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13 votes

Why eigen values and poles of a system are equivalent?

Let's consider a discrete-time state space model (the derivation for a coninuous-time system is completely analogous): $$\begin{align}\mathbf{q}[n+1]&=\mathbf{Aq}[n]+\mathbf{b}x[n]\\ y[n]&=\...
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  • 79.3k
11 votes
Accepted

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 ...
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  • 39.6k
8 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" ...
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7 votes

When Is a Kalman Filter Different from a Moving Average?

I won't add any equations, I will just add some intuition. I will also limit my self for Additive Gaussian White Noise. Now, in that case the Kalman filter can written as a Least Squares problem to ...
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  • 39.6k
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 (...
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  • 61
5 votes

When Is a Kalman Filter Different from a Moving Average?

as the first answer (with the most votes) says, the kalman filter is better in Any case when signal is changing. Notice the problem statement These use the algorithm to estimate some constant voltage. ...
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  • 720
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 ...
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  • 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-...
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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}...
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  • 21.8k
4 votes

When Is a Kalman Filter Different from a Moving Average?

Another take: The Kalman Filter lets you add more information about how the system you're filtering works. In other words, you can use a signal model to improve the output of the filter. Sure, a ...
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  • 21.8k
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 ). ...
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  • 1,072
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)...
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  • 26.6k
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 ...
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  • 8,151
3 votes
Accepted

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 ...
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  • 2,636
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 &&...
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  • 738
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, ...
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  • 4,852
3 votes
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Expectation maximization of moving average with binary source input

I think your E-step is correct(only one term missing in the last expression: $-N\ln\sigma_w$ ). To obtain the M-step you have to differentiate with respect to all your parameters. You don't have to ...
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3 votes

Can a state space model have changing state size over time?

Short answer: Yes but no. Long answer: If a system matrix is rectangular then it means either; There are more number of states than the number of their derivatives which is not meaningful. If you ...
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  • 502
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 \\ \...
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  • 26.6k
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{...
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  • 8,151
2 votes

Why is my discretized transfer function unstable when my discretized state-space model is stable?

You're using a high order model so you might be running into numerical precision issues. Make sure the realization of model_state_space is minimal. You can do this with ...
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2 votes

kalman filter with time-varying noise?

1) It depends on what you call the standard Kalman filter -- I will call the equations in the picture below to be the "standard Kalman filter". You can easily derive an expression for the Kalman ...
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  • 1,071
2 votes
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Doubt in state space representation

Your FIR state space representation seems to be doing too much. The way I would write it is to have the process noise is $\epsilon(t)$ as your input, and your state as two time-delayed copies of it: ...
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  • 21.8k
2 votes
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How to find the output signal of a filter using state space matrices?

We don't know anything about your system, but general rule is: $$\begin{array} &y[n] &= \mathbf{C}x[n] + \mathbf{D}u[n]\\ x[n+1] &=\mathbf{A}x[n] + \mathbf{B}u[n] \\ \end{array}$$ Where $...
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  • 10.4k
2 votes
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How to represent the nonlinear model as a state space in Unscented Kalman Filter

Rather than write $$ x_t = Ax_{t-1} + f(u_{t-1},\mathbf{w}) $$ as the state update equation, I'd write: $$ \xi = \left[ \begin{array}{c} x_t\\ u_t \end{array} \right] $$ and then $$ \xi_t = g(\...
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  • 21.8k
2 votes

Why is it necessary to have two state variables

Let's rewrite your system as $$ m \ddot{x}(t) + b \dot{x}(t) + k x(t) = f(t) $$ then you can see what you're saying: Everything I need to know is in $x(t)$! But is it really? State space systems ...
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  • 21.8k
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 $$ \...
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  • 738
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: ...
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2 votes
Accepted

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....
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