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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18 views

Nonlinear state space representation involving derivatives

For the follwing nonlinear system: $$\dot x_{1} = 2/b \cdot [(c_{1}-u_{1}) \dot u_{1} + (r_{1}-u_{2}) \dot u_{2}]x_{1} \\ \dot x_{2} = 2/b \cdot [(c_{2}-u_{1}) \dot u_{1} + (r_{2}-u_{2}) \...
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Hamilton-Jacobi equation for L2 stablity

I want to minimize the following functional: $\int_{0}^{\infty} \gamma^2 \frac{u^Tu}{2}-\frac{y^Ty}{2}\, dx$ subject to $\dot x = f(x) + G(x)u$ and $y(t)=h(x)$. At first I define the Hamiltonian: H = ...
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56 views

Stability with input same as with nonlinear function?

Assume there is a dynamical system $$ \frac{d x(t)}{dt} = A \cdot x(t) + q(x(t)) $$ and that $A$ is stable and that $q$ is a nonlinear and very complicated function. We only know $q$ is smooth and ...
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stability of solution of ode

I'm new to ODE and currently working on an optimal control problem. I'm trying to calculate the trajectories of a wheeled robot. I can steer it only continously but in theory it makes also sense to ...
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40 views

Understanding Controllability Matrix

Consider \begin{equation*} \dot{x} = Ax + Bu,\quad x \in\mathbb{R}^n,\ u \in\mathbb{R}^m,\quad A \in\mathbb{R}^{n\times n},\ B \in \mathbb{R}^{n\times m} \end{equation*} \begin{equation*} \text{Rank}(...
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1answer
47 views

Deriving a matrix inequality related to stability theory

I am reading a paper related to control theory and struggle to understand a matrix inequality derivation that is briefly introduced by the author: Given that we have the following equations: $ x(k+1)=...
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23 views

Proof of Controllability Matrix

I have searched for the proof of the following theorem but wasn't satisfied with what I found, \begin{equation} \dot{x}(t) = Ax(t) + Bu(t),\quad x(t) \in \mathbb{R}^n \text{ and } u(t) \in \mathbb{R}^...
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1answer
44 views

How am I suppose to estimate the next state vector if the model have internal integration? - Kalman filter

Assume that we have a state space model with no integration (no poles at 1) $$x(k+1) = Ax(k) + Bu(k)\\y(k) = Cx(k)$$ And we know our kalman gain matrix $K$. To compute the next state $\hat x(k+1)$, ...
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1answer
40 views

Driving the state of a discrete system to zero in one step

I have the following system of difference equations: $\textbf{x}(k+1) = A \textbf{x}(k) + \textbf{b} u(k)$ where: $A = \begin{bmatrix} 1 & 2 \\ 3 & \alpha \end{bmatrix} $ and, $\mathbf{b} =...
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34 views

Control process with long deadtime

I am a programmer who lacks the mathematics side of things. I need help writing out a SOPDT equation and a Smith Predictor using simple math. I did not go to school for this and understand that there ...
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34 views

Stabilizing controls in linear quadratic regulator

I am studying a linear quadratic control problem with discounting. For $\gamma \in (0,1)$, $Q \succeq 0$ and $R \succ 0$ and linear dynamics $s_{t+1}=As_t + B a_t$, let the total cost starting in ...
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how many equation do I need to build state space equation of a rlc circuit [closed]

Hi guys I've a question about control theory and circuits. If I have a simple rlc circuit, is there a way to know how many node/ loop equations do I need to write to get the state space equation? For ...
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27 views

Equivalence of Lyapunov equation for continuous and discrete case

I am currently studying the original discrete/continous equivalence proof of the Lyapunov equation by Rice 1967. Continuous case: $A^\star L + L A = -C$ for $C \succcurlyeq 0$ Discrete case: $A^\...
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Root locus method, deriving the position of the asymptote centroid

In the Root Locus Method, the linear asymptotes are centered at a point on the real axis given by $$ \sigma_A=\frac{\sum_{j=1}^n(-p_j)-\sum_{i=1}^M(-z_i)}{n-M}, $$ where $p_j$ are the $j$th open-loop ...
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Help with this transfer function of a geared servo motor

I need to find the Transfer Function. I have these mathematical model equations (already with Laplace applied): 1) $U(s) = Kp\left( \theta I(s) - \theta_{o}(s) \right)$ 2) $U(s) = RI(s) + LsI(s) + ...
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Why does the Z transform represent a delay?

I'm studying the Z-transform. I recently did by hand the Z transform of an discrete impulse delayed $\mathcal{z}\{\delta[n-k]\} = z^{-k}$ I get that this means that any signal can be represented as ...
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28 views

Moore Penrose inverse of the end-point map

We consider the end-point map $\mathcal{E}$ of a nonlinear control system: $$ \dot{x}(t) = f(t,x(t), u(t)), \quad x\in \mathbb{R}^n,\ u\in \mathcal{U}\subset L^{2}([0,T], \mathbb{R}^m) $$ starting ...
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1answer
23 views

Discrete LTI systems with complex inputs?

I'm reading and pondereing about the convolution sumation, properties and how this is related to discrete LTI systems. I'm using the book Signals and Systems by Alan V. Oppenheim, and on the chapter ...
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22 views

Derivation of internal dynamics of nonlinear system in order to derive Byrnes-Isidori Normal Form

I have a nonlinear system (Ball & Beam) which is described by the following equations of motion: $$ \ddot{y} + \frac{mg}{a} \sin(θ) -\frac{m}{a}y\dot{θ}^2 = 0 $$ $$ \ddot{θ} + \frac{2m}{b}y\dot{...
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Restructuring Ill-conditioned problem for better numeric results

I'm not sure if I'm asking in the right place since this is kind of a field overlap but let's see. I have a dynamic equation system and want to optimize a subset of it's parameters. The algorithm ...
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1answer
45 views

Joint Spectral Radius Relation

Let $\theta: \mathbb{N} \rightarrow \Sigma$ a switching signal, $\Sigma=\{1,\dots,m\}$ where $m$ is an integer and $ m \ge 2$, and let $\mathcal A=\{A_\sigma \in \mathbb{R}^{n\times n} |\sigma \in \...
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1answer
66 views

Linearization of a three tank system resulting in singularity

This question is very closely related to another question previously asked here on Math Stackexchange, where linearization of a three-tank-system is discussed. After linearization you would get at ...
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24 views

Lyapunov Candidate function to derive parameter estimation law

I have a system and a reference model represented in state space in the following form: \begin{gather} \dot{x} = Ax+Bu \\\ u = -Kx+k_rr \\\ K,k_r : constants - controller \ gains \\ A_m = A-BK \\\ \...
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1answer
41 views

A discrete-time system is positively stabilizable iff (A,B) is stabilizable and the eigenvalues are in the open unit disk

I'm currently busy with some research in positive controllability and positive stabilizability for discrete time systems. In an article of M.E. Evans and D.N.P. Murthy ("Controllability of discrete-...
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1answer
25 views

Necessary stability condition for a second order discrete time system $x(k+2) = Ax(k+1) + Bx(k)$

Let $x(k) \in \mathbb{R}^n$, $A \in \mathbb{R}^{n\times n}$, $B \in \mathbb{R}^{n\times n}$. Consider the following discrete time system: $$x(k+2) = Ax(k+1) + Bx(k)$$ where $x(1) = Ax(0)$ and $x(0) \...
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18 views

Unscented Kalman filter and measurement function

Suppose I have a nonlinear system with states $x$ and measurements $z$. I don't have a measurement function for $z=h(x)+n$ where n is my gaussian noise term. However I do have the sufficient ...
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1answer
69 views

Problem understanding Chetaev theorem

I am studying control theory, and I am focusing on the Lyapunov stability. In particular, I am looking the Chetaev theorem, but I have some problems understanding it well. I know that the Cheatev ...
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2answers
46 views

Absolutely continuous and almost everywhere solution of a controlled dynamical system

This is a doubt i had while reading the first chapter form the book Mathematical Control Theory, Jerzy Zabczyk, in the first chapter, page $11$ the author talks about solution for a class of ...
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17 views

Computing the Transfer Function of open-loop-gain equation (Z-Transform)

I am designing a closed-loop control system. I need to figure out how I should design the compensator so that the system is stable for a wide range of inputs. The problem is, the open-loop-gain ...
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83 views

Why does the Lyapunov criterion only gives sufficient conditions for stability?

I am studying stability for control systems, and I have written in the notes of my professor that the Lyapunov Criterion only gives sufficient conditions for stability, and not necessary and ...
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108 views

Proving that similarity transformation of state-space preserves the Euclidean norm

I am aware that the state space realization of a dynamical system is not unique. So if we have a dynamical system: $\dot{x} = Ax + Bu$ $y = Cx$ Then we can write it as $\begin{bmatrix} \dot{x}\\y \...
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How do I find the integral of a distribution?

I have the following distribution: $\Delta (x)=\begin{Bmatrix} \begin{pmatrix} x_1\\ -x_2\\ 0 \end{pmatrix}&\begin{pmatrix} x_1\\ 0\\ -x_3 \end{pmatrix} \end{Bmatrix}$ and I have found that ...
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68 views

Non linear system -control problem

How can i solve this problem in MATLAB? I have this non linear system: $\frac{dx_1}{dt}=5sin(6t)-x_1$ $\frac{dx_2}{dt}=3x_1x_2-2x_2+1$ $y=x_2$ Initial condition are the following: $x(0)=[-2,-1....
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1answer
44 views

What does it mean that a distribution is integrable?

I am studying geometric control theory, and I am focusing on the Frobenius theorem. I have seen that it gives sufficient and necessary conditions for integrability of a distribution, but I am having ...
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1answer
29 views

Breakaway point of Root Locus not as expected

I have the following Open Loop Transfer Function: $$H_{ol}s = \frac{k(s+4)}{(s+1)(s+2)(s+3)}$$ To find the breakaway point of the unmatched poles we find the characteristic equation of the closed ...
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1answer
65 views

What is a distribution

I am studying the basics of geometrical control theory, and I am struggling with some concepts. At the moment I am studyng the concept of distribution. So far I have understood that a distribution is ...
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1answer
37 views

Find input for a control system such that system reaches a desired state $x_L$ at time $t=L$.

Given is the equation in state space of a control system $$\dot{x}(t)=Ax(t)+Bu(t)$$ $x(t)$ is the state vector of length $n$, $A$ is a square matrix of $n\times n$, and $B$ is a vector of length $n$. ...
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27 views

Optimal control: Feedback control

Suppose the state of a system is given by $X_t$ such that $X_0 = x \in (0,1)$ and $\dot X_t = [1 - \alpha_t C] X_t (1-X_t)$ where $C > 1$ and $\alpha_t \in [0,1]$. An agent chooses $(\alpha_t)_{t ...
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43 views

All equivalent inverse LQR problems

Inspired by this question I wondered if it is possible to fully parameterize the inverse optimal control problem. So given a stabilizing state feedback policy $$ u(t) = -K\,x(t), \tag{1} $$ for a ...
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23 views

Extract input transfer function

Can someone explain me how to figure out the $U(s)$ equation for the following control system ? i've tried to follow this procedure: $Y(s) =\frac { G(s)C(s)+D(s) }{1+ G(s)C(s)+D(s)}$ where $D(s)$ ...
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1answer
76 views

Problem understanding invariant subspaces and foliations

I am studying control theory and I am starting the concept of geometric control theory. As a prerequisite to this, I am studying the concet of invariant subspaces, and I having some troubles ...
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28 views

Transfer Function of Electrical System

I am trying to find the transfer function $\frac{V_{out}(s)}{V_{in}(s)}$ of an electrical system that operates according to the diagram below. $V_{in}$ is switched between 0 and 13 Volts (i.e $V_{in}(...
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2answers
31 views

Help with Modal canonical form linear systems.

I'm trying to find the modal canonical form of a linear system, the book I'm using is Linear systems theory and design by Chen, but has no clear explanation about this topic, and I'm having a hard ...
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32 views

How does directional derivative work?

Consider a vector field $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, I have that the directional derivative (or Lie derivative) can be defined as: $L_f = \sum_{i=1}^{n}f_i \frac{d}{dx}$ which as ...
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1answer
45 views

Problem understanding passage from explicit representation to implicit representation

I am studying control theory and I am having difficulties understanding a concept. Consider the following relationships, which represent the input to state behavior and the input output behavior ...
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1answer
50 views

How to solve LQR if regulating outputs instead of states?

Consider the state space system given by $$ \dot{x} = Ax(t) + Bu(t)$$ $$ y = Cx(t) + Du(t)$$ The standard LQR cost is given by $$ J = \int \big( x(t)^T Q x(t) + u(t)^T R u(t) \big)dt$$ Instead, ...
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1answer
84 views

Why for a linear system, the stability for a generic equlibrium point is equivalent to the stability of the origin?

I am studying the concept of stability for linear and for nonlinear systems. While studying the stability for a linear system I found this definition from the notes of my professor: for a linear ...
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1answer
43 views

Linear System Controllability?

From wikipedia, Consider the continuous linear system: $$\mathbf{\dot x} (t) = A(t)\mathbf{x}(t) + B(t)\mathbf{u}(t)$$ There exists a control $u$ from state $x_0$ at time $t_0$ to state $x_1$ at time ...
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1answer
81 views

Difference between I-state-space controller and PI-controller

Im wondering about the difference of an I-state-space controller and an (simple) PI-Controller? As I know, you get the advantage to place any dynamic with the state-space-control (if the manipulating ...

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