7 votes
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What sensors can be fused using the Kalman Filter framework

Remark: I will answer this using the Linear framework of the Kalman Filter but the idea is the same. The Kalman Filter basically propagate and fuses Gaussian Distributions in order to calculate the ...
Royi's user avatar
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6 votes

State space physical meaning

As you pointed out, there are many state-space realizations of one particular transfer function. The reason is that a transfer function only represents the input-output behavior of a system (...
Gab's user avatar
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5 votes
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Bilinear transformation of continuous time state space system

I've had the same question last week, but I've managed to find how to derive it (getting rid of those $z$ terms is indeed tricky). I will give here detailed demonstration of how to arrive to the ...
Klaz's user avatar
  • 181
5 votes

System classification: unit-time delay

You are right that a distributed system could be "something like a transmission line". Note that the system $$y(t)=x(t-T)\tag{1}$$ is a simple model of a transmission line, where just a frequency-...
Matt L.'s user avatar
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5 votes
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Linear Time-Invariant system without State-space form

This seems like a homework problem, but I'll bite. The thing with a state-space representation is that it needs to have finite dimension: $$ x_{k+1} = \mathbf{A} x_k + \mathbf{B} u_k\\ y_k = \mathbf{C}...
Peter K.'s user avatar
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5 votes
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Intuition for $\mathbf{P} = \mathbf{0}$ in steady-state when $\mathbf{Q} = \mathbf{0}$ (Kalman filter)

We each have different life experiences to fuel our intuition, but try this one out: Let $\mathbf A = 1$ and $\mathbf Q = 0$, and $\mathbf C = 1$ -- i.e., the actual state variable just doesn't change,...
TimWescott's user avatar
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4 votes
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Discrete State Space Model - Why Are We Calculating $ x \left[ k + 1 \right] $ Instead of $ \dot{\boldsymbol{x}} \left( t \right) $?

I will ask you something that will give you intuition. How would you calculate the Gradient of an image? Image is a discretization of reality, so how would you estimate the gradient of the "Reality" ...
Royi's user avatar
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4 votes

Discrete State Space Model - Why Are We Calculating $ x \left[ k + 1 \right] $ Instead of $ \dot{\boldsymbol{x}} \left( t \right) $?

Hi: I've been wondering about the same exact thing myself and the light bulb finally turned on a few days ago when I went back to Kalman's 1960 paper. ( I've read it many times but not recently ). ...
mark leeds's user avatar
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4 votes
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What is wrong with my residue partial expansion method? (Transfer Function into State-Space Modal/Diagonal Form)

The discrepancy between your derivation and matlab's computation results because of a convention mismatch you used during the partial fraction expansion: Given that the function to be expanded is $H(s)...
Fat32's user avatar
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4 votes

What sensors can be fused using the Kalman Filter framework

Are there types of measurements that are not compatible for sensor fusion? Can any measurement be fused to better inform the underlying model? Any sensor that gives you more information about the ...
TimWescott's user avatar
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3 votes
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Got stack in calculating state-space representation

Model your deterministic input as a $2 \times 1$ matrix : $$ \begin{bmatrix} u \\ T_a \\ \end{bmatrix} $$ Then your state space equations will be: $$ \begin{bmatrix} \dot x_1 \\ \dot x_2 \\ \...
Fat32's user avatar
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3 votes
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$N$ point moving average filters in state space

Yes sure they are LTI. Let $A$ be the $(L-1)\times (L-1)$ shift matrix $$ A := \begin{pmatrix}0 & 1 & 0 && \dots & 0\\0 & 0 & 1 & 0 &\dots &0\\\vdots &&...
LJSilver's user avatar
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3 votes
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MATLAB: Implementing Least Squares Estimator for a Given Model

The equation you're trying to solve is $$ \mathbf{y}=\mathbf{X}\mathbf{h}, $$ where $\mathbf{h}$ is your unknown. The matrix $\mathbf{X}$ is going to have a time-shifted structure that reflects the ...
David's user avatar
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3 votes
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Control design: under what conditions can closed-loop poles be placed arbitrarily?

Unnecessary additional information that might help From a pragmatic point of view, there might be a problem when trying to control a given plant using only its transfer function $G_p(s)$. In general, ...
Tendero's user avatar
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3 votes
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How to linearize this state space model and write it in discrete form?

You have $$ f\left(\mathbf x, u\right) = \begin{bmatrix}\frac{-1}{T}\tau+\frac{K}{T} u \\ \frac{\tau}{mr} \\ 0 \end{bmatrix} \tag a $$ From which you (eventually) derive $$ \mathbf {A}_d=\begin{...
TimWescott's user avatar
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3 votes
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State space transformation

Start with $z(t) = \begin{bmatrix} \dot \theta & \theta & \dot x \end{bmatrix}^T$. Take it's derivative: $$ \frac d {dt} z(t) = \begin{bmatrix}\dot \theta \\ \theta \\ \dot x\end{bmatrix} = \...
TimWescott's user avatar
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3 votes
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Why do they say that complex exponentials are eigenfunctions of LTI systems, when there are still transient responses?

Note that you only get a transient if you switch the system (or the input) on at a certain finite point in time. If the input $x(t)=e^{s_0t}$ has existed forever, the output is given by the ...
Matt L.'s user avatar
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2 votes
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Discrete time non-linear time invariant system dynamics descriptions (state-space or input-output relationship)

Well, any input-output representation obviously admits a state-sapce form. for your equation in $y[k]$ you can easily construct one as follows. Create a "shift" system (an integrator chain) as $$ \...
LJSilver's user avatar
  • 768
2 votes

Bilinear transformation of continuous time state space system

I wanted to add to this some specifics on the MATLAB implementation. Continuous time system: $$\dot{x} = A x + B u$$ Laplace transform: $$sx = Ax + Bu$$ Tustin discretization: $$\frac{\alpha(z-1)}{z+1}...
Siva's user avatar
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2 votes
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Learner level information on Kalman filtering for different input kinds

I am most convinced you should try Hidden Markov Chains (which i am not explaining here), just justifying its use: Kalman Filter: continuous state space, discrete observations https://stats....
Brethlosze's user avatar
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2 votes

MATLAB: Implementing Least Squares Estimator for a Given Model

If X is your design matrix then the matlab implementation of Ordinary Least Squares is: h_hat = X'*X\(X'*y); I attempted to answer your other question here: ...
Chad Sexington's user avatar
2 votes

A State space model for discrete Sine wave Using kalma filter

You could use a nonlinear Kalman filter, such as the extended Kalman filter (EKF), and track the phase and frequency as your state variables. In this case, your Kalman filter is essentially acting ...
Robert L.'s user avatar
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2 votes
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Python toolboxes for state-space estimation via subspace estimation

You can use pyvib to do frequency based subspace identification. Beware that there is no estimation of the initial state. It is possible to do optimization of the identified model, if the data is not ...
Paw's user avatar
  • 36
2 votes

Kalman Filter EM Estimation of Covariances

I think there is a bug in your code. In KalmanFilter(), observation_matrices=H is probably not what we want. By the reference in ...
Hans's user avatar
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2 votes

Kalman Filter EM Estimation of Covariances

Actually, I have found something. Indeed, it is a complicated problem to estimate transition and observation covariances. In Wikedia it says: https://en.wikipedia.org/wiki/Kalman_filter#...
ABK's user avatar
  • 171
2 votes

How to model state space for complex valued system correctly in SIMULINK (MATLAB)?

You can split up the real and imaginary part of the state into their own seperate states. Namely by defining $x_r=\mathrm{Re}(x)$, $x_i=\mathrm{Im}(x)$, $A_r=\mathrm{Re}(A)$, $A_i=\mathrm{Im}(A)$, $...
fibonatic's user avatar
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2 votes
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Fundamental questions about state-space and Kalman filters

Starting at the top and working my way down. Good questions, by the way! So basically state and output is modelled as gaussian distributions that have slowly changing means (to be estimated), ...
Peter K.'s user avatar
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2 votes
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How does Overlap-Add work for IIR filter?

According to wiki, it seems that overlap add technique usually works for FIR filters. I hasn't read the link you posted and the matrix form for IIR before. But when it comes to buffer for IIR, I have ...
Po-wei Huang's user avatar
2 votes

State-space filters in fixed-point?

Do a similarity transform on the state-space equation*. I'm assuming your original system is $$\begin{matrix} x_n = A x_{n-1} + B u_n \\ y_n = C x_n \end{matrix}$$ Let $T = \begin{bmatrix} 1 & 0 \...
TimWescott's user avatar
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