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Questions tagged [control-theory]

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The desired trajectory of the output 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 should manipulate the inputs to the system to obtain the desired effect on the output of the system

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Is there always at least one attracting set in an invariant set?

(Edited) Suppose we have an invariant set for a dynamical system. Can we claim there exists an attracting set (including the strange attractors) in the invariant set? If so, is there any specific ...
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Solution for first order ODE with discontinuous right hand side

I'm now approaching for the first time to first order differential equations with discontinuous right hand side. Let $A\subset \mathbb{R}\times\mathbb{R}^n$ and $f=f(t, x):A\to \mathbb{R}^n$. Fix $(...
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15 views

Eigenvalues of normalized vs unnormalized Laplacian of weighted digraph

Let $G$ be a weighted digraph. What is the connection between the eigenvalues of the normalized and unnormalized Laplacians of $G$. I think there is no explicit connection. We can at most find some ...
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1answer
24 views

How to decompose a MIMO system into a multiplication minimum and nonminimum phase part

suppose the transfer function of m-input-m-output system is $$G\left( s \right) = \left[ {\begin{array}{*{20}{c}} {{g_{11}}\left( s \right)}& \cdots &{{g_{1m}}\left( s \right)}\\ \vdots & ...
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Smooth endpoint map

Consider a control system $$ \dot x(t) =f(x(t),u(t)) \qquad (\star) $$ where $f$ is a smooth vector field and $x\in \mathbb{R}^n$. The endpoint mapping is defined by $$ E:\mathbb{R}^n\times \mathbb{...
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46 views

How to identify a system with no overshoot in terms of poles?

I'm trying to find a standard in order to identify a system with no overshoot in term of poles. I know that there is something related to dominant poles. I was looking into a particular 4th-order ...
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2answers
42 views

Stablility of a linearized time-delay system

I have a linearized time-delay system as follows: $$\frac{\mathrm d X}{\mathrm d t} = a[X(t)-X^*] + b [X(t-R) - X^*], $$ where $a$, $b$ are constants, $R$ is the constant delay, and $X^*$ is the ...
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Static behaviour and dynamic behaviour of a system

I have a system with the static behavior: $y(t)=a+by_{0}(t)$ where $y(t)$ is the output and $y_{0}(t)$ is the input. The dynamic behavior of this system is: $G(s)=\frac{K}{1+Ts}$ To this equation ...
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21 views

Q learning for more than one goal state

I have recently implemented a reinforcement learning (TD method) problem consisting of 19 states and 2 actions (Increase/decrease relative to previous time step action) with one goal state. Now I want ...
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is this function regular?

I am learning nonsmooth analysis for discontinuous dynamical systems. An important concept in this field is the regularity of functions. A function $f$ is regular at $x$ if its right directional ...
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21 views

How to find the transfer function for advection equation?

The Problem: I have to find the transfer functions of the two equations in my system and of the whole system but I am not sure about something and as I haven't had mathematics lessons for my three ...
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1answer
29 views

A simple quadratic optimizer for only constraints on input

I'm going to implement an quadratic optimizer with C for embedded systems. I will do that because I need speed. But I have some trouble to find a quadratic optimizer for C that works with embedded ...
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1answer
60 views

Property of interconnected feedback systems

In the figure you can see the statespace form of a feedback interconnection system. Very quick question: is there a reason they have taken $D_1=0$ and $D_2=0$? It makes workings a lot easier but I ...
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2answers
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Is there any rule of thumb when it comes to selecting control/predict horizon for MPC?

I have a simple question: Is there any rule of thumb when it comes to selecting control/predict horizon for MPC? Normaly I set control and predict horizon equals, but I have heard that's not good ...
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1answer
38 views

Continuous vs. Discrete State-Space Model

Time-invariant continuous model: $\dot{x}(t) = Ax(t)+Bu(t)$ $y(t) = Cx(t)+Du(t)$ Time-invariant discrete model: $x_{k+1} = Ax_{k}+Bu_{k}$ $y_{k} = Cx_{k}+Du_{k}$ Why does the continuous model ...
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1answer
70 views

Controllability of cascade connection of two systems

I have two linear control systems that are represented by their state space models $$\left( \begin{array}{c|c} A_1 & B_1 \\ \hline C_1 & D_1 \\ \end{array} \right), \left( \begin{array}{c|c}...
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Controlabiliy of discrete augmented systems

The statement is: given a discrete MIMO system $(\Phi, \Gamma)$ and a time delay of $n_d$ input delays, $n_d \in \mathbb{N}$, then is the augmented system controllable i.e. stabilizable? I have a hard ...
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33 views

Matlab function which will perform the analysis of my system with respect to Control Theory.

$x`(t)=Ax(t)+Bu(t)$ $y`(t)=Cx(t)$ I have three matrices. For example, the matrices are: And using only one Matlab function I would like to get the following information: 1 Eigenvalues of the ...
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43 views

Is sliding mode control complete in mathematics?

Consider a double integrator control system $$\begin{cases}\dot{x}_1 = x_2,\\\dot{x}_2=u,\end{cases}$$ where $\dot{x} := \mathrm{d}x/\mathrm{d}t$. Then we apply the sliding mode control $u = -\beta(...
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How to find out which pole goes to which infinity zero in root locus

Branches are colored in MATLAB's rlocus function. How can it be easily determined, without using a software, that which pole-zero are at the both ends of a branch? ...
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24 views

Find initial conditions for vector space evolution in a continous-time system

as stated in the title, given an n x n matrix A, I have to find all the initial conditions from which the state evolves into a vector space of dimension ...
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30 views

What are the optimal investment strategies for dynamically reallocating assets with different risks?

Let's consider the following simplified model. Suppose you start with some money and invest for a fixed amount of time T. You have one risky asset and one risk free asset to choose from. The risky ...
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23 views

Computing partial derivative of certain line integrals

Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $\mathbf{w}$, and (2) input data ${\bf x} \in {\mathbb R}^d$. Fix $i \in [d]$, consider the following ...
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1answer
27 views

Understanding negative definite/semidefinite functions [closed]

I'm working on control theory and have some difficulty understanding if a function is negative definite or semidefinite. Given the system $\dot{x_1} = -x_2^2$ $\dot{x_2} = -x_1^2x_2 + x_1^3 - x_2$ ...
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37 views

Optimal control problem with a path constraint which involves controls at two distinct time points

I am faced with an optimal control problem in continuous time which includes a path constraint which involves controls at two distinct points in time. I do not know how to approach this problem. I do ...
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29 views

Transfer Function, what does s Stand for

Trying to understand what the variable s is for transfer functions (if there is a common accepted use of it). In the problem space I am working in, control theory, I believe I have seen definitions of ...
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1answer
22 views

Automatic control of linear system with non independent inputs

I'm currently trying to (re)learn automatic control 2 years after having finished college, and I'm having a hard time. I'm trying to control a simplistic lunar lander in a 2d space. The only way to ...
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1answer
427 views

How to solve $\dot{x}=f(x)/||f(x)||$?

How to solve the following ODE? \begin{equation} \frac{d}{dt}x=\frac{f(x)}{\|f(x)\|}, \end{equation} where $x: \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f: ...
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1answer
48 views

Exosystem for reference generation

Recently I have read a paper where an LTI system of the form $$ \begin{align} \dot{x}_p &= A_p x_p + B_p u \\ y &= C_p x_p \end{align}\tag{1} $$ for the control plant was considered. In ...
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55 views

Lyapunov Stability for a Nonlinear, Time-varying system

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems. Say we are given a nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_2(t)[...
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1answer
59 views

Properties of Positive Real Functions

I am trying to understand the properties of positive real (PR) and strictly positive real (SPR) transfer functions. If given a transfer function I know how to determine whether or not the function is ...
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1answer
69 views

What is the difference between controllability and reachability?

Here are the two problem statements I'm trying to understand: Reachability. The reachability problem is to “find the set of all the final states $x(T)$ reachable starting from a given initial state $...
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1answer
29 views

A state-space representation of an integro-differential equation implies a false statement

I would like to convert the equation $\ddot{y}+\int_0^t y(\tau)d\tau=0$ to state-space representation. Below, I present my attempt, which seems to be contradicting, and then ask my question at the end....
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Why does $x[k] = k^k$ not have a Z-transform?

The signal $x[k] = k^k, k=1,2,3...$ does not have a Z-transform. Why? The definition of the Z-transform is: $$X[z] = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$
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1answer
35 views

Construction of a linear system stratifying certain requirements

Construct a linear system $$ \begin{cases} \dot x = Ax + Bu \\ y=Cx \end{cases} $$ that satisfies all the following requirements: 4th order 1 input & 1 output Unstable Stabilizable ...
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1answer
48 views

Equilibrium points of an inverted pendulum on a cart

Question Find the equilibrium points of the inverted pendulum on a cart system whose dynamics equation are given by $$ \begin{bmatrix} (M+m) & -m l \cos\theta\\ -m l \cos\theta & (J+m l^2) \...
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1answer
25 views

Closed form solution for Double Integrators dynamics

I have a relatively specific problem. Consider a linear system, $\dot{x}(t) = Ax(t)+Bu(t)$, where $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times 1}$ are the constant matrix and ...
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0answers
19 views

LQG with bias rejection for quadcopter attitude control

We're trying to design an attitude controller for a quadcopter. The system dynamics are given: $$ \boldsymbol{\dot{q}} = \frac{1}{2} \boldsymbol{q} \otimes \begin{pmatrix} 0 \\ \vec{\omega} \end{...
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0answers
24 views

Choosing a complex square root

First off, I'm not sure if this question belongs here so I apologize if it's out of place. I'm working with a system of coupled linear partial differential equations in space and time with which I'd ...
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1answer
33 views

Questions about LQG with full information

I have implemented LQG in MATLAB software. But, now I do not know how to determine the value of optimal cost. Each way of calculating cost, returns a different value. Which one should I trust to ...
2
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0answers
30 views

Given a system of functions, find a system of differential equations which describe that system

Inspired by modelling phenomena in biology, I'm wondering whether there has been mathematical study on the following question: Given some $\mathbf{X}(t) \in \mathbb{R}^n$, find $f = f(X_1, \dots, X_n)...
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0answers
20 views

Why are Integral Controllers never used on their own? [closed]

I've noticed in a lot of cases where we have a plant with a large steady-state error that meets every other specification (overshoot, settling time) apart from the steady-state, that we always use a ...
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2answers
57 views

How to check the stability of a neural network?

Is there any way or theory to prove that a neural network is stable? For instance, when I use an NN to learn from a dataset, how could I prove that the NN's learning will converge and give the ...
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0answers
27 views

ordering of variables for computing the Jacobian and eigenvalues

I'm a engineering student (i.e. no solid foundations on "true" mathematics), sorry if my question is silly. When I was computing the Jacobian to study the stability of equilibria points on power ...
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1answer
37 views

optimal control to minimise a path

I'm having issues solving this problem. Here is what I have tried so far. $$ u=\dot {x_1} + x_1 $$ $$ J= \frac{1}{2} \int_{0}^{t_1}((2x_1)^2+2\dot {x_1} x_1 + \dot {(x_1)^2})dt$$ Can I proceed and ...
2
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1answer
52 views

LaSalle for time varying systems

I am looking for an explanation, why LaSalles theorem is in general not applicable to time varying systems. Can someone provide an example system with $$ \dot{x}(t) = A(t)x(t) \tag{1} $$ I.e., why ...
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1answer
26 views

Bode of Controller and Sensitivity Functions in a State Feedback Controller System

I have a system: $A$, $B$ and $C$ forms the state space representation of the system. The system has a state feed back controller with an integral controller. I want to draw bode plot of controller,...
1
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1answer
51 views

Show that if a linear dynamical equation is controllable at $t_0$, then it is controllable at any $t<t_0$.

Consider a $n$-dimentional $p$-input equation: $$\dot{x}=Ax+Bu$$ where $A$ and $B$ are constant $n\times n$ and $n\times p$ real matrices. By definition, the latter state equation is said to be ...
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0answers
26 views

Time-Discretize a Linear Quadratic OCP (Bolza Function)

I'm trying to formulate an optimal control problem based off of this given Minimum-Time Cost Function: $$J(t_f)=\frac{1}{2}[x(t_f)-x_{des}(t_f)]^TP_f[x(t_f)-x_{des}(t_f)] + \frac{1}{2}\int_{t_0}^{t_f}(...
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1answer
46 views

Lyapunov stability of 4x4 matrix.

Consider the following continuous-time state space representation of the form: $\frac{d}{dx}x(t) = Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t\in \mathbb{R}^{+}$ $A=\begin{bmatrix}-1&3&0&0\\-...