Questions tagged [steady-state]

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

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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 ...
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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: ...
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Stability of a system of nonlinear difference equations

My problem: I have a system of two nonlinear difference equations. How do I analyze the stability of the steady state? (Or where to start the analysis at least?) $$ q_1(x_{1n},x_{1,n+1},x_{1,n+2},x_{...
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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) = −\beta \left(2d \...
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Are these steady states of non linear dynamic system actually steady states?

I have the following non linear dynamic system in discrete time: \begin{equation} x_{t+1} = \frac{1}{1 + \exp\left(- \beta \left( 2 d \left(c + \frac{(1 - c)}{1 + a (1 - x_{t}) d}\right) - b - d \...
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How to solve the Ordinary differential equation with denominator having functional variable

How to solve the Second order Ordinary differential equation with denominator having functional variable x''(y)+x'(y)-(x(y)-z(y))/(x(y)+z(y))=0
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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 ...
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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? ...
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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 ...
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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 ...
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1 vote
1 answer
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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 ...
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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, ...
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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? ...
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Steady-state error of operator splitting schemes

Reaction-diffusion equations are typically solved by splitting the problem into two sub-problems: diffusive transport (D) reactions/chemistry (R) Operator splitting (OS) schemes are used to retain ...
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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 ...
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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 ...
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Fraction of Time Markov Chains

I need help regarding the ending of a question. Suppose that if it has rained for the past three days, then it will rain today with probability 0.8; if it did not rain for any of the past three days, ...
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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 ...
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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 ...
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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 ...
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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 ...
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3 votes
1 answer
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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 ...
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Example of a system of N coupled non-linear first order ODEs having a stable fixed point

Let $\vec{u}(t) = (u_1(t), \dots, u_N(t))$ be a well-behaved function of $t \in \mathbb{R}$ that takes values in $\mathbb{R}^N$ and satisfies: \begin{equation} \frac{\mathrm{d} \vec{u}}{\mathrm{d}t} = ...
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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 ...
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Setting up differential-difference equations

I've spent some time trying to wrap my head around differential-difference equations, I've found this question from my University's past paper library: In this question we aren't given much, we don't ...
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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}(\...
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1 vote
1 answer
316 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 ...
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1 vote
1 answer
89 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 $\...
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1 answer
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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 ...
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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 ...
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2 votes
1 answer
88 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 ...
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-2 votes
1 answer
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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 ...
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0 votes
1 answer
126 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, ...
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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 ...
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1 answer
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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 ...
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1 vote
2 answers
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Why is the Steady State Response described as steady state despite being multiplied to a negative exponential?

I'm evaluating Newton's Temperature Model $$dT/dt = k(T_e-T)$$ to find the response of the system.$T_e$ and $T$ are both functions of t. The response evaluates to $$T=e^{-kt}\int e^{ks} T_e(s)ds + ...
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1 answer
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Sequential dynamical system

I would like to understand the following example based on the following definition Definition: Alternatively, one can choose to update the states of the variables according to some fixed update order, ...
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Solving Differential Equations for Steady States

I have been looking at the following equations in an article* and wanted to know how the $E(∞)$ was derived. By substituting $\gamma E$ for $A$ and factorising it is easy to see how $E(∞)$ can equal 0 ...
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1 answer
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Equilibrium point stability given the derivative

Given a system described by the following equation: $$y'' + y'^4 + y'^2*u + y^3 = 0 $$ where $y(0) = 0$ and $ y'(0) \neq 0 $ , what is the stability of the equilibrium point? The eq. point is $0$ and $...
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1 vote
0 answers
68 views

Steady state error of Nonunity-feeback Systems

I am reading the book "Autotamic Control Systems" by Farid Golnaraghi and Benjamin C.Kuo, Tenth Edition. In the book: Figure 1 Consider the nonunity-feedback system above, where r(t) is ...
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  • 111
1 vote
2 answers
99 views

Proving that non-absorbing Markov States have steady state probability of $0$

Suppose that I have a Markov chain that has absorbing states. Since there are absorbing states, lets group the Markov matrix into four blocks: the submatrix all states in the absorbing region(s) $A$, ...
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5 votes
0 answers
191 views

Steady state of diffusion-advection on the torus

Let $P$ be a positive scalar function and $\mathbf{v}(\mathbf{x})$ is an assigned smooth vector field. The quantity $P(t,\mathbf{x})$ evolves according to a transport equation of the kind $$ \...
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What would be the steady state of x in following system X -k-> 2X, 2X -k*v(t)-> ∅ when v(t) = vo (const)?

I have a system of a mass action type. $\require{AMScd}$ \begin{CD} X @>{\text{$k$}}>> 2X, 2X @>{\text{$k\cdot v(t)$}}>> ∅ \end{CD} When $v(t) = v_0(const)$, what is the stable ...
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1 vote
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Interpreting the reciprocal of the reproductive ratio, $R_0$

$R_0$ is the average number of secondary cases arising from a single infectious individual in a fully susceptible population. In many of the compartmental models for epidemiology, the parameter $\...
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1 answer
143 views

What is the steady state probability?

A lot of board games involve rolling dice and moving around a cyclical board. Monolopy is the most common example. On the 16 position board below, the player’s piece was on the bottom row as depicted ...
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1 answer
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Board game: steady state probability of being in starting location (GO)

Considering a board game which has 20 positions only. There are 2 fair dice that can be rolled and the player moves in a clockwise position over the board positions. Considering that the dice are fair ...
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209 views

Application of PDE for solving steady state heat problem.

A thin rectangular homogeneous thermally conducting plate occupies the region $0 \leq x \leq a$, $0 \leq y \leq b$. The edge $y = 0$ is held at temperature $Tx(x − a)$, where T is a constant and the ...
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I could use some direction or help to see if I am on the right path for this problem.

The problem is to solve the PDE: $$ \frac{\partial^2 u}{\partial t^{2}} - \frac{\partial}{\partial x}\left(K(x)\frac{\partial u}{\partial x}\right) $$ subject to the following boundary conditions:...
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What's the difference between indeterminacy and explosiveness in the context of dynamic systems?

One important question to ask is if the model has as unique stable (asymptotically stationary) solution (determinacy) or multiple solutions (indeterminacy). But what's the difference between ...
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1 answer
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Time to reach equilibrium in convection-diffusion

I have a convection-diffusion PDE in the form of: $$\frac{{\partial x}}{{\partial t}} = w\frac{{{\partial ^2}x}}{{\partial {z^2}}} - \frac{{\partial x}}{{\partial z}}$$ Assuming I know the initial ...
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