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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45 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
78 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
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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 ...
7
votes
2answers
79 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 ...
0
votes
2answers
178 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
45 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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0answers
92 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 ...
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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 ...
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1answer
26 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 $u_{2}=y_{1}$, ...
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 ...
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vote
1answer
56 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, $x_0$ ...
3
votes
0answers
27 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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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 $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
4
votes
1answer
54 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
66 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 ...
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1answer
38 views

Showing that three systems define the same behaviour

I have three input/state/output representations of the form $$\begin{cases}\frac{d}{dt}x=Ax+Bu \\ y=Cx\end{cases}$$ with the three systems given by: $$A_{1}=\begin{bmatrix} -1 & 0 \\ 0 & -2 ...
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1answer
56 views

Hamiltonian question, find optimal controller, simple question

An object is moving with accordance to Newton's laws: $\begin{pmatrix} \dot{y} \\ \dot {v} \end{pmatrix} = \begin{pmatrix} v \\ u \end{pmatrix}$ where $y$ is the objects location and $v$ is its speed. ...
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0answers
18 views

Is it necessary to divide by the pivot entry when using Routh's Array?

I have read that multiplying a row by a positive constant doesn't change the end result of computing Routh's Array, doesn't this mean that dividing each entry by the respective pivot is not necessary? ...
0
votes
1answer
56 views

Math major transferred to Electrical Engineering, trying to bridge the gap [closed]

I did a minor in mathematics a couple years ago and the non-engineering (i.e. rigorous) math I have been exposed to were two proper courses in prob and statistics, 2 courses in real analysis and 2 ...
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0answers
20 views

Stochastic Control, Variable Coefficients

Here is my problem (I'll present it in terms of two maximizations, for simplicity): Given two cost functions, solve simultaneously $V_1(y_1) = {min_{u_1 \in \mathcal{A}(y)}} J(y_1, u_1)$ $V_2(y_1) ...
0
votes
1answer
44 views

Explain what the teacher did - system of ode, control theory.

There are a few things I'm not clear about in her solution and would appreciate a short explanation. We are given the system $\dot{x}=-ax+bu$. with an initial value $x(0)=x_0$. We want to find a ...
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1answer
64 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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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 $H^*$ ...
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1answer
87 views

Finding the input/output representation

I have a system of differential equations which can be written as $R(\frac{d}{dt})w=0$, where $$R(\xi)=\begin{bmatrix}6-5\xi+\xi^{2} & -3+\xi \\ 2-3\xi+\xi^{2} & -1+\xi\end{bmatrix}$$ I want ...
2
votes
1answer
54 views

Controllability of internal subsystem and input-outpout controllability

I am trying to prove that the state equation $\dot{x} = \begin{bmatrix} A_{11} A_{12}\\A_{21} A_{22}\end{bmatrix}x + \begin{bmatrix} B_{1}\\ 0\end{bmatrix}u$ is controllable if and only if the pair ...
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0answers
103 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 ...
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vote
1answer
20 views

Symmetric Matrix Sign equivalence and rank of extenden matrix

I have the following matrix strict inequality: where $X,Y,A,B \in \mathbb{R}^{n\times n}$, $X,Y$ are symmetric matrices $(X=X^T, Y=Y^T)$, there are no conditions imposed on $A$, nor on $B$ ...
0
votes
1answer
43 views

Solving the matrix differential equation $\dot \Delta P(t) = (A + P(t)C^{T}R^{-1}C)\Delta P(t) + \Delta P(t)(A^{T} + C^{T}R^{-1}CP(t))$

Here $P, \Delta P \in \mathbb{R}^{N X N}$ The initial condition $\Delta P(0)$ is given and the dynamics of $P(t)$ is known. $ A,C,Q,R$ are constant matrices of compatible dimensions. Since it is a ...
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votes
1answer
64 views

Singular transfer function matrix and System singularity

I have a linearized dynamic system that can be summarized as: [ΔY] = [A][ΔX] The transfer function matrix, [A], is singular for steady state. My question is ...
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1answer
46 views

Barrier certificate for simple system

I'm trying to prove that for the system $x' \leq -x$, if the system starts with $x \leq 0$, then it is always the case that $x \leq 0$. I'm sure that there are easy ways to prove this, but I'm ...
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49 views

How to get the peak value of a second order system that has non-zero initial condition with step input?

It is known that the peak value of a second order system which is excited with a step input can be expressed by $y(t_p)=1+e^{({-\pi\zeta}/{\sqrt{1-\zeta^2}})}$ In my case, the initial conditions are ...
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1answer
48 views

How to apply Picard-Lindelof existence and uniqueness theorem for autonomous LTI dynamical system $\dot x = Ax$?

In nonlinear dynamical system, we have the picard-lindelof existence and uniqueness principle which guarantees existence of unique solution to problem of the type $\dot x = f(x,t), x(0) = x_o$ ...
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votes
1answer
54 views

Why do high-frequency dynamics quickly go away in a step response?

As we know, a step input hits all the frequencies of a dynamical system. However, my professor told me today that the high-frequency response is only present for a short time at the very start, and ...
3
votes
2answers
78 views

When is a system called linear?

In real time systems / control engineering we have to solve exercises like this: Check if the following systems are linear: 1) $0.2\ddot{x}(t) - (t^2 + 2t -1) x(t) = 3 w(t)$ 2) ...
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votes
1answer
77 views

Asymptotic stability of cascade control

For many systems, it seems to be common practice to stack controllers on top of each other. For example, in a quadcopter, one first builds an attitude controller, then builds a velocity controller ...
13
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139 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 ...
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0answers
18 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 ...
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1answer
50 views

Expression for polynomial of companion matrix

I am rather stuck on an exercise concerning the companion/controllability matrix (the exercise stems from a course in control theory). Given the companion matrix \begin{equation} ...
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1answer
29 views

Finding an admissible control such that $(0,0) \to (1,e^2)$

Verify the controllability of the system $(A,B)$, for $$A=\begin{pmatrix}1&0\\1&-1\end{pmatrix}, \; B=\begin{pmatrix}1\\1\end{pmatrix}$$ Find a control $u \in L([0,1];R)$ such that ...
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0answers
33 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 ...
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29 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) ...
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votes
2answers
98 views

Observability of a System in State Space form

as we all know, each system has infinite state space representations. My question is, whether the observability/controllability (and other common properties) are a properties of the SYSTYEM or ...
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vote
1answer
48 views

$X= \int_0^S e^{(S-s)A} B u(s) ds \Rightarrow X= \int_0^T e^{(T-s)A} B \bar{u}(s) ds$

Consider the ODE system $$X'(t) = AX(t)+Bu(t)$$ where $X(t) \in R^n, \; A \in R^{n \times n} \text{ and } B \in R^{n \times m}$. In control theory, we define the set of states reachable as $$A(0,T) = ...
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1answer
136 views

Functional optimization: maximize a double integral where the functional appears twice

Please help me solve the following optimization problem. Suppose that you have to choose a function $U: [0,1]\mapsto [0,1],$ which must be nondecreasing ($U'\geq 0$) to maximize the following ...
3
votes
1answer
29 views

Find piecewise constant function u for $X'(t)=AX(t) + Bu(t)$ and $X(t)=\begin{pmatrix}10 \\0 \end{pmatrix}$ for some T

Consider the system $$x''(t)=u(t)$$ such that $x(0)=100, \; x'(0)=50$. Find a function $u$ piecewise constant such that $x(T)=0, \; x'(T)=10$ for a time $T$ Using the control theory language, it is ...
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0answers
21 views

Robust Control of a Linear System with Input Saturation

The following system is considered: $$\dot{x}=Ax+Bu+Qw$$ Where, $x\in R^2$ is the state vector,$u\in R^1$ is the control input and $w\in R^1$ is an unmatched disturbance signal. The goal is to ...
0
votes
1answer
64 views

Control Theory: Why is $A+BK$ called a closed loop system?

Given a control system $\dot x = Ax + Bu$ and $y = Cx$. Suppose we use state feedback to create $u = +Kx$ where $K$ is the gain matrix. Subbing into above equation, we have $\dot x = Ax + Bu = Ax + ...
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vote
2answers
124 views

Can someone explain the meaning of asymptotically stability in the following definition?

I am taking this from an online course note. Given a system $\dot x = Ax$, we say that: The origin is asymptotically stable if $x(t) \to 0$ as $t \to \infty \thinspace \forall x(0)$ I am ...
2
votes
0answers
45 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' ...
2
votes
1answer
109 views

What is $H^\infty$ norm and why is it used in control theory?

Can anyone knowledgable elaborate on what exactly is a $H^\infty$ norm and why it is used in control theory instead of some other norms? Thanks!