Method and system for empirical modeling of time-varying, parameter-varying, and nonlinear systems via iterative linear subspace computation

• US 8,898,040 B2
• Filed: 09/03/2010
• Issued: 11/25/2014
• Est. Priority Date: 09/03/2009
• Status: Active Grant

• Current Assignee: Adaptics, Inc.
• Original Assignee: Adaptics, Inc.
• Inventors: Larimore, Wallace E.
• Primary Examiner(s): Saxena, Akash
• Application Number:
  US12/875,456
• Publication Number:
  US 20110054863A1
• Time in Patent Office: 1,544 Days
• Field of Search: 703/2
• US Class Current: 703/2
• CPC Class Codes: G06F 17/18 for evaluating statistical data , e.g. average values, frequency distributions, probability functions, regression analysis forecasting specially adapted for a specific administrative, business or logistic context G06Q10/04
• CPC Class Codes: G06F 2111/10 Numerical modelling
• CPC Class Codes: G06F 2218/00 Aspects of pattern recognition specially adapted for signal processing
• CPC Class Codes: G06F 30/23 using finite element methods [FEM] or finite difference methods [FDM]
• CPC Class Codes: G06F 9/48 Program initiating; Program switching, e.g. by interrupt

Abstract

Methods and systems for estimating differential or difference equations that can govern a nonlinear, time-varying and parameter-varying dynamic process or system. The methods and systems for estimating the equations may be based upon estimations of observed outputs and, when desired, input data for the equations. The methods and systems can be utilized with any system or process that may be capable of being described with nonlinear, time-varying and parameter-varying difference equations and can used for automated extraction of the difference equations in describing detailed system or method behavior for use in system control, fault detection, state estimation and prediction and adaptation of the same to changes in a system or method.

Claims

1. A method of reducing aircraft wing flutter by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, comprising:

sensing, with a sensor, wing flutter on an aircraft;

storing sensed wing flutter data on a memory;

performing at least one of a nonlinear and linear autoregressive with inputs ((N)ARX) model fitting with one of a predetermined autoregressive with inputs (ARX) model and a predetermined nonlinear ARX (NARX) model to the stored wing flutter data, the fitting comprising;

performing a parameter estimation using stored aircraft wing flutter input and output data determined from a predetermined iteration of an algorithm for subspace identification with a processor, at least one of a set of ARX models of increasing order with a specified maximum order or a set of linear regression problems in terms of NARX models of increasing order and monomial degrees with a specified maximum order and degree, comprising;

performing a model comparison, with a processor, to compute an Akaike'"'"'s Information Criterion (AIC) of model fits for at least each ARX order and each NARX order and degree;

selecting a model that minimizes the AIC for at least one of a set of predetermined ARX models with a minimum AIC and a set of predetermined NARX models with a minimum AIC, wherein if more than one model achieves the desired minimum AIC, then selecting the ARX model or NARX model that further minimizes the number of estimated parameters that is also computed in the AIC computation;

performing a state space model fitting of a state space dynamic model of wing flutter that is parametric in its scheduling parameters, with a processor, using the ARX or NARX model selected as minimizing AIC, the state space model fitting comprising;

performing a corrected future calculation, by a processor, the corrected future calculation determining one or more corrected future outputs of wing flutter data through prediction and subtraction of an effect of one or more future inputs of wing flutter data on future outputs of the algorithm;

determining estimates of states with values whose elements are ordered as their predictive correlation for the future by performing a canonical variate analysis (CVA), with a processor, between corrected future outputs and past augmented inputs;

selecting one of a state order that minimizes the AIC or the lowest order of state orders that minimize the AIC;

inputting the estimates of states into one or more state equations;

performing a linear regression calculation on the one or more state equations to determine matrix coefficients of the state equations, and

providing a dynamic model of aircraft wing flutter in the form of state equations with linear parameter varying matrix coefficients as functions of the scheduling parameters to extend subspace identification methods to LPV and nonlinear parameter varying systems with general scheduling functions.

2. The method of reducing aircraft wing flutter by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of claim 1, wherein the ARX model fitting has a linear regression problem that assumes a maximum ARX order to be considered.

3. The method of reducing aircraft wing flutter by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions of claim 1, wherein the fitting of the ARX models further comprises:

a specified ARX order lagp wherein, for the specified ARX order lagp, and for every time t greater than lagp, the prediction of outputs y_t using an autoregression of past outputs v<sub>t−

i</sub> for 0<

i<

lagp+1, and an exogenous moving average comprising past inputs u<sub>t−

i</sub>, and further augmented past inputs ρ

<sub>t−

i</sub>u<sub>t−

i</sub> and augmented state estimates ρ

<sub>t−

i</sub>x<sub>t−

i</sub> augmented respectively by Kronecker products that are similarly time shifted past scheduling functions ρ

<sub>t−

i</sub> for −

1<

i<

lagp+1.