# Co-observability |6 January 2022|tags: math.OC

A while ago Prof. Jan H. van Schuppen published his book Control and System Theory of Discrete-Time Stochastic Systems. In this post I would like to highlight one particular concept from the book: (stochastic) co-observability, which is otherwise rarely discussed.

We start with recalling observability. Given a linear-time-invariant system with , : one might wonder if the initial state can be recovered from a sequence of outputs . (This is of great use in feedback problems.) By observing that , , one is drawn to the observability matrix Without going into detectability, when is full-rank, one can (uniquely) recover (proof by contradiction). If this is the case, we say that , or equivalenty the pair , is observable. Note that using a larger matrix (more data) is redudant by the Cayley-Hamilton theorem (If would not be full-rank, but by adding “from below” it would somehow be full-rank, one would contradict the Cayley-Hamilton theorem.). Also note that in practice one does not “invert” but rather uses a (Luenberger) observer (or a Kalman filter).

Now lets continue with a stochastic (Gaussian) system, can we do something similar? Here it will be even more important to only think about observability matrices as merely tools to assert observability. Let be a zero-mean Gaussian random variable with covariance and define for some matrices and the stochastic system We will assume that is asymptotically (exponentially stable) such that the Lyapunov equation describing the (invariant) state-covariance is defined: Now the support of the state is the range of .

A convenient tool to analyze will be the characteristic function of a Gaussian random variable , defined as . It can be shown that for a Gaussian random variable  . With this notation fixed, we say that is stochastically observable on the internal if the map is injective on the support of (note the ). The intuition is the same as before, but now we want the state to give rise to an unique (conditional) distribution. At this point is seems rather complicated, but as it turns out, the conditions will be similar to ones from before. We start by writing down explicit expressions for , as we find that for the observability matrix corresponding to the data (length) of , a matrix containing all the noise related terms and a stacked vector of noises similar to . It follows that is given by , for . Injectivity of this map clearly relates directly to injectivity of . As such (taking the support into account), a neat characterization of stochastic observability is that .

Then, to introduce the notion of stochastic co-observability we need to introduce the backward representation of a system. We term system representations like and “forward” representations as . Assume that , then see that the forward representation of a system matrix, denoted is given by . In a similar vein, the backward representation is given by . Doing the same for the output matrix yields and thereby the complete backward system Note, to keep and fixed, we adjust the distribution of . Indeed, when is not full-rank, the translation between forward and backward representations is not well-defined. Initial conditions cannot be recovered.

To introduce co-observability, ignore the noise for the moment and observe that , , and so forth. We see that when looking at observability using the backward representation, we can ask if it possible to recover the current state using past outputs. Standard observability looks at past states instead. With this in mind we can define stochastic co-observability on the interval be demanding that the map is injective on the support of (note the ). Of course, one needs to make sure that is defined. It is no surprise that the conditions for stochastic co-observability will also be similar, but now using the co-observability matrix. What is however remarkable, is that these notions do not always coincide.

Lets look at when this can happen and what this implies. One reason to study these kind of questions is to say something about (minimal) realizations of stochastic processes. Simply put, what is the smallest (as measured by the dimension of the state ) system that gives rise to a certain output process. When observability and co-observability do not agree, this indicates that the representation is not minimal. To get some intuition, we can do an example as adapted from the book. Consider the scalar (forward) Gaussian system for . The system is stochastically observable as and . Now for stochastic co-observability we see that , as such the system is not co-observable. What this shows is that behaves as a Gaussian random variable, no internal dynamics are at play and such a minimal realization is of dimension .

For this and a lot more, have a look at the book!