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

Empirical Model Reduction of Controlled Nonlinear Systems

In this paper we introduce a new method of model reduction for nonlinear systems with inputs and outputs. The method requires only standard matrix computations, and when applied to linear systems results in the usual balanced truncation. For nonlinear systems, the method makes used of the Karhunen-Lo`eve decomposition of the state-space, and is an extension of the method of empirical eigenfunctions used in fluid dynamics. We show that the new method is equivalent to balanced-truncation in the linear case, and perform an example reduction for a nonlinear mechanical system

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

Empirical differential Gramians for nonlinear model reduction

In this paper, we present an empirical balanced truncation method for nonlinear systems whose input vector fields are constants. First, we define differential reachability and observability Gramians. They are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory), and it is difficult to find them as functions of the initial state and input. The main result of this paper is to show that for a fixed state trajectory, it is possible to compute the values of these Gramians by using impulse and initial state responses of the variational system. Therefore, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations, differently from conventional nonlinear balancing methods. We further develop an approximation method, which only requires trajectories of the original nonlinear systems

On the empirical balanced truncation for 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

A Unified Software Framework for Empirical Gramians

A common approach in model reduction is balanced truncation, which is based on Gramian matrices classifying certain attributes of states or parameters of a given dynamic system. Initially restricted to linear systems, the empirical Gramians not only extended this concept to nonlinear systems but also provided a uniform computational method. This work introduces a unified software framework supplying routines for six types of empirical Gramians. The Gramian types will be discussed and applied in a model reduction framework for multiple-input multiple-output systems

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

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