Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The external input of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller ...

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37 views

Z-transform and$ H_2$ space

The following is from the preliminaries of a paper. Let $\mathbb{D} = \{z \in \mathbb{C} : |z|<1\}$ be the unit disc of complex numbers. A function $G:(\mathbb{C} \cup \{\infty\})\backslash \...
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1answer
72 views

Transfer function of differential eqaution

I'm trying to find out the transfer function of simple differential equation: $$a_0\dot y + a_1y=b_0x+b_1$$ The problem is i have no idea what to do with $b_1$. If we apply the Laplace transform ...
2
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1answer
349 views

Transfer function for double cart system

System: Define X2 = Y2; I've described the system with the following diff equation: $$f_{tot} = m_1\ddot{x_1} + k(x_2-x_1)+m_2\ddot{x_2}+B(\dot{x_2}-\dot{x_1})$$ where m1, m2, k and B are Cart ...
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0answers
29 views

Does the iteration $e_i^\top x_{t+1} = \max_j e_i^{\top} (\alpha A^j x_{t} + b^j)$ converge?

Given a constant $0 < \alpha < 1$, a matrix $A \in R^{n \times n}$ and a vector $b \in \mathbb{R}^n$, it is well-known that a sufficient condition for the iteration $x_{t+1} = \alpha A x_t + b$ ...
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1answer
314 views

Direct vs Indirect Learning Control

What is the difference between direct and indirect learning control? I found the following comments on direct and indirect control in this paper by Wang, Gao, and Doyle: "Survey on iterative learning ...
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44 views

Upper bound for affine differential equation

Let $\frac{dx}{dt} = a x + b$ be a stable affine differential equation where $a \in \mathbb{R}^-,b \in \mathbb{R}$ and let $c \in \mathbb{R}^-, d \in \mathbb{R}^+$. How can we determine a maximum ...
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1answer
312 views

Understanding controllability indices

I'm teaching myself linear control systems through various online materials and the book Linear Systems Theory and Design by Chen. I'm trying to understand controllability indices. Chen says that ...
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3answers
198 views

If $\|u\|^2 = c^2$, then $\|\dot x\|^2 \to a $ for $\ddot x = -\dot x + u(t)$?

I am dealing with the next simple equation $$ \ddot x = -\dot x + u(t), $$ where $u, x\in\mathbb{R}^m$, with $m \geq 1$, and I am wondering if for $\|u\|^2 = c^2 > 0$ then $\|\dot x\|^2\to a$, ...
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1answer
243 views

What does linearization do in MATLAB's PID tuner?

I noticed that the PID tuner from MATLAB has a linearizatin step before tuning. What does this linearization step do? And why we have to linearize a model in PID tuning?
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2answers
338 views

Convolution of unit impulse with impulse response

I have a question that's been bothering me. If we can convolve any arbitrary input with a system's impulse response to get the system's total response for that input, then if we convolve the impulse ...
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1answer
77 views

Controllability of a pair of matrices

If the pair $(A,B)$ is controllable, then is the pair $(A^{2},(A+I)B)$ controllable? The question becomes more interesting if there exists $(A,B)$ is uncontrollable, but the pair $(A^{2},(A+I)B)$ ...
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52 views

Stochastic control with stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ and smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$, how can we optimise the ...
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2answers
548 views

Effect of a simple pole vs complex conjugate poles

If $H(s)$ is a transfer function and it has just one pole in $s = p$, $p \in \mathbf{R}$, $$H(s) = \displaystyle \frac{H_0}{(s - p)}$$ the frequency response is $20 \log_{10} |H(j\omega)|$. With $\...
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0answers
120 views

(Still open) Convergence of a certain matrix product representing a class of piecewise linear dynamical systems.

Consider a discrete-time dynamical system of the following kind. Assume $x(t) \in \mathbb{R}^n$. $x(t+1) = A_{\Omega(x(t))} x(t)\quad$ where $\quad A_i = \left [ \begin{array}{c|c} 1 & 0\\ \hline ...
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2answers
870 views

PID controller convergence

Is there any material anywhere on convergence of PID controllers? Ie, if we formalize the "plant process" in some way, like $y_{t+1} = f(x_t,y_t)$ (in other words, the process value at a given time ...
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1answer
106 views

What is the Laplace transform transfer function of affine expression $\dot x = bu + c$?

For the one dimensional case, with $a, b, c$ being real constants, $u$ being the system input, $x$ the state, what is the Laplace transfer function of: $$\dot x = bu + c$$ Ideally I'm looking for ...
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66 views

Problem with understanding theorem on Riccati Equation.

`The matrices $A,B,C,D,X$ are real, square, $n \times n$. I have trouble understanding theorem 7.1.2 from Lancaster & Rodman "Algebraic Riccati Equations". The part that I understand is as ...
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2answers
52 views

fourier transform integral, parseval's theorem?

I have a fourier transform which is $$X(jω)=\frac{\cos(2ω)}{ω^2+ω+1}$$ and I am trying to calculate the value of the integral: $$∫x(t)dt \ \ \ \ \ \ x \in (-\infty, \infty)$$. I was thinking I ...
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1answer
35 views

Does there exist a notation for the set of poles of a function f(x)?

For eigenvalues we have a really nice notation $\sigma$ denoting the spectrum of this matrix i.e. the set of all eigenvalues. Before knowing $\sigma$, I just used $eigs(A)$ to denote the set of ...
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0answers
141 views

Jacobian Matrix of 6DOF Body (with IMU)

I am trying to derive the analytical Jacobian for a system that is essentially the equations of motion of a body (6 degrees of freedom) with gyro and accelerometer measurements. This is part of an ...
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45 views

Achieving asymptotic tracking of a nonlinear system with bounded input

I have the following nonlinear, continuous-time ODE \begin{equation} \dot{x}=K-Lq-q^2u, \end{equation} where the constant values $K$ and $L$ are strictly positive real numbers, the state $q$ and the ...
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1answer
40 views

Help figuring out output signal of LTI system.

Would greatly appreciate any help in figuring out the output signal of my discrete time LTI system. My input signal is cos(ωn) and my frequency response is H(e^jω)=(1+e^−jω)/2.
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44 views

$Ax = b$ & $Ax + b$

Ask a dumb question but confuse me long time. The following is what I know: 1st case $Ax = b$ is an affine set in $x$,i.e. $\{x | Ax = b\}$, and it is linear in $x$. 2nd case $ f(x) = Ax + b$ ...
3
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1answer
44 views

How do you find the state space representation of $G(s) = \frac {1}{s^2+s+1}$

Let $G(s) = \frac {1}{s^2+s+1}$ be the transfer function of the system Then $Y(s)(s^2+s+1) = U(s)$ Therefore $y'' + y' + y = u$ After this step, how should I set up my state transition variable $x$ ...
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1answer
47 views

Reference request: Controllable and Observable form for transform function

I came across some online material a year ago that claimed that a the ABCD matrix of a transfer function $$G(z) = \frac{b_1 z+b_2}{z^2+a_1z + a_2}$$ can be directly computed from the coefficients of ...
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1answer
180 views

Linear algebra of state space representation won't be linear (superposition theorem)…

After answering a question about calculating the state space representation of a circuit with 3 sources in it (the circuit is there), I had a doubt - while checking, it became clear there is something ...
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1answer
36 views

Relationship with trace and asymptotic stability in control theory [closed]

What is the relationship between $\mathrm{tr}(\exp(tA) \exp(tA^\ast))$ and asymptotic stability in control theory ?
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174 views

$L^2$ Bounds for Markov Chains.

Consider a non-negative, square stochastic $n \times n$ matrix $P$ (rows sum to one, $P$ is ergodic). We are interested in characterizing the set of $n \times n$ invertible matrices $A$ such that we ...
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39 views

solver for non-convex matrix optimization with convex constraints

So here is the problem: $\max_{D} ~~ \|A+BD\|$ subject to $\|D\|<1$ (any norm you like) where matrices A and B are given. The cost function is evidently convex as well as the constraint, but ...
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2answers
436 views

Model Predictive Control

I have a few confusions about Model Predictive Control (MPC). Since they are all minor questions related to the same category, I ask them under one topic. In an article, the cost function is defined ...
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0answers
66 views

How do I flow in a circle on a set of vector fields?

Consider a set of complete vector fields on a manifold. They each have an associated one parameter group of diffeomorphisms related to the generated flow. What is a necessary and sufficient ...
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1answer
241 views

Equivalence of controllability and reachability in discrete time systems

I am trying to prove that the statements; $\Sigma_d$ is controllable, $\Sigma_d$ is reachable, The pair $(A,B)$ is controllable (in other words $<A|\ im\ B>=\mathcal{X}).$ are equivalent ...
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1answer
61 views

How to prove the both identity (matrix)

I read a paper, and the paper use the following identities (that hold true in any ring) $(I+AB)^{-1}A = A(I+BA)^{-1}$ $(I+AB)^{-1} = I - A(I+BA)^{-1}B$ Any way to prove this? How to open the ...
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1answer
546 views

Convolution between impulse response

I read a paper, and am confused about the following: Suppose $W$ is an operator with impulse response (IR) $w$. And suppose $w^n$ is the IR of $W^n$. My question is the following: ...
3
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0answers
195 views

Dynamic programming problem for discrete linear time varying system

I'm working on a linear time varying discrete(LTV) multi input multi output(mimo) system. I formulate the problem description in the following way $$x_i(k+1) = x_i(k)\cdot A_i(k) + B_i(k)\cdot u_i(k)$...
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0answers
51 views

Solving simple decision-making model over multiple periods

Consider the following model. Each period t=0,1,..., an agent makes an effort $x\in R_+$ to solve a problem. The value from solving the problem is $V>0$. The relationship between effort and ...
3
votes
1answer
61 views

What is an example of a map not satisfying this rank condition?

Definition: Consider a Lie Group $G$ and a set of right invariant vector fields on $G$, denoted $\Gamma$. A point $y \in G$ is called normally accessible from a point $x \in G$ by $\Gamma$ if there ...
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2answers
161 views

Sufficiently rich signals

I know that a signal is sufficiently rich of order $n$ when it "includes" at least $\dfrac{n}{2}$ different frequencies. This is intuitive when we are talking about a sine but what about other kind of ...
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1answer
67 views

Notation: Polynomial of the Differential Operator

I having difficulty with some notation relating to control theory. Given that $H(s)$ is a strictly proper, scalar transfer function (i.e. a ratio of polynomial functions with a higher degree in the ...
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1answer
89 views

Understanding block diagrams

If I have block diagram with input $X(s)$ that goes to a block with $\frac{1}{s + 2}$ in it and then by way of $w(s)$ to a block with $s$ in it, and finally to the output $Y(s)$, how do I find the ...
0
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1answer
130 views

MATLAB Feedback

I am trying to use the feedback function in matlab and for the most part I understand it. But I came across this syntax: [x1 x2] = feedback(sys1, sys2, 1, 1, -1); ...
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1answer
97 views

Matlab calculate output

I'm trying to write a matlab function that takes in a transfer function and the input so it can calculate the output. So far, based on this information under I have the following piece of matlab code:...
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1answer
229 views

What is the difference between disturbance and noise for dynamic systems

In most references from dynamic system theory, the following linear continuous dynamic system is considered. $$\frac{\text{d}x(t)}{\text{d}t}=Ax(t)+Bu(t)+Dd_{1}(t)\quad (1)$$ $$y(t)=Cx(t)+Ed_{2}(t) \...
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1answer
46 views

Project a function on a space?

The problem I'm solving is $\begin{cases} & \dot{x}_{1} = -u \\ & \dot{x}_{2} = 4x_{1} \end{cases}$ $x_{1}(0) = x_{2}(0)=0, |u| \leq 3, t \in [0;2], u^{0}(t)\equiv0, J[u] = -2x_{1}(2) + ...
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1answer
1k views

Simulink model from a nonlinear State Space

I have the nonlinear state space already constructed in MuPAD as shown: u is the input and y is the output. What is the best way for me to take this to Simulink?
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2answers
548 views

Black's formula and feedback system stability

Consider a hypothetical system with open-loop transfer function $G(s)$. Place it in positive feedback with unit gain. (That is, take its output and directly add it to its input.) The closed-loop ...
3
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0answers
41 views

Infimum of radially unbound functional

I am having difficulty following a proof about balls (subsets) of radially unbounded functionals. Let $U$ be a Banach Space. Let the space of admissible controls $U_{ad}\subset U \ne \emptyset$ be ...
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1answer
71 views

Show what happens to indentations around poles on imaginary axis when acted on by a conformal map

Can someone provide a link to a proof or motivate here (not looking for a rigorous proof) of a very important result in complex analysis, particularly in applications to control systems engineering: ...
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2answers
237 views

Control Theory Textbook

I'm looking for a good textbook or series of lecture notes for learning about sampled data control theory. I'm a relative beginner in this area, so I'm looking for a gentle introduction. I'm ...
3
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1answer
83 views

(Possible) application of Sarason interpolation theorem

This question is related to the following Wikipedia article on Nevanlinna–Pick interpolation. At the end it has been written as Pick–Nevanlinna interpolation was introduced into robust control by ...