Identification and estimation for models described by differential-algebraic equations

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The work described in the thesis is strongly applications oriented.

Introduction

Many similarities have been found in applying modelling and control techniques to problems arising in fields that appear very different. For example, the areas of neurophysiology, respiratory gas exchange processes, electro-optic sensor systems, helicopter flight-control, hydro-electric power generation and surface ship or underwater vehicles appear to have little in common. However, closer examination shows that they have many similarities in terms of the types of problem that are presented, both in modelling and in system design.

In addition to nonlinear behaviour; most models of these systems involve significant uncertainties or require important simplifications if the model is to be used in a real-time application such as automatic control. One recurring theme, that is important both in the modelling work described and for control applications, is the additional insight that can be gained through the dual use of time-domain and frequency-domain information. One example of this is the importance of coherence information in establishing the existence of linear or nonlinear relationships between variables and this has proved to be valuable in the experimental investigation of neuromuscular systems and in the identification of helicopter models from flight test data.

Frequency-domain techniques have also proved useful for the reduction of high-order multi-input multi-output models. Another important theme that has appeared both within the modelling applications and in research on nonlinear control system design methods, relates to the problems of optimisation in cases where the associated response surface has many local optima. Finding the global optimum in practical applications presents major difficulties and much emphasis has been placed on evolutionary methods of optimisation both genetic algorithms and genetic programming in providing usable methods for optimisation in design and in complex nonlinear modelling applications that do not involve real-time problems.

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Another topic, considered both in the context of system modelling and control, is parameter sensitivity analysis and it has been found that insight gained from sensitivity information can be of value not only in the development of system models e. A technique has been developed based on sensitivity analysis for the semi-automatic tuning of cascade and feedback controllers for multi-input multi-output feedback control systems. This tuning technique has been applied successfully to several problems. Inverse systems also receive significant attention in the thesis. These systems have provided a basis for theoretical research in the control systems field over the past two decades and some significant applications have been reported, despite the inherent difficulties in the mathematical methods needed for the nonlinear case.

Inverse simulation methods, developed initially by others for use in handling-qualities studies for fixed-wing aircraft and helicopters, are shown in the thesis to provide some important potential benefits in control applications compared with classical methods of inversion.

New developments in terms of methodology are presented in terms of a novel sensitivity based approach to inverse simulation that has advantages in terms of numerical accuracy and a new search-based optimisation technique based on the Nelder-Mead algorithm that can handle inverse simulation problems involving hard nonlinearities.

Engineering applications of inverse simulation are presented, some of which involve helicopter flight control applications while others are concerned with feed-forward controllers for ship steering systems. The methods of search-based optimisation show some important advantages over conventional gradient-based methods, especially in cases where saturation and other nonlinearities are significant.

The final discussion section takes the form of a critical evaluation of results obtained using the chosen methods of system identification, parameter estimation and optimisation for the modelling and control applications considered. Zhuk , R. Shorten, T.

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Tchrakian, S. Moore and R. Robust traffic density estimation using Discontinuous Galerkin formulation of a macroscopic model Tigran T.


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Tchrakian, Sergiy Zhuk and Alberto C. Nogueira Jr. Becker, L. Horesh, A.

Sergiy Zhuk - Publications

Aravkin, E. It is based on an extension of the Galerkin projection method. The extended method models. All proposed methods preserve symmetry, positivity and quadratic invariants for the Riccati equations, and non-stationary Lyapunov functions.

Modeling and estimation for control

In addition, an efficient and numerically. Time evolution of reachability sets for dynamical systems Sergiy Zhuk University of Edinburgh seminar , Symplectic Mobius integrators for LQ optimal control problems J. Frank and S. Ragnoli, S. Zhuk , M. Zayats and M. Hartnett ECC , Taking into account a priori estimates for the projection error, we apply the minimax method to construct the state estimate for the reduced model that gives us, in turn, the estimate of the Fourier-Galerkin coefficients associated with a solution of the original macroscopic model. We illustrate our approach with a numerical example which demonstrates its shock capturing capability using only sparse measurements, and under high uncertainty in initial conditions.

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Infinite horizon optimal control and stabilizability for linear descriptor systems Mihaly Petreczky and Sergiy Zhuk preprint , Abstract In this paper we construct an infinite horizon LQ optimal controller for a linear stationary differential-algebraic equation DAE and prove that stabilizability from the given initial state is necessary and sufficient for the controller existence.

We also solve the problem of finite horizon LQ optimal control. Our approach is based on ideas from linear geometric control theory that allows one to represent solutions of DAEs as outputs of LTI system.


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It is applicable for generic DAEs without imposing additional regularity assumptions. The extended method models projection coefficients,. The technique is the combination of supplementary forcing on. Minimax filtering for sequential aggregation: Application to ensemble forecast of ozone analyses Mallet, Vivien and Nakonechny, Alexander and Zhuk, Sergiy Journal of Geophysical Research: Atmospheres 19 , Wiley Online Library, Abstract Abstract [1] This paper presents a new algorithm for sequential aggregation of an ensemble of forecasts.

At any forecasting step, the aggregation consists of 1 computing new weights for the ensemble members represented by different numerical models and 2 forecasting. Infinite horizon optimal control and stabilizability of linear descriptor systems Petreczky, Mihaly and Zhuk, Sergiy and Douai, France arXiv preprint arXiv By using generalized Kalman duality and geometric control we prove that the finite infinite horizon observer exists if and only if the DAE is observable detectable.

Remarkably, the regularity for the DAE is not required. Minimax filtering for sequential aggregation: application to ensemble forecast of ozone analysis Vivien Mallet, Alexander Nakonechny and Sergiy Zhuk Journal of Geophysical Research , The assimilation method produces analyses whenever new observations become available.

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At the same time, an ensemble of given simulations is generated in order to forecast future time steps. The objective of EFA is to forecast the upcoming analyses to be generated with the future observations with a linear combination of the ensemble of forecasts. The weights of the linear combination depend on the ensemble simulation, on time and on the state component. Hence, for any forecast variable and at any location in space, EFA linearly combines the ensemble of forecasts to produce one single forecast of the upcoming analysis for the given variable at the given location.

In previous work, aggregation was carried out by machine learning algorithms that are well adapted to operational forecasting as they enjoy robustness properties. The approach is indeed applied on the French national air quality forecasting platform, Prevair. Nevertheless, the method does not provide uncertainty estimations. Instead of machine learning, we propose to apply a minimax filter on the aggregation weights, which allows us to preserve the forecast performance while introducing uncertainty estimation along with the forecasts.

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The approach forecasts the upcoming analyses, which are the best a posteriori representation of the real state in some sense, significantly better than any model. A Kalman filter can also be applied on the aggregation weights; we will discuss how it compares to the minimax filter in this context. The method will be illustrated for the forecast of peak ozone concentration fields over Europe, based on an ensemble of chemistry-transport models.

When dealing with environmental applications, it allows to study the tracking of clouds on satellite. Inverse problems for linear ill-posed differential-algebraic equations with uncertain parameters Zhuk, Sergiy Discrete Contin. Syst , , Abstract Abstract. This paper describes a minimax state estimation approach for linear differential-algebraic equations DAEs with uncertain parameters.

The approach addresses continuous-time DAEs with non-stationary rectangular matrices and uncertain bounded deterministic. Inverse problems for ill-posed linear differential-algebraic equations with variable coefficients Sergiy Zhuk Int. Minimax state estimation for linear discrete-time differential-algebraic equations Sergiy Zhuk Automatica J. IFAC 46 , , Estimation of the states of a dynamical system described by linear equations with unknown parameters Sergiy Zhuk Ukrainian Mathematical Journal , Springer, State estimation for dynamic system described by linear equation with closed operator in Hilbert space Sergiy Zhuk Differential Equations and Topology , On some properties of linear mapping induced by linear descriptor differential equation Sergiy Zhuk Modern analysis and applications , Closedness and normal solvability of an operator generated by a degenerate linear differential equation with variable coefficients Sergiy Zhuk Nonlinear Oscillations , , Springer, Minimax estimates for linear differential-algebraic equations Sergiy Zhuk Doctoral dissertation , Taras Shevchenko University of Kyiv, Minimax estimation for linear descriptor differential equations Sergiy Zhuk Journal of Applied and Computational Mathematics , Springer,

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