Regression-optimization control of production process with dynamic inputs
Dynamic control of a production process of a manufacturing system is facilitated, where the control process includes receiving runtime input data for multiple input variables of the production process. The production process is represented, at least in part, by a physics-based expression, with at least one term of the physics-based expression being a function of two or more input variables of the production process. The control process includes determining coefficient and bias terms for a dynamic linear model connecting the multiple input variables and an output of the production process, where the terms are based, at least in part, on the input variables. The dynamic linear model and determined coefficient and bias terms are provided in an optimization model to generate a regression-optimization model which determines an optimized value of a control variable for the production process, which is used in facilitating control of the production process.
1. A computer-implemented method of facilitating control of a production process of a manufacturing system, the computer-implemented method comprising:
receiving, by at least one processor, runtime input data for multiple input variables of the production process, the production process represented, at least in part, by a physics-based expression, with at least one term of the physics-based expression being a function of two or more input variables of the multiple input variables of the production process;
determining, by the at least one processor based on the physics-based expression, coefficient and bias terms for a dynamic linear model connecting the multiple input variables of the production process to an output of the production process, the coefficient and bias terms of the dynamic linear model being based, at least in part, on input variables of the multiple input variables;
providing, by the at least one processor, the dynamic linear model with the determined coefficient and bias terms in an optimization model to generate a regression-optimization model which determines an optimized value of a control variable for the production process; and
facilitating control, at least in part, of the production process using the determined optimized value of the control variable.
2. The computer-implemented method of claim 1 , further comprising:
obtaining the dynamic linear model from, at least in part, the physics-based expression, the physics-based expression being a complex multi-step expression that connects the multiple input variables of the production process and the output of the production process, where at least one input variable of the multiple input variables is a dynamic input variable; and
wherein the providing comprises embedding the dynamic linear model with the determined coefficient and bias terms in the optimization model.
3. The computer-implemented method of claim 1 , wherein the optimization model can be solved by using methods from the group consisting of: mixed integer linear programming-based (MTLP) methods, gradient-based non-linear optimization (NLOpt) methods, and derivative-free optimization (DFO methods.
4. The computer-implemented method of claim 1 , wherein the dynamic linear model comprises:
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where:
x 1 . . . x m x are primary variables;
z 1 . . . z m z are secondary variables;
w 1 z , w m z z , w 1 x , w m x x , are dependent on observed variables, which can be the following form:
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where:
{a i,j } i=1, . . . l;j=1, . . . k and {b i,j } i=1, . . . n;j=1, . . . m are from a group consisting of: a number; an index of a respective input variable; and a dictionary of the above fraction, with l, k, n, and m being positive integers.
5. The computer-implemented method of claim 1 , wherein the at least one term of the physics-based expression is a function of two or more observed input variables of the multiple input variables of the production process.
6. The computer-implemented method of claim 1 , wherein determining, by the at least one processor based on the physics-based expression, coefficient and bias terms of the dynamic linear model includes mapping the physics-based expression to the dynamic linear model.
7. The computer-implemented method of claim 1 , wherein the determining comprises converting terms of the physics-based expression into a specified expression, the specified expression having a numerator and a denominator where each term can be decomposed into a defined form, the defined form being from a group consisting of: a constant×(an observed input variable); a product of two or more observed input variables; and a constant×(a product of two or more observed input variables).
8. The computer-implemented method of claim 1 , wherein one or more input variables of the multiple input variables of the production process change from one production run to another production run of the production process, resulting in one or more of the coefficient and bias terms of the dynamic linear model changing dynamically from the one production run to the other production run.
9. The computer-implemented method of claim 1 , wherein providing, by the at least one processor, the dynamic linear model with the determined coefficient and bias terms in the optimization model includes reading in at runtime the dynamic linear model with the determined coefficient and bias terms into the optimization model to generate the regression-optimization model which determines the optimized value of the control variable for the production process.
10. A computer system for facilitating control of a production process of a manufacturing system, the computer system comprising:
a memory; and
at least one processor in communication with the memory, wherein the computer system is configured to perform a method, said method comprising:
receiving, at the at least one processor, runtime input data for multiple input variables of the production process, the production process represented, at least in part, by a physics-based expression, with at least one term of the physics-based expression being a function of two or more input variables of the multiple input variables of the production process;
determining, by the at least one processor based on the physics-based expression, coefficient and bias terms for a dynamic linear model connecting the multiple input variables of the production process to an output of the production process, the coefficient and bias terms of the dynamic linear model being based, at least in part, on input variables of the multiple input variables;
providing, by the at least one processor, the dynamic linear model with the determined coefficient and bias terms in an optimization model to generate a regression-optimization model which determines an optimized value of a control variable for the production process; and
facilitating control, at least in part, of the production process using the determined optimized value of the control variable.
11. The computer system of claim 10 , further comprising:
obtaining the dynamic linear model from, at least in part, the physics-based expression, the physics-based expression being a complex multi-step expression that connects the multiple input variables of the production process and the output of the production process, where at least one input variable of the multiple input variables is a dynamic input variable; and
wherein the providing comprises embedding the dynamic linear model with the determined coefficient and bias terms in the optimization model.
12. The computer system of claim 10 , wherein the dynamic linear model comprises:
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1
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+
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x
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+
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where:
x 1 . . . x m x are primary variables;
z 1 . . . z m z are secondary variables;
w 1 z , w m z z , w 1 x , w m x x , are dependent on observed variables, which can be the following form:
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where:
{a i,j } i=1, . . . l;j=1, . . . k and {b i,j } i=1, . . . n;j=1, . . . m are from a group consisting of: a number; an index of a respective input variable; and a dictionary of the above fraction, with l, k, n, and m being positive integers.
13. The computer system of claim 10 , wherein the at least one term of the physics-based expression is a function of two or more observed input variables of the multiple input variables of the production process.
14. The computer system of claim 10 , wherein determining, by the at least one processor based on the physics-based expression, coefficient and bias terms of the dynamic linear model includes mapping the physics-based expression to the dynamic linear model.
15. The computer system of claim 10 , wherein the determining comprises converting terms of the physics-based expression into a specified expression, the specified expression having a numerator and a denominator where each term can be decomposed into a defined form, the defined form being from a group consisting of: a constant×(an observed input variable); a product of two or more observed input variables; and a constant×(a product of two or more observed input variables).
16. The computer system of claim 10 , wherein one or more input variables of the multiple input variables of the production process change from one production run to another production run of the production process, resulting in one or more of the coefficient and bias terms of the dynamic linear model changing dynamically from the one production run to the other production run.
17. A computer program product for facilitating control of a production process of a manufacturing system, the computer program product comprising:
one or more computer-readable storage media and program instructions embodied therewith, the program instructions being readable by a processing circuit to cause the processing circuit to perform a method comprising:
receiving runtime input data for multiple input variables of the production process, the production process represented, at least in part, by a physics-based expression, with at least one term of the physics-based expression being a function of two or more input variables of the multiple input variables of the production process;
determining, by the at least one processor based on the physics-based expression, coefficient and bias terms for a dynamic linear model connecting the multiple input variables of the production process to an output of the production process, the coefficient and bias terms of the dynamic linear model being based, at least in part, on input variables of the multiple input variables;
providing, by the at least one processor, the dynamic linear model with the determined coefficient and bias terms in an optimization model to generate a regression-optimization model which determines an optimized value of a control variable for the production process; and
facilitating control, at least in part, of the production process using the determined optimized value of the control variable.
18. The computer program product of claim 17 , wherein the dynamic linear model comprises:
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=
w
1
z
z
1
+
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+
w
m
z
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z
m
z
+
w
1
x
x
1
+
…
+
w
m
x
x
x
m
x
+
b
where:
x 1 . . . x m x are primary variables;
z 1 . . . z m z are secondary variables;
w 1 z , w m z z , w 1 x , w m x x , are dependent on observed variables, which can be the following form:
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(
a
1
,
1
·
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)
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]
where:
{a i,j } i=1, . . . l;j=1, . . . k and {b i,j } i=1, . . . n;j=1, . . . m are from a group consisting of: a number; an index of a respective input variable; and a dictionary of the above fraction, with l, k, n, and m being positive integers.
19. The computer program product of claim 17 , wherein the determining comprises converting terms of the physics-based expression into a specified expression, the specified expression having a numerator and a denominator where each term can be decomposed into a defined form, the defined form being from a group consisting of: a constant×(an observed input variable); a product of two or more observed input variables; and a constant×(a product of two or more observed input variables).
20. The computer program product of claim 17 , wherein one or more input variables of the multiple input variables of the production process change from one production run to another production run of the production process, resulting in one or more of the coefficient and bias terms of the dynamic linear model changing dynamically from the one production run to the other production run.