Model predictive control with uncertainties

• US 9,983,554 B2
• Filed: 11/25/2014
• Issued: 05/29/2018
• 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
• Primary Examiner(s): Ali, Mohammad
• Assistant Examiner(s): Sanders, Joshua
• Application Number:
  US14/552,788
• Publication Number:
  US 20160147203A1
• Time in Patent Office: 1,281 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 dynamics, wherein the method uses a processor coupled with stored instructions implementing the method, wherein the instructions, when executed by the processor carry out at least some steps of the method, 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, and constraints on the state of the machine, wherein the operation is controlled online over a plurality of iterations, each iteration comprises;

determining a current state of the machine using measurements of outputs of the machine controlled with a previous control input determined for a previous iteration;

updating a parameter of a model of the machine dynamics to reduce a prediction error between the current state and a state estimated using the model of the machine dynamics, and wherein the parameter of the model of the machine dynamics represents a physical quantity of the machine;

optimizing the cost function to produce a control input, wherein the cost function includes a first term related to a performance of the machine and a second term related to improving estimation of the parameter of the model of the machine dynamics, wherein the second term is weighted by a function of the prediction error, and wherein the second term includes an information functional of a predicted error covariance of the parameter of the model of the machine dynamics; and

controlling the machine using the control input.

2. The method of claim 1, wherein the function of the prediction error is a nonnegative, nondecreasing function.

3. The method of claim 1, wherein the model of the machine dynamics 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 parameter of the model of the machine dynamics is updated such that the updated value of the parameter is within the range of values.

4. The method of claim 3, 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.

5. The method of claim 4, wherein the model of the machine dynamics is updated recursively such that the parameters of the model of the machine dynamics are within the convex combination of linear models and the convex combinations of additive disturbances.

6. The method of claim 4, wherein the updating the model of machine dynamics 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.

7. The method of claim 3, wherein the constraints on the control inputs include 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 parameter of the model of machine dynamics within the range defined by the uncertainty model.

8. The method of claim 4, 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 dynamics defined by the nominal and the uncertainty models.

9. The method of claim 3, 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 dynamics defined by the nominal and the uncertainty models.

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

11. The method of claim 10, 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 control-invariant subset of the states of the machine are contained in the union of the convex components.

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

13. A method for controlling an operation of a machine according to a model of the machine dynamics 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, wherein the method uses a processor coupled with stored instructions implementing the method, wherein the instructions, when executed by the processor carry out at least some steps of the method, comprising:

controlling iteratively the operation of the machine with control inputs determined using the model of the machine dynamics based on an optimization of a cost function, wherein the optimization is subject to control-invariant constraints on the operation of the machine including constraints on the control inputs and the state of the machine, 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 the parameter of the model of the machine dynamics represents a physical quantity of the machine, wherein the operation is controlled online over a plurality of iterations, each iteration comprises;

determining a current state of the machine using measurements of outputs of the machine controlled with a previous control input determined for a previous iteration;

updating a parameter of a model of the machine dynamics to reduce a prediction error between the current state and a state estimated using the model of the machine dynamics, such that the updated value of the parameter is within the range of values;

optimizing the cost function to produce a control input, wherein the cost function includes a first term related to a performance of the machine and a second term related to improving estimation of the parameter of the model of the machine dynamics, wherein the second term is weighted by a function of the prediction error, and wherein the second term includes an information functional of a predicted error covariance of the parameter of the model of the machine dynamics; and

controlling the machine using the control input.

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

a memory to store 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 operation of the machine including constraints on the control inputs and the state of the machine, wherein the parameter of the model of the machine dynamics represents a physical quantity of the machine; and

at least one processor to

determine a current state of the machine using measurements of outputs of the machine;

update the parameter of the model of the machine dynamics to reduce a prediction error between the current state and a state estimated using the model of the machine dynamics, such that the updated value of the parameter is within the range of values;

optimize the cost function to produce a control input, wherein the cost function includes a first term related to a performance of the machine and a second term related to improving estimation of the parameter of the model of the machine dynamics, wherein the second term is weighted by a function of the prediction error, and wherein the second term includes an information functional of a predicted error covariance of the parameter of the model of the machine dynamics; and

control the machine using the control input.

15. The control system of claim 14, 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.