Tagged Questions
0
votes
0answers
17 views
How to make future trajectory to be prescribed trajectory
i do not understand the role of control theory about how it make trajectory to be prescribed trajectory.
because if future path's parameters are known and input into system, does it mean that it will ...
0
votes
1answer
19 views
which book teaches analysis of nyquist, bode and rlocus diagram
would like to use knots to get a formula for nyquist diagram however, no crossing, and have no experience in analysis of graph related to control, as i have no books mentioning this
and i observe ...
0
votes
1answer
25 views
Show that a set is positively invariant set
Consider the system:
$$\frac{dx_1}{dt} = -x_1$$
$$\frac{dx_2}{dt} = (x_1x_2 - 1)(x_2)^3 + (x_1x_2 - 1 + (x_1)^2)x_2$$
Show that $T= \{x\in \Bbb{R}^2\mathbin{|}x_1x_2 \geq 2\}$ is a positively ...
0
votes
0answers
40 views
Proof - dichotomy(R. Datko) [closed]
I need some help for a proof. I'm studying about dichotomy and I need the proof for the following theorem (my teacher suggested me that is Datko's theorem.)
A system is exponential dichotomic if and ...
0
votes
0answers
12 views
Phase Plane of Digital Systems
I have a nonlinear digital system which can not become differential equation with subtracting the states and deviding them by the time difference on account of being nonlinear. Therefore, I want to ...
1
vote
1answer
64 views
Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems
Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz,
\begin{equation}
\begin{split}
\dot{x}_1 ...
0
votes
0answers
33 views
References that discuss systems of ODEs on the non-negative orthant of $\mathbb{R}^n$?
Does anyone know of any references discussing initial value problems on the non-negative orthant? More specifically, consider the initial value problem
$\frac{dx}{dt}=f(x),\quad\quad ...
3
votes
0answers
102 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 ...
1
vote
2answers
72 views
Stability analysis $\dot{x}=-\gamma x + \alpha$
Suppose that $\alpha(t)$ is an infinitesimal as t goes to infinity, i.e., $\lim_{t\rightarrow\infty}\alpha(t)$=0.
Consider the ODE
$$
\dot{x}(t)=-\gamma x(t) + \alpha(t), \quad \gamma>0
$$
Can we ...
1
vote
1answer
55 views
Choose $\mathbf{B}$ such that eigenvalues are un/controllable
I have the state space system $\dot{x} = Ax + Bu$
with $A = \begin{bmatrix} 1 & -5 \\ -5 & 1\end{bmatrix}$.
I have to find a $B$ vector such that the system has $\lambda = 6$ as controllable ...
0
votes
1answer
224 views
Solving lyapunov equation, Matlab has different solution, why?
I need to solve the lyapunov equation i.e. $A^TP + PA = -Q$. With $A = \begin{bmatrix} -2 & 1 \\ -1 & 0 \end{bmatrix}$ and $Q = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$.
Hence...
...
1
vote
2answers
244 views
Why do we want to know the poles and zeros of a linear system?
I know that I already asked this kind of question on the website, but meanwhile I have much more knowledge about the subject and ready to describe my real problem with enough background information I ...
2
votes
3answers
374 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
0answers
41 views
Question regarding continuous time systems
If $S_u$ is the set of all solutions to the continuous time problem x(dot) = Ax + Bu and $\phi_h :(R^n)^R to (R^n)^N$ be an operator which maps every state x into the sequence (x(0), ...
1
vote
1answer
42 views
Derivation of a formula for discrete time case
Given is the following discrete system
$$\begin{align*}
&x(k + 1) = Ax(k) + Bu(k)\\
&x(0) = x_0\;.
\end{align*}$$
How do we prove that the explicit solution formula for $x(k)$ (analogously ...
1
vote
0answers
89 views
generalizing superfunctions of entire functions
Let $z$ be complex and $x$ real.
Define $f(z,0) = f(z)$ where $f(z)$ is an entire function.
Define $f(z,x)$ as the $x$ th superfunction of $f(z)$.
We know that $f(z,x-1) = f(f^{-1}(z,x)+1)$ where ...
2
votes
3answers
101 views
Weird condition for a transfer function
I am reading a paper that presents the following system, represented by a second-order transfer function:
$G(s) = \frac{K\times(1+0.036s)}{(1+0.0018s)(1+as)}$,
where the gain $K$ is a known ...
1
vote
1answer
66 views
Parametric uncertainty in conditional term of piecewise nonlinear dynamical system
Consider a Hammerstein nonlinear dynamical system of the form
$\mathbf{\dot{x}} = \mathbf{Ax} + \mathbf{Bu}$,
where the non-linearity is in the control term $\mathbf{u}$, and has a piecewise ...
2
votes
1answer
107 views
When are attracting sets invariant?
Consider a control system of the form $\dot{x}(t) = f(x(t), u(t))$ where $u(t)$ is the control input, $x \in \mathbb{R}^{n}$, $u \in \mathbb{R}^{m}$. Assume $f$ is Lipschitz continuous so that the ...
2
votes
0answers
43 views
Estimating the input to a system from a system state
[ Cross-posted to: http://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ]
I have a system for which I have obtained a non-linear time-varying ...
0
votes
2answers
130 views
Feedback characteristics of nonlinear dynamical systems
I am trying to understand the following article:
A. Lahellec, S. Hallegatte, J.-Y. Grandpeix, P. Dumas and S. Blanco, Europhys.
Lett. 81, 60001 (2008).
In the beginning, it was quite easy to follow:
...
1
vote
1answer
202 views
Property of dynamical system and transformation
EDIT2: After some discussion here's the original problem:
Let M be a n-D manifold and $\dot x=F(x)u_1, F\in \mathbb{R}^{n\times m}, u_1 \in \mathbb{R}^{m}$ be a control system evolving on M (F is the ...
0
votes
1answer
68 views
Best method for this example to get from transfer function to state space
I have this system here:
In this example the state space representation $ \frac{dx}{dt} = Ax + bu $ and the corresponding transition matrix $\Phi(t)$ is asked for.
So to get the state space, I ...
0
votes
2answers
109 views
System matrix of a 2nd order state space representation
I am completely stuck on this:
The 2nd order system should be in this form: $\frac{dx}{dt}=Ax$ where A is the system matrix.
$$x(t) = \begin{pmatrix} 2-e^{-t} \\ 1+2e^{-t} \end{pmatrix}$$
$$x(t=0) =: ...
1
vote
1answer
274 views
For what values does this system show BIBO stability?
I got this system state representation:
$$\begin{align}
\frac{dx}{dt} &= ...
1
vote
1answer
90 views
Is this the correct way to get the state space representation of this system?
In this exercise the state space representation of the imaged system is asked for.
$$G_1(s) = \frac{s-1}{s+2} = 1 - \frac{3}{s+2} G_2(s)=\frac{1}{s-1}$$
I can see that $G_1(s)$ is "able to leap" ...
1
vote
2answers
124 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:
$$ ...
1
vote
1answer
299 views
How to obtain a possible state space representation of this 2nd order transfer function?
I have this 2nd order transfer function:
$$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$$
...
0
votes
1answer
168 views
How to obtain the state matrix of this trajectory?
Continuous-time LTI case.
I have a problem getting the state matrix of this trajectory.
One element of the state matrix is known.
$$ A = \begin{pmatrix} a & 4 \\c & d \end{pmatrix}
$$
I ...
