# 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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### Restructuring Ill-conditioned problem for better numeric results

I'm not sure if I'm asking in the right place since this is kind of a field overlap but let's see. I have a dynamic equation system and want to optimize a subset of it's parameters. The algorithm ...
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### Unscented Kalman filter and measurement function

Suppose I have a nonlinear system with states $x$ and measurements $z$. I don't have a measurement function for $z=h(x)+n$ where n is my gaussian noise term. However I do have the sufficient ...
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### Problem understanding Chetaev theorem

I am studying control theory, and I am focusing on the Lyapunov stability. In particular, I am looking the Chetaev theorem, but I have some problems understanding it well. I know that the Cheatev ...
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### Absolutely continuous and almost everywhere solution of a controlled dynamical system

This is a doubt i had while reading the first chapter form the book Mathematical Control Theory, Jerzy Zabczyk, in the first chapter, page $11$ the author talks about solution for a class of ...
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### Computing the Transfer Function of open-loop-gain equation (Z-Transform)

I am designing a closed-loop control system. I need to figure out how I should design the compensator so that the system is stable for a wide range of inputs. The problem is, the open-loop-gain ...
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### Why does the Lyapunov criterion only gives sufficient conditions for stability?

I am studying stability for control systems, and I have written in the notes of my professor that the Lyapunov Criterion only gives sufficient conditions for stability, and not necessary and ...
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### What does it mean that a distribution is integrable?

I am studying geometric control theory, and I am focusing on the Frobenius theorem. I have seen that it gives sufficient and necessary conditions for integrability of a distribution, but I am having ...
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### Breakaway point of Root Locus not as expected

I have the following Open Loop Transfer Function: $$H_{ol}s = \frac{k(s+4)}{(s+1)(s+2)(s+3)}$$ To find the breakaway point of the unmatched poles we find the characteristic equation of the closed ...
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### What is a distribution

I am studying the basics of geometrical control theory, and I am struggling with some concepts. At the moment I am studyng the concept of distribution. So far I have understood that a distribution is ...
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### Find input for a control system such that system reaches a desired state $x_L$ at time $t=L$.

Given is the equation in state space of a control system $$\dot{x}(t)=Ax(t)+Bu(t)$$ $x(t)$ is the state vector of length $n$, $A$ is a square matrix of $n\times n$, and $B$ is a vector of length $n$. ...
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### Help with Modal canonical form linear systems.

I'm trying to find the modal canonical form of a linear system, the book I'm using is Linear systems theory and design by Chen, but has no clear explanation about this topic, and I'm having a hard ...
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### How does directional derivative work?

Consider a vector field $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, I have that the directional derivative (or Lie derivative) can be defined as: $L_f = \sum_{i=1}^{n}f_i \frac{d}{dx}$ which as ...
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### Problem understanding passage from explicit representation to implicit representation

I am studying control theory and I am having difficulties understanding a concept. Consider the following relationships, which represent the input to state behavior and the input output behavior ...
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### How to solve LQR if regulating outputs instead of states?

Consider the state space system given by $$\dot{x} = Ax(t) + Bu(t)$$ $$y = Cx(t) + Du(t)$$ The standard LQR cost is given by $$J = \int \big( x(t)^T Q x(t) + u(t)^T R u(t) \big)dt$$ Instead, ...
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### Why for a linear system, the stability for a generic equlibrium point is equivalent to the stability of the origin?

I am studying the concept of stability for linear and for nonlinear systems. While studying the stability for a linear system I found this definition from the notes of my professor: for a linear ...
From wikipedia, Consider the continuous linear system: $$\mathbf{\dot x} (t) = A(t)\mathbf{x}(t) + B(t)\mathbf{u}(t)$$ There exists a control $u$ from state $x_0$ at time $t_0$ to state $x_1$ at time ...