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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Conceptual problems when minimizing a simple functional

I have a problem with what seems a very simple functional maximization. Let's define: $$ J[z]=\int \left( u(z)-\frac{\dot z^2}{2} \right) dt $$ Where $u(z)=-z^2+5$. The problem is to find $$ \arg\...
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63 views

Prove that the equation $Ax+Bu=0$ has a solution $u$ for every $x$

We know that $\text{rank}(\lambda I-E, F)= 2n$ (full row rank) for all $\{\lambda\in\mathbb{C} \mid \Re(\lambda) \ge 0\}$ where E=$\left(\begin{matrix} A_{n\times n} & 0_{n\times n} \\ ...
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71 views

LTI Multi-input Control System. Proof that controllability holds given a state feedback.

The question is : Prove that (A, B) is controllable if and only if (A + BK, B) is controllable for all K. My proof thus far: Let $u=kx +v$ Consider the Im(Qc) = Im(B) + (A+BK)*Im(B) + ... + (A+BK)^...
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40 views

Reference Request: investigation of higher order dynamical systems

In dynamical system and control theory, people usually investigate into system of the type $$\dot x = f(x,u)$$ Is there any references to looks into the theory of higher order dynamical systems of ...
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32 views

Routh Stability criterion

To check for the numbers of poles lying in the right side of the s-plane in a causal system, why do we check for the sign change only in first column ?
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120 views

Geometric interpretation of Q in Lyapunov's equation

Lyapunov's equation says: given any $Q > 0$ ($Q$ positive definite) there is $P > 0$ such that $A^T P + P A + Q = 0$ if and only if for $\frac{dx(t)}{dt}=A x(t)$ it is the case that the real ...
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73 views

How to prove the convergence in such a case?

$$ \dot x(t)=Ax(t)+Bu(t),\quad t\in[0,T]\\ x(0)=x_0\\ u\in L^\infty([0,T],\mathbb R^m)\text{ and }x\in L^\infty([0,T],\mathbb R^n) $$ It is known that $\lim_{a\to 0}||u_a-u_0||_{L^2}=0$, where $u_0$ ...
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79 views

Boundedness of the input

$$ \dot x(t)=Ax(t)+Bu(t),\quad t\in[0,T]\\ x(0)=x_0\\ x(t)\text{ is bounded on }t\in[0,T]\\ A\text{ and }B\text{ are given constants},\quad B\neq 0 $$ My question: Is $u(t)$ also bounded on $t\in[...
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22 views

Prove of disprove the asymmetry of a bode phase plot around the resonant frequency.

I am asked to prove or disprove the following: The bode phase plot for G(j$\omega$) given by G($j\omega) = a/(s+a)$ with a>0 is asymmetric with respect to (a,$-\pi/4$). I know how to derive the ...
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22 views

Proving an equivalence relation $(A_1,B_1)$ ~ $(A_2,B_2)$

Let $(A_1,B_1),(A_2,B_2) \in \Bbb R^{n\times n} \times \Bbb R^{n \times m}$. We say that $(A_1,B_1)$ and $(A_2,B_2)$ are similar written $(A_1,B_1)$ ~ $(A_2,B_2)$, if there exists $S \in$ GL (n, $\Bbb ...
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1answer
34 views

In the LQ control problem why is $2PA = A^{T}P + PA$

I do not understand why the equation below holds assuming $A$ and $P$ are both square matrices in $\Bbb{R}^{n*n}$ and $P$ is symmetric and positive definite (i.e. $ P = P^{T} $ and $ x^{T}Px > 0 $) ...
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594 views

How to control a nonlinear system with a PID controller

Is the design of a PID controller for a nonlinear system different from for a linear system? [I think math.stackexchange.com is the most suited SE.]
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29 views

Solving Differential Equation about rate of infected computers

I am having some trouble solving this differential equation for the rate of infected computers in a botnet at time t $$\frac{\mathrm{d}x }{\mathrm{d} t} = \frac{1}{c\nu (1-x) + \beta x(1-x) - \gamma ...
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24 views

Quadratic form of Kroenecker products of skew-symmetric matrices

I am trying to understand under which conditions on $P=P^\top>0$ , $C=C^\top$, the following quadratic form is zero: $$ x^\top \left( D U^\top \frac{L-L^\top}{2} U \otimes PC \right)x = 0 $$ ...
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24 views

Question regarding two properties of controllable subspace in control theory

In Mathematical Control Theory II: Behavioral Systems and Robust Control Two claims: $\langle A+BK | im B \rangle = \langle A | im B \rangle$ $\langle A | im B \rangle$ is the smallest $A$-...
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48 views

Values of a matrix for which a system represents stable or unstable systems

For $\alpha$, $\beta$, $\gamma$ $\in\mathbb{R}$, consider the system $$\frac{d}{dt}x=\underbrace{\begin{bmatrix}0 & \alpha & \gamma & 0 \\ -\alpha & 0 & 0 & 0 \\ 0 & 0 &...
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29 views

Finding the maximum of step response for a given transfer funtion

Assuming that the following transfer function is given: $$F(s)=\frac{\Sigma_{k=0}^m b_k s^k}{\Sigma_{k=0}^n a_k s^k}$$ $$m\le n$$ Lets say $g(t)$ is the step response to this transfer function. ...
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33 views

Displacement of differential equation

In [Forni and Sepulchre, arXiv:1305.3456] the authors state that given the differential equation \begin{equation} \dot{x}=f(x,u),\quad (1) \end{equation} where $f:\mathbb{R}^{n\times m}\to\mathbb{R}^...
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32 views

Phase and frequency locked loop

In electronics equipment, a unit named phase-locked loop (PLL) is used. Simply speaking, it adjusts the phase $p_r$ of a reference signal like $r(t)=sin(f_r*t+p_r)$ with constant frequency $f_r$ to ...
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34 views

Controllability of a system with point matrices

Consider the scalar nonlinear system $$\frac{d}{dt}x=\underbrace{\sin x+u}_{f(x,u)}$$ with equilibrium point $(x^{\ast},u^{\ast})=(0,0)$ We have $\frac{\partial f}{\partial x}=\cos x$ and $\frac{\...
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34 views

Imaginary results when calculating equilibrium points of a 2nd order system

The following system is given: $$ \frac{d\textbf{x}}{dt}=\begin{bmatrix} -x_1^2+x_2 \\ -x_1-x_2^2 \end{bmatrix} $$ I want to calculate the equilibrium points of the system and did that as follows: $$ ...
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42 views

Linearisation of nonlinear system of equations

Consider $$\begin{aligned}\frac{d}{dt}x_{1} & =x_{2} \\ \frac{d}{dt}x_{2} & =-x_{1}-(\alpha+x_{1}^{2})x_{2}\end{aligned}$$ with $\alpha\ne 0$ and equilibrium $x^\ast=0$. Then $f(x_{1},x_{2})=...
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1answer
54 views

Control theory - what method is used to find the discrete time control system here

We are given a car model: $$\dot x = V\cos(a) \quad \dot y = V\sin(a) \quad \dot a = u$$ $V$ some arbitrary number Make an (first order) approximation $$\dot x = V \quad \dot y = Va \quad \dot a = ...
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42 views

What are the steps to write a control function?

I have a system defined like this: $$\delta\delta x(t) = \rho(t) \cos(\alpha(t)) \\ \delta\delta y(t) = -g + \rho(t) \sin(\alpha(t))$$ I need to write a control function to calculate $\rho(t)$ and $\...
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47 views

No local optima in quantum control?

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation: $\frac{d x(t)}{dt} = F(x,u)$ one can consider the ...
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84 views

Modified Z-transform

Can somebody tell me where I do a mistake in derivation of modified (or advanced) Z-transform of digital parabolic sequence i. e $$f(k) = (k\cdot T)^2,$$ where $T$ is the sampling period and $k$ is ...
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1answer
35 views

Calculate equilibrium points of a 2nd order system

The following 2nd order system is given: $$ \frac{d\textbf{x}}{dt}=\begin{bmatrix} -6 & -\frac{2}{\pi} \\ 0 & \frac{1}{2} \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}+\begin{bmatrix}\...
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84 views

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 ...
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254 views

Eigenvalues of matrix with linear dependent rows

Why is it that, if a matrix has a linear dependent row, e.g.: $$ A=\begin {pmatrix} 1 \ 2 \\ 2 \ 4 \end {pmatrix}, $$ at least one eigenvalue is zero?
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1answer
47 views

Lyapunov invariant set for affine systems

Given a linear system $\dot{x}=Ax$ such that the real part of every eigenvalue of $A$ is less than $0$, Lyapunov's equation $A^T P + P A = -Q$ with $Q$ being any suitably sized positive definite ...
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129 views

Why no Forward Dynamic Programming in stochastic case?

Dynamic programming usually works "backward" - start from the end, and arrive at the start. This works both when there is and when there isn't uncertainty in the problem (e.g. some noise in the state)....
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38 views

Controllability matrix cannot be defined $\implies$ the system cannot be controllable?

Let $B\in\mathbb{R}^{n\times m}$ and $A=\lambda I$, for $\lambda\in\mathbb{R}$. I want to show that a necessarily condition for the controllability of $(A,B)$ is $m\ge n$. I assume this means for $$A=...
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27 views

Defining two matrices to eliminate latent variables

So I have the systems $$\Sigma_{1}:p_{1}(\frac{d}{dt})y_{1}=q_{1}(\frac{d}{dt})u_{1}$$ $$\Sigma_{2}:p_{2}(\frac{d}{dt})y_{2}=q_{2}(\frac{d}{dt})u_{2}$$ And the equations $u_{2}=y_{1}$, $u_{1}=u+y_{...
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0answers
26 views

Time-dependent inequalities in optimal controller

I need to build the optimal controller, i.e. one that maximizes: $J = \int_{0}^{t_f} f(u) \mathrm{d}t$ For the following time-dependent system: $\dot{x} = g(x, u, t)$, $x(t) \geq l(t)\; \forall t \...
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1answer
56 views

Convexity of the set

$$ \dot x(t)=Ax(t)+Bu(t)\\ x(t_0)=x_0\\ x(t_f)=x_f\\ a\leq x\leq b $$ $x(t)$ is driven by $u(t)$ to satisfy the above differential equation, two boundary conditions, and the inequality. Here, $x_0$ ...
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35 views

Sinusoidal steering on SO(3) Lie group

This is part of an exercise from Shankar's book Nonlinear Systems: Analysis, Stability, and Control, Problem 8.11, p.381, which is a satellite control problem. I rephrase my question as follows. ...
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38 views

Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
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How would I solve the following equation, which is similar to an algebraic Riccati equation or a nonlinear sylvester equation?

I have the following matrix equation that I would like to solve for $X$: $0 = AX + XB + XCX + D$ In general, $X$ will be rectangular, with $(m\times n)$ dimensions. So if I write the equation out ...
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75 views

Extracting information from Singular Value Decomposition.

I am currently working on a heat pump system. The problem involves multiple inputs and outputs. During self study I came across the SVD technique, and learned that it can relate orthogonal inputs to ...
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38 views

Showing that three systems define the same behaviour

I have three input/state/output representations of the form $$\begin{cases}\frac{d}{dt}x=Ax+Bu \\ y=Cx\end{cases}$$ with the three systems given by: $$A_{1}=\begin{bmatrix} -1 & 0 \\ 0 & -2 ...
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1answer
57 views

Hamiltonian question, find optimal controller, simple question

An object is moving with accordance to Newton's laws: $\begin{pmatrix} \dot{y} \\ \dot {v} \end{pmatrix} = \begin{pmatrix} v \\ u \end{pmatrix}$ where $y$ is the objects location and $v$ is its speed. ...
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19 views

Is it necessary to divide by the pivot entry when using Routh's Array?

I have read that multiplying a row by a positive constant doesn't change the end result of computing Routh's Array, doesn't this mean that dividing each entry by the respective pivot is not necessary? ...
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1answer
59 views

Math major transferred to Electrical Engineering, trying to bridge the gap [closed]

I did a minor in mathematics a couple years ago and the non-engineering (i.e. rigorous) math I have been exposed to were two proper courses in prob and statistics, 2 courses in real analysis and 2 ...
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20 views

Stochastic Control, Variable Coefficients

Here is my problem (I'll present it in terms of two maximizations, for simplicity): Given two cost functions, solve simultaneously $V_1(y_1) = {min_{u_1 \in \mathcal{A}(y)}} J(y_1, u_1)$ $V_2(y_1) =...
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1answer
45 views

Explain what the teacher did - system of ode, control theory.

There are a few things I'm not clear about in her solution and would appreciate a short explanation. We are given the system $\dot{x}=-ax+bu$. with an initial value $x(0)=x_0$. We want to find a ...
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1answer
70 views

Understanding how Nehari's problem connects with robust stabiliziation and Nevanlinna-Pick

I'm reading Young's "An Introduction to Hilbert space". In chapter 15 he writes about robust stabilization in control theory and ends with that this boils down to an interpolation problem called the ...
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40 views

Minimum norm matrix to stablize a linear system

Suppose all of the eigenvalues of $A$ locate strictly on the right half plane. $(A,B)$ is controllable, $H$ is symmetric and strictly positive definite. I wonder is there a optimal solution $H^*$ ...
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1answer
90 views

Finding the input/output representation

I have a system of differential equations which can be written as $R(\frac{d}{dt})w=0$, where $$R(\xi)=\begin{bmatrix}6-5\xi+\xi^{2} & -3+\xi \\ 2-3\xi+\xi^{2} & -1+\xi\end{bmatrix}$$ I want ...
2
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1answer
56 views

Controllability of internal subsystem and input-outpout controllability

I am trying to prove that the state equation $\dot{x} = \begin{bmatrix} A_{11} A_{12}\\A_{21} A_{22}\end{bmatrix}x + \begin{bmatrix} B_{1}\\ 0\end{bmatrix}u$ is controllable if and only if the pair $(...
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103 views

Bounding reachability of damped harmonic oscillator using barrier certificates

I'm trying to prove that, under certain conditions, a damped harmonic oscillator that starts on one side of the equilibrium remains on that side of the equilibrium. More precisely, consider the ...