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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Which constant matrices $A$ have the property that $\dot{x} =Ax+Bu$ is controllable for every non zero $B$?

Which constant matrices $A$ have the property that $\dot{x} =Ax+Bu$ is controllable for every non zero $B$? Now to check controllability I usually check the rank of the matrix $$R = \left( B\ \ \ \ ...
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76 views

Show that the system $\Sigma(SAS^{-1},SB,CS^{-1},D)$ is observable/controllable iff $\Sigma(A,B,C,D)$ is observable/controllable

I am given the two linear systems: \begin{eqnarray} \Sigma_1: \dot{x}&=&Ax+Bu\\ y&=&Cx+Du \end{eqnarray} and \begin{eqnarray} \Sigma_2: \dot{x}&=&\bar{A}x+\bar{B}u\\ ...
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1answer
54 views

$A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that $\exists p\in\Bbb R ^m$ s.t. $(A,Bp)$ is controllable iff $(A,B)$ is controllable

Let $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that there exists a vector $p\in \Bbb R ^m$ such that $(A,Bp)$ is controllable iff $(A,B)$ is controllable. Here when I say $(A,B)$ ...
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51 views

Can you find an expression for $F_{12}(t,\tau)$ in terms of $F_{11}(t,\tau)$ and $F_{22}(t,\tau)$?

I have a problem with this...I can not figure out how to solve it..! can you help me? thank you!! Show that if $A(t)$ is partitioned as $$ A(t) = \begin{pmatrix} A_{11}(t) & A_{12}(t) \\ ...
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1answer
75 views

Optimal consumption policy

I start with an initial capital C and at the beginning of day $n=1,...,N$ I observe the random variable $X_n$, where $\mathbb E X_n=\mu_n$. The $X_n$ are independent. I also choose $c_n$ on day $n$, ...
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78 views

Another machine problem

This is linked to my previous question Discounted optimization problem I have difficulties again finding a formula for $F(x)$. We consider a machine at time $t$ which is in state $x$. The machine ...
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1answer
101 views

Found transition matrix and state matrix

I tried to found a solution for this problem but I can't! Any suggestion?Thank you. Given that A is a 2x2 matrix and that dx/dt=Ax(t) suppose that x(0)=[1 ; -3] implies x(t)=[e^-3t ; -3e^-3t] and ...
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1answer
46 views

Help finding the transfer matrix for this system

Find the transfer function of the following system: \begin{eqnarray} \dot{x}_1&=&ax_1+bx_2 + u\\ \dot{x}_2 &=&-bx_1 +ax_2\\ \dot{x}_3&=&cx_3\\ y &=& x_1+x_3 ...
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51 views

Non linear state equation show that nominal output is 1

Do you help me with this? Consider the nonlinear state equation $\dot{x} = \begin{bmatrix} u \\ u x_1 -x_3 \\ x_2 - 2x_3 \end{bmatrix}$ $y = x_2 - 2x_3$ with nominal initial state $x^* = ...
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69 views

Linearize a nonlinear system

I haven't really encountered control theory before. Could you possibly recommend some papers/textbooks that start with the very basics of the field? I have a couple of months to get to grips with the ...
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1answer
35 views

Check if a system admits solutions of period 2

I have the following problem. Let $r \geq 0$ be a parameter in the discrete time system $x(k + 1) = r − rx(k)$. Verify whether there exist $r \geq 0$ such that this system admits solutions of period ...
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2answers
127 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
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1answer
181 views

Converting a direction vector to an angle

I have a machine that controls its own velocity via the use of a fixed thruster mounted on its rear. The machine knows its current velocity, and it knows the velocity it must attain. By subtracting ...
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1answer
74 views

I have reached a rut in my understanding of control systems. How do I cross this?

A little background here. I'm an undergrad in the final year. I have decided academia as my career path. My grades are not high but my research caliber is good and I have ongoing projects that are ...
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1answer
303 views

State space representation of transfer function $ 1/s $

I have three questions here: 1) I'm looking for a method to compute state-space equations by hand when given a transfer function, so far I think the best I've found is here. 2) Here's an example ...
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2answers
40 views

Having trouble forming the initial matrices for a positioning problem

The question asks me to solve the positioning problem where: $$ \dot{x_1} = x_2 $$ $$ \dot{x_2} = u_1 \in U_{bb} $$ $$ x_1(0) = - \text{X} (<0) $$ $$ x_2 (0) = 0 $$ $$ x_1(t_1) = 0$$ $$ x_2(t_1) = ...
2
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1answer
165 views

Basic example of system controllability

Ok so I'm having a tough time understanding an example in my book, there is a great deal of hand-waving and I'm having a hard time following. I'd like to see the example worked out if possible, with ...
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0answers
37 views

How can I prove this theorem about differential inclusions?

Consider the following differential equations with initial conditions at time $t_0$ specified: $\dot{x}_1 = f_1(x_1,t) ; \,\,\,x_1: [t_0,T]\to\mathbb{R}^n, ...
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2answers
189 views

Matrix representation of a co-domain restriction of a linear operator

Consider the finite-dimensional linear operator: $\mathcal{A}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3},$ with $Ax=y,$ $A=\left[\begin{array}{ccc} 1 & 0 & 1\\ 1 & -2 & -1\\ 0 & 1 ...
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56 views

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.Suggestion: take u to be the suitable cut off version of ...
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211 views

Show that Bellman-Gronwall's inequality

I'm trying to prove the theorem general of the Bellman-Gronwall's inequality: Assume that $u(t)$ be real valued non - negative continuous function, and such that $$u(t)\le u(\tau ...
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92 views

Please someone tell me the name of this algebra trick or where I can learn how to do it..

Okay so my algebra knowledge is pretty guff.. I am taking a control systems class and pretty much all the questions I am expected to revise, are about doing this algebraic manipulation and I don't ...
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2answers
146 views

Prove that $x \equiv 0$ of $\dot{x}(t)=a(t)x$ is Uniformly Asymptotically Stable

I have a problem: Consider the scalar equation: $$\dot{x}(t)=a(t)x \tag{I}$$ where $a(t) \in C(\mathbb{R}^+)$. Prove that $x \equiv 0$ of $(I)$ is Uniformly Asymptotically Stable iff ...
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278 views

Show that System $(I)$ is stable iff $X(t)$ is bounded.

I have a theorem: For a linear homogeneous system: $$\dfrac{dx}{dt}=A(t)x \tag{I}$$ Where $A(t)=(a_{ij}(t))_{n \times n} \in C(\mathbb{R}^+,\mathbb{R}^{n \times n})$ Suppose that $X(t)$ be the ...
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1answer
197 views

Analyse closed loop transfer function

I have a transfer function from $x_c$ to $x$ $ \dfrac{x_c}{x} = \dfrac{k}{s + k} $ And I want to analyse the stability and find the best possible value for k. I've tried to convert the closed loop ...
2
votes
1answer
114 views

Compute $e^{tA}$

When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(. Here's my problem: For the system of equations: $$\begin{cases} & \text{ } ...
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2answers
104 views

Determine the stability of $(x,y)=(0,0)$

Determine the stability of $(x,y)=(0,0)$: 1/$$\bf{\begin{cases} & \mathrm{ } \dot{x}= -2x-y+2xy^2-3x^3\\ & \mathrm{ } \dot{y}= \dfrac{x}{3}-y-x^2y-7y^3 \end{cases} \tag {1}}$$ 2/ ...
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108 views

Construct the Mikhailov hodograph for the equation $f(z)=z^3+z^2+z+2$.

Construct the Mikhailov hodograph for the equation $$f(z)=z^3+z^2+z+2$$ Here's my solution: We have $$f(i\omega)=(-\omega^2+2)+i(-\omega^3+\omega)$$. We consider $Ref(i \omega)=0$ and $Re \omega ...
3
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1answer
155 views

Construct a Liapunov function for this system

Construct a Liapunov function for the system (Determine the stability of $x \equiv 0$): I have an example:$$\begin{cases} & \mathrm { } \dot{x}= -x^3+xy^2\\ & \mathrm { } \dot{y}= ...
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1answer
476 views

Determine the stability property of the critical point at the origin ($x=y=0$) for the following system.

Determine the stability property of the critical point at the origin ($x=y=0$) for the following system: I have an example: $$\begin{cases} & \mathrm{ } \dot{x}= \tan(y-x)\\ & \mathrm{ } ...
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1answer
68 views

If all the solutions of $\frac{dy}{dt}=A(t)y$ are bounded, then all the solutions of $\frac{dz}{dt}=[A(t)+B(t)]z$ are bounded

In the book by Richard Bellman, Stability theory of differential equations Theorem 6 (p.43) If all the solutions of $\frac{dy}{dt}=A(t)y$ are bounded, then all the solutions of ...
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106 views

Show that all the roots of $\frac{dx}{dt}=A(t)x$ are bounded in $[t_0, \infty)$.

For real system of equations$$\frac{dx}{dt}=A(t)x,(1)$$ where $A(t) \in C[t_0, +\infty)$. Prove that if $\int_{t_0}^{\infty} \|A(t_1)+A^T(t_1)\|< +\infty$ then all the roots of (1) are bounded in ...
3
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2answers
144 views

Show that root $x\equiv 0$ of $\dfrac{dx}{dt}=F(t,x)$ is uniformly stable (uniformly asymptotically stable)

I have a problem: For the system of equations: $$\bf \dfrac{dx}{dt}=F(t,x) \tag 1$$ where $F$ is continuous in $I \times D \subset\mathbb{R}\times \mathbb{R}^n$ and $F(t,0)\equiv0$, ...
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0answers
139 views

Prove that all the solutions of (2): $\frac{dy}{dt}=A(t)y+f(t)$ are bounded in $ \left[t_0,+\infty \right )$

I have a problem: Assume that system (1): $$\dfrac{dx}{dt}=A(t)x$$ is stable, where $A(t) \in C\left [t_0,+\infty \right )$, when $t \to \infty$ and $$\begin{cases} & \mathrm{ } ...
2
votes
2answers
103 views

Prove that, every solution of the scalar system: $\dfrac{dx}{dt}=y,\dfrac{dy}{dt}=-\dfrac{2y}{t},(t \ge 1) $ is bounded in domain $[1, +\infty)$

EDITED I have a problem: Prove that, every solution of the scalar system: $$\dfrac{dx}{dt}=y,\dfrac{dy}{dt}=-\dfrac{2y}{t},(t \ge 1) $$ is bounded in domain $[1, +\infty)$, but this system's not ...
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33 views

On the controllability function (minimising a functional)

Consider a system of ODEs $$\dot{x}(t)=f(x(t))+g(x(t))u(t),$$ where $f:\mathbb{R}^n\to\mathbb{R}^n$ and $g:\mathbb{R}^n\to\mathbb{R}^{n\times m}$ are smooth. Let $L:\mathbb{R}^n\times ...
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Stability theory: Every solution of the scalar equation: $\ddot{x}+\left [a+b(t) \right ]x=0$ is bounded in $\left [t_0, +\infty \right )$.

EDITED Prove that: If $a>0$ and $$\int_{t_0}^{\infty} |b(t_1)|\mathrm{d}t_1<+\infty$$ then every solution of the scalar equation: $$\ddot{x}+\left [a+b(t) \right ]x=0$$ is bounded in $\left ...
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1answer
385 views

A good reference on optimal control theory

Ok, so I am reading about decision making and I came across this subject. Fortunately it has a Wiki, but the point is I want to see some examples, and learn to solve regular problems of this field. ...
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113 views

Stability of linear systems with singular state matrix

Given a linear time invariant system $\dot X(t) = AX(t)$ where $X \in {R^{n \times 1}}$ and $A \in {R^{n \times n}}$ is a singular matrix ($A$ has at least one zero eigenvalue). How can I study the ...
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1answer
5k views

Kalman Filter to determine position and attitude from 6DOF IMU (accelerometer + gyroscope)

I'm going to describe the problem I'm trying to solve and walk through what I understand so far about the Kalman Filter. I have an IMU which gives me the following measurements every time interval t: ...
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1answer
77 views

online learning to maximize profit

I have a software which takes input as investment and gives the output as return on a particular stock. Now profit metric $x_i$ is defined as the ratio of return $g_i$ to maximum possible return ...
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2answers
118 views

How does the singularity of a system matrix affect the system's stability?

What can be said about system stability, given a singular system matrix below? \begin{align} A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ...
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1answer
70 views

On impulsive optimal control with functions of not bounded variation

I have the following optimal control problem $$ J=\int_0^TF(t,y_1(t),y_2(t))dt \to \min, $$ subject to \begin{align} &\dot y_1(t) = f(t,y_1(t),y_2(t)) + g(t)\nu(t),\\ &\dot y_2(t) = ...
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1answer
130 views

Linear time varying into linear time invariant.

My original problem, is to transform Linear time varying systems of the form , for example: $$\begin{bmatrix}\dot{x1} \\ \dot{x2} \end{bmatrix} = \begin{bmatrix} -3t^2 & 0 \\ 6t^5 & -6t^2 ...
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1answer
30 views

Minimization problem

For Which positive value(s) of $x$ the following function is most minimum $f(x) = x^2 + ax +c$ [ where $a ,c > 0$ ] [note : I know there is no positive $x$ for which $f(x)$ is minimum but I ...
0
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2answers
133 views

Closed loop stability

Regarding the Lyapunov stability, we check if a nonlinear system stays near the equilibrium point or approaches to e.p. as time goes to infinity, when it is disturbed. Let's assume that we have a ...
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1answer
100 views

Finding Transfer functions for linearised systems

I'm using Nise for my control systems class. Finding a linearised system is all gravy baby, but when it comes to finding the transfer function Nise does some stuff which confounds me: See page 6/7, ...
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1answer
226 views

Transfer functions for systems with no input but with feedback

This is a system of differential equations with a single loop feedback(piecewise-function):(I have also drawn a control system diagram below for this very system) $$\frac{\mathrm{dx} }{\mathrm{d} ...
2
votes
1answer
88 views

Time minimum optimal trajectory of a point in 2D

I am trying to find the optimal control of the following problem. We have a material point $x(t)\in\mathbb{R}^2$ with mass $m=1$. It can accelerate in any direction with maximum acceleration of $1$, ...
2
votes
1answer
203 views

Exponentials of a matrix

I just was working with matrix exponentials for solving problems in control theory. Suppose $A $ is a square matrix. How can we interpret $A_1 = e^ {\textstyle-A\log(t) }$, where $\log$ is natural ...