I have been doing this exercise for long time, and I am never satisfied by the result. Now, I try again. What is observability? As process controllers, we all have in mind the definitions of the books. For example we, process controllers, can easily recall, by heart, the definition for linear systems. Given a dynamic model M describing a process (how the hell do I put equations in this blog?)
(being x the vector state of dimension N, u the input, and y the vector of the measurements with dimension m, for simplicity m=1), the system is observable with the model M and the measurement y if the observability matrix O, dim(O)=NxN
is full rank (i.e. rank(O)=N). And generally, the story ends with the students calculating O, then its rank. If this rank is full, then they may implement the kalman filter borrowing some Matlab routine. For nonlinear systems, the story is similar but, instead of having the matrices A, B, C, we have nonlinear maps, which make us invoke the Lie derivatives for the calculation of O. And again, if O is full rank, ok, observability holds.
But then again, what is observability? What does it tell? Well, I would say that observability is a property of the model that, if satisfied, allows the available process measurements together with the model, to fully reconstruct the state of the system. I try to make the concept more clear making an example. Let me take the non isothermal CST reactor with first order rate for the reaction A–>B (Fig. 1)
Fig. 1 The CSTR
described by the model equations:
You may remember that its plane phase, in case of multiple equilibrium point (E1, E2, E3) looks a little bit like Fig. 2.
Fig. 2 (Not very accurate-) Plane phase of the CSTR with multiple steady states E1, E2, E3.
Let me assume that the system is travelling in a trajectory that, from the initial condition x0=[T0, C0], is going towards the equilibrium point E1 (Fig 3),
Fig. 3 The system is approaching the stable steady state E1 following this state trajectory
or in other words: concentration and temperature in the tank are varying until the stable steady state E1 is reached. At the time t, the state of the system is described by the coordinates x(t)=[T(t),C(t)] (Fig. 4).
The state of the system at the time t
If only temperature measurements are available, the output map is y=T. In the plain phase y=T is a straight line (Fig. 5). This measurement alone gives only partial information about the state. Infact, it does not say anything about concentration.
Fig. 5 In the plane phase, y=T is an horizontal line.
However, if we take the first Lie derivative (the CSTR model is nonlinear) of the output map, we can draw a second curve in the phase plane ( ẏ = ∂h/∂x= fT (T,C) ) (Fig. 6). One can note that, in order to draw this curve, the model equation fT is required.
Fig. 6 Another curve can be drawn in the phase plane by using the information given by fT (T,C)
If observability holds, the information carried by this second curve ẏ = fT (T,C) significantly differs from the one deriving from the first straight line y=T (i.e. ẏ = fT (T,C) is not coincident nor parallel to the straight line). If observability holds, the intersection between the two curves (Fig. 7) represents the state of the system at the time t (Fig.8):
Fig. 7 The intersection between y=T and ẏ = fT (T,C)
Fig, 8 The intersection between y=T and ẏ = fT (T,C) and the state trajectory
For a two dimensional system (like the considered CSTR), two different curves are necessary to identify the state of the system. This is why the 2×2 observability matrix O is required to have rank 2. If O was rank deficient, it would mean that the second curve is parallel or coincident to the first one, and the intersection point (i.e. the state of the system) does not exist. In other words the system is not observable.
I know that there are more rigorous definitions of observability, but they are too abstracted for me. I much more like to think about observability in the way I just described. I need to thank prof. Jesus Alvarez, who taught me the observability concepts this way during my stay in Mexico City in 2013.
Coming next: detectability!