment, causality is nature's way of doing its thing. Causality says that if you have a system $y(t) = h\left(x(t), t\right)$, and it is causal, then $y(t_0)$ is ...
TimWescott's user avatar
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1 vote

RC high pass filter differential equation

In an RC high-pass filter, the output is taken across the resistor instead of the capacitor. Otherwise, if the output was taken across the capacitor, the RC high-pass filter would start to work as an ...
Karakoncolos's user avatar
1 vote

From transfer function to differential equation

The solution to the differential equation is given by the sum of a particular solution and the solution of the homogeneous differential equation. The particular solution is a solution to the non-...
Matt L.'s user avatar
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1 vote
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From transfer function to differential equation

Given a transfer function $$G_v(s) = \frac{k_v}{1 + sT} \tag{1}$$ the corresponding LCCDE, with $y(t)$ being the solution, and $x(t)$ being the input, will be $$ T ~\dot{y}(t) + y(t) = k_v ~x(t) \tag{...
Fat32's user avatar
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1 vote

Particular Solution to Difference Equation

The solution you came up with is the correct homogeneous solution (i.e., when $x[n]=0$). Thing is, this is a non-homogeneous difference equation, and its solutions are of the form $$y[n]=y_h[n]+y_p[n]=...
cjferes's user avatar
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1 vote

Impulse response if initial conditions are given

The reason you're having trouble with this is because the impulse response of a system tells you something considerably different from a system's behavior when it is starting with non-zero initial ...
TimWescott's user avatar
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1 vote

Finding the system output by convolution

You can't compute the output by convolving the input with the system's impulse response if the initial conditions are non-zero. The reason is simple: there is no information about the initial ...
Matt L.'s user avatar
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1 vote
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Proof for the solution of homogenous difference equation

If you substitute your second equation back into the first equation, you obtain $$\begin{align}\sum_{k=0}^{N}a_k\sum_{m=1}^NA_mz_m^{n-k}&=\sum_{m=1}^NA_mz_m^n\sum_{k=0}^Na_kz_m^{-k}\end{align}\...
Matt L.'s user avatar
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1 vote

Difference equation with variable coefficients in MATLAB

When $n=0$ you may need to know $y[-1]$. The current output $y(n)$ depends on the current input $x(n)$ and previous output $y(n-1)$ scaled.
jomegaA's user avatar
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