A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (p. 117-126).(cont.) Finally, we present projection schemes which result in improved accuracy of the reduced order TPWL models, as well as discuss approaches leading to guaranteed stable and passive TPWL reduced-order models.In this study we discuss the problem of Model Order Reduction (MOR) for a class of nonlinear dynamical systems. In particular, we consider reduction schemes based on projection of the original state-space to a lower-dimensional space e.g. by using Krylov methods. In the nonlinear case, however, applying a projection-based MOR scheme does not immediately yield computationally efficient macromodels. In order to overcome this fundamental problem, we propose to first approximate the original nonlinear system with a weighted combination of a small set of linearized models of this system, and then reduce each of the models with an appropriate projection method. The linearized models are generated about a state trajectory of the nonlinear system corresponding to a certain 'training' input. As demonstrated by results of numerical tests, the obtained trajectory quasi-piecewise-linear reduced order models are very cost-efficient, while providing superior accuracy as compared to existing MOR schemes, based on single-state Taylor's expansions. In this dissertation, the proposed MOR approach is tested for a number of examples of nonlinear dynamical systems, including micromachined devices, analog circuits (discrete transmission line models, operational amplifiers), and fluid flow problems. The tests validate the extracted models and indicate that the proposed approach can be effectively used to obtain system-level models for strongly nonlinear devices. This dissertation also shows an inexpensive method of generating trajectory piecewise-linear (TPWL) models based on constructing the reduced models 'on-the-fly', which accelerates simulation of the system response. Moreover, we propose a procedure for estimating simulation errors, which can be used to determine accuracy of the extracted trajectory piecewise-linear reduced order models.by MichaÅ Jerzy RewieÅski.Ph.D

Autonomous Volterra algorithm for steady-state analysis of nonlinear circuits

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Precise computer controlled positioning of robot end effectors using force sensors

A major problem in space applications of robotics and docking of spacecraft is the development of technology for automated precise positioning of mating components with smooth motion and soft contact. To achieve the above objective, a design method was developed for optimally placing the closed-loop poles of a discretized robotic control system at exact prescribed locations inside the unit circle of the complex z-plane. The design method combines the merits of the pole placement and the linear quadratic design approaches. The proposed design procedure is based on the assignment of one real eigenvalue or two complex conjugate (or real) eigenvalues at each design step. The method involves solutions of simple algebraic equations and this is considered to be efficient for on-line or off-line computations. Also, two methods for the linearization of the nonlinear model of a robotic manipulator were presented. Since automatic control of multi-degree freedom robotic manipulators involves high nonlinear equations of systems, a pilot project was proposed involving the control of a one-dimensional system. This simple system can be readily implemented for testing the concepts and algorithms

Projection-Based Reduced Order Modeling for Spacecraft Thermal Analysis

This paper presents a mathematically rigorous, subspace projection-based reduced order modeling (ROM) methodology and an integrated framework to automatically generate reduced order models for spacecraft thermal analysis. Two key steps in the reduced order modeling procedure are described: (1) the acquisition of a full-scale spacecraft model in the ordinary differential equation (ODE) and differential algebraic equation (DAE) form to resolve its dynamic thermal behavior; and (2) the ROM to markedly reduce the dimension of the full-scale model. Specifically, proper orthogonal decomposition (POD) in conjunction with discrete empirical interpolation method (DEIM) and trajectory piece-wise linear (TPWL) methods are developed to address the strong nonlinear thermal effects due to coupled conductive and radiative heat transfer in the spacecraft environment. Case studies using NASA-relevant satellite models are undertaken to verify the capability and to assess the computational performance of the ROM technique in terms of speed-up and error relative to the full-scale model. ROM exhibits excellent agreement in spatiotemporal thermal profiles (<0.5% relative error in pertinent time scales) along with salient computational acceleration (up to two orders of magnitude speed-up) over the full-scale analysis. These findings establish the feasibility of ROM to perform rational and computationally affordable thermal analysis, develop reliable thermal control strategies for spacecraft, and greatly reduce the development cycle times and costs

Predictive control using an FPGA with application to aircraft control

Alternative and more efficient computational methods can extend the applicability of MPC to systems with tight real-time requirements. This paper presents a “system-on-a-chip” MPC system, implemented on a field programmable gate array (FPGA), consisting of a sparse structure-exploiting primal dual interior point (PDIP) QP solver for MPC reference tracking and a fast gradient QP solver for steady-state target calculation. A parallel reduced precision iterative solver is used to accelerate the solution of the set of linear equations forming the computational bottleneck of the PDIP algorithm. A numerical study of the effect of reducing the number of iterations highlights the effectiveness of the approach. The system is demonstrated with an FPGA-inthe-loop testbench controlling a nonlinear simulation of a large airliner. This study considers many more manipulated inputs than any previous FPGA-based MPC implementation to date, yet the implementation comfortably fits into a mid-range FPGA, and the controller compares well in terms of solution quality and latency to state-of-the-art QP solvers running on a standard PC

Adaptive Multi-Rate Wavelet Method for Circuit Simulation

In this paper a new adaptive algorithm for multi-rate circuit simulation encountered in the design of RF circuits based on spline wavelets is presented. The ordinary circuit differential equations are first rewritten by a system of (multi-rate) partial differential equations (MPDEs) in order to decouple the different time scales. Second, a semi-discretization by Rothe's method of the MPDEs results in a system of differential algebraic equations DAEs with periodic boundary conditions. These boundary value problems are solved by a Galerkin discretization using spline functions. An adaptive spline grid is generated, using spline wavelets for non-uniform grids. Moreover the instantaneous frequency is chosen adaptively to guarantee a smooth envelope resulting in large time steps and therefore high run time efficiency. Numerical tests on circuits exhibiting multi-rate behavior including mixers and PLL conclude the paper

PLAWE: A piecewise linear circuit simulator using asymptotic waveform evaluation

Ankara : Department of Electrical and Electronics Engineering and the Institute of Engineering and Science of Bilkent University, 1994.Thesis (Ph.D.) -- Bilkent University, 1994.Includes bibliographical references leaves 73-81.A new circuit simulation program, PLAWE, is developed for the transient analysis of VLSI circuits. PLAWE uses Asymptotic Waveform Evaluation (AWE) technique, which is a new method to analyze linear(ized) circuits, and Piecewise Linear (PWL) approach for DC representation of nonlinear elements. AWE employs a form of Pade approximation rather than numerical integration techniques to approximate the response of linear(ized) circuits in either the time or the frequency domain. AWE is typically two or three orders of magnitude faster than traditional simulators in analyzing large linear circuits. However, it can handle only linear(ized) circuits, while the transient analysis problem is generally nonlinear due to the presence of nonlinear devices such as diodes, transistors, etc.. We have applied the AWE technique to the transient simulation of nonlinear circuits by using static PWL models for nonlinear elements. But, finding a good static PWL model which fits well to the actual i — v characteristics of a nonlinear device is not an easy task and in addition, static PWL modelling results in low accuracy. Therefore, we have developed a dynamic PWL modeling technique which uses SPICE models for nonlinear elements to enhance the accuracy of the simulation while preserving the efficiency gain obtained with AWE. Hence, there is no modelling problem and we can adjust the accuracy level by varying some parameters. If the required level of accuracy is increased, more simulation time is needed. Practical examples are given to illustrate the significant improvement in accuracy. For circuits containing especially weakly nonlinear devices, this method is typically at least one order of magnitude faster than HSPICE. A fast and convergent iteration method for piecewise-linear analysis of nonlinear resistive circuits is presented. Most of the existing algorithms are applicable only to a limited class of circuits. In general, they are either not convergent or too slow for large circuits. The new algorithm presented in this thesis is much more efficient than the existing ones and can be applied to any piecewise-linear circuit. It is based on the piecewise-linear version of the Newton-Raphson algorithm. As opposed to the NewtonRaphson method, the new algorithm is globally convergent from an arbitrary starting point. It is simple to understand and it can be easily programmed. Some numerical examples are given in order to demonstrate the effectiveness of the presented algorithm in terms of the amount of computation.Topçu, SatılmışPh.D

Optimal Subharmonic Entrainment

For many natural and engineered systems, a central function or design goal is the synchronization of one or more rhythmic or oscillating processes to an external forcing signal, which may be periodic on a different time-scale from the actuated process. Such subharmonic synchrony, which is dynamically established when N control cycles occur for every M cycles of a forced oscillator, is referred to as N:M entrainment. In many applications, entrainment must be established in an optimal manner, for example by minimizing control energy or the transient time to phase locking. We present a theory for deriving inputs that establish subharmonic N:M entrainment of general nonlinear oscillators, or of collections of rhythmic dynamical units, while optimizing such objectives. Ordinary differential equation models of oscillating systems are reduced to phase variable representations, each of which consists of a natural frequency and phase response curve. Formal averaging and the calculus of variations are then applied to such reduced models in order to derive optimal subharmonic entrainment waveforms. The optimal entrainment of a canonical model for a spiking neuron is used to illustrate this approach, which is readily extended to arbitrary oscillating systems