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 time, a controller should manipulate the inputs to the system to obtain the desired effect on the output of the system

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42 views

Proving input-to-state stability for a set of equilibria

Consider the following nonlinear system: \begin{align} \dot{x} &= -axy \\ \dot{y} &= axy - by \end{align} It's easy to check that this system has a continuum of equilibria (of the form $(x,0)$)...
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How do I build a state space model of a given phenomenon?

I am a control engineering student and I would like to create a state space model of a given real world problem. The idea is to consider something that can be experienced in the real world, for ...
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41 views

stability of closed-loop nonlinear system

I would like to understand if it is possible to demonstrate the stability of this closed-loop nonlinear system $$a b -K_1 u = a K_2 \dot{a}$$ where $a$ is the variable I am trying to control, $u$ is ...
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Solving Nash equilibrium

Suppose you are given a payoff matrix dimensions m*n in which player A has m strategies and player B has n strategies and in each cases it results in a different outcome, so how could you find the ...
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Kalman filter: observability

A standard state-space for Kalman filter is: \begin{align} x_{t+1}&= F x_{t} + Gw_t\\ y_t&= Hx_t + v_t. \end{align} We know that there exists a similarity transform (i.e., an invertible matrix)...
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Passivity of a dynamical system

Let a dynamical system , in particular a RLC circuit. I want to study the passivity of this. In general the notion of $\textbf{passivity}$ comes out form the statics physics for which by a ...
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39 views

Fixed-point theorem on $[0,1]$ applications in stochastic control

I was looking at this very old paper on a specialized case of the Knaster-Tarski fixed-point theorem and its stochastic analogue. I'm wondering if there are some applications, if any, on stochastic ...
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Why does $H_{y'}(x,y,y',p) = 0$ and $H_{y'y'}(x,y,y',0) \leq 0 $ imply $y$ is an optimal curve?

I am reading Liberzon's Calculus of Variations and Optimal Control theory, and trying to understand the Maximum principle. It is currently going over the Hamiltonian formulation. I will define some of ...
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Math topics/subjects that would contribute to a deeper understanding of Engineering concepts [closed]

I want to dig deeper into the physical meaning, assumptions and derivations of the mathematical models and formulas I took in mechatronics engineering when I was a student. I don't know where to start,...
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Lyapunov theorem: is there an extension to when $\dot V$ is nd but $V$ is psd?

The well known Lyapunov theorem says given an autonomous system $$\dot x = f(x)$$ with equilibrium $0 = f(\overline x)$, if we can find a function that is positive definite (pd) $V > 0, \forall x ...
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70 views

What class of differential equations are these?

In my research I came across the type of differential equation described below. I've looked in introductory ODE and PDE books for a treatment of this type of equations, but I had no success. So any ...
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How to go back from the stabilization of the pendulum around $\theta=\frac{\pi}{2}$ to stabilization around $\theta=0$

Let the pendulum equation: $\ddot \theta+\sin{\theta}+b\dot\theta=cu$, that in state space ($x_1=\theta$, $x_2=\dot \theta$) representation will be: $\begin{cases} \dot x_1=x_2\\ \dot x_2=-\sin{x_1}-...
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How to solve this nested feedback loop

I have a nested feedback loop that looks like this. There is an input $X(s)$, output $Y(s)$ and a plant with a function of $$F(s)=\frac{R^3+R^2sL}{s^2R^2L-R^3s}$$ The input $X(s)$ will subtract the ...
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Indirect Lyapunov method for exponential stability

Le the following non linear system: $\begin{cases} \dot x_1=x_2\\ \dot x_2=-x_1^3-x_2 \end{cases}$ with $V(x)=\frac{1}{4}x_1^3+\frac{1}{2}x_2^2$ by Lyapunov Theorem I have proven that the origin is an ...
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Asymptotical stability of the origin of a non linear system

Let the system $\begin{cases} \dot x_1=-x_1-x_1^3+x_2\\ \dot x_2=-x_1-x_2 \end{cases}$ The origin is in my opinion asymptotically stable since: let $V(x)=\frac{1}{2}(x_1^2+x_2^2)$, that is positive ...
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Optimal Control with State Constraints: Capture Basin is Closed under a Convexity Assumption.

I'm currently reading the paper "Reachability and minimal times for state constrained nonlinear problems without any controllability assumption" from Bokanowski, Forcadel, and Zidani. We ...
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Asymptotic controllability of linear in phase system

Suppose we have $\dot x= F(t, u)x$ for $x\in\Bbb R^n$ and $F$ a continuous matrix in $t$ and $u$. In fact, in my case, there exists a smooth $f(t)$ s.t. $F(t, u)$ is invertible whenever $u \neq f(t)$. ...
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Linearized input affine system

For a control problem given by the nonlinear dynamics $\dot x=f(x,u)$, we may express the state space by declaring $z=(x,u)^⊤$ with $\dot u = v$ and writing $\begin{bmatrix}\dot x \\ \dot{u}\end{...
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Feedback-linearization of a pendulum around $\delta_1\neq 0$

In Non Linear Control Theory (Khalil) it is proposed the stabilization of the pendulum equation by feedback linearization. In particular let $\delta_1\neq 0$ we want by $u=u_{SS}+u_\delta$ to ...
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Show the asymptotical stability at the origin of a non linear system

Show that at the origin the following system has an asymptotically stable point: $\begin{cases} \dot{x_1}=-\phi_1(x_1)+\phi_2(x_2)\\ \dot{x_2}=\phi_1(x_1)-\phi_2(x_2)\\ \end{cases}$ ,with $x_i^2\...
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Relationship between structured singular value and spectral radius

I am following the proof of Lemma 3.7 in the following paper on the complex structured singular value (ssv)(https://authors.library.caltech.edu/75933/1/musurvey.pdf) and am trying to show that $\mu_{\...
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Given only a system transfer function, after changing it using state feedback, can we know if the resulting system is controllable? Observable?

Practice test question Given $$ G(s) = \frac{(s-2)(s-5)}{(s+1)(s-3)(s+4)} $$ Use state feedback to change it to $$ \frac{(s-5)}{(s+1)(s+4)}$$ Is the resulting system controllable? Observable? If no, ...
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A puzzling KKT for LMI vs. scalar constraint

I am trying to understand the KKT conditions for LMI constraints in order to solve my original question in KKT conditions for $\max \log \det(X)$ with LMI constraints. In the meantime, I found a much ...
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68 views

Normal form of a non linear system

Let the system: $\begin{cases} \dot{x_1}=a\sin{x_2}\\ \dot{x_2}=-x_1^2+u\\ y=x_1 \end{cases}$ that is: $\begin{cases} \dot{x}= \begin{bmatrix} a\sin{x_2}\\ -x_1^2 \end{bmatrix}+\begin{bmatrix} 0\\ 1 \...
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2answers
50 views

Find the lyapunov function to prove the asymptotic stability

Let's the following non linear system: $\begin{cases} \dot{x_1}=x_2&\\ \dot{x_2}=-x_1^3&\\ \end{cases}$ determine if the origin is asymptotically stable and in this case if it is globally ...
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2answers
107 views

Lyapunov function for a non linear 3 dimensional system

How it is possible to find a Lyapunov function for the following system? \begin{cases} \dot {x_1}=x_2+x_3 & \\ \dot {x_2}=-\sin x_1-x_3 & \\ \dot {x_3}=-\sin x_1+x_2 & \end{...
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How to measure quantity difference between a nonlinear system of equations and its linearization?

I faced such a problem. I have a nonlinear system for control synthesis and I should compare not only my controllers but also a linear version of my system to describe the legitimacy of this ...
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1answer
102 views

Stability of a discrete-time closed-loop

I modeled a dynamic system like so: $$ \dot{y} = au, $$ i.e. as an integrator. Every $T = 0.1$ seconds, the measurement is updated and remains constant in between. This motivated me to model it as a ...
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1answer
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Sparsity of $L^1$ Norm Regularization vector

Consider a generic optimal control problem $$ \min\limits_{u(t)}\int\limits_0^T f(x(t), u(t))\, dt + \mathcal{R}(u) $$ where the evolution of $x(t)$ is given by a generic ODE system $\dot x(t) = g(x(t)...
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64 views

Which way of solving from nonlinear control to choose?

I have a nonlinear system: \begin{cases} x'=f(x)+u \\ y=f(x) \end{cases} where $f(x)$ - gradient of some one-extremal function (for example $f=e^{-(x)^2}$), i.e. $\frac{df}{dx}$. Task: I want ...
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1answer
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Is there a program that can take matrices of variables and compute the inverse?

I have a control theory / optimal control problem that is formulated like this: $A(x)\dot x = f(x,u)$ Where A(x) is a matrix containing state variables. And I want to isolate $\dot x$ by doing this: $\...
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Effective boundary and Pareto optimal solution. Multi-Objective oprimization

I am starting to learn Control Theory, and start to explore Multi-objective optimization, I cant find in the net examples of how to solve such tasks, may someone advice how to solve this kind of ...
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How to discretize these equations of motion in ordinary differential equations.

I'm trying to learn how a lap time simulator works so I can start working on my own version. What I'm struggling with at the moment is how the partial differential equations are derived. I'm writing ...
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System identification of closed loop dynamics?

I have a reference input and measured output, r and y in the figure. (Data has been acquired in closed loop). I'm interested in identifying the whole closed loop dynamics (not just the plant), the ...
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1answer
47 views

My ODE system's state consists only of the differences in the location and velocities of two particles. How can I solve for the absolute values?

I have a linear ODE describing the relative motion of two particles ($T$ and $M$) in 3 dimensions. I have the individual accelerations for $T$ and $M$, but the location and velocity coordinates are ...
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1answer
52 views

How can I find a Lyapunov function for this control system?

I have the non-linear control system: $$ y_1'=y_2^3 $$ $$y_2'=-y_1+u,$$ where $$u \in [-1,1]$$ and I was wondering what's a good Lyapunov function and a good candidate for $u$?
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Finding transfer function of a temperature model, complexity in algebra.

Here is the equations, I'm trying to find a transfer function relating Ti (internal temperature, output) with Ta (outside temperature, input), I know the differential equation will equate to a first ...
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3answers
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Showing $\lim_{\nu\to\infty}\ln(\coth(\frac\nu2))\to2e^{-\nu}$ and $ \lim_{\nu\to0}\ln(\coth(\frac\nu2))\to-\ln(\frac\nu2)$

I am struggling to derive a couple limits I have come across in a paper I am reading. Both involve the natural log of the hyperbolic cotangent. The paper seems to be saying the two terms trend the ...
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Interactions (MIMO systems)

Page According to the page above, an LTI MIMO system could be affected by many interactions. I have two questions about this page: Is it described the system using differential equations (as well as ...
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Time-invariance of a MIMO system

According to my book, given a MIMO system with n inputs and m outputs (corresponding to the n inputs), the model itself is said to be time-invariant if a time shifting of inputs causes a time shifting ...
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Difference between stopping times in deterministic optimal control

I am currently reading the book "Controlled Markov Processes and Viscosity solutions" by Fleming and Soner. In the first chapter about deterministic controls, they introduce a ...
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48 views

special finite time models

based on Haimo manuscript the below equation is finite time: $$ \dot x(t) = - {\rm sign}(x(t)) \tag1 $$ and based on the manuscript many other equation are proposed, such as terminal sliding mode,...
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Numerical solution of HJB (Hamilton-Jacobi-Bellman equations) in practice

I am struggling to understand what the use of HJB equations in order to solve optimal control problems is in practice. I found the following approach in the book "Stochastic Controls" by ...
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37 views

Linearized system: Value of the δu(t)

To obtain the response of the nonlinear system for any input, Jacobian linearization procedure can be applied. According to procedure, we have to change the state variables by using values of ...
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Why $\operatorname{rank}(C)=r<n$ means that there are $r-n$ vectors vertical to every column of $C$?

So I am studying a solved example we did in control systems class but I struggle a little bit with the linear algebra involved: We are given a system (S), $x \in R^n and A: n \times n$ that: $ (S) = ...
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1answer
85 views

Solving state-space function with using of Runge-Kutta method

I need to implement my own integration routine that will take state space function $f$, free variable $t$, and initial state $x(0)$ as input and produce the solution $x(t)$ as output. I thought that ...
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1answer
27 views

Boundedness of optimal input of scalar system

Context: Given is the scalar system $$\dot{x}=-ax+u$$ with $x(0)=x_0$ and $|u(t)|\leq M$ for a constant $a$ and a positive constant $M$ along with cost criterion $$J(u)=x(1)+\frac{1}{2}\int_{0}^{1}u^2(...
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1answer
45 views

Complex Analysis - Derivation of Bode Integral

I am trying to work through the steps deriving the Bode Integral in the following paper: https://www.sciencedirect.com/science/article/pii/S2405896315025045 I am stuck on page 261, left column, on the ...
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1answer
42 views

Open Loop Transfer Function and Stability?

Given the following open loop transfer function $$G_{OL} = \frac{k}{(s+1)^2(0.5s + 1)^2}e^{-\tau s}$$ Let $k = k_{max}/2$. Let $\tau_{max}$ denote the value of the parameter $\tau$ such that the ...
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19 views

What is the difference between smooth and analytic of admissible control?

In the affine nonlinear control system, $\dot{x} = f(x) +\sum g_i(x)u_i$, if drift vector field $f$ and control vector field $g$ is analytic, the accessibility rank condition does imply ...

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