# 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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### Asymptotic stability by comparison with another system

Consider a nonlinear system of the form $$\dot{x} = f(x)x,$$ with $x \in \mathbb{R}^n$ and $f: \mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$. I know that there exists an asymptotically stable ...
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### Control a non-affine system

I have a question about the control of a non-affine system. Here is my system $\dot{x} = a(u) + b(u) . u$ \label{a_beta} a(u) = 0.22 \left( \frac{116(u^3 + 1)-4.06 \lambda }{(\lambda +...
1 vote
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### How to compute equilibrium of a discrete time system?

I know that a continuous time system $\dot{x}= f(x)$ has an equilibrium at $x_0$ s.t $f(x_0)=0.$ But I cannot translate the same idea in discrete-time systems since there we are given the system as a ...
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### Uniqueness of optimal control in infinite horizon LQR

Consider the following discrete finite horizon LQR problem with dynamics $$x_{t+1} = Ax_t + B u_t$$ and cost matrices $Q$ and $R$. The goal is to find a linear control $K$ which optimizes \begin{...
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### Show that if $(A, B)$ is controllable then the D.T described by $x(k+1) = Ax(k-1) + Bu(k)$ with initial states $x(0), x(1)$ is also controllable

I show this statement on some lecture notes : It is obvious that if $(A, B)$ is controllable then the D.T with initial conditions $x(0), x(1)$ described by $$x(k+1)= Ax(k-1) + Bu(k)$$ is also ...
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1 vote
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### About a property of solutions to the delay equation $\frac{dx}{dt} = Ax(t) + F(t,\,x_t)$ in a Banach space $E$

I'm dealing with the equation \begin{equation*} \tag{1} \frac{dx}{dt} = Ax(t) + F(t, \, x_t),\end{equation*} in which $A$ is the generator of a $C_0$-semigroup $(T(t))_{t \, \geqslant \, 0}$ on a ...
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### Convert continuous system of equations into difference system

I have a differential system of equations like below: \begin{equation*} \left\{ \begin{aligned} & \dot{X} (t) = A_1X(t) + B_1U(t) \\ & \dot{Y} (t) = A_2Y(t) + B_2U(t)+CX(t) ...
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### Kalman Filter -- handling large covariance matrices with principal-component-like structures

I understand that the estimation of the covariance matrices are the important part of the Kalman filter. However in my use case my covariance matrices are really big but with a pretty neat factor ...
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### How to approximate any line segment within a circular region using the minimum number of connected rotating axes

This problem arises from my personal experience in developing a game mod. At that time, I wanted to create a drone system for vehicles, but due to the limitations of the game itself, I could only ...
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### Observer for LTI system with linear inequality constraints for state

I have searched quite a bit about this topic but only found methods that consider equality constraints. Consider LTI system \begin{align} \frac{d}{dt}x&=Ax+Bu\\ y&=Cx+Du \end{align} with ...
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1 vote
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### Proving that the $H_\infty$ norm for scalar transfer functions is a norm

I am trying to prove that the $H_\infty$ norm of a scalar transfer function, $h$ (with no poles on the imaginary axis), defined as $$||h||_\infty := \sup_\omega |i\omega|$$ is a norm, i.e. it ...
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1 vote
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### Adding time delay compensation to LQR

Is it as straightforward to add time delay compensation to LQR that uses full-state feedback, as it is with PID that uses error-based feedback? PID can use eg Smith predictor, but when I search for ...
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### Indexability of Restless Multi-Armed Bandit Problem

While reading a book on Restless multi-armed bandit problem (specifically Ch.6 of J.C. Gittins, 2011: Multi‐Armed Bandit Allocation Indices), the idea of indexability comes up yet I find it hard to ...
1 vote
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### Nyquist plot stability analysis

I'm a bit confused with the stability analysis using Nyquist plots. Suppose I have the following open loop transfer function: $G(s) = \frac{s-1}{s^2(s+2)^2}$ I want to make a stability analysis under ...
1 vote
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### Laplace Transform of a Second Order Systems Response to Triangular Pulse

I've been trying to derive the time domain response of a second order system to triangular pulse input using Laplace Transformation but even if I seem to be able to derive it Simulink simulations ...
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### Find Gain K and Time constant K of a system from the time response

There is a given system $\frac{K}{sT + 1}$ of order 1. The responses are in the image below and the 2 inputs are $u1(t) = 1(t)$ and $u_2(t) = \sqrt{2} \cdot \sin(\omega_2 t)$. How can I find the K and ...
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### Optimal control problem with Hamiltonian linear in control

Let's consider the following deterministic optimal control problem, where $c(t)$ is the control, and $x(t)$ and $y(t)$ are the state variables: \begin{align} J(t) = \inf_{c(t)} \ &\int_0^\infty e^{...
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### ROC of a fractional order system in respective to its poles

My understanding about fractional order system is that it can have poles on the right hand side of imaginary axis in s-plane and yet being stable. (statement 1) There are other theorems about ...
1 vote
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### Model Reference Adaptive Control for Linear Algebraic Plants

This is a homework problem from my adaptive control course: Given the plant $y_p = a_pu(t)$ ($a_p\neq 0$) and the reference model $y_m = a_mr(t)$, where $r(t)$ is bounded and continuous. Design a ...
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### numerically solving for the fixed points of a system of nonlinear ODEs

I was looking at an excellent lecture series on Robotics by Russ Tedrake, and he discusses Linear Quadratic Control (LQR) for system of nonlinear differential equations. So as he suggests, robots are ...
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### Is a system, that is globally asymptotically stable for any constant input also input-to-state stable? [closed]

I am referring to the ISS definition by Sontag of ${\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}$ I understand that 0-GAS is a necessary condition for ISS. But is GAS for all ...
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### Cylindrical Water Reservoir Model and System Behavior [closed]

System Description We're looking for the mathematical model of a cylindrical water reservoir. The reservoir is characterized by three variables: The inlet flow, $Q_e(t)$, at time t The outlet flow, ...
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### Deterministic optimisation problem with inequality constraint

Let's consider the following deterministic constrained optimisation problem, where $c(t)$ is the control, and $x(t)$ and $y(t)$ are the state variables: \begin{align} J(t) = \inf_{c(t)} \ &\int_0^\...
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### Mapping the zeros of a transfer function in S domain and Z domain

The $s$ and $z$ domain variables are connected through the expression $z = e^{sT}$, in which, $s$ is the Laplace variable and $T$ is the sampling period. However, I have found no rigorous method that ...
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