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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89 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 ...
4
votes
2answers
137 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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2answers
274 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
161 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
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1answer
106 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{ } ...
1
vote
2answers
96 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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0answers
70 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
votes
1answer
146 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}= ...
2
votes
1answer
399 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{ } ...
2
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1answer
53 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: http://www.mediafire.com/download/gqlo8iqa5b4pd95/Richard_Bellman_Stability_theory_of_differential_equations_2008.djvu Richard_Bellman_Stability_theory_of_differential_equations__2008 ...
4
votes
2answers
103 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
votes
2answers
133 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 $F(t,0)\equiv0$, ...
3
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0answers
119 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
101 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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1answer
30 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 ...
3
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4answers
184 views

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
251 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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0answers
85 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
3k 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: ...
0
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1answer
73 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 ...
2
votes
2answers
107 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 ...
2
votes
1answer
65 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) = ...
0
votes
1answer
98 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 ...
0
votes
1answer
28 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
votes
2answers
125 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 ...
0
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1answer
87 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, ...
0
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1answer
183 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
77 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
188 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 $A_1 = e^ {\large-A\log(t) }$, where $\log$ is natural ...
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2answers
345 views

How can I, as a future mathematician, contribute most to Smart Grid research?

After I've finished my Master's degree in mathematics, I too want to use my powers for good. One endeavour I consider good is the pursuit of the design and implementation of a Smart Grid which will, ...
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1answer
66 views

Controllability on nonlinear systems

Dynamic system in the book (chapter 6, page 67 in http://www.me.berkeley.edu/ME237/6_cont_obs.pdf) $$ \begin{cases} \dot{x}_{1}=x_{2}^{2}\\ \dot{x}_{2}=u\end{cases} $$ so $$ ...
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1answer
130 views

Question about inverse-variance weighting

Suppose we want to make inference on an unobserved realization $x$ of a random variable $\tilde x$, which is normally distributed with mean $\mu_x$ and variance $\sigma^2_x$. Suppose there is another ...
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2answers
113 views

Stability of unit feedback LTI system (s-1)/(s(s+1)) vs. Nyquist Criterion

Consider a unit feedback system $$ X(s) = \frac{G(s)}{1+G(s)} $$ where the open loop transfer function of the system is $$ G(s) = \frac{s-1}{s(s+1)} $$ Open loop Bode & Nyquist plots: ...
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1answer
83 views

which book teaches analysis of nyquist, bode and rlocus diagram

would like to use knots to get a formula for nyquist diagram however, no crossing, and have no experience in analysis of graph related to control, as i have no books mentioning this and i observe ...
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0answers
130 views

Solving this optimal control problem

The problem is: $max \int_{-1}^1 (tx - u^2) dt$ where $\dot{x} = x + u^2, u(t) \in [0,1]$ for every $t \in [-1, 1]$ End points: $x(-1) = 0, x(1) = e^2 - e^{1 + \frac{1}{e}} $ I need to find an ...
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0answers
54 views

restricting the domain of the unbounded operator

Can anyone give me a a restricted non-void set of bounded inputs which results in bounded outputs, though the operator is not bounded on the whole space (consider L2-space). One can consider a simple ...
2
votes
0answers
63 views

Find u that minimizes the integral mean

How do I find $u : [0,\infty) \to \mathbb{R}^m$ that minimizes \begin{equation} J(u(\cdot)) = \lim_{t \to \infty} \frac{1}{t} \int_0^t L(x(\tau),u(\tau)) d \tau, \end{equation} subject to ...
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1answer
120 views

Positioning problem optimal control

Consider the positioning problem: $\dot x_1 = x_2$, $\dot x_2 = u$ with $x_1(0)=0,x_2(0)=X, X>0$. show that the bang-bang control switch can be employed to steer the system to the origin. Find the ...
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1answer
135 views

Optimal control

Consider the growth equation: $ \dot{x} = tu $, with $x(0)=0$ and $x(1)=1$, and with the cost function: $ J= \int_0^1 u^2 dt $. Show that $u^*=3t$ is a successful control, with $x^*=t^3$ and $J^*=3$ ...
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0answers
42 views

improving fluid analysis on queuing problems

Discrete-queuing models are hard to solve computationally and can become easily intractable with an increased number of state-action pairs. Markov decision processes can be employed to come up with ...
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83 views

optimal control -Taylor expansion - PDE problem

I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point. For the given control ...
3
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1answer
344 views

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 ...
2
votes
2answers
267 views

Finding the steady state error in the Laplace domain

I have the following block diagram: Now I like to find the steady state error for theta_ref being a step input and for several values of n, Td, K1 and K2. For the moment we can assume all gains ...
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1answer
32 views

Pontryagin Principle for Random Process

Can Pontryagin's principle be written for minimizing the expected cost in the case of a stochastic process which is controlled? (cost at each time is function of state and action)?? I have not come ...
5
votes
1answer
82 views

Linearization of $ m \dfrac{dy^2}{dt^2} = u(t) - C_d \left( \dfrac{dy}{dt} \right)^2-mg $

$$ m \frac{dy^2}{dt^2} = u(t) - C_d \left( \frac{dy}{dt} \right)^2-mg $$ where $$\begin{align*} y(t)&=\text{missile altitude}\\ u(t)&= \text{force}\\ m&= \text{mass}\\ C_d&= ...
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vote
2answers
93 views

Does negative third derivative imply negative first derivative?

Does negative third derivative imply negative first derivative? For a system, the negative of the derivative of the Lyapunov function means the system is stable. How about the negative third ...
2
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2answers
137 views

What is the difference between regulator and stabilization

What is the difference between regulator and stabilization in control theory don't they both minimize the disturbance to the system? could answer be elaborated from the view of state and output?
20
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5answers
1k 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 ...
1
vote
1answer
88 views

Nyquist criterion

When using the Nyquist stability criterion, amplitude-frequency characteristic etc. we go from the Laplace image $G(s)$ to $G(j\omega )$. By definition of the Laplace transform, $s=\sigma + j\omega$. ...
0
votes
1answer
52 views

Can linearization of a function around $x=0$ show whether first derivative is positive or negative?

As title says, can linearization of a function $f(x)$ (by the method of taylor series around $x=0$) show whether first derivative of the function ($df/dx$) is positive or negative at $x=0$? And.. ...