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 $$x$$.

Time Invariance

Here's the simplest way I know how to formally check for time invariance:

• First, take every $$t$$ in the expression and replace it with $$t-t_d$$
• Second take every $$y(\Theta)$$ and $$x(\Phi)$$ (where $$\Theta$$ and $$\Phi$$ just represent whatever the argument currently looks like) and replace their arguments with $$y(\Theta-t_d)$$ and $$x(\Phi-t_d)$$
• If those two expression do not represent the same thing then the system is time varying.

For example:

$$\begin{align*} y(t)&=\int_t^{t+5}x(\tau)\,d\tau\\ 1:&~t\rightarrow t_d & y(t-t_d)&=\int_{t-t_d}^{t-t_d+5}x(\tau)\,d\tau\\ 2:&~y(\Theta)\rightarrow y(\Theta-t_d), x(\Phi)\rightarrow x(\Phi-t_d) & y(t-t_d)&=\int_t^{t+5}x(\tau-t_d)\,d\tau \end{align*}$$

These may look a little different, but if you notice that the limits of the integral will become arguments of $$x(\tau)$$ you can see that they are really the same expression. With a transformation of variables in the second expression of:

$$\begin{align*} \gamma&=\tau-t_d\\ d\gamma&=d\tau\end{align*}$$

we can get the second expression to be

$$\begin{align*} 2:&~y(\Theta)\rightarrow y(\Theta-t_d), x(\Phi)\rightarrow x(\Phi-t_d) & y(t-t_d)&=\int_{\gamma=t-t_d}^{\gamma=t-t_d+5}x(\gamma)\,d\gamma\\ ~&~ & ~&=\int_{\gamma=t-t_d}^{\gamma=t+5-t_d}x(\gamma)\,d\gamma \end{align*}$$

which is the same as the first expression.

On the other hand, let's look at:

$$\begin{align*} y(t)&=\int_{0}^{t}x(2\tau)\,d\tau\\ 1:&~t\rightarrow t_d & y(t-t_d)&=\int_{0}^{t-t_d}x(2\tau)\,d\tau\\ 2:&~y(\Theta)\rightarrow y(\Theta-t_d), x(\Phi)\rightarrow x(\Phi-t_d) & y(t-t_d)&=\int_{0}^{t}x(2\tau-t_d)\,d\tau \end{align*}$$

With a transformation of variables in the second expression of:

$$\begin{align*} 2\beta&=2\tau-t_d\\ \beta&=\tau-t_d/2\\ d\beta&=d\tau \end{align*}$$

we can get

$$\begin{align*} 2:&~y(\Theta)\rightarrow y(\Theta-t_d), x(\Phi)\rightarrow x(\Phi-t_d) & y(t-t_d)&=\int_{\beta=-t_d/2}^{\beta=t-t_d/2}x(2\beta)\,d\beta \end{align*}$$

which is not the same as the first expression; this system is time-varying (for two reasons: there is a specific time involved with the 0 limit and there is a time scaling other than unity with the $$2\tau$$)

Main thing to look for that identify a system as time-varying are:

• $$t$$ that end up outside of the arguments of $$x()$$ or $$y()$$
• scalings on $$t$$ other than unity (any negative scaling and any positive scaling other than 1)

Note that time shifts are inherently time invariant.

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) however may be stable if the model is a differential equation relating $$y(t)$$ and its derivatives to $$x(t)$$ to its derivatives.
  • $$y(t)=\frac{dx(t)}{dt}$$ is unstable
  • $$\frac{dy(t)}{dt}+y(t)=x(t)$$ is stable
  • $$\frac{dy(t)}{dt}-y(t)=x(t)$$ is unstable
  • $$\frac{dy(t)}{dt}+y(t)=2\frac{dx(t)}{dt}+x(t)$$ is stable
  • $$\frac{dy(t)}{dt}-y(t)=2\frac{dx(t)}{dt}+x(t)$$ is unstable
• 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

Memorylessness

A system is memoryless if $$y(t)$$ depends only on $$x(t)$$ at time $$t$$ and does not need information about either $$y(t)$$ or $$x(t)$$ in the past or future. Note that if there are functions of $$t$$ that are not arguments of $$y$$ or $$x$$ those do not impact memorylessness.

• Memoryless systems:
  • $$y(t)=Kx(t)$$
  • $$y(t)=Kx(t) + t + 1$$
  • $$y(t)=\cos(t+1)\,x(t)$$
• Systems with memory:
  • $$y(t)=\int_{-\infty}^{t}x(\tau)\,d\tau$$
  • $$y(t)=x(t-1)$$
  • $$y(t)=x(t)+y(t-1)$$
• Jury is still out:
  • Derivatives...

Causal

A system is causal if $$y(t)$$ depends only on $$x(t)$$ and/or $$y(t)$$ at or before time $$t$$ and does not need information about either $$y(t)$$ or $$x(t)$$ in the future. Note that if there are functions of $$t$$ that are not arguments of $$y$$ or $$x$$ those do not impact causality. All memoryless systems are causal by definition. All non-causal systems have memory by definition. Causal systems may or may not also be memoryless.

• Causal systems:
  • $$y(t)=Kx(t)$$
  • $$y(t)=Kx(t) + t + 1$$
  • $$y(t)=\cos(t+1)\,x(t)$$
  • $$y(t)=\int_{-\infty}^{t}x(\tau)\,d\tau$$
  • $$y(t)=x(t-1)$$
  • $$y(t)=x(t)+y(t-1)$$
• Non-causal systems:
  • $$y(t)=\int_{-\infty}^{t+1}x(\tau)\,d\tau$$
  • $$y(t)=\int_{0}^{t}x(\tau)\,d\tau$$
  • $$y(t)=x(t+1)$$
  • $$y(t)=x(t+1)+y(t-1)$$
• Jury is still out:
  • Derivatives...again...
    • $$\frac{dy(t)}{dt}=\lim_{h\rightarrow 0}\frac{y(t)-y(t-h)}{h}$$ looks causal
    • $$\frac{dy(t)}{dt}=\lim_{h\rightarrow 0}\frac{y(t+h)-y(t)}{h}$$ looks non-causal
    • $$\frac{dy(t)}{dt}=\lim_{h\rightarrow 0}\frac{y(t+h)-y(t-h)}{2h}$$ looks non-causal