Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The external input 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 ...

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12 views

Bandwidth from a transfer function

Currently I'm having problem wrapping my head around the following. Suppose you have a dynamical system described by the transfer function $$ G(s)=\frac{as}{(s+b)(s+c)} $$ depending on the variables ...
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22 views

How to interpret multi-conditional piecewise functions.

I'm trying to simulate hysteresis and the its inverse for a control problem. This is a model found in [Tao & Kokotovic, Adaptive Control Systems with Actuator and Sensor Nonlinearities] to ...
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10 views

Let $C\in\mathbb{R}^{p\times n},~A\in\mathbb{R}^{n\times n}$. When does $[C; CA^m]$ have rank $n$?

Let $C\in\mathbb{R}^{p\times n},~A\in\mathbb{R}^{n\times n}$, where $A$ is a nonsigular matrix. Define the following set of $\bar{m}$ matrices: $$O_m = \begin{bmatrix} C \\ CA^m ...
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12 views

Finding range of values for gain for which dominant time constant of stable system is less than 0.2s

I have a system with transfer function W(s) = K * 1/(s*(s+2)^2). Based on its root locus, I need to find range of values for gain K for which dominant time constant of stable system is less than 0.2s. ...
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32 views

Matrix transformation for linear state-space systems

In http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/lecture-notes/MIT6_241JS11_lec12.pdf on pages 11-12 it is said: For a stable ...
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17 views

asymptotic stability with exact feedback and feedback with measurement errors

I'm trying to show global asymptotic stability (GAS) for a system with a feedback controller. I managed to show GAS for the perfect feedback signal, that is $$\gamma(x(t)) = K_p x_1(t) + K_d ...
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53 views

Optimal control with state-dependnet solution

I'm trying to solve the following control problem $$ \begin{eqnarray*} \max & & \int_{0}^{T}\sum_{i=1}^{2}-c_{i}(x_{i}-u_{i})^{+}\\ s.t. & & ...
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29 views

What is the domain and codomain of a transfer function? [closed]

Let's say I have the transfer function- $\textbf{H}(j\omega)=\cfrac{1}{1+j\omega RC}$ Where does this function map to and from, and can it be plotted visually?
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20 views

Demonstration involving inequality of traces of product of psd matrix

Let, $ \forall i \in [1, N]: P_i \in \mathbb{R}^{n \times n}, P_i \succ 0, w_i \in \mathbb{R}, \bar{P} = \sum_{i=1}^N w_i P_i$. Then, I want to demonstrate that $ \sum_{i=1}^N w_i ...
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1answer
38 views

Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence

Let $$ A = \begin{bmatrix} \lambda_1^{p_1} & \lambda_2^{p_1} & \cdots & \lambda_n^{p_1} \\ \lambda_1^{p_2} & \lambda_2^{p_2} & \cdots & \lambda_n^{p_2} \\ ...
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1answer
17 views

Python - CVXOPT: What exactly should I check for G when "Rank(A) < p or Rank([G; A]) < n” exception is thrown?

I am new to using the CVXOPT module for Python and would definitely appreciate any illumination as to why the exception is thrown for my problem. (Also my first time posting a problem anywhere, so ...
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1answer
19 views

LQR problem with interaction term between state and control

Consider the usual linear process $x_{t+1} =Ax_t+Bu_t+Cw_{t+1}$ where $w_t$ is an independent and identically distributed $N(0,I)$ process. The objective is $$ V=E\sum_{t=0}^\infty \beta^t(x_t'Qx_t + ...
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1answer
25 views

Stuck relating the input to output (Transfer function)

i want to find the transfer function of a differential equation (given below) $\ddot\theta = a [ ([b\times Xin] - bk\dot\theta) - \ddot\theta] - c\phi $ (where $\phi$ and $\theta$ are time ...
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3answers
50 views

Mathematics for Guidance, navigation and control

I'm finishing my math degree this week and have been looking for some subject to practice and study on my own while I'm doing some work as a programmer. I'm interested in getting my master's later but ...
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19 views

Transfer function unity and output function poles are related?

By solving a few examples, I found the pattern that, given a differential equation in $y(t)$ and $x(t)$, where $y(t)$ can be called the input and $x(t)$ the output, if we make the condition that $y(t) ...
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66 views

How to use Newton's method for finding fixed points in Poincare maps.

As a homework I have to reproduce the numerical method given in the paper. Where there's the system $$ \dot{u}=f(u)+s(t)\\\\u=(u_1,u_2,u_3)\in\mathbb{R}^3$$ and ...
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2answers
42 views

Say wheter an LTI system is controllable or not, without the controllable matrix

I have matrix $\mathbf{A}=\begin{bmatrix}2 & 1 & 0 & 0 \\ 0 & 2 & 0 &0 \\ 0 & 0 & -1 &0 \\0 & 0 & 0 &-2\end{bmatrix}$ and vector ...
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2answers
26 views

Maximizing the Nullity of a Symbolic Gram Matrix

I have a symbolic gram matrix, that is, a matrix $AA^T$ with some entries being variables. I would like to find a solution for my variables which maximizes the nullity of this matrix, or equivalently, ...
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1answer
65 views

Inverse Vectorization Vec^-1

Hope that you will find this post in good health. I am Mr.Adnan from Pakistan with research background in Control systems. I am working on one problem in which Hadamard weights are using. During ...
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22 views

Distributed control problem which involves the p-Laplacian operator

Someone could help me to deduce the optimality system for the optimal control problem: \begin{align} &\min_{u\in L^{2}(\Omega)} ...
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53 views

A question about the “state-transition-matrix” of a physical system,

Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
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52 views

What is the difference between “state-transition-matrix” and a transition matrix?

What's the difference between a state-transition-matrix and a transition matrix (say, for an ergodic Markov Chain) that is typically taught in a basic probability theory course? This is the first ...
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1answer
40 views

Compactness and positive invariance of set under flow of ODEs

Given a system of ODEs, $$x'=y$$ $$y'=x-x^3-y$$ $$x(0)=x_0$$ $$y(0)=y_0,$$ also given a set $S=\{(x,y):V(x,y)\le k, x>0\}$, $V(x,y)=-\frac{x^2}{2}+\frac{x^4}{4}+\frac{y^2}{2}$, where ...
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18 views

Does the existence of a Algebraic Riccati Equation implies the existence of an functional minimization?

Let $\forall k\ge 0. V_k(x)$ be the value function related to the recursive optimization problem $ J(x_0) = \underset{u}{\inf} \sum_{k=0}^{N-1} x_k^T Q x_k + u_k^T R u_k + x_{N}^T P_N x_N \\ s.t. ...
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0answers
12 views

How to determine the transition probability in Sequential Importance Sampling (SIS) for Particle Filter

Given a state-space model \begin{align} x_k &= f_k(x_{k-1}, v_{k-1}),\\ z_k &= h_k(x_k, w_k), \end{align} where $x_k \in {\mathbb R}^{n}$ and $y_k \in {\mathbb R}^{m}$ are the system state ...
2
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1answer
26 views

Help with multivariable transfer function

I am looking to find the transfer function from w to z in this loop. I have been trying for a while looking all the relationships but just don't know how to express w in terms of r,d and n and then ...
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1answer
34 views

How to calculate the transfer function from a group of equations below?

The group of equations below describe the relationship of variables from a circuit(C stands for capicator, L is for inductor etc.). And the equation at the bottom shows how the transfer function is ...
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1answer
49 views

Can the root locus of a minimum phase plant become unstable?

I have a discrete system for which the root locus equation is given as: $$A(z) + K\cdot B(z) = 0$$ They are such that $A(0) = 1, B(0) > 0$, and $K>0$. $\frac B A$ is minimum phase and a ...
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0answers
40 views

Hartman-Grobman Theorem - Necessary?

The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields ...
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2answers
48 views

Stability criteria for linear systems with auxiliary variables

Classical texts for control theory show the linear system $\dot x=A \,x$, is stable if the real parts of the eigenvalues are negative. Does the same criteria apply for a system of the following ...
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23 views

Relationship between delay and fractional order differential equation

There are control systems with delay [1] There are differential equations with fractional order: $$D^\alpha x(t)=f(t,x(t))$$ I am wondering why we see control systems with delay and fractional ...
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1answer
28 views

How to obtain a stabilization problem in linear system with controller?

The scheme of system: The equasion after Laplace transform: $$Y(p) = \frac{PID(p)\cdot H(p)}{1 + PID(p)\cdot H(p)} Y^d(p)$$ Now I want to make inverse Laplace transform and then plot $y(t)$, but ...
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1answer
29 views

Centroid Root Locus

I can't figure out how to find the root locus centroid for the poles of this simple equation in a positive feedback system. $$ H(s)=\frac{s}{s^2+3s+1} $$ I have read in many places that the centroid ...
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26 views

How to Optimize PID Gains Non-Heuristically

In robotics, variables that control processes (usually voltage output to actuators) are continually adjusted by a PID (Proportional Integral Derivative) Control algorithm to improve the result of a ...
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20 views

Routh stability criterion ranges of k

Hi I need to find the ranges of k and $\beta$ such that the steady state error will be less than 10% for unit step input%. I'm currently at the point where I know that the steady state error is given ...
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1answer
22 views

steady state output transfer function

Hi If I'm given an input: $$r(t) = 2 \sin(3t).$$ And If I'm supposed to get the steady state response $$y(t)$$ for the given transfer function $$G(s) = \frac{1}{(s+1)(s+1)}$$ Is the following ...
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0answers
15 views

Root loci transfer function

Hi I asked a question earlier about this problem: $$T(s) = \frac{G(s)}{1+G(s)H(s)}$$ Where $$H(s) = \frac{s}{s+1}$$ $$G(s) = \frac{k(s+4)}{(s+2)(s^2+s+6)}$$ . Poles resulting: $$-1, -2, -0.5 + ...
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1answer
34 views

Transfer function Root loci

Hi I was given a closed system as: $$T(s) = \frac{G(s)}{1+G(s)H(s)}$$ Where $$H(s) = \frac{s}{s+1}$$ $$G(s) = \frac{k(s+4)}{(s+2)(s^2+s+6)}$$ This is where I'm unsure of. When I calculated the ...
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29 views

How to discretize state space with uniform grid

Let us consider a general continuous time stochastic differential equation represented by *dx* = A(x)dt + B(x)udt + $\sigma$ dw where A(x) represent the ...
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35 views

Optimisation/Control in lieu of knowing the future …

I’m interested to hear about the methods that would be best applied to an optimization/control problem. Context: I’m looking at the concept of ‘peak shaving’ in power networks, which are achieved by ...
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29 views

Trajectories for a Second Order System with negative real eigenvalues

Consider the Second-order System $$ \dot{\mathbf{x}} = A \mathbf{x} $$ $a_1$ and $a_2$ are positive real numbers, with $a_1 > 2*a_2$. The matrix A is given as $$A = \begin{pmatrix} ...
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28 views

Polar form of the Fourier transform of $\sin(t)$

I'm studying signal processing, and I came across the Fourier transform of sin(t). It ends up being a purely imaginary (dirac delta) impulse pair. But when considering the frequency domain ...
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3answers
50 views

modal truncation of state space system while preserving certain eigenvalues

Given a state space system $(A,B,C)$: $$\dot{x}=Ax+Bu\\y=Cx$$ Is there any method to obtain a reduced system $(A_r,B_r,C_r)$, where $$\dot{x}=A_rx+B_ru\\y=C_rx,$$ such that the eigenvalues of $A_r$ ...
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22 views

Controllability problem for linear systems

I need to prove: The pair (A,B) is controllable iff (A-BK,B) is controllable. A,B and K are the real matrices: nxn, nx1, 1xn, respectively. Thank you.
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36 views

Matched pole zero discretization

There are several techniques to discretize continuous-time transfer functions to discrete-time transfer functions. Some of them, such as, zero-order-hold, forward euler or Tustin, are well known. ...
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40 views

Mathematical control and differential equations

The differential equation $$\frac{dx}{dt}= Ax(t)+ Bu(t)+ f(t,x(t))+ g(t,u(t))$$ subjected to $x(0)=a$ defined on $[0, t], t>0$, where $u$ is a control and $x$ is a state. Does this differential ...
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23 views

How ca I discretize nonlinear ode ODEs?

I am trying to discretize a nonlinear set of ordinary differential equations. Usually I got good results with Euler, nevertheless, in this case Euler is not fitting the system response for a constant ...
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1answer
28 views

How to write a transfer function (in Laplace domain) from a set of linear differential equations?

Provided I have a system of linear differential equations (in time domain) such as: $$\begin{cases} \dot{x}(t)=Ax(t)+By(t)+Cz(t)\\ \dot{y}(t)=A'x(t)+B'y(t)+C'z(t)\\ \dot{r}(t)=B''y(t)\\ \end{cases}$$ ...
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19 views

About the limitations of linear analysis in comparison with Lyapunov

Consider the connection of two electrical circuits. Both circuits, Z1 and Z2, are stable and only one of them is non-passive. I.e., the eigenvalues are located in the LHP but $\Re e\{Z_{2}(j\omega)\}$ ...
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1answer
44 views

Minimal time trajectory optimization for body in 2D with waypoints

I am trying to control a spacecraft moving in 2D space through a number of waypoints as fast as possible. The spacecraft has directional acceleration (with an upper limit on its magnitude, and a limit ...