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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Question about nested conditional expectation.

Question 1: I am interested in finding out, why the following is or is not the case: $\mathbb{E}\left[ \mathbb{E} \left[( X_k|(Y_0,Y_1,...,Y_{k-1}) )|( Y_k|(Y_0,Y_1,...,Y_{k-1}) ...
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13 views

Final value of 1/(( s+2 )² * (s² - s + 1)) in the time domain

The original question is given as $$\frac {d^3y}{dt^3}+y=u=(1-t)e^{-2t}$$ The initial value y(0) = 0 and the same for all derivatives of y. Determine Y(s) What happens to u(t) and y(t) when ...
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9 views

How to find the $H_2$ norm of MIMO system in matlab

$K$ is given. $A,B_1, B_2, C_1, C_2, D$ are also given. I want to find the following: $P_{11}, P_{12}, P_{21}, P_{22}$ And use the resulte in 1., to find the $H_2$ norm of the system: ...
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15 views

Frequency response for a non-linear sytem(saturation) having a linear part in series to a sinusoidal input [on hold]

Could you help me get a solution to the problem stated as below? I know that a forcing function(r=1.5sinwt) will lead to the phenomenon called jump resonance. Can I use the quasi-linear approach ...
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1answer
46 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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30 views

How to find a partial derivative with respect to a matrix?

Let we have a $2\times2$ matrix $A=\begin{bmatrix}a_1&a_2\\a_3&a_4\end{bmatrix}$, a $1\times2$ matrix $C$, and a $2\times1$ matrix $X$. How can we calculate derivative of $CAX$ with respect to ...
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1answer
27 views

What exactly is a random disturbance in control theory?

I'm embarrassed to even ask but... I frequently see this word used in articles about dynamical systems, but not until now have I questioned what it really is. I understand that it is opposite of a ...
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8 views

Multiplication between anticausality or causality [on hold]

When reading the paper, I meet a question as following: Assumption: 1. discrete-time 2. Many systems: causal: $S_1,S_2,...$ anticausal: $S_a,S_b,...$ Question: 1. Is the multiplication between causal ...
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24 views

stabilizable ,detectable and regulator

Assume that $(A_2,B_2)$ is stabilizable and $(C,A)$ is detectable then there exist aregulator if the equation $TA_1-A_2T-B_2V=A_3$ $D_1+D_2T+EV=0$ have solution (T,V).if $A_1$ is antisatable the ...
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1answer
18 views

Kronecker Product and state space

I am reading a paper, and one step of it seems like the following: If $S_1 = $ Then, $I \otimes S_1$ = How to show it? (Suppose dimension of all of them are correct. $I$ is the ...
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12 views

Multiplication of transfer function

If I have the following: How do I show the following: $P_{11} = G_{11} + G_{12}\hat Y\tilde MG_{21}$ is: I am stuck in this complicated system. Or, the other simpler one: ...
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1answer
14 views

Control theory: when does $G(s) = \frac{1}{P_\lambda(A)}$

In other words, under what condition is the system transfer function G(s) = Y(s)/U(s) equivalent to the reciprocal of the characteristic equation of the $A$ matrix in state space realization?
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1answer
35 views

Is multiplication commutative in the laplace domain?

I'm studying control theory and saw this picture explaining some of the basic rules. My question is if we could also say that Y(s) = (G2(s) * G1(s)) * U(s) Or Y(s) = U(s) * G2(s) * G1(s) I'm ...
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1answer
32 views

Shortest path in the plane under derivative constraint

A colleague posed a toy problem to me today that degenerates to finding the curve y(x) of shortest length than connects two points in the plane (WLOG: y(0) = 0, y(a) = b), such that y'(0) = 0. This ...
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14 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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42 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 ...
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1answer
32 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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1answer
20 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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3answers
173 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
29 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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25 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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23 views

Masters thesis topics in iterative learning control (ILC)?

Looking for ideas on thesis topics in iterative learning control or repetitive control. One topic of interest thus is far is ILC of discrete-time nonlinear systems. Any thoughts or recommendations ...
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30 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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2answers
43 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
39 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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26 views

Control theory: Why doesn't the separation principle hold in nonlinear control theory?

It is widely known in control that separation principle is one of the best tool for pole placement and design of stabilizing controller in linear system. Many results also note the inability of ...
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2answers
37 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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104 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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31 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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1answer
64 views

Controllability of a linear time invariant system, whose A matrix is formed by Jordan Blocks

I am studying for a linear system theory exam later on this week. The professor has recommended some problems in order to practice and prepare for the exam. This is one of them that I'm trying to ...
2
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1answer
107 views

Derivation of the Mexican wave as Mean field equilibrium of an ergodic control problem

I'm dealing with a Mean Field system introduced by Gueant, Lasry, Lions in their "Mean Field Games and Applications" which admits as solution the so-called Mexican wave. Without going into the details ...
3
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1answer
72 views

Design control law

Consider the function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ which is continuously differentiable and strongly convex. Let $x^*$ to be the unique global minimizer of $f$. Assume that $L_1\|x\| ...
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2answers
557 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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3answers
79 views

showing any controllable system can be put in 'controller' form.

I am looking at the proof of the following theorem: Let $\dot{x} =Ax + Bu$ be a controllable single input system, where $\Delta_A:= \det(\lambda I -A) = \lambda^n + a_1\lambda^{n-1} + \ldots + ...
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1answer
56 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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0answers
31 views

Recover a specific solution from the general solution of the Riccati equation.

Consider the equation $XAX - AX = 0$, where $A,X$ are square $n \times n$ real matrices. We know $A$ and assume for simplicity it is diagonable. We want to solve the equation for $X$. We have $XAX - ...
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2answers
41 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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1answer
27 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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2answers
30 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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0answers
45 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 ...
21
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5answers
2k views

What is the mathematical foundation of Control Theory?

There is a question which I'm wondering again and again in recent months. I have taken courses like Elementary Differential Equations, Signals and Systems, Linear Control Systems, General Theory of ...
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5answers
8k views

Why use a Kalman filter instead of keeping a running average

I've been trying to understand Kalman filters. Here are some examples that have helped me so far: http://bilgin.esme.org/BitsBytes/KalmanFilterforDummies.aspx ...
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5answers
2k views

Control / Feedback Theory

I am more interested in the engineering perspective of this topic, but I realize that fundamentally this is a very interesting mathematical topic as well. Also, at an introductory level they would be ...
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0answers
24 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
36 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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2answers
158 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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34 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
36 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 $x$ ...
2
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1answer
36 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
84 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 ...