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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Contrability to zero state

We have LTI system $\dot{x}(t)=Ax(t)+Bu(t)$, where the matrix $A$ is Hurwitz and $(A,B)$ is controllable. I need to show that $\forall \beta >0$ and $\forall$ initial states $x(0)$, $\exists$ an ...
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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 ...
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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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47 views

Sufficiently rich signals

I know that a signal is sufficiently rich of order $n$ when it "includes" at least $\dfrac{n}{2}$ different frequencies. This is intuitive when we are talking about a sine but what about other kind of ...
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1answer
33 views

Notation: Polynomial of the Differential Operator

I having difficulty with some notation relating to control theory. Given that $H(s)$ is a strictly proper, scalar transfer function (i.e. a ratio of polynomial functions with a higher degree in the ...
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45 views

Understanding block diagrams

If I have block diagram with input $X(s)$ that goes to a block with $\frac{1}{s + 2}$ in it and then by way of $w(s)$ to a block with $s$ in it, and finally to the output $Y(s)$, how do I find the ...
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1answer
26 views

MATLAB Feedback

I am trying to use the feedback function in matlab and for the most part I understand it. But I came across this syntax: [x1 x2] = feedback(sys1, sys2, 1, 1, -1); ...
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23 views

Matlab calculate output

I'm trying to write a matlab function that takes in a transfer function and the input so it can calculate the output. So far, based on this information under I have the following piece of matlab ...
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47 views

What is the difference between disturbance and noise for dynamic systems

In most references from dynamic system theory, the following linear continuous dynamic system is considered. $$\frac{\text{d}x(t)}{\text{d}t}=Ax(t)+Bu(t)+Dd_{1}(t)\quad (1)$$ $$y(t)=Cx(t)+Ed_{2}(t) ...
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13 views

Project a function on a space?

The problem I'm solving is $\begin{cases} & \dot{x}_{1} = -u \\ & \dot{x}_{2} = 4x_{1} \end{cases}$ $x_{1}(0) = x_{2}(0)=0, |u| \leq 3, t \in [0;2], u^{0}(t)\equiv0, J[u] = -2x_{1}(2) + ...
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48 views

Simulink model from a nonlinear State Space

I have the nonlinear state space already constructed in MuPAD as shown: u is the input and y is the output. What is the best way for me to take this to Simulink?
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If a linearized system is not observable, can the nonlinear system still be observable?

I know that if a linearized system is not controllable then the nonlinear system might still be controllable. Does this also hold for observability? I am talking about control-affine systems.
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Black's formula and feedback system stability

Consider a hypothetical system with open-loop transfer function $G(s)$. Place it in positive feedback with unit gain. (That is, take its output and directly add it to its input.) The closed-loop ...
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26 views

Upper bound on Lyapunov equation solution

We know from literature on the Lyapunov equation that there exists a unique, symmetric, positive definite solution for the matrix $P$ in the equation $$A'P+PA=-I,$$ where $A$ is a Hurwitz matrix. Is ...
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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 ...
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1answer
44 views

Show what happens to indentations around poles on imaginary axis when acted on by a conformal map

Can someone provide a link to a proof or motivate here (not looking for a rigorous proof) of a very important result in complex analysis, particularly in applications to control systems engineering: ...
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2answers
52 views

Control Theory Textbook

I'm looking for a good textbook or series of lecture notes for learning about sampled data control theory. I'm a relative beginner in this area, so I'm looking for a gentle introduction. I'm ...
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24 views

Basic measure theory: Composite functions and points of nondifferentiability

I have a function $V(x(t))$. $x(t)$ is continuous, but not everywhere differentiable w.r.t. $t$. What can we say about $V$ at these points of non-differentiability? To explain I have included some ...
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39 views

(Possible) application of Sarason interpolation theorem

This question is related to the following Wikipedia article on Nevanlinna–Pick interpolation. At the end it has been written as Pick–Nevanlinna interpolation was introduced into robust control by ...
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1answer
34 views

Determine system controllability based on solutions to the state equation with zero input

Given is a single input single output, time invariant state space system. \begin{equation} x(t) = \left(\begin{array}{r} 5 \\ -1 \\ 4\end{array}\right)e^{-2t} \end{equation} \begin{equation} x(t) = ...
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48 views

Controllability of a linear time invariant system, whose A matrix is formed by Jordan Blocks

I am studying for a linear system theory exam later on this week. The professor has recommended some problems in order to practice and prepare for the exam. This is one of them that I'm trying to ...
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34 views

Modelling a transfer function for plant/system empirically

In an attempt to learn about PID controllers, I'm designing a small desktop thermal control system. I have a power resistor mounted to a heatsink, with a thermister placed nearby to measure ...
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36 views

Simulating a system with known impulse response

I want to simulate a system whose impulse response is like the following: $$h(t) = e^{-at} \sin(t)$$ The graph of which should look similar to the plot below: I want to simulate the output for ...
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1answer
45 views

State space and linearization

I have a question about state space representation. How can I represent an equation in which I have only the second and first derivatives? For example where $u$ is the control input. If I put ...
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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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1answer
34 views

State space system gives different bode plot then transfer function matrix

I have a discrete state space system with matrices $A$,$B$,$C$ and $D$ with sampling period $T_s$. I can either create a state space system, sys1 = ss(A,B,C,D,Ts), ...
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26 views

Why does the denominator of TF change in MATLAB when multiplying by proportional gain?

Trying to simulate a unity feedback closed loop system with gain of $K$ Let's say I want a proportional gain of $K = 5$. My plant's TF is $G(s) = \frac{10}{s^2+2s+1}$. I thought that $KG(s) = ...
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1answer
25 views

Is the pair controllable/observable?

The matrices $Q\in\mathbb R^{n\times n}$ and $G\in\mathbb R^{n\times n}$ are both symmetric positive semidefinite, $A\in\mathbb R^{n\times n}$ is invertible. Moreover, $(A,G)$ is controllable, and ...
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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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1answer
25 views

Simulating a controlled dynamical system

I am try to simulate a controlled dynamical system of the form $$\dot{x}=f(x,\phi(x)),$$ where $\phi$ is the controller. To do so, I am using Octave (an open source version of Matlab). My commands ...
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1answer
27 views

Comparing controllers using Bode plot

I know that Bode plot is used when determining the stability of the open loop system. But is it possible to compare controllers using Bode plot? In my example I have a process $1/Ls$ and a PI ...
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31 views

Controllable and observable

The square matrices $A$ is invertible, $Q$ and $G$ symmetric positive semidefinite. Moreover, $(A,G)$ is controllable, and $(Q,A)$ is observable. I have the following question Is $(-A,-G)$ ...
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1answer
29 views

Transversality conditions in optimal control with non-linear final pay-off

I have a doubt regarding transversality condition in the case of a non linear final pay-off. For instance, I need to solve with the Pontryagin maximum principle the following optimization problem ...
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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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1answer
28 views

Show that the system is controllable (i.e. prove P has full rank)

Given the matrix: $$A = \begin{pmatrix}m&1&0&0&0\\ 0&m&1&0&0&...\\ 0&0&m&1&0&...\\ 0&0&0&m&1&...&\\ ...
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1answer
43 views

Determine the transfer function and step response of the state from the variation of parameters formula and the output from the transfer function.

Let $A = [-1 0; 0 −2] , B = [ 0; 1] , C = [1; 0]^T , D = 0$ be a state space realization. Determine the transfer function. Determine the step response of the state from the variation of parameters ...
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1answer
50 views

Lyapunov linearized stability analysis

I have this system: $\dot x=-(x-1)(x-2)^2$ I'm asked to find the equilibria and to study the stability using: i) linearization ii) appropriate Lyapunov function How should I linearize the system? ...
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2answers
28 views

How to periodically estimate states of a LTI if the output is measured irregularly?

How can I periodically estimate the states of a discrete linear time-invariant system in the form $$\vec{x}(k+1)=\textbf{A}\vec{x}(k)+\textbf{B}\vec{u}(k)$$ ...
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39 views
0
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1answer
32 views

Barbalat's Lemma

I have this problem to solve: Use Barbalat’s Lemma to show that $lim_{t→∞} x_1(t) = 0$ for the system: $\dot x= − x_1 + x_1 x_2 $ $\dot x_2= − \gamma x_1^2$ , where $\gamma > 0$. Can we you ...
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1answer
34 views

Designing linear systems to respond to particular kinds of oscillations

Say that I have a linear system which is being perturbed by an oscillating signal of a single frequency, of the form $$ \dot{\vec{x}}(t) = A\vec{x} + B \sin(\alpha t), $$ where $B$ is a vector of 1s ...
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39 views

Solving the matrix equation $D P = D P A D$ for stochastic matrices.

Here, I call any real matrix with positive entries with rows summing to one a stochastic matrix (it need not be square). $D,A,P$ are stochastic. $P$ of size $n \times n$ is given. $D$ of size $k ...
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1answer
41 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?
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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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1answer
46 views

Relation between Riccati Algebraic Equation and optimization problem

Reading this page: http://www.mathworks.com/help/robust/ug/minimizing-linear-objectives-under-lmi-constraints.html I got stuck in the result that says it can be show that minimizing Trace of X (a ...
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26 views

Optimal stopping problem

Consider the OU process: $dX_t = -X_tdt + dW_t$, $X_0 = 0$. Compute the optimal stopping time for the following problem: $$v = \sup_{\tau} E[|X_{\tau}| - \tau]$$ So far I have set $L\phi = 0$, ...
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1answer
30 views

State space system, with a state space system as feedback

I have two state space systems. Now I want to compute the state space system where the first state space system is the input of the other... $$M_1 = \begin{cases}\dot{x}_1 = A_1 x_1 + B_1 u_1 \\ y_1 ...
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1answer
63 views

Use linear quadratic regulator to minimize output error

I would like to create an Infinite-horizon, continuous-time LQR with a cost functional defined as $$J = \int_{0}^\infty \left( e^T Q e + u^T R u \right) dt$$ where e is the states' error $x-x_d$, ...
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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 $Beta(\alpha,1)$ ...