Eigenvalue assignment in linear optimal-control systems via reduced-order models

The stochastic optimal control of linear systems with time-varying and partially observable parameters is synthesized under noisy measurements and a quadratic performance criterion. The structure of the regulator is given, and the optimal solution is reduced to a two-point boundary-value problem. Comments on the numerical solution by appropriate integration schemes is included.

In this paper, we show that a linear system with distributed state delay can be approximated by a linear delay-free system that has the same state dimension as the original system if a so-called smallness condition holds. The smallness condition is an inequality which the maximum lag and norms of the system matrices have to satisfy. The eigenvalues of the approximate system correspond to the dominant eigenvalues of the original system with distributed delay. We provide a numerically stable, iterative algorithm to compute the state matrix of the approximate system. Furthermore, we show that, based on the proposed approximation, the stabilisation, pole placement and setpoint tracking control problems of the addressed class of distributed delay systems can be performed using methods developed for delay-free systems. Simulation results are provided to show the applicability of the proposed approximation and control design method.

The robust eigenvalue assignment of systems subject to parameter perturbations is addressed. Based on some essential properties of induced norms and matrix measures, the authors derive some sufficient conditions which ensure the assignment of the system's eigenvalues in the specified region irrespective of the system perturbations. The robustness bounds for eigenvalue assignment are obtained without the need to solve the Lyapunov equation. An example is given to illustrate the effectiveness and ease of the proposed analysis methods. >

The tensor-train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions of certain types of parametrized partial differential equations. For many of these problems, computing the solution explicitly would require an infeasible amount of memory and computational time. While the TT format makes these problems tractable, iterative techniques for solving the PDEs must be adapted to perform arithmetic while maintaining the implicit structure. The fundamental operation used to maintain feasible memory and computational time is called rounding, which truncates the internal ranks of a tensor already in TT format. We propose several randomized algorithms for this task that are generalizations of randomized low-rank matrix approximation algorithms and provide significant reduction in computation compared to deterministic TT-rounding algorithms. Randomization is particularly effective in the case of rounding a sum of TT-tensors (where we observe 20\times speedup), which is the bottleneck computation in the adaptation of GMRES to vectors in TT format. We present the randomized algorithms and compare their empirical accuracy and computational time with deterministic alternatives.

The bronchial tree plays a main role in the human respiratory system because the air distribution throughout the lungs and gas exchange with blood occur in the airways whose dimensions vary from several centimeters to micrometers. Organization of about 60,000 conducting airways and 33 million respiratory airways in a limited space results in a complex structure. Due to this inherent complexity and a high number of airways, using target-oriented dimensional reduction is inevitable. In addition, there is no general reduced-order model for various types of problems. This necessitates coming up with an appropriate model from a variety of different reduced-order models to solve the desired problem. Lumped formulation, trumpet, or typical path model of whole or parts of bronchial tree are frequently used reduced-order models. On the other hand, using any of these models results in underestimation of flow heterogeneity leading to inaccurate prediction of the systems whose mechanisms depend on the fluid heterogeneity. In this study, a simple robust model combining mechanistic and non-mechanistic modeling approaches of the bronchial tree is proposed which overcomes the limitations of the previous reduced-order models and gives the same results of a detailed mechanistic model for the first time. This model starts from an accurate multi-branching model of conducting and respiratory airways (i.e., the base model) and suggests a proxy model of conducting airway and reduced-order model of respiratory airways based on the base model to significantly reduce computational cost while retaining the accuracy. The combination of these models suggests various reduced-order surrogate models of the human bronchial tree for different problems. The applications and limitations of each reduced-order model are also discussed. The accuracy of the proposed model in the prediction of fluid heterogeneity has been examined by the simulation of multi-breath inert gas washout because the alveolar slope is the reflection of fluid heterogeneity where the computational time decreases from 121h (using the base model) to 4.8s (using the reduced-order model). A parallel strategy for solving the equations is also proposed which decreases run time by 0.18s making the model suitable for real-time applications. Furthermore, the ability of the model has been evaluated in the modeling of asthmatic lung as an instance of abnormal lungs, and in the modeling of O2-CO2 exchange as an instance of nonlinear reacting systems. The results indicate that the proposed model outperforms previous models based on accuracy, robustness, and run time.

For a single-output linear time-invariant system, output controllability is shown to be a necessary and sufficient condition for the existence of linear state variable feedback that freely assigns the eigenvalues of the closed-loop system that are observable from the output. An application of this result to the single-output linear servomechanism shows that the ability of such a system to track with zero steady-state error and an arbitrary transient response is equivalent to output controllability of the plant, and is independent of the class of command signals and disturbances.

In this paper a new model order reduction of linear continuous time system is proposed. An improved technique of Intelligent Search Evolution algorithm (ISEA) is used for generating the reduced order model. The proposed order reduction technique have the advantages of obtaining optimal model order reduction with less no of iterations and faster convergence with less computational time. The feasibility of proposedmethodis explained with the help of Himmelblau function and model order reduction. The efficacy of the proposed order reduction technique is illustrated through a numerical example and results are compared with other existing model order reduction technique. The reduced order model generated by using ISEA will maintain the characteristics of original system very closely.

formula omitted] model reduction for uncertain switched linear discrete-time systems

The aim of this work is to build a numerical software library based on the \(\tau \)-method to solve differential problems using MATLAB. The \(\tau \)-method can be very effective in the solution of certain type of these problems, and therefore, the existence of a numerical library for its dissemination is of major importance. Furthermore, the method has been used for the solution of particular problems but has not yet been explored as a general technique. Focus will be on stability issues, namely those issued from the solution of algebraic linear systems required for the process. Additionally, preconditioners for the solution with the \(\tau \)-method will be tackled, with emphasizes on incomplete LU factorizations and (block) Jacobi preconditioners. We also propose an iterative approach, build upon an LU factorization over a moderate initial size, generating better approximations and providing a priori error estimate at each iteration. Numerical results enlightening the efficiency of the proposed methods will be presented.

Addressing climate change through carbon capture and storage (CCS) technologies necessitates advanced computational methodologies for subsurface CO2 storage monitoring. This study focuses on the Illinois Basin Decatur Project (IBDP), a CCS demonstration pilot aimed at CO2 injection into a deep saline reservoir. We introduce a novel framework combining Dynamic Mode Decomposition (DMD), a data-driven model reduction technique, with direct data assimilation to streamline the calibration of CO2 plume evolution models. This approach enhances rapid tracking and overcomes the computational challenges of traditional high-fidelity numerical reservoir simulations known as the full-order model (FOM). Using DMD, we analyzed five distinct FOM simulation cases of the IBDP with varied permeability in the Mt. Simon section to develop reduced-order models (ROMs). These ROMs utilize three state variables: reservoir pressure, CO2 plume saturation, and bottom-hole pressure (BHP) from a CO2 injection well alongside multi-level pressures from a monitoring well derived from the FOM and the actual field data respectively. Initial FOM simulation cases assessed the impact of permeability multipliers on pressure responses. We then transformed these into ROMs using DMD, preserving essential dynamics. Linear interpolation between permeabilities and DMD outputs—modes and eigenvalues—established relationships for rapid BHP prediction under different scenarios. Employing a Kalman filter, we optimized a global permeability multiplier, using the ROMs, to align measured and simulated BHP values, ensuring model calibration. The final calibrated FOM was further decomposed to a DMD-based ROM, enabling quick, accurate predictions, significantly reducing computational time from hours to minutes. Utilizing an ROM derived through DMD, we achieved an order of 160 reduction in computational time (from 8 hours to just 3 minutes) for a 3-year historical CO2 injection period modeled with 547,000 cells of the FOM. The ROM demonstrated remarkable fidelity, with a mean absolute error of 1.46 psi for pressure and 3.7e-05 for CO2 plume saturation, effectively capturing the dynamics of the full-order model. This substantial decrease in computational time illustrates an advantageous trade-off between speed and accuracy, optimizing the potential for long-term forecasting and monitoring of CO2 sequestration. Incorporating the IBDP as a case study, this research contributes a significant advancement to reservoir simulation practices, offering a potent, efficient tool for CCS monitoring. By integrating DMD for ROM construction with precise data assimilation-based calibration, the study provides a comprehensive solution for swift and accurate subsurface CO2 plume tracking, essential for the successful implementation of CCS projects and the broader effort to mitigate climate change impacts.

Reduced-order models (ROM), which are based on the Volterra theory for nonlinear systems, for the evaluation of nonlinear unsteady aerodynamic forces are presented. The ROMs provide a means for rapid evaluation of frequency-domain generalized aerodynamic forces, which can then be used in traditional flutter analysis schemes to calculate flutter characteristics about nonlinear steady flows. Two ROMs are formulated, an impulse-type ROM that is based on the convolution of ROM kernels with the input signal, and a step-type ROM that is based on convolution with the derivative of the input signal. Linear, first-, and second-order kernels are identified for these two ROMs from direct CFD impulse and step responses. The ROM methodology is demonstrated with the heave and elastic modes of the AGARD 445.6 wing. It was found that the accuracy of the CFD-based impulse response is dependent on the choice of input amplitude and computational time step, and that the impulse-type kernels are highly sensitive to inaccuracies in the impulse responses used for their identification. The step-type ROM was found to be robust, and resulted in good predictions of direct responses. The introduction of second-order kernels did not significantly improve the predictions, indicating a difficulty in performing true nonlinear identification. The use of first-order step-type ROM offered a significant computational time saving compared to the full CFD frequency response analysis.

In this thesis we analyze and develop methods based on model order reduction (MOR) for the solution of optimization problems constrained by time-dependent partial differential equations (PDEs). The methods combine a direct solution approach with model reduction via proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). The reduced-order models (ROMs) are used to approximate the high-dimensional dynamic systems originating from a spatial discretization of a PDE. However, when used in an optimization algorithm, conventional POD/DEIM ROMs often lack the ability to give adequate approximations of the gradient. We propose methods for a suitable enhancement of the ROMs for the optimization purpose which are based on the inclusion of derivative information in the POD and DEIM subspaces. We distinguish two types of error between quantities evaluated with the high-dimensional model and its ROM approximation in dependency on the optimization variable q: The reconstruction error which is evaluated with the same q0 which is used constructing the ROM and the prediction error which assesses approximations at q with a ROM constructed at q0 different to q. The novel reconstruction results we present include estimates for solutions of the adjoint equation and the sensitivity equations as well as for the gradient of the objective function. Based on the estimates we explain how the POD and DEIM bases should be extended with either adjoint or sensitivity information. The enhanced ROMs allow control of the reconstruction error for the objective and its gradient up to machine precision. Moreover, we propose a POD prediction estimate for the objective of the optimization problem in a neighborhood of q where the ROM is constructed. In case of sensitivity-extended POD and DEIM bases we give an analogous result for solutions of the states. The derivative-extended ROMs are then used to develop adaptive algorithms for the solution of optimal control and parameter estimation problems which results in great runtime improvements for the optimization while ensuring high approximation quality of the solution of the original problem. For the parameter estimation case a novel a posteriori error estimate is proposed which assesses the quality of suboptimal solutions obtained with the ROM. A further fundamental contribution is a discussion of discretize-then-optimize (DTO) vs. optimize-then-discretize (OTD) approaches in the context of MOR for optimization. We analyze advantages and disadvantages of both approaches and discuss to which extent our methods exhibit properties of either strategy. We also give examples of representative optimization problems in which standard POD/DEIM ROMs show an inacceptable behavior and can be successfully solved by derivative-extended ROMs. We have further implemented the developed methods emphasizing an efficient realization which is important for the investigation of the MOR potential. We showcase the practical performance of the proposed algorithms and the superiority of derivative-extended over conventional ROMs on two academic and one industry-relevant application which exhibit a variety of challenges for the model reduction approach in optimization.

In this paper, we present four projection-based model order reduction algorithms grounded in Laguerre functions, specifically tailored for linear systems with quadratic outputs. The first algorithm utilises a Galerkin projection strategy, where the projection matrix is construced from the Laguerre expansion coefficient vectors of the system's state impulse response. The resulting reduced-order model accurately matches selected Laguerre expansion coefficients of the original system's impulse response output. The remaining three algorithms implement Petrov-Galerkin projections combined with the low-rank square root balancing method. While these methods share an identical low-rank factor for the controllability Gramian, they differ in their selection of the observability Gramian, which is chosen from either the linearised system, the quadratic-bilinearized system, or the original linear system with quadratic outputs. A notable advantage of these algorithms is their ability to avoid solving large-scale Lyapunov equations. Instead, all low-rank factors are directly constructed from the Laguerre expansion coefficient vectors via recurrence formulas, substantially improving computational efficiency. The effectiveness and performance of the proposed algorithms are demonstrated through a series of numerical experiments.

NAGUMO CONDITIONS FOR ORDINARY DIFFERENTIAL EQUATIONS

The partial eigenvalue assignment problem concerns reassigning afew of undesired eigenvalues of a linear system to suitably chosenlocations and keeping the other large number of eigenvalues andeigenvectors unchanged (no spill-over). This paper considers thepartial eigenvalue assignment problem with time delay robustness.A time delay robustness measure is presented by analyzing thesensitivity of the assigned eigenvalues with respect to timedelay. The problem is formulated as an unconstrained minimizationproblem with the cost function involving the time delay robustnessmeasure. A numerical algorithm with analytical formulation of thegradient for the cost function is provided. A numerical example isincluded to show the effectiveness of the proposed method.