Low-order Linear System Research Articles

Learning of Causal Observable Functions for Koopman-DFL Lifting Linearization of Nonlinear Controlled Systems and Its Application to Excavation Automation

Effective and causal observable functions for low-order lifting linearization of nonlinear controlled systems are learned from data by using neural networks. While Koopman operator theory allows us to represent a nonlinear system as a linear system in an infinite-dimensional space of observables, exact linearization is guaranteed only for autonomous systems with no input, and finding effective observable functions for approximation with a low-order linear system remains an open question. Dual-Faceted Linearization uses a set of effective observables for low-order lifting linearization, but the method requires knowledge of the physical structure of the nonlinear system. Here, a data-driven method is presented for generating a set of nonlinear observable functions that can accurately approximate a nonlinear control system to a low-order linear control system. A caveat in using data of measured variables as observables is that the measured variables may contain input to the system, which incurs a causality contradiction when lifting the system, i.e., taking derivatives of the observables. The current work presents a method for eliminating such anti-causal components of the observables and lifting the system using only causal observables. The method is applied to excavation automation, a complex nonlinear dynamical system, to obtain a low-order lifted linear model for control design.

Learning Output Reference Model Tracking for Higher-Order Nonlinear Systems with Unknown Dynamics

This work suggests a solution for the output reference model (ORM) tracking control problem, based on approximate dynamic programming. General nonlinear systems are included in a control system (CS) and subjected to state feedback. By linear ORM selection, indirect CS feedback linearization is obtained, leading to favorable linear behavior of the CS. The Value Iteration (VI) algorithm ensures model-free nonlinear state feedback controller learning, without relying on the process dynamics. From linear to nonlinear parameterizations, a reliable approximate VI implementation in continuous state-action spaces depends on several key parameters such as problem dimension, exploration of the state-action space, the state-transitions dataset size, and a suitable selection of the function approximators. Herein, we find that, given a transition sample dataset and a general linear parameterization of the Q-function, the ORM tracking performance obtained with an approximate VI scheme can reach the performance level of a more general implementation using neural networks (NNs). Although the NN-based implementation takes more time to learn due to its higher complexity (more parameters), it is less sensitive to exploration settings, number of transition samples, and to the selected hyper-parameters, hence it is recommending as the de facto practical implementation. Contributions of this work include the following: VI convergence is guaranteed under general function approximators; a case study for a low-order linear system in order to generalize the more complex ORM tracking validation on a real-world nonlinear multivariable aerodynamic process; comparisons with an offline deep deterministic policy gradient solution; implementation details and further discussions on the obtained results.

Localized method of approximate particular solutions for solving unsteady Navier–Stokes problem

The localized method of approximate particular solutions (LMAPS) is proposed to solve two-dimensional transient incompressible Navier–Stokes systems of equations in primitive variables. The equations contain the Laplacian operator. In avoiding ill-conditioning problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear system within local supporting domain in which five nearest neighboring points and multiquadrics are used for interpolation. Then local matrices are reformulated in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The method is assessed on driven cavity problem and flow around cylinder. The numerical experiments show that the newly developed LMAPS is suitable for solving incompressible Navier–Stokes equations with high accuracy and efficiency.

Local RBFs Based Collocation Methods for Unsteady Navier-Stokes Equations

AbstractThe local RBFs based collocation methods (LRBFCM) is presented to solve two-dimensional incompressible Navier-Stokes equations. In avoiding the ill-conditioned problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear systems within local supporting domain. Then, we reformulate local matrix in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The numerical experiments have shown that the developed LRBFCM is suitable for solving the incompressible Navier-Stokes equations with high accuracy and efficiency.

Integral sliding mode based optimal composite nonlinear feedback control for a class of systems

This paper presents an optimization method of designing the integral sliding mode (ISM) based composite nonlinear feedback (CNF) controller for a class of low order linear systems with input saturation. The optimal CNF control is first designed as a nominal control to yield high tracking speed and low overshoot. The selection of all the tuning parameters for the CNF control law is turned into a minimization problem and solved automatically by particle swarm optimization (PSO) algorithm. Subsequently, the discontinuous control law is introduced to reject matched disturbances. Then, the optimal ISM-CNF control law is achieved as the sum of the optimal CNF control law and the discontinuous control law. The effectiveness of the optimal ISM-CNF controller is verified by comparing with a step by step designed one. High tracking performance is achieved by applying the optimal ISM-CNF controller to the tracking control of the micromirror.

Local method of approximate particular solutions for two-dimensional unsteady Burgers’ equations

The local method of approximate particular solutions (LMAPS) is first proposed to solve Burgers’ equations. In avoiding the ill-conditioning problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear systems within the local supporting domain. Then the global solutions can be obtained by reformulating the local matrix in the global and sparse matrix. The obtained large sparse linear systems are directly solved instead of using more complicated iterative method. The numerical experiments have shown that the proposed method is suitable for solving the two-dimensional unsteady Burgers’ equations with high accuracy and efficiency.

A comparison study of the LMAPS method and the LDQ method for time-dependent problems

This paper compares numerical solutions of spatial-temporal partial differential equations based on two RBF-based meshless methods: the local method of approximate particular solutions (LMAPS) and the local RBFs-based DQ method (LDQ). To avoid the ill-conditioned problems of the global version, the weighting coefficients at the supporting points are determined by solving low-order linear systems instead of large dense linear systems. The Runge–Kutta method is adopted for time stepping schemes. The numerical experiments have shown that the LMPAS method and the LDQ method are capable of solving the initial boundary value problem for spatial-temporal partial differential equations with high accuracy and efficiency.

Modeling SOFC plants on the distribution system using identification algorithms

Abstract To determine the potential impacts of fuel cells on future distribution system, dynamic models of fuel cells should be created, reduced in order, and scattered throughout test feeders. This paper presents the implementation of an efficient method for computing low-order linear system models of solid oxide fuel cells (SOFCs) from time domain simulations. The method is the Box–Jenkins algorithm for calculating the transfer function of a linear system from samples of its input and output.

Optimal Control Systems

1R6. Optimal Control Systems. - DS Naidu (Idaho State Univ, Pocatello ID). CRC Press LLC, Boca Raton FL. 2003. 433 pp. ISBN 0-8493-0892-5. $99.95.Reviewed by I Kolmanovsky (Sci Res Lab, MD-2036, Ford Motor Co, 2101 Village Rd, Dearborn MI 48124).The book, Optimal Control Systems, by DS Naidu provides an introduction to key concepts and methods of the optimal control theory for deterministic continuous-time and discrete-time finite-dimensional dynamic systems. The material of the book is based on the author’s experience teaching graduate level optimal control courses and has grown out of the lecture notes used by the author in these courses. After an Introduction (the first chapter of the book), the second chapter describes in detail the main ideas in the calculus of variations (culminating in the study of Euler-Lagrange equations and second variation-based Legendre conditions) and seamlessly transitions to the treatment of the optimal control problems. The Hamiltonian function is introduced early on which subsequently features prominently in the treatment of the Pontryagin’s maximum principle. The end result of Chapter 2 is a solution to several types of optimal control problems (such as fixed end-time, with no control constraints) using the principles of the calculus of variations but stated in the formalism that will later be useful for more general optimal control problems. These developments are already sufficient for the in-depth treatment of Linear Quadratic Optimal Control problem for continuous-time systems, which is one of the central tools in the control theory. This treatment is a subject of Chapters 3 and 4 where the finite horizon and infinite horizon cases, set point and trajectory tracking, fixed-end point regulator system, Linear Quadratic Regulator with a specified degree of stability, and other topics are considered in detail. Frequency domain properties of the linear quadratic regulator are analyzed and the celebrated results on one half to infinity gain margin and 60 degrees phase margin are derived. Chapter 5 treats similar issues as in Chapters 2-4 but for discrete-time systems. In Chapter 6, the treatment of constrained control problems begins with the introduction of the Pontryagin maximum principle for these systems (using the notion of admissible variations) and dynamic programming techniques. The latter are also used to illustrate an alternative approach to deriving continuous-time and discrete-time linear quadratic regulator solutions. Time-optimal control problems are treated in Chapter 7 including the analysis of the number of switchings in the optimal “bang-bang” control laws for linear systems. Other special optimal control problems (energy and fuel-optimal control problems with integral costs) including the case of singular controls are touched upon as well. The developments are completed by considering the point wise-in-time state constraints and describing how they can be treated using either the penalty function method or the slack variable method. The appendices help make the book self-contained by reviewing some of the key results in linear algebra and state space analysis of linear systems, and by providing the listings of the relevant Matlab files (available electronically from the author). The book is very readable and careful attention has certainly been given to provide the reader with a comprehensive set of tools without overburdening with the complex mathematical details and notations (although the key mathematical ideas are, indeed, well exposed and the logic underlying the developments is transparent and thorough). It should be accessible to a wide audience including graduate students and practitioners from engineering and other fields. Optimal Control Systems can be utilized both as a textbook in the introductory courses in the optimal control theory, or as a quick reference by practicing engineers. The use of Matlab/Simulink to treat the examples (including the snippets of the actual code), historical remarks about the scientists behind the major discoveries in the optimal control theory, recipe-style summaries of the key methodologies, exercise problems at the end of each of the chapters are undoubtedly the strong points of this book which significantly contribute to its pedagogical value. On a somewhat more critical side, most of the examples in the book are, in fact, of relatively low order linear systems. This can actually be viewed as an advantage in that the in-depth treatment of these examples is possible so that the main ideas can be illustrated rather well. At the same time, the treatment of more of higher order nonlinear system examples in combination with a more extended introduction into the numerical methods of the optimal control could have been beneficial for the readers to gain an additional insight into how the underlying techniques can be applied.

Performance comparison of three identification methods for the analysis of electromechanical oscillations

This paper describes the results of a study to evaluate the performance of three identification methods for the study of low frequency electromechanical oscillations. The three identification methods considered are: the Steiglitz-McBride algorithm; the eigensystem realization algorithm; and the Prony method. The identification methods are used to identify low order linear systems of power systems modeled in standard transient stability programs. This is accomplished by processing the system response to a simple probing pulse. The frequency domain characteristics of several identified systems are compared using three power systems with lightly damped electromechanical modes.

Identification of low-order linear power system models from EMTP simulations using the Steiglitz-McBride algorithm

This paper presents the implementation of an efficient method for computing low order linear system models of power systems from time domain simulations. The method is the identification algorithm initially introduced by K. Steiglitz and L.E. McBride for calculating the transfer function of a linear system from samples of its input and output. The paper describes the original algorithm, added extensions, and issues addressed for its practical implementation. The applicability of the method is demonstrated by computing a low-order equivalent linear system of a detailed three-phase power system, modeled on an Electromagnetic Transients Program (EMTP). The paper discusses practical implementation issues.

Human ocular counterrolling induced by varying linear accelerations.

Ocular counterrolling (OCR) has previously been studied using static head tilt or continuous rotation about the line of sight as a stimulus to the otolith organs. This study presents the first measurements of OCR in humans induced by linear accelerations. Dynamic measurements of the response to lateral linear acceleration indicate the eye movements to be on the order of 2 degrees for 0.2 g peak acceleration, 0.2 Hz sinusoidal acceleration. These values are consistent with static OCR studies. The dynamics of the response are similar to a low order linear system with a dominant time constant of 0.33 s. A previous model predicts a time constant of 0.32 s. Sinusoidal oscillation at 0.2, 0.4, and 1.0 Hz with a 0.2 g peak acceleration showed good agreement with the model in both gain and phase. The question of amplitude linearity remains unsettled. This otolithocular reflex, over short periods at least, appears to be stationary in the statistical sense.

On-line identification of sensory systems using pseudorandom binary noise perturbations

A technique of on-line identification of linear system characteristics from sensory systems with spike train or analog voltage outputs was developed and applied to the semicircular canal. A pseudorandom binary white noise input was cross-correlated with the system's output to produce estimates of linear system unit impulse responses (UIRs), which were then corrected for response errors of the input transducers. The effects of variability in the system response characteristics and sensitivity were studied by employing the technique with known linear analog circuits. First-order unit afferent responses from the guitarfish horizontal semicircular canal were cross-correlated with white noise rotational acceleration inputs to produce non-parametric UIR models. In addition, the UIRs were fitted by nonlinear regression to truncated exponential series to produce parametric models in the form of low-order linear system equations. The experimental responses to the white noise input were then compared with those predicted from the UIR models linear convolution, and the differences were expressed as a percent mean-square-error (%MSE). The average difference found from a population of 62 semicircular canal afferents was relatively low mean and standard deviation of 10.2 +/- 5.9 SD%MSE, respectively. This suggests that relatively accurate inferences can be made concerning the physiology of the semicircular canal from the linear characteristics of afferent responses.

Fundamental properties of the impulse response of low-order linear systems†

The effect of pole-zero geometry on the impulse response of low-order linear systems has been established, and a theorem presented for the existence of sign changes in the response of three-pole, one-zero systems. Impulse domain approximations are investigated, and a straightforward parameter estimation procedure developed. The paper is a contribution to the knowledge required for the exploitation of pseudorandom binary sequences for system dynamic testing.