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 ...

learn more… | top users | synonyms

1
vote
1answer
16 views

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 ...
0
votes
0answers
20 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 $t \in \mathbb{R}^{+}$.
1
vote
1answer
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 ...
1
vote
0answers
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 $\dot{x}=f(x,t)$ ...
0
votes
0answers
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. ...
0
votes
0answers
21 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 ...
0
votes
0answers
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
1
vote
1answer
32 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 ...
5
votes
0answers
49 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 $$ ...
1
vote
0answers
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 ...
0
votes
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?
1
vote
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 ...
-2
votes
0answers
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 $N \times N$ ...
0
votes
1answer
499 views

Using overshoot and settling time formula to determine pole location?

Is it possible to use the formula for overshoot and settling to determine where where ones pole should. by using the overshoot and settling time formula i mean, using it to define what $\zeta$ and ...
0
votes
0answers
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 ...
0
votes
1answer
45 views

Why do high-frequency dynamics quickly go away in a step response?

As we know, a step input hits all the frequencies of a dynamical system. However, my professor told me today that the high-frequency response is only present for a short time at the very start, and ...
1
vote
0answers
49 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 ...
0
votes
1answer
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
3
votes
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 ...
0
votes
1answer
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 ...
1
vote
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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 . ...
1
vote
1answer
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 ...
0
votes
0answers
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?
5
votes
0answers
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 ...
7
votes
2answers
199 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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
2answers
169 views

eigenvalues of the sum of a diagonal matrix and a skew-symmetric matrix

Suppose $A$ is a skew-symmetric matrix (i.e., $A+A^{\top}=0$) and $D$ is a diagonal matrix. Under what conditions, $A+D$ is a Hurwitz stable matrix?
11
votes
3answers
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
0
votes
1answer
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}\\ ...
0
votes
1answer
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: $$ \frac{d\mathbf{x}}{dt}=\mathbf{A}\mathbf{x}+\pmatrix {1 \\ -1}u$$ $$y=\pmatrix {2 & -2}\mathbf{x}$$ ...
0
votes
1answer
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 $$ ...
1
vote
1answer
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} \\ ...
0
votes
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) + ... + ...
0
votes
1answer
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 ?
1
vote
2answers
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 ...
0
votes
1answer
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 ...
3
votes
2answers
140 views

Derivation of the Mexican wave as Mean field equilibrium of an ergodic control problem

I'm dealing with a Mean Field system introduced by Gueant, Lasry, Lions in their "Mean Field Games and Applications" which admits as solution the so-called Mexican wave. Without going into the details ...
1
vote
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 ...
0
votes
1answer
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 ...
1
vote
0answers
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: $$ x^\top \left( D U^\top \frac{L-L^\top}{2} U \otimes PC \right)x = 0 $$ ...
0
votes
0answers
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 ...
0
votes
1answer
617 views

In what cases are the eigenvalue equal to the pole points?

I have a transfer function in form of a matrix and want to determine the stability of the whole system. Now I'm wondering if I need to calculate the pole points or the eigenvalue. A friend of mine ...
0
votes
1answer
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 ...