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17

There are a lot of books out there, but if you are interested in Control and Signal Processing, I strongly suggest you take a look a Stephen Boyd Lectures from standford: http://www.youtube.com/watch?v=bf1264iFr-w There's the first one, the entire course is really valuable and he is a great Teacher. Appart from That here's a good list of my preferred ...


13

The order of system gives the number of memory storing elements in the circuit. What you have written above is a KVL equation of the circuit. But when you determine the order of the system you need to find the transfer function of the system. Laplace domain is normally used. The highest power of $s$ in the denominator in a strictly proper transfer function ...


7

Let's consider a discrete-time state space model (the derivation for a coninuous-time system is completely analogous): $$\begin{align}\mathbf{q}[n+1]&=\mathbf{Aq}[n]+\mathbf{b}x[n]\\ y[n]&=\mathbf{c}^T\mathbf{q}[n]+dx[n]\tag{1} \end{align}$$ where $x[n]$ is the input, $y[n]$ is the output, and $\mathbf{q}[n]$ is the state vector. Taking the $\...


6

Hmmmmmmmmm, interesting question. Since you want to use the second derivative as your criteria, it would seem that you would want to have the maximum second derivative absolutie value for as short of a duration as possible. This would suggest piecing together parabolas, matching the first derivatives at the joints. How to do this algorithmically will take ...


5

Decibels (dB) are used to represent a power ratio with a logarithmic scale. Specifically, a power ratio can be expressed in dB as follows: $$ R|_{dB} = 10 \log_{10}R = 10 \ \log_{10}\frac{P_1}{P_2} $$ What does a negative number of dB mean? Manipulate the above equation a bit: $$ R|_{dB} = 10 \log_{10}R $$ $$ \frac{R|_{dB}}{10} = \log_{10}R $$ $$ 10^{\...


5

$$R\frac{di}{dt}-\frac1Ci=\frac{de_i}{dt}.$$ Isn't that a nice first order differential equation ?


5

You cannot make conclusions about the stability of a system by only considering its transfer function evaluated on the imaginary axis $s=j\omega$. Replacing $s$ by $j\omega$ in the transfer function only makes sense for a stable system, otherwise you get a function of $\omega$ that does not describe the system, but another (stable) system. Let me explain ...


5

Causality is a necessary condition for realizability. Stability (or, at least, marginal stability) is also important for a system to be useful in practice. For linear time-invariant (LTI) systems, which are fully characterized by their transfer function, we get realizability constraints on the transfer function. For continuous-time LTI systems, if we work ...


5

The problem is that you took the derivative of the function $$\hat{x}_u(t)=2e^{-3t}-e^{-4t}\tag{1}$$ whereas using the Laplace transform you implicitly assumed that $x_u(t)$ equals zero for $t<0$: $$x_u(t)=\hat{x}_u(t)u(t)=(2e^{-3t}-e^{-4t})u(t)\tag{2}$$ where $u(t)$ is the unit step function. If you take the derivative of $(2)$ then you'll get the ...


5

First, combine your two variable set of first order differential equations into a single variable second order one. $$ \frac{d^2y}{dt^2} = c \frac{dx}{dt} = acy + bc $$ $$ \frac{d^2y}{dt^2} - acy = bc $$ Solve: $$ y(t) = C_1 e^{\sqrt{ac} \cdot t} + C_2 e^{-\sqrt{ac} \cdot t} - \frac{b}{a} $$ $$ x(t) = \int{ ( ay(t) + b ) dt } $$ $$ x(t) = C_1 \frac{a}{...


5

Two principles here: When dealing with a differential equation, you define intermediate state variables so everything is in terms of first derivatives. This system is nonlinear, so the state-space equations won't be in terms of matrices. Applying these principles, we define a state vector: $$ \mathbf x = [x_1, x_2]^T, $$ where: $$ x_1 = y \\ x_2 = \dot y $$...


4

Imagine that you're heating (or cooling) a home with a modern furnace (or air conditioner). the reference or set point is the temperature that you set your thermostat to be. the feedback signal is the actual temperature that is measured with some kinda thermometer. the actual value that you are trying to control, whether it's temperature or the position ...


4

In electronics, a Schmitt trigger converts an analog signal to a digital signal by using two different thresholds (limits as you call them), one for an output state change from 0 to 1 and another for a change from 1 to 0. The first is somewhat higher so that if the input fluctuates between those two thresholds, then there will be no change in the output. The ...


4

It's important to realize that in practice many types of systems are used, and only some of them can be regarded as (approximately) LTI. The (didactical) advantage of treating LTI systems in a basic signals and systems course is the elegance and relative simplicity of the underlying theory. Stability and causality are easily checked, and the input-output ...


4

A signal is a physical quantity (e.g. voltage) carrying information, or a set of values (e.g. samples in discrete case) of the given function for different values of the underlying independent variable. In the diagram above $\epsilon(t)$ is a mathematical modeling (i.e. function) that characterizes the set of inputs (i.e. scalers) going into the controller....


4

If I understand your question correctly then you're asking under which conditions the inverse $\mathcal{Z}$-transform of a given function $F(z)$ exists. Since the inverse $\mathcal{Z}$-transform is just a Laurent expansion of $F(z)$ around $z=0$, the inverse transform exists if $F(z)$ can be developed in a Laurent series about $z=0$. This is the case if $F(z)...


4

If your system is an ideal differentiator with input-output relation $$y(t)=\frac{dx(t)}{dt}\tag{1}$$ then its transfer function is $$H(s)=\frac{Y(s)}{X(s)}=s\tag{2}$$ From the equation in your question you obtain for its step response $$A(s)=1\tag{3}$$ which in the time domain corresponds to a Dirac delta impulse: $$a(t)=\delta(t)\tag{4}$$ This is ...


3

FIR filters are fairly common in some areas of control theory. As they usually incur a lot of added phase/time-delay, they are not really usable in the feedback path of regular control systems, but they are useful when the added phase/time-delay is not affecting the system in an adverse way, or when the particular phase response and time-delay is desired. ...


3

A great book to learn about digital signal processing in general ist "Understanding digital signal processing" from "Richard Lyons". Amazon-Link. It's easy to read and requiers no previous knowledge. You'll get comfortable with discrete signals, FFT, filters and so on very fast. All the mathematics that you need is described inside this book (you'll need no ...


3

This is definitively not a satisfying answer, but what you call "adaptive" in combination with a pi-controller is commonly called gain-scheduling or lpv-gain-scheduling (lpv = linear parameter varying). Maybe that helps you find a book.


3

If you know the exact response of the camera, you can convert the brightness samples of each pixel to a linear intensity scale and perform the averaging there. That will make your whole problem intensity linear and should solve all your problems. However, I would strongly recommend using a more advanced exposure algorithm. For example you could introduce a ...


3

Because every signal can be decomposed into a linear series of scaled impulses that are shifted in time. Thus applying a linear time-invariant system on each impulse in the series and then summing the results again will give the same result as applying it on the whole signal. Therefore, one only needs to know how the system responds to an impulse to be able ...


3

There's a very simple way to check controllability, indeed if you define the reachability matrix $$ R = \begin{pmatrix}B & AB & \dots & A^{n-1}B\end{pmatrix} $$ then the reachable subspace is the image of R. Hence to check complete controllability you just have to check that $R$ is full rank. First, I think there's an error in the question, $B$ ...


3

For symplicity, consider the SISO linear system \begin{align*} \dot x &= ax +bu\\ y &= cx \end{align*} with $x$,$u$ and $y$ taking values in $\mathbb R$. Assume that you want to stabilize the system by making $y$ to converge to some fixed set point $\bar y$. Then necessarely there must be a point $(\bar x,\bar u)$ such that \begin{align*} 0 &= a\...


3

Well, the two systems differ only at low frequencies. In fact is you define $$ R_L = \dfrac{\tau_zs+1}{\tau_ps+1},\qquad R_I = \dfrac{\tau_zs+1}{\tau_ps} $$ you have that $R_I(j\omega)\underset{\omega\to\infty}{\to}R_L(j\omega)$. Therefore, you can expect that the behavoiur of the closed-loop system differs only at lower frequencies. What does the PI do? ...


3

As is usual with these sort of questions, the answer is "it depends". For example, suppose $$ F(x) = x $$ then the loop is immediate. For example, suppose $$ F(x) = x/2 $$ then the loop never occurs unless $x = 0$. The same will be true for more complex $F$s. For example $$ F(x) = \mathbf{A} x $$ where $\mathbf{A}$ is a matrix. If $\mathbf{A}$ is a ...


3

They are system norms, a metric that you can compare two different systems in terms of their generalized gain and spread. You can look these up no need for attaching physical motivation. You don't. They are for assesing and using in the optimization programs. Yes, but only if you know what you are doing. Can be. But again, they are for assessment and ...


3

You're forgetting the discontinuity at $t=0$. Your step response is actually $$Y(t)=(1+e^{-0.1t})u(t)\tag{1}$$ where $u(t)$ is the unit step function. Differentiating $(1)$ gives $$h(t)=-0.1e^{-0.1t}u(t)+(1+e^{-0.1t})\delta(t)=-0.1e^{-0.1t}u(t)+2\delta(t)\tag{2}$$ where $\delta(t)$ is the Dirac delta impulse. Convolving $(2)$ with a unit step gives the ...


3

You can make a discretized regularized linear equation system. $\bf d$ is the original signal $\bf v$ is what we add to the signal, (the additive change) $\bf v+d$ is the result. So there are two things we want: We want the result to have as small second derivative as possible. We want the change to be as small as possible. $$\min_{\bf v}\left\{\|{\bf D_2}...


3

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" if you're given only the image? In the case above we use Finite Differences to approximate the continuous derivative. So what actually is approximating $ \dot{...


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