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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22
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5answers
2k 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 ...
15
votes
5answers
10k 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 ...
11
votes
2answers
905 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
11
votes
2answers
580 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 ...
10
votes
3answers
2k 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 ...
9
votes
6answers
2k 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 ...
9
votes
2answers
149 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 ...
8
votes
2answers
163 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 ...
7
votes
2answers
155 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 ...
7
votes
1answer
229 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 ...
6
votes
2answers
414 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
1k 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.
5
votes
2answers
1k 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 ...
5
votes
1answer
87 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&= ...
4
votes
3answers
193 views

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 $\|\dot x\|^2\to a$, ...
4
votes
2answers
108 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 ...
4
votes
2answers
626 views

PID controller convergence

Is there any material anywhere on convergence of PID controllers? Ie, if we formalize the "plant process" in some way, like $y_{t+1} = f(x_t,y_t)$ (in other words, the process value at a given time ...
4
votes
2answers
147 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 ...
4
votes
2answers
285 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 ...
4
votes
1answer
190 views

Difference between Variation of Calculus problems and Control Theory problems?

Variation of Calculus seems to have problems without the control with variables such as state and time. Then again Control Theory problems seems to have problems with one extra variable that is ...
4
votes
1answer
363 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
47 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
1answer
2k 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 ...
4
votes
0answers
63 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 ...
4
votes
0answers
159 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{ } ...
3
votes
3answers
2k 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
4answers
201 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 ...
3
votes
3answers
238 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
1answer
570 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) = ...
3
votes
3answers
84 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 + ...
3
votes
2answers
1k 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
321 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
1answer
39 views

How do you find the state space representation of $G(s) = \frac {1}{s^2+s+1}$

Let $G(s) = \frac {1}{s^2+s+1}$ be the transfer function of the system Then $Y(s)(s^2+s+1) = U(s)$ Therefore $y'' + y' + y = u$ After this step, how should I set up my state transition variable $x$ ...
3
votes
1answer
56 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 ...
3
votes
2answers
174 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
1answer
139 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
101 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 ...
3
votes
2answers
46 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
115 views

Prove $0$ is an exponentially stable equilibrium of the system $x'=f(x)+g(x)$ if $f(0)=g(0)=0$

Besides the conditions in the title, we have: $0$ is an exponential equilibrium of the system $y'=f(y)$ $|g(x)|\leq \mu|x|,\forall x \in \mathbb{R}^n$ $\mu$ is sufficiently small! What I have ...
3
votes
1answer
161 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
145 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
122 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
204 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
368 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
2answers
751 views

Derivation of the Riccati Differential Equation

I am attempting to derive the Riccati Equation for linear-quadratic control. The original equation is: $-\partial V/\partial t = \min_{u(t)} \{x^TQx + u^TRu + \partial V^T/\partial x(Ax + Bu) \}$ $x ...
3
votes
1answer
773 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
59 views

Transfer function of differential eqaution

I'm trying to find out the transfer function of simple differential equation: $$a_0\dot y + a_1y=b_0x+b_1$$ The problem is i have no idea what to do with $b_1$. If we apply the Laplace transform ...
3
votes
1answer
29 views

Poles and zeros of a system matrx

While I am reading lecture notes on Poles and Zeros of MIMO systems, I find the following example, which is not clear for me. $$ H(s) =\pmatrix{1 & \dfrac{1}{s-3} \\ 0 & 1 } $$ The ...
3
votes
3answers
86 views

Rank of the matrix product $C e^{At} B$

Let $A \in \mathbb R^{n \times n}$. Fix $m<n$ and let $B \in \mathbb R^{n \times m}$, $C \in \mathbb R^{(m-1) \times n}$ be two matrices with full rank. What I am interested in is the matrix ...
3
votes
1answer
94 views

Stability of $\dot{x}=-A(t)x$

I already have an ODE of $A(t)$, that is $\dot{A}=-G(A(t)-A^*)$, where $G$ and $A^*$ are constant positive definite matrices. Thus I can deduce that $A(t)$ exponentially converge to $A^*$. Now I take ...