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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Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
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73 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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13 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 ...
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62 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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30 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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87 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{ } ...
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61 views

Stability of linear systems with singular state matrix

Given a linear time invariant system $\dot X(t) = AX(t)$ where $X \in {R^{n \times 1}}$ and $A \in {R^{n \times n}}$ is a singular matrix ($A$ has at least one zero eigenvalue). How can I study the ...
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57 views

Find u that minimizes the integral mean

How do I find $u : [0,\infty) \to \mathbb{R}^m$ that minimizes \begin{equation} J(u(\cdot)) = \lim_{t \to \infty} \frac{1}{t} \int_0^t L(x(\tau),u(\tau)) d \tau, \end{equation} subject to ...
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49 views

How verification argument really works?

Let $C(u,s)$ be cost functional for an admissible control $u$ with initial state of the system being $s$. Our aim is the solution of the following problem: $$\inf_u E(C(u,s))$$ We defined the value ...
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169 views

how to obtain state space diagram and state space model for transfer function

How do we obtain the state space diagram and state space model for transfer function for example the question is given How to draw state variable diagram for the given transfer function ...
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74 views

Condition for controllability of matrix pairs

I have a question regarding how determine if a matrix pair is controllable. If $A$ and $b$ are given by A = [$\lambda_1 0 0 ...., 0 \lambda_20 0 0,...,00...\lambda_n$] (matrix) and $b = [b_1 b_1 ...
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46 views

Question regarding continuous time systems

If $S_u$ is the set of all solutions to the continuous time problem x(dot) = Ax + Bu and $\phi_h :(R^n)^R to (R^n)^N$ be an operator which maps every state x into the sequence (x(0), ...
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155 views

Matrices made negative semidefinite, but not simultaneously

Consider matrices $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, such that $A$ has at least $1$ strictly positive eigenvalue. Let $X_1, X_2 \in \mathbb{R}^{n \times n}$ such that ...
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30 views

Prove that A is marginally stable iff there exist a $P$ $\in$ $S^n$, $P \succ 0$ such that $A^T + PA \leq 0$

Asymptotic stability, which means that all eigenvalues of A are in the open left half plane is easily proven. See the scan in the attachment. However, in the book the proof for the second case where ...
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35 views

Kalman Filter application to non-linear system.

I want to use the Kalman filter to have a better estimate of the state of a system which I know its equations of motion: ...
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26 views

Relationship between output and poles/zeros in the complex plane

Context There are lots of videos online which explain the time domain equivalent of poles depending on their place in the complex plane, but it's only useful for the simplest examples for which we ...
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33 views

analytic solution to structured algebraic Riccati equation

In solving a model I have written down for a research paper, I am left with two Algebraic Riccati Equations, that is I need to solve for $X$ in the equation \begin{align*} X = A^\top (X + XB(R + ...
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40 views

Continuous time optimisation question- find value function and optimal control

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 ...
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35 views

Can $x'(t)\leq \mu |x(t)|,\forall t\geq 0$ imply $x(t)<\mu |e^{-\mu t}|$?

Assume $\mu>0$ If it can't, is it possible to give a good exponential bond(better with negative constant in front of t.)?
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27 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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58 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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25 views

On the controllability function (minimising a functional)

Consider a system of ODEs $$\dot{x}(t)=f(x(t))+g(x(t))u(t),$$ where $f:\mathbb{R}^n\to\mathbb{R}^n$ and $g:\mathbb{R}^n\to\mathbb{R}^{n\times m}$ are smooth. Let $L:\mathbb{R}^n\times ...
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38 views

improving fluid analysis on queuing problems

Discrete-queuing models are hard to solve computationally and can become easily intractable with an increased number of state-action pairs. Markov decision processes can be employed to come up with ...
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73 views

optimal control -Taylor expansion - PDE problem

I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point. For the given control ...
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162 views

Question about proof of bounded real lemma

My question is: it is possible to proof the bounded real lemma for $H_\infty$ performance with the following procedure? The $H_\infty$ performance is defined as: \begin{align} \parallel ...
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107 views

Identifiability of a state space system

I'm trying to solve assignment 4E.5 from this sheet (ship steering dynamics). My question are: Do I need to perform the Laplace Transform in order to check for identifiability? The state space model ...
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49 views

What's the shooting algorithm for the mass-spring problem (ode)?

I have the following problem : $$ \begin{aligned} \frac{d x(t)}{dt} &= y(t)\\ \frac{d y(t)}{dt} &= -x(t)+y(t)(1-x(t)^2)+u(t) \end{aligned} $$ with the initial condition $(x,y) = (0,0)$. Those ...
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96 views

generalizing superfunctions of entire functions

Let $z$ be complex and $x$ real. Define $f(z,0) = f(z)$ where $f(z)$ is an entire function. Define $f(z,x)$ as the $x$ th superfunction of $f(z)$. We know that $f(z,x-1) = f(f^{-1}(z,x)+1)$ where ...
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605 views

Partial differentiation of vector to find Jacobian (extended Kalman filter)

I am working through some coursework on self-tuning control and part of one of the questions requires the use of the extended Kalman filter for joint parameter and state estimation. For completeness, ...
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115 views

What is the difference between various kalman filters?

What is the difference between additive and multiplicative kalman filters, as well as some other kinds? I'm also looking for reference texts and articles that describe the algorithms, so ...
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41 views

Determine Gain using root locus

I have this closed loop transfer function $T(s) = \frac{KG(s)}{1+KG(s)}$ Where G(s) is given, and K is Gain. I've to calculate the gain for which the damping ratio is 0.707. I've done that by ...
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Designing a state feedback law for a nonholonomic system

Consider the set \begin{equation*} A_r=\left\{(e_x,e_y,L)\in\mathbb{R}^3:e_x=e_y=0,L(t)=\sqrt{\dfrac{\mu}{p_0^3}}t,t\in\mathbb{R}_{\geq0}\right\} \end{equation*} I have been trying to design a state ...
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27 views

How to control a System that tends to Zero

For a design project in a Control course, my classmates and I must create a Controller that steers an unknown system to a given trajectory within certain constraints. The system is given to us in ...
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29 views

control engineering transfer function vibration

What does vibrational mode even mean? How do you tell it from the poles?
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29 views

How to compute the Jacobian Matrix of the next system?

I have a little problem with notation and I do not know how to work it out. I have the next system $$ \dot x = A(x)x, $$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has ...
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13 views

Stein's theorem and the solution of Q=H-B*HB

Stein's (1952) Theorem 1 says: "A necessary and sufficient condidion that B is convergent is that there exists a positive definite Hermitian matrix H, for which H-B*HB is positive definite." ...
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19 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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Root counter of a closed loop system

How can we determine the root locus counter of the following closed loop system $$ s^3 + K_2s^2+K_1s+K_1 = 0 $$ where $K_1$ and $K_2$ are parameters which vary from $0$ to $\infty$ . I have tried ...
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64 views

Draw the block-diagram of the system for a satellite and study the closed-loop response of the system to a unit step input for a PD controller

I'm trying to resolve this exercize. Any ideas? Thanks for your help! Consider the example of a satellite attitude system , for which the vehicle dynamics are governed by the differential equation ...
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21 views

DE: $pU^2 = (rx-1)U' + Ur + \frac{1}{2}\sigma^2U''$?

I have been trying to solve the following ordinary differential equation that results from a problem in stochastic control theory. $U$ is a function of $x$. $pU^2 = (rx-1)U' + Ur + ...
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36 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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Under what circumstance should I use a Continuous time Markov Chain instead of a discrete time Markov Chain?

Why should I use one over the other, if I can basically reduce the small time-interval $h$ to be small enough that it simulates continuity? I guess this question is somewhat analogous to control ...
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125 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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80 views

Stability of the optimal control steady state

I am struggling with a pretty much complicated optimal control problem, which I solve in Mathematica. The optimal controls are second order delayed, which makes it unclear to me how to analyse the ...
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84 views

Unit adjoint eigenvectors

Q.What are unit adjoint eigenvectors? Below I give the context where I found the mathematical term 'adjoint eigenvectors': $$\vec{f_u}.\vec{e_s}=\vec{f_s}.\vec{e_u}=0$$ so that by resolving a ...
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24 views

How to make future trajectory to be prescribed trajectory

i do not understand the role of control theory about how it make trajectory to be prescribed trajectory. because if future path's parameters are known and input into system, does it mean that it will ...
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93 views

Solving this optimal control problem

The problem is: $max \int_{-1}^1 (tx - u^2) dt$ where $\dot{x} = x + u^2, u(t) \in [0,1]$ for every $t \in [-1, 1]$ End points: $x(-1) = 0, x(1) = e^2 - e^{1 + \frac{1}{e}} $ I need to find an ...
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50 views

restricting the domain of the unbounded operator

Can anyone give me a a restricted non-void set of bounded inputs which results in bounded outputs, though the operator is not bounded on the whole space (consider L2-space). One can consider a simple ...
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35 views

MATLAB plotting issue

I posted this question on StackOverflow, but did not get any answers, so hopefully this will work better. Is anyone familiar with plotting in the Matlab SISOtool? For some reason, I cannot access the ...
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240 views

Getting Transfer Function from a Block Diagram

Question: I'm a student who has to extract transfer functions from block diagrams quite often. It would help if there was a graphical tool where I could manipulate block diagrams and see their ...