7
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 ...
- 20.5k
6
votes
Accepted
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 ...
- 191
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 (...
- 75
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-...
- 92.4k
5
votes
Accepted
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}...
- 26k
5
votes
Accepted
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,...
- 13.3k
4
votes
Accepted
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" ...
- 20.5k
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 ). ...
- 1,117
4
votes
Accepted
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)...
- 28.4k
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 ...
- 13.3k
4
votes
Accepted
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 ...
- 92.4k
3
votes
Accepted
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 \\ \...
- 28.4k
3
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:
$$\frac{\alpha(z-1)}{z+1}...
- 31
3
votes
Accepted
$N$ point moving average filters in state space
Yes sure they are LTI. Let $A$ be the $(L-1)\times (L-1)$ shift matrix
$$
A := \begin{pmatrix}0 & 1 & 0 && \dots & 0\\0 & 0 & 1 & 0 &\dots &0\\\vdots &&...
- 768
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 ...
- 2,891
3
votes
Accepted
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, ...
- 5,040
3
votes
Accepted
States transformation of the bilinear transform
The Tustin approximation is concerned with transfer functions, i.e. relations between inputs and outputs. In state space representation
$$ \dot{\mathbb{x}}(t) = A \mathbb{x}(t) + B \mathbb{u}(t) $$
$$ ...
- 2,393
3
votes
Accepted
How to linearize this state space model and write it in discrete form?
You have
$$
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{...
- 13.3k
3
votes
Accepted
State space transformation
Start with $z(t) = \begin{bmatrix} \dot \theta & \theta & \dot x \end{bmatrix}^T$. Take it's derivative:
$$
\frac d {dt} z(t) = \begin{bmatrix}\dot \theta \\ \theta \\ \dot x\end{bmatrix} = \...
- 13.3k
2
votes
Accepted
Discrete time non-linear time invariant system dynamics descriptions (state-space or input-output relationship)
Well, any input-output representation obviously admits a state-sapce form. for your equation in $y[k]$ you can easily construct one as follows. Create a "shift" system (an integrator chain) as
$$
\...
- 768
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....
- 1,420
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: ...
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 ...
- 2,232
2
votes
Accepted
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 ...
- 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 ...
- 121
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#...
- 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 $x_r=\mathrm{Re}(x)$, $x_i=\mathrm{Im}(x)$, $A_r=\mathrm{Re}(A)$, $A_i=\mathrm{Im}(A)$, $...
- 994
2
votes
Accepted
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), ...
- 26k
2
votes
Accepted
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 ...
- 457
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