Set-membership approach and Kalman observer based on zonotopes for discrete-time descriptor systems

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper proposes a set-membership state estimator and a zonotopic Kalman observer for discrete-time descriptor systems. Both approaches are developed in a set-based context considering system disturbances, measurement noise, and unknown inputs. This set-membership state estimation approach determines the set of consistent states with the model and measurements by constructing a parameterized intersection zonotope. Two methods to minimize the size of this intersection zonotope are provided: one inspired by Kalman filtering and the other based on solving an optimization problem involving a series of linear matrix inequalities. Additionally, we propose a zonotopic Kalman observer for discrete-time descriptor systems. Moreover, the relationship between both approaches is discussed. In particular, it is proved that the zonotopic Kalman observer in the current estimation type is equivalent to the set-membership approach. Finally, a numerical example is used to illustrate and compare the effectiveness of the proposed approaches.Peer ReviewedPostprint (author's final draft

Distributed Set-Based Observers Using Diffusion Strategy

Distributed estimation is more robust against single points of failure and requires less communication overhead compared to the centralized version. Among distributed estimation techniques, set-based estimation has gained much attention as it provides estimation guarantees for safety-critical applications and copes with unknown but bounded uncertainties. We propose two distributed set-based observers using interval-based and set-membership approaches for a linear discrete-time dynamical system with bounded modeling and measurement uncertainties. Both algorithms utilize a new over-approximating zonotopes intersection step named the set-based diffusion step. We use the term diffusion since our intersection of zonotopes formula resembles the traditional diffusion step in the stochastic Kalman filter. Our new zonotopes intersection takes linear time. Our set-based diffusion step decreases the estimation errors and the size of estimated sets and can be seen as a lightweight approach to achieve partial consensus between the distributed estimated sets. Every node shares its measurement with its neighbor in the measurement update step. The neighbors intersect their estimated sets constituting our proposed set-based diffusion step. We represent sets as zonotopes since they compactly represent high-dimensional sets, and they are closed under linear mapping and Minkowski addition. The applicability of our algorithms is demonstrated by a localization example. All used data and code to recreate our findings are publicly availabl

Parameters uncertainties and error propagation in modified atmosphere packaging modelling

IATE Axe 5 : Application intégrée de la connaissance, de l’information et des technologies permettant d’accroître la qualité et la sécurité des aliments Publication Inra prise en compte dans l'analyse bibliométrique des publications scientifiques mondiales sur les Fruits, les Légumes et la Pomme de terre. Période 2000-2012. http://prodinra.inra.fr/record/256699International audienceMathematical models are instrumental tools to predict gas (O2 and CO2) evolution in headspaces of Modified Atmosphere Packaging (MAP). Such models simplify the package design steps as they allow engineers to estimate the optimal values of packaging permeability for maintaining the quality and safety of the packed food. However, these models typically require specifying several input parameter values (such as maximal respiration rates) that are obtained from experimental data and are characterized by high uncertainties due to biological variation. Although treating and modelling this uncertainty is essential to ensure the robustness of designed MAPs, this subject has seldom been considered in the literature. In this work, we describe an optimisation system based on a MAP mathematical model that determines optimal permeabilities of packaging, given certain food parameters. To integrate uncertainties in the model while keeping the optimisation computational burden relatively low, we propose to use an approach based on interval analysis rather than the more classical probabilistic approach. The approach has two advantages: it makes a minimal amount of unverified assumption concerning uncertainties, and it requires only a few evaluations of the model. The results of these uncertainty studies are optimal values of permeabilities described by fuzzy sets. This approach was conducted on three case studies: chicory, mushrooms and blueberry. Sensitivity analysis on input parameters in the model MAP was also performed in order to point out that parameter influences are dependent on the considered fruit or vegetable. A comparison of the interval analysis methodology with the probabilistic one (known as Monte Carlo) was then performed and discussed

Advances in Reachability Analysis for Nonlinear Dynamic Systems

Systems of nonlinear ordinary differential equations (ODEs) are used to model an incredible variety of dynamic phenomena in chemical, oil and gas, and pharmaceutical industries. In reality, such models are nearly always subject to significant uncertainties in their initial conditions, parameters, and inputs. This dissertation provides new theoretical and numerical techniques for rigorously enclosing the set of solutions reachable by a given systems of nonlinear ODEs subject to uncertain initial conditions, parameters, and time-varying inputs. Such sets are often referred to as reachable sets, and methods for enclosing them are critical for designing systems that are passively robust to uncertainty, as well as for optimal real-time decision-making. Such enclosure methods are used extensively for uncertainty propagation, robust control, system verification, and optimization of dynamic systems arising in a wide variety of applications. Unfortunately, existing methods for computing such enclosures often provide an unworkable compromise between cost and accuracy. For example, interval methods based on differential inequalities (DI) can produce bounds very efficiently but are often too conservative to be of any practical use. In contrast, methods based on more complex sets can achieve sharp bounds, but are far too expensive for real-time decision-making and scale poorly with problem size. Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative while maintaining high efficiency by exploiting redundant model equations that are known to hold for all trajectories of interest (e.g., linear relationships among chemical species in a reaction network that hold due to the conservation of mass or elements). These linear relationships are implied by the governing ODEs, and can thus be considered redundant. However, these advances are only applicable to a limited class of system in which pre-existing linear redundant model equations are available. Moreover, the theoretical results underlying these algorithms do not apply to redundant equations that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear redundant equations, etc. This dissertation continues a line of research that has recently achieved very promising bounding results using methods based on differential inequalities. In brief, the major contributions can be divided into three categories: (1) In regard to algorithms, this dissertation significantly improves existing algorithms that exploit linear redundant model equations to achieve more accurate and efficient enclosures. It also develops new fast and accurate bounding algorithms that can exploit nonlinear redundant model equations. (2) Considering theoretical contributions, it develops a novel theoretical framework for the introduction of redundant model equations into arbitrary dynamic models to effectively reduce conservatism. The newly developed theories have more generality in terms of application. For example, complex nonlinear constraints that involve states, time derivatives of the system states, and time- varying inputs are allowed to be exploited. (3) A new differential inequalities method called Mean Value Differential Inequalities (MVDI) is developed that can automatically introduce redundant model equations for arbitrary dynamic systems and has a second-order convergence rate reported the first time among DI-based methods

Reachability analysis and deterministic global optimization of differential-algebraic systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 447-460).Systems of differential-algebraic equations (DAEs) are used to model an incredible variety of dynamic phenomena. In the chemical process industry in particular, the numerical simulation of detailed DAE models has become a cornerstone of many core activities including, process development, economic optimization, control system design and safety analysis. In such applications, one is primarily interested in the behavior of the model solution with respect variations in the model inputs or uncertainties in the model itself. This thesis addresses two computational problems of general interest in this regard. In the first, we are interested in computing a guaranteed enclosure of all solutions of a given DAE model subject to a specified set of inputs. This analysis has natural applications in uncertainty quantification and process safety verification, and is used for many important tasks in process control. However, for nonlinear dynamic systems, this task is very difficult. Existing methods apply only to ordinary differential equation (ODE) models, and either provide very conservative enclosures or require excessive computational effort. Here, we present new methods for computing interval bounds on the solutions of ODEs and DAEs. For ODEs, the focus is on efficient methods for using physical information that is often available in applications to greatly reduce the conservatism of existing methods. These methods are then extended for the first time to the class of semi-explicit index-one DAEs. The latter portion of the thesis concerns the global solution of optimization problems constrained by DAEs. Such problems arise in optimal control of batch processes, determination of optimal start-up and shut-down procedures, and parameter estimation for dynamic models. In nearly all conceivable applications, there is significant economic and/or intellectual impetus to locate a globally optimal solution. Yet again, this problem has proven to be extremely difficult for nonlinear dynamic models. A small number of practical algorithms have been proposed, all of which are limited to ODE models and require significant computational effort. Here, we present improved lower-bounding procedures for ODE constrained problems and develop a complete deterministic algorithm for problems constrained by semi-explicit index-one DAEs for the first time.by Joseph Kirk Scott.Ph.D