9
votes
2answers
95 views

Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For ...
0
votes
0answers
40 views

Conversion of continuous, linear stochastic system to discrete, LQR/LQG

I have the standard stochastic, linear time varying system $dx(t) = (A(t)x(t) + B(t)u(t))dt + G(t)dw(t) $ with $x(t_0) = x_0$ with quadratic cost $J = x(t_F)^TQ_Fx(t_F) + \int_{t_0}^{t_F}\left( ...
1
vote
0answers
24 views

How to solve Bellman's optimal equation from the first principle

How to solve the following set (finite) of equations $$ v_*(s) = \max_{a\in A(s)} \sum_{s'} p(s'|s,a) [r(s,a,s') + \gamma v_*(s')]$$ $p$ and $r$ functions are given.
1
vote
0answers
24 views

Extension of Schur-Cohn for quadratic matrix equation

Starting from a quadratic in $z\in\mathbb{C}$ with real scalar coefficients $b,c$: $z^2 + bz+c=0$ and using the Schur-Cohn recursion, I can get the following conditions on $a,b,c$ such that $|z|\leq ...
0
votes
0answers
32 views

How to find a transformation matrix T?

(1.) Suppose there is matrix C of dimension "p by n" where p is less then n i.e. p I want to know is there any particular way exist to find transformation matrix T of dimension "n by n" such that ...
1
vote
1answer
29 views

Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
3
votes
1answer
72 views

No repeated eigenvalues or the real part of any eigenvalue is not zero

I have an $n$ x $n$ matrix $M=\begin{bmatrix}-1 & -1\\ \frac{1}{2} & 0 & -\frac{1}{2}\\ & \ddots & \ddots & \ddots\\ & & \frac{1}{2} & 0 & -\frac{1}{2}\\ ...
2
votes
0answers
36 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
3answers
78 views

Rank of the matrix product $C e^{At} B$

Let $A \in \mathbb R^{n \times n}$. Fix $m<n$ and let $B \in \mathbb R^{n \times m}$, $C \in \mathbb R^{(m-1) \times n}$ be two matrices with full rank. What I am interested in is the matrix ...
2
votes
1answer
73 views

How to prove these relations

I have following three basic recurrence relations $$\mathcal U_{k+1}=A(I+\mathcal U_{k}Q)^{-1}\mathcal U_{k}A^{\mathrm T}+G\\ \mathcal V_{k+1}=\mathcal V_{k}(I+Q\mathcal U_{k})^{-1}A^{\mathrm T}\\ ...
2
votes
1answer
39 views

Full rank of the matrix $Ce^{At}B$

Let $A \in \mathbb R^{n \times n}$. Fix $m <n$ and let $B \in \mathbb R^{n \times m}$, $C \in \mathbb R^{(m-1) \times n}$ be two matrices with full rank. I want to find algebraic conditions (which ...
0
votes
1answer
23 views

making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
0
votes
1answer
51 views

Prove that $(A,B)$ is uncontrollable $\Longleftrightarrow$ $\exists P$ $\in$ $\mathbb{R}^{nxn}$, $P \neq 0$: $PA - AP = 0$, $PB=0$

In my course advanced system Theory I had the following question: Prove the following equivalence for the pair $(A,B)$ $\in$ $\mathbb{R}^{nxn}$ x $\mathbb{R}^{nxm}$: $(A,B)$ is uncontrollable ...
1
vote
0answers
35 views

Prove that A is marginally stable iff there exist a $P$ $\in$ $S^n$, $P \succ 0$ such that $A^T + PA \leq 0$

Asymptotic stability, which means that all eigenvalues of A are in the open left half plane is easily proven. See the scan in the attachment. However, in the book the proof for the second case where ...
0
votes
1answer
38 views

Determine value a in system matrix

I'm trying to solve the following problem: "Look at the image of trajectories of a linear, time-invariant system with the form: $\frac{d\textbf x}{dt}=\textbf {Ax}:$ Determine possible eigenvectors ...
0
votes
0answers
36 views

How to compute the Jacobian Matrix of the next system?

I have a little problem with notation and I do not know how to work it out. I have the next system $$ \dot x = A(x)x, $$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has ...
3
votes
0answers
46 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 ...
1
vote
0answers
46 views

analytic solution to structured algebraic Riccati equation

In solving a model I have written down for a research paper, I am left with two Algebraic Riccati Equations, that is I need to solve for $X$ in the equation \begin{align*} X = A^\top (X + XB(R + ...
3
votes
1answer
68 views

Calculate state transition matrix

I have a question to the following problem: "There is a linear, time-invariant System with the form $\frac{d\mathbf{x}}{dt}=\mathbf{A}x$. The Eigenvalues of the matrix $A$ are $s_1=-1$ and $s_2=-2$, ...
2
votes
1answer
60 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: ...
2
votes
1answer
64 views

Relationship between controllability and null space of A, B

For the dynamic system $\dot x = Ax + Bu$ There's a saying that this system is controllable when $Ker(B) \in Ker(A)$, which means that $u$ have the control in every dimension of $x$. I have no ...
2
votes
2answers
67 views

Which constant matrices $A$ have the property that $\dot{x} =Ax+Bu$ is controllable for every non zero $B$?

Which constant matrices $A$ have the property that $\dot{x} =Ax+Bu$ is controllable for every non zero $B$? Now to check controllability I usually check the rank of the matrix $$R = \left( B\ \ \ \ ...
0
votes
1answer
37 views

Help finding the transfer matrix for this system

Find the transfer function of the following system: \begin{eqnarray} \dot{x}_1&=&ax_1+bx_2 + u\\ \dot{x}_2 &=&-bx_1 +ax_2\\ \dot{x}_3&=&cx_3\\ y &=& x_1+x_3 ...
0
votes
2answers
84 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
0
votes
1answer
199 views

State space representation of transfer function $ 1/s $

I have three questions here: 1) I'm looking for a method to compute state-space equations by hand when given a transfer function, so far I think the best I've found is here. 2) Here's an example ...
2
votes
1answer
122 views

Basic example of system controllability

Ok so I'm having a tough time understanding an example in my book, there is a great deal of hand-waving and I'm having a hard time following. I'd like to see the example worked out if possible, with ...
1
vote
2answers
118 views

Matrix representation of a co-domain restriction of a linear operator

Consider the finite-dimensional linear operator: $\mathcal{A}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3},$ with $Ax=y,$ $A=\left[\begin{array}{ccc} 1 & 0 & 1\\ 1 & -2 & -1\\ 0 & 1 ...
0
votes
1answer
47 views

Can linearization of a function around $x=0$ show whether first derivative is positive or negative?

As title says, can linearization of a function $f(x)$ (by the method of taylor series around $x=0$) show whether first derivative of the function ($df/dx$) is positive or negative at $x=0$? And.. ...
3
votes
2answers
874 views

Solution of a Sylvester equation?

I'd like to solve $AX -BX + XC = D$, for the matrix $X$, where all matrices have real entries and $X$ is a rectangular matrix, while $B$ and $C$ are symmetric matrices and $A$ is formed by an outer ...
4
votes
2answers
1k views

Root Locus Diagrams - “Breakaway Point”

Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
0
votes
1answer
95 views

find the general control function [duplicate]

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $A = \begin{pmatrix} 3 & 2 & 2 \\ -1 & ...
3
votes
2answers
135 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
0
votes
0answers
34 views

Determine general form of control function andthus show this coul have been achieved earlier [duplicate]

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
3
votes
3answers
1k views

Poles and Zeros of Linear Systems

This period I follow a course in System and Control Theory. This is all about linear systems $$\frac{dx}{dt}= Ax + Bu $$ $$y = Cx + Du $$ where A,B,C,D are matrices, and x, u and y are vectors. ...
2
votes
1answer
81 views

Quadratic bounds

Consider the quadratic form $x^T(B^TP + PB)x$ where is $B=DA$, where $D$ is diagonal with positive or nonnegative entries, and $A$ is Hurwitz. Now consider a symmetric positive definite matrix $Q$ and ...
1
vote
1answer
165 views

Solution of matrix equation or matrix inequality

When there exist a positive definite solution $S$ of the following matrix equation or matrix inequality: $ SA^{T}+AS+\alpha S-\beta BB^{T}=0 $, that is, what condition on $A,B, \alpha, \beta$ can ...
1
vote
1answer
60 views

Quadraticize a generic function

I have a generic function: $g(s,u)$. Now I want to have a local approximation near the point $(s^{\star}, u^{\star})$ in the quadratic form $$s^{T} Q s + u^{T} R u$$ to apply an optimal control ...
1
vote
1answer
67 views

Modification of the continuous time Lyapunov Equation

For my research I have been working with different continuous-time Lyapunov equations of the form \begin{equation} M R + R M^\text{T} = G \end{equation} where all matrices are real and $n\times n$ ...
4
votes
1answer
280 views

Solving for specific entries in a Lyapunov Equation

Let $A$ be a $2n\times 2n$ real matrix with the following structure \begin{equation} A = \left(\begin{matrix} 0 & -I \\ K & S \end{matrix}\right) \end{equation} with all sub-matrices of size ...
2
votes
0answers
156 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 ...
3
votes
2answers
308 views

Practical physics problem about distance between lines.

I am a software engineer and I’m developing a soccer game. I have a solution for this problem based on Newton law and I’m using Newton method to solve the equation I got. I’m here because I think ...
2
votes
1answer
830 views

Convert a linear difference equation into a controllable state-space model

I have the following linear difference equation (which is an discrete-time SISO ARX model): $y(k)+\sum_{i=1}^{n}a_iy(k-i)=\sum_{i=1}^{n}b_iu(k-i)$ and I need to transform it in an equivalent ...
2
votes
1answer
462 views

Derivation of the Riccati Differential Equation

I am attempting to derive the Riccati Equation for linear-quadratic control. The original equation is: $-\partial V/\partial t = \min_{u(t)} \{x^TQx + u^TRu + \partial V^T/\partial x(Ax + Bu) \}$ $x ...
1
vote
2answers
596 views

Decomposition of a unitary matrix via Householder matrices

If $U$ is unitary, how can I show that there exist $w_{1},w_{2},...,w_{k}\in \mathbb{C}^{n}$, $k\leq n$, and $\theta_{1},\theta_{2},...,\theta_{n}\in \mathbb{R}$ such that $U=U_{w_{1}}U_{w_{2}}\cdots ...
8
votes
3answers
2k views

If matrices $A$ and $B$ commute, $A$ with distinct eigenvalues, then $B$ is a polynomial in $A$

If $A\in M_{n}$ has $n$ distinct eigenvalues and if $A$ commutes with a given matrix $B\in M_{n}$, how can I show that $B$ is a polynomial in $A$ of degree at most $n-1$? I think first I need to show ...