Krylov subspaces from bilinear representations of nonlinear systems

Purpose – The paper is aimed at the development of novel model reduction techniques for nonlinear systems. Design/methodology/approach – The analysis is based on the bilinear and polynomial representation of nonlinear systems and the exact solution of the bilinear system in terms of Volterra series. Two sets of Krylov subspaces are identified which capture the most essential part of the input-output behaviour of the system. Findings – The paper proposes two novel model-reduction strategies for nonlinear systems. The first involves the development, in a novel manner compared with previous approaches, of a reduced-order model from a bilinear representation of the system, while the second involves reducing a polynomial approximation using Krylov subspaces derived from a related bilinear representation. Both techniques are shown to be effective through the evidence of a standard test example. Research limitations/implications – The proposed methodology is applicable to so-called weakly nonlinear systems, where both the bilinear and polynomial representations are valid. Practical implications – The suggested methods lead to an improvement in the accuracy of nonlinear model reduction, which is of paramount importance for the efficient simulation of state-of-the-art dynamical systems arising in all aspects of engineering. Originality/value – The proposed novel approaches for model reduction are particularly beneficial for the design of controllers for nonlinear systems and for the design and analysis of radio-frequency integrated circuits

Model reduction of weakly nonlinear systems

In general, model reduction techniques fall into two categories — moment —matching and Krylov techniques and balancing techniques. The present contribution is concerned with the former. The present contribution proposes the use of a perturbative representation as an alternative to the bilinear representation [4]. While for weakly nonlinear systems, either approximation is satisfactory, it will be seen that the perturbative method has several advantages over the bilinear representation. In this contribution, an improved reduction method is proposed. Illustrative examples are chosen, and the errors obtained from the different reduction strategies will be compared

Empirical balanced truncation of nonlinear systems

Novel constructions of empirical controllability and observability gramians for nonlinear systems for subsequent use in a balanced truncation style of model reduction are proposed. The new gramians are based on a generalisation of the fundamental solution for a Linear Time-Varying system. Relationships between the given gramians for nonlinear systems and the standard gramians for both Linear Time-Invariant and Linear Time-Varying systems are established as well as relationships to prior constructions proposed for empirical gramians. Application of the new gramians is illustrated through a sample test-system.Comment: LaTeX, 11 pages, 2 figure

Balanced truncation of perturbative representations of nonlinear systems

The paper presents a novel approach for a balanced truncation style of model reduction of a perturbative representation of a nonlinear system. Empirical controllability and observability gramians for nonlinear systems are employed to define a projection matrix. However, the projection matrix is applied to the perturbative representation of the system rather than directly to the exact nonlinear system. This is to achieve the required increase in efficiency desired of a reduced-order model. Application of the new method is illustrated through a sample test-system. The technique will be compared to the standard approach for reducing a perturbative representation of a nonlinear system

Krylov Subspace Model Order Reduction for Nonlinear and Bilinear Control Systems

The use of Krylov subspace model order reduction for nonlinear/bilinear systems, over the past few years, has become an increasingly researched area of study. The need for model order reduction has never been higher, as faster computations for control, diagnosis and prognosis have never been higher to achieve better system performance. Krylov subspace model order reduction techniques enable this to be done more quickly and efficiently than what can be achieved at present. The most recent advances in the use of Krylov subspaces for reducing bilinear models match moments and multimoments at some expansion points which have to be obtained through an optimisation scheme. This therefore removes the computational advantage of the Krylov subspace techniques implemented at an expansion point zero. This thesis demonstrates two improved approaches for the use of one-sided Krylov subspace projection for reducing bilinear models at the expansion point zero. This work proposes that an alternate linear approximation can be used for model order reduction. The advantages of using this approach are improved input-output preservation at a simulation cost similar to some earlier works and reduction of bilinear systems models which have singular state transition matrices. The comparison of the proposed methods and other original works done in this area of research is illustrated using various examples of single input single output (SISO) and multi input multi output (MIMO) models

Optimal control and robust estimation for ocean wave energy converters

This thesis deals with the optimal control of wave energy converters and some associated observer design problems. The first part of the thesis will investigate model predictive control of an ocean wave energy converter to maximize extracted power. A generic heaving converter that can have both linear dampers and active elements as a power take-off system is considered and an efficient optimal control algorithm is developed for use within a receding horizon control framework. The optimal control is also characterized analytically. A direct transcription of the optimal control problem is also considered as a general nonlinear program. A variation of the projected gradient optimization scheme is formulated and shown to be feasible and computationally inexpensive compared to a standard nonlinear program solver. Since the system model is bilinear and the cost function is not convex quadratic, the resulting optimization problem is shown not to be a quadratic program. Results are compared with other methods like optimal latching to demonstrate the improvement in absorbed power under irregular sea condition simulations. In the second part, robust estimation of the radiation forces and states inherent in the optimal control of wave energy converters is considered. Motivated by this, low order H∞ observer design for bilinear systems with input constraints is investigated and numerically tractable methods for design are developed. A bilinear Luenberger type observer is formulated and the resulting synthesis problem reformulated as that for a linear parameter varying system. A bilinear matrix inequality problem is then solved to find nominal and robust quadratically stable observers. The performance of these observers is compared with that of an extended Kalman filter. The robustness of the observers to parameter uncertainty and to variation in the radiation subsystem model order is also investigated. This thesis also explores the numerical integration of bilinear control systems with zero-order hold on the control inputs. Making use of exponential integrators, exact to high accuracy integration is proposed for such systems. New a priori bounds are derived on the computational complexity of integrating bilinear systems with a given error tolerance. Employing our new bounds on computational complexity, we propose a direct exponential integrator to solve bilinear ODEs via the solution of sparse linear systems of equations. Based on this, a novel sparse direct collocation of bilinear systems for optimal control is proposed. These integration schemes are also used within the indirect optimal control method discussed in the first part.Open Acces

On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning

This paper studies and analyzes a preconditioned Krylov solver for Helmholtz problems that are formulated with absorbing boundary layers based on complex coordinate stretching. The preconditioner problem is a Helmholtz problem where not only the coordinates in the absorbing layer have an imaginary part, but also the coordinates in the interior region. This results into a preconditioner problem that is invertible with a multigrid cycle. We give a numerical analysis based on the eigenvalues and evaluate the performance with several numerical experiments. The method is an alternative to the complex shifted Laplacian and it gives a comparable performance for the studied model problems

Model Order Reduction in Porous Media Flow Simulation and Optimization

Subsurface flow modeling and simulation is ubiquitous in many energy related processes, including oil and gas production. These models are usually large scale and simulating them can be very computationally demanding, particularly in work-flows that require hundreds, if not thousands, runs of a model to achieve the optimal production solution. The primary objective of this study is to reduce the complexity of reservoir simulation, and to accelerate production optimization via model order reduction (MOR) by proposing two novel strategies, Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD-DEIM), and Quadratic Bilinear Formulation (QBLF). While the former is a training-based approach whereby one runs several reservoir models for different input strategies before reducing the model, the latter is a training-free approach. Model order reduction by POD has been shown to be a viable way to reduce the computational cost of flow simulation. However, in the case of porous media flow models, this type of MOR scheme does not immediately yield a computationally efficient reduced system. The main difficulty arises in evaluating nonlinear terms on a reduced subspace. One way to overcome this difficulty is to apply DEIM onto the nonlinear functions (fractional flow, for instance) and to select a small set of grid blocks based on a greedy algorithm. The nonlinear terms are evaluated at these few grid blocks and interpolation based on projection is used for the rest of them. Furthermore, to reduce the number of POD-DEIM basis and the error, a new approach is integrated in this study to update the basis online. In the regular POD-DEIM work flow all the snapshots are used to find one single reduced subspace, whereas in the new technique, namely the localized POD-DEIM, the snapshots are clustered into different groups by means of clustering techniques (k-means), and the reduced subspaces are computed for each cluster in the online (pre-processing) phase. In the online phase, at each time step, the reduced states are used in a classifier to find the most representative basis and to update the reduced subspace. In the second approach in order to overcome the issue of nonlinearity, the QBLF of the original nonlinear porous media flow system is introduced, yielding a system that is linear in the input and linear in the state, but not in both input and state jointly. Primarily, a new set of variables is used to change the problem into QBLF. To highlight the superiority of this approach, the new formulation is compared with a Taylor's series expansion of the system. At this initial phase of development, a POD-based model reduction is integrated with the QBLF in this study in order to reduce the computational costs. This new reduced model has the same form as the original high fidelity model and thus preserves the properties such as stability and passivity. This new form also facilitates the investigation of systematic MOR, where no training or snapshot is required. We test these MOR algorithms on the SPE10 and the results suggest twofold runtime speedups for a case study with more than 60,000 grid blocks. In the case of the QBLF, the results suggests moderate speedups, but more investigation is needed to accommodate an efficient implementation. Finally, MOR is integrated in the optimization work flow for accelerating it. The gradient based optimization framework is used due to its efficiency and fast convergence. This work flow is modified to include the reduced order model and consequently to reduce the computational cost. The water flooding optimization is applied to an offshore reservoir benchmark model, UNISIM-I-D, which has around 38,000 active grid blocks and 25 wells. The numerical solutions demonstrate that the POD-based model order reduction can reproduce accurate optimization results while providing reasonable speedups