0
votes
0answers
29 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 ...
0
votes
1answer
34 views

Can I always put a system in modal form?

Given the transfer function of a system, can I always put the system in modal form? Are there exceptions?
0
votes
1answer
38 views

Can you find an expression for $F_{12}(t,\tau)$ in terms of $F_{11}(t,\tau)$ and $F_{22}(t,\tau)$?

I have a problem with this...I can not figure out how to solve it..! can you help me? thank you!! Show that if $A(t)$ is partitioned as $$ A(t) = \begin{pmatrix} A_{11}(t) & A_{12}(t) \\ ...
0
votes
1answer
165 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
94 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 ...
2
votes
1answer
99 views

Compute $e^{tA}$

When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(. Here's my problem: For the system of equations: $$\begin{cases} & \text{ } ...
2
votes
1answer
165 views

Exponentials of a matrix

I just was working with matrix exponentials for solving problems in control theory. Suppose $A $ is a square matrix. How can we interpret $A_1 = e^ {\large-A\log(t) }$, where $\log$ is natural ...
1
vote
0answers
161 views

Question about proof of bounded real lemma

My question is: it is possible to proof the bounded real lemma for $H_\infty$ performance with the following procedure? The $H_\infty$ performance is defined as: \begin{align} \parallel ...
3
votes
2answers
700 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 ...
0
votes
1answer
103 views

Solve $AXB=X^\top$

Suppose that $X$ is an $m\times n$ matrix, $A$ and $B$ are $n\times m$ matrices. How can you solve $$AXB=X^\top.$$ Is there an explicit formulation of $X$ in terms of $A$ and $B$ that makes the ...
0
votes
1answer
91 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
134 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 ...
0
votes
1answer
421 views

Calculating the Kalman decomposition of a matrix?

If we are given the matrices $$ A = \begin{bmatrix} -2 & 3 & 4 & 1\\ 1 & 6 & 6 & 3\\ 5 & 6 & 6 & 4\\ 0 & -17 & ...
2
votes
0answers
74 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 ...
5
votes
1answer
407 views

Smith normal form of a polynomial matrix.

I have the following matrix: $P(s) :=$ \begin{bmatrix} s^2 & s-1 \\ s & s^2 \end{bmatrix} How does one compute the Smith normal form of this matrix? I can't quite grasp the algorithm.
1
vote
1answer
152 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
2answers
241 views

State transform from one state space representation to another

I have a state space representation, system S1, in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ This system is transformed with the state transform $$x=T z$$ into the system S2: $$ ...
3
votes
1answer
549 views

How to obtain a possible state space representation of this 2nd order transfer function?

I have this 2nd order transfer function: $$G(s) = \frac{2}{s} + \frac{1}{s+2}$$ And I need to find a possible state space representation in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ ...
1
vote
1answer
272 views

How to obtain the state matrix of this trajectory?

Continuous-time LTI case. I have a problem getting the state matrix of this trajectory. One element of the state matrix is known. $$ A = \begin{pmatrix} a & 4 \\c & d \end{pmatrix} $$ I ...
1
vote
2answers
527 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 ...
4
votes
2answers
753 views

The nature of eigenvectors of a given eigenvalue

I am working on a problem where I have an ($n \times n $) matrix A and an eigenvalue of A, $\lambda$, where $\lambda$ has geometric multiplicity 1. The right and left eigenvectors of A corresponding ...