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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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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220 views

Stability of linear systems with singular state matrix

Given a linear time invariant system $\dot X(t) = AX(t)$ where $X \in {R^{n \times 1}}$ and $A \in {R^{n \times n}}$ is a singular matrix ($A$ has at least one zero eigenvalue). How can I study the ...
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32 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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47 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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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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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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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
70 views

Understanding how Nehari's problem connects with robust stabiliziation and Nevanlinna-Pick

I'm reading Young's "An Introduction to Hilbert space". In chapter 15 he writes about robust stabilization in control theory and ends with that this boils down to an interpolation problem called the ...
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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
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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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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682 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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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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36 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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17 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 + 2....
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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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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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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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1answer
44 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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4k views

What is the physical meaning of Bode plot in case of unstable system?

I know that from the mathematical point of view it doesn't matter if we plot Bode diagram of stable or unstable system. It's just a function of complex value. However from the physical point of view, ...
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350 views

how to derive the canonical form of a transfer second order equation?

How to derive the canonical form of the second order transfer function?? $$\frac{(\omega_n)^2}{s^2+2\zeta\omega_ns + (\omega_n)^2}$$
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302 views

how to obtain state space diagram and state space model for transfer function

How do we obtain a state-space realization and a block diagram of a given transfer function? Consider the transfer function $$\frac{C(s)}{R(s)}=\frac{5s}{3s^{2}+3s+1}$$ Steps for solution are $$\...
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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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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
50 views

“Dictionary” of linearizations for nonlinear dynamical system

I have recently jumped on a control project that involves predicting output of a nonlinear system given some input. The team has used $N$ training input/output relationships to build a 'dictionary' ...
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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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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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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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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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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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1answer
57 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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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
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
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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1answer
107 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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188 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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101 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
219 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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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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1answer
544 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, ...