Control and Estimation Oriented Model Order Reduction for Linear and Nonlinear Systems

Optimization based controls are advantageous in meeting stringent performance requirements and accommodating constraints. Although computers are becoming more powerful, solving optimization problems in real-time remains an obstacle because of associated computational complexity. Research efforts to address real-time optimization with limited computational power have intensified over the last decade, and one direction that has shown some success is model order reduction. This dissertation contains a collection of results relating to open- and closed-loop reduction techniques for large scale unconstrained linear descriptor systems, constrained linear systems, and nonlinear systems. For unconstrained linear descriptor systems, this dissertation develops novel gramian and Riccati solution approximation techniques. The gramian approximation is used for an open-loop reduction technique following that of balanced truncation proposed by (Moore, 1981) for ordinary linear systems and (Stykel, 2004) for linear descriptor systems. The Riccati solution is used to generalize the Linear Quadratic Gaussian balanced truncation (LQGBT) of (Verriest, 1981) and (Jonckheere and Silverman, 1983). These are applied to an electric machine model to reduce the number of states from $>$100000 to 8 while improving accuracy over the state-of-the-art modal truncation of (Zhou, 2015) for the purpose of condition monitoring. Furthermore, a link between unconstrained model predictive control (MPC) with a terminal penalty and LQG of a linear system is noted, suggesting an LQGBT reduced model as a natural model for reduced MPC design. The efficacy of such a reduced controller is demonstrated by the real-time control of a diesel airpath. Model reduction generally introduces modeling errors, and controlling a constrained plant subject to modeling errors falls squarely into robust control. A standard assumption of robust control is that inputs/states/outputs are constrained by convex sets, and these sets are ``tightened'' for robust constraint satisfaction. However, robust control is often overly conservative, and resulting control strategies cannot take advantage of the true admissible sets. A new reduction problem is proposed that considers the reduced order model accuracy and constraint conservativeness. A constant tube methodology for reduced order constrained MPC is presented, and the proposed reduced order model is found to decrease the constraint conservativeness of the reduced order MPC law compared to reduced order models obtained by gramian and LQG reductions. For nonlinear systems, a reformulation of the empirical gramians of (Lall et al., 1999) and (Hahn et al., 2003) into simpler, yet more general forms is provided. The modified definitions are used in the balanced truncation of a nonlinear diesel airpath model, and the reduced order model is used to design a reduced MPC law for tracking control. Further exploiting the link between the gramian and Riccati solution for linear systems, the new empirical gramian formulation is extended to obtain empirical Riccati covariance matrices used for closed-loop model order reduction of a nonlinear system. Balanced truncation using the empirical Riccati covariance matrices is demonstrated to result in a closer-to-optimal nonlinear compensator than the previous balanced truncation techniques discussed in the dissertation.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/140839/1/riboch_1.pd

Maximum Entropy/Optimal Projection (MEOP) control design synthesis: Optimal quantification of the major design tradeoffs

The underlying philosophy and motivation of the optimal projection/maximum entropy (OP/ME) stochastic modeling and reduced control design methodology for high order systems with parameter uncertainties are discussed. The OP/ME design equations for reduced-order dynamic compensation including the effect of parameter uncertainties are reviewed. The application of the methodology to several Large Space Structures (LSS) problems of representative complexity is illustrated

emgr - The Empirical Gramian Framework

System Gramian matrices are a well-known encoding for properties of input-output systems such as controllability, observability or minimality. These so-called system Gramians were developed in linear system theory for applications such as model order reduction of control systems. Empirical Gramian are an extension to the system Gramians for parametric and nonlinear systems as well as a data-driven method of computation. The empirical Gramian framework - emgr - implements the empirical Gramians in a uniform and configurable manner, with applications such as Gramian-based (nonlinear) model reduction, decentralized control, sensitivity analysis, parameter identification and combined state and parameter reduction

Statistical Learning For System Identification, Estimation, And Control

Despite the recent widespread success of machine learning, we still do not fully understand its fundamental limitations. Going forward, it is crucial to better understand learning complexity, especially in critical decision making applications, where a wrong decision can lead to catastrophic consequences. In this thesis, we focus on the statistical complexity of learning unknown linear dynamical systems, with focus on the tasks of system identification, prediction, and control. We are interested in sample complexity, i.e. the minimum number of samples required to achieve satisfactory learning performance. Our goal is to provide finite-sample learning guarantees, explicitly highlighting how the learning objective depends on the number of samples. A fundamental question we are trying to answer is how system theoretic properties of the underlying process can affect sample complexity. Using recent advances in statistical learning, high-dimensional statistics, and control theoretic tools, we provide finite-sample guarantees in the following settings. i) System Identification. We provide the first finite-sample guarantees for identifying a stochastic partially-observed system; this problem is also known as the stochastic system identification problem. ii) Prediction. We provide the first end-to-end guarantees for learning the Kalman Filter, i.e. for learning to predict, in an offline learning architecture. We also provide the first logarithmic regret guarantees for the problem of learning the Kalman Filter in an online learning architecture, where the data are revealed sequentially. iii) Difficulty of System Identification and Control. Focusing on fully-observed systems, we investigate when learning linear systems is statistically easy or hard, in the finite sample regime. Statistically easy to learn linear system classes have sample complexity that is polynomial with the system dimension. Statistically hard to learn linear system classes have worst-case sample complexity that is at least exponential with the system dimension. We show that there actually exist classes of linear systems, which are hard to learn. Such classes include indirectly excited systems with large degree of indirect excitation. Similar conclusions hold for both the problem of system identification and the problem of learning to control

Large Scale Data Assimilation with Application to the Ionosphere-Thermosphere.

Data assimilation is the process of merging measurement data with a model to estimate the states of a system that are not directly measured. By means of data assimilation, we can expand the effectiveness of limited measurements by using the model and, at the same time, increase the accuracy of model estimates using the measurements. In this dissertation, we survey and develop data assimilation algorithms that are applicable to large-scale nonlinear systems. Very high order dynamics, nonlinearity, and input uncertainties are addressed since they characterize the problems associated with large-scale data assimilation. Specifically, we focus on developing the data assimilation algorithms for the ionosphere-thermosphere using the Global Ionosphere-Thermosphere Model (GITM). For developing computationally tractable algorithms, we obtain finite-horizon optimal reduced-order estimators for time-varying linear systems, and, subsequently, develop linear suboptimal reduced-complexity estimators. The suboptimal estimators are based on localization and the reduced-rank square root of the error covariance. To deal with nonlinearity, we use the unscented Kalman filter and ensemble Kalman filter. We apply suboptimal reduced-complexity algorithms developed for linear systems based on the unscented Kalman filter. Also, we develop the ensemble-on-demand Kalman filter, which can be used for the special case of a single global disturbance, and which avoids propagating the ensemble members for all of the time steps. Furthermore, we show that the ensemble size of the ensemble Kalman filter does not have to be unnecessarily large if the statistics of the disturbance sources are identified. Finally, we apply the ensemble-on-demand Kalman filter and ensemble Kalman filter to data assimilation based on GITM for uncertain solar EUV flux and geomagnetic storm conditions, respectively. We present data assimilation results, through extensive numerical investigations using simulated measurements. While performing simulations, we observe that poor correlations between states should be set to zero to avoid filter instability. In addition, ionosphere and thermosphere measurements can be used together with an appropriate region of data injection to guarantee overall good estimation performance. With those constraints, we show that good estimation results can be obtained using a small ensemble size for each ensemble filter.Ph.D.Aerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60758/1/iskim_1.pd

Sensor Placement Algorithms for Process Efficiency Maximization

Even though the senor placement problem has been studied for process plants, it has been done for minimizing the number of sensors, minimizing the cost of the sensor network, maximizing the reliability, or minimizing the estimation errors. In the existing literature, no work has been reported on the development of a sensor network design (SND) algorithm for maximizing efficiency of the process. The SND problem for maximizing efficiency requires consideration of the closed-loop system, which is unlike the open-loop systems that have been considered in previous works. In addition, work on the SND problem for a large fossil energy plant such as an integrated gasification combined cycle (IGCC) power plant with CO2 capture is rare.;The objective of this research is to develop a SND algorithm for maximizing the plant performance using criteria such as efficiency in the case of an estimator-based control system. The developed algorithm will be particularly useful for sensor placement in IGCC plants at the grassroots level where the number, type, and location of sensors are yet to be identified. In addition, the same algorithm can be further enhanced for use in retrofits, where the objectives could be to upgrade (addition of more sensors) and relocate existing sensors to different locations. The algorithms are developed by considering the presence of an optimal Kalman Filter (KF) that is used to estimate the unmeasured and noisy measurements given the process model and a set of measured variables. The designed algorithms are able to determine the location and type of the sensors under constraints on budget and estimation accuracy. In this work, three SND algorithms are developed: (a) steady-state SND algorithm, (b) dynamic model-based SND algorithm, and (c) nonlinear model-based SND algorithm. These algorithms are implemented in an acid gas removal (AGR) unit as part of an IGCC power plant with CO2 capture. The AGR process involves extensive heat and mass integration and therefore, is very suitable for the study of the proposed algorithm in the presence of complex interactions between process variables

Effective Statistical Control Strategies for Complex Turbulent Dynamical Systems

Control of complex turbulent dynamical systems involving strong nonlinearity and high degrees of internal instability is an important topic in practice. Different from traditional methods for controlling individual trajectories, controlling the statistical features of a turbulent system offers a more robust and efficient approach. Crude first-order linear response approximations were typically employed in previous works for statistical control with small initial perturbations. This paper aims to develop two new statistical control strategies for scenarios with more significant initial perturbations and stronger nonlinear responses, allowing the statistical control framework to be applied to a much wider range of problems. First, higher-order methods, incorporating the second-order terms, are developed to resolve the full control-forcing relation. The corresponding changes to recovering the forcing perturbation effectively improve the performance of the statistical control strategy. Second, a mean closure model for the mean response is developed, which is based on the explicit mean dynamics given by the underlying turbulent dynamical system. The dependence of the mean dynamics on higher-order moments is closed using linear response theory but for the response of the second-order moments to the forcing perturbation rather than the mean response directly. The performance of these methods is evaluated extensively on prototype nonlinear test models, which exhibit crucial turbulent features, including non-Gaussian statistics and regime switching with large initial perturbations. The numerical results illustrate the feasibility of different approaches due to their physical and statistical structures and provide detailed guidelines for choosing the most suitable method based on the model properties

Proceedings of the Workshop on Applications of Distributed System Theory to the Control of Large Space Structures

Two general themes in the control of large space structures are addressed: control theory for distributed parameter systems and distributed control for systems requiring spatially-distributed multipoint sensing and actuation. Topics include modeling and control, stabilization, and estimation and identification

Feedback control of parametrized PDEs via model order reduction and dynamic programming principle

In this paper, we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of basis functions, we combine recent basis generation methods and parameter partitioning techniques. Furthermore, we present a novel technique to construct non-uniform grids in the reduced domain, which is based on statistical information. Finally, we discuss numerical examples to illustrate the effectiveness of the proposed methods for PDEs in two space dimensions