JP6686338B2 - Control system, control system design method, and program - Google Patents

Description

  The present invention relates to a control system, a control system design method, and a program.

  One-loop controllers such as ON-OFF control and PID control have been widely applied in the industrial world. Also, many control design methods for these controllers have been developed. These control designs are based on classical control theory.

  On the other hand, research on control design methods using modern control theory has progressed since the 1960s, and the results have been applied to industry. In the 1980s, control design focusing on model error such as robust control called post-modern control theory became mainstream.

  Robust control is a new control theory that combines classical control and modern control. Robust control is to introduce a new concept “uncertainty (Δ)” that expresses ambiguity and incompleteness of a model, and consider the uncertainty to design a control system that satisfies stability and control performance at the same time. Is possible. Robust control is a very effective control theory for designers who are worried about uncertainties such as model errors and disturbances. However, since robust control theory is built on classical control theory and modern control theory, background knowledge of the two theories is required. Therefore, if the design is performed without properly understanding the characteristics of the robust control theory and the controlled object, the control performance may be inferior to the PID control that is a typical example of the classical control. PID control is still mainstream in the industrial world because it is one of the control methods that has the highest “effect” of control performance and stability with respect to the “cost” of knowledge level, time, and equipment required for design. It is thought to be because it is one.

  In this regard, Japanese Patent Laying-Open No. 2003-195905 (Patent Document 1) proposes a control device and a temperature controller that realize a robust stability by calculating a PID control parameter that satisfies the robust stability in the robust control theory. However, Patent Literature 1 focuses only on robust stability and does not consider robust performance for evaluating control performance. For this reason, the design becomes conservative, and sufficient control performance cannot be guaranteed. Moreover, in patent document 1, the model predictive control law is also incorporated. An approach to improve the control performance can be considered by adjusting the weighting coefficient of the model predictive control law. However, by combining all three control theories, the points and policies of control design become vague, and as a result it is difficult to obtain the desired performance and stability.

  Further, JP-A-9-85407 (Patent Document 2) proposes a molten steel level control method in a mold in a continuous casting machine. Patent Document 2 includes three control methods of PID control, observer control, and robust control, and by switching the three control methods depending on the conditions of the respective threshold values of the pouring speed, the residual molten steel amount in the ladle, and the tundish weight. We propose highly accurate molten steel level control. However, in order to design this control system, background knowledge of three control methods is required, and since it is necessary to design three types of controllers in advance, the engineering cost is tripled. Furthermore, since all three control methods are based on different control theory systems, there is a problem in that there is a risk of causing a sudden change in the controller output due to switching of the control methods and destabilization of the system.

JP, 2003-195905, A JP, 9-85407, A

  As described above, in order to apply the design method shown in the background art, knowledge of modern control technology is required in addition to classical control, and design is difficult unless both skills are mastered.

  A main invention in the present application aims to provide a design method in which the stability and control performance of a system are taken into consideration while maintaining PID control or equivalent simplicity.

One of the means for solving the above problems is a feedback controller that performs feedback control of the controlled object based on a deviation between an output from the controlled object and a predetermined target value, and a nominal modeled model of the controlled object. the output of the plant model, the output from the controlled object, with the deviation is input, output and disturbance feedback element input and output are added to the control target of the feedback controller, comprising Ru controls In the system, the disturbance feedback element is defined as a transfer function of any one of a coefficient gain, a first-order time delay element, an integration element, or a differentiation element, and a parameter of the transfer function is introduced into the control target. Robust stability considering the given uncertainty and the nominal performance of the nominal plant model The value of the relation formula showing the robust performance of the control system, it is a value smaller than the predetermined value indicating the internal stability of the control system, characterized by.

  In addition, the problem disclosed by the present application and the solution to the problem will be clarified by the description of the mode for carrying out the invention, the description of the drawings, and the like.

  According to the present invention, it is possible to enhance stability and control performance by introducing a disturbance feedback element while taking advantage of the characteristics of feedback control such as PID control which is most popular in the industrial world.

FIG. 3 is a block diagram showing functions of a feedback control system including a disturbance feedback element in the embodiment of the present invention. FIG. 6 is a block diagram showing a function of robust disturbance feedback control having a multiplicative uncertainty (Δ) to an output system in an embodiment of the present invention. FIG. 3 is a diagram showing a ΔN structure and a ΔPL structure in an embodiment of the present invention. FIG. 3 is a diagram showing a concept of an algorithm for searching a parameter L of a disturbance feedback element in the embodiment of the present invention. FIG. 6 is a diagram showing an example of robust stability (RS), nominal performance (NP), and robust performance (RP) in the frequency domain in an embodiment of the present invention. In one embodiment of the present invention, it is a flow chart which shows a search algorithm of L which is a parameter of a disturbance feedback element for robust disturbance feedback control. FIG. 7 is a diagram showing details of a search procedure for a first-order lag system parameter L in FIG. 6 in the embodiment of the present invention. FIG. 7 is a diagram showing an example of an adjustment result of the L search algorithm in the embodiment of the present invention.

  At least the following matters will be made clear by the description in the present specification and the accompanying drawings. In the following description, “control system” and “control system” (or simply “system”) are used interchangeably.

=== Outline of control system ===
A block diagram of a disturbance feedback control system used in this embodiment is shown in FIG. As shown in FIG. 1, the control system according to this embodiment includes a disturbance feedback element 4 (transfer function) in a feedback control system including a controlled object 1 (transfer function G), a controller 2 (transfer function K), and a direct feedback loop. L) is added.

The feedback control system is, for example, a PID control system, and the difference between the output y (including the disturbance d) of the controlled object 1 and the set value r is input to the controller 2, and the controller 2 outputs a signal u _l . Further, the difference ε between the output y _n of the nominal plant model 3 (transfer function G _n ) and the output y of the controlled object 1 is input to the disturbance feedback element 4. The output u _d of the disturbance feedback element 4 is added to the output u _l of the controller 2 in the adder 7, and the signal u (= u _d + u _l ) thus added is input to the controlled object 1. In the following description, components of the control system, such as the controller K, may be represented by transfer functions.

  The closed loop transfer function of this control system is expressed by the following equation.

In the transfer function of such a control system, when the transfer function G of the controlled object 1 and the transfer function G _n of the nominal plant 3 completely match (G = G _n ), the above (formula 1) is expressed as follows. .

Here, the first term of (Equation 2) is the transfer function of the set value response (input r to output y), and the second term is the disturbance response transfer function (input d to output y).

From (Equation 2), under the condition of G = G _n , L disappears from the first term, so it is understood that the disturbance feedback element L works effectively only for the disturbance response of the second term. Furthermore, since L is on the denominator side of the transfer function, it can be seen that the larger the value of L, the more the disturbance signal d is canceled out. The above is the basic features of the disturbance feedback control.

<< Introduction of Robust Control Theory >>
The robust control theory is introduced into the disturbance feedback control system shown in FIG. As described above, robust control theory introduces uncertainty Δ into the model.Therefore, there are various representation methods for this uncertainty Δ, such as multiplicative or additive, input system or output system, depending on the control target or control purpose. is there. Here, robust disturbance feedback control having multiplicative uncertainty and output system uncertainty is used, but the method of expressing uncertainty Δ in the present invention is not limited.

  A block diagram of a robust disturbance feedback control system with multiplicative uncertainty and output system uncertainty is shown in FIG. Note that FIG. 3 shows a ΔPL structure constructed by organizing the input / output signals of FIG. The details will be described below.

In FIG. 2, the controlled object G of FIG. 1 is replaced with a model including a nominal plant model G _n , a weighting function Wo and an uncertainty Δ, and a weighting function W _{P of the} output y is introduced, as described later in detail. Has been done.

For such a control system, the output signal y ′ from the nominal plant model 11 (transfer function G _n ) is represented by the following equation.

Further, the uncertainty yΔ of the output signal of the controlled object 11 is calculated by using the weighting function Wo for the output system and the multiplicative uncertainty,

By substituting (Equation 3) into (Equation 4), the following equation is obtained.

However, u _l is an output from the controller K, u _d is an output from the feedback element L, and u is a control signal obtained by adding u _l and u _d (u = u _{l +} u _d ).

u is expressed by the following equation using the first equation of (Equation 3) and y ′ ₌ G _n u.

Also, using the relation of u = u _{l +} u _d , u _l is expressed as follows.

Also, the following is obtained from the relational expression of u _l = Ke and e '= W _p e.

However, W _p is a weighting function for weighting the output error e.

  Then, the model error ε is shown below.

Since y _{n =} G _n u _l and y ₌ d + uΔ + y ′, (Equation 9) can also be written as follows.

By substituting the second equation of (Equation 3) and (Equation 7) into (Equation 10), the following equation is obtained.

<< ΔPL structure and ΔN structure >>
By organizing the robust disturbance feedback control structure as described above, the closed loop system of FIG. 2 shows the elements other than the uncertainty Δ and the disturbance feedback element L collectively as P, as shown in FIG. 3B. It can be rewritten like a ΔPL structure. Then, based on FIG. 3B, the following relational expressions are obtained from (Expression 4), (Expression 7), (Expression 8), and (Expression 11).

From equation (12), (uΔ, d, u d) and (yΔ, e ', ε) following P matrix showing the relationship between is obtained.

Here, P ₁₁ is a 2 × 2 matrix, P ₁₂ is a 2 × 1 matrix, P ₂₁ is a 1 × 2 matrix, and P ₂₂ is a 1 × 1 matrix.

  Next, the N matrix is obtained from this P matrix. That is, by representing elements other than Δ in the ΔPL structure shown in FIG. 3B by N, the control system of FIG. 3B is rewritten by the ΔN structure of FIG. 3A. Then, an N matrix showing this ΔN structure is derived, and using this N matrix, an evaluation index of robust control theory, robust stability (Robust Stability: RS), nominal performance (Nominal Performance: NP), and robust performance (Robust Performance: RP) shall be described.

  Specifically, when the linear fractional transformation of the P matrix is performed, the N matrix is represented by the following formula.

Further, by rearranging (Equation 14), the N matrix is represented by the following equation.

However, S and T and S _L and T _L satisfy the following (formula 16) and (formula 17), respectively.

<< conditions for robust stability, nominal performance, robust performance >>
Using the above results, we derive the conditions of robust stability, nominal performance, and robust performance.

First, robust stability (RS) and nominal performance (NP) are obtained from the diagonal elements of (Equation 15). In the element N ₁₁ in the first row and first column of the N matrix, the norm || N ₁₁ || <1 is the condition of RS, and therefore the following equation is obtained. When the N matrix is internally stable, it is called Nominal Stability (NS).

Further, in the element N ₂₂ in the second row and second column of the N matrix, the norm || N ₂₂ || <1 is the condition of NP, and therefore the following equation is obtained.

Robust performance (RP) is derived using such NP. In the first place, NP indicates the performance condition based on the nominal plant model, which does not consider the uncertainty (Δ). On the other hand, RP is a performance condition considering uncertainty Δ. Therefore, by adding the worst case uncertainty | Δ | = 1 to the NP condition shown in (Equation 19), the RP condition can be obtained as follows.

After all, RP is expressed by the sum of RS and NP, and the sum is defined to be less than 1 (see the following Expression 21).

The above (Equation 21) is the robust performance (RP) condition of the control method of the present invention. The present invention utilizes this RP condition to provide an algorithm for searching for a parameter L that satisfies RP <1.

  By the way, when the usual H∞ control theory is used without applying the control method of the present invention, a ΔPK structure is obtained, and Robust Control Toolbox provided by control design software such as Matlab (trademark) is used. The controller K that satisfies the robust performance will be designed. The relational expression (including the P matrix) and the N matrix showing the ΔPK structure in this case are represented by the following expressions.

However, S and T are a sensitivity function and a complementary sensitivity function represented by 1 / (1 + PK) and PK / (1 + PK), respectively.

<< L search algorithm >>
A search algorithm for the disturbance feedback element L will be described with reference to FIGS. FIG. 4 is a diagram showing the concept of the L search algorithm. FIG. 5 is a diagram showing an example of robust stability (RS), nominal performance (NP), and robust performance (RP) in the frequency domain. FIG. 6 is a flowchart showing the L search procedure. FIG. 7 shows details of the procedure for searching L as the first-order lag element in FIG. FIG. 8 is a diagram showing an example of the adjustment result of the L search algorithm.

As shown in FIG. 4, the concept of the L search algorithm in the present embodiment is to adjust L contained in the transfer functions of S _L and T _L to obtain two peak values of RP in the frequency domain. It is shaping into a value. One of the two peaks of RP is derived from NP and the other peak is derived from RS. Since the two peak values are in a trade-off relationship with each other, RS decreases as NP decreases, and NP increases as RS decreases. For example, digging a hole in a sandy area with a scoop to lower it is similar to making a pile of sand removed with a scoop elsewhere.

  Therefore, in the present embodiment, as shown in FIG. 4, by adjusting L like a scoop, the RP in the frequency domain like a sand is formed into a desired value (RP <1).

  FIG. 5 illustrates the gains of RS, NP, and RP in the frequency domain under the condition of L = 0. As shown in FIG. 5, RP (solid line) has peak gains derived from RS (dashed line) and NP (dashed line). Further, in FIG. 5, it can be seen that the RP at L = 0 does not satisfy the robust performance condition RP <1 because the peak gain derived from NP is larger than 1.

  Here, it is desirable that L has a simple structure. Therefore, in the procedure of the present embodiment, first, L is defined as a coefficient gain and a search is performed. The search ends when RP <1 is achieved with L defined by the coefficient gain. On the other hand, if RP <1 is not achieved by the coefficient gain L, redefine L as a first-order lag element, an integral element, or a derivative element and search again. This is a rough L search flow.

  Specifically, the L search algorithm is composed of five steps, as shown in FIG. The details of each step will be described below.

・ First step: Step S1
In the first step, initial setting is performed. For example, the nominal plant model G _n , the model g including uncertainty, the controller K, and the frequency domain to be evaluated are set. Here, the model g including the uncertainty is represented by the following, for example.

However, Δ _m and Δ are uncertainties.

  These initial settings may be given, for example, based on plant specifications, measurement results, and the like.

-Second step: Step S2
In the second step, the cutoff frequency is set. Further, based on the various parameters set in the first step, the parameters a ₀ , a ₁ , ..., _An , b ₀ , b included in the numerator and denominator of the following two weighting functions W _o and W _p are given. ₁ is determined.

・ Third step: Step S3
In the third step, the maximum gain L _max0 of RP at L = 0 is calculated. Then, this maximum gain L _max0 is set as the initial value of the L gain search.

-Fourth step: Steps S4-S8
In the fourth step, L is defined as a gain and the search is executed. That is, the L gain is gradually increased and repeated calculation is performed until RP <1 is satisfied or the set number of repetitions l _max is reached.

For example, it is defined as L = L _max0 × 10 ^l−1 (l = 1, 2, 3, ... L _max ). First, l = 1 is set (step S4), and the maximum gain of the RP formed by each L is calculated. Among them, L, which forms the smallest maximum gain, is set as the midpoint of the next trial number. Then, if L showing the minimum robust performance exists at the inner point of the search range, the search range is reduced at an arbitrary ratio, and if L exists on the boundary of the search range, The search range is expanded at a rate of (step S6).

Then, in step S7, it is determined whether or not RP <1 is satisfied. If this condition is satisfied, the process ends. If not satisfied, L is increased in step S8, the process returns to step S5, and calculation is performed again. If RP <1 is not satisfied even after calculation up to the number of iterations l _max, proceed to the next fifth step.

・ Fifth step: Steps S9 to S14
In the fifth step in the present embodiment, the search is executed with the transfer function as the first-order lag element L (s) defined by the following. The transfer function L (s) may be defined as an integral element or a differential element.

Here, k _L is the gain of the transfer function L (s), k _Lmin is the lower limit of k _L , k _Lmax is the upper limit of k _L , τ _L is the time constant of transfer function L (s), and τ _Lmin is τ _{L is} the lower limit value, and τ _Lmax is the upper limit value of τ _L.

L as a first-order lag element is designated and searched in a two-dimensional search area consisting of the k _L axis and the τ _L axis. In the present embodiment, as shown in FIG. 7, the search points are divided on a logarithmic scale, and the maximum gain of RP is calculated for all the search points on the created two-dimensional mesh.

Specifically, in step S11, when the number of trials is the kth time (k = 1, 2 ...), gains k _Lbest and τ _Lbest that minimize the maximum gain of RP are selected from all search points. Ask for. Then, such gains k _Lbest and τ _Lbest are set as the midpoints of the search area in the next trial (k + 1 time). The following Expressions 27 and 28 show this.

In addition, if L that shows the minimum robustness exists at the inner point of the search range, the search range is reduced at an arbitrary ratio, and if L exists on the boundary of the search range, Expand the search range by a percentage.

Then, in step S12, it is determined whether or not RP <1 is satisfied. If this condition is not satisfied, the (k + 2) th trial is performed. On the other hand, when this condition is satisfied, in step S13, the absolute value of the error between the _two peak gains y ₁ and y ₂ of the RP when the maximum gain of the RP in each trial becomes the minimum is previously determined. It is judged whether it is less than the set allowable error E. That is, the sufficiency of the following (Formula 29) is determined.

When this condition is satisfied, the process ends. If the condition is not satisfied, steps S10 to S13 are repeated until the number of trials reaches i _max . Then, if (Equation 29) is not satisfied even when the number of trials reaches i _max , the processing ends.

  The above is the L search algorithm of the present invention. FIG. 8 shows an example of the adjustment result by the algorithm in this embodiment. It can be confirmed that the result satisfying RP <1 is obtained.

  By determining the disturbance feedback element L in this manner and designing the control system using this disturbance feedback element L, a control system having robust performance can be obtained. The procedure for searching for the disturbance feedback element L described above, the procedure for designing a control system using such L, and the function of the designed control system are executed as a program of a computer including a CPU, a ROM, and a RAM. It

  As described above, in the present embodiment, when designing the control system, the stability and control performance of the system are taken into consideration via the parameter L even though the PID control or an equivalent simple configuration is used. Further, the parameter L is expressed as a transfer function of a coefficient gain or a time delay element, for example, a first-order delay element, and has a very simple structure. Therefore, a control system with stability and control performance is designed based on the basic knowledge of classical control such as PID control without detailed knowledge of classical control and modern control technology necessary for robust control design. It is possible.

  Further, it is easy to introduce the present embodiment into an existing feedback control system. Furthermore, it is also possible to reduce the burden and risk in terms of technology and cost when a control system designer or plant operator introduces this embodiment.

  The present embodiment is suitable for a system having strong uncertainty and nonlinearity, and a system in which it is difficult to implement an expensive control system. As a control purpose, it is suitable for systems that prioritize disturbance suppression. For example, it is suitable for a control system such as constant superheat control of a refrigeration cycle.

As described above, the feedback controller 12 that performs the feedback control of the controlled object 11 based on the deviation between the output y from the controlled object 11 and the predetermined target value r, and the output of the nominal plant model 13 that models the controlled object 11 The deviation ε between y _n and the output y from the controlled object 11 is input, and the output u _d is added to the output u _l of the feedback controller 12 to be input to the controlled object 11. , And the parameter L of the disturbance feedback element 14 is determined to have a predetermined stability and control performance under the uncertainty Δ introduced in the controlled object 11.

  According to this embodiment, it is possible to enhance the stability and control performance of the system by newly introducing the disturbance feedback element L while making the most of the characteristics of feedback control such as PID control which is most popular in the industrial world. It is possible. As described above, the present embodiment is designed based on the existing feedback control technology and control structure. Therefore, when the control designer or the plant operator introduces the new technology, the technical and cost burdens are reduced. It also has the effect of lowering the risk.

  Further, since the disturbance feedback element 14 is defined as a transfer function such as a coefficient gain, a first-order time delay element, an integral element, or a derivative element, the disturbance feedback element 14 has a very simple structure. It can be installed in inexpensive control equipment.

  In addition, the parameter L of the disturbance feedback element 14 is determined based on the peak value represented by the relational expression representing the robust performance of the control system, so that the parameter L that satisfies the robust performance can be easily determined. it can. This facilitates the design of the entire control system.

  Further, the remaining control system excluding the uncertainty Δ and the disturbance feedback element 13 is represented by a predetermined model P, and the parameter L of the disturbance feedback element 13 is based on the uncertainty Δ, the disturbance feedback element 13, and the predetermined model P. By determining, the parameter L that satisfies the robust performance can be easily determined, and the control system can be easily designed.

  It should be noted that the above-described embodiment is for facilitating the understanding of the present invention and is not for limiting the interpretation of the present invention. The present invention can be modified and improved without departing from the spirit thereof, and the present invention includes equivalents thereof.

11, 13 Nominal plant model 12 Controller 14 Disturbance feedback element 18 Uncertainty 19, 21 Weighting function

Claims (4)

A feedback controller that performs feedback control of the controlled object based on the deviation between the output from the controlled object and a predetermined target value,
The deviation of the output of the nominal plant model that models the controlled object and the output from the controlled object is input, and the output is added to the output of the feedback controller and the disturbance is input to the controlled object. A feedback element,
A control system for Ru provided with,
The disturbance feedback element is
It is defined as a transfer function of any one of coefficient gain, first-order time delay element, integral element, or derivative element,
The parameters of the transfer function are
Robust stability after considering the uncertainty introduced into the controlled object, and the nominal performance of the nominal plant model, the value of the relational expression representing the robust performance of the control system represented by the sum of the, A value that is smaller than a predetermined value that indicates the internal stability of the control system,
Control system characterized by. A method of designing a control system including a feedback controller that performs feedback control of the controlled object based on a deviation between an output from the controlled object and a predetermined target value,
A nominal plant model that models the controlled object, and a deviation between the output of the nominal plant model and the output from the controlled object is input, and the output is added to the output of the feedback controller to the controlled object. introducing a, a disturbance feedback element to be input,
A step of introducing the uncertainty on the controlled object,
Defining the disturbance feedback element as a transfer function of any one of coefficient gain, first-order time-delay element, integral element, or derivative element;
Setting the values of the parameters of the transfer function,
Robust stability after considering the uncertainty introduced into the controlled object, and the nominal performance of the nominal plant model, the value of the relational expression representing the robust performance of the control system represented by the sum of, Calculating using the values of the parameters of the transfer function,
Searching for the value of the parameter of the transfer function that makes the calculated value of the relational expression smaller than a predetermined value indicating the internal stability of the control system by changing the value of the parameter of the transfer function. When,
A method for designing a control system , comprising : A method of designing a control system according to claim 2, wherein
In the defining step,
The disturbance feedback element is defined as a transfer function of coefficient gain,
Determining whether or not there is a parameter value of the transfer function that makes the value of the relational expression smaller than the predetermined value,
In the determining step, when it is determined that the value of the parameter of the transfer function that makes the value of the relational expression smaller than the predetermined value does not exist, the disturbance feedback element is a first-order time delay element, an integration element, Or a step of redefining any one of the differential elements as a transfer function,
Further including,
The setting step, the calculating step, and the searching step are performed for the transfer function redefined in the redefining step,
A method for designing a control system characterized by: A program for designing a control system including a feedback controller that performs feedback control of the controlled object based on a deviation between an output from the controlled object and a predetermined target value,
A nominal plant model that models the controlled object, and a deviation between the output of the nominal plant model and the output from the controlled object is input, and the output is added to the output of the feedback controller to the controlled object. A disturbance feedback element to be input, and a procedure for introducing
A procedure for introducing uncertainty into the controlled object;
A procedure for defining the disturbance feedback element as a transfer function of any one of coefficient gain, first-order time delay element, integral element, or derivative element,
A step of setting the value of the parameter of the transfer function,
Robust stability after considering the uncertainty introduced into the controlled object, and the nominal performance of the nominal plant model, the value of the relational expression representing the robust performance of the control system represented by the sum of the, A procedure for calculating using the values of the parameters of the transfer function,
A procedure for searching for a value of the parameter of the transfer function, which makes the calculated value of the relational expression smaller than a predetermined value indicating the internal stability of the control system, by changing the value of the parameter of the transfer function. When,
A program that causes a computer to execute.