Questions tagged [systems-theory]
[STUB] The study of abstract organization of phenomenon, usually under the assumption of self-regulatory feedback.
19
questions
0
votes
1answer
17 views
Choosing desired self-conjugate closed-loop pole locations
If i am given:
A = [1 -2; 0.5 -1];
B = [2; 2];
C = [-1 1];
D = 0;
\zeta = 0.707;
\omega_0 = 2 rad/s;
I need to design a state controller for the system such that ...
2
votes
1answer
39 views
How to show that the dimension of the output subspace $D_K$ is $m$?
I have some problems with a homework question, and would appreciate if I can get some help.
Consider a single-input-single-output system
$$\dot{x} = Ax + Bu \\
y = Cx$$
and its corresponding transfer ...
2
votes
1answer
83 views
What is the finite eigenvalue of a controlled system as $\epsilon \rightarrow 0$?
Consider a state space system (A, B) as follows:
$$A = \begin{bmatrix}
a_1 & a_2 \\
-a_2 & 0 \\
\end{bmatrix},
B= \begin{bmatrix}
0 \\
1 \\
\end{bmatrix}$$
where $a_1$ ...
0
votes
0answers
35 views
How to get the transfer function out of a Nyquist diagram
If there is given a Nyquist diagram, what are the steps to find out what kind of transfer function belongs to it?
I have an example of a diagram with a multiple choice question, but would really like ...
0
votes
1answer
139 views
Is a full rank square matrix necessarily a positive definite matrix?
(Skagestad, 2005) states the following conclusion in page 128.
"The system $(A, B)$ is state controllable if and only if the Gramian matrix $W(t)$ has full rank (and thus is positive definite) for ...
2
votes
1answer
142 views
Property of interconnected feedback systems
In the figure you can see the statespace form of a feedback interconnection system.
Very quick question: is there a reason they have taken $D_1=0$ and $D_2=0$?
It makes workings a lot easier but I ...
3
votes
1answer
332 views
Controllability of cascade connection of two systems
I have two linear control systems that are represented by their state space models
$$\left(
\begin{array}{c|c}
A_1 & B_1 \\
\hline
C_1 & D_1 \\
\end{array}
\right),
\left(
\begin{array}{c|c}...
1
vote
2answers
2k views
What is the significance of unstable zeros in control system?
I am new to Systems Theory, I recently learnt about inertia of a polynomial which gives the number of stable, antistable and imaginary zeros. I am just trying to understand if there is a point in ...
1
vote
1answer
1k views
Relationship between $L_\infty$ and $L_2$ norms in time-domain and frequency-domain
To find a way of bringing together a minimization problem defined in the time-domain with systems described in the frequency domain via transfer functions, and thus bridge the gap between norms ...
1
vote
0answers
37 views
Is there a word for taking a statistical model, and working backwards to the algebraic proof?
As an example, I'm trying to work backwards from the statistics generated from the Monty Hall Problem and figure out the variables and their relation to each other.
The specifics of the Monty Hall ...
2
votes
1answer
155 views
Do equal eigenvalues in a SS matrix imply algebraic equivalence?
In state space (SS) theory, the concept algebraic equivalence means, for matrix $A\in\mathbb{R}^{n\times n}$, column vector $B\in \mathbb{R}^n$ and row vector $C\in \mathbb{R}^n$, there is an ...
1
vote
2answers
7k views
find out that system is time-invariant or not
Is this system linear and time-invariant?
$$y(t) = ā3x(2t ā 2) + x(t)$$
I found this that is not time-variant but I am not sure.
4
votes
3answers
203 views
Dynamical systems, causality and derivatives order
Talking about input/output representation of a dynamical system, the professor said that the equation(s) involved must satisfy this condition in order for the system to be qualified as "causal":
...
0
votes
1answer
184 views
How to show that Im R is an A-invariant subspace?
Suppose we have $A=\begin{pmatrix}4 & -4 & 2 \\ 3 & -3 & 2 \\ -3 & 2 & -3 \end{pmatrix}$, and $\operatorname{Im} R = \operatorname{basis}\{ \begin{pmatrix}1 & 4 & -4\...
0
votes
1answer
28 views
Dynamic real-time system problem
I am struggling with a systems theory problem, the task is as follows:
u(t) -> H(s) -> y(t)
H(s) being the transfer function
$$
H(s) = H(s) = \frac{s+1}{s(s+2)^{2}}
$$
$$
u(t) = e^{-5t}
$$
So ...
3
votes
1answer
308 views
Noncausal dynamical system
The differential equation
$$a_ny(t)^{(n)} + \dots + a_0y(t)^{(0)} = b_mu(t)^{(m)} + \dots + b_0u(t)^{(0)} $$
with $a_i,b_i \in \mathbb{R}$ and $y,u:\mathbb{R}\to\mathbb{R}$ describes a time-...
0
votes
1answer
866 views
Equivalence of controllability and reachability in discrete time systems
I am trying to prove that the statements;
$\Sigma_d$ is controllable,
$\Sigma_d$ is reachable,
The pair $(A,B)$ is controllable (in other words $<A|\ im\ B>=\mathcal{X}).$
are equivalent for ...
1
vote
2answers
258 views
determine if the system is time-invariant
I need to determine if the following system is time-invariant or not, and I'm a bit unsure about it.
$$y(t)= \int_{-\infty}^{t-2} \tau \cdot x(2\tau)d\tau $$
3
votes
2answers
7k views
How do I find transfer function of a discrete-time system when its state-space form is given?
I read this and this Wikipedia pages, but both of them are explaining continuous-time systems. My question is about discrete-time case.
For example, given the state-space equations of the second ...
