Observability of Hybrid Systems
Project Summary
Hybrid systems are dynamical systems that exhibit both continuous and discrete behaviors. Such systems have a wide range of applications, including systems biology, computer vision, flight control systems, etc. While there is a vast amount of literature on analysis, verification, stability, reachability, and controller design for hybrid systems, there are relatively fewer results available on observability of hybrid systems.

Observability refers to the study of the conditions under which it is possible to uniquely infer the state of a dynamical system from measurements of its inputs and outputs. Most of our research has concentrated on a class of hybrid systems called Switched Linear Systems (SLSs), which are a collection of linear models connected by switching between a number of discrete states or modes, that is

for discrete time: x(k+1) = A(k) x(k) + B(k) u(k), y(k) = C(k) x(k)

and for continuous-time: x'(t) = A(t) x(t) + B(t) u(t), y(t) = C(t) x(t)

For this class of systems, the observability problem can be stated as follows: given the parameters of the constituent linear systems, find conditions on these parameters under which one can uniquely recover the states (continuous and discrete) of the system. We have investigated these questions for both autonomous and non-autonomous SLS. The sections below summarize our main results.

Observability of Autonomous Jump Linear Systems
In [2], we studied conditions on the observability of discrete-time autonomous SLSs, that is SLSs with no input. In this case, the output of the system can be expressed as Y = O x0 where O is called the observability matrix.

In the case of an autonomous linear system, the observability matrix does not depend on the discrete state trajectory. Hence, the system is observable if and only if the observability matrix of the system is full rank: rank (O) = n where n is the order of the linear system. This is the well known rank test for observability of linear systems.

For autonomous SLSs, the observability matrix depends on the switching between different linear systems, hence O = O(w) . In [2], we proposed several definitions of observability, depending on whether we are interested in recovering the continuous state, the discrete state, or both. We showed that, under the condition that consecutive switches are separated by a minimum dwell time of 4n, the observability of a SLS can be characterized by simple rank tests on the constituting linear systems. Generally, these rank tests are of the form: rank( [O ,O'] ) = 2n for all i neq j = 1, ... , n. In other words, this condition requires that the observability matrices of any pair of linear systems be full rank.

In [1], we obtained similar results for the observability of continuous-time switched linear systems. The main difference between continuous-time and discrete-time, is that in the continuous case the separation between consecutive switches can be arbitrarily small, while in the continuous case the minimum dwell time depends on the order of the system.

Observability of Non-Autonomous Jump Linear Systems
We are currently in the process of extending our research to a more general class of SLSs with inputs [3]. In this case, the input-output map is of the form Y=O x0+T U. We propose several definitions of observability, which enable us to recover only the discrete states (Mode Observability) or both the continuous and discrete states (Observability). In addition, our definitions consider the possibility of recovering the states of the system based on finding some inputs ([Mode] Observability) or recovering the states for all inputs (Strong [Mode] Observability). Then for each definition, we derive conditions on the structural parameters of the constituent linear systems which make it possible to uniquely recover the states of the system.

In the case of an autonomous linear system, the matrix T does not depend on the discrete state trajectory. Hence, we may substract the effect of the input from the output. As a consequence, the system is observable if and only if the observability matrix of the system is full rank: rank (O) = n where n is the order of the linear system. This is the same condition as in the autonomous case. In the case of SLSs, the T matrix depends on the discrete state, T = T (w), thus the input does have an effect on whether the continuous and the discrete states can be uniquely determined. In this case, our observability conditions are simple rank tests which exploit the geometry of the observability subspaces and can be related to the Markov parameters of the individual linear systems.

For example, in mode observability, which refers to the possibility of uniquely recovering the discrete state for some inputs, we need that at least one of the following conditions hold:

rank ( [O ,O'] ) = 2n, or rank ( [O ,O' ,T -T' ] ) > rank ( [O ,O'] ).

Another example is strong observability, which in general means uniquely recovering both continuous and discrete states of the system. In this case, we need the following two conditions be simultaneously satisfied:

rank ( [O ,O'] ) = 2n, and rank ( [O ,O' ,T -T' ] ) = rank ( [O ,O'] ) + rank ( T -T' ).

We have also investigated conditions under which we can determine the switching times for some inputs (Detection) or uniquely recover the switching times for all inputs (Strong Detection). We show that the conditions for [Strong] Detection are weaker than the ones obtained for the [Strong] Observability. Similar conditions can be obtained in the case of continuous-time SLSs.

Publications
[1]
Hybrid Systems: Computation and Control, pages 526-539, Springer Verlag, 2003.
[2]
IEEE Conference on Decision & Control, pages 3614-3619, 2002.
[3]
Submitted to IEEE Conference on Decision & Control, 2008.