Logarithmic Concave Observers

Abstract: The problem of (online) state estimation for dynamical systems arises frequently in control. The well known Kalman Filter comprises a coherent theory for the case of linear systems with Gaussian noise, but as soon as either condition is relaxed the picture becomes much less clear. This thesis investigates the case when process disturbances and measurements are relaxed from Gaussian to log-concave. The range of systems that can be analyzed is broadened while still retaining enough structure that many desirable properties are preserved. Strongly log-concave functions are introduced as a means to quantify the Gaussian-like properties of log-concave functions. The main contribution of the thesis is two fundamental theorems, one giving a bound on covariance and the other describing how (strong) log-concavity is preserved and propagated. Applying the theorems to log-concave observers, they are found to have much in common with the Kalman Filter. It is shown that a Kalman Filter can be constructed that gives a conservative bound on the error covariance of the log-concave Bayesian Observer. Event based control is one case where measurements are far from (uncorrelated) Gaussian, but often log-concave. An example of control and state estimation for such a system is pursued throughout the thesis. Using proper consideration a Kalman Filter is found that gives a reasonable approximation of the optimal log-concave observer.