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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81 views

Is there a formalized control-theory-like approach for uncontrollable systems?

I'm basically trying to control a system to achieve a given set of outputs, but I don't actually have enough inputs to control all the outputs. Is there any formalized theory on how to achieve the set ...
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2answers
53 views

Why does this phase calculation go to 180 instead of 90?

This is all coming from the following video I am studying from http://www.youtube.com/watch?v=XSS6L42ce88 So I am working from this system $$ G(s)\,=\,\frac{4}{s^{2}+s+2}$$ and the video states the ...
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0answers
23 views

Is the ordered list of controllability indices invariant, or the unordered list?

The list of controllability indices of a linear time-invariant system is invariant under state feedback and change of variables. What is invariant exactly though: the ordered list, or the unordered ...
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1answer
83 views

Calculate step response from impulse response LTI-system.

Can someone please give me a few pointers on how to calculate the step response for an LTI system with this impulse response?? \begin{equation} h[n] = 2^nu[n]. \end{equation}
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3answers
200 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?
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17 views

Controlled system question optimisation

Consider the controlled system $x_{t+1} = x_t + u_t + 3\epsilon_{t+1}$, where the $\epsilon_t$ are independent $N(0,1)$ variables. The instantaneous cost at time t is $x_t^2 + 2u_t^2$. Assuming that ...
3
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1answer
96 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 $s_2=-2$, ...
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1answer
59 views

Optimal control question: find value function and optimal input

If we have a continuous-time system with a scalar state variable, plant equation $$\dot{x}= u,$$ and cost function $$Q\int_o^h u^2 dt + x(h)^2,$$ then by writing the dynamic programming equation in ...
4
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1answer
41 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 ...
2
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1answer
98 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: ...
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1answer
25 views

Stability of system with gain K

It is asked to sketch the root loci of the system . I have done that . Then it is asked to determine the stability of the system as function of $K$ . After plotting I see that for $K>0$ it is ...
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1answer
22 views

Determination of the modulus of continuity

I'm trying to prove the uniqueness of the viscosity solution of an Hamilton-Jacobi-Bellman equation. Thanks to a classical result, I'm left to check if it exist a modulus of continuity $\omega_1$ --- ...
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1answer
103 views

Relationship between controllability and null space of A, B

For the dynamic system $\dot x = Ax + Bu$ There's a saying that this system is controllable when $Ker(B) \in Ker(A)$, which means that $u$ have the control in every dimension of $x$. I have no ...
2
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1answer
49 views

Second order control system

We have the open loop transfer function $$ G(s)H(s) = \frac{k(s+2)}{s^2+2s+3}$$ There were three parts to this question : a. The value of $k$ for which repetitive roots occur b. The range of $k$ ...
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1answer
193 views

Proof of Impulse response of a BIBO stable system

I was wondering if anyone here could expand upon or provide the proof of the boxed theorem which I have shown in the image below? Any help would be greatly appreciated.
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1answer
49 views

Frequency response of Continous-time system

Not sure where to start on this one: $$H(s)={(s-j\omega_0)(s+j\omega_0)\over(s+\omega_0\cos\theta+j\omega_0\sin\theta)\left(s+\omega_0\cos\theta-j\omega_0\sin\theta\right)}$$ Sketch the frequency ...
2
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1answer
98 views

stability of a closed-loop system

We have $$G(s)H(s) = \frac{Ke^{-s}}{s(s^2+5s+9)}$$ We have to determine the maximum value of K for the closed-loop system to be stable . We have to do it using the Routh hurwitz criterion . I ...
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1answer
165 views

Block Diagram Reduction

i am trying to simplify this systems block diagram. I calculated something but I am not sure about it, is my reduction true? Thank you.
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2answers
42 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 ...
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1answer
141 views

Basic question about my understanding of the Lyapunov equation

Consider the system $\dot{x}(t) =Ax(t)$ where $A \in \Bbb R^{n\times n}$. Now let $P$ be a symmetric matrix and define $V(x) = x^T Px$. Then $V(x)$ satisfies $$\frac{d}{dt}V(x) = -x^TQ x,$$ where $Q = ...
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1answer
80 views

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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1answer
178 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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1answer
81 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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1answer
104 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, ...
2
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1answer
164 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 ...
2
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0answers
76 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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1answer
74 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( ...
2
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1answer
62 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: $$ \frac{\partial}{\partial t}x(t)= A(t)x(t), \: x(\tau)=x_0 $$ ...
2
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2answers
68 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
27 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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1answer
44 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
176 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
28 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
84 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) + ...
3
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1answer
107 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 ...
2
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2answers
78 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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1answer
67 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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0answers
34 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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1answer
49 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) \\ ...
3
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1answer
71 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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0answers
77 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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1answer
95 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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1answer
44 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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0answers
50 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
64 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
29 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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2answers
105 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
160 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
70 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
243 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 ...