Training a model of a non-linear process

1. A method for training a parametric universal nonlinear dynamic approximator of a nonlinear process, the method comprising:

identifying process inputs and outputs (I/O);

determining an order for a parameterized dynamic model comprised in the parametric universal nonlinear dynamic approximator, wherein the order specifies the number of parameters for the parameterized dynamic model, and wherein the parameters of the parameterized dynamic model are not inputs or outputs of the nonlinear process;

determining a structure for a nonlinear approximator comprised in the parametric universal nonlinear dynamic approximator for modeling dependencies of the parameters of the parameterized dynamic model upon operating conditions of the nonlinear process;

collecting data for the identified process I/O;

determining constraints on behavior of the parametric universal nonlinear dynamic approximator from prior knowledge, including one or more constraints for the nonlinear approximator for modeling dependencies of the one or more parameters of the parameterized dynamic model;

formulating an optimization problem for training the nonlinear approximator;

executing an optimization algorithm to train the nonlinear approximator subject to the determined constraints by solving the optimization problem, thereby determining the dependencies of the parameters of the parameterized dynamic model upon operating conditions of the process, wherein outputs of the nonlinear approximator are not outputs of the nonlinear process;

verifying compliance of the parametric universal nonlinear dynamic approximator with the specified constraints; and

storing the trained nonlinear approximator and the parameterized dynamic model, wherein the stored nonlinear approximator and the parameterized dynamic model compose a trained parametric universal nonlinear dynamic approximator;

wherein the trained parametric universal nonlinear dynamic approximator is usable to optimize and control the nonlinear process.

2. The method of claim 1 , wherein said verifying the compliance of the parametric universal nonlinear dynamic approximator with the specified constraints comprises one or more of:

using interval arithmetic over the global input region; or

using interval arithmetic with input-region partitioning.

3. The method of claim 1 , wherein said determining the order comprises:

executing the optimization algorithm to determine an optimal order of the parameterized dynamic model.

4. The method of claim 1 ,

Wherein said executing the optimization algorithm to determine the optimal order of the parameterized dynamic model and said executing the optimization algorithm to determine dependencies of the parameters of the parameterized dynamic model are performed concurrently.

5. The method of claim 1 ,

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.

6. The method 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; or

an empirical model.

7. The method of claim 1 , wherein the nonlinear approximator comprises a universal nonlinear approximator.

8. The method of claim 1 , wherein the parameterized dynamic model comprises a multi-input (MIMO) dynamic model implemented with a set of difference equations.

9. The method of claim 8 , wherein the set of difference equations comprises a set of discrete time polynomials.

10. The method of claim 1 , further comprising:

coupling the parametric universal nonlinear dynamic approximator to the nonlinear process or a representation of the nonlinear process; and

performing said collecting data, said determining constraints, said formulating an optimization problem, said executing an optimization algorithm, said verifying compliance, and said storing the trained nonlinear approximator and the parameterized dynamic model, in real time in an iterative manner to adaptively train the parametric universal nonlinear dynamic approximator.

11. The method of claim 10 , wherein said formulating an optimization problem comprises determining or modifying an objective function.

12. The method of claim 10 , wherein the nonlinear approximator includes a feedback loop, and wherein the feedback lop is operable to provide output of the nonlinear approximator from a previous cycle as input to the nonlinear approximator for a current cycle.

13. The method of claim 12 , wherein the 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; or

empirical data.

14. A computer accessible memory medium that stores program instructions for training a parametric universal nonlinear dynamic approximator of a nonlinear process, wherein the program instructions are executable by a processor to perform:

identifying process inputs and outputs (I/O);

determining an order for a parameterized dynamic model comprised in the parametric universal nonlinear dynamic approximator, wherein the order specifies the number of parameters for the parameterized dynamic model, and wherein the parameters of the parameterized dynamic model are not inputs or outputs of the nonlinear process;

determining a structure for a nonlinear approximator comprised in the parametric universal nonlinear dynamic approximator for modeling dependencies of the parameters of the parameterized dynamic model upon operating conditions of the nonlinear process;

collecting data for the identified process I/O;

determining constraints on behavior of the parametric universal nonlinear dynamic approximator from prior knowledge, including one or more constraints for the nonlinear approximator for modeling dependencies of the one or more parameters of the parameterized dynamic model;

formulating an optimization problem for training the nonlinear approximator;

executing an optimization algorithm to train the nonlinear approximator subject to the determined constrains by solving the optimization problem, thereby determining the dependencies of the parameters of the parameterized dynamic model upon operating conditions of the process, wherein outputs of the nonlinear approximator are not outputs of the nonlinear process;

verifying compliance of the parametric universal nonlinear dynamic approximator with the specified constraints; and

storing the trained nonlinear approximator and the parameterized dynamic model, wherein the stored nonlinear approximator and the parameterized dynamic model compose a trained parametric universal nonlinear dynamic approximator;

wherein the trained parametric universal nonlinear dynamic approximator is usable to optimize and control the nonlinear process.

15. The memory medium of claim 14 , wherein said verifying the compliance of the parametric universal nonlinear dynamic approximator with the specified constraints comprises one or more of:

using interval arithmetic over the global input region; or

using interval arithmetic with input-region partitioning.

16. The memory medium of claim 14 , wherein said determining the order comprises:

executing the optimization algorithm to determine an optimal order of the parameterized dynamic model.

17. The memory medium of claim 14 ,

wherein said executing the optimization algorithm to determine the optimal order of the parameterized dynamic model and said executing the optimization algorithm to determine dependencies of the parameters of the parameterized dynamic model are performed concurrently.

18. The memory medium of claim 14 ,

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.

19. The memory medium of claim 14 , wherein the program instructions are further executable to perform:

coupling the parametric universal nonlinear dynamic approximator to the nonlinear process or a representation of the nonlinear process; and

performing said collecting data, said determining constraints, said formulating an optimization problem, said executing an optimization algorithm, said verifying compliance, and said storing the trained nonlinear approximator and the parameterized dynamic model, in real time in an iterative manner to adaptively train the parametric universal nonlinear dynamic approximator.

20. A system for training a parametric universal nonlinear dynamic approximator of a nonlinear process, the system comprising:

means for identifying process inputs and outputs (I/O);

means for determining an order for a parameterized dynamic model comprised in the parametric universal nonlinear dynamic approximator, wherein the order specifies the number of parameters for the parameterized dynamic model, and wherein the parameters of the parameterized dynamic model are not inputs or outputs of the nonlinear process;

means for determining a structure for a nonlinear approximator comprised in the parametric universal nonlinear dynamic model upon operating conditions of the nonlinear process;

means for collecting data for the identified process I/O;

means for determining constraints on behavior of the parametric universal nonlinear dynamic approximator from prior knowledge, including one or more constraints for the nonlinear approximator for modeling dependencies of the one or more parameters of the parameterized dynamic model;

means for formulating an optimization problem for training the nonlinear approximator;

means for executing an optimization algorithm to train the nonlinear approximator subject to the determined constraints by solving the optimization problem, thereby determining the dependencies of the parameters of the parameterized dynamic model upon operating conditions of the process, wherein outputs of the nonlinear approximator are not outputs of the nonlinear process;

means for verifying compliance of the parametric universal nonlinear dynamic approximator with the specified constraints; and

means for storing the trained nonlinear approximator and the parameterized dynamic model, wherein the stored nonlinear approximator and the parameterized dynamic model compose a trained parametric universal nonlinear dynamic approximator;

wherein the parametric universal nonlinear dynamic approximator is usable to optimize and control the nonlinear process.