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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15
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4answers
642 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 ...
9
votes
2answers
476 views

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 ...
8
votes
3answers
1k views

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 ...
7
votes
4answers
694 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 ...
7
votes
4answers
3k 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 ...
7
votes
2answers
557 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
6
votes
2answers
262 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, ...
5
votes
1answer
79 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&= ...
5
votes
1answer
412 views

Smith normal form of a polynomial matrix.

I have the following matrix: $P(s) :=$ \begin{bmatrix} s^2 & s-1 \\ s & s^2 \end{bmatrix} How does one compute the Smith normal form of this matrix? I can't quite grasp the algorithm.
4
votes
2answers
755 views

The nature of eigenvectors of a given eigenvalue

I am working on a problem where I have an ($n \times n $) matrix A and an eigenvalue of A, $\lambda$, where $\lambda$ has geometric multiplicity 1. The right and left eigenvectors of A corresponding ...
4
votes
1answer
247 views

Solving for specific entries in a Lyapunov Equation

Let $A$ be a $2n\times 2n$ real matrix with the following structure \begin{equation} A = \left(\begin{matrix} 0 & -I \\ K & S \end{matrix}\right) \end{equation} with all sub-matrices of size ...
4
votes
1answer
33 views

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 ...
4
votes
2answers
750 views

Root Locus Diagrams - “Breakaway Point”

Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
3
votes
3answers
904 views

Poles and Zeros of Linear Systems

This period I follow a course in System and Control Theory. This is all about linear systems $$\frac{dx}{dt}= Ax + Bu $$ $$y = Cx + Du $$ where A,B,C,D are matrices, and x, u and y are vectors. ...
3
votes
3answers
110 views

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?
3
votes
2answers
708 views

Solution of a Sylvester equation?

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 ...
3
votes
1answer
227 views

How can I efficiently sketch a Nyquist diagram?

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 ...
3
votes
2answers
48 views

significance of zeros of a transfer function?

In control theory, the poles of a transfer function give information about the stability and behavior of a system. I'm not sure and can't find anywhere what the significance of the zeros of a ...
3
votes
2answers
75 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 ...
3
votes
1answer
59 views

Calculate state transition matrix

I have a question to the following problem: "There is a linear, time-invariant System with the form $\frac{d\mathbf{x}}{dt}=\mathbf{A}x$. The Eigenvalues of the matrix $A$ are $s_1=-1$ and $s_2=-2$, ...
3
votes
2answers
258 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 ...
3
votes
2answers
36 views

Horizon selection

$$x(t+1) =Ax(t)+Bu(t)=\left(\begin{matrix}0&0&0.8\\1&0&0\\0&1&0 \end{matrix}\right)x(t)+\left(\begin{matrix}1\\0\\0 \end{matrix}\right)u(t)$$ Basically the excercise is to find ...
3
votes
1answer
127 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}= ...
3
votes
2answers
86 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
134 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
3
votes
1answer
94 views

Upper bound concerning Snell envelope

Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the habitual ...
3
votes
1answer
166 views

Maximum principle for a control with mixed constraints

Consider the dynamical system $\mathbf{x}' = f(\mathbf{x,u},t)$ where $\mathbf{x} \in \mathbb{R}^2, \mathbf{u} \in \mathbb{R}^3$ and $f$ has no explicit time dependence. As conventional, $\mathbf{x}$ ...
3
votes
2answers
295 views

Practical physics problem about distance between lines.

I am a software engineer and I’m developing a soccer game. I have a solution for this problem based on Newton law and I’m using Newton method to solve the equation I got. I’m here because I think ...
3
votes
1answer
549 views

How to obtain a possible state space representation of this 2nd order transfer function?

I have this 2nd order transfer function: $$G(s) = \frac{2}{s} + \frac{1}{s+2}$$ And I need to find a possible state space representation in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ ...
3
votes
1answer
68 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 $n$, ...
3
votes
1answer
257 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 ...
3
votes
1answer
69 views

Estimating the input to a system from a system state

[ Cross-posted to: http://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ] I have a system for which I have obtained a non-linear time-varying ...
3
votes
0answers
41 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
3
votes
0answers
73 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 ...
3
votes
1answer
121 views

minimization problem on differential equations - optimal control

I am trying to minimize an time-integral of a linear function with respect to differential equations. The problem is formally defined as follows: Given $\lambda< \mu_1, \mu_2$ fixed ...
3
votes
1answer
118 views

Verification for maximum principle

Given optimal control problem $$ \dot x = f(t,x(t),u(t)), \quad x(0) = x_0,\\ J(u) = \int_0^T f^0(t,x(t),u(t))dt \to \min, $$ we can apply Pontryagin's maximum principle to get a necessary condition ...
2
votes
4answers
166 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 ...
2
votes
4answers
85 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 ...
2
votes
1answer
250 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
votes
1answer
165 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 ...
2
votes
3answers
132 views

Weird condition for a transfer function

I am reading a paper that presents the following system, represented by a second-order transfer function: $G(s) = \frac{K\times(1+0.036s)}{(1+0.0018s)(1+as)}$, where the gain $K$ is a known ...
2
votes
1answer
39 views

Second order control system

We have the open loop transfer function $$ G(s)H(s) = \frac{k(s+2)}{s^2+2s+3}$$ There were three parts to this question : a. The value of $k$ for which repetitive roots occur b. The range of $k$ ...
2
votes
2answers
60 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 + ...
2
votes
2answers
169 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 ...
2
votes
2answers
2k views

Why do we want to know the poles and zeros of a linear system?

I know that I already asked this kind of question on the website, but meanwhile I have much more knowledge about the subject and ready to describe my real problem with enough background information I ...
2
votes
1answer
325 views

Existence of global solution of Riccati equation

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) = ...
2
votes
1answer
55 views

stability of a closed-loop system

We have $$G(s)H(s) = \frac{Ke^{-s}}{s(s^2+5s+9)}$$ We have to determine the maximum value of K for the closed-loop system to be stable . We have to do it using the Routh hurwitz criterion . I ...
2
votes
1answer
210 views

The yacht race problem from L.C.Evans's PDE book 2nd edition chapter 10

There is one yacht starting at $(x_1,0)$ when $t=0$, which is sailing toward positive direction of $x$-axis with a constant velocity $b_1$, another yacht is starting at $(0,x_2)$ when $t=0$, and is ...
2
votes
1answer
59 views

Pole-zero cancellation Paradox

Suppose we have an open-loop transfer function $$G(s) = \frac{1}{s(s+a)(s+b)}$$ If we plot the root locus for the closed-loop system we will get roughly something like this : Now the question is ...
2
votes
1answer
44 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: ...