Linear Systems 101 - Part 1 - (De)motivation for Continuous-time Systems


I was thinking that it would be awesome to publish some material regarding Control Systems and the subject I study, Markov Jump Linear Systems. For the former subject, there is a lot of material on the web. The latter subject can be quite complex and difficult to understand for a beginner. Here, I am gonna try to write brief posts explaning these subjects (and many more) in a nutshell.

So, let us begin with a basic example to motivate this subject. Consider a RC circuit and the associated differential equation of the voltage on the capacitor:

\begin{equation}RC \dot v_c  + v_c  = v_i\end{equation}

where vi is an input signal applied to the circuit. This equation is easily obtained through Kirchhoff's Law , Ohm's Law and the equation of the current across the capacitor. We assume here that R and C are constant. Let us assume that the capacitor is initially charged and that no input is applied to the system so that 

\begin{equation}v_c(0) = v_{c0}, v_i(t) = 0, t \geq 0.\end{equation}

Thus, we can directly get the response by integration so that

\begin{equation} d v_c / v_c = - dt/RC \implies v_c(t) = e^{-t/RC}v_{c0}, t \geq 0 , \end{equation} where we impose throughout this series that all signals are 0 for t < 0.

Here let us introduce some basic definitions:

  • vi is the system input: an external signal which modifies the system behavior
  • vc is the system state: an internal signal which describes the system behavior
If we are able to measure vc, we say that vc is the measured ouput (oh yeah?). If instead we could only measure the cap current ic, then ic would be our measured output. It depends on the available information.

Whenever we set the system input to zero, as done in this example, then the response vc(t) is called natural (or unforced) response. In this case, we are interested in knowing the state trajectory  vc(t) (or path, or values) from t=0 on. Will it increase indefinitely (an unstable system)? Will it decrease to 0 (a stable system)? Will it start oscillating? As it turns out, we can know that by analysing the argument of the exponencial function in vc(t). Since R and C are positive constants, and the time is nonnegative, we note that

\begin{equation}v_c(t) = \frac{v_{c0}}{e^{t/RC}} \to 0 \text{ as } t \to \infty\end{equation} 

That is pretty cool. This equations tells us that a charged capacitor can be discharged through a resistor, as we do in practice! We call the steady state response of a system as the behavior when t gets ``big enough (we may discuss what that means when t gets big enough for us). 

But what is in between t=0 and t ``big enough''? This region is called the transient response of the system. It can get quite complicated depending on the system. For us, as we can see from vc(t), it is just an exponential decay whose timing depends on the so-called time constant:
\begin{equation}\tau = RC\end{equation}
The time constant dictates how fast (or slow) the system goes from its initial condition to the origin. For the interested, you can have a look at the curves in RC circuit or plot this voltage on Wolfram Alpha.

Can I use these definitions in other types of systems? Yes, sure you can with some care. As it turns out, the analysis can become a little more complicated as we are going to see on the next posts. What we can postulate here right now is that any unforced first order scalar linear system is described by
\begin{equation}\dot x = -a x, x(0) = x_0,\end{equation}
whose time response is given by
\begin{equation}x(t) = e^{-at} x_0, t\geq0\end{equation} 
In the next post, we are going to deepen our understanding on first order linear systems. BTW, this ``first ordem'' I just mentioned is due to the highest derivative degree of the differential equation describing the system, which in this case is one.


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