Questions tagged [system-identification]

is related to control theory, dynamic optimization and statistical methods to build mathematical models based on some measured data.

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Differential Equation Parameter Identification

Problem Description: I want to find what the change in displacement $y$ and velocity $\dot{y}$ is in response to a change in the frequency represented as a scalar parameter $\mathrm{p}$ (i.e. $\frac{\...
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31 views

How do I know if $u(t)$ and $y(t)$ is linear to each other?

Assume that we have input $u (t) $ and output $y (t) $ from transfer function $G (s) $. How do I know if input $u (t) $ and output $y (t) $ is linear to each other if the model $G (s) $ is unknown?
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14 views

System identification

someone can explain me how to identify a model of an unknown system of this general form: $y(k)=y(k-1)\theta_1 +u(k-1)\theta_2$ I've to figure out the the $\theta$ value in order to reconstruct the ...
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Can a vector of random variables be separated into dependent and independent variation?

Is it possible to uniquely decompose a vector $\underset{L \times 1}{x}$ of $L$ random variables into dependent and independent sources of variation? Suppose we know the distribution $P_x$ of a mean-...
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Use neural network to create a difference equation?

Got this idea. Assume that we have a output data $y (k) $ and input data $y (k) $ and we want to create an difference equation. $$ay (k) + by (k-1) + cy (k-2) + \dots + Q_n y (k-n) = du (k) + eu (...
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1answer
7 views

Finding discrete matrix $A$ from time continuous data - System identification

I'm seeking a way to find discrete $A$ from $P_k = [CA^kB, CA^kM]$ inside a hankel matrix. I can show you an example to find $A$ if we only have $CA^kB$ for $i = 1, 2, 3, 4, 5, \dots, n$ Assume that ...
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28 views

Observer Kalman Filter Identification - Why does my markov parameters jump so much?

First of all. This is not a programming question. I do not requesting help about programming. I'm requesting help about if I have made correct math steps here. Second, I got some issues with my web ...
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How can I determine $A, B, C, D, K$ from subspace? System identification

Assume that we want to find this model: $$x(k+1) = Ax(k) + Bu(k) + Ke(k)\\ y(k) = Cx(k) + Du(k)$$ And we want to set it up in this problem form: $$ \begin{bmatrix} x(1) & x(2) & x(3) & \...
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20 views

Least-Squares System Identification of combined waveshaper + FIR system

I have a system that consists of a polynomial waveshaper $s(x) = s_0 + s_1x + s_2x^2 + \dots + s_lx^l$ followed by a causal FIR filter with coefficients $h = \begin{bmatrix} h_0 & h_{1} & h_{...
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37 views

How can I estimate a kalman gain matrix from system identification method?

Assume that we have the input data $u$ and output data $y$ and we want to estimate a state space model $$\dot x = Ax + Bu\\ y = Cx$$ But we want also to find the kalman gain matrix $K$. I found a ...
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29 views

A priori structural identifiability analysis of a system of ordinary differential equation using differential algebra

Consider an ordinary differential equation model of a dynamic system: $\dot{x} = f(x,u,p)$ $y = g(x,p)$ where $x$ is the n-dimensional state vector, $u$ is the r-dimensional input vector, $p$ is ...
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131 views

Akaike's Final Prediction Error (FPE)

Considering a zero mean weak (wide-sense) stationary $p^\mathrm{th}$ order AR process $\eta(t)$, such that \begin{equation*} \eta(t) = \sum_{k=1}^p \alpha_k \eta(t-k) + \varepsilon(t),\quad \text{...
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21 views

Parameter identifiability and collinearity

I am working on a simple differential equation model (with 8 parameters) and try to estimate the parameters regarding datas. I tried MCMC methods but the chains do not seem to converge, even after a ...
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38 views

Is there a way to check true/false stability in a discrete transfer/state space model?

Assume that we have a discrete transfer function $H(z)$ and a discrete state space model $x(k+1) = Ax(k) + Bu(k)$. I know how to check the stability, but computing the eigenvalues from the ...
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25 views

Is chaning the reference gain a good control strategy? Feed forward control with system identification?

Assume that we have a estimated system: $$\hat G(s)$$ And we want the system $\hat G(s)$ follow the reference $r(t)$. If we add an input $u(t)$ to $\hat G(s)$ we will get a output response: $$y(t) =...
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How to estimate a delay?

Assume that we are using recursive least squares to estimate a transfer function $$A(q)y(t) = B(q)u(t)$$ But the input $u(t)$ is delayed with $d$ time, eg: $$A(q)y(t) = B(q)u(t-d)$$ How can I find ...
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241 views

Who is Ho in the Ho-Kalman algorithm?

It is mentioned in the following article https://cacm.acm.org/news/210107-in-memoriam-rudolf-kalman-19302016/fulltext that (with attribution which I suspect is wrong) He also worked with Yu-Chi ...
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49 views

Avoid computing the inverse - Extended Least Square

The extended least square estimates this polynomial equation: $$A(q)y(t) = B(q)y(t) + C(q)e(t)$$ By using: $$\epsilon(t) = y(t) - \phi^T(t-1)\hat \theta(t-1)$$ $$\hat \theta(t) = \theta(t-1) + P(t)...
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221 views

Batch Least squares

I need to solve in Matlab a Least squares problem, \begin{equation} \begin{bmatrix} x^2 \\ ux \\ u^2 \end{bmatrix}^T\begin{bmatrix} H_{xx} \\ 2H_{xu} \\ H_{uu} \end{bmatrix} = y \end{equation} ...
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71 views

linear model matrix identification with least squares

I need to do a linear model identification using least squared method. My model to identify is a matrix $A$. My linear system is: $[A]_{_{n \times m}} \cdot [x]_{_{m \times 1}} = [y]_{_{n \times 1}}$ ...
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1answer
148 views

System Identification in a Closed-loop feedback system

I have an unstable system that doesn't have a useable output when open-loop excitation is applied. Subsequently I've used a controller to control its output. I want to use the system identification ...
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3answers
72 views

Suppose $\dot{x}=Ax$, where $A \in \mathbb{R}^{2\times2}$. Determine $A$ from the initial and final states

Two experiments were done: $x(0)= [1,1]^T$ and $x(1)=[4, -2]^T$ $x(0)= [1, 2]^T$ and $x(1)=[5, -2]^T$ How can I find the $A$ matrix? I know how to find $e^{At}$, for $t=1$, using these $2$ ...
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220 views

How can I find the initial state vector from state space model?

Assume that we have input $u(k)\in \Re $ and output $y(k) \in \Re$ and we estimate the black box model by using subspace identification method. $$x (k+1) = Ax (k) + Bu(k) $$ $$y (k) = Cx (k) $$ If ...
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1answer
71 views

Model reduction of estimated state space models - System identification

Assume that we have a dynamical model in form of this simple transfer function $$G(s) = \frac{1}{2s^2 + 5s + 4}$$ G = tf(1, [2 5 4]) We do a step response with ...
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63 views

How can I reduce noise from measurement without a Kalman Filter?

I'm going to create an adaptive Model Predictive Controller (MPC). The model is a state space model. Due to noise, it's very difficult to determine the model order. I'm using subspace identification ...
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1answer
159 views

How can I compute recursive QR-factorization?

I wonder if it's possible to find the $Q$ and $R$ matrices from this QR-equation with only compute QR at one time only: $$A = QR$$ if, the first column of $A$ got removed and then a new column got ...
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484 views

Is there any rule of thumb when it comes to selecting control/predict horizon for MPC?

I have a simple question: Is there any rule of thumb when it comes to selecting control/predict horizon for MPC? Normaly I set control and predict horizon equals, but I have heard that's not good ...
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55 views

sample covariance matrix

Suppose two covariance function estimators, with the same formula except for a coefficient. Then make two sample covariance matrix(SCM) from each of the functions. Why should these matrices differ in ...
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Recommended book on “Dynamic Model Learning” related to “System Identification”?

I need to take a course on so-called "Dynamic Model Learning" related to "System Identification" + "Machine Learning stuff". But my backrgound is more on Computer Science and Electrical Engineering ...
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1answer
247 views

Getting differential equation from Boyle's law

I'm trying to simulate pipe installation with known amount of air in Simulink and design a PI controller for it. I was able to construct equations which reflects well my real installation. My pressure ...
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1answer
67 views

Nonlinear system identification with known model

I would like to ask for suggestions regarding system identification with known model structure, but without known parameters. The model is a model of a physical system, it can be assumed that it is ...
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193 views

What's the point of having a reference model in Model Reference Adaptive Control?

The MRAC(or called MRAS where S = 'System') controller is called a adaptive controller. I don't know why, because it have no system identification process loop. But what's the point to having a ...
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103 views

How to derive a recursive version of a regularised cost function

I am to derive a recursive version of the following cost function and examine for which choice of D can we have a estimator windup $V(\theta) = \frac{1}{2}\sum_{t=1}^n(y(t)-\phi(t)^T\theta)^2 + \...
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1answer
148 views

System identification of a resonant system

I want to write a matlab script that would identify a system from it's inputs and outputs. I have so far had good results with simple systems, but with this slightly more complex one I'm not able to ...
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Grey-box system ID for a set of DAEs

I have a system of 1st order odes given by $$ \dot{x_1}(t) = \alpha_1 f_1(x_1,t) + \beta_1 u(t) \\ \dot{x_2}(t) = \alpha_2 f_2(x_2,t) + \beta_2 u(t) $$ They are constrained by an algebraic equation ...
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1answer
50 views

Identification or characterisation of time-varying parameter of a first order ODE

I have a first order ode given by $$\dot{x}(t) = \alpha f(x,t) + \beta u(t)$$ where $\left(\alpha,\beta\right)\in \mathbb{R}$ are known constants (i.e. parameters). Starting from a rich set of ...
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39 views

Identification of part of RHS of a standard ODE from input-output simulation data

From first principles, I know that the best reduced-order mathematical model that describes my complicated physical system is $$\dot{x}(t) = \alpha f(u,t) + \beta u(t)$$ where $\left(\alpha,\beta\...
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1answer
437 views

I can't get my Recursive Least Square algorithm work - What have I miss?

I'm want to do a recursive least square algorithm but I can't get it to work. If you don't know what recursive least square algorithm is. Well, it just ordinary least square, but it's an algorithm ...
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70 views

How should I interprent the Recursive Least Square algorithm?

This is the recursive least square algorithm. $$\hat{\theta}(t) = \hat{\theta}(t-1) + K(t)(y(t) - \phi^T(t)\hat{\theta}(t-1)) \\ K(t) = P(t)\phi(t) \\ P(t) = (I-K(t)\phi^T(t))P(t-1)$$ As I know, I ...
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1answer
728 views

MIT rule VS Lyapunov design - Adaptive Control

I wonder what's the difference between MIT rule and Lyapunov design when it comes to adaptive control law? As I get it, MIT rule is more like "transfer function"-based control system and Lyapunov ...
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1answer
191 views

Structured transfer function estimation in MATLAB?

I have some input-output data for a dynamical system (input = stimulus, output = observation). Assuming that this system is linear time-invariant, I am trying to estimate a transfer function $H(s)$ of ...
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62 views

How do I get the amplitude of a variable sine function?

Let's assume that we have a dynamical system $$G(s) = \frac{Y(s)}{U(s)}$$ And we have measure the input $u(t)$ And we have measure the output $y(t)$. Here we can see that if we increase the ...
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Can I use least square method to estimate a Box-Jenkins model?

I have the known measurement: Noise $e(t)$, output $y(k)$, input $u(k)$. I need to find the Box-Jenkins model: $$y[k] = \frac{B(q)}{F(q)}u[k] + \frac{C(q)}{D(q)}e[k]$$ Where $B, F, C, D$ are ...
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39 views

Estimate a transfer function from an arbitrary response?

Assume that we have a step answer which look like this: This step answer is from a difference equation. Called a discrete ODE: $$y[k] + a_1y[k - 1] + a_2y[k - 2] = b_1u[k] $$ Which can be a ...
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1answer
124 views

How do I find the noise for ARMAX model - System identification

A discrete transfer function can be described like this: $$ y[k] = \frac{B(q^{-1})}{F(q^{-1})}u[k]$$ Where can be a polynomal expression: $$B(q^{-1}) = b_0+ b_1q^{-1} + b_2q^{-2} + b_3q^{-3} + \dots ...
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417 views

Estimate a state space model by using Singular Value Decomposition

This article is from 1989 and it contains an algorithm how to estimate a state space model from arbitrary input and output. The algorithm begin with to create two hankel matrices. $$H_1 = \begin{...
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3answers
153 views

Identify a transfer function from frequency data?

Assume that I want to identify a transfer function from frequency data. Assume that I have a transfer function step response which look like this: By experience, I would say that this step response ...
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53 views

How can I interpret the discrete impulse response measurement?

Assume that we have our discrete transfer function $G(e^{j\omega_k})$ where, $k = \frac{\pi k}{M} , k = 0, \dots , M$. If we want to find Markov-parameters $g_k$ from the impulse response $$g_k = \...
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90 views

How do I find $L_{22}$ from LQ-decomposition when using Hankel matrices

Assume that we have input vector $y_k \in \Re^{pxN}, u_k \in \Re^{mxN}$. The vectors can be interpreted as: $$u_k = (u_0 , u_1, u_2, u_3, \dots , u_{N-1})$$ $$y_k = (y_0 , y_1, y_2, y_3, \dots , y_{...
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95 views

Identify a state space model from measured inputs and outputs

I'm using subspace identification to identify a black-box state space model. To identify I follow these steps: We have measured a vector of inputs $u_k \in \Re^{m}$ and outputs $y_k \in \Re^{l}$. For ...