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

Control theory deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to the desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality.

2484 questions
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What is the difference between controllability and reachability?

Here are the two problem statements I'm trying to understand: Reachability. The reachability problem is to “find the set of all the final states $x(T)$ reachable starting from a given initial state $x(t_0)$”. Controllability. The controllability…
Lod
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How/why is the relative degree of a transfer function related to the causality of the system it represents?

A transfer function can be classified as strictly proper, proper or improper depending on its relative degree, i.e. the difference between the degree of the polynomial in the denominator and the degree of the polynomial in the numerator. If a…
jvf
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Examples of Lie Brackets exposing unexpected control spaces

Today I showed some students how Lie Brackets have the interesting ability to reveal hidden trajectories which can be assembled through motion primitives, resulting in non-holonomic approximations of holonomic movement. Parallel parking is, of…
Kenn Sebesta
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What is the physical meaning of Bode plot in case of unstable system?

I know that from the mathematical point of view it doesn't matter if we plot Bode diagram of stable or unstable system. It's just a function of complex value. However from the physical point of view, Bode plot shows steady response to a sinusoidal…
Rafal Golcz
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Why can't an improper transfer function be realized?

A major result in control system theory is that a transfer function, $$G\left( s \right) = \frac{{Y\left( s \right)}}{{U\left( s \right)}}$$ has a state space realization if and only if the degree of $Y(s)$ is less than or equal to the degree of…
ITA
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How to control a nonlinear system with a PID controller

Is the design of a PID controller for a nonlinear system different from for a linear system? [I think math.stackexchange.com is the most suited SE.]
Karlo
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Design control law

Consider the function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ which is continuously differentiable and strongly convex. Let $x^*$ to be the unique global minimizer of $f$. Assume that $L_1\|x\| \leq \nabla f(x) \leq L_2 \|x\|$. Consider the…
user147687
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Can I use control theory when the output is stochastic and has delayed reactions to the controller?

I am a complete novice in control theory. My understanding of control theory is that it can be used to adjust parameters of a system based on feedback to reach some desired state. It sounds like a useful set of tools for a problem that I have, but…
timleathart
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Is the following inside of outside the Nyquist curve?

Consider the following Nyquist plot, Is the green dot outside it or inside it? To me this would be outside the interior of the plot. Is this correct?
user561840
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Lyapunov functions with tracking?

Lyapunov functions are used to investigate stability of an eqilibrium (often the control reference $r(t) = 0$)... but what happens if tracking control is considered (i.e. $r(t)$ is non constant but a function of time)? How would I treat the…
SampleTime
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Pole-zero cancellation Paradox

Suppose we have an open-loop transfer function $$G(s) = \frac{1}{s(s+a)(s+b)}$$ If we plot the root locus for the closed-loop system we will get roughly something like this : Now the question is when I add a new zero to the system which is at $-a$…
abkds
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Horizon selection

$$x(t+1) =Ax(t)+Bu(t)=\left(\begin{matrix}0&0&0.8\\1&0&0\\0&1&0 \end{matrix}\right)x(t)+\left(\begin{matrix}1\\0\\0 \end{matrix}\right)u(t)$$ Basically the excercise is to find the minimum value of $N$ such that…
Karlis Olte
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showing any controllable system can be put in 'controller' form.

I am looking at the proof of the following theorem: Let $\dot{x} =Ax + Bu$ be a controllable single input system, where $\Delta_A:= \det(\lambda I -A) = \lambda^n + a_1\lambda^{n-1} + \ldots + a_{n-1}\lambda + a_n$. Show this system is isomorphic to…
Slugger
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How to synthetize a controller $\dot{u} = F x + G u$ which stabilizes $\dot{x} = Ax + Bu$?

$\textbf{Introduction}$: I study linear control theory. Among strategies, we begin with vector field $Ax + Bu$, $A \in M_{n^2}(\mathbb{R})$, $B \in M_{n \times m}(\mathbb{R})$, and synthesize a controller $u$ given by product $F x$ such that matrix…
User 42
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Can an input stabilize two parallel systems with different initial conditions but the same dynamics?

Assume two parallel systems of the form $\dot{x}=f(x_t)+u_t$ and $\dot{y}=f(y_t)+u_t$. The input that is added to these two systems is the same. Also assume that $f$ is globally unstable, meaning starting at any initial condition $x_0$ except the…
Amir Khazraei
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