Mario Eduardo Villanueva

Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs

This paper presents a framework for constructing and analyzing enclosures of the reachable set of nonlinear ordinary differential equations using continuous-time set-propagation methods. The focus is on convex enclosures that can be characterized in terms of their support functions. A generalized differential inequality is introduced, whose solutions describe such support functions for a convex enclosure of the reachable set under mild conditions. It is shown that existing continuous-time bounding methods that are based on standard differential inequalities or ellipsoidal set propagation techniques can be recovered as special cases of this generalized differential inequality. A way of extending this approach for the construction of nonconvex enclosures is also described, which relies on Taylor models with convex remainder bounds. This unifying framework provides a means for analyzing the convergence properties of continuous-time enclosure methods. The enclosure techniques and convergence results are illustrated with numerical case studies throughout the paper, including a six-state dynamic model of anaerobic digestion.

Evaluation of criteria for bioreactor comparison and operation standardization for mammalian cell culture

Development of bioprocesses with mammalian cell culture deals with different bioreactor types and scales. The bioreactors might be intended for generation of cell inoculum and production, research, process development, validation, or transfer purposes. During these activities, not only the difficulty of up and downscaling might lead to failure of consistency in cell growth, but also the use of different bioreactor geometries and operation conditions. In such cases, criteria for bioreactor design and process transfer should be carefully evaluated in order to select appropriate cultivation parameters.

Guaranteed parameter estimation of non-linear dynamic systems using high-order bounding techniques with domain and CPU-time reduction strategies

This paper is concerned with guaranteed parameter estimation of nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding—or approximating as closely as possible—the set of all possible parameter values such that the predicted values of certain outputs match their corresponding measurements within prescribed error bounds. A set-inversion algorithm is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a given threshold on the approximation level is met. Such exclusion tests rely on the ability to bound the solution set of the dynamic system for a finite parameter subset, and the tightness of these bounds is therefore paramount; equally important in practice is the time required to compute the bounds, thereby defining a trade-off. In this paper, we investigate such a tradeoff by comparing various bounding techniques based on Taylor models with either interval or ellipsoidal bounds as their remainder terms. We also investigate the use of optimization-based domain reduction techniques in order to enhance the convergence speed of the set-inversion algorithm, and we implement simple strategies that avoid recomputing Taylor models or reduce their expansion orders wherever possible. Case studies of various complexities are presented, which show that these improvements using Taylor-based bounding techniques can significantly reduce the computational burden, both in terms of iteration count and CPU time.

A validated integration algorithm for nonlinear ODEs using Taylor models and ellipsoidal calculus

This paper presents a novel algorithm for bounding the reachable set of parametric nonlinear differential equations. This algorithm is based on a first-discretize-then-bound approach to enclose the reachable set via propagation of a Taylor model with ellipsoidal remainder, and it accounts for truncation errors that are inherent to the discretization. In contrast to existing algorithms that proceed in two phases-an a priori enclosure phase, followed by a tightening phase-the proposed algorithm first predicts a continuous-time enclosure and then seeks a maximal step-size for which validity of the predicted enclosure can be established. It is shown that this reversed approach leads to a natural step-size control mechanism, which no longer relies on the availability of an a priori enclosure. Also described in the paper is an open-source implementation of the algorithm in ACADO Toolkit. A simple numerical case study is presented to illustrate the performance and stability of the algorithm.

Bounding the Solutions of Parametric ODEs: When Taylor Models Meet Differential Inequalities

Abstract This article presents a new method for computing Taylor models of the solutions of parametric ODEs, based on the theory of differential inequalities. Rather than bounding the solutions directly using interval analysis, the idea is to bound the remainder term in a Taylor series expansion of these solutions, which leads to a high-order convergence rate. A practical procedure for propagating the Taylor model estimators over a given time horizon is described. The methodology is illustrated by the case study of a Lotka-Volterra system.

Stable set-valued integration of nonlinear dynamic systems using affine set-parameterizations

Many set-valued integration algorithms for parametric ordinary differential equations (ODEs) implement a combination of Taylor series expansion with either interval arithmetic or Taylor model arithmetic. Due to the wrapping effect, the diameter of the solution-set enclosures computed with these algorithms typically diverges to infinity on finite integration horizons, even though the ODE trajectories themselves may be asymptotically stable. This paper starts by describing a new discretized set-valued integration algorithm that uses a predictor-validation approach to propagate generic affine set-parameterizations, whose images are guaranteed to enclose the ODE solution set. Sufficient conditions are then derived for this algorithm to be locally asymptotically stable, in the sense that the computed enclosures are guaranteed to remain stable on infinite time horizons when applied to a dynamic system in the neighborhood of a locally asymptotically stable periodic orbit (or equilibrium point). The key requireme...

Chebyshev model arithmetic for factorable functions

This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden.

Ellipsoidal Arithmetic for Multivariate Systems

Abstract The ability to determine enclosures for the image set of nonlinear functions is pivotal to many applications in engineering. This paper presents a method for the systematic construction of ellipsoidal extensions of factorable functions. It proceeds by lifting the ellipsoid to a higher dimensional space for every atom operation in the function’s directed acyclic graph, thereby accounting for dependencies. We present theoretical results regarding the quadratic Hausdorff convergence of the computed enclosures. Moreover, we propose an efficient implementation, whereby the shape matrix of the lifted ellipsoid is stored in sparse format, and every atom operation corresponds to a sparse update in that matrix. We illustrate these developments with two numerical examples.

Criteria for bioreactor comparison and operation standardisation during process development for mammalian cell culture

BackgroundDevelopment of bioprocesses for animal cells has to dealwith different bioreactor types and scales. Bioreactorsmight be intended for generation of cell inoculum andproduction, research, process development, validation ortransfer purposes. During these activities, not only thedifficulty of up- and downscaling might lead to failureof consistency in cell growth, but also the use of differ-ent bioreactor geometries and operation conditions. Insuch cases, the criteria for bioreactor design and processtransfer should be carefully evaluated in order to avoidan erroneous transfer of cultivation parameters.In this work, power input, mixing time, impeller tipspeed, and Reynolds number have been compared sys-tematically for the cultivation of the human cell lineAGE1.HN

Robust MPC via minmax differential inequalities

This paper is concerned with tube-based model predictive control (MPC) for both linear and nonlinear, input-affine continuous-time dynamic systems that are affected by time-varying disturbances. We derive a min-max differential inequality describing the support function of positive robust forward invariant tubes, which can be used to construct a variety of tube-based model predictive controllers. These constructions are conservative, but computationally tractable and their complexity scales linearly with the length of the prediction horizon. In contrast to many existing tube-based MPC implementations, the proposed framework does not involve discretizing the control policy and, therefore, the conservatism of the predicted tube depends solely on the accuracy of the set parameterization. The proposed approach is then used to construct a robust MPC scheme based on tubes with ellipsoidal cross-sections. This ellipsoidal MPC scheme is based on solving an optimal control problem under linear matrix inequality constraints. We illustrate these results with the numerical case study of a spring-mass-damper system.