Matrix generation technique and plant control technique

Abstract

In this disclosure, equations to be solved in the model predictive control are transformed by using an off-line algebraic simplification method into a matrix operational expression representing a product of a coefficient matrix and a vector regarding solution inputs within a control horizon is equal to a function vector regarding target values of output states and the output states. The size of the coefficient matrix is reduced compared with the conventional matrix. Then, the matrix operational expression is solved in an online plant control apparatus with present output states and present target values of the output stats of a plant to be controlled, by the direct method, to output the solution to the plant.

What is claimed is: 
     
         1 . A computer-readable, non-transitory storage medium storing a program for causing a computer to execute a process, the process comprising:
 converting state equations, which represent a relationship between solution output states and solution inputs of a plant to be controlled, and initial condition expressions of output states into first linear expressions representing the solution output states at each step by the output states and the solution inputs;   converting first conditional expressions representing a relationship between a first Lagrange multiplier and a partial differentiation of a Hamiltonian with respect to the output states, and relational expressions representing a relationship between a value of the first Lagrange multiplier after a predictive horizon and a function regarding differences between target values of the output states and the solution output states after the predictive horizon into second linear expressions representing the first Lagrange multiplier at each step by the target values and the solution output states, wherein the Hamiltonian relates to a sum of a first function regarding a weighting addition of a square of an error between the target values of the output states and the output states and a square of inputs, a product of the first Lagrange multiplier and the state equations, and a second function regarding a second Lagrange multiplier and constraint conditions;   converting the first linear expressions and the second linear expressions into third linear expressions representing the first Lagrange multiplier at each step by the target values, the solution inputs and the output states, by substituting the first linear expressions into the second linear expressions;   converting the third linear expressions and second conditional expressions that are partial differentiations of the Hamiltonian with respect to the inputs and represents a relationship among the first Lagrange multiplier, the second Lagrange multiplier and the solution inputs, into fourth linear expressions representing a relationship at each step among the target values, the solution inputs and the output states; and   transforming the fourth linear expressions into a matrix operational expression representing a product of a coefficient matrix and a vector regarding the solution inputs within a control horizon is equal to a function vector regarding the target values and the output states to obtain the coefficient matrix and the function vector.   
     
     
         2 . The computer-readable, non-transitory storage medium as set forth in  claim 1 , wherein the process further comprises generating a first constraint vector, a second constraint vector, a third constraint vector and a fourth constraint vector, and
 wherein the first constraint vector is a vector that includes, for each time, first constant terms in fifth linear expressions obtained by approximating, by the solution inputs, the second Lagrange multipliers associated with lower limit values of the inputs included in the constraint conditions, and the first constant terms include the lower limit values of the inputs included in the constraint conditions and the solution inputs at a corresponding time, which were calculated one sampling time before, and   wherein the second constraint vector includes, for each time, first coefficients of the solution input in the fifth linear expressions, and the first coefficients include the lower limit values of the inputs included in the constraint conditions, and the solution inputs at the corresponding time, which were calculated one sampling time before, and   wherein the third constraint vector is a vector that includes, for each time, second constant terms in sixth linear expressions obtained by approximating, by the solution inputs, the second Lagrange multipliers associated with upper limit values of the inputs included in the constraint conditions, and the second constant terms include the upper limit values of the inputs included in the constraint conditions and the solution inputs at the corresponding time, which were calculated one sampling time before, and   wherein the fourth constraint vector includes, for each time, second coefficients of the solution inputs in the sixth linear expressions, and the second coefficients include the upper limit values of the inputs included in the constraint conditions, and the solution inputs at the corresponding time, which were calculated one sampling time before.   
     
     
         3 . A matrix generation apparatus, comprising:
 a data storage unit storing (a) state equations, which represent a relationship between solution output states and solution inputs of a plant to be controlled, (b) initial condition expressions of output states, (c) first conditional expressions representing a relationship between a first Lagrange multiplier and a partial differentiation of a Hamiltonian with respect to the output state, (d) relational expressions representing a relationship between a value of the first Lagrange multiplier after a predictive horizon and a function regarding differences between target values of the output states and the solution output states after the predictive horizon, and (e) second conditional expressions that are partial differentiations of the Hamiltonian with respect to the inputs and represents a relationship among the first Lagrange multiplier, the second Lagrange multiplier and the solution inputs, wherein the Hamiltonian relates to a sum of a first function regarding a weighting addition of a square of an error between the target values of the output states and the output states and a square of inputs, a product of the first Lagrange multiplier and the state equations, and a second function regarding a second Lagrange multiplier and constraint conditions; and   a processing unit configured to execute a process comprising:
 reading out the state equations and the initial condition expressions from the data storage unit to convert the state equations and the initial condition expressions into first linear expressions representing the solution output states at each step by the output states and the solution inputs ; 
 reading out the first conditional expressions and the relational expressions from the data storage unit to convert the first conditional expressions and the relational expressions into second linear expressions representing the first Lagrange multiplier at each step by the target values and the solution output states; 
 converting the first linear expressions and the second linear expressions into third linear expressions representing the first Lagrange multiplier at each step by the target values, the solution inputs and the output states, by substituting the first linear expressions into the second linear expressions; 
 reading out the second conditional expressions from the data storage unit to convert the third linear expressions and the second conditional expressions into fourth linear expressions representing a relationship at each step among the target values, the solution inputs and the output states; and 
 transforming the fourth linear expressions into a matrix operational expression representing a product of a coefficient matrix and a vector regarding the solution inputs within a control horizon is equal to a function vector regarding the target values and the output states to obtain the coefficient matrix and the function vector. 
   
     
     
         4 . A matrix generation method, comprising:
 converting, by using a computer, state equations, which represent a relationship between solution output states and solution inputs of a plant to be controlled, and initial condition expressions of output states into first linear expressions representing the solution output states at each step by the output states and the solution inputs;   converting, by using the computer, first conditional expressions representing a relationship between a first Lagrange multiplier and a partial differentiation of a Hamiltonian with respect to the output states, and relational expressions representing a relationship between a value of the first Lagrange multiplier after the predictive horizon and a function regarding differences between target values of the output states and the solution output states after the predictive horizon into second linear expressions representing the first Lagrange multiplier at each step by the target values and the solution output states, wherein the Hamiltonian relates to a sum of a first function regarding a weighting addition of a square of an error between the target values of the output states and the output states and a square of inputs, a product of the first Lagrange multiplier and the state equations, and a second function regarding a second Lagrange multiplier and constraint conditions;   converting, by using the computer, the first linear expressions and the second linear expressions into third linear expressions representing the first Lagrange multiplier at each step by the target values, the solution inputs and the output states, by substituting the first linear expressions into the second linear expressions;   converting, by using the computer, the third linear expressions and second conditional expressions that are partial differentiations of the Hamiltonian with respect to the inputs and represents a relationship among the first Lagrange multiplier, the second Lagrange multiplier and the solution inputs, into fourth linear expressions representing a relationship at each step among the target values, the solution inputs and the output states; and   transforming, by using the computer, the fourth linear expressions into a matrix operational expression representing a product of a coefficient matrix and a vector regarding the solution inputs within a control horizon is equal to a function vector regarding the target values and the output states to obtain the coefficient matrix and the function vector.