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I started learning dynamical system a couple of weeks ago and the lecture tried to define what is a SISO system in the last lecture.

The lecture wasn't very clear and did not give a formal definition or some examples,

My question is: What does the "time independence" axiom in SISO systems states ? (an example in addition will be greatly appriciated)

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What axiom you're talking about? In the state-space model $\dot{x} = Ax + Bu, y = Cx + Du,$ IIRC the system is time-invariant iff $A, B, C, D$ are constant w.r.t. time. – user2468 Aug 25 '12 at 2:46
up vote 1 down vote accepted Linear Time Invariant systems are the simplest category to study. The right-hand side (the generator of the dynamics) only depend on state and input but not time, hence they're time invariant.

Whether the systems are defined on continuous time (ODEs) or discrete time (difference equations) - but typically continuous state- they are amenable to Laplace or z transform methods, in other words can turn dynamical systems into algebraic equations.

time invariant systems also include nonlinear systems, which cannot be treated as LTIs unless linearized or through special methods.

Up the complexity ladder are systems where time enters into the right hand side periodically.

If time enters the equations arbitrarily, then the system can be truly complex.

As Rene' says, SISO are just in contrast to MIMO (Google: MIMO antenna). This concept is independent of time-invariance. SISO are scalar i/o systems, while MIMO systems are vector i/o.

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A SISO system is simply a Single Input Single Output dynamical system. I don't see how "time independence" plays a role in this definition. For that matter, I am not sure what you mean by the "time independence" axiom.

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It may be easier for someone to answer your query if you include the definition you are referring to, and subsequently identify the time dependence axiom. – Michael Albanese Sep 20 '12 at 22:42

To add to the above, your lecturer probably included time-invariance in the definition of SISO systems both to be able to later introduce the concept of a transfer function and for historical reasons. As pointed out above, SISO systems have nothing to do with time-invariance (or for that matter linearity); a SISO system solely refers to a mapping from an $1$ dimensional "input signal" $u:[t_0,+\infty)\rightarrow\mathbb{R}$ to a $1$ dimensional "output signal" $[t_0,+\infty)\rightarrow\mathbb{R}$. This is in contrast with a MIMO system that refers to a mapping of an $m$ dimensional signal, $u:[t_0,+\infty)\rightarrow\mathbb{R}^m$, to a $p$ dimensional signal, $y:[t_0,+\infty)\rightarrow\mathbb{R}^p$.

This said, the SISO terminology was first used to refer to linear-time invariant (LTI) SISO systems. Frequency domain descriptions of LTI systems (transfer functions for SISO systems and matrices of transfer functions for MIMO systems) have been around for a long time (1930s or earlier for SISO systems, 1950s/60s for MIMO systems). These critically rely on linearity of the systems, thus no analogue exists for non-linear systems. There have been some attempts to generalise these notions over the last two or three decades to time-varying systems, however these generalisations are no where as widely used and studied. For this reason, some people in control and systems theory use interchangeably the terms SISO and LTI SISO.

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