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Questions tagged [control-theory]

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The desired trajectory of the output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over ...

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34 views

Jordan form of simple 2x2 matrix

Considere the following transfer function: $\frac{1}{s^2+1}$ Calculate the Jordan form, real Jordan form and determine if this system is Lyapunov stable? My approach: The system's $A$ matrix is: $\...
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28 views

A Fact on the Transfer Function of a Linear System

While doing some work today, I found that the transfer function of the linear system $(A,B,C,D)$ is equal to the upper Schur complement of the $(n+p)\times (n+m)$ block matrix $$ M(s) = \begin{...
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1answer
29 views

Why is $\lim_{t\to\infty}e^{st}\to 0$ iff $\Re\{s\}<0$?

For a complex variable $s=a+ib$ and the condition $\Re\{s\}<0$ we have the limit $$\lim_{t\to\infty}e^{st}\to 0$$ Question: Why no condition for the imaginary part? If $\Re\{s\}>0$ we have $a&...
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stability of a matrix in control theory

let $\dot x=Ax$ and $S$ be an orthogonal basis of matrix $A$ i.e. $AS=0$ and let $S_\perp$ be the orthonormal complement of $S$. Is $S_{\perp}^T A S_\perp$ an stable matrix?
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33 views

How can I add find the gain from root locus and poles?

I try to find the P-gain from a root locus plot where I know the poles. Assume that we got a reference model: $$G(s) = \frac{\omega_n^2}{s^2 + 2\zeta \omega_n s + \omega_n^2 }$$ Where $\zeta$ and $\...
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1answer
32 views

State response given $A$ matrix and eigenvector matrix $M$ without using $M^{-1}$

This question is from a System Theory test without answers or solutions: Consider the system $\dot{x}=Ax$, where $A=\begin{bmatrix}-3&1&2&-1&1&0\\2&-2&0&2&-2&...
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1answer
43 views

Disprove/Prove Existence of Periodic Solution for Autonomous ODE

Consider the system $\dot x = x^2 + y^2 -1$ and $\dot y = y - 2xy$. I am new in this field. I draw the vector filed and I saw that there is no obvious periodic solution. How can I prove/disprove the ...
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1answer
21 views

How to sketch the phase trajectory of the system denoted by the block diagram

How to sketch the phase trajectory of the system denoted by the block diagram? I built the following Simulink model but I couldn't figure out what to represent the rate feedback $s+a$ with. ...
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0answers
26 views

Proof of marginal stability

Given a homogeneous continuous linear time invariant system: $$ \frac{dx(t)}{dt} = Ax(t) \;,\; A \in \mathbb{R}^{n\times n},\; x(t) \in \mathbb{R}^{n},\; t \ge 0 $$ Is there any reference (book or ...
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1answer
24 views

Injectivity trajectory to set singular control to zero

Consider a control system of the form $$ \dot x(t) = X(x(t)) + u(t)\, Y(x(t)) \qquad \qquad (*) $$ where $X,Y$ are two smooth vector fields, and $u$ is the (bounded measurable) control function. ...
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17 views

Relationship of accelerometer and velocity and position state space? [closed]

I have a accelerometer and I’m measuring distance and velocity by integration. I want to build a state space model of this. How would I go about it? I know $a = \mathrm{d}v/\mathrm{d}x \cdot \mathrm{...
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1answer
74 views

Interpretation of Differential Algebraic Equation

Considering the Differential Algebraic Equation (DAE) of the form $$\dot{x}=f(x,y)$$ $$0=g(x,y)$$ We can use the Implicit function theorem to conclude that as long as the Jacobian $ \frac{\partial g(...
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50 views

Is derivation more stable than integration?

As the question says it: is derivation more stable as a function than integration? And if so, how to prove it? During my university days I remember a professor mentioned this, but at that time I didn'...
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22 views

Reference: control theory

I'm looking for suggestions on books or reviews on control theory, if possible written in a fairly modern language and not excessively technical on the math side. In particular, I would need the ...
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1answer
53 views

Orthogonal to a vector space

Consider the control system $$ \dot x(t) = X(x(t)) + u(t)\, Y(x(t)) $$ where $X,Y$ are two smooth vector fields and $u$ is a given function. Consider a reference trajectory $x(.)$ defined on $[0,T]$...
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2answers
38 views

Series of Specific Matrix Products

I'm currently facing a problem where I'm not sure whether a closed form solution exists. Suppose you have two real matrices $A \in \mathbb{R}^{nxn}$ and $B \in \mathbb{R}^{nxm}$ (which arise from a ...
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1answer
15 views

Stability proof of nominal MPC with terminal cost and constraint

While going trough these slides, I wasn't able to make sense of the following on slide 32: (if only providing the url to the slides is not ok I will edit the question, but doing it like that saves a ...
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2answers
55 views

Matrix to the power t.

Compute the matrix $A^t$ for the following cases: $A_1=\begin{bmatrix}0&0\\0&1\end{bmatrix}, \quad A_2=\begin{bmatrix}-1&0\\0&-2 \end{bmatrix}, \quad A_3=\begin{bmatrix}0&1\\0&...
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1answer
34 views

Control of systems modelled by PDE

I come from an aerospace engineering background. Currently, doing PhD in the same field, specialization in Control and Dynamical systems. I am looking for recommendations on how to study PDEs ...
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2answers
58 views

question about span

Let $\dot x = Ax$ and $S$ be an orthogonal basis of null space of $A$. Does it imply that $x$ converge to a point in $\operatorname{span}\{S\}$?
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$H_\infty$ norm convergence (control systems)

So I have a question about proving the convergence (or rather a bound) of an optimization problem. I want to minimize the following function: $$\gamma := \left\|\frac{Q[A-B(x)C]}{\Re \{A \}} \right\|...
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19 views

Rate of convergence: Adaptive system

Consider the following dynamical system \begin{align} \dot{x}_1&=-ax_1 + w^T(t)x_2,\quad x_1\in\mathbb{R}^1 \\ \dot{x}_2 &= -w(t)x_1, \quad x_2\in\mathbb{R}^n \end{align} where $a>0$ and $w(...
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31 views

MIMO transformation to controller canonical form

I am unable to prove a result concerning MIMO linear dynamical systems. Let $$ \dot{X} = A\cdot X + B\cdot U$$ be a linear time invariant dynamical system, with $A \in \mathbb{R}^{n\times n}$, $B \in \...
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31 views

Derivative of Discrete Algebraic Riccati Equation

We have the Discrete Algebraic Riccati Equation: $$ \mathbf{X} = \mathbf{A}^T\mathbf{X}\mathbf{A} - (\mathbf{A}^T\mathbf{X}\mathbf{B})(\mathbf{R} + \mathbf{B}^T\mathbf{X}\mathbf{B})^{-1}(\mathbf{B}^T\...
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from multi input to single input in linear dynamical systems

I am working with some linear multi input dynamical systems. There is a result here which reduces the problem to single input linear systems. Given the following linear system: $$ \dot{X} = A\cdot X + ...
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1answer
64 views

Lyapunov equation.

This question is from a system theory test without answers or solutions: Let the following two cases be given $A) \quad A=\begin{bmatrix}-2&1\\-1&0\end{bmatrix} \quad $and$ \quad C=\begin{...
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0answers
32 views

How fast do controlled parameters follow the controlling ones?

Consider the two roots (blue) of $f(z) = z^2 - e^{i\varphi}$ that "follow" $\varphi$ (black) with half speed (from this MSE-question): Are there comparable statements of a similar kind: About the ...
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1answer
23 views

Discrete time impulse response

The following question is from a System Theory exam whitout answers or solutions: Which of the following discrete-time state-space model (A,B,C) of the form $x(t+1)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \...
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23 views

How to get Laplace transform of $V(t)=(1-\frac{2\Delta R_b}{3R_{b0}})(1-\frac{\Delta R_b}{R_{b0}RC}t)$

In a paper the following time-domain formula is given: $$ V(t)=\left(1-\frac{2\Delta R_b}{3R_{b0}}\right)\left(1-\frac{\Delta R_b}{R_{b0}RC}t\right) $$ which is said to be identical to the following ...
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39 views

Lyapunov, asymptotic and BIBO stability of $4$ given systems

Which of the systems are Lyapunov, asymptotically or BIBO stable: $1) \quad \left[ \begin{array}{c|c} A & B\\ \hline C & \end{array} \right]=\left[ \begin{array}{ccc|c} -2&1&0&2\\ ...
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1answer
115 views

Real integration theorem (Laplace transform)

Real Integration Theorem This theorem establishes the relationship between the Laplace transform of a function and that of its integral. It states that $$ \mathscr{L}\left[ \int_0^t f(t) \...
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1answer
81 views

Role of the weight matrix $M$ in $x^T M u$ in the LQR cost function

I wonder what the role of the weight matrix $M$ is in the performance index $$J = \int_0^{t_f}{\left( x^T Q x + u^T R u + x^T M u \right) \mathrm d t}$$ for an optimal control problem where $$\dot ...
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0answers
19 views

Z transform of real-time input to discrete transfer function

I am trying to implement a first order discrete transfer function to model a small centrifugal pump to control its output flow rate. I started with a continuous first order transfer function which ...
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64 views

Routh-Hurwitz criterion for systems with delay

Is there an analogue of Routh-Hurwitz criterion for systems with time-delay? In other words, given the open-loop transfer function $$G(s) = e^{-sT} \frac{P(s)}{Q(s)}$$ where $P$ and $Q$ are ...
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2answers
54 views

Bounding the solution to a Riccati equation

I have the following continuous-time matrix Riccati equation $$A X + X A' - X b b' X + Q = 0$$ where $Q>0$, $A$ is a diagonal matrix with strictly negative eigenvalues and $b$ is a (column) ...
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1answer
21 views

Stability of Discrete Time state space system with eigenvalues 0, 1/2 and 1.

This question is from a system theory exam without answers. So I was wondering if my resoning is correct. Consider the discrete-time state-space realization $x(t+1)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \...
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1answer
26 views

Rewrite (Laplace) transfer function

For a sinusoidal oscillator I've constructed the following transfer function: $$H(s)=\frac{sC_2R_1}{s^2C_2C_1R_2R_1 + s(C_2R_2+C_1R_1)+1-sC_2R_1\frac{R_f}{R_b}}$$ I expect I should be able to rewrite ...
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1answer
28 views

Transfer function given A,B,C, diagonal and eigenvectors.

Consider the continuous-time state-space representation $\frac{d}{dt}x(t)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t \in R^+,$ with the matrices given by $A =\begin{bmatrix}-4&-5&0&5\\-4&...
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2answers
40 views

Determine Jordan block size.

The following question is from as System Theory test. Let the system matrix $A$ be given as $A = \begin{bmatrix} 0&0&0&1\\0&-1&1&3\\0&1&-1&-1\\0&-1&1&2 ...
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2answers
41 views

Impulse response to Jordan form.

Which matrix $A$ in real Jordan from is such that, for suitable choices of the matrices $B$ and $C$, continuous-time state-space model $(A,B,C)$ of the form $\frac{d}{dt}x(t)=Ax(t)+Bu(t), \quad y(t)=...
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0answers
73 views

On the choice of general Lyapunov functions for discontinuous control

Let us consider the following dynamical system: $$ \dot{X} = A\cdot X + B\cdot u$$ where $X,B \in \mathbb{R}^{n\times 1}$. The considered system is linear, but I think that $A\cdot X$ can be ...
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1answer
32 views

minimal time to reach given state. number of time steps unclear.

Consider the discrete-time state-space realization $$x(t+1)=Ax(t)+Bu(t), \qquad y(t)=Cx(t)$$ with $$A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 2 & 0 & 0 \end{bmatrix}, \...
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1answer
45 views

Stability of system of differential equations.

The following question is from a System Theory test without answers or solutions. Let a continuous-time LTI system be given by the following differential equations: $\frac{d^2}{dt^2}y_1(t)+4\frac{d}{...
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1answer
56 views

Discretization of continuous-time state-space system.

This question is from a Systems Theory test without answers or solutions. Consider the folowing continuous-time state-space system $\dot{x}=Ax+Bu, \quad y=Cx.$ The continuous-time system given above ...
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1answer
28 views

Can I build an adaptive controller by using an ODE solver and a 3D graphics engine? [closed]

Let's assume that you're using a 3D graphics engine with built in physics. You create a inverted pendelum in a 3D designing software, e.g Blender, and then import the model into your 3D grapics engine ...
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1answer
47 views

Transfer function is $0$?

Given the continuous time state space model: $\dot{x}(t)=Ax(t)+Bu(t)$, $\quad y(t)=Cx(t), \quad t\in R^{+}$ with: $\left[ \begin{array}{c|c} A & B \\ \hline C & \\ \end{array} \right]$...
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0answers
37 views

What's wrong with this robust control scheme?

I'm learning how to control a double integrator with $H_\infty$. my model is simply $ \dot{r} = v $ $ \dot{v} = F/m $ $ r(t_0) = 0$ m, $v(t_0) = 0 $ m/s, $m = 1000 $ kg so I want to be able to ...
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1answer
53 views

The gradient of the Frobenius norm under a similarity transformation

I am looking for the gradient of the following cost function $||T^{-1}AT|| + ||T^{-1}BT||$ with respect to $T$. $A$, $B$ are real square matrices. $T$ is a coordinate change of corresponding ...
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1answer
54 views

Determining eigenvalues with limited information

The following question is from a System Theory test with only answers (no solutions). Maybe someone here knows how to tackle it. Consider the discrete time system $$x(k+1) = Ax(k)$$ with a matrix $$...
2
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1answer
37 views

Instability of a system subject to periodic perturbation (cont'd)

This is a follow-up to this question. Consider the following 2-dimensional system $$ \dot{x}(t) = A(t)x(t) \quad x(0)\in\mathbb{R}^2, $$ where $A(t)$ is a 2-dimensional time-varying matrix. Suppose ...