Tagged Questions
0
votes
1answer
21 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
130 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 ...
3
votes
0answers
78 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
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 ...
2
votes
3answers
360 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
38 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
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 ...
1
vote
1answer
48 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
50 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
159 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
148 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
223 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 ...
1
vote
1answer
344 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 ...
1
vote
1answer
225 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
355 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 ...
7
votes
2answers
723 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 ...
