Questions tagged [systems-theory]

[STUB] The study of abstract organization of phenomenon, usually under the assumption of self-regulatory feedback.

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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 ...
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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 ...
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117 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}...
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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 ...
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1answer
647 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 ...
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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 ...
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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 ...
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find out that system is time-invariant or not

Is this system linear and time-invariant? $$y(t) = −3x(2t − 2) + x(t)$$ I found this that is not time-variant but I am not sure.
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163 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": ...
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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\...
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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 ...
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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-...
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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 ...
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158 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. $$y(t)= \int_{-\infty}^{t-2} \tau \cdot x(2\tau)d\tau $$
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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 ...