Linear system

In control engineering, a linear system may be thought of as a dynamical system which relates a certain set of signals (the input signals) to another set of signals (the output signals) in a linear fashion. Here an input signal refers to a signal that can be interpreted as entering the system while an output signal is one which can be interpreted as leaving the system. This is the definition of a linear system in an input output formalism in which signals are assumed to always be classifiable as either an input or an output. Since it is not clear that this distinction between signals is generic to every system, it is debatable whether the input output formalism is the most appropriate way of thinking about systems. This motivated the development of an alternative formalism known as the behavioral appoach to systems theory which focuses on trajectories of the system rather than on inputs and outputs. Nonetheless, there is a similar definition of a linear system within the latter formalism.

The linearity property of a linear system makes it more amenable to mathematical analysis. For instance, the linear equations describing the system can often be explicitly solved. Therefore, it is the most extensively studied type of system in the literature and this has led to the development of key system theoretic concepts, such as observability, controllability, detectability and stabilizability, which were subsequently generalized to other types of systems, such as nonlinear systems.

Formal definition

In this section a formal definition of a linear system will be given in the input output formalism.

Let $T\subset \mathbb {R}$, $U\subset \mathbb {R} ^{m}$ and $Y\subset \mathbb {R} ^{n}$, where $m,n$ are positive integers. Denote ${\mathcal {U}}$ as the set of admissible functions $u:T\rightarrow U$ which can act as input signals, and denote ${\mathcal {Y}}$ as the set of admissible functions $y:T\rightarrow Y$ which can act as output signals. Admissibility here means that the functions satisfy any additional conditions which may be dictated by the system (such as differentiability or integrability conditions). Then a linear system L is simply a linear map $L:{\mathcal {U}}\rightarrow {\mathcal {Y}}$.

A map L from ${\mathcal {U}}$ to ${\mathcal {Y}}$ is said to be linear if $L(ax+by)=aL(x)+bL(y)$ for all $a,b\in \mathbb {R}$ and for all $x,y\in {\mathcal {U}}$

References

1. H. Kwakernaak and R. Sivan, Modern Signals and Systems, Upper Saddle River, NJ: Prentice Hall, 1991.
2. W. Brogan, Modern Control Theory (3 ed.), Upper Saddle River, NJ: Prentice Hall, 1991.
3. K. Ogata, Modern Control Engineering (2 ed.), Lebanon, IN: Prentice Hall, 1990.
4. J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems Theory: A Behavioral Approach (2 ed.), ser. Texts in Applied Mathematics Vol. 26, Springer, 2007.