Tagged Questions

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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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 ...
16k 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 http://www.scipy.org/Cookbook/...
149 views

Level sets of solution to Nonlinear PDE

I work on Stochastic Control theory and BSDE's for my research. In my research, I characterized the set I am interested in as the level set of a function which is a viscosity solution to nonlinear PDE ...
1k views

Difference between Bellman and Pontryagin dynamic optimization?

Can someone please explain the difference between dynamic optimization via the Bellman equation and dynamic optimization via Pontryagin's maximization principle? Thanks
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Lyapunov stability question (Arnold's trivium)

V.I. Arnold put the following question in his Mathematical trivium: Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization? It puzzled me for a ...
3k 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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If matrices $A$ and $B$ commute, $A$ with distinct eigenvalues, then $B$ is a polynomial in $A$

If $A\in M_{n}$ has $n$ distinct eigenvalues and if $A$ commutes with a given matrix $B\in M_{n}$, how can I show that $B$ is a polynomial in $A$ of degree at most $n-1$? I think first I need to show ...
171 views

Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For ...
174 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 ...
540 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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I'm trying to calculate $e^{At}$. Where do I go wrong?

Let $$A=\begin{bmatrix}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -4 & 0\end{bmatrix}$$ I want to determine $e^{At}$. I tried it using ...
240 views

Rigorous mathematical treatments of engineering topics

I started out as an engineering student and got interested in mathematics. So after some point (Rigorous analysis and linear algebra, some real analysis, basic measure theory and topology etc.) I ...
288 views

Checking the stability of an equilibrium point

I have the linearization of a non-linear system about an equilibrium point as follows $$\dot x = (-A+M)x,$$ where $x\in\mathbb{R}^3$, $A$ is a positive definite matrix and $M$ has its eigenvalues ...
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Reference request: “initial” PDE control

I'm somewhat familiar with the ideas of boundary and internal (source) control for PDE (in particular, for wave equations) but am wondering if the following type of problem is classified/studied; if ...
122 views

Two definitions of uniformly observable, are they equivalent?

When I do my research, I found that there are two definitions of uniformly observable, I can't help thinking are they equivalent? These two definitions are listed as follows. For a linear stochastic ...
139 views

Eigenvalues of a matrix written in controllable canonical form

Let the following equation represent a stable (marginally) dynamical system in discrete time domain $$\mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k$$ ...
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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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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 $\|\dot x\|^2\to a$, ...
120 views

How to use mathematically the I and D of a PID controller

I am trying to mathmatically understand how the $P$, $I$, and $D$ parameters work on a system, quite having a hard time doing so. I've only been able to show that the Steady State Error (SSE) never ...
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An introduction to control systems

I am looking for an introduction on control systems in the context of engineering, but treated from a more mathematical point of view. Does anybody have a good reference?
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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 ...
763 views

Consider a Riccati differential equation $$\dot P + A(t)^{T}P + PA(t) -PB(t)R(t)B(t)^{T}P + Q(t) = 0,\;\;\; P(t_0) = P_0 = P_0^{T} \geqslant 0$$ where $Q(t) = Q(t)^{T} \geqslant 0$, $R(t) = R(t)^{... 3answers 54 views Relation between controllability and stabilization of a system Suppose i have a control system which is described as: $$\left\{ \begin{array}{c} \dot{x}(t)=Ax(t)+Bu(t)\\ y(t)=Cx(t)+Du(t) \end{array} \right.$$ and i know it is controllable. I use a state ... 1answer 85 views (Possible) application of Sarason interpolation theorem This question is related to the following Wikipedia article on Nevanlinna–Pick interpolation. At the end it has been written as Pick–Nevanlinna interpolation was introduced into robust control by ... 2answers 125 views Reference for LPV controls I am looking for a good mathematical introduction to LPV (Linear Parameter Varying) methods in control theory. I would like it to be more on the mathematical side of things, instead of something aimed ... 3answers 97 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 + a_{n-1}\...
I'd like to solve $AX -BX + XC = D$, for the matrix $X$, where all matrices have real entries and $X$ is a rectangular matrix, while $B$ and $C$ are symmetric matrices and $A$ is formed by an outer ...
I have the following transfer function: $$P(s) = \frac{3}{(s-1)(s+2)(s+3)}, s= j\omega$$ I got the starting and endpoints: $$\omega_0 = -\frac{1}{2}, \omega_\infty = 0$$ When I split the ...