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

• US 7,630,868 B2
• Filed: 10/29/2007
• Issued: 12/08/2009
• Est. Priority Date: 06/29/2000
• Status: Expired due to Term

• Current Assignee: Aspentech Corporation
• Original Assignee: Aspen Technology Inc (Emerson Electric Company)
• Inventors: Treiber, S. Steven, Guiver, John P., Turner, Paul, Lines, Brian
• Primary Examiner(s): Rodriguez; Paul L
• Assistant Examiner(s): Thangavelu; Kandasamy
• Application Number:
  US11/978,320
• Publication Number:
  US 20080071394A1
• Time in Patent Office: 771 Days
• Field of Search: 703/2, 703/12, 703/13, 703/6, 219/497, 700/29, 700/31, 700/44, 700/37, 706/25, 706/10, 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 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 parameterized dynamic model, configured to model the nonlinear process, wherein the parameterized dynamic model comprises one or more parameters that are not inputs or outputs of the nonlinear process; and

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

wherein the parametric universal nonlinear dynamic approximator is configured to predict process outputs necessary for predictive control and optimization of the nonlinear process 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 parameterized dynamic model based on the process operating conditions; and

operating the parameterized dynamic model to;

receive the values of the one or more parameters;

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, 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; and

store the one or more predicted process outputs;

wherein the parametric universal nonlinear dynamic approximator is configured to be coupled to the nonlinear process or a representation of the nonlinear process and the nonlinear process is configured to receive the one or more process inputs and produce the one or more process outputs;

wherein the nonlinear approximator and the parameterized dynamic model of the parametric universal nonlinear dynamic approximator are configured to be trained in an integrated manner by an optimization process that is configured to (i) determine model errors based on the one or more process outputs and the one or more predicted process outputs; and

(ii) adaptively train the parametric universal nonlinear dynamic approximator in an iterative manner using the model errors and the optimization process; and

wherein, in training the parametric universal nonlinear dynamic approximator in an iterative manner using the model errors and the optimization process, the optimization process is configured to;

identify process inputs and outputs (I/O);

determine an order of the parameterized dynamic model, wherein the order specifies a number of parameters comprised in the parameterized dynamic model;

collect data representing process operating conditions;

determine 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;

formulate an optimization problem;

execute an optimization algorithm to determine the dependencies of the parameters of the parameterized dynamic model upon operating conditions of the nonlinear process subject to the determined constraints by solving the optimization problem, thereby training the nonlinear approximator; and

verify compliance of the parametric universal nonlinear dynamic approximator with the determined constraints.

2. The memory medium of claim 1, wherein, in verifying the compliance of the parametric universal nonlinear dynamic approximator with the determined constraints, the optimization process is configured to:

use interval arithmetic over the global input region; and

/or

use interval arithmetic with input-region partitioning.

3. The memory medium of claim 1, wherein in determining the order of the parameterized dynamic model, the optimization process is configured to:

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

4. The memory medium of claim 1, wherein the optimization process is configured to determine the order of the parameterized dynamic model and train the nonlinear approximator concurrently.

5. The memory medium of claim 1, wherein in formulating the optimization problem, the optimization process is configured to determine or modify an objective function.

6. The memory medium of claim 1, wherein, in solving the optimization problem, the optimization process is configured to solve for an objective function subject to the determined constraints.

7. The memory medium of claim 1, wherein, after being trained, the overall behavior of the parametric universal nonlinear dynamic approximator is consistent with prior knowledge of the nonlinear process.

8. 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.

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

10. The memory medium of claim 1, wherein the nonlinear approximator includes a feedback loop, and wherein the feedback loop is configured to provide output of the nonlinear approximator from a previous cycle as input to the nonlinear approximator for a current cycle.

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

12. The memory medium of claim 11, wherein the set of difference equations comprises a set of discrete time polynomials.

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

the nonlinear process;

or

a representation of the nonlinear process.

14. The memory medium of claim 13, 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.

15. The memory of medium claim 1, wherein the parametric universal nonlinear dynamic approximator is configured to be coupled to the nonlinear process, wherein the parametric universal nonlinear dynamic approximator is further configured to be coupled to a control process, wherein the control process is configured 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 parameterized dynamic model of the nonlinear process comprised in the parametric universal nonlinear dynamic approximator upon operating condition 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) 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.

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

identifying process inputs and outputs (I/O);

determining, using a computer, an order for a parameterized dynamic model comprised in the parametric universal nonlinear dynamic approximator, wherein the order specifies a 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, using the computer, 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, using the computer, data for the identified process I/O;

determining, using the computer, 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, using the computer, an optimization problem for training the nonlinear approximator;

executing, using the computer, 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, using the computer, compliance of the parametric universal nonlinear dynamic approximator with the determined constraints;

storing, using the computer, 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; and

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

17. The method of claim 16, wherein said verifying the compliance of the parametric universal nonlinear dynamic approximator with the determined constraints comprises one or more of:

using interval arithmetic over the global input region;

or

using interval arithmetic with input-region partitioning.

18. The method of claim 16, wherein said determining the order comprises:

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

19. The method of claim 16, 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.

20. The method of claim 16, wherein formulating the optimization problem comprises:

determining an objective function; and

wherein solving the optimization problem comprises;

solving for the objective function subject to the determined constraints.

21. 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 a 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 approximator for modeling dependencies of the parameters of the parameterized 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 determined 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 dynamic approximator;

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

22. 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;

a1) initializing, using the computer, 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 by;

identifying process inputs and outputs (I/O);

determining an order of the parameterized dynamic model, wherein the order specifies a number of parameters comprised in the parameterized dynamic model;

collecting data representing process operating conditions;

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;

executing an optimization algorithm to determine the dependencies of the parameters of the parameterized dynamic model upon operating conditions of the nonlinear process subject to the determined constraints by solving the optimization problem, thereby training the nonlinear approximator; and

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

a2) executing, using the computer, 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

a3) initializing, using the computer, 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 function for optimization of the nonlinear process;

g) operating, using the computer, 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.

23. The method of claim 22, further comprising:

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

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

24. The method of claim 23, wherein said modifying the optimization problem comprises modifying one or more of:

constraints;

an objective function;

model parameters;

optimization parameters; and

optimization data.

25. 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 a

     1) 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 by;

identifying process inputs and outputs (I/O);

determining an order of the parameterized dynamic model, wherein the order specifies a number of parameters comprised in the parameterized dynamic model;

collecting data representing process operating conditions;

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;

executing an optimization algorithm to determine the dependencies of the parameters of the parameterized dynamic model upon operating conditions of the nonlinear process subject to the determined constraints by solving the optimization problem, thereby training the nonlinear approximator; and

verifying compliance of the parametric universal nonlinear dynamic approximator with the determined constraints

means for a2) 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 a3) initializing the parameterized dynamic model with the determined initial values for 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 function for optimization of the nonlinear process;

means for g) operating the nonlinear processing 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.

26. A 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 state space model for predictive control and optimization of a nonlinear process, comprising;

a state space dynamic model, configured to model the nonlinear process, wherein the state space dynamic model comprises one or more coefficients that are not inputs or outputs of the nonlinear process; and

a nonlinear approximator, configured to model dependencies of the one or more coefficients of the state space dynamic model upon operating conditions of the nonlinear process;

wherein the state space model is configured to predict process outputs necessary for predictive control and optimization of the nonlinear process 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 coefficients of the state space dynamic model based on the process operating conditions; and

operating the state space dynamic model to;

receive the values of the one or more coefficients;

receive the one or more process inputs;

generate one or more predicted process outputs based on the received values of the one or more coefficients and the received one or more process inputs, wherein the one or more predicted process outputs are dependent on the one or more process inputs, the values of the one or more coefficients, and past values of the process inputs and outputs; and

store the one or more predicted process outputs;

wherein the state space model is configured to be coupled to the nonlinear process or a representation of the nonlinear process and the nonlinear process is configured to receive the one or more process inputs and produce the one or more process outputs;

wherein the nonlinear approximator and the state space dynamic model of the state space model are configured to be trained in an integrated manner by an optimization process that is configured to (i) determine model errors based on the one or more process outputs and the one or more predicted process outputs; and

(ii) adaptively train the state space model in an iterative manner using the model errors and the optimization process; and

wherein, in training the state space model in an iterative manner using the model errors and the optimization process, the optimization process is configured to;

identify process inputs and outputs (I/O);

determine an order of the state space dynamic model, wherein the order specifies a number of coefficients comprised in the state space dynamic model;

collect data representing process operating conditions;

determine constraints on behavior of the state space model from prior knowledge, including one or more constraints for the nonlinear approximator for modeling dependencies of the one or more coefficients of the state space dynamic model;

formulate an optimization problem;

execute an optimization algorithm to determine the dependencies of the coefficients of the state space dynamic model upon operating conditions of the nonlinear process subject to the determined constraints by solving the optimization problem, thereby training the nonlinear approximator; and

verify compliance of the state space model with the determined constraints.

27. A computer implemented method for training a state space model of a nonlinear process, the method comprising:

identifying, using a computer, process inputs and outputs (I/O);

determining, using the computer, an order for a state space dynamic model comprised in the state space model, wherein the order specifies a number of coefficients for the state space dynamic model, and wherein the coefficients of the state space dynamic model are not inputs or outputs of the nonlinear process;

determining, using the computer, a structure for a nonlinear approximator comprised in the state space model for modeling dependencies of the coefficients of the state space dynamic model upon operating conditions of the nonlinear process;

collecting, using the computer, data for the identified process I/O;

determining, using the computer, constraints on behavior of the state space model from prior knowledge, including one or more constraints for the nonlinear approximator for modeling dependencies of the one or more coefficients of the state space dynamic model;

formulating, using the computer, an optimization problem for training the nonlinear approximator;

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

verifying, using the computer, compliance of the state space model with the determined constraints; and

storing, using the computer, the trained nonlinear approximator and the state space dynamic model, wherein the stored nonlinear approximator and the state space dynamic model compose a trained state space model;

wherein the trained state space model is usable to optimize and control the nonlinear process.

28. A system for training a state space model of a nonlinear process, the system comprising:

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

means for determining an order for a state space dynamic model comprised in the state space model, wherein the order specifies a number of coefficients for the state space dynamic model, and wherein the coefficients of the state space dynamic model are not inputs or outputs of the nonlinear process;

means for determining a structure for a nonlinear approximator comprised in the state space model for modeling dependencies of the coefficients of the state space 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 state space model from prior knowledge, including one or more constraints for the nonlinear approximator for modeling dependencies of the one or more coefficients of the state space 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 coefficients of the state space 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 state space model with the determined constraints; and

means for storing the trained nonlinear approximator and the state space dynamic model, wherein the stored nonlinear approximator and the state space dynamic model compose a trained state space model; and

wherein the state space model is usable to optimize and control the nonlinear process.

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

a) initializing, using a computer, a state space model to a current status of the nonlinear process, comprising process inputs and outputs, said initializing comprising;

a1) initializing, using the computer, inputs to a nonlinear approximator comprised in the state space model, wherein the nonlinear approximator is trained to model dependencies of one or more coefficients of a state space dynamic model of the nonlinear process comprised in the state space model upon operating conditions of the nonlinear process by;

identifying process inputs and outputs (I/O);

determining an order of the state space dynamic model, wherein the order specifies a number of coefficients comprised in the state space dynamic model;

collecting data representing process operating conditions;

determining constraints on behavior of the state space model from prior knowledge, including one or more constraints for the nonlinear approximator for modeling dependencies of the one or more coefficients of the state space dynamic model;

formulating an optimization problem;

executing an optimization algorithm to determine the dependencies of the coefficients of the state space dynamic model upon operating conditions of the nonlinear process subject to the determined constraints by solving the optimization problem, thereby training the nonlinear approximator; and

verifying compliance of the state space model with the determined constraints;

a2) executing the trained nonlinear approximator to determine initial values for the one or more coefficients of the state space dynamic model based on the current status of the nonlinear process; and

a3) initializing the state space dynamic model with the determined initial values for the one or more coefficients;

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 state space model 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 function for optimization of the nonlinear process;

g) operating, using the computer, 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.

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

means for a) initializing a state space model to a current status of the nonlinear process, comprising process inputs and outputs, comprising;

a

     1) means for initializing inputs to a nonlinear approximator comprised in the state space model, wherein the nonlinear approximator is trained to model dependencies of one or more coefficients of a state space dynamic model of the nonlinear process comprised in the state space model upon operating conditions of the nonlinear process by;

identifying process inputs and outputs (I/O);

determining an order of the state space dynamic model, wherein the order specifies a number of coefficients comprised in the state space dynamic model;

collecting data representing process operating conditions;

determining constraints on behavior of the state space model from prior knowledge, including one or more constraints for the nonlinear approximator for modeling dependencies of the one or more coefficients of the state space dynamic model;

formulating an optimization problem;

executing an optimization algorithm to determine the dependencies of the coefficients of the state space dynamic model upon operating conditions of the nonlinear process subject to the determined constraints by solving the optimization problem, thereby training the nonlinear approximator; and

verifying compliance of the state space model with the determined constraints;

a2) means for executing the trained nonlinear approximator to determine initial values for the one or more coefficients of the state space dynamic model based on the current status of the nonlinear process;

a3) means for initializing the state space dynamic model with the determined initial values for one or more coefficients;

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 state space model 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 function for optimization of the nonlinear process;

means for g) operating the nonlinear processing 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.