Computer method and apparatus for constraining a non-linear approximator of an empirical process

• US 8,296,107 B2
• Filed: 11/10/2009
• Issued: 10/23/2012
• Est. Priority Date: 06/29/2000
• Status: Active Grant

• Current Assignee: Aspentech Corporation
• Original Assignee: Aspen Technology (Telefonaktiebolaget LM Ericsson)
• Inventors: Turner, Paul, Guiver, John P., Lines, Brian, Treiber, S. Steven
• Primary Examiner(s): Thangavelu, Kandasamy
• Application Number:
  US12/615,754
• Publication Number:
  US 20100057222A1
• Time in Patent Office: 1,078 Days
• Field of Search: 703/2, 703/12, 703/13, 700/29, 700/31, 700/37, 219/497, 706/10, 706/25, 702/185
• US Class Current: 703/2
• CPC Class Codes: G05B 13/027 using neural networks only
• CPC Class Codes: G05B 17/02 electric

Abstract

A constrained non-linear approximator for empirical process control is disclosed. The approximator constrains the behavior of the derivative of a subject empirical model without adversely affecting the ability of the model to represent generic non-linear relationships. There are three stages to developing the constrained non-linear approximator. The first stage is the specification of the general shape of the gain trajectory or base non-linear function which is specified graphically, algebraically or generically and is used as the basis for transfer functions used in the second stage. The second stage of the invention is the interconnection of the transfer functions to allow non-linear approximation. The final stage of the invention is the constrained optimization of the model coefficients such that the general shape of the input/output mappings (and their corresponding derivatives) are conserved.

Claims

1. A non-transitory computer readable memory medium that stores computer executable instructions that when executed by a processor perform model predictive control and optimization of a nonlinear process by implementing

a parametric universal nonlinear dynamic approximator for predictive control and optimization of a nonlinear process, comprising:

a dynamic parameterized model, operable to model the nonlinear process, wherein the dynamic parameterized model receives one or more parameters that are not inputs or outputs of the nonlinear process, and

wherein the one or more parameters are outputs of an explicit mapping to a parameter space; and

a nonlinear approximator, operable to explicitly model dependencies of the one or more parameters of the dynamic parameterized model upon operating conditions of the nonlinear process;

wherein the parametric universal nonlinear dynamic approximator is operable to predict process outputs necessary for predictive control and optimization of the nonlinear process, wherein actual measurements of at least one of the process outputs do not exist, by;

operating the nonlinear approximator to;

receive one or more process operating conditions, including one or more process inputs; and

generate values for the one or more parameters of the dynamic parameterized model based on the process operating conditions; and

provide the values for the one or more parameters to the dynamic parameterized model; and

operating the dynamic parameterized model to;

receive the values of the one or more parameters from the nonlinear approximator;

receive the one or more process inputs;

generate one or more predicted process outputs based on the received values of the one or more parameters and the received one or more process inputs; and

store the one or more predicted process outputs;

wherein the parametric universal nonlinear dynamic approximator is operable to be coupled to the nonlinear process, wherein the parametric universal nonlinear dynamic approximator is further operable to be coupled to a control process, wherein the control process is operable to;

a) initialize a parametric universal nonlinear dynamic approximator to a current status of the nonlinear process, comprising process inputs and outputs, by;

initializing inputs to a nonlinear approximator comprised in the parametric universal nonlinear dynamic approximator, wherein the nonlinear approximator is trained to model dependencies of one or more parameters of a dynamic parameterized model of the nonlinear process comprised in the parametric universal nonlinear dynamic approximator upon operating conditions of the nonlinear process;

executing the trained nonlinear approximator to determine initial values for the one or more parameters of the dynamic parameterized model based on the current status of the nonlinear process; and

initializing the parameterized dynamic model with the determined initial values for the one or more parameters;

b) formulate an optimization problem, including specifying an objective function for optimization of the nonlinear process;

c) generate a profile of manipulated variables for the nonlinear process over a control horizon in accordance with the specified objective function for optimization of the nonlinear process;

d) operate the parametric universal nonlinear dynamic approximator in accordance with the generated profile of manipulated variables, thereby generating predicted outputs for the nonlinear process;

e) determine a deviation of the predicted outputs from a desired behavior of the nonlinear process;

f) repeat b)- e) one or more times to determine an optimal profile of manipulated variables in accordance with the specified objective function for optimization of the nonlinear process;

g) operate the nonlinear process in accordance with the optimal profile of manipulated variables, thereby generating process output; and

repeat a)- g) one or more times to dynamically control the nonlinear process.

2. The memory medium of claim 1, wherein the one or more predicted process outputs are dependent on the one or more process inputs, the values of the one or more parameters, and past values of the process inputs and outputs.

3. The memory medium of claim 2, wherein the nonlinear approximator and the dynamic parameterized model of the parametric universal nonlinear dynamic approximator are operable to be trained in an integrated manner by an optimization process.

4. The memory medium of claim 3, wherein the parametric universal nonlinear dynamic approximator 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 adaptively train the parametric universal nonlinear dynamic approximator in an iterative manner using the model errors and an optimizer.

5. The memory medium of claim 2, wherein, after being trained, the overall behavior of the parametric universal nonlinear dynamic approximator is consistent with prior knowledge of the nonlinear process.

6. The memory medium 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 memory medium of claim 1, wherein the nonlinear approximator comprises a universal nonlinear approximator.

8. The memory medium 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.

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

10. The memory medium of claim 9, wherein the set of difference equations comprises a set of discrete time polynomials.

11. The memory medium of claim 9, wherein the one or more process inputs are received from one or more of:

the nonlinear process;

or

a representation of the nonlinear process.

12. The memory medium of claim 11, 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.

13. A non-transitory computer readable memory medium that stores computer executable instructions that when executed by a processor perform training a parametric universal nonlinear dynamic approximator of a nonlinear process, by implementing:

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.

14. The memory medium of claim 13, 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.

15. The memory medium of claim 13, wherein said determining the order comprises:

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

16. The memory medium of claim 13, 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.

17. The memory medium of claim 13,

wherein formulating the optimization problem comprises:

determining an objective function; and

wherein solving the optimization problem comprises;

solve for the objective function subject to the determined constraints.

18. The memory medium of claim 13, 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.

19. A computer implemented method for controlling a nonlinear process, the method comprising:

a) initializing, using a computer, a parametric universal nonlinear dynamic approximator to a current status of the nonlinear process, comprising process inputs and outputs, said initializing comprising;

initializing inputs to a nonlinear approximator comprised in the parametric universal nonlinear dynamic approximator, wherein the nonlinear approximator is trained to model dependencies of one or more parameters of a parameterized dynamic model of the nonlinear process comprised in the parametric universal nonlinear dynamic approximator upon operating conditions of the nonlinear process;

executing the trained nonlinear approximator to determine initial values for the one or more parameters of the parameterized dynamic model based on the current status of the nonlinear process; and

initializing the parameterized dynamic model with the determined initial values for the one or more parameters;

b) formulating, using the computer, an optimization problem, including specifying an objective function for optimization of the nonlinear process;

c) generating, using the computer, a profile of manipulated variables for the nonlinear process over a control horizon in accordance with the specified objective function for optimization of the nonlinear process;

d) operating, using the computer, the parametric universal nonlinear dynamic approximator in accordance with the generated profile of manipulated variables, thereby generating predicted outputs for the nonlinear process;

e) determining, using the computer, a deviation of the predicted outputs from a desired behavior of the nonlinear process;

f) repeating b)-e) one or more times to determine an optimal profile of manipulated variables in accordance with the specified objective for the nonlinear process;

g) operating the nonlinear process in accordance with the optimal profile of manipulated variables, thereby generating process output; and

repeating a)-g) one or more times to dynamically control the nonlinear process.

20. The method of claim 19, further comprising:

h) modifying the optimization problem based on the input to the model;

wherein said repeating a)-g) comprises repeating a)-h).

21. The method of claim 20, wherein said modifying the optimization problem comprises modifying one or more of:

constraints;

an objective function;

model parameters;

optimization parameters; and

optimization data.

22. The method of claim 19, 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.

23. The method of claim 19, wherein the nonlinear approximator comprises a universal nonlinear approximator.

24. The method of claim 19, 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.

25. The method of claim 19, wherein the parameterized dynamic model comprises a multi-input, multi-output (MIMO) dynamic model implemented with a set of difference equations.

26. The method of claim 25, wherein the set of difference equations comprises a set of discrete time polynomials.

27. The method of claim 25, wherein the one or more process inputs are received from one or more of:

the nonlinear process;

or

a representation of the nonlinear process.

28. The method of claim 27, 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.

29. The method of claim 19, wherein said repeating a)-g) one or more times to dynamically control the nonlinear process further comprises:

adapatively training the parametric universal nonlinear dynamic approximator in an iterative manner.

30. A non-transitory computer readable memory medium that stores computer executable instructions that when executed by a processor perform controlling a nonlinear process, by implementing:

a) initializing a parametric universal nonlinear dynamic approximator to a current status of the nonlinear process, comprising process inputs and outputs, said initializing comprising;

initializing inputs to a nonlinear approximator comprised in the parametric universal nonlinear dynamic approximator, wherein the nonlinear approximator is trained to model dependencies of one or more parameters of a parameterized dynamic model of the nonlinear process comprised in the parametric universal nonlinear dynamic approximator upon operating conditions of the nonlinear process;

executing the trained nonlinear approximator to determine initial values for the one or more parameters of the parameterized dynamic model based on the current status of the nonlinear process; and

initializing the parameterized dynamic model with the determined initial values for the one or more parameters;

b) formulating an optimization problem, including specifying an objective function for optimization of the nonlinear process;

c) generating a profile of manipulated variables for the nonlinear process over a control horizon in accordance with the specified objective function for optimization of the nonlinear process;

d) operating the parametric universal nonlinear dynamic approximator in accordance with the generated profile of manipulated variables, thereby generating predicted outputs for the nonlinear process;

e) determining a deviation of the predicted outputs from a desired behavior of the nonlinear process;

f) repeating b)-e) one or more times to determine an optimal profile of manipulated variables in accordance with the specified objective for the nonlinear process;

g) operating the nonlinear process in accordance with the optimal profile of manipulated variables, thereby generating process output; and

repeating a)-g) one or more times to dynamically control the nonlinear process.

31. The memory medium of claim 30, wherein the program instructions are further executable to perform:

h) modifying the optimization problem based on the input to the model;

wherein said repeating a)-g) comprises repeating a)-h).

32. The memory medium of claim 31, wherein said modifying the optimization problem comprises modifying one or more of:

constraints;

an objective function;

model parameters;

optimization parameters; and

optimization data.

33. A system for controlling a nonlinear process, the system comprising:

means for a) initializing a parametric universal nonlinear dynamic approximator to a current status of the nonlinear process, comprising process inputs and outputs, comprising;

means for initializing inputs to a nonlinear approximator comprised in the parametric universal nonlinear dynamic approximator, wherein the nonlinear approximator is trained to model dependencies of one or more parameters of a parameterized dynamic model of the nonlinear process comprised in the parametric universal nonlinear dynamic approximator upon operating conditions of the nonlinear process;

means for executing the trained nonlinear approximator to determine initial values for the one or more parameters of the parameterized dynamic model based on the current status of the nonlinear process;

means for initializing the parameterized dynamic model with the determined initial values for the one or more parameters;

means for b) formulating an optimization problem, including specifying an objective function for optimization of the nonlinear process;

means for c) generating a profile of manipulated variables for the nonlinear process over a control horizon in accordance with the specified objective function for optimization of the nonlinear process;

means for d) operating the parametric universal nonlinear dynamic approximator in accordance with the generated profile of manipulated variables, thereby generating predicted outputs for the nonlinear process;

means for e) determining a deviation of the predicted outputs from a desired behavior of the nonlinear process;

means for f) repeating b)-e) one or more times to determine an optimal profile of manipulated variables in accordance with the specified objective for the nonlinear process;

means for g) operating the nonlinear process in accordance with the optimal profile of manipulated variables, thereby generating process output; and

means for repeating a)-g) one or more times to dynamically control the nonlinear process.