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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23 views

How do you find the state space representation of $G(s) = \frac {1}{s^2+s+1}$

Let $G(s) = \frac {1}{s^2+s+1}$ be the transfer function of the system Then $Y(s)(s^2+s+1) = U(s)$ Therefore $y'' + y' + y = u$ After this step, how should I set up my state transition variable $x$ ...
2
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1answer
32 views

Reference request: Controllable and Observable form for transform function

I came across some online material a year ago that claimed that a the ABCD matrix of a transfer function $$G(z) = \frac{b_1 z+b_2}{z^2+a_1z + a_2}$$ can be directly computed from the coefficients of ...
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1answer
66 views

Linear algebra of state space representation won't be linear (superposition theorem)…

After answering a question about calculating the state space representation of a circuit with 3 sources in it (the circuit is there), I had a doubt - while checking, it became clear there is something ...
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1answer
21 views

Relationship with trace and asymptotic stability in control theory

What is the relationship between $\mathrm{tr}(\exp(tA) \exp(tA^\ast))$ and asymptotic stability in control theory ?
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+50

$L^2$ Bounds for Markov Chains.

Consider a non-negative, square stochastic $n \times n$ matrix $P$ (rows sum to one, $P$ is ergodic). We are interested in characterizing the set of $n \times n$ invertible matrices $A$ such that we ...
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20 views

solver for non-convex matrix optimization with convex constraints

So here is the problem: $\max_{D} ~~ \|A+BD\|$ subject to $\|D\|<1$ (any norm you like) where matrices A and B are given. The cost function is evidently convex as well as the constraint, but ...
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0answers
24 views

Exponential convergence of controlled variables

I am reading a paper and I don't understand why, after some math they say that the controlled variables $$ \dot{\psi}_{13} $$ and $$ \dot{\psi}_{23} $$ converge exponentially. This is the paragraph ...
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20 views

what exactly is a center invariant subspace?

I'm studying robust control, and I have a matrix with two invariant subspaces. One is stable, which I assume is spanned from the eigenvectors with real part less than zero, and the other is a center ...
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2answers
41 views

Model Predictive Control

I have a few confusions about Model Predictive Control (MPC). Since they are all minor questions related to the same category, I ask them under one topic. In an article, the cost function is defined ...
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0answers
56 views

How do I flow in a circle on a set of vector fields?

Consider a set of complete vector fields on a manifold. They each have an associated one parameter group of diffeomorphisms related to the generated flow. What is a necessary and sufficient ...
0
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1answer
27 views

Equivalence of controllability and reachability in discrete time systems

I am trying to prove that the statements; $\Sigma_d$ is controllable, $\Sigma_d$ is reachable, The pair $(A,B)$ is controllable (in other words $<A|\ im\ B>=\mathcal{X}).$ are equivalent ...
2
votes
1answer
31 views

How to prove the both identity (matrix)

I read a paper, and the paper use the following identities (that hold true in any ring) $(I+AB)^{-1}A = A(I+BA)^{-1}$ $(I+AB)^{-1} = I - A(I+BA)^{-1}B$ Any way to prove this? How to open the ...
0
votes
1answer
42 views

Convolution between impulse response

I read a paper, and am confused about the following: Suppose $W$ is an operator with impulse response (IR) $w$. And suppose $w^n$ is the IR of $W^n$. My question is the following: ...
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0answers
99 views

Dynamic programming problem for discrete linear time varying system

I'm working on a linear time varying discrete(LTV) multi input multi output(mimo) system. I formulate the problem description in the following way $$x_i(k+1) = x_i(k)\cdot A_i(k) + B_i(k)\cdot ...
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0answers
21 views

How to control acceleration in a PI-controller?

Inputting a big desired position in this system will cause the desired speed to go from 0 to maximum. How can I avoid this and make it accelerate instead?
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0answers
37 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 ...
3
votes
1answer
42 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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2answers
52 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 ...
0
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1answer
41 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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1answer
55 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 ...
0
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1answer
29 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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1answer
31 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 ...
0
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1answer
59 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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votes
1answer
17 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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1answer
76 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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14 views

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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2answers
89 views

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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0answers
30 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 ...
3
votes
0answers
25 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 ...
0
votes
1answer
47 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: ...
0
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2answers
69 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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29 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 ...
3
votes
1answer
45 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 ...
2
votes
1answer
37 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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0answers
52 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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54 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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0answers
39 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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vote
1answer
52 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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0answers
43 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
42 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), ...
0
votes
1answer
28 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) = ...
0
votes
1answer
29 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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0answers
76 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), ...
0
votes
1answer
31 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 ...
0
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1answer
37 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 ...
0
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1answer
32 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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vote
1answer
36 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 ...
2
votes
0answers
33 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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vote
1answer
31 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&...&\\ ...
0
votes
1answer
51 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 ...