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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2answers
640 views

PID controller convergence

Is there any material anywhere on convergence of PID controllers? Ie, if we formalize the "plant process" in some way, like $y_{t+1} = f(x_t,y_t)$ (in other words, the process value at a given time ...
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1answer
75 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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0answers
37 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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vote
2answers
45 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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2answers
39 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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1answer
32 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 ...
2
votes
0answers
96 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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0answers
30 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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1answer
36 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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0answers
38 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$ ...
3
votes
1answer
40 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
votes
1answer
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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vote
1answer
111 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 ...
1
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1answer
30 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 ?
8
votes
2answers
164 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 ...
0
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0answers
34 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
32 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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0answers
24 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 ...
1
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2answers
147 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
62 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
votes
1answer
70 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
42 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
349 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
125 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
28 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
46 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
54 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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votes
2answers
77 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
votes
1answer
53 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 ...
1
vote
1answer
66 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
votes
1answer
49 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); ...
0
votes
1answer
68 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
votes
1answer
118 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) ...
0
votes
1answer
28 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) + ...
0
votes
1answer
332 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?
0
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0answers
33 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
229 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
50 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
35 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
52 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: ...
1
vote
2answers
137 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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0answers
35 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
56 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
55 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) = ...
0
votes
1answer
68 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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0answers
163 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
62 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 ...
1
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1answer
121 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
44 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 ...
1
vote
1answer
79 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), ...