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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Show exponential stability quadratic form

Please help me with the following proof: Suppose $\dot x=f(x(t))$ and suppose that we have: $$ \frac{d}{dt}\left( x(t)^TPx(t) \right)\le -x(t)^TQx(t) $$ where $P$ and $Q$ are symmetric ...
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19 views

Does every $\mathcal{L}_{2}$ signal have bounded $\mathcal{L}_{2}$ derivative?

Let a real signal $f(t) \in \mathcal{L}_{2}$. Does it always imply that $\dot{f}(t) \in \mathcal{L}_{2}$? It is assumed that $\dot{f}(t)$ exists for all $t \in \mathbb{R}^{+}$.
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37 views

perturbation of exponentiolly stable system

consider the following system on $\Bbb{R}^n$ $\dot{x} = f(x,t)+g(x,t) $$ $$ $$ $ $ (*) $ assume that f(0,t)=g(0,t) = 0 and 1. 0 is an exponentiolly stable equilibrium of $\dot{x}=f(x,t)$ ...
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16 views

Is constant system a Causal System?

Is y(t) = 1 a causal system? From the definition of causal systems , a causal system is a system where the output depends on past and current inputs. Here the system doesn't depend on any input. ...
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53 views

Reachability from non-zero initial state?

I have the following system: $$ \dot x(t)=\begin{bmatrix} -2 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}x(t)+\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}u(t) $$ The ...
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20 views

LTI system: solving for the time at which a system state reaches a given value

Suppose I have the following Linear Time Invariant (LTI) system: \begin{equation} \dot{x}(t) = Ax(t) + Bu(t) \end{equation} where $x(t)=\begin{bmatrix}x_1(t) & x_2(t) &\ldots ...
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10 views

How to calculate mean of busy-cylce

Let $\{X(t)\}$ be birth-death process on a finite state {0,1,2} with non negative birth rates $(\lambda_0,\lambda_1)$and death rates $(\mu_1,\mu_2)$. Suppose $\mathbb{P}(X(0)=0)=1$ and $s_0=\inf ...
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31 views

Clarke's generalized gradient formula computed on functions defined on open sets

In the book [1], Clarke et al. define the generalized gradient for a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ as follows. 8.1. Theorem (Generalized Gradient Formula). Let ...
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28 views

Why we cannot apply pole placement for the following system?

$$ \begin{cases} \dot{x}=\begin{pmatrix} -2 & 0 \\ 0 & -2 \\ \end{pmatrix} x+\begin{pmatrix} 2 \\ 2 \\ \end{pmatrix} u \\ y=Cx \end{cases} $$ How can I show that if $p_1 \neq -2$ and ...
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47 views

Minimum infinity norm control problem

I am having trouble understanding Example 2 of section 5.9 of Luenberger's Optimization by Vector Space Methods. The problem is to select a current $u(t)$ on $[0,1]$ to drive a motor governed by $$ ...
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35 views

Matrix comparison depend on one scalar variable

Let $A$, $B$, $K$ be $n\times n$, $n\times m$ and $m\times n$ matrix respectively. $\alpha_i>0$ is a scalar. Consider the following matrix: $H_i=\int_0^\infty e^{(A+\alpha_i ...
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23 views

If system's all the eigenvalue decrease, the peak point of output becomes smaller and the lowest point becomes larger.

The system is a linear time-inviarant system as $$d\mathbf{T}/dt = \mathbf{A}\mathbf{T}(t) + \mathbf{B}$$ where $\mathbf{T},\mathbf{B}$ is an $N\times 1$ vector and $\mathbf{A}$ is an $N \times N$ ...
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32 views

Kalman Filter, observable system

I have been trying to understand well the Kalman Filter recently and there is one fact which I haven't seen proved rigorously yet. If the system is observable, then the covariance of the error ...
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48 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
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22 views

Discrete time system - phase response

I have a question about a discrete time filter. All I have is the pole-zero plot and I have to calculate the impulse, phase and magnitude response. To make this a proper fraction I used polynomial ...
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1answer
75 views

Stability of sampled-data systems using Lyapunov functions

For continuous systems, Lyapunov functions provide a general technique to establish stability. For example, the simple system $x' = -x$, a Lyapunov function is $V(x) = \frac{1}{2}x^2$. It is easy to ...
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2answers
52 views

What is the difference between an impulse response and a transferfunction?

An imupulse response, is the output you get when you apply an impulse, like a delta dirac function, to your system (only for LTI?). By knowing the impulse response you know the system. The ...
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41 views

LQR Problem Minimization Prove

Given J for an LQR Problem is $J = \frac{1}{2} \int {z_1}^T \hat Q z_1 + v^T Q_{22} v\,dt $ where $\hat Q$ above is given as $\hat Q = Q_{11} - Q_{12}{Q^{-1}}_{22}Q_{21}$ is minimized if we use ...
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3answers
45 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 ...
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1answer
30 views

Explain: Independent Column variables as Linear combination of free variables.

I am reading a book on control systems and stuck on a text in it. We have $Sx = 0$ where S $\in$ $\Re^{m x n}$ and is full rank i.e Rank of S = m. $x \in \Re^n$ Now it states that "Exactly m ...
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25 views

Find the normal form of this function

A second order control theory function looks like: $$\text{H}_{(s)}=\frac{\text{K}_p}{\frac{1}{\omega_0^2}\cdot s^2+\frac{2\beta}{\omega_0}\cdot s+1}$$ Now I've got the function, with ...
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28 views

How does state transition matrix works

Suppose I have a simple vehicle moving in 2D. The state vector for the vehicle is X=[x y vx vy ax ay], that is, it contains the position (x,y), the velocity (vx, vy) and the acceleration (ax, ay) of ...
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41 views

What is the smallest positive value of K which makes the closed-loop system unstable?

You are given a transfer function $\displaystyle G(s)=\frac{1.81K(s+20)}{(s^3+10s^2+32s+32)}$. This system is connnected with unity negative feedback. I've tried so many things but I can't do it . ...
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41 views

Are integro-differential equations considered dynamical systems?

A definition of the dynamical system is that: $\phi:R \times E \to E$ is a dynamical system where $\phi \in C^1$, $E$ open subset of $\mathbb{R}^n$, and if $\phi_t(x) = \phi(t,x)$, then ...
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1answer
16 views

PID tuning for system with parameter

How can I tune a PID to control a system with a parameter? Can I know beforehand how to change the PID parameters in function of the value of the system parameter?
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14 views

Should the performance of a PID controller be independent of the input?

Assume you designed a PID controller to let a given system track a unit step. Will the controlled system exhibit the same behaviour with regard to step inputs with different amplitudes?
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2answers
30 views

How is a state disturbance matrix constructed?

Consider the system: $\dot{x}$ = Ax + Bu y = Cx + Du Where x contains 4 states, we have 2 inputs $u = \begin{bmatrix}u_1\\u_2\end{bmatrix}$ and A, B, C & D are known. Now if 2 separate noise ...
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1answer
37 views

How to drive a vehicle (limited by acceleration) on a flat ground to a given point as fast as possible?

So I have a function $\mathbf{x}(t): \mathbb{R} \rightarrow \mathbb{R}^2$, which is supposed to mean the path of the vehicle (time mapped to position). The initial conditions $\mathbf{x}(0)$ and ...
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107 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 ...
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31 views

Calculate state transition matrix with one left and right eigenvector

How is it possible to calculate the state transition matrix of the following LTI-System: $$ \frac{d\mathbf{x}}{dt}=\mathbf{A}\mathbf{x}+\pmatrix {1 \\ -1}u$$ $$y=\pmatrix {2 & -2}\mathbf{x}$$ ...
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30 views

Linearization of a Matrix Inequality with Quadratic Terms

I derived the following matrix inequality for finding a stabilizing observer-based feedback gain $K$. $$ \begin{bmatrix}-\lambda P& \tilde{A}^T & 0\\ \tilde{A} &-P^{-1} & \tilde{B}\\ ...
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42 views

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 $$ ...
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58 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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2answers
47 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) + ... + ...
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38 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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27 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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94 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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72 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 ...
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2answers
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 ...
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19 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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20 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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30 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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93 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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26 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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20 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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1answer
18 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 ...
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35 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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25 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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30 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 ...
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18 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 ...