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Questions tagged [steady-state]

For questions about steady states in systems theory, which are unchanging in time.

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How to prove that the convergence to a steady state exists if the change in transition probabilities are sufficiently slow?

Let's say we have a transition matrix $Q_{n}$ for each time step $n$ of a discrete Markov process, where it doesn't stay stationary for all time steps. I want to prove that if the change between each ...
magg13__'s user avatar
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1 answer
100 views

Find Gain K and Time constant K of a system from the time response

There is a given system $\frac{K}{sT + 1}$ of order 1. The responses are in the image below and the 2 inputs are $u1(t) = 1(t)$ and $u_2(t) = \sqrt{2} \cdot \sin(\omega_2 t)$. How can I find the K and ...
sneha_jerin's user avatar
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Differential equation and limit as an equilibrium

I'm trying to understand the following theorem: Theorem: Suppose that $x=x(t)$ is a solution of $\dot{x}=F(x)$ where $F$ is continuous. Suppose that $x(t)$ approaches a finite limit $\textbf{a}$ as $t ...
Maximilian's user avatar
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What is the steady state distribution of this Poisson process with non-constant rate?

I am looking for the steady state distribution of the following Poisson process: $$d x(t) = -k_1(x(t)-k_2)dt + k_3dN(t)$$ where $k_1$, $k_2$ and $k_3$ are constants and the rate $\lambda(x)$ of the ...
user1031129's user avatar
4 votes
1 answer
104 views

Why do we require positive recurrence for a Markov chain to have steady states?

Theorem 4.1 of the book Introduction to Probability Models (10th edition) by Sheldon Ross states that an "irreducible ergodic" Markov chain has limiting probabilities that exist. And ergodic ...
Rohit Pandey's user avatar
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How do you find the steady periodic solution of this ODE?

Find the steady state periodic solution of the differential equation: $$ y^{''}+16y = f(t)$$ where $f(t)$ is an odd function of period $2\pi$ such that $$f(t) = t $$ $$ 0 \leq t < \pi $$
spooky99's user avatar
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Lower bound for probability of birth-death process at 0

Consider a birth-death process with birth transition rate of 1 and death transition rate of $r + \gamma$ at every state $r \in \mathbb{N}$. Can we come up with an lower bound efor the steady-state ...
Alireza Amani's user avatar
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30 views

Existence of steady state in nonlinear systems

I have a quite general and naive question. Is the existence of a steady state in a nonlinear system well-defined? In linear systems, e.g. $$\frac{\mathrm{d}f}{\mathrm{d}t} = Af,$$ the steady-state ...
Rudolf Smorka's user avatar
1 vote
2 answers
150 views

Find the expected number of tosses to win a game

A friend and I play a game. We each start with two coins. We take it in turns to toss a coin; if it comes down heads, we keep it, if tails, we give it to the other. I always go first, and the game ...
drobin's user avatar
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What's differences between steady-state Kalman filter and Time-varying Kalman filter?

I understand that the steady-state Kalman filter is more computationally efficient because the noise in a measurement equation and the shock in a state equation have a constant variance over the time ...
user14261785's user avatar
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How to find the steady state in this system when one variable is irrelevant for equilibrium?

I have a nonlinear system of differential equations for functions $x(t)$ and $y(t)$ with parameters $\epsilon$ and $\lambda$ $$\frac{\text dx}{\text dt}=(1-x)\cdot[1-\epsilon x-2\lambda xy(1-x)]$$ $$\...
BioPhysicist's user avatar
3 votes
1 answer
257 views

Steady State Temperature Distribution in a Rectangular Plate [closed]

We need to solve the following : $$ \nabla^2๐‘ข=0, 0\leq ๐‘ฅ\leq๐‘Ž, 0\leq y\leq b $$ satisfying the boundary conditions $$ ๐‘ข(0,y)=0, 0\leq y\leq b \\ ๐‘ข(๐‘ฅ,0)=๐‘ข(๐‘ฅ,๐‘)=0, 0\leq ๐‘ฅ\leq ๐‘Ž \\ ๐‘ข_x(a,๐‘ฆ)=...
Ankit Kumar's user avatar
2 votes
1 answer
130 views

Find conserved quantities with respect to parameters in PDE

I have a standard, steady-state convection-diffusion PDE given as: $$ -p_1 \nabla^2 c(x,y) + \vec{v} \cdot \vec{\nabla}c(x,y) = f(x,y) $$ With $\vec{v} := \frac{p_2}{\sqrt{2}}\langle 1,1\rangle$. Here ...
David G.'s user avatar
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Showing an endemic steady state is stable

I need to show that the steady state of this non-dimensional model is stable using minimal algebra however I am not sure how to approach this without long lines of working. The model is: $$\frac{dS}{...
user00134857693's user avatar
1 vote
1 answer
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Cannot find non-trivial analytic equilibrium to ODE system, despite plots appearing to tend to steady state.

I have a system of 6 non-linear, ordinary differential equations: $$ \frac{dA}{dt} = r_A(1-f_A)m(N_1,k_{N_1})A - bAC $$ $$ \frac{dB}{dt} = r_B(1-f_B)(pm(N_1,k_{N_1})+m(N_2,k_{N_2}))B + b(A+B)C $$ $$ \...
G-Shillcock's user avatar
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How do we determine which steady state a given set of initial condition approaches in a system?

If we have an ordinary differential equation system with multiple steady states, how do we determine which steady state a given set of initial condition approaches in a system?
Brian Nguyen's user avatar
4 votes
0 answers
51 views

Find steady state of AIDS epidemic model

AIDS epidemic in a homosexual population The following diagram shows the AIDS epidemic in a homosexual population: Then the model can be described by $$ \begin{gathered} d X / d t=B-\mu X-\lambda c X ...
WhyMeasureTheory's user avatar
1 vote
1 answer
473 views

Biological interpretation of equilibrium points

To answer the question what an equilibrium point biologically means, I found the following knowledge on internet and I made a gist out of it. When we say $N^*$ is an equilibrium point, we understand ...
Manjoy Das's user avatar
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1 vote
1 answer
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What do the (high) values in a controllability matrix mean?

Consider a linear state space system $ \dot{x} = Ax + B$, with $x$ being a vector of state variables, and $A$ and $B$ being known matrices. I checked the controllability matrix of the system $Co = [B \...
user313866's user avatar
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Stability of two consecutive equilibrium points

I was studying the existence of two species in a ecosystem. I was thinking if there could be two consecutive stable equilibrium points. I don't have a valid proof in this regard. But if geometrically ...
Manjoy Das's user avatar
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2 votes
1 answer
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Discrepancy: steady state of scaled Brownian motion on sphere is not a harmonic function

Brief Background If you are familiar with Brownian motion in the sphere via SDEs in local coordinates/heat kernel, you can easily skip to the question. Otherwise, here is a brief background, which I ...
Nap D. Lover's user avatar
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Doubt in Example 4.3.4 of Brémaud's Markov Chains book

I'm reading Brรฉmaud's Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues 2nd edition and I'm not following example 4.3.4 in page 159. Here's what's in it: Let $A$ be a square matrix of ...
Yagger's user avatar
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2 votes
1 answer
90 views

LQR definitions

I have to define the choice of parameters I have chosen to create an LQR controller for a drone, and I have written the following: High penalties in the Q matrix mean that the state will try to ...
Michael Adlerstein's user avatar
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183 views

Markov chain converges to the same steady state for different initial probability vectors.

I was asked to write a code to simulate the following Markov chain, and find the PMF of the random variable $X$: The code I've written for simulating the given Markov chain: ...
nothatcreative5's user avatar
1 vote
1 answer
116 views

If steady states of a dynamic system exist only as limits, are they actually steady states?

I have a nonlinear dynamic model in discrete time. A simplified version of my dynamic system is: \begin{equation} x_{t+1} = \frac{1}{1 + \exp(f(x_t))} \end{equation} where $$f(x_t) = โˆ’\beta \left(2d \...
Esperanta's user avatar
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Exsitence of stationary distribution for M/M/1 with non-homogeneous poisson arrival rate

Consider an M/M/1 queue where arrivals occur at rate $\lambda(t)$ according to a Poisson process at time $t$ and move the process from state $i$ to $i+1$, and service times have an exponential ...
one user's user avatar
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Periodic Summation Response. How to separate its Transient and Steady-State Expression?

Background My question comes from here, it's a response of 1st order LPF RC circuit from an arbitrary periodic input. How to determine the transient response of a circuit to causal periodic inputs? ...
Unknown123's user avatar
1 vote
0 answers
47 views

Regarding an equation in an iteration method

I am going through a paper on an iterative method related to discrete optimal regulator, G. Hewer, "An iterative technique for the computation of the steady state gains for the discrete optimal ...
Shab's user avatar
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1 answer
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Terminology for stability of equilibria in systems with discrete state-space

What is the proper terminology to describe the stability of a fixed point in a system with a discrete state-space? The states compose a high dimensional discrete torus, and the fixed point in question ...
an instance's user avatar
1 vote
1 answer
188 views

Fixed point versus point with slope $0$

When solving a first order differential equation $$\frac{dy}{dx} = f(y)$$ for the fixed points (or steady states), we set the differential equation to $0$ and solve for the values of $y$ that are ...
An Ignorant Wanderer's user avatar
1 vote
0 answers
50 views

Eikonal equation with zero right-hand side

I have the eikonal equation: $$\left|\frac{du}{dx}(x)\right| = 0, \, x\in \Omega, \quad u(x) = f(x), \, x\in \Gamma.$$ Does the above have the same solution as: $$\frac{du}{dx}(x) = 0, \, x\in \Omega, ...
lightxbulb's user avatar
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1 answer
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Do some nonlinear PDE steady state solutions depend on initial conditions (non unique)?

I was told by a colleague that for some nonlinear PDEs the initial conditions can change the steady-state solution. So can a stable steady solution depend on the initial conditions for nonlinear PDEs? ...
E. Nerney's user avatar
2 votes
0 answers
28 views

Separating a harmonically driven ODE into oscillating and non-oscillating components

Consider the first-order ODE $$ x'(t) = f(t) \cos(\omega t) - \gamma x(t) $$ whose exact solution is $$ x(t) = x(0) e^{-\gamma t} + \int_0^t e^{-\gamma (t-t')} \cos(\omega t') f(t')~dt' $$ For simple ...
Endulum's user avatar
  • 266
3 votes
1 answer
169 views

Stability of steady states using the Jacobian (linear approximation)

I'm studying the stability of steady states by means of the eigenvalues of $J$. So far the criteria is this: All eigenvalues $\gt 0 \implies$ unstable All eigenvalues $\lt 0 \implies$ stable. In ...
Cate's user avatar
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258 views

Taking the steady state of a matrix

I'm finding the steady state of this matrix \begin{bmatrix}0&1/3&0&1/4\\1/2&0&1/3&1/4\\0&1/3&0&1/2\\1/2&1/3&2/3&0\end{bmatrix} I know I first have to ...
V N's user avatar
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1 vote
0 answers
23 views

How to describe stochastic steady states?

Imagine I have the following system of ODEs, with $x\equiv x(t),y\equiv y(t)$, $$ \dot{x}=f(x,y)\\ \dot{y}=g(x,y) $$ If $f$ and $g$ are not stochastic, having the system converge into a steady state ...
sam wolfe's user avatar
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1 vote
0 answers
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Deriving the state transition diagram for Markov Process

I'm preparing for an exam and one of the preparation questions for the exam is the following: Consider a computer system with TWO processors and NO waiting queue. Out of the two processors, one is ...
juimdpp's user avatar
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3 votes
2 answers
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When does a PDE have a steady-state solution?

I just started studying different types of PDEs and solving them with various boundary and initial conditions. Generally, when working on class assignments the professors will somewhat lead us to the ...
Mjoseph's user avatar
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3 votes
1 answer
450 views

Natural-Forced and Transient-SteadyState pairs of solutions

We have the following circuit, where, $u(0)=V_{0}$. The ode that describes this circuit that has $V_{s}$ as input and the voltage $u(t)$ of the capacitor as output is the following: $\dot{u} + \tau u ...
Alex Kps Bdc's user avatar
1 vote
0 answers
27 views

Solving/approximating 2-D Markov chain

I came across the following continuous time Markov chain and would like to know if there is a good way to solve/approximate steady state distribution of the states. Note that the horizontal right and ...
Andrew Yao's user avatar
1 vote
0 answers
215 views

Steady State Solution of a 1D Heat Equation

I was given the 1D heat equation $\frac{\partial u}{\partial t}=u+\frac{\partial^2 u}{\partial x^2}$ with the boundary conditions of $0 < x < \pi$ , $u(0,t)=0$ , $\frac{\partial u}{\partial x}(\...
Andrew Igdal's user avatar
1 vote
1 answer
994 views

Steady-state solution of the 1D heat equation with source term and nonhomogeneous Neumann boundary conditions

I am trying to solve the steady-state solution of the 1D heat equation with a known source/sink term and non-homogeneous Neumann boundary conditions, however I am not sure if an analytical solution ...
farronait's user avatar
1 vote
1 answer
549 views

Steady state solution for a differential equation

Consider the differential equation $\frac{dy}{dt}=ry(1-\frac{y}{a})(1-2by+y^2)$ with $0<a<b$. I wish to answer the two following questions: (1) Find the steady states for the model (2) Plot $\...
user avatar
0 votes
1 answer
504 views

Probability of being in each transient state in an absorbing Markov chain, given that you are not in an absorbing state?

I have tried to derive a generalized answer to this question, but don't know how to check my work. To clarify, I am asking for the probability of being in each transient state of an absorbent Markov ...
Eli Smith's user avatar
0 votes
0 answers
23 views

Constant to power of t in steady-state

I am wondering how to get the steady-state for the following Euler equation. I know that we can get rid of time in subscripts. However, here I have a constant (a) to the power of $t$. Does anyone know ...
Giordano's user avatar
2 votes
1 answer
272 views

How to add a non-zero mean to the equation of state of Kalman filter

The measurement data of the laser gyro is used to establish the noise random process model, and then the Kalman filter state equation is established through the model parameters. First, remove the ...
dengm155's user avatar
-2 votes
1 answer
59 views

how to calculate the steady state mc linear equations

i have the following Markov chains equations and not sure how to solve those: a=.2 a +.5b+0.6d b=.1a+.1b+.2d c=0.7a+.1c d=.4b+.9c+.2d pi=a+b+c+d how do you approach that? i tried -.8a+.5b+.6d=0 but ...
vanetoj's user avatar
0 votes
1 answer
637 views

Can non-irreducible Markov chains converge?

I know that Markov chains that are irreducible and aperiodic are guaranteed to converge and have an invariant distribution, but can a non-irreducible one do too? If so, what would be an example? Also, ...
Melanie's user avatar
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1 vote
0 answers
143 views

Stability of the dynamic system

I have a function ${\dot{\varphi } = \gamma - F(\varphi )}$ (where $\varphi$ - is 2${\pi }$-periodic function) and graph of function $F(\varphi)$. So it's needed to research this graph (to find the ...
Jane's user avatar
  • 63
0 votes
1 answer
459 views

Find a state-space model for the plant

Consider the plant shown in the figure,plant. Find a state-space model for the plant. I know, I have to use the transfer function to get the state-state model. here I have a confusion about the ...
dilru's user avatar
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