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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95 views

What is the Laplace transform transfer function of affine expression $\dot x = bu + c$?

For the one dimensional case, with $a, b, c$ being real constants, $u$ being the system input, $x$ the state, what is the Laplace transfer function of: $$\dot x = bu + c$$ Ideally I'm looking for ...
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40 views

Recover a specific solution from the general solution of the Riccati equation.

Consider the equation $XAX - AX = 0$, where $A,X$ are square $n \times n$ real matrices. We know $A$ and assume for simplicity it is diagonable. We want to solve the equation for $X$. We have $XAX - ...
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59 views

Problem with understanding theorem on Riccati Equation.

`The matrices $A,B,C,D,X$ are real, square, $n \times n$. I have trouble understanding theorem 7.1.2 from Lancaster & Rodman "Algebraic Riccati Equations". The part that I understand is as ...
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44 views

fourier transform integral, parseval's theorem?

I have a fourier transform which is $$X(jω)=\frac{\cos(2ω)}{ω^2+ω+1}$$ and I am trying to calculate the value of the integral: $$∫x(t)dt \ \ \ \ \ \ x \in (-\infty, \infty)$$. I was thinking I ...
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35 views

Does there exist a notation for the set of poles of a function f(x)?

For eigenvalues we have a really nice notation $\sigma$ denoting the spectrum of this matrix i.e. the set of all eigenvalues. Before knowing $\sigma$, I just used $eigs(A)$ to denote the set of ...
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119 views

Jacobian Matrix of 6DOF Body (with IMU)

I am trying to derive the analytical Jacobian for a system that is essentially the equations of motion of a body (6 degrees of freedom) with gyro and accelerometer measurements. This is part of an ...
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43 views

Achieving asymptotic tracking of a nonlinear system with bounded input

I have the following nonlinear, continuous-time ODE \begin{equation} \dot{x}=K-Lq-q^2u, \end{equation} where the constant values $K$ and $L$ are strictly positive real numbers, the state $q$ and the ...
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40 views

Help figuring out output signal of LTI system.

Would greatly appreciate any help in figuring out the output signal of my discrete time LTI system. My input signal is cos(ωn) and my frequency response is H(e^jω)=(1+e^−jω)/2.
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41 views

$Ax = b$ & $Ax + b$

Ask a dumb question but confuse me long time. The following is what I know: 1st case $Ax = b$ is an affine set in $x$,i.e. $\{x | Ax = b\}$, and it is linear in $x$. 2nd case $ f(x) = Ax + b$ ...
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42 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 $x$ ...
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42 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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161 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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36 views

Relationship with trace and asymptotic stability in control theory [closed]

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

$L^2$ 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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38 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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262 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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64 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 ...
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163 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 ...
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1answer
56 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 ...
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1answer
490 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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186 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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47 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 ...
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61 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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117 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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60 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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79 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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99 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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87 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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1answer
188 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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39 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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714 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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385 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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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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67 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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183 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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1answer
74 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
72 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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1answer
77 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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70 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
169 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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50 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
104 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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1answer
33 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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61 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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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
42 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
78 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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1answer
43 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
124 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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99 views

Eigenvalues 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} ...