(48c) Higher-Order Inclusion Techniques for Guaranteed Parameter Estimation of Nonlinear Dynamic Systems

Among the available techniques to account for uncertainty in parameter estimation, guaranteed parameter estimation aims to determine all the parameter values of a model that are consistent with the measurements under given uncertainty scenarios [1]. Our focus in this presentation is on nonlinear dynamic systems and we consider the case that the uncertainty enters the estimation problem in the form of bounded measurement errors. Set-inversion algorithms, which approximate such parameter sets by a box partition at an arbitrary precision, were first developed for algebraic models in the early 1990s by Moore [2] and Jaulin and Walter [3] using interval analysis; and not long after were these algorithms extended to dynamic systems using ODE bounding techniques [4,5]. Nonetheless, seldom can any estimation problem with more than a few uncertain parameters be tackled with such algorithms. The main computational bottleneck for guaranteed parameter estimation in higher-dimensional dynamic systems appears to be the ability to compute tight bounds on parametric solutions of the dynamic system.

In the first part of the presentation, we review recent developments in ODE bounding techniques [6] based on Taylor models that enjoy higher-order convergence properties and we illustrate the benefits of these techniques for our problem. Next, we introduce optimization-based domain reduction techniques in order to enhance the convergence speed of the set-inversion algorithm as well as simple strategies that avoid recomputing the ODE bounds wherever possible. A challenging case study in anaerobic digestion is presented for a model describing complex liquid-gas transfer and pH self-regulation mechanisms and featuring multiple time scales. The results demonstrate that the proposed improvements allow tackling guaranteed parameter estimation in up to seven parameters within reasonable computational times.

References:
[1] Walter, E. (ed.) (1990). Parameter Identifications with Error Bound, Mathematics & Computers in Simulation, vol. 32. Elsevier.
[2] Moore, R.E. (1992). Parameter sets for bounded-error data. Mathematics & Computers in Simulation, vol. 34 (2), p. 113-119.
[3] Jaulin, L. and Walter, E. (1993). Set inversion via interval analysis for nonlinear bounded-error estimation. Automatica, vol. 29 (4), pp. 1053-1064.
[4] Jaulin, L. (2002). Nonlinear bounded-error state estimation of continuous-time systems. Automatica, vol. 38, pp. 1079-1082.
[5] Raissi, T., Ramdani, N. and Candau, Y. (2004). Set membership state and parameter estimation for systems described by nonlinear differential equations. Automatica, vol. 40, pp. 1771-1777.
[6] Villanueva, M.E., Houska, B. and Chachuat, B. (2013). Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs. Journal of Global Optimization, in revision.