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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16 views

Bode Sensitivity Integral

The bode sensitivity integral is well known for linear control systems. To state it in simplistic terms, Any system $L$ with relative degree $2$ or more satisfies the following equation for the ...
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23 views

Projection of periodic trajectories

Let $(\bar x(t),\bar u(t)),\, t\in [0,1]$ solution of $$ \left \{ \begin{array}{l} \dot x_1 (t) = u(t)\, f(x(t)) \\ \dot x_2 (t) = u(t)\\ x_1(0) = 0, x_1(1)=1 \\ x_2(0) = x_2(1) \end{array} \right. $$ ...
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1answer
17 views

Lyapunov function guarantees local exponential stability

can someone give me a proof of http://nptel.ac.in/courses/101108047/module13/Lecture%2031.pdf page 15? Suppose all conditions for asymptotic stability are satisfied. In addition to it, suppose $\...
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20 views

Find parameter $k_f$ for which steady-state error equals $0$

$u(t)=t, G_0(s)=\frac{k_p}{s}, G_R(s)=k_f$ Find parameter $k_f$ for which steady-state error equals 0 Finding $E(s)$ $$E(s) = U(s)*\frac{1}{1+\frac{k_p}{s}*k_f} = \frac{1}{s^2+k_pk_f*s}$$ Steady-...
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1answer
29 views

Is there a strategy for discrete control of a system with dynamics near sample rate?

I'm trying to control a system where the controller sample rate is physically fixed and the plant has significant dynamics on the same order as the sample rate. I understand that one would prefer to ...
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1answer
20 views

Fourth order filter with assigned poles

I need a bit of reverse engineering here, basic one. I want to design a 4th order filter and indeed I already did it but I don't remember how I did it. I know that my poles need to be in $p1, p2, ...
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31 views

What is the difference between the infinity norm of a transfer function and the infinity norm of a matrix

I am studying robust control system, and get confused with the following two definitions of infinity norm. (G(jw) is the transfer function of a MIMO system) [1] $$\left \| G \right \| _\infty = \max \...
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47 views

How do I get this dynamics to follow this trajectory?

Given this dynamics: $$x_{k+1} = x_k + a\cos(u_k)$$ $$y_{k+1} = y_k + a\sin(u_k)/\cos(x_k) $$ I want the input that would make this system ($x$ and $y$ states) follow this trajectory (lemniscate): $...
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1answer
49 views

Lyapunov function suggestion for a system

Can you please suggest a Lyapunov function to prove the stability of the following system: \begin{equation} \dot x=-\frac{\partial f(x)}{\partial x}-a \lambda - \lambda P u\\ \dot \lambda=(a+Pu)^\top ...
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2answers
46 views

Lyapunov equation, uniqueness, stability.

With respect to $\dot{x}(t)=Ax(t)+Bu(t)$, and $u=-Kx(t)$, and $K=R^{-1}B^{\rm T}P$, where P solution of $\tilde{A}^{\rm T}P+P\tilde{A}+Q+PBR^{-1}B^{\rm T}P$, where $\tilde{A}=A-BK$. I believe that, ...
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21 views

How to programatically solve the optimal control problem?

I have to programatically (write a program) find a control function $u(\cdot)$ to minimize the following functional: $$ J(u,x) = \int_0^T { f_0(x(t), u(t), t)}dt + \Phi(x(0)) \rightarrow \min$$ ...
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53 views

How to describe orbits of vector fields?

Let $X(x)=Ax$ be the linear vector field in $\mathbb{R}^3$ defined by the matrix $$A= \begin{pmatrix} 0&-1&0\\ 1&0&0\\ 0&0&0 \end{pmatrix} $$ and $Y$ be the constant vector ...
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1answer
40 views

System control of an induction heated system

Previous Question i understand i asked my previous question regarding this topic the wrong way. thou before i had time to respond the question was closed. On the other side i couldn't rephrase my ...
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1answer
45 views

Is the controllability gramian always positive definite?

I am trying to understand the balanced truncation algorithm and have some trouble distinguishing between controllability matrix and controllability gramian. If my understanding is correct, a linear ...
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1answer
33 views

Compute H-infinity norm in Matlab

Please can someone write a command in Matlab for calculating $H_{\infty}$ norm for the following system: $$\frac{d}{dt}z(t)=Az(t)+Bu(t)+Fw(t)$$ $$y(t)=Cz(t)+Du(t)$$ where $A$, $B$, $C$, $D$, and $F$ ...
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1answer
38 views

Control of Nonlinear Cascaded systems

For control of cascaded linearized system, my objective is to design a stabilizing controller. For stability and performance analysis of such structures, I have been trying to find a book where such ...
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1answer
28 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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23 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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1answer
23 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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49 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 $$\gamma(x(t)) = K_p x_1(t) + K_d x_2(t).$$...
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57 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. & & \dot{x}_{i}=\alpha_{i}-\beta_{i}\min(u_{...
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1answer
33 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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21 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 \operatorname{tr}\...
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1answer
64 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} \\ \lambda_1^{p_3}...
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1answer
26 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
25 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
29 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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4answers
69 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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21 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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101 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 $s(t)=(0,0,\sum_{k=0}^{\infty}d\delta(t-...
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2answers
51 views

Say whether an LTI system is controllable or not, without the controllability 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 $\mathbf{b}=\begin{...
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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
77 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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25 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)} J(u)=\frac12\int_{\Omega}|y_{u}-y_{d}|^{2}dx+\frac{\nu}{2}\int_{\...
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57 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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62 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
41 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 $-\frac{1}{4}&...
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20 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. x_{...
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14 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
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
35 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
50 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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45 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 form:...
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26 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
30 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 $y^...
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1answer
31 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 ...