(373k) Analysis of the Transient and Limit Distributions of Stochastic Linear Systems with Zonotopic Support Sets

Authors: 
Villanueva, M. E., ShanghaiTech
Houska, B., ShanghaiTech
The theory of stochastic linear discrete-time processes has been of paramount importance in control theory ever since its early days (Kalman, 1960; Bertsekas and Shreve, 1978) . For instance, it provides the theoretical basis for the classical linear quadratic Gaussian (LQG) controller. In the most basic setting, the elements of the disturbance sequence and the initial value of a linear discrete-time system, are modeled as Gaussian random variables.

Unfortunately, more often than not, real systems are subject to physical limitations. Hence, one may be interested in the behavior of stochastic systems whose process noise has bounded support. For asymptotically stable systems, it is easy to assert that in the limit, the probability distribution of the state can not possibly be Gaussian. This is a consequence of the bounded-input-bounded-ouput lemma. This lemma states that, under mild assumptions, the reachable set of a system with bounded inputs remains bounded (Blanchini and Miani, 2008). Hence, the state's distribution can not be Gaussian.

This talk addresses the problem of computing and approximating the transient and limit distributions of stochastic linear systems, whose process noise sequence consists of independent and identically uniformly distributed on given compact zonotopes. In this context, the contributions of this talk are two-fold:

  • First, we develop explicit formulas for the cumulants of both the transient and limit distributions of the state of the system. These formulas are constructed using a novel class of k-symmetric Lyapunov recursions, which subsumes the well known matrix Lyapunov recursion.
  • Once the moments or cumulants are known, one can recover the underlying probability distribution by means of a Gram-Charlier Expansion (GCE) (Hald, 2000; Withers and Nadarajah, 2014). Although, GCEs are well known, their construction remains a challenge. In particular, standard GCEs are known to diverge for problems of practical interest (Cramer, 1957). Furthermore, a standard construction, relying on inverse Fourier operators (Kendall and Stuart, 1969), can not be directly applied in our setting, since the cumulant generating function is not an entire function. In order to address these problems, we introduce a nonstandard) generalized Gram-Charlier Expansion for the transient and limit distributions of the system's state. This GCE is constructed via Chebyshev polynomials of the second kind and expanded with respect to a Wigner semicircle distribution. Furthermore, we provide a convergence result for this expansion, which holds under a reachability condition for the system.

The applicability of the approach is illustrated by means of a methodology for synthesizing feedback controllers for stochastic linear systems subject to joint chance- and state- constraints.

References:

Bertsekas, D.P. and Shreve, S.E., (1978). Stochastic optimal control: the discrete time case. Academic press.

Kalman, R. E., (1960). A new approach to linear filtering and prediction problems. Transactions of the ASME Journal of Basic Engineering, 82:35–45.

Blanchini. F. and Miani, S., (2008) Set-Theoretic Methods in Control. Birkhäuser.

Hald, A., (2000). The early history of the cumulants and the Gram-Charlier series. International Statistical Review, 68(2):137–153.

Withers C.S. and Nadarajah S., (2014). The dual multivariate Charlier and Edgeworth expansions. Statistics & Probability Letters, 87:76–85.

Cramer, H., (1957). Mathematical methods of statistics. Princeton University Press.

Kendall, M.G. and Stuart, A.. (1969). The Advanced Theory of Statistics. Volume 1 (3rd Edition), Griffin.