JPH09297604A - Generation method and automatic generation device for state equation - Google Patents

Abstract

PROBLEM TO BE SOLVED: To shorten the time required for production of a state equation by calculating the state variable of an input variable and the partial derivative concerning an operation variable from a state variable and an input variable and generating a linear state equation and an output equation from the partial derivative. SOLUTION: When the operational elements corresponding to the state variable, operation variable and output variable of a state equation given from an input part 4 are designated to the block diagram data given from an input part 3, an automatic generation part 5 of the state equation traces a signal from the start points set at the operational elements corresponding to the state and output variables through the operational elements corresponding to the state and operation variables in the upstream direction. Thus, the part 5 calculates the state variable of an input variable and the partial derivative concerning the operation variable from the relevant state and output variables and then generates a linear state equation and an output equation from the partial derivative. As a result, the time required for production of the state equation can be extremely shortened.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for generating a linear state equation used for designing a multivariable control system, and an apparatus for automatically generating such a state equation from input block diagram data and arithmetic elements. Regarding

[0002]

2. Description of the Related Art In designing a control system, a linear equation of state represented by the following first and second equations is very effective as a mathematical model representing the dynamic behavior of a plant to be controlled. is there.

DX / dt = AX + BU (1) Y = CX + DU (2) However, in the first and second equations, X is a state variable that represents the state of the plant and is n-dimensional. A vector, U is an operation variable for operating the pant and is an m-dimensional vector, and Y is an output variable to be controlled and is an l-dimensional vector. Also, A,
B, C, and D are (n × n), (n × m), and (l ×), respectively.
n), (l × m) -dimensional constant coefficient matrix.

However, an actual plant model is rarely represented by a linear equation of state such as the first and second equations, and is generally represented by a nonlinear equation of state. For example, in a boiler device, its steam characteristics are represented by a combination of a continuity equation, an energy conservation equation, a momentum conservation equation, and a vapor state equation, and the mathematical model is constituted by a nonlinear differential equation and a nonlinear algebraic equation. The number of mathematical expressions is 250 or more.

Conventionally, such a conversion from a plant model to a state equation is performed by consolidating a required number of equations from the plant model as shown in FIG.
/Dt=f(x1,x2,...,xn,u1,...,um) and obtain the partial derivative ∂f _i / ∂x _j of this nonlinear equation of state The linear state equations shown in Equations 1 and 2 are derived, and this is solved with a differential equation analysis system to obtain the time response calculation result, and block diagram data is created from the same plant model, and this is analyzed with the block diagram. The system is used to solve the problem, obtain the time response calculation result, and confirm the validity of the state equation from both calculation results.

When using such a method, it goes without saying that an error is likely to occur in the process of converting the mathematical expression, and it takes time to check the error. In addition, analysis by the original model is indispensable for checking the equation of state and verifying the designed control method, but for that purpose, it is necessary to create separate data for simulation in addition to this calculation, which requires twice the labor. There is a problem such as.

[0007]

If the linear equations of state as shown in the first and second equations are obtained, it is established in the field of control engineering such as optimal regulator theory, robust control theory, and adaptive control theory. Advanced control systems can be designed using the theory obtained. Many other control system design methods are also based on the linear state equations shown in the first and second equations. Further, by examining the eigenvalues of the matrix A, various data can be obtained regarding the dynamic characteristics of the plant. Such a linear equation of state is an indispensable mathematical model when studying a control system, and development of a method and a device for generating it in a short time is desired.

The present invention has been made to solve the above problems, and its purpose is to provide a linear state indispensable for studying a control system based on block diagram data created from a plant model. It is to provide a method and apparatus for automatically generating equations.

[0009]

In order to achieve the above-mentioned object, the present invention relates to an algebraic operation element such as addition, subtraction, multiplication and division, and a dynamic time-dependent operation such as integration, first-order delay, etc. regarding a method of generating a state equation. The state of the state equation for the block diagram data composed of individual operation elements such as operation elements and external inputs that generate outputs corresponding to time, and the connection data that represents the signal flow between each operation element. When the calculation element corresponding to each variable, operation variable, output variable is specified,
By tracing the signal in the upstream direction from the calculation element corresponding to the state variable and the output variable to the calculation element corresponding to the state variable and the manipulated variable, the deviation of the input variable of the state variable and the output variable with respect to the state variable and the manipulated variable can be obtained. Find the derivative,
A linear state equation and an output equation are generated from this partial derivative.

With respect to the automatic generator of the state equation, algebraic calculation elements such as addition, subtraction, multiplication and division, dynamic calculation elements depending on time such as integration and first-order delay, and external inputs for generating outputs corresponding to time The input part of the block diagram data composed of individual calculation elements and the connection data representing the flow of signals between the calculation elements, and the calculation variables corresponding to the state variables, manipulated variables, and output variables of the state equation. When the block diagram and the operation element are input to the input section and each of the input sections, the operation corresponds to the state variable and the operation variable in the upstream direction with a signal starting from the operation element corresponding to the state variable and the output variable. The partial derivatives with respect to the state variables and the manipulated variables of the input variables of the state variables and output variables are obtained by tracing to the elements, and linear state equations and output equations are generated from these partial derivatives. A generation unit of the state equation that the output of the coefficient matrix of the state equation is a calculation result, and the configuration of and a storage device for files coefficient matrix output state equation from the output unit.

That is, in the block diagram data shown in FIG. 1, the state variable x _i representing the dynamic behavior of the controlled object can be associated with a dynamic element such as a first-order lag or a second-order lag,
The output variables y _i, which are variables to be controlled, can correspond to dynamic elements or algebraic elements. The manipulated variables u _i used to control the plant can also correspond to dynamic or algebraic elements.

It is now assumed that the following formulas (3) and (4) are derived from the given block diagram data regarding the state variables x _i and the output variables y _i .

Dx _i / dt = f _i (x ₁ , x ₂ , ・ ・ ・ ・ ・, x _n , u _l , ・ ・ ・ ・ ・, u _m ) (3) y _i = g _i (x ₁ , x ₂ , ..., x _n , u _l , ..., u _m ) (4) where f _i and g _i are non-linear functions.

If the third and fourth equations are Taylor-expanded on the assumption that minute variations of x _i , y _i , and u _i are obtained, the linear forms of the fifth and sixth equations can be obtained.

[0015]

[Equation 1]

Δx _i , Δy _i , and Δu _i are state variables,
If the output variable and the manipulated variable are used, the first and second equations can be created from the fifth and sixth equations. Here, the i-th row and j-th column element a _ij of the matrix A is ∂f _i / ∂ in the fifth equation.
is equal to x _j .

However, in an actual plant such as a boiler, the variables included in the model are as many as 2000 to 3000, and the differential equations and algebraic equations are mixed, so that the above third and fourth equations are virtually derived. Difficult to do.

On the other hand, noting that the block diagram is a directed graph, the partial derivative between variables can be numerically obtained by tracing the signal flow in the opposite direction. By using this method, ∂f / ∂x, ∂f / ∂u, ∂g / ∂x, ∂ of the fifth and sixth equations can be obtained without using the third and fourth equations.
g / ∂u can be directly determined, and A, B, C, D of the first and second equations can be automatically created in the computer.

The block diagram data for the dynamic characteristic simulation usually includes a model that has not been used for the control device or control system design that has been used so far. Even in this case, as shown in FIG. 1, the element for which the state equation is to be created is designated and the partial derivative between the designated elements is used to convert the block diagram data in an arbitrary range into a linear state equation. It is possible. If a linear equation of state is obtained in this way, a general-purpose block diagram analysis system will be used thereafter to apply an advanced regulator theory, robust control theory, adaptive control theory, etc. to a sophisticated control system. Can be designed.

[0020]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS FIG. 2 shows an apparatus for automatically generating a state equation according to the present invention. As is apparent from the figure, the automatic state equation generation apparatus of this example includes a display means 1 such as a graphic display for displaying input data and calculation results, and a workstation or personal computer for performing various processes and calculations. And the like, and a storage unit 7 such as a hard disk device. The calculation unit 2 corresponds to the block diagram data input unit 3 and the state variables, operation variables, and output variables of the state equation. There is provided an input unit 4 of a calculation element, an automatic generation unit 5 of a linear state equation which is a calculation target, and an output unit 6 of a coefficient matrix of a state equation which is a calculation result.

FIG. 3 shows an internal processing flow of the calculating means 2. As is clear from this figure, in the present example, the generation of the linear state equation is performed by (S1) reading the block diagram data, (S2) reading the state variables, the operation variables, and the output variables, and (S3) The state equation is created using the data, and (S4) the coefficient matrix that is the calculation result is output to the storage means 7.

The input processing of the block diagram data in step S1 depends on the data format of the block diagram data, but generally, the one shown in the following formula (7) is widely used.

Variable name: = operator (input variable 1, ..., Input variable k ₁ , $ coefficient 1, ..., Coefficient k ₂ ); (7) In FIG. An example of block diagram data is shown. Since the input variable indicates the element on the upstream side, the calculation content of each element and the signal flow can be expressed by the expression like the above-mentioned expression 7. In the block diagram data input unit 3, the data of FIG. 4 is input, and {variable name, operator, input variable 1, ..., Input variable k ₁ , $ coefficient 1, ..., Coefficient k ₂ } Memorize in pairs. To speed up the process, variable names and input variables can be coded and stored.

In step S2 of FIG. 3, the state variables, operation variables, and output variables of the state equation are associated with the elements of the block diagram data. FIG. 5 shows an example of the designation format. The input unit 4 for state variables, manipulated variables, and output variables inputs these data and stores them as a set of {state variable, variable name of block diagram element}.

In step S3 of FIG. 3, a state equation is created according to the processing flow of FIG. In the block diagram data, external input variables and dynamic variables hold values at the initial point. Therefore, in order to calculate the values of the algebraic elements based on these external input variables and dynamic variables, the calculation is first ordered in step S11. This can be determined by starting from the external input variable and the dynamic variable at the initial time point and searching for which variable the variable is used as the input variable. Then, step S12
After setting the initial value to the dynamic variable in step S13, the process proceeds to step S13, and the value of the algebraic variable z _i is calculated according to the calculation order determined in step S11 with the external input variable and the dynamic variable at the initial point as starting points. To do. At the same time,
The partial differential coefficient (element partial differential coefficient) ∂φ _i / ∂z _j related to the input variable is obtained for each variable. However, φ _i represents the calculation formula of the variable i.

In steps S14 to S16, the partial derivatives on the right sides of the third and fourth equations are obtained. The partial derivative ∂f _i / ∂x _{j on} the right side of the third equation can be easily determined by the method shown in FIG. 7 if the partial derivative related to the input variable of the dynamic variable is obtained.

FIG. 7 shows a processing method focusing on elm3. First, a table that can store numerical values for all variables is prepared, and "1" is set in the position corresponding to the input variable elm2 of the dynamic variable elm3. Next, the signal is traced upstream, and the deviation coefficients "1" and "-1" of elm1 and elm4 with respect to elm2 are set at the positions of elm1 and elm4, which are the input variables of elm2. elm1 is an instrumental variable, and terminates the search there. On the other hand, upstream variables elm4 is x _1, the partial derivative .differential.H ₂ / ∂ a value "-1" position elm4 the position of _{x 1 (elm3) = ∂h 2} / ∂x 1
Set the product of. elm3 is a dynamic variable, and the processing ends here. The above calculation means that the following chain rule of partial differentiation is executed, and elm is set at the position of each variable.
The partial derivative of each variable for 3 is stored.

[0028]

[Equation 2]

On the other hand, since x ₁ (elm3) is expressed by the equation 8 below, the equation of state is expressed by equation 9 below.

[0030]

(Equation 3)

∂f ₁ / ∂ in the above equations 9 to 5
x ₁ , ∂f ₁ / ∂x ₂ , ... Can be specifically determined as shown in the following tenth formula.

[0032]

(Equation 4)

∂f ₁ / ∂x ₁ is the 1st row and 1st column of the coefficient matrix A shown in the above formula 1, and ∂f ₁ / ∂x ₂ is the 1st row and 2nd column of the coefficient matrix A and ∂f ₁ / ∂u ₁ is set in the first row, first column of the coefficient matrix B shown in the first expression, and ∂f ₁ / ∂u ₂ is set in the first row, second column of the coefficient matrix B. That is, by this, the first row of the coefficient matrices A and B shown in the first equation can be determined. And
By repeating this for all state variables and output variables as shown in FIG. 6, the coefficient matrices A, B, C, and D of the first and second equations can be completed.

The coefficient matrices A, B, C and D are output to the storage means 7 of FIG. 2 in step S4 of FIG. In this case, it is more preferable that the coefficient matrices A, B, C and D be output as character type data so that they can be easily used in other control system design systems and dynamic characteristic analysis systems.

[0035]

As described above, according to the present invention,
Since linear state equations that have a great effect on control system design and dynamic characteristic analysis can be automatically created from nonlinear block diagram data, the state equation creation time can be greatly shortened and state equation creation errors can be prevented. can do. As the nonlinear block diagram data that is the basis of the linear equation of state, the data used for the dynamic characteristic simulation can be used as it is. Therefore, the data can be shared and the original model can be used by using the block diagram simulator. And the linear equation of state model can be easily compared, and the control system can be efficiently designed.

[Brief description of drawings]

FIG. 1 is an explanatory diagram of a state equation generation method according to the present invention.

FIG. 2 is a configuration diagram of a state equation generation device according to an exemplary embodiment.

FIG. 3 is a flowchart showing a procedure for generating a state equation.

FIG. 4 is an explanatory diagram showing an example of block diagram data.

FIG. 5 is an explanatory diagram showing an example of designation of state variables, operation variables, and output variables.

FIG. 6 is a flowchart showing a procedure executed by an automatic generation unit of a state equation.

FIG. 7 is an explanatory diagram showing a method of calculating a partial derivative in a block diagram.

FIG. 8 is a flowchart showing a conventional state equation generation method.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 display means 2 computing means 3 block diagram data input section 4 input section for state variables, manipulated variables, output variables 5 automatic generation section for state equations 6 coefficient matrix output section for state equations 7 storage means

Claims (2)

[Claims] 1. An algebraic arithmetic element such as addition, subtraction, multiplication and division, and integration.
Time-dependent dynamic operation elements such as next-delay, individual operation elements such as external inputs that generate output corresponding to time,
When the operation elements corresponding to the state variable, operation variable, and output variable of the state equation are specified for the block diagram data composed of the connection data representing the signal flow between the operation elements, the state variable And the partial derivative with respect to the state variable and the manipulated variable of the input variable of the state variable and the output variable by tracing the signal in the upstream direction from the computation element corresponding to the output variable to the computation element corresponding to the state variable and the manipulated variable. And a linear equation of state and an output equation are generated from this partial derivative. 2. An algebraic arithmetic element such as addition, subtraction, multiplication and division, and integration.
Time-dependent dynamic operation elements such as next-delay, individual operation elements such as external inputs that generate output corresponding to time,
An input unit of block diagram data composed of connection data representing the flow of signals between the respective arithmetic elements, an input unit of arithmetic elements corresponding to state variables, manipulated variables and output variables of the state equation, and When the block diagram data and the arithmetic element are input to the input unit, the signal is traced upstream from the arithmetic element corresponding to the state variable and the output variable to the arithmetic element corresponding to the state variable and the manipulated variable. By calculating the partial derivatives with respect to the input variables of the state variables and output variables and the manipulated variables, the state equation generator that generates linear state equations and output equations from these partial derivatives, and the state that is the operation result An automatic generation of a state equation, comprising: an output unit for the coefficient matrix of the equation; and a storage device for storing the coefficient matrix for the state equation output from the output unit. Equipment.