Tagged Questions
1
vote
0answers
28 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
129 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
61 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
49 views
find the general control function
Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference
where:
$A = \begin{pmatrix}
and:
So far I have caculated the controlability matrix to be
$
C
the system is ...
1
vote
2answers
83 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
74 views
Calculating the Kalman decomposition of a matrix?
If we are given the matrices
$$
A = \begin{bmatrix}
-2 & 3 & 4 & 1\\
1 & 6 & 6 & 3\\
5 & 6 & 6 & 4\\
0 & -17 & ...
2
votes
0answers
46 views
Condition for controllability of matrix pairs
I have a question regarding how determine if a matrix pair is controllable.
If $A$ and $b$ are given by
A = [$\lambda_1 0 0 ...., 0 \lambda_20 0 0,...,00...\lambda_n$] (matrix) and $b = [b_1 b_1 ...
2
votes
0answers
110 views
Smith normal form of a polynomial matrix.
I have the following matrix: $P(s) :=$
\begin{bmatrix}
s^2 & s-1 \\
s & s^2
\end{bmatrix}
How does one compute the Smith normal form of this matrix? I can't quite grasp the algorithm.
1
vote
1answer
107 views
Solution of matrix equation or matrix inequality
When there exist a positive definite solution $S$ of the following matrix equation or matrix inequality:
$
SA^{T}+AS+\alpha S-\beta BB^{T}=0
$,
that is, what condition on $A,B, \alpha, \beta$ can ...
0
votes
0answers
251 views
Conversion of PID controller components with state feedback into single transfer function and discrete state-space form
I've been wrestling with this problem for about a week now, as a part of a year-long project. We're designing a controller for a specific reactor based on a model. After looking at this for a while, I ...
1
vote
2answers
120 views
State transform from one state space representation to another
I have a state space representation, system S1, in the form of:
$$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$
This system is transformed with the state transform $$x=T z$$
into the system S2:
$$ ...
1
vote
1answer
286 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:
$$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$
...
0
votes
1answer
163 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
354 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
$U=U_{w_{1}}U_{w_{2}}\cdots ...
4
votes
2answers
346 views
The nature of eigenvectors of a given eigenvalue
I am working on a problem where I have an ($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 ...
