Optimization techniques for error bounds of ODEs

Fehlerschranken von Anfangswertproblemen mit unbestimmten Anfangsbedingungen werden herkömmlicherweise mit Hilfe von Intervallanalysis berechnet, allerdings mit mäßigem Erfolg. Die traditionelle Herangehensweise führt zu asymptotischen Fehlerabschätzungen, die nur gültig sind, wenn die maximale Schrittweite gegen Null geht. Jedoch benötigt eine effiziente Approximation größtmögliche Schrittweiten, ohne die Genauigkeit zu mindern. Neue Entwicklungen in der globalen Optimierung ermöglichen es, das Finden von Fehlerschranken als globales Optimierungsproblem aufzufassen. Das ist insbesondere wichtig im Fall, dass die Differentialgleichungen oder die Anfangsbedingungen bedeutende Unschärfen enthalten. Es wurde ein neuer Solver - DIVIS (Differential Inequality based Validated IVP Solver) - entwickelt, um die Fehlerschranken für Anfangswertprobleme mit Hilfe von Fehlerabschätzungen und Optimierungstechniken zu berechnen. Die Idee dabei ist, die Fehlerabschätzung von Anfangswertproblemen durch elliptische Approximation zu berechnen. Die validierten Zustandseinschliessungen werden mit Hilfe von Differentialungleichungen berechnet. Die Konvergenz dieser Methode hängt von der Wahl geeigneter Vorkonditionierer ab. Das beschriebene Schema wurde in MATLAB und AMPL implementiert. Die Ergebnisse wurden mit VALENCIA-IVP, VNODE-LP und VSPODE verglichen.Error bounds of initial value problems with uncertain initial conditions are traditionally computed by using interval analysis but with limited success. Traditional analysis only leads to asymptotic error estimates valid when the maximal step size tends to zero, while efficiency in the approximation requires that step sizes are as large as possible without compromising accuracy. Recent progress in global optimization makes it feasible to treat the error bounding problem as a global optimization problem. This is particularly important in the case where the differential equations or the initial conditions contain significant uncertainties. A new solver DIVIS (Differential Inequality based Validated IVP Solver) has been developed to compute the error bounds of initial value problems by using defect estimates and optimization techniques. The basic idea is to compute the defect estimates of initial value problems by using outer ellipsoidal approximation. The validated state enclosures are computed by applying differential inequalities. Convergence of the method depends upon a suitable choice of preconditioner. The scheme is implemented in MATLAB and AMPL and the resulting enclosures are compared with VALENCIA-IVP, VNODE-LP and VSPODE

Verified global optimization for estimating the parameters of nonlinear models

Nonlinear parameter estimation is usually achieved via the minimization of some possibly non-convex cost function. Interval analysis allows one to derive algorithms for the guaranteed characterization of the set of all global minimizers of such a cost function when an explicit expression for the output of the model is available or when this output is obtained via the numerical solution of a set of ordinary differential equations. However, cost functions involved in parameter estimation are usually challenging for interval techniques, if only because of multi-occurrences of the parameters in the formal expression of the cost. This paper addresses parameter estimation via the verified global optimization of quadratic cost functions. It introduces tools for the minimization of generic cost functions. When an explicit expression of the output of the parametric model is available, significant improvements may be obtained by a new box exclusion test and by careful manipulations of the quadratic cost function. When the model is described by ODEs, some of the techniques available in the previous case may still be employed, provided that sensitivity functions of the model output with respect to the parameters are available

Dynamic Modeling and Stability Analysis of Stochastic Multi-Physical Systems Applied to Electric Power Systems

[ES] La naturaleza aleatoria que caracteriza algunos fenómenos en sistemas físicos reales (e.g., ingeniería, biología, economía, finanzas, epidemiología y otros) nos ha planteado el desafío de un cambio de paradigma del modelado matemático y el análisis de sistemas dinámicos, y a tratar los fenómenos aleatorios como variables aleatorias o procesos estocásticos. Este enfoque novedoso ha traído como consecuencia nuevas especificidades que la teoría clásica del modelado y análisis de sistemas dinámicos deterministas no ha podido cubrir. Afortunadamente, maravillosas contribuciones, realizadas sobre todo en el último siglo, desde el campo de las matemáticas por científicos como Kolmogorov, Langevin, Lévy, Itô, Stratonovich, sólo por nombrar algunos; han abierto las puertas para un estudio bien fundamentado de la dinámica de sistemas físicos perturbados por ruido. En la presente tesis se discute el uso de ecuaciones diferenciales algebraicas estocásticas (EDAEs) para el modelado de sistemas multifísicos en red afectados por perturbaciones estocásticas, así como la evaluación de su estabilidad asintótica a través de exponentes de Lyapunov (ELs). El estudio está enfocado en EDAEs d-index-1 y su reformulación como ecuaciones diferenciales estocásticas ordinarias (EDEs). Fundamentados en la teoría ergódica, es factible analizar los ELs a través de sistemas dinámicos aleatorios (SDAs) generados por EDEs subyacentes. Una vez garantizada la existencia de ELs bien definidas, hemos procedido al uso de técnicas de simulación numérica para determinar los ELs numéricamente. Hemos implementado métodos numéricos basados en descomposición QR discreta y continua para el cómputo de la matriz de solución fundamental y su uso en el cálculo de los ELs. Las características numéricas y computacionales más relevantes de ambos métodos se ilustran mediante pruebas numéricas. Toda esta investigación sobre el modelado de sistemas con EDAEs y evaluación de su estabilidad a través de ELs calculados numéricamente, tiene una interesante aplicación en ingeniería. Esta es la evaluación de la estabilidad dinámica de sistemas eléctricos de potencia. En el presente trabajo de investigación, implementamos nuestros métodos numéricos basados en descomposición QR para el test de estabilidad dinámica en dos modelos de sistemas eléctricos de potencia de una-máquina bus-infinito (OMBI) afectados por diferentes perturbaciones ruidosas. El análisis en pequeña-señal evidencia el potencial de las técnicas propuestas en aplicaciones de ingeniería.[CA] La naturalesa aleatòria que caracteritza alguns fenòmens en sistemes físics reals (e.g., enginyeria, biologia, economia, finances, epidemiologia i uns altres) ens ha plantejat el desafiament d'un canvi de paradigma del modelatge matemàtic i l'anàlisi de sistemes dinàmics, i a tractar els fenòmens aleatoris com a variables aleatòries o processos estocàstics. Aquest enfocament nou ha portat com a conseqüència noves especificitats que la teoria clàssica del modelatge i anàlisi de sistemes dinàmics deterministes no ha pogut cobrir. Afortunadament, meravelloses contribucions, realitzades sobretot en l'últim segle, des del camp de les matemàtiques per científics com Kolmogorov, Langevin, Lévy, Itô, Stratonovich, només per nomenar alguns; han obert les portes per a un estudi ben fonamentat de la dinàmica de sistemes físics pertorbats per soroll. En la present tesi es discuteix l'ús d'equacions diferencials algebraiques estocàstiques (EDAEs) per al modelatge de sistemes multifísicos en xarxa afectats per pertorbacions estocàstiques, així com l'avaluació de la seua estabilitat asimptòtica a través d'exponents de Lyapunov (ELs). L'estudi està enfocat en EDAEs d-index-1 i la seua reformulació com a equacions diferencials estocàstiques ordinàries (EDEs). Fonamentats en la teoria ergòdica, és factible analitzar els ELs a través de sistemes dinàmics aleatoris (SDAs) generats per EDEs subjacents. Una vegada garantida l'existència d'ELs ben definides, hem procedit a l'ús de tècniques de simulació numèrica per a determinar els ELs numèricament. Hem implementat mètodes numèrics basats en descomposició QR discreta i contínua per al còmput de la matriu de solució fonamental i el seu ús en el càlcul dels ELs. Les característiques numèriques i computacionals més rellevants de tots dos mètodes s'illustren mitjançant proves numèriques. Tota aquesta investigació sobre el modelatge de sistemes amb EDAEs i avaluació de la seua estabilitat a través d'ELs calculats numèricament, té una interessant aplicació en enginyeria. Aquesta és l'avaluació de l'estabilitat dinàmica de sistemes elèctrics de potència. En el present treball de recerca, implementem els nostres mètodes numèrics basats en descomposició QR per al test d'estabilitat dinàmica en dos models de sistemes elèctrics de potència d'una-màquina bus-infinit (OMBI) afectats per diferents pertorbacions sorolloses. L'anàlisi en xicotet-senyal evidencia el potencial de les tècniques proposades en aplicacions d'enginyeria.[EN] The random nature that characterizes some phenomena in the real-world physical systems (e.g., engineering, biology, economics, finance, epidemiology, and others) has posed the challenge of changing the modeling and analysis paradigm and treat these phenomena as random variables or stochastic processes. Consequently, this novel approach has brought new specificities that the classical theory of modeling and analysis for deterministic dynamical systems cannot cover. Fortunately, stunning contributions made overall in the last century from the mathematics field by scientists such as Kolmogorov, Langevin, Lévy, Itô, Stratonovich, to name a few; have opened avenues for a well-founded study of the dynamics in physical systems perturbed by noise. In the present thesis, we discuss stochastic differential-algebraic equations (SDAEs) for modeling multi-physical network systems under stochastic disturbances, and their asymptotic stability assessment via Lyapunov exponents (LEs). We focus on d-index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Supported by the ergodic theory, it is feasible to analyze the LEs via the random dynamical system (RDSs) generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important numerical and computational features of both methods are illustrated through numerical tests. All this investigation concerning systems modeling through SDAEs and their stability assessment via computed LEs finds an appealing engineering application in the dynamic stability assessment of power systems. In this research work, we implement our QR-based numerical methods for testing the dynamic stability in two types of single-machine infinite-bus (SMIB) power system models perturbed by different noisy disturbances. The analysis in small-signal evidences the potential of the proposed techniques in engineering applications.Mi agradecimiento al estado ecuatoriano que, a través del Programa de Becas para el Fortalecimiento y Desarrollo del Talento Humano en Ciencia y Tecnología 2012 de la Secretaría Nacional de Educación Superior, Ciencia y Tecnología (SENESCYT), han financiado mis estudios de doctorado.González Zumba, JA. (2020). Dynamic Modeling and Stability Analysis of Stochastic Multi-Physical Systems Applied to Electric Power Systems [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/158558TESI

Verified integration of differential equations with discrete delay

Many dynamic system models in population dynamics, physics and control involve temporally delayed state information in such a way that the evolution of future state trajectories depends not only on the current state as the initial condition but also on some previous state. In technical systems, such phenomena result, for example, from mass transport of incompressible fluids through finitely long pipelines, the transport of combustible material such as coal in power plants via conveyor belts, or information processing delays. Under the assumption of continuous dynamics, the corresponding delays can be treated either as constant and fixed, as uncertain but bounded and fixed, or even as state-dependent. In this paper, we restrict the discussion to the first two classes and provide suggestions on how interval-based verified approaches to solving ordinary differential equations can be extended to encompass such delay differential equations. Three close-to-life examples illustrate the theory

Global optimisation for dynamic systems using novel overestimation reduction techniques

The optimisation of dynamic systems is of high relevance in chemical engineering as many practical systems can be described by ordinary differential equations (ODEs) or differential algebraic equations (DAEs). The current techniques for solving these problems rigorously to global optimality rely mainly on sequential approaches in which a branch and bound framework is used for solving the global optimisation part of the problem and a verified simulator (in which rounding errors are accounted for in the computations) is used for solving the dynamic constraints. The verified simulation part is the main bottleneck since tight bounds are difficult to obtain for high dimensional dynamic systems. Additionally, uncertainty in the form of, for example, intervals is introduced in the parameters of the dynamic constraints which are also the decision variables of the optimisation problem. Nevertheless, in the verified simulation the accumulation of trajectories that do not belong to the exact solution (overestimation) makes the state bounds overconservative and in the worst case they blow up and tend towards ±∞. In this thesis, methods for verified simulation in global optimisation for dynamic systems were investigated. A novel algorithm that uses an interval Taylor series (ITS) method with enhanced overestimation reduction capabilities was developed. These enhancements for the reduction of the overestimation rely on interval contractors (Krawczyk, Newton, ForwardBackward) and model reformulation based on pattern substitution and input scaling. The method with interval contractors was also extended to Taylor Models (TM) for comparison purposes. The two algorithms were tested on several case studies to demonstrate the effectiveness of the methods. The case studies have a different number of state variables and system parameters and they use uncertain amounts in some of the system parameters and initial conditions. Both of the methods were also used in a sequential approach to address the global optimisation for dynamic systems problem subject to uncertainty. The simulation results demonstrated that the ITS method with overestimation reduction techniques provided tighter state bounds with less computational expense than the traditional method. In the case of the forward-backward contractor additional constraints can be introduced that can potentially contribute significantly to the reduction of the overestimation. Similarly, the novel TM method with enhanced overestimation reduction capabilities provided tighter bounds than the TM method alone. On the other hand, the optimisation results showed that the global optimisation algorithm with the novel ITS method with overestimation reduction techniques converged faster to a rigorous solution due to the improved state bounds

Verified interval enclosure techniques for robust gain scheduling controllers

In real-life applications, dynamic systems are often subject to uncertainty due to model simplifications, measurement inaccuracy or approximation errors which can be mapped to specific parameters. Uncertainty in dynamic systems can come either in stochastic forms or as interval representations. The latter is applied if the uncertainty is bounded as it will be done in this paper. The main idea is to find a joint approach for an interval-based gain scheduling controller while simultaneously reducing overestimation by enclosing state intervals with the least amount of conservativity. The robust and/ or optimal control design is realized using linear matrix inequalities (LMIs) to find an efficient solution and aims at a guaranteed stabilization of the system dynamics over a predefined time horizon. A temporal reduction of the widths of intervals representing worst-case bounds of the system states at a specific point of time should occur due to asymptotic stability proven by the employed LMI-based design. However, for commonly used approaches in the computation of interval enclosures, those interval widths seemingly blow up due to the wrapping effect in many cases. To avoid this, we provide two interval enclosure techniques — an exploitation of cooperativity and an exponential approach — and discuss their applicability taking into account two real-life applications, a high-bay rack feeder and an inverse pendulum

Global Optimisation for Dynamic Systems using Interval Analysis

Engineers seek optimal solutions when designing dynamic systems but a crucial element is to ensure bounded performance over time. Finding a globally optimal bounded trajectory requires the solution of the ordinary differential equation (ODE) systems in a verified way. To date these methods are only able to address low dimensional problems and for larger systems are unable to prevent gross overestimation of the bounds. In this paper we show how interval contractors can be used to obtain tightly bounded optima. A verified solver constructs tight upper and lower bounds on the dynamic variables using contractors for initial value problems (IVP) for ODEs within a global optimisation method. The solver provides guaranteed bound on the objective function and on the first order sensitivity equations in a branch and bound framework. The method is compared with three previously published methods on three examples from process engineering

On the computation of output bounds on parallel inputs pharmacokinetic models with parametric uncertainty

Pharmacokinetic models are of utmost importance in drug and medical research. The class of parallel inputs models consists of two or more linear chains connected together in parallel. It has been used to represent pharmacokinetic processes in which the input shows effects on the output with different delays in time. Due to physiological variability, the exact values of the model parameters are uncertain, but they can be bounded by intervals. In this case, the computation of output bounds can be posed as the solution of an initial value problem (IVP) for ordinary differential equations (ODEs) with uncertain initial conditions. However, current methods may produce a significant overestimation. In this paper, a new method to minimise overestimation when using the parallel inputs model is proposed and applied to two cases: subcutaneous insulin absorption for artificial pancreas research, and the study of the double-peak phenomenon observed for certain drugs. Our proposal consists in performing a model reduction in conjunction with analytical solutions of the input chains and a monotonicity analysis of model states and parameters. This method allows obtaining tighter output bounds with low computational cost compared to the latest techniques.This work was partially supported by the Spanish Ministerio de Ciencia e Innovacion through Grant DPI-2010-20764-C02, and by the Universitat Politecnica de Valencia through Grant PAID-05-09-4334.De Pereda Sebastián, D.; Romero Vivó, S.; Bondía Company, J. (2013). On the computation of output bounds on parallel inputs pharmacokinetic models with parametric uncertainty. Mathematical and Computer Modelling. 57:1760-1767. https://doi.org/10.1016/j.mcm.2011.11.031S176017675