Model Predictive Control with Uncertainties

• US 20160147203A1
• Filed: 11/25/2014
• Published: 05/26/2016
• Est. Priority Date: 11/25/2014
• Status: Active Grant

• Current Assignee: Mitsubishi Electric Research Laboratories, Inc. (Mitsubishi Electric Corporation)
• Original Assignee: Mitsubishi Electric Research Laboratories, Inc. (Mitsubishi Electric Corporation)
• Inventors: Di Cairano, Stefano, Weiss, Avishai
• Granted Patent: US 9,983,554 B2
• Time in Patent Office: Days
• CPC Class Codes: G05B 13/042 in which a parameter or coefficient is automatically adjusted to optimise the performance

Abstract

A method controls iteratively the operation of the machine with control inputs determined using the model of the machine based on an optimization of a cost function subject to constraints on the control inputs. A current iteration of the method includes determining a current state of the machine after the controlling with a previous control input determined for a previous iteration by optimizing a previous cost function using a previous model of the machine and determining a current model of the machine to reduce a difference between the current state and a state estimated using the previous model of the machine. The cost function is updated during the current iteration based on a difference between the previous model and the current model to produce a current cost function. A current control input for the controlling at the current iteration is determined using the current model and the current cost function.

Claims

1. A method for controlling an operation of a machine according to a model of the machine, comprising:

controlling iteratively the operation of the machine with control inputs determined using the model based on an optimization of a cost function subject to constraints on the control inputs, wherein at least one current iteration comprises;

determining a current state of the machine after the controlling with a previous control input determined for a previous iteration by optimizing a previous cost function using a previous model of the machine;

determining a current model of the machine to reduce a difference between the current state and a state estimated using the previous model of the machine;

updating the cost function based on a difference between the previous model and the current model to produce a current cost function; and

determining a current control input for the controlling at the current iteration using the current model and the current cost function, wherein steps of the methods are performed by a processor of a controller controlling the operation of the machine.

2. The method of claim 1, wherein the cost function includes a first term for determining a first value for a control input according to an objective of the operation and includes a second term for determining a second value for the control input for reducing the difference between a future state of the machine and a future state estimated with the current model of the machine, such that the optimization optimizes a combination of the first term and the second term, wherein the updating the cost function includes changing a weight of the second term in the combination.

3. The method of claim 2, wherein the second term of the cost function includes a function of the difference between the current state of the machine and a state of the machine estimated using the current machine model from a previous machine state and a previous machine control input, wherein the updating comprises:

determining the difference between the previous and the current models; and

updating the second term of the cost function with the determined difference.

4. The method of claim 2, wherein the first term of the cost function is related to a performance of the machine, and the second term is related to improving estimation of the parameters of the model of the machine, wherein the second term is weighted by reliability of the current model of the machine measured as a nonnegative, nondecreasing function of a prediction error of the current model of the machine determined from the previous machine state, the previous machine control input and the current machine state.

5. The method of claim 1, wherein the model of the machine includes a nominal model defining relationships among parameters of the model and an uncertainty model defining a range of values for at least one parameter of the model, and wherein the current model is determined such that a current value of the parameter of the current model is within the range of values.

6. The method of claim 5, wherein the uncertainty model is extended such that a combination of the nominal model and the uncertainty model is included into a convex combination of linear models with a convex combinations of additive disturbances.

7. The method of claim 6, wherein the current model of the machine is determined recursively such that the parameters of the current model are within the convex combination of linear models and the convex combinations of additive disturbances.

8. The method of claim 6, wherein the determining the current model comprises:

determining a combination vector of the linear models and a combination vector of the additive disturbances;

projecting the combination vector of the linear models in the convex combination of the linear models; and

projecting the combination vector of the additive disturbances in the convex combination of the additive disturbances.

9. The method of claim 5, wherein the constraints on the control inputs include control-invariant constraints on the control inputs selected such that any value of a control input satisfying the control-invariant constraints maintains a state of the machine in a control-invariant subset of states satisfying constraints on the operation of the machine, wherein for any state of the machine within the control-invariant subset there is an admissible control input satisfying the control-invariant constraints and maintaining the state of the machine within the control-invariant subset for all values of the parameter of the model within the range defined by the uncertainty model.

10. The method of claim 6, wherein the constraints on the control inputs include control-invariant constraints determined from the convex combination of the linear models and the convex combination of the additive disturbances by backward reachability iterations that guarantee that for any state into a control-invariant subset of states satisfying constraints on the operation of the machine, there exists a control input such that the control-invariant constraints are satisfied during the operation for all values of the parameters of the model of the machine defined by the nominal and the uncertainty models.

11. The method of claim 5, wherein the constraints on the control inputs include stability constraints converging the state of the machine to a target value for all values of the parameters of the model of the machine defined by the nominal and the uncertainty models.

12. The method of claim 1, wherein the stability constraints include a control Lyapunov function of the machine.

13. The method of claim 12, wherein the control Lyapunov function is an infinity-norm control Lyapunov function that satisfies a feasible value reduction test for all states of the machine satisfying the control-invariant constraints comprising of:

selecting rows of a matrix describing the infinity-norm control Lyapunov function;

determining convex components of the states of the machine that satisfy, for positive and negative conditions, a feasible value reduction on the rows of the matrix describing the infinity-norm control Lyapunov function for at least one input of the machine satisfying the control-invariant constraints;

determining a union of the convex components;

verifying that all the states of the machine in the the control-invariant subset of the states of the machine are contained in the union of the convex components.

14. The method of claim 1, wherein the optimization is solved by numerical optimization algorithms.

15. A method for controlling an operation of a machine according to a model of the machine including a nominal model defining relationships among parameters of the model and an uncertainty model defining a range of values for at least one parameter of the model, comprising:

controlling iteratively the operation of the machine with control inputs determined using the model of the machine based on an optimization of a cost function, wherein the optimization is subject to control-invariant constraints on the control inputs selected such that any value of the control input satisfying the control-invariant constraints maintains a state of the machine in a control-invariant subset of states satisfying constraints on the operation of the machine, wherein for any state of the machine within the control-invariant subset there is an admissible control input satisfying the control-invariant constraints and maintaining the state of the machine within the control-invariant subset for all values of the parameters of the model within the range defined by the uncertainty model, wherein at least one current iteration comprises;

determining a current state of the machine resulted from the controlling with a previous control input determined for a previous iteration by optimizing a previous cost function using a previous model of the machine;

determining a current model of the machine to reduce a difference between the measured current state and a state estimated using the previous model of the machine, such that a current value of the parameter of the current model is within the range of values;

updating the cost function based on a difference between the previous and the current models, wherein the cost function includes a first term for determining a first value for the control input according to an objective of the operation and includes a second term for determining a second value for the control input for reducing the difference between the current state of the machine and a state estimated with a model of the machine, such that the optimization optimizes a combination of the first term and the second term, wherein the updating the cost function includes changing a weight of the second term in the combination; and

determining a current control input for the controlling at the current iteration using the current model and the current cost function, wherein steps of the methods are performed by a processor of a controller controlling the operation of the machine.

16. The method of claim 15, wherein the second term of the cost function includes a function of the difference of the current state of the machine and the state of the machine estimated using the current machine model from a previous machine state and a previous machine control input:

determining the difference between the previous and the current models; and

updating the second term of the cost function with the determined difference.

17. A control system for controlling iteratively an operation of a machine according to a model of the machine, comprising:

a memory storing the model of the machine including a nominal model defining relationships among parameters of the model and an uncertainty model defining a range of values for at least one parameter of the model, and the constraints on the machine; and

at least one processor for executing modules of the controller comprising;

a control input module for determining a current control input for the controlling during a current iteration by optimizing a current cost function subject to constraints on the current control input using a current model of the machine;

a model learning module for determining the current model of the machine, such that a current value of the parameter of the current model is with the range of values and reduces a difference between a current state of the machine resulted from the controlling with a previous control input determined for a previous iteration by optimizing a previous cost function using a previous model of the machine and an estimated state estimated using the previous model of the machine; and

a cost function module updating the previous cost function using a difference between the current state of the machine and the state of the machine estimated using the current machine model from a previous machine state and a previous machine control input.

18. The control system of claim 17, wherein the current cost function includes a first term for determining a first value for the current control input according to an objective of the operation and includes a second term for determining a second value for the current control input for reducing the difference between the current state and the state estimated according to the current model of the machine, such that the optimization optimizes a combination of the first term and the second term, wherein the updating the cost function includes changing a weight of the second term in the combination.

19. The control system of claim 18, wherein the second term of the cost function is updated using a function of the difference between the current state of the machine and the state of the machine estimated using the current machine model from a previous machine state and a previous machine control input.

20. The control system of claim 17, wherein the constraints on the control inputs include control-invariant constraints on the control inputs selected such that any value of a control input satisfying the control-invariant constraints maintains a state of the machine in a control-invariant subset of states satisfying constraints on the operation of the machine, wherein for any state of the machine within the control-invariant subset there is an admissible control input satisfying the control-invariant constraints and maintaining the state of the machine within the control-invariant subset for all values of the parameter of the model within the range defined by the uncertainty model.