Parametric universal nonlinear dynamics approximator and use

Abstract

System and method for modeling a nonlinear process. A combined model for predictive optimization or control of a nonlinear process includes a nonlinear approximator, coupled to a parameterized dynamic or static model, operable to model the nonlinear process. The nonlinear approximator receives process inputs, and generates parameters for the parameterized dynamic model. The parameterized dynamic model receives the parameters and process inputs, and generates predicted process outputs based on the parameters and process inputs, where the predicted process outputs are useable to analyze and/or control the nonlinear process. The combined model may be trained in an integrated manner, e.g., substantially concurrently, by identifying process inputs and outputs (I/O), collecting data for process I/O, determining constraints on model behavior from prior knowledge, formulating an optimization problem, executing an optimization algorithm to determine model parameters subject to the determined constraints, and verifying the compliance of the model with the constraints.

1 . A combined model for predictive optimization or control of a nonlinear process, comprising: 
 a nonlinear approximator; and    a parameterized dynamic model, coupled to the nonlinear approximator, wherein the parameterized dynamic model is operable to model the nonlinear process;    wherein the nonlinear approximator is operable to: 
 receive one or more process inputs; and  
 generate one or more parameters for the parameterized dynamic model;  
   wherein the parameterized dynamic model is operable to: 
 receive the one or more parameters;  
 receive the one or more process inputs; and  
 generate one or more predicted process outputs based on the received one or more parameters and the received one or more process inputs; and  
   wherein the one or more predicted process outputs are useable to analyze and/or control the nonlinear process.    
     
     
         2 . The combined model of  claim 1 , wherein the combined model is operable to be trained to model the nonlinear process in an integrated manner, and wherein the nonlinear approximator and the parameterized dynamic model are trained together substantially concurrently.  
     
     
         3 . The combined model of  claim 2 , 
 wherein the combined model is operable to be trained to model the nonlinear process in an integrated manner by an optimization process; and    wherein the optimization process is operable to perform an optimization algorithm to determine model parameters for the parameterized dynamic model.    
     
     
         6 . The combined model of  claim 3 , 
 wherein the combined model is operable to be coupled to the nonlinear process or a representation of the nonlinear process;    wherein the nonlinear process is operable to receive the one or more process inputs and produce the one or more process outputs;    wherein the optimization process is operable to determine model errors based on the one or more process outputs and the one or more predicted process outputs; and    wherein the optimization process is operable to train the combined model in an iterative manner using the model errors and an optimizer.    
     
     
         7 . The combined model of  claim 6 , wherein, in training the combined model in an iterative manner using the model errors and an optimizer, the optimization process is operable to: 
 identify process inputs and outputs (I/O);    collect data for process inputs and outputs I/O;    determine constraints on model behavior from prior knowledge;    formulate an optimization problem;    execute an optimization algorithm to determine model parameters subject to the determined constraints by solving the optimization problem; and    verify the compliance of the model with the specified constraints.    
     
     
         8 . The combined model of  claim 7 , wherein, in verifying the compliance of the model with the specified constraints, the optimization process is operable to: 
 use interval arithmetic over the global input region; and/or use interval arithmetic with input-region partitioning.    
     
     
         9 . The method of  claim 7 , wherein in executing an optimization algorithm to determine model parameters, the optimization process is operable to: 
 execute the optimization algorithm to determine an optimal order of the model.    
     
     
         10 . The combined model of  claim 7 , 
 wherein the optimization process is further operable to: 
 determine an order of the model; and  
   wherein, in executing the optimization algorithm to determine model parameters, the optimization process is operable to: 
 execute the optimization algorithm to determine optimal parameters of the model based on the determined order of the model.  
   
     
     
         11 . The combined model of  claim 7 , 
 wherein, in formulating the optimization problem, the optimization process is operable to determine or modify an objective function.    
     
     
         12 . The combined model of  claim 7 , 
 wherein, in solving the optimization problem, the optimization process is operable to solve an objective function subject to the determined constraints.    
     
     
         13 . The combined model of  claim 2 , 
 wherein, after being trained, the overall behavior of the combined model is consistent with a priori knowledge of the nonlinear process.    
     
     
         14 . The combined model of  claim 1 , wherein the nonlinear approximator comprises one or more of: 
 a neural network;    a support vector machine;    a statistical model;    a parametric description of the nonlinear process;    a Fourier series model; and    an empirical model.    
     
     
         15 . The combined model of  claim 1 , wherein the nonlinear approximator comprises a universal nonlinear approximator.  
     
     
         16 . The combined model of  claim 1 , wherein the nonlinear approximator includes a feedback loop, and wherein the feedback loop is operable to provide output of the nonlinear approximator from a previous cycle as input to the nonlinear approximator for a current cycle.  
     
     
         17 . The combined model of  claim 1 , wherein the parameterized dynamic comprises a multi-input, multi-output (MIMO) dynamic model implemented with a set of difference equations.  
     
     
         18 . The combined model of  claim 17 , wherein the set of difference equations comprises a set of discrete time polynomials.  
     
     
         19 . The combined model of  claim 17 , wherein the one or more process inputs are received from one or more of: 
 the nonlinear process; and    a representation of the nonlinear process.    
     
     
         20 . The combined model of  claim 19 , wherein the a representation of the nonlinear process comprises one or more of: 
 a first principles model;    a statistical model;    a parametric description of the nonlinear process;    a Fourier series model;    an empirical model; and    empirical data.    
     
     
         21 . The combined model of  claim 1 , wherein the combined model is operable to be coupled to the nonlinear process, wherein the combined model is further operable to be coupled to a control process, wherein the control process is operable to: 
 a) initialize the model to a current status of the nonlinear process;    b) determine parameters of the model, including manipulated variables;    c) generate a profile of manipulated variables;    d) operate the model in accordance with the generated profile of manipulated variables, thereby generating a model response;    e) determine a deviation of the model response from a desired behavior;    f) repeat c)-e) one or more times to determine an optimal profile of manipulated variables;    g) operate the nonlinear process in accordance with the optimal profile of manipulated variables, thereby generating process output; and    h) provide the nonlinear process output as input to the model; and    i) repeat a)-h) one or more times to dynamically control the nonlinear process.    
     
     
         22 . A method for training a model of a nonlinear process, the method comprising: 
 identifying process inputs and outputs (I/O);    collecting data for process inputs and outputs I/O;    determining constraints on model behavior from prior knowledge;    formulating an optimization problem;    executing an optimization algorithm to determine model parameters subject to the determined constraints by solving the optimization problem; and    verifying the compliance of the model with the specified constraints.    
     
     
         23 . The method of  claim 22 , wherein said verifying the compliance of the model with the specified constraints comprises one or more of: 
 using interval arithmetic over the global input region; and    using interval arithmetic with input-region partitioning.    
     
     
         24 . The method of  claim 22 , wherein said executing an optimization algorithm to determine model parameters comprises: 
 executing the optimization algorithm to determine an optimal order of the model.    
     
     
         25 . The method of  claim 22 , further comprising: 
 determining an order of the model; and    wherein said executing an optimization algorithm to determine model parameters comprises: 
 executing the optimization algorithm to determine optimal parameters of the model based on the determined order of the model.  
   
     
     
         26 . The method of  claim 22 , 
 wherein the model comprises a parametric universal nonlinear dynamics approximator (PUNDA) model, comprising: 
 a nonlinear approximator; and  
 a parameterized dynamic model, coupled to the nonlinear approximator, wherein the parameterized dynamic model is operable to model the nonlinear process; and  
   wherein, after said verifying, the overall behavior of the PUNDA model is consistent with the prior knowledge.    
     
     
         27 . The method of  claim 22 , 
 wherein formulating the optimization problem comprises: 
 determining an objective function; and  
   wherein solving the optimization problem comprises: 
 solving the objective function subject to the determined constraints.  
   
     
     
         28 . A system for training a model of a nonlinear process, the system comprising: 
 means for identifying process inputs and outputs (I/O);    means for collecting data for process inputs and outputs I/O;    means for determining constraints on model behavior from prior knowledge;    means for formulating an optimization problem;    means for executing an optimization algorithm to determine model parameters subject to the determined constraints by solving the optimization problem; and    means for verifying the compliance of the model with the specified constraints.    
     
     
         29 . A method for controlling a nonlinear process, the method comprising: 
 a) initializing the model to a current status of the nonlinear process;    b) determining parameters of the model, including manipulated variables;    c) generating a profile of manipulated variables;    d) operating the model in accordance with the generated profile of manipulated variables, thereby generating a model response;    e) determining a deviation of the model response from a desired behavior;    f) repeating c)-e) one or more times to determine an optimal profile of manipulated variables;    g) operating the nonlinear process in accordance with the optimal profile of manipulated variables, thereby generating process output; and    h) providing the nonlinear process output as input to the model; and repeating a)-h) one or more times to dynamically control the nonlinear process.    
     
     
         30 . The method of  claim 29 , further comprising: 
 i) modifying the optimization problem based on the input to the model;    wherein said repeating a)-h) comprises repeating a)-i).    
     
     
         31 . The method of  claim 30 , wherein said modifying the optimization problem comprises modifying one or more of: 
 constraints;    an objective function;    model parameters;    optimization parameters; and    optimization data.    
     
     
         32 . A system for controlling a nonlinear process, the system comprising: 
 means for a) initializing the model to a current status of the nonlinear process;    means for b) determining parameters of the model, including manipulated variables;    means for c) generating a profile of manipulated variables;    means for d) operating the model in accordance with the generated profile of manipulated variables, thereby generating a model response;    means for e) determining a deviation of the model response from a desired behavior;    means for f) repeating c)-e) one or more times to determine an optimal profile of manipulated variables;    means for g) operating the nonlinear process in accordance with the optimal profile of manipulated variables, thereby generating process output; and    means for h) providing the nonlinear process output as input to the model; and    means for repeating a)-h) one or more times to dynamically control the nonlinear process.