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

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.

What is claimed is: 
     
         1 . A method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, comprising:
 sensing, with a sensor, dynamic system data;   storing sensed dynamic system 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 dynamic system data, the fitting comprising:
 performing a parameter estimation using stored dynamic system 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 dynamic system operation 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 dynamic system data through prediction and subtraction of an effect of one or more future inputs of dynamic system 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 dynamic system data 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 modelling a dynamic system 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 modelling a dynamic system 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 y t−i  for 0<i<lagp+1, and an exogenous moving average comprising past inputs u t−i , and further augmented past inputs   
       
         
           
             
               
                   
               
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       and augmented state estimates 
       
         
           
             
               
                   
               
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       augmented respectively by Kronecker products that are similarly time shifted past scheduling functions ρ t−i  for −1<i<lagp+1. 
     
     
         4 . The method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of  claim 1 , wherein the dynamic system is an engine. 
     
     
         5 . The method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of  claim 4 , wherein the engine is an automotive engine. 
     
     
         6 . The method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of  claim 4 , wherein the engine is a turbine engine. 
     
     
         7 . The method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of  claim 1 , wherein the dynamic system is a chemical process. 
     
     
         8 . The method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of  claim 7 , wherein the chemical process occurs in a stirred tank reactor. 
     
     
         9 . The method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of  claim 7 , wherein the chemical process occurs in one or more distillation columns. 
     
     
         10 . The method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of  claim 1 , wherein the dynamic system is an aircraft. 
     
     
         11 . The method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of  claim 1 , wherein the dynamic system is an automobile. 
     
     
         12 . The method of modelling a dynamic system by extending subspace identification methods to linear parameter varying (LPV) and nonlinear parameter varying systems with general scheduling functions, of  claim 1 , wherein the dynamic system is a ship.