2.3: Linear Systems

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Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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Next we consider linearity. Roughly speaking, a system is linear if its behavior is scale-independent; a result of this is the superposition principle. More precisely, suppose that \(y_1(t) = F[u_1(t)]\) and \(y_2(t) = F[u2(t)]\). Then linearity means that for any two constants \(\alpha_1\) and \(\alpha_2\),

\[y(t) = \alpha_1 y_1 (t) + \alpha_2 y_2 (t) = F[ \alpha_1 u_1 (t) + \alpha_2 u_2 (t)]\, . \]

A simple special case is seen by setting \(\alpha_2 = 0\):

\[y(t) = \alpha_1 y_1 (t) = F[ \alpha_1 u_1 (t)]\, , \]

making clear the scale-invariance. If the input is scaled by \( \alpha_1\), then so is the output. Here are some examples of linear and nonlinear systems:

\begin{align*} y(t) &= c \dfrac{du} {dt} && \text{(linear and time-invariant)} \\[4pt] y(t) &= \displaystyle \int\limits_{0}^{t} u(t_1)\, dt_1 && \text{(linear but not time-invariant)} \\[4pt] y(t) &= 2u^2(t) && \text{(nonlinear but time-invariant)} \\[4pt] y(t) &= 6u(t) && \text{(linear and time-invariant).} \end{align*}

Linear, time-invariant (LTI) systems are of special interest because of the powerful tools we can apply to them. Systems described by sets of linear, ordinary or differential differential equations having constant coefficients are LTI. This is a large class! Very useful examples include a mass \(m\) on a spring \(k\), being driven by a force \(u(t)\):

\[m y''(t) + ky(t) = u(t)\, ,\]

where the output \(y(t)\) is interpreted as a position. A classic case of an LTI partial differential equation is transmission of lateral waves down a half-infinite string. Let \(m\) be the mass per unit length, and \(T\) be the tension (constant on the length). If the motion of the end is \(u(t)\), then the lateral motion satisfies

\[m \frac {\partial^2 y(t,x)}{\partial t^2} = T \frac {\partial^2 y(t,x)} {\partial x^2} \]

with \(y(t, x = 0) = u(t)\). Note that the system output \(y\) is not only a function of time but also of space in this case.