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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Is Lyapunov equation always solvable with A as a negative definite matrix?

Given a negative definite matrix $A$ and $Q=I$, is the Lyapunov equation for $P$, that is, $PA+A^TP=-I$ always solvable? what kind of form does the solution have? I will appreciate if examples can be ...
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323 views

Local stability + global attractivity = global asymptotic stability?

I was wondering how could I prove such a property stated in [Angeli, 2004]. For instance, consider the system $\dot{x}=f(x)$, where $f:\mathbb{R}^n\to\mathbb{R}^n$ is Lipschitz continuous. Claim. ...
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90 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 ...
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124 views

Continuity of solution of Riccati equation with negative source term

Let $t_0 > 0$ and consider the scalar Riccati differential equation $ p'(t) + 2 a(t) p(t) - r(t) \, p(t)^2 + q = 0 \; , $ with initial condition $ p(t_0) = 0 \; , $ in which $a$ is a function, ...
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183 views

Lyapunov stabilty, elementary question

Let’s say I have a system 1/(T1s+1) or any other n-th order polynomial and a PI controller (KP and TI). I already know that the system is stable but for, let’s say, educational purposes (not ...
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83 views

Cellular Automata Method to Reach Equilibrium on a Network of Numbers

Here's a puzzle that I'm curious if anyone can attack with a simple method without resorting to simulation. I can tell you the answer (well, an answer) from running some programs. But I'm writing a ...
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84 views

Given a system $\dot{x}(t)=Ax(t) + Bu(t)$ find $S$ such that $s^{-1}AS=\bar{A}$, $S^{-1}B=B$ is in controller form.

I am given the system $\dot{x}(t)=Ax(t) + Bu(t)$ where $$A = \left( \begin{matrix} -1 & 0& 2\\ 0 & -3 & 0 \\ 1&0&0 \end{matrix} \right), \quad B = \left( ...
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66 views

Exponential and uniform stability

I really don't understand how resolve this exercise...with Lyapunoff? can someone help me? Thanks Consider the state equation: $$ \frac{\partial}{\partial t}x(t)= A(t)x(t), \: x(\tau)=x_0 $$ ...
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3answers
83 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 + ...
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1answer
30 views

whether a Gramin matrics is symetric or not?

A Gramin matrix is defined as $$ G^c(t_0,t_f)=\int_{t_0}^{t_f} \exp\bigl((t_0-t)A\bigr) BB^T \Bigl(\exp\bigl((t_0-t)A)\Bigr)^T \,dt$$ where ${}^T$ is for transpose of matrices. How can i prove it?
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45 views

Bode phase calculus with frequency to $0$ and $\infty$ in a quadratic term

The tangent to calculate is: $$\phi=-\tan^{-1}\left[\frac{2\zeta\frac{\omega}{\omega_n}}{1-(\frac{\omega}{\omega_n})^2}\right]$$ Where $\omega_n$ is constant. If $\omega=0 \to\phi=0$ ...
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5answers
185 views

Does Interpolation/Extrapolation the crucial thing happening in our brain while driving a motor vehichle?

Does Interpolation/Extrapolation is the crucial thing happening in our brain while driving a motor vehichle? What I'd like to know is the mathematics happeing while we are behind the wheel. PS : I am ...
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1answer
33 views

Extensions to linear control with output constraints

Does anybody know which extensions to the linear controller exist that can cope with constraints in the output value and its derivative? Usually, the plant being controlled have some limits and I ...
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1answer
120 views

Affine Non Autonomous State Space system

Normally We all know the state space model of the the form der(x) = F*x(t)+G*u(t) y = H*x(t)+J*u(t). However I came across a state space model which has the following form der(x) = F*x(t)+G*u(t) + ...
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1answer
113 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 ...
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82 views

Which constant matrices $A$ have the property that $\dot{x} =Ax+Bu$ is controllable for every non zero $B$?

Which constant matrices $A$ have the property that $\dot{x} =Ax+Bu$ is controllable for every non zero $B$? Now to check controllability I usually check the rank of the matrix $$R = \left( B\ \ \ \ ...
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79 views

Show that the system $\Sigma(SAS^{-1},SB,CS^{-1},D)$ is observable/controllable iff $\Sigma(A,B,C,D)$ is observable/controllable

I am given the two linear systems: \begin{eqnarray} \Sigma_1: \dot{x}&=&Ax+Bu\\ y&=&Cx+Du \end{eqnarray} and \begin{eqnarray} \Sigma_2: \dot{x}&=&\bar{A}x+\bar{B}u\\ ...
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54 views

$A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that $\exists p\in\Bbb R ^m$ s.t. $(A,Bp)$ is controllable iff $(A,B)$ is controllable

Let $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that there exists a vector $p\in \Bbb R ^m$ such that $(A,Bp)$ is controllable iff $(A,B)$ is controllable. Here when I say $(A,B)$ ...
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51 views

Can you find an expression for $F_{12}(t,\tau)$ in terms of $F_{11}(t,\tau)$ and $F_{22}(t,\tau)$?

I have a problem with this...I can not figure out how to solve it..! can you help me? thank you!! Show that if $A(t)$ is partitioned as $$ A(t) = \begin{pmatrix} A_{11}(t) & A_{12}(t) \\ ...
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76 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 $n$, ...
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78 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 ...
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101 views

Found transition matrix and state matrix

I tried to found a solution for this problem but I can't! Any suggestion?Thank you. Given that A is a 2x2 matrix and that dx/dt=Ax(t) suppose that x(0)=[1 ; -3] implies x(t)=[e^-3t ; -3e^-3t] and ...
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47 views

Help finding the transfer matrix for this system

Find the transfer function of the following system: \begin{eqnarray} \dot{x}_1&=&ax_1+bx_2 + u\\ \dot{x}_2 &=&-bx_1 +ax_2\\ \dot{x}_3&=&cx_3\\ y &=& x_1+x_3 ...
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51 views

Non linear state equation show that nominal output is 1

Do you help me with this? Consider the nonlinear state equation $\dot{x} = \begin{bmatrix} u \\ u x_1 -x_3 \\ x_2 - 2x_3 \end{bmatrix}$ $y = x_2 - 2x_3$ with nominal initial state $x^* = ...
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1answer
72 views

Linearize a nonlinear system

I haven't really encountered control theory before. Could you possibly recommend some papers/textbooks that start with the very basics of the field? I have a couple of months to get to grips with the ...
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1answer
38 views

Check if a system admits solutions of period 2

I have the following problem. Let $r \geq 0$ be a parameter in the discrete time system $x(k + 1) = r − rx(k)$. Verify whether there exist $r \geq 0$ such that this system admits solutions of period ...
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131 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
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1answer
181 views

Converting a direction vector to an angle

I have a machine that controls its own velocity via the use of a fixed thruster mounted on its rear. The machine knows its current velocity, and it knows the velocity it must attain. By subtracting ...
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1answer
75 views

I have reached a rut in my understanding of control systems. How do I cross this?

A little background here. I'm an undergrad in the final year. I have decided academia as my career path. My grades are not high but my research caliber is good and I have ongoing projects that are ...
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1answer
309 views

State space representation of transfer function $ 1/s $

I have three questions here: 1) I'm looking for a method to compute state-space equations by hand when given a transfer function, so far I think the best I've found is here. 2) Here's an example ...
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2answers
40 views

Having trouble forming the initial matrices for a positioning problem

The question asks me to solve the positioning problem where: $$ \dot{x_1} = x_2 $$ $$ \dot{x_2} = u_1 \in U_{bb} $$ $$ x_1(0) = - \text{X} (<0) $$ $$ x_2 (0) = 0 $$ $$ x_1(t_1) = 0$$ $$ x_2(t_1) = ...
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1answer
171 views

Basic example of system controllability

Ok so I'm having a tough time understanding an example in my book, there is a great deal of hand-waving and I'm having a hard time following. I'd like to see the example worked out if possible, with ...
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37 views

How can I prove this theorem about differential inclusions?

Consider the following differential equations with initial conditions at time $t_0$ specified: $\dot{x}_1 = f_1(x_1,t) ; \,\,\,x_1: [t_0,T]\to\mathbb{R}^n, ...
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2answers
195 views

Matrix representation of a co-domain restriction of a linear operator

Consider the finite-dimensional linear operator: $\mathcal{A}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3},$ with $Ax=y,$ $A=\left[\begin{array}{ccc} 1 & 0 & 1\\ 1 & -2 & -1\\ 0 & 1 ...
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56 views

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.Suggestion: take u to be the suitable cut off version of ...
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214 views

Show that Bellman-Gronwall's inequality

I'm trying to prove the theorem general of the Bellman-Gronwall's inequality: Assume that $u(t)$ be real valued non - negative continuous function, and such that $$u(t)\le u(\tau ...
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4answers
92 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 ...
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146 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 ...
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284 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 ...
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203 views

Analyse closed loop transfer function

I have a transfer function from $x_c$ to $x$ $ \dfrac{x_c}{x} = \dfrac{k}{s + k} $ And I want to analyse the stability and find the best possible value for k. I've tried to convert the closed loop ...
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1answer
114 views

Compute $e^{tA}$

When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(. Here's my problem: For the system of equations: $$\begin{cases} & \text{ } ...
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2answers
108 views

Determine the stability of $(x,y)=(0,0)$

Determine the stability of $(x,y)=(0,0)$: 1/$$\bf{\begin{cases} & \mathrm{ } \dot{x}= -2x-y+2xy^2-3x^3\\ & \mathrm{ } \dot{y}= \dfrac{x}{3}-y-x^2y-7y^3 \end{cases} \tag {1}}$$ 2/ ...
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108 views

Construct the Mikhailov hodograph for the equation $f(z)=z^3+z^2+z+2$.

Construct the Mikhailov hodograph for the equation $$f(z)=z^3+z^2+z+2$$ Here's my solution: We have $$f(i\omega)=(-\omega^2+2)+i(-\omega^3+\omega)$$. We consider $Ref(i \omega)=0$ and $Re \omega ...
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1answer
157 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}= ...
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1answer
494 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{ } ...
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1answer
69 views

If all the solutions of $\frac{dy}{dt}=A(t)y$ are bounded, then all the solutions of $\frac{dz}{dt}=[A(t)+B(t)]z$ are bounded

In the book by Richard Bellman, Stability theory of differential equations Theorem 6 (p.43) If all the solutions of $\frac{dy}{dt}=A(t)y$ are bounded, then all the solutions of ...
4
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2answers
107 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
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2answers
144 views

Show that root $x\equiv 0$ of $\dfrac{dx}{dt}=F(t,x)$ is uniformly stable (uniformly asymptotically stable)

I have a problem: For the system of equations: $$\bf \dfrac{dx}{dt}=F(t,x) \tag 1$$ where $F$ is continuous in $I \times D \subset\mathbb{R}\times \mathbb{R}^n$ and $F(t,0)\equiv0$, ...
4
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0answers
145 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{ } ...
2
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2answers
103 views

Prove that, every solution of the scalar system: $\dfrac{dx}{dt}=y,\dfrac{dy}{dt}=-\dfrac{2y}{t},(t \ge 1) $ is bounded in domain $[1, +\infty)$

EDITED I have a problem: Prove that, every solution of the scalar system: $$\dfrac{dx}{dt}=y,\dfrac{dy}{dt}=-\dfrac{2y}{t},(t \ge 1) $$ is bounded in domain $[1, +\infty)$, but this system's not ...