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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25 views

What is an example of a map not satisfying this rank condition?

Definition: Consider a Lie Group $G$ and a set of right invariant vector fields on $G$, denoted $\Gamma$. A point $y \in G$ is called normally accessible from a point $x \in G$ by $\Gamma$ if there ...
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23 views

Infimum of radially unbound functional

I am having difficulty following a proof about balls (subsets) of radially unbounded functionals. Let $U$ be a Banach Space. Let the space of admissible controls $U_{ad}\subset U \ne \emptyset$ be ...
3
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64 views

Design control law

Consider the function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ which is continuously differentiable and strongly convex. Let $x^*$ to be the unique global minimizer of $f$. Assume that $L_1\|x\| ...
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53 views

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 ...
3
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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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120 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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50 views

Moment Generating Function of a Beta random variable.

After getting some excellent help on this problem in the statistics SE, I am reformuluating my question. Let me know if I should just delete it and ask a new one. Let $V$ be a $Beta(\alpha,1)$ ...
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38 views

Reconstruction of state covariance from output covariance

Let us be given an LTI system $$ \frac{d}{dt} x (t) = A x(t), \;\; x(0)=x_0 \\ y(t) = Cx(t) $$ where $x_0$ is a random vector (e.g. uncertainty). Then it is known that the expectation $\mathbb ...
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42 views

Reachable set using constant control input

Consider a linear time-invariant control system given by the differential equation \begin{align*} \dot{x}(t) &= Ax(t) + Bu(t), \;\; x(0) = x_0, \end{align*} where $x\in \mathbb{R}^n$ and $A,B$ ...
2
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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 ...
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75 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 ...
2
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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 $(A,B)$ ...
2
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90 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 ...
2
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63 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 ...
2
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65 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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56 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 ...
2
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205 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 ...
2
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92 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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48 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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157 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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42 views

Difference equation of second order system with zero

I saw from lecture notes that difference equation of a first order system is like this: (1) (2) (3) (4) 1. What happens between (3) and (4)? It looks like inverse Z-transform but according ...
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74 views

Control Function with solution and fixed initial data on time interval, critical point of a cost functional?

Let $u(t)$ be a solution of the ODE $u''(t)+tu'(t) + u(t) = f(t)$ on the time interval $[0,T]$, with fixed initial data $u(0)=u_0$, $u'(0) = u_1$ where $f(t)$ is a control function. Find $f(T), ...
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25 views

singular values of a matrix written in controllable canonical form

Let the following equation represent a stable(marginally) dynamical system in discrete time domain \begin{equation} \mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k \end{equation} ...
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50 views

Solving Optimal Control with non linear cost function

I am trying to solve the Kermak Mc-Kendrick SIR model using a non linear cost function, but I am stuck on how to possibly solve it. I need to find an optimal control $u(t)$ in $[0,T]$ that minimize: ...
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43 views

Stability of transfer functions with internal delay

I would like to know what the best method is for finding stability of transfer functions that have internal delays. Basically I have a transfer function of the form: $\frac{f(s) e^{-st}}{g(s) + h(s) ...
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44 views

Inverse of State-space representation

Ask two questions from a paper (2012 ACC): Consider the plant: Let X be the stabilizing solution of the Riccati equation: where . Define the LQR gain by . The transfer matrix has a ...
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0answers
33 views

Weird derivative computation

I found the following formulas in a control theory textbook : $$s(x,t)=\left(\frac{d}{dt}+\lambda\right)^{(n-1)}\varepsilon $$ where $\varepsilon(t)=T\left(\frac{e(t)}{p(t)}\right)$ and ...
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43 views

Constructing error state kalman filter

I am trying to construct an error state kalman filter for GPS/INS integration using simulated data and I am having problem on a few steps. My error state vector is $\delta x = [\delta\alpha \, ...
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32 views

Prospect of research in some stochastic optimization/approximation field

This question is a not a technical one. Sorry for that. As I am new to the area of stochastic optimization/control, I want to know the active prospect of research in the following areas 1) ...
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28 views

Extension of Schur-Cohn for quadratic matrix equation

Starting from a quadratic in $z\in\mathbb{C}$ with real scalar coefficients $b,c$: $z^2 + bz+c=0$ and using the Schur-Cohn recursion, I can get the following conditions on $a,b,c$ such that $|z|\leq ...
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41 views

Function with bounded derivative as ODE

Given a function $x(t)$, I am looking for a function $y(t)$ which closely follows $x(t)$ except that its derivative must be bounded by a constant $c$, i.e. $\dot{y} \leq c$. Is there a way to describe ...
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44 views

designing a controller for an unstable plant?

How do would anyone make a controller for a system which is $G(s) = \frac{(s-2)}{(s-1)(s-6)}$ I do not see how this system can ever become stable, without using pole/zero cancelation. So how ...
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0answers
55 views

Bode plot of an unstable system?

I am a bit confused on how to sketch a bode plot for an unstable system? (being a/all pole(s) lies on RHP). I tried plotting it in matlab, but it doesn't resemble the output i was expecting using ...
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25 views

Control stability problem

Design controllers $u_1$ depending on $x$ and $u_2$ depending on $y$ such that the following system is exponentially stable: $$\dot x = A_1 x + B_1 u_1 + C_1 y \\ \dot y = A_2 y + B_2 u_2 + C_2 x ...
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37 views

Nyquist diagram of transfer function

Transfer function of a system is given as $$G(s) = \frac{100(s+5)}{s^2(s+3)(s^2+4)}$$ Sketch the Nyquist diagram and find if the system is stable. Also find the gain margin and phase margin. Please ...
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35 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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77 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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36 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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0answers
55 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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0answers
33 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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164 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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84 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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42 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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84 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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0answers
222 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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0answers
125 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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97 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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700 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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137 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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26 views

Solving simple decision-making model over multiple periods

Consider the following model. Each period t=0,1,..., an agent makes an effort $x\in R_+$ to solve a problem. The value from solving the problem is $V>0$. The relationship between effort and ...