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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29 views

Find the normal form of this function

A second order control theory function looks like: $$\text{H}_{(s)}=\frac{\text{K}_p}{\frac{1}{\omega_0^2}\cdot s^2+\frac{2\beta}{\omega_0}\cdot s+1}$$ Now I've got the function, with $a,r,k\in\...
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40 views

How does state transition matrix works

Suppose I have a simple vehicle moving in 2D. The state vector for the vehicle is X=[x y vx vy ax ay], that is, it contains the position (x,y), the velocity (vx, vy) and the acceleration (ax, ay) of ...
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65 views

What is the smallest positive value of K which makes the closed-loop system unstable?

You are given a transfer function $\displaystyle G(s)=\frac{1.81K(s+20)}{(s^3+10s^2+32s+32)}$. This system is connnected with unity negative feedback. I've tried so many things but I can't do it . I'...
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53 views

Are integro-differential equations considered dynamical systems?

A definition of the dynamical system is that: $\phi:R \times E \to E$ is a dynamical system where $\phi \in C^1$, $E$ open subset of $\mathbb{R}^n$, and if $\phi_t(x) = \phi(t,x)$, then $\...
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31 views

PID tuning for system with parameter

How can I tune a PID to control a system with a parameter? Can I know beforehand how to change the PID parameters in function of the value of the system parameter?
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16 views

Should the performance of a PID controller be independent of the input?

Assume you designed a PID controller to let a given system track a unit step. Will the controlled system exhibit the same behaviour with regard to step inputs with different amplitudes?
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2answers
39 views

How is a state disturbance matrix constructed?

Consider the system: $\dot{x}$ = Ax + Bu y = Cx + Du Where x contains 4 states, we have 2 inputs $u = \begin{bmatrix}u_1\\u_2\end{bmatrix}$ and A, B, C & D are known. Now if 2 separate noise ...
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2answers
65 views

How to drive a vehicle (limited by acceleration) on a flat ground to a given point as fast as possible?

So I have a function $\mathbf{x}(t): \mathbb{R} \rightarrow \mathbb{R}^2$, which is supposed to mean the path of the vehicle (time mapped to position). The initial conditions $\mathbf{x}(0)$ and $\dot{...
5
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0answers
122 views

Two definitions of uniformly observable, are they equivalent?

When I do my research, I found that there are two definitions of uniformly observable, I can't help thinking are they equivalent? These two definitions are listed as follows. For a linear stochastic ...
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1answer
35 views

Calculate state transition matrix with one left and right eigenvector

How is it possible to calculate the state transition matrix of the following LTI-System: $$ \frac{d\mathbf{x}}{dt}=\mathbf{A}\mathbf{x}+\pmatrix {1 \\ -1}u$$ $$y=\pmatrix {2 & -2}\mathbf{x}$$ $$...
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40 views

Linearization of a Matrix Inequality with Quadratic Terms

I derived the following matrix inequality for finding a stabilizing observer-based feedback gain $K$. $$ \begin{bmatrix}-\lambda P& \tilde{A}^T & 0\\ \tilde{A} &-P^{-1} & \tilde{B}\\ ...
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47 views

Conceptual problems when minimizing a simple functional

I have a problem with what seems a very simple functional maximization. Let's define: $$ J[z]=\int \left( u(z)-\frac{\dot z^2}{2} \right) dt $$ Where $u(z)=-z^2+5$. The problem is to find $$ \arg\...
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1answer
65 views

Prove that the equation $Ax+Bu=0$ has a solution $u$ for every $x$

We know that $\text{rank}(\lambda I-E, F)= 2n$ (full row rank) for all $\{\lambda\in\mathbb{C} \mid \Re(\lambda) \ge 0\}$ where E=$\left(\begin{matrix} A_{n\times n} & 0_{n\times n} \\ ...
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2answers
74 views

LTI Multi-input Control System. Proof that controllability holds given a state feedback.

The question is : Prove that (A, B) is controllable if and only if (A + BK, B) is controllable for all K. My proof thus far: Let $u=kx +v$ Consider the Im(Qc) = Im(B) + (A+BK)*Im(B) + ... + (A+BK)^...
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1answer
40 views

Reference Request: investigation of higher order dynamical systems

In dynamical system and control theory, people usually investigate into system of the type $$\dot x = f(x,u)$$ Is there any references to looks into the theory of higher order dynamical systems of ...
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33 views

Routh Stability criterion

To check for the numbers of poles lying in the right side of the s-plane in a causal system, why do we check for the sign change only in first column ?
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131 views

Geometric interpretation of Q in Lyapunov's equation

Lyapunov's equation says: given any $Q > 0$ ($Q$ positive definite) there is $P > 0$ such that $A^T P + P A + Q = 0$ if and only if for $\frac{dx(t)}{dt}=A x(t)$ it is the case that the real ...
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1answer
73 views

How to prove the convergence in such a case?

$$ \dot x(t)=Ax(t)+Bu(t),\quad t\in[0,T]\\ x(0)=x_0\\ u\in L^\infty([0,T],\mathbb R^m)\text{ and }x\in L^\infty([0,T],\mathbb R^n) $$ It is known that $\lim_{a\to 0}||u_a-u_0||_{L^2}=0$, where $u_0$ ...
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2answers
79 views

Boundedness of the input

$$ \dot x(t)=Ax(t)+Bu(t),\quad t\in[0,T]\\ x(0)=x_0\\ x(t)\text{ is bounded on }t\in[0,T]\\ A\text{ and }B\text{ are given constants},\quad B\neq 0 $$ My question: Is $u(t)$ also bounded on $t\in[...
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22 views

Prove of disprove the asymmetry of a bode phase plot around the resonant frequency.

I am asked to prove or disprove the following: The bode phase plot for G(j$\omega$) given by G($j\omega) = a/(s+a)$ with a>0 is asymmetric with respect to (a,$-\pi/4$). I know how to derive the ...
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1answer
22 views

Proving an equivalence relation $(A_1,B_1)$ ~ $(A_2,B_2)$

Let $(A_1,B_1),(A_2,B_2) \in \Bbb R^{n\times n} \times \Bbb R^{n \times m}$. We say that $(A_1,B_1)$ and $(A_2,B_2)$ are similar written $(A_1,B_1)$ ~ $(A_2,B_2)$, if there exists $S \in$ GL (n, $\Bbb ...
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1answer
35 views

In the LQ control problem why is $2PA = A^{T}P + PA$

I do not understand why the equation below holds assuming $A$ and $P$ are both square matrices in $\Bbb{R}^{n*n}$ and $P$ is symmetric and positive definite (i.e. $ P = P^{T} $ and $ x^{T}Px > 0 $) ...
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767 views

How to control a nonlinear system with a PID controller

Is the design of a PID controller for a nonlinear system different from for a linear system? [I think math.stackexchange.com is the most suited SE.]
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1answer
29 views

Solving Differential Equation about rate of infected computers

I am having some trouble solving this differential equation for the rate of infected computers in a botnet at time t $$\frac{\mathrm{d}x }{\mathrm{d} t} = \frac{1}{c\nu (1-x) + \beta x(1-x) - \gamma ...
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0answers
24 views

Quadratic form of Kroenecker products of skew-symmetric matrices

I am trying to understand under which conditions on $P=P^\top>0$ , $C=C^\top$, the following quadratic form is zero: $$ x^\top \left( D U^\top \frac{L-L^\top}{2} U \otimes PC \right)x = 0 $$ ...
0
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1answer
24 views

Question regarding two properties of controllable subspace in control theory

In Mathematical Control Theory II: Behavioral Systems and Robust Control Two claims: $\langle A+BK | im B \rangle = \langle A | im B \rangle$ $\langle A | im B \rangle$ is the smallest $A$-...
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1answer
51 views

Values of a matrix for which a system represents stable or unstable systems

For $\alpha$, $\beta$, $\gamma$ $\in\mathbb{R}$, consider the system $$\frac{d}{dt}x=\underbrace{\begin{bmatrix}0 & \alpha & \gamma & 0 \\ -\alpha & 0 & 0 & 0 \\ 0 & 0 &...
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29 views

Finding the maximum of step response for a given transfer funtion

Assuming that the following transfer function is given: $$F(s)=\frac{\Sigma_{k=0}^m b_k s^k}{\Sigma_{k=0}^n a_k s^k}$$ $$m\le n$$ Lets say $g(t)$ is the step response to this transfer function. ...
0
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1answer
33 views

Displacement of differential equation

In [Forni and Sepulchre, arXiv:1305.3456] the authors state that given the differential equation \begin{equation} \dot{x}=f(x,u),\quad (1) \end{equation} where $f:\mathbb{R}^{n\times m}\to\mathbb{R}^...
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1answer
32 views

Phase and frequency locked loop

In electronics equipment, a unit named phase-locked loop (PLL) is used. Simply speaking, it adjusts the phase $p_r$ of a reference signal like $r(t)=sin(f_r*t+p_r)$ with constant frequency $f_r$ to ...
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1answer
34 views

Controllability of a system with point matrices

Consider the scalar nonlinear system $$\frac{d}{dt}x=\underbrace{\sin x+u}_{f(x,u)}$$ with equilibrium point $(x^{\ast},u^{\ast})=(0,0)$ We have $\frac{\partial f}{\partial x}=\cos x$ and $\frac{\...
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34 views

Imaginary results when calculating equilibrium points of a 2nd order system

The following system is given: $$ \frac{d\textbf{x}}{dt}=\begin{bmatrix} -x_1^2+x_2 \\ -x_1-x_2^2 \end{bmatrix} $$ I want to calculate the equilibrium points of the system and did that as follows: $$ ...
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1answer
42 views

Linearisation of nonlinear system of equations

Consider $$\begin{aligned}\frac{d}{dt}x_{1} & =x_{2} \\ \frac{d}{dt}x_{2} & =-x_{1}-(\alpha+x_{1}^{2})x_{2}\end{aligned}$$ with $\alpha\ne 0$ and equilibrium $x^\ast=0$. Then $f(x_{1},x_{2})=...
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1answer
54 views

Control theory - what method is used to find the discrete time control system here

We are given a car model: $$\dot x = V\cos(a) \quad \dot y = V\sin(a) \quad \dot a = u$$ $V$ some arbitrary number Make an (first order) approximation $$\dot x = V \quad \dot y = Va \quad \dot a = ...
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42 views

What are the steps to write a control function?

I have a system defined like this: $$\delta\delta x(t) = \rho(t) \cos(\alpha(t)) \\ \delta\delta y(t) = -g + \rho(t) \sin(\alpha(t))$$ I need to write a control function to calculate $\rho(t)$ and $\...
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47 views

No local optima in quantum control?

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation: $\frac{d x(t)}{dt} = F(x,u)$ one can consider the ...
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0answers
88 views

Modified Z-transform

Can somebody tell me where I do a mistake in derivation of modified (or advanced) Z-transform of digital parabolic sequence i. e $$f(k) = (k\cdot T)^2,$$ where $T$ is the sampling period and $k$ is ...
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1answer
35 views

Calculate equilibrium points of a 2nd order system

The following 2nd order system is given: $$ \frac{d\textbf{x}}{dt}=\begin{bmatrix} -6 & -\frac{2}{\pi} \\ 0 & \frac{1}{2} \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}+\begin{bmatrix}\...
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87 views

I'm trying to calculate $e^{At}$. Where do I go wrong?

Let $$A=\begin{bmatrix}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -4 & 0\end{bmatrix}$$ I want to determine $e^{At}$. I tried it using ...
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2answers
279 views

Eigenvalues of matrix with linear dependent rows

Why is it that, if a matrix has a linear dependent row, e.g.: $$ A=\begin {pmatrix} 1 \ 2 \\ 2 \ 4 \end {pmatrix}, $$ at least one eigenvalue is zero?
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1answer
47 views

Lyapunov invariant set for affine systems

Given a linear system $\dot{x}=Ax$ such that the real part of every eigenvalue of $A$ is less than $0$, Lyapunov's equation $A^T P + P A = -Q$ with $Q$ being any suitably sized positive definite ...
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156 views

Why no Forward Dynamic Programming in stochastic case?

Dynamic programming usually works "backward" - start from the end, and arrive at the start. This works both when there is and when there isn't uncertainty in the problem (e.g. some noise in the state)....
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1answer
38 views

Controllability matrix cannot be defined $\implies$ the system cannot be controllable?

Let $B\in\mathbb{R}^{n\times m}$ and $A=\lambda I$, for $\lambda\in\mathbb{R}$. I want to show that a necessarily condition for the controllability of $(A,B)$ is $m\ge n$. I assume this means for $$A=...
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1answer
27 views

Defining two matrices to eliminate latent variables

So I have the systems $$\Sigma_{1}:p_{1}(\frac{d}{dt})y_{1}=q_{1}(\frac{d}{dt})u_{1}$$ $$\Sigma_{2}:p_{2}(\frac{d}{dt})y_{2}=q_{2}(\frac{d}{dt})u_{2}$$ And the equations $u_{2}=y_{1}$, $u_{1}=u+y_{...
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0answers
26 views

Time-dependent inequalities in optimal controller

I need to build the optimal controller, i.e. one that maximizes: $J = \int_{0}^{t_f} f(u) \mathrm{d}t$ For the following time-dependent system: $\dot{x} = g(x, u, t)$, $x(t) \geq l(t)\; \forall t \...
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vote
1answer
57 views

Convexity of the set

$$ \dot x(t)=Ax(t)+Bu(t)\\ x(t_0)=x_0\\ x(t_f)=x_f\\ a\leq x\leq b $$ $x(t)$ is driven by $u(t)$ to satisfy the above differential equation, two boundary conditions, and the inequality. Here, $x_0$ ...
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0answers
36 views

Sinusoidal steering on SO(3) Lie group

This is part of an exercise from Shankar's book Nonlinear Systems: Analysis, Stability, and Control, Problem 8.11, p.381, which is a satellite control problem. I rephrase my question as follows. ...
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38 views

Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
4
votes
1answer
56 views

How would I solve the following equation, which is similar to an algebraic Riccati equation or a nonlinear sylvester equation?

I have the following matrix equation that I would like to solve for $X$: $0 = AX + XB + XCX + D$ In general, $X$ will be rectangular, with $(m\times n)$ dimensions. So if I write the equation out ...
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0answers
79 views

Extracting information from Singular Value Decomposition.

I am currently working on a heat pump system. The problem involves multiple inputs and outputs. During self study I came across the SVD technique, and learned that it can relate orthogonal inputs to ...