Linear Time-invariant Systems

Linearity

A system is linear if it can be defined by linear differential equations. In particulars, if the functions $\mathbf{f}$ and $g$ in its state-space model are linear functions of the state variables $\mathbf{x}(t)$ and input $u(t)$.

Time-Invariance

A system is said time-invariant if it can be by differential equations with constant coefficients. In particular, if the functions $\mathbf{f}$ and $g$ in its state space model do not depend on the time $t$ explicitly.

• See some examples in Time-invariant and time varying systems

Assume that a time-invariant system has zero initial conditions, and zero input generates zero output. If input $u(t)$ produces output $y(t)$, then input $u(t-\tau )$ for all $\tau \ge 0$.

Form

A LTI system has the following form of state space model:

$$\begin{array}{rl}\dot{\mathbf{x}}& =\mathbf{Ax}(t)+\mathbf{b}u(t)\\ y(t)& =\mathbf{cx}(t)+du(t)\end{array}$$

where $\mathbf{A}\in {\mathbb{R}}^{n\times n}$, $\mathbf{b}\in {\mathbb{R}}^{n\times 1}$, $c\in {\mathbb{R}}^{1\times n}$, and $d\in \mathbb{R}$ are constant matrices.

State Machine Interpretation

Linear time-invariant systems can be seen as a state machine where $S={\mathbb{R}}^m,X={\mathbb{R}}^1,Y={\mathbb{R}}^n$, and $f$ and $g$ are linear functions of their input. In discrete time, they can be described as a linear difference equation, like

$$y[t]=3y[t-1]+6y[t-2]+5x[t]+3x[t-2]$$

where $y[t]$ is $y$ at time $t$. LTI systems can be implemented using state to store relevant previous input and output information.

Recurrent Neural Networks are a lot like a non-linear version of LTIs.