Method and apparatus for preconditioned predictive control

Patent metadata

Field	Value
Publication number	US-10613490-B2
Application number	US-201815888143-A
Country	US
Kind code	B2
Filing date	Feb 5, 2018
Priority date	Feb 5, 2018
Publication date	Apr 7, 2020
Grant date	Apr 7, 2020

Abstract

A predictive controller for controlling a system subject to constraints including equality and inequality constraints on state and control variables of the system, includes an estimator to estimate a current state of the system using measurements of outputs of the system and a controller to solve, at each control step, a matrix equation of necessary optimality conditions to produce a control solution and to control the system using the control solution to change a state of the system. The matrix equation includes a block-structured matrix having a constraint Jacobian matrix of the equality constraints of the system. The controller determines the control solution iteratively using two levels of iterations including a first level of iterations that selects active inequality constraints for each point of time within a control horizon, updates the constraint Jacobian matrix, with a low-rank update for a change in the set of active inequality constraints, to include the equality constraints and the active inequality constraints, and updates a preconditioning matrix, with a low-rank factorization update, in response to the low-rank update of the constraint Jacobian matrix. The second level of iterations solves the matrix equation with the updated constraint Jacobian matrix using the updated preconditioning matrix to produce the control solution.

First claim

We claim:

1. A predictive controller for controlling a system subject to constraints including equality and inequality constraints on state and control variables of the system, comprising:
an estimator to estimate a current state of the system using measurements of outputs of the system; and
a controller to solve, at each control step, a matrix equation of necessary optimality conditions to produce a control solution and to control the system using the control solution to change a state of the system, wherein the matrix equation includes a block-structured matrix having a constraint Jacobian matrix of the equality constraints of the system, wherein the controller determines the control solution iteratively using two levels of iterations including a first level of iterations and a second level of iterations performed within an iteration of the first level of iterations,
wherein the first level of iterations, performed until a first termination condition is met,
selects active inequality constraints for each point of time within a control horizon,
updates the constraint Jacobian matrix, with a low-rank update for a change in the set of active inequality constraints, to include the equality constraints and the active inequality constraints, and
updates a preconditioning matrix, with a low-rank factorization update, in response to the low-rank update of the constraint Jacobian matrix,

wherein the second level of iterations, performed until a second termination condition is met, solves the matrix equation with the updated constraint Jacobian matrix using the updated preconditioning matrix to produce the control solution.

2. The predictive controller of claim 1 , wherein an inequality constraint is active at a point of time when a constraint function of the inequality constraint is equal to a constraint bound of the inequality constraint at the point of time, wherein the first level of iterations selects the set of active inequality constraints based on the current state of the system and a current sequence of control inputs.

3. The predictive controller of claim 2 , wherein the first termination condition includes a feasibility of the control solution and an optimality of the control solution,
wherein the controller tests the optimality and the feasibility of the control solution and exits the first level of iterations when the first termination condition is met, otherwise,
wherein the control solution is not optimal, the controller removes the active inequality constraint having a negative dual variable that prevents optimality of the control solution and repeats the first level of iterations, and otherwise,
wherein the control solution is not feasible, the controller updates the current sequence of control inputs with feasible portion of the control solution, adds a blocking inequality constraint in the set of active inequality constraints, and repeats the first level of iterations.

4. The predictive controller of claim 3 , wherein the controller, in response to adding the blocking inequality constraint, adds a row in the constraint Jacobian matrix with a first function of coefficients of the blocking inequality constraint and updates the preconditioning matrix with a second function of the coefficients of the blocking inequality constraint.

5. The predictive controller of claim 4 , wherein the controller, in response to removing the active inequality constraint, removes a row in the constraint Jacobian matrix that corresponds to the removed active inequality constraint and updates the preconditioning matrix with a reverse of the second function.

6. The predictive controller of claim 1 , wherein each change in the set of active inequality constraints corresponds to only one constraint being either added or removed, which leads to a rank-one factorization update for the preconditioning matrix, in response to the rank-one update of the constraint Jacobian matrix.

7. The predictive controller of claim 1 , wherein the preconditioning matrix is updated or evaluated using a reduced arithmetic precision, compared to the arithmetic precision that is used for the matrix equation.

8. The predictive controller of claim 1 , wherein the preconditioning matrix has a block-tridiagonal structure, wherein each block is positive definite, wherein the controller updates the factorization of the preconditioning matrix while preserving the block-tridiagonal structure.

9. The predictive controller of claim 8 , wherein the controller updates the factorization of the preconditioning matrix using a low-rank update of a block-tridiagonal Cholesky factorization.

10. The predictive controller of claim 1 , wherein, for a current control step, the controller initializes the Jacobian matrix and the factorization of the preconditioning matrix with values of the Jacobian matrix and the factorized preconditioning matrix determined during a previous control step.

11. The predictive controller of claim 1 , wherein the preconditioning matrix is an augmented Lagrangian block-diagonal matrix, and wherein the second level of iterations solves the matrix equation using a minimal residual method.

12. The predictive controller of claim 1 , wherein the preconditioning matrix is a Schur complement block-diagonal matrix, and wherein the second level of iterations solves the matrix equation using a minimal residual method.

13. The predictive controller of claim 12 , wherein the controller regularizes a Hessian of the matrix of the matrix equation to produce a positive definite approximation of the Hessian of the matrix, and wherein the controller determines elements of the Schur complement block-diagonal matrix based on the approximation of the Hessian of the matrix.

14. The predictive controller of claim 1 , wherein the preconditioning matrix is a constraint preconditioner, and wherein the second level of iterations solves the matrix equation using a projected conjugate gradient method.

15. The predictive controller of claim 14 , wherein the controller regularizes a Hessian of the matrix of the matrix equation to produce a positive definite approximation of the Hessian of the matrix, and wherein the controller determines elements of the constraint preconditioner based on the approximation of the Hessian of the matrix.

16. The predictive controller of claim 1 , wherein the control system is a vehicle, and wherein the controller determines an input to the vehicle based on the control solution, wherein the input to the vehicle includes one or a combination of an acceleration of the vehicle, a torque of a motor of the vehicle, and a steering angle.

17. A vehicle including the predictive controller of claim 1 .

18. The method of claim 1 , wherein the preconditioning matrix has a block-tridiagonal structure, wherein each block is positive definite, wherein the method updates the factorization of the preconditioning matrix while preserving the block-tridiagonal structure.

19. The method of claim 18 , wherein the controller updates the factorization of the preconditioning matrix using a low-rank update of a block-tridiagonal Cholesky factorization.

20. A method for controlling a system subject to constraints including equality and inequality constraints on state and control variables of the system, wherein the method uses a processor coupled with stored instructions implementing the method, wherein the instructions, when executed by the processor carry out steps of the method, comprising:
estimating a current state of the system using measurements of outputs of the system;
s

Assignees

• Mitsubishi Electric Res Laboratories Inc

Inventors

• Quirynen Rien
• Kniazev Andrei
• Di Cairano Stefano

Classifications

• G06F16/23

  Updating · CPC title

• G05B13/048 Primary

  using a predictor · CPC title

• B25J9/1607

  Calculation of inertia, jacobian matrixes and inverses · CPC title

• G06F17/11

  for solving equations {, e.g. nonlinear equations, general mathematical optimization problems (optimization specially adapted for a specific administrative, business or logistic context G06Q10/04)} · CPC title

• G06F17/16

  Matrix or vector computation {, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization (matrix transposition G06F7/78)} · CPC title