System Properties

Here are some shortcuts to determining some properties of systems.

Linearity

The simplest formal way I know to check for linearity of a system is to calculate the following:

$$\begin{align*} x_1&\rightarrow y_1\\ x_2&\rightarrow y_2\\ ax_1+bx_2&\rightarrow???\end{align*}$$

If the latter leads to $$ay_1+by_2$$ for all constants $$a,b$$ then the system is linear.

Integrals and derivatives are inherently linear since:

$$\begin{align*} \int \left(ax_1(\Theta)+bx_2(\Theta)\right)\,d\tau,\end{align*}$$

where $$\Theta$$ is whatever the argument of $$x$$ is in the integral, can be written as the linear combination:

$$\begin{align*} a\int x_1(\Theta)\,d\tau+b\int x_2(\Theta)\,d\tau\end{align*}$$

and

$$\begin{align*} \frac{d}{dt}\left(ax_1(\Theta)+bx_2(\Theta)\right)\end{align*}$$

can be written as the linear combination:

$$\begin{align*} a\frac{d}{dt}\left(x_1(\Theta)\right)+ b\frac{d}{dt}\left(x_2(\Theta)\right)\end{align*}$$

Things that are inherently nonlinear are:

• Additive constants like $$y(t)=x(t)+1$$
• Powers of $$x$$ or $$y$$ other than the first like $$y(t)=x^2(t)$$ or $$y(t)=x(t)-y^3(t-1)$$ of $$y(t)=1/x(t)$$
• Products of $$x$$ and $$y$$ like $$y(t)=x(t)\,y(t-1)$$
• Trig functions of the input or output such as $$y(t)=\cos\left(x(t)\right)$$ or $$\sin\left(y(t)\right)=x(t)$$

Note that time scales, time shifts, and multiplicative constants (in addition to integrals and derivatives) are inherently linear, though if any of the nonlinearities above is included, the system becomes nonlinear. For instance, $$y(t)=\int_0^tx^2(\tau)\,d\tau$$ is nonlinear, not because of the integration but because of the square on $$x$$.

Stability

The formal check for stability is to determine, if $$x(t)\leq M$$ for all time $$t$$ and some finite constant $$M$$, does $$y(t)\leq N$$ for all $$t$$ for some finite (potentially different) constant $$N$$.

Inherently unstable systems include:

• Some inverses $$y(t)=\frac{1}{x(t)}$$
  • Though not all - for example, $$y(t)=\frac{1}{x^2(t)+1}$$ is stable for real-valued $$x(t)$$
• Certain trig functions (tan, sec, csc, cot)
• Derivatives (infinite in the presence of a step)
• Functions of $$t$$ not in the argument of $$x$$ or $$y$$ can be problems:
  • $$y(t)=t\,x(t)$$ (linear, time-varying, non-stable)
  • $$y(t)=x(t)+t$$ (nonlinear, time-varying, non-stable)

They may not always be a problem though:

• $$y(t)=\cos(t)\,x(t)$$ (linear, time-varying, stable)
• $$y(t)=x(t)+\sin(t)$$ (nonlinear, time-varying, stable)

• Integrals with increasing distance between the limits of integration; that is, something like$$\begin{align*} y(t)=\int_0^tx(\tau)\,d\tau\end{align*}$$ is unstable since the integration window grows with t, whereas$$\begin{align*} y(t)=\int_{t-2}^{t+2}x(\tau)\,d\tau\end{align*}$$ is stable since the integration window is always 4 units.

Time shifts, time scales, scalar multiples, and additive constants are all inherently stable