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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14
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149 views

Level sets of solution to Nonlinear PDE

I work on Stochastic Control theory and BSDE's for my research. In my research, I characterized the set I am interested in as the level set of a function which is a viscosity solution to nonlinear PDE ...
6
votes
0answers
140 views

Eigenvalues of a matrix written in controllable canonical form

Let the following equation represent a stable (marginally) dynamical system in discrete time domain \begin{equation} \mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k \end{equation} ...
5
votes
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 ...
4
votes
0answers
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 ...
4
votes
0answers
157 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)....
4
votes
0answers
52 views

Stochastic control with stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ and smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$, how can we optimise the ...
4
votes
0answers
68 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
4
votes
0answers
184 views

Prove that all the solutions of (2): $\frac{dy}{dt}=A(t)y+f(t)$ are bounded in $ \left[t_0,+\infty \right )$

I have a problem: Assume that system (1): $$\dfrac{dx}{dt}=A(t)x$$ is stable, where $A(t) \in C\left [t_0,+\infty \right )$, when $t \to \infty$ and $$\begin{cases} & \mathrm{ } \...
3
votes
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. ...
3
votes
0answers
197 views

Dynamic programming problem for discrete linear time varying system

I'm working on a linear time varying discrete(LTV) multi input multi output(mimo) system. I formulate the problem description in the following way $$x_i(k+1) = x_i(k)\cdot A_i(k) + B_i(k)\cdot u_i(k)$...
3
votes
0answers
41 views

Infimum of radially unbound functional

I am having difficulty following a proof about balls (subsets) of radially unbounded functionals. Let $U$ be a Banach Space. Let the space of admissible controls $U_{ad}\subset U \ne \emptyset$ be ...
3
votes
0answers
82 views

Another machine problem

This is linked to my previous question Discounted optimization problem I have difficulties again finding a formula for $F(x)$. We consider a machine at time $t$ which is in state $x$. The machine ...
3
votes
0answers
160 views

Condition for controllability of matrix pairs

I have a question regarding how determine if a matrix pair is controllable. If $A$ and $b$ are given by A = [$\lambda_1 0 0 ...., 0 \lambda_20 0 0,...,00...\lambda_n$] (matrix) and $b = [b_1 b_1 ......
2
votes
0answers
33 views

Observation model for Kalman Filter

I'm having a problem in finding the Jacobian of an observation model of a camera in a 3D orbital scenario. There is a satellite and a landmark, the positions of which are described respectively with ...
2
votes
0answers
191 views

Solving one Systems Equations, Research Level Questions?!

We have three equations as follows: $A_p=F \cos(\alpha+ \phi) - \mu N^{'}_{S_1} - \mu N_{S_1} - W \sin \theta = m ( \ddot x - r \ddot \theta (\sin (\gamma+ \phi)) )$ $B_p=F \cos(\alpha+ \phi) - \mu ...
2
votes
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 \...
2
votes
0answers
31 views

Controllability of $x' = Ax + Bu(t)$ implies controllability of $\left \{ \begin{matrix} x' = Ax + By \\ y'=u(t) \end{matrix} \right.$

Suppose that the system $$x'(t)=Ax(t)+Bu(t)$$ is controllable in $R^n$, where $A$ is $n \times n$, $B$ is $ m \times n$ and $u(t)$ is $m \times 1$ Show that the system $$\left \{ \begin{matrix} x'(t) ...
2
votes
0answers
61 views

Stability for a infinite dimensional dynamical system

Suppose I have a infinite dimensional dynamical system as $\dot{x_n}=Ax$ where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find ...
2
votes
0answers
145 views

Jacobian Matrix of 6DOF Body (with IMU)

I am trying to derive the analytical Jacobian for a system that is essentially the equations of motion of a body (6 degrees of freedom) with gyro and accelerometer measurements. This is part of an ...
2
votes
0answers
92 views

Moment Generating Function of a Beta random variable.

After getting some excellent help on this problem in the statistics SE, I am reformuluating my question. Let me know if I should just delete it and ask a new one. Let $V$ be a $Beta(\alpha,1)$ ...
2
votes
0answers
47 views

Reconstruction of state covariance from output covariance

Let us be given an LTI system $$ \frac{d}{dt} x (t) = A x(t), \;\; x(0)=x_0 \\ y(t) = Cx(t) $$ where $x_0$ is a random vector (e.g. uncertainty). Then it is known that the expectation $\mathbb E[x(t)...
2
votes
0answers
61 views

Reachable set using constant control input

Consider a linear time-invariant control system given by the differential equation \begin{align*} \dot{x}(t) &= Ax(t) + Bu(t), \;\; x(0) = x_0, \end{align*} where $x\in \mathbb{R}^n$ and $A,B$ ...
2
votes
0answers
28 views

Controlled system question optimisation

Consider the controlled system $x_{t+1} = x_t + u_t + 3\epsilon_{t+1}$, where the $\epsilon_t$ are independent $N(0,1)$ variables. The instantaneous cost at time t is $x_t^2 + 2u_t^2$. Assuming that $...
2
votes
0answers
103 views

Cellular Automata Method to Reach Equilibrium on a Network of Numbers

Here's a puzzle that I'm curious if anyone can attack with a simple method without resorting to simulation. I can tell you the answer (well, an answer) from running some programs. But I'm writing a ...
2
votes
0answers
75 views

Find u that minimizes the integral mean

How do I find $u : [0,\infty) \to \mathbb{R}^m$ that minimizes \begin{equation} J(u(\cdot)) = \lim_{t \to \infty} \frac{1}{t} \int_0^t L(x(\tau),u(\tau)) d \tau, \end{equation} subject to \begin{...
2
votes
0answers
103 views

How verification argument really works?

Let $C(u,s)$ be cost functional for an admissible control $u$ with initial state of the system being $s$. Our aim is the solution of the following problem: $$\inf_u E(C(u,s))$$ We defined the value $...
2
votes
0answers
74 views

What's the shooting algorithm for the mass-spring problem (ode)?

I have the following problem : $$ \begin{aligned} \frac{d x(t)}{dt} &= y(t)\\ \frac{d y(t)}{dt} &= -x(t)+y(t)(1-x(t)^2)+u(t) \end{aligned} $$ with the initial condition $(x,y) = (0,0)$. Those ...
2
votes
0answers
50 views

Question regarding continuous time systems

If $S_u$ is the set of all solutions to the continuous time problem x(dot) = Ax + Bu and $\phi_h :(R^n)^R to (R^n)^N$ be an operator which maps every state x into the sequence (x(0), x(h),....,(x(ih),....
2
votes
0answers
159 views

Matrices made negative semidefinite, but not simultaneously

Consider matrices $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, such that $A$ has at least $1$ strictly positive eigenvalue. Let $X_1, X_2 \in \mathbb{R}^{n \times n}$ such that $...
2
votes
0answers
189 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 ...
1
vote
0answers
21 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-...
1
vote
0answers
55 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): $...
1
vote
0answers
50 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 ...
1
vote
0answers
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 ...
1
vote
0answers
48 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 ...
1
vote
0answers
53 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. ...
1
vote
0answers
43 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 ...
1
vote
0answers
31 views

Force Control on a Tether

On the ground, I have a winch driven by a synchronous motor to which I can command a torque input $u$. Parameters of this winch, such as friction $d$ and moment of inertia $J$ are well understood and ...
1
vote
0answers
37 views

perturbation of exponentiolly stable system

consider the following system on $\Bbb{R}^n$ $\dot{x} = f(x,t)+g(x,t) $$ $$ $$ $ $ (*) $ assume that f(0,t)=g(0,t) = 0 and 1. 0 is an exponentiolly stable equilibrium of $\dot{x}=f(x,t)$ 2.$|...
1
vote
0answers
39 views

Matrix comparison depend on one scalar variable

Let $A$, $B$, $K$ be $n\times n$, $n\times m$ and $m\times n$ matrix respectively. $\alpha_i>0$ is a scalar. Consider the following matrix: $H_i=\int_0^\infty e^{(A+\alpha_i BK)^Tt}(\alpha_iI+\...
1
vote
0answers
83 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
1
vote
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: $$ x^\top \left( D U^\top \frac{L-L^\top}{2} U \otimes PC \right)x = 0 $$ ...
1
vote
0answers
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. ...
1
vote
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 ...
1
vote
0answers
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 $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
1
vote
0answers
80 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 ...
1
vote
0answers
40 views

Minimum norm matrix to stablize a linear system

Suppose all of the eigenvalues of $A$ locate strictly on the right half plane. $(A,B)$ is controllable, $H$ is symmetric and strictly positive definite. I wonder is there a optimal solution $H^*$ ...
1
vote
0answers
104 views

Bounding reachability of damped harmonic oscillator using barrier certificates

I'm trying to prove that, under certain conditions, a damped harmonic oscillator that starts on one side of the equilibrium remains on that side of the equilibrium. More precisely, consider the ...
1
vote
0answers
19 views

If state is reachable in time T_1, then it is reachable in time $T > T_1$

Consider a Linear Time System with the admissble control set $$U = \left\{ u: R \rightarrow R^m \;|\;\text{u is integrable in any finite interval} \right\} $$. Show that, if starting on $x_0=0$ we ...
1
vote
0answers
34 views

How to understand “maximal ”, bounded“, and ”complete" for solutions of a (hybrid) dynamical system?

I am trying to read some text books about hybrid dynamical system, in which maximal solution, bounded solution, and complete solution are mentioned frequently. The following passage is a description ...