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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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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18 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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11 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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33 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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1answer
54 views

What is the difference between moving-horizon DP and MPC?

What is the difference between moving-horizon DP (dynamic programming) and MPC (modelbased predictive control)? In both cases, the system input at time $t$ is determined by solving a finite-time ...
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54 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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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

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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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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1answer
18 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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1answer
105 views

Estimating the input to a system from a system state

[ Cross-posted to: http://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ] I have a system for which I have obtained a non-linear time-varying ...
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1answer
8k views

Kalman Filter to determine position and attitude from 6DOF IMU (accelerometer + gyroscope)

I'm going to describe the problem I'm trying to solve and walk through what I understand so far about the Kalman Filter. I have an IMU which gives me the following measurements every time interval t: ...
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0answers
186 views

What is the Difference between Various Kalman Filters?

What is the difference between additive and multiplicative kalman filters, as well as some other kinds? I'm also looking for reference texts and articles that describe the algorithms, so ...
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0answers
96 views

How to Construct an Error State Kalman Filter?

I am trying to construct an error state kalman filter for GPS/INS integration using simulated data and I am having problem on a few steps. My error state vector is $\delta x = [\delta\alpha \, ...
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1answer
641 views

Using overshoot and settling time formula to determine pole location?

Is it possible to use the formula for overshoot and settling to determine where where ones pole should. by using the overshoot and settling time formula i mean, using it to define what $\zeta$ and ...
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1answer
217 views

Verification for maximum principle

Given optimal control problem $$ \dot x = f(t,x(t),u(t)), \quad x(0) = x_0,\\ J(u) = \int_0^T f^0(t,x(t),u(t))dt \to \min, $$ we can apply Pontryagin's maximum principle to get a necessary condition ...
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20 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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71 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 ...
2
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1answer
500 views

Inverse of State-space representation

Ask two questions from a paper (2012 ACC): Consider the plant: Let X be the stabilizing solution of the Riccati equation: where . Define the LQR gain by . The transfer matrix has a ...
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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
66 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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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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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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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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0answers
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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0answers
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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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. ...
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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0answers
13 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 ...
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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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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
40 views

Feedback in Sampled Data System

I have a feedback loop as shown in figure. Broken link represents sampling and signal with asterisk as superscript represents discrete signal. ZOH is the abbreviation for zero order hold operation. ...
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1answer
30 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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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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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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1answer
23 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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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 ...