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Questions tagged [stability]

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1answer
62 views

Can a Fourier Transform exist even if the j$\omega$ axis is not in the Region of Convergence in it's Laplace Transform

A couple of confusions have been occurred. The Signal I'm considering is f(t) = sin(t)*u(t) Fourier Transform of it can be derived. $-i \pi (\delta (\omega -1)-\delta (\omega +1))$ According to my ...
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1answer
30 views

Determining asymptotic stability using transfer function?

In an exam task, I am asked to determine the transfer function of the following direct-time system and decide whether it's stable. I think this system is canonical and the amplifiers 'on top' ...
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0answers
28 views

Verify stability of the system [closed]

A DT system is described as follows: $$y[n] = T(x[n]) = x[n] + \sum_{l=1}^\infty (0.9^l(x[n-l]+x[n+l])) $$ I have found that the Discrete-time necessary and sufficient condition for stability is. $$ ...
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2answers
21 views

Discrepancy in stability conditions when calculating via RH criterion and Nyquist criteria

I have the following open loop transfer function for a unity feedback system. $$G(s)=\frac{K(s+20)^2}{s^3}$$ 1.When using RH criterion it can be easily proved that the closed loop transfer function ...
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1answer
14 views

Routh's stability criterion: zeros of the auxiliary polynomial and of its derivative

When using RH criterion and using the auxiliary equation special case, which of the following is true? The auxiliary equation $A(s)=0$ gives some(or all) of the symmetrical poles. The differentiated ...
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1answer
37 views

BIBO Stability in Z-domain

I'd really appreciate it if someone could please explain to me the condition for a LTI system to be BIBO stable, in z-domain. I have a background in control, and in linear control for example, if we ...
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0answers
59 views

Eigenvalues of a product of matrices with specific structures

I'm working on a multichannel feedback system with open-loop frequency response $\mathbf{A}(\omega) = \mathbf{B}(\omega)(\mathbf{C}(\omega)-\mathbf{D}(\omega))$. The time-domain convolution matrices ...
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2answers
69 views

Stability of system with poles inside unit circle - conflict with differential equation

I am trying to understand why a system with a single pole inside the unit circle is stable. For example, take a system with one pole at $z=\frac{1}{2}$. The literature says the system is stable. As a ...
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1answer
62 views

Is $\cos(x[n])$ stable? [closed]

I am highly confused about the stability of $\cos(x[n])$. If we provide a bounded input such as $x[n]=u[n]$, the output is bounded. Now if we provide an unbounded signal $x[n]=\delta[n]$, the output ...
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2answers
89 views

Realization of a filter based on its transfer function

How can we check whether the filter is realizable given its transfer function and What are the parameters the realization depends on? Here is an example: Show that a filter with transfer function ...
0
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1answer
46 views

Simple question about Sinc()

Suppose I use the ideal lowpass filter eq to attempt creating a IIR filter using it : $$ h[n] = \frac{\sin(\omega_c \, n)}{\pi \, n}, \qquad -\infty < n < \infty $$ Obviously, I can't really (...
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0answers
26 views

Estimating a discrete summer with constrained input bandwidth

I have a discrete-time system which can be described as: $$ Y_m = \sum_{r=-N_g}^{R-1+N_g} c_r x[R(m-1) + r] $$ The unknowns are $c_k$ but I know that they have the following approximate behavior: $$...
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1answer
86 views

BIBO Stability for system with no poles

I have two questions regarding systems with no poles: Why does a system with no poles have a finite support? Why if the system has a finite support it means that it is BIBO stable?
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1answer
54 views

Find a stable transfer function $G(z)$ such that $|G(z)| = |H(z)|$

Consider the following causal IIR transfer function: $$ H(z) = \frac{2z^3 - 4z^2 + 9}{(z-3)(z^2+z+0.5)} $$ Is $H(z)$ a stable function? If it is not stable, find a stable transfer function $G(z)...
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1answer
125 views

Determine if $ y[n] = ny[n-1] + x[n]$ is linear time invariant and BIBO stable

Check if the following system is linear time invariant and BIBO stable.. $$ y[n] = ny[n-1] + x[n] $$ for $n\ge 0$. We are also given that the system is at rest (i.e. $y[−1] = 0$). I know that to ...
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0answers
26 views

How to design a *desired signal* for Lyapunov stability analysis?

How to design a desired signal for Lyapunov stability analysis? Like given in this paper: http://journals.tubitak.gov.tr/elektrik/issues/elk-15-23-3/elk-23-3-8-1212-29.pdf
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3answers
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Why does stability only consider, whether the system is bounded? That doesn't mean the response is smooth?

Why does (BIBO) stability only consider, whether the system is bounded? That doesn't mean the response is smooth? That is, one could have system whose values are bounded. Still it could exhibit "...
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1answer
75 views

Absolute Integrable Sinc function

How do we prove that $$\int_{-\infty}^{\infty}\bigg|\dfrac{\sin t}{\pi t}\bigg|dt\to \infty$$ This comes in the context of stability of LTI system with impulse response $h(t) = \dfrac{\sin t}{\pi t}$. ...
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1answer
372 views

How to conclude LTI, causality and BIBO stability of a system represented by a differential equation?

I have started to learn about systems represented by differential equations in Oppenheim's Signals & Systems, and I got really confused about it. I am trying to understand how I can show that a ...
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0answers
186 views

BIBO Stability of an impulse response

I am trying to prove if $h_1[n] = 2^{-n} u[n]$ and $h_2[n] = (0.1)^n$ are BIBO stable or not, but I'm not sure if these analysis are correct. I'm having some trouble to understand this specific ...
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1answer
326 views

Does “improper” imply that a system cannot be stable and causal?

This answer and the comments in it made me wonder whether the following statement is true: If a transfer function is improper, then that system cannot be causal and stable at the same time. I had ...
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0answers
39 views

numerical instability while solving a physical model

I am trying to solve an estimation problem and my physical model look smth like this: $A'/A^2 = B'/B^2 + C'/C^2$, from which I have measured $A$ and a reference for $B'$. I want to try adaptive ...
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57 views

Frequency Response Question on LSI System

Someone please explain me the question its seems very complicated to me. I just want know what the question asking and how to solve it i dont want fully solved solution. Thank you!!
1
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1answer
62 views

Generay way to find Stability of system

I am trying to find a general way to find system stability.I have applied these methods and struck in confusion For eg: If $$y(t)= \int_{-\infty}^{t} x(\tau) \sin(4\tau) d\tau$$ then find whether the ...
0
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1answer
182 views

Stability of open-loop transfer function from its Nyquist plot

I am facing a confusion on understanding system open-loop transfer function stability from its Nyquist plot. According to the formula, for open loop transfer function stability: $$Z=N+P=0$$ where $N$ ...
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2answers
244 views

Determine the stability of a system without using the $\mathcal Z$-transform (described by a difference equation)

For example, let's say a causal LTI System is described by the following equation: $$y[n] - ay[n-1] = x[n] - bx[n-1],\quad n \in Z$$ Is there a way to determine (in this case) the stability of the ...
0
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1answer
592 views

Marginal Stability based on Poles

We know that a discrete-time system with a (Z-transform) transfer function that has a pole of magnitude 1 (i.e. $|z|=1$ is a pole of the transfer function) is marginally stable if the pole at $z=1$ is ...
0
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1answer
84 views

Time-invariance, causality and stability of $h(t)$ of four given systems

Question: The impulse response functions of four linear systems $S_1,\ S_2,\ S_3,\ S_4$ are given respectively by \begin{align} h_1(t)&=1\\ h_2(t)&=u(t)\\ h_3(t)&=\frac{u(t)}{(t+1)}...
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1answer
72 views

BIBO Stability of a piecewise function

I have a function $y$ defined as $$ y(t) = \left\{\begin{array} ~t~ \mbox{ where} |t| \le 3\\ 0~ \mbox{ otherwise} \end{array} \right . $$ With a system defined as $$G(t) = ty(t), $$ is it BIBO ...
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1answer
72 views

Transient response of system with single pole $0 \le p < 1$

$G(z) = \frac{1-p}{z-p}$ If the value of p satisfies $ 0 \leq p < 1$ there are no oscillations in the transient response. Question: Why is that $\uparrow$ true? I know roughly what a ...
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1answer
50 views

How to determine the poles from a graph

From my knowledge of stability, I understand that if the function approaches a finite number then the system will be stable. Thus if a system is stable its poles will be on the left of the $j\omega$-...
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2answers
85 views

When inverting a transfer function, solving for the input using the output does the causality status change

suppose $y(n)=ax(n-1)+bx(n-2)+\dots$ ($y$ is the output and $x$ the input). What happens if I want to solve $x(n)$ from $y(n)$? Z transform: $$Y(z)=G(z)X(z)\tag{1}$$ then $$X(z)=\frac{1}{G(z)...
1
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1answer
63 views

Calculating Stability of Phaser Feedback Loop

I have been doing some experiments with phaser programs. I really like setting the phaser feedback to really high levels, such that the effect is really metallic and resonant, almost like a physical ...
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0answers
39 views

Quadcopter's simulation vs. empirical response

As part of my PhD, I am designing a controller for a quadcopter. However, I was able to tune 3 PID controllers to control and stabilize the roll, pitch, and yaw channels of the quadcopter, which is ...
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2answers
100 views

Stable digital filter with leading zeroes

I'm looking construct a stable pole-only filter where the feedback coefficients start with a block of zeroes, i.e. \begin{align} a_0 &= 1\\ a_i &= 0, \textrm{ for}\ 1 \le i \lt k\\ a_i &\...
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2answers
87 views

why exponential term neglected in equation?

where does that exponential term gone, is this because it is a constant term or it has to do something with stablity?
3
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1answer
401 views

causality of the system $y[n] = x(2n)$

Can somebody please tell me why the system $y[n] = x(2n)$ is non-causal ? I know that causal systems depend on the past and present values of input and this system satisfies the condition. So why is ...
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0answers
50 views

How to find maximum gain for second order recursive filter

Suppose I have the following filter: \begin{equation} \ y[n] + a_1 y[n-1] + a_2 y[n-2] = x[n] \end{equation} with complex poles $p$ and $p^*$ so $ a_1^2-4a_2<0 $. This gives: \begin{equation} \ p,p^...
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1answer
5k views

Position of poles and Stability in $z$ domain

We know in Laplace Transform, if the poles lie on the left of $j\omega$ axis, we can say the system is stable. Similarly can we comment on the stability based on poles position in $\mathcal Z$-...
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1answer
60 views

can $\frac{1}{H(z)}$ be causal and stable? [duplicate]

if we have linear phase FIR filter $H(z)$ which is causal and stable can $\frac{1}{H(z)}$ be causal and stable ? can it be causal without been stable ?
1
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1answer
312 views

Linearity, Causality and Stability of a System

Consider a system: $$ y[n] = y[n-1] + u[n], $$ where $y[n]$ is the output and $u[n]$ is the unit step function. Is this system causal, linear, time-invariant and stable ? My attempt at the ...
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1answer
463 views

How to check if the poles of transfer function are stable?

$$H(z)=\frac{az}{[z-(b+ja)][z-(b-ja)]}$$ The two poles are looking like this: $z_{\inf, 1}=b+ja, z_{\inf, 2}=b-ja$ I know they must be inside the unit circle: $|z_{\inf,i}|<1$ So I replace the ...
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1answer
51 views

Is my answer incomplete? Checking the stability of a system

Yesterday, during my exam, I had the following exercise: Given $$H(s) = \frac{1}{s^2+2s+4}$$ check if it's stable. which was supposed to be the hardest (since it was the last one). From my ...
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2answers
338 views

BIBO stability of $y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$

How can I prove that the LTI system with (output $y(t)$, input $x(t)$) $$y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$$ is BIBO (bounded-input/bounded output) stable?
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1answer
165 views

Definition of minimum-phase system

I saw a couple of definitions for minimum-phase in different textbooks and I'm trying to understand what the implication of each of them. The first definition I saw was: An invertible system which ...
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1answer
97 views

Stable gain nyquist plot

From Modern Control Engineering 5th Edition page 469 The transfer function of a plant controller by a proportional controller is given by $$G(s) = \frac{K(s+0.5)}{s^3 + s^2 + 1}$$ In the book $G(j\...
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1answer
142 views

Under what conditions do the phase margin and Nyquist criteria give the same results?

When designing feedback systems, I often evaluate stability by thinking about phase margin: the closed loop system $$T(s) = \frac{L(s)}{1+L(s)}$$ is stable if $L(s)$ has positive phase margin, i.e., $...
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1answer
680 views

Causality and ROC of a stable LTI system

So I am looking at a stable LTI system whose input is $x[n]$ and output is $y[n]$. The equation relating the two is here: $$ y[n-1]-\frac{10}{3}y[n]+y[n+1]=x[n] $$ I was able to compute its system ...
3
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1answer
371 views

Allpass Filters - Causal and Stable

So I have been learning about how to test systems for causality and stability but I am confused about the implications on their unit circle representation. Would it be safe to say that a causal and ...
0
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1answer
180 views

Behavior of tanh IIR filters

If we insert a tanh function (or any other activation function) between the feedback summation and the unit delays, how will such an IIR filter behave for values of $|a| < $1 and $a > ±1$ ? The ...