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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

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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 ...
Trb2's user avatar
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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 $\dot{x} = a(u) + b(u) . u$ \begin{equation}\label{a_beta} a(u) = 0.22 \left( \frac{116(u^3 + 1)-4.06 \lambda }{(\lambda +...
Ehsan Aslmostafa's user avatar
1 vote
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41 views

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 ...
tonythestark's user avatar
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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{...
sdevlin's user avatar
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2 answers
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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 ...
tonythestark's user avatar
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 ...
user405919's user avatar
3 votes
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Discrete time linear quadratic regulator - uniqueness of feedback gain given Riccati solution

Suppose we have a controllable discrete time linear system \begin{align*} x_{t+1} = Ax_t + Bu_{t} \end{align*} In order to design a stabilizing LQR controller with respect to supply rate $l(x,u)=x^{\...
Sampath Kumar's user avatar
3 votes
1 answer
73 views

What is wrong with this proof that a linear, bounded, time invariant operator on $L_p$ must be a convolution?

I'm trying to understand if this is true and how to prove it, "If $T$ is a bounded, time invariant operator on $L_p(\mathbb{R})$, then $T$ is a convolution operator.'' Here's an attempt at a ...
travelingbones's user avatar
4 votes
2 answers
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Relative degree of a transfer function

Consider the (scalar) transfer function $$T(s)=\frac{p_ms^m+p_{m-1}s^{m-1}+\cdots+p_1s+p_0}{s^n+q_{n-1}s^{n-1}+\cdots+q_1s+q_0},$$ with $p_m\neq0$ and $m<n.$ The coefficients $p_i$ and $q_i$ are ...
Kian Shah's user avatar
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Mathematical interpretation and Fuzzy logic interpretation of d err/dt or change of error

Thank you for the possibility to ask a question. I am new at this forum. Currently I am scratching the surface of fuzzy logic with the idea to go deeper an deeper. From calculus, I understand that a ...
user3245276's user avatar
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63 views

Approximation of z-transform transfer function

It is common in control theory to approximate a transfer function neglecting the high order terms, in example, a transfer function with two poles: $$ P=\frac{1}{(as+1)(bs+1)}=\frac{1}{abs^2+(a+b)s+1}...
The Newbie Toad's user avatar
4 votes
2 answers
173 views

How to find an exact solution for $X=X^T \in \mathbb{R}^{n \times n}$ satisfying $AX=XA^T$ and $B=XC^T$

Assuming that I know that the following pair of equations has an exact solution: $$\exists X=X^T \in \mathbb{R}^{n \times n}: AX=XA^T,\ B=XC^T$$ For some matrices $A \in \mathbb{R}^{n \times n}$, $B \...
user9413641's user avatar
2 votes
2 answers
70 views

Space state representation from given differetial equation

I have the ODE: $\ddot{\alpha}+4\dot{\alpha}+3\alpha=2u-\dot{u}.$ Define $y=[\alpha\ \ \ \dot{\alpha}]^T$. Find $A,B,C,D$ such that: $$\dot{x}=Ax+Bu$$ $$y=Cx+Du$$ When $y=\alpha$, I get: $$A= \begin{...
lg123's user avatar
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3 votes
1 answer
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Observable and unobservable subspace

In control theory, the set of unobservable states is defined as $\operatorname{Ker}(W)$, where $W$ is the observability Gramian. Thus, the set of unobservable states is also a subspace. Is it correct ...
tut's user avatar
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1 answer
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Definition of the value function in control theory

The following definition stems from the notes "An Introduction to Mathematical Optimal Control Theory" by Lawrence C. Evans (they are available online for free). In defining the value ...
Littlejacob2603's user avatar
1 vote
1 answer
105 views

Stable Kalman Filter estimator with given covariance matrices

I asked this question a while back. Essentially considering the follow basic Kalman Filter, following the Wikipedia convention. \begin{equation} \begin{split} x_k &= F_kx_{k-1} + B_k u_k +w_k\\ ...
Taylor Fang's user avatar
1 vote
1 answer
39 views

Parameter control problem derived via Dirichlet principle / variational formulation of Poisson equation / Lagrange multipliers

The below is a distilled-down version of a more involved problem I am looking at. Suppose for simplicity that $\Omega = B_R(0) \subset \mathbb{R}^2$ and $R$ is very large. Let us further define $f_s\,\...
Pink and Floyd's user avatar
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1 answer
40 views

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) ...
Nrits's user avatar
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8 votes
1 answer
222 views

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 ...
Taylor Fang's user avatar
2 votes
0 answers
47 views

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 ...
S PLATEX's user avatar
4 votes
0 answers
140 views

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 ...
user3137490's user avatar
1 vote
1 answer
52 views

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 ...
Siva's user avatar
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1 vote
2 answers
107 views

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 ...
J B's user avatar
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0 answers
26 views

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 ...
Brain Lim's user avatar
1 vote
2 answers
69 views

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 ...
lord voldemort's user avatar
1 vote
0 answers
40 views

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 ...
Ahmet Burak's user avatar
1 vote
0 answers
72 views

A question about Wiener filter based on Linear Estimation by Kailath

In my linear estimation class based on the textbook Linear Estimation by Kailath, we went through the process of finding LLSE of $\hat{x}(t+\lambda)$ for fixed $\lambda$ given $\{y(\tau)|-\infty<\...
monad's user avatar
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0 votes
1 answer
109 views

Differential Equation to Linear State-Space Model

I derived a differential equation for a system that I am studying that takes the following form: $$\dot{x}-au_1(x-x_0)=bu_2+cu_1$$ This was derived from expanding a non-linear differential equation as ...
b7031719's user avatar
1 vote
0 answers
58 views

State feedback control with some states left uncontrolled [closed]

In short I have a linear MIMO system: $$\dot{\textbf{x}}=\textbf{Ax}+\textbf{Bu}$$ Is it possible to use state feedback control in some way so that some states are left uncontrolled? A bit more ...
Daniel Varga's user avatar
1 vote
0 answers
46 views

How to determine if my control system is following the most controllable direction?

I have a linear discrete time control system $x (t) = A.x(t-1) + B.u$. I computed the controllability matrix ($C$) of this system and did singular vector decomposition (SVD) of the $C$ matrix, $USV_h =...
Pranjal Garg's user avatar
2 votes
0 answers
56 views

Relationship discrete algebraic Riccati and discrete Lyapunov equation

Suppose that $K$ is the optimal control for an LQR problem with inputs $(A,B,Q,R)$, i.e.: $$ K = -(B^\top P B + R)^{-1} B^\top P A$$ where $P$ solves the discrete algebraic Riccati equation: $$P = A^\...
sdevlin's user avatar
  • 21
3 votes
1 answer
72 views

how to show global asymptotic stability with $V(x)=f(x)^{T}Pf(x)$ as a lyapunov function.

consider the system $f(x)=\dot{x}$ with $f(0)=0$, $f(x)$ is continuously differentiable. $f(x)$ can be written as $f(x)=\int_{0}^{1}\frac{\partial f}{\partial x}(x\sigma)x\partial\sigma$ (The first ...
TiredMechanicalEng's user avatar
-1 votes
1 answer
65 views

Pure complex equilibrium point, can I conclude anything about its stability? [closed]

In a dynamic system: \begin{align} \ddot{x} &= -D_2 \dot{y}^2 - R_2 + F_2 + D_1\dot{x}^2 + R_1 - F_1 \\ \ddot{y} &= -D_3 \dot{z}^2 - R_3 + F_3 + D_2\dot{y}^2 + R_2 - F_2 \\ \ddot{z} &= -...
sagaz_malin's user avatar
1 vote
1 answer
57 views

The difference of accessibility distribution between linear systems and nonlinear systems

I get confused about the difference in accessibility between linear and nonlinear systems. For nonlinear systems with the form \begin{equation} \dot{x}(t)=f(x)+\sum_{i=1}^m g_i(x)u(t). \end{equation} ...
Rui Tachibana's user avatar
3 votes
0 answers
125 views

When does an optimal input sequence for a discrete-time system exist?

Suppose an LTI discrete-time system is given by the equations $$ x_{k+1} = Ax_k + Bu_k,\\ y_{k} = Cx_k + Du_k $$ with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\...
Benjamin Tennyson's user avatar
1 vote
0 answers
76 views

Integral representation for system with Lyapunov-like inequality [closed]

Consider the system $\dot x = f(x)$ with $f(0)=0$, f continuously differentiable and $\frac{\partial f}{\partial x}(x)^TP+P\frac{\partial f}{\partial x}(x)\preceq-I$ for all x and some $P=P^T\succ0$. ...
BeNavon's user avatar
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0 votes
0 answers
29 views

truncating a system and plotting it on bode

Let system G(s) be: $$ G(s)=\sum_{i=0}^{10}\frac{(-1)^i}{(2i+1)^2}\frac{\omega_i}{s^2+2\zeta_i \omega_i s+\omega_i ^2}$$ $$\omega_i=\frac{(2i+1)\pi}{T}\, T=1\, \zeta_i=0.2$$ Its impulse response is an ...
Rice's user avatar
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0 votes
0 answers
32 views

direction of poles of 2 cascading systems

Given the following system: $$G(s)=G_2(s)G_1(s)=\begin{bmatrix} > \begin{array}{c|c} A_2&B_2\\ \hline C_2&D_2 \end{array} > \end{bmatrix} \begin{bmatrix} \begin{array}{c|c} A_1&...
adir's user avatar
  • 1
0 votes
1 answer
101 views

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 ...
sneha_jerin's user avatar
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0 answers
227 views

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^{...
NC520's user avatar
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0 answers
31 views

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 ...
Anders's user avatar
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1 vote
0 answers
58 views

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 ...
ArGenya's user avatar
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0 votes
1 answer
86 views

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 ...
krishnab's user avatar
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4 votes
1 answer
170 views

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 ...
LCG's user avatar
  • 51
0 votes
1 answer
44 views

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, ...
Knowledge Seeker's user avatar
0 votes
0 answers
60 views

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^\...
NC520's user avatar
  • 341
0 votes
0 answers
73 views

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 ...
Saeed's user avatar
  • 61
0 votes
1 answer
41 views

Multiplication of a time-domain sinusoid to a s-domain (Laplace) signal?

I am confused between the transformations between the time-domain and the frequency domain. I have a signal y(t) which is a sum of multiple sinusoids. I band-pass filter this signal to extract one ...
Ayush Sharma's user avatar
1 vote
0 answers
62 views

What is the intuition behind exponentials in the frequency domain signifying delay in a signal?

Question is in the title. I am trying to teach myself control theory and it involves a lot of taking things as truth without much intuition or proof. In particular, I'm trying to understand how $e^{s \...
rocksNwaves's user avatar
4 votes
1 answer
116 views

Limiting behaviour of state vector under delayed (open-loop) inputs

Given a discrete-time LTI system as $$x_{i+1} = A x_{i} + B u_{i}$$ and suppose the feedback law $u_{i} = \mathcal{K}(x_{i})$ assymtotically stabilizes the closed-loop system. Now, consider the ...
Bart Wolleswinkel's user avatar

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