JP2014153949A - Non-linear pid controller using gmdh and gmdh weighting factor determination method - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a non-linear PID controller capable of operating with a small capacity of memory.SOLUTION: A non-linear PID controller (10A) comprises: a deviation arithmetic unit (12) for calculating a deviation between an output value and a target value of a non-linear system (100) to be controlled; a derivative arithmetic unit (14) for calculating a derivative value of the deviation; a GMHD-PID adjustment unit (16A) for calculating a proportional gain, integral gain, and derivative gain associated with PID control of the non-linear system from three values which are a target value, deviation, and derivative value on the basis of GMDH theory; and a PID control unit (18) for performing PID control on the non-linear system by the calculated proportional gain, integral gain, and derivative gain on the basis of the deviation.

Description

  The present invention relates to a PID controller, and more particularly, to a nonlinear PID controller that performs PID control of a nonlinear system.

  In industrial processes, PID control is still widely used because of its simple control structure. However, many real systems have non-linearity, and it is difficult to always obtain a good control result by the control using a fixed PID parameter. As an effective control method for such a nonlinear system, a CMAC-PID controller using a Cerebellar Model Articulation Controller (CMAC) which is a kind of neural network (NN) has been proposed (for example, Non-Patent Document 1). However, the fact that online learning is necessary to acquire optimal control parameters and that a large amount of memory is required to realize the control system are major obstacles to practical use. CMAC-FRIT has already been reported as a solution to the former problem (see, for example, Non-Patent Document 2). This is because, by incorporating the FRIT (Fictious Reference Iterative Tuning) method, which directly calculates control parameters from the closed loop data of the target system, into CMAC learning, it is possible to learn CMAC offline without building a system emulator. It is a technique that can be done.

Ryota Kurosumi and Toru Yamamoto: "A design of intelligent PID control system using cerebellar computation model", IEEJ Transactions. C, Vol.125, No.4, 607-615, (2005) Wakiya, Onishi, Yamamoto: “Design of CMAC-PID controller using FRIT for nonlinear systems”, Transactions of the Society of Instrument and Control Engineers, Vol.48, No.12 (2012)

  Since the CMAC-PID controller uses a CMAC having a three-dimensional input space for each parameter of the PID, the load table becomes enormous. For this reason, when the CMAC-PID controller is realized by a microcomputer or the like, a large-capacity memory is required to store the CMAC load table. However, mounting a large-capacity memory in a microcomputer is disadvantageous in terms of cost.

  In view of the above problems, an object of the present invention is to solve the problem of the memory capacity of the CMAC-PID controller, and to provide a nonlinear PID controller that can operate with a smaller capacity memory.

  A nonlinear PID controller according to an aspect of the present invention includes a deviation calculating unit that calculates a deviation between an output value of a nonlinear system to be controlled and a target value, a differential calculating unit that calculates a differential value of the deviation, and the target GMHD-PID adjustment unit that calculates a proportional gain, an integral gain, and a differential gain according to PID control of the nonlinear system based on a GMDH (Group Method of Data Handling) theory from three values of the value, the deviation, and the differential value And a PID control unit that performs PID control of the nonlinear system based on the calculated proportional gain, integral gain, and differential gain based on the deviation.

  According to this, gain adjustment related to PID control is performed using GMDH. GMDH is one of the network models, and for example, CMAC constructed through learning can be approximately expressed as a nonlinear model. In GMDH, only the weight coefficient needs to be stored in the memory, so that the necessary memory capacity can be reduced. In addition, the calculation cost for calculating the PID gain during operation can be greatly reduced.

  Specifically, the GMHD-PID adjustment unit individually calculates each gain from the three values in each GMDH network, and each GMDH network is formed by hierarchically connecting adderline operators. .

  More specifically, each of the GMDH networks includes first and second adderline operators to which any two of the three values are input, the output of the first adderline operator, and the second adderline operation. And a third adderline operator to which the child output is input.

  Further, according to another aspect of the present invention, the method for determining the weighting factor of GMDH in the nonlinear PID controller described above separately determines a proportional gain, an integral gain, and a differential gain related to the PID control of the nonlinear system from the three values. Each CMAC (Cerebellar Model Articulation Controller) to be calculated on the computer is constructed on a computer, the CMAC is trained to perform PID control of the nonlinear system, and based on the input / output sample data of each learned CMAC, The weighting coefficient of each GMDH network for calculating each gain from the three values is determined.

  In the GMDH weighting factor determination method, two adderline operators to which any two of the three values are input are arranged in the first hierarchy, and outputs of the two adderline operators in the first hierarchy are input. For each of the three GMDH networks in which the adderline operators are arranged in the second hierarchy, the weighting factors of the two adderline operators in the first hierarchy are determined based on the input / output sample data, and the 1 After determining the weighting factors of the two adderline operators in the hierarchy, the weighting factors of the second line of the adderline operators are determined again based on the input / output sample data, and the weighting factors of all the adderline operators are determined. After the determination, the respective contribution ratios of the three GMDH networks are calculated, and one of the three GMDH networks having the higher contribution ratio is calculated. Select, a weighting factor of the selected GMDH network is determined as the weighting coefficient of each GMDH network for calculating the respective gain from the 3 values.

  According to the present invention, the problem of the memory capacity of the CMAC-PID controller is solved, and a nonlinear PID controller that can operate with a smaller capacity memory can be realized.

Block diagram of CMAC-PID controller Configuration diagram of CMAC-PID adjustment unit Schematic diagram showing CMAC CMAC-FRIT block diagram Block diagram of nonlinear PID controller (GMDH-PID controller) according to an embodiment of the present invention Schematic diagram showing the GMDH network Schematic diagram showing the structure of the adaline operator Graph showing the static characteristics of the Billilinear model Graph showing control result by fixed PID controller The graph which shows the control result by a CMAC-PID controller The graph which shows transition of the PID gain corresponding to the control result of FIG. The graph which shows the control result by a GMDH-PID controller The graph which shows transition of the PID gain corresponding to the control result of FIG. The graph which shows another control result by a fixed PID controller The graph which shows the control result of the generalization capability verification by a CMAC-PID controller The graph which shows the control result of the generalization capability verification by a GMDH-PID controller The graph which shows transition of the PID gain corresponding to the control result of FIG. The graph which shows transition of the PID gain corresponding to the control result of FIG.

  A nonlinear PID controller according to an embodiment of the present invention first generates a CMAC-PID controller, learns the CMAC-PID controller, and then generates a CMAC by approximating it with a nonlinear function based on the GMDH theory. Is done. Hereinafter, a generation process and configuration of a nonlinear PID controller according to an embodiment of the present invention will be described with reference to the drawings. It should be noted that the same components are denoted by the same reference numerals and description thereof will not be repeated.

≪Design of CMAC-PID control system≫
As a control law, a speed type PID control law given by the following equation is considered.

Here, u (t) is a control input, and e (t) indicates a control deviation defined by the following equation. K _P (t), K _I (t), and K _D (t) are a proportional gain, an integral gain, and a differential gain at each time, respectively.

In the formula (2), r (t) and y (t) indicate a target value and a system output, respectively. Further, Δ is a difference operator and is defined by Δ: = 1−z ⁻¹ . Z ⁻¹ represents a delay operator and means z ⁻¹ e (t) = e (t−1). In the CMAC-PID controller, the PMAC gains K _P (t), K _I (t), and K _D (t) at time t are adjusted by the CMAC controller.

<CMAC>
A block diagram of the CMAC-PID controller is shown in FIG. The CMAC-PID controller 10 includes a deviation calculator 12 that calculates e (t) by subtracting y (t) from r (t), and a differential calculation that calculates a differential value Δe (t) of e (t). CMAC− for calculating K _P (t), K _I (t), and K _D (t) from the unit 14 (corresponding to the difference operator described above) and r (t), e (t), Δe (t) A PID adjustment unit 16 and a PID control unit 18 that performs PID control of the nonlinear system 100 with K _P (t), K _I (t), and K _D (t) calculated based on e (t). Yes.

As shown in FIG. 2, the CMAC-PID adjustment unit 16 includes three CMACs 162 that learn each of K _P (t), K _I (t), and K _D (t).

  FIG. 3 shows a specific structure of CMAC. First, when an input of (3, 6) is given to the input space, 8, 9, 3 are calculated from the load table based on the label set {B, F, J}, {c, g, k}. Refer to it and output a total of 20. In response to this input, for example, when the teacher signal is 14, learning is performed in addition to a value obtained by dividing the difference between the output and the teacher signal by the number of load tables, that is, (14-20) / 3. . Here, a two-dimensional input space has been described for the sake of simplicity of explanation, but the input space of the CMAC 162 is configured by three dimensions of r (t), e (t), and Δe (t), and As shown, it is discretized into N load tables W. Refer to Non-Patent Document 1 for details.

<Offline learning with CMAC-FRIT>
For offline learning of CMAC, FRIT, which is one of model-free control design methods, is applied. In FRIT, it is possible to directly calculate the control parameter using one experiment data (closed loop data). FIG. 4 shows a block diagram of offline learning of CMAC using FRIT. In FIG. 4, r ₀ (t), e ₀ (t), u ₀ (t), and y ₀ (t) indicate operation data obtained by the initial experiment. Furthermore, tilde r (t) indicates a pseudo reference input and is expressed by the following equation.

Here, C (z ⁻¹ ) represents a polynomial of the PID controller. In CMAC-FRIT, offline learning is performed on the CMAC load necessary for calculating the PID gain according to the following adjustment rule.

In the formula (5), η _P , η _I , and η _D represent learning coefficients. J (t) is an evaluation function expressed by the following equation.

In the equation (6), y _r (t) is an output from the reference model G _m (z ⁻¹ ) represented by the equation (7).

Here, P (z ⁻¹ ) is a characteristic polynomial of the reference model and is expressed by the following equation.

(9) T _s is the sampling time in a formula, sigma, [delta] is the rising characteristics of the respective control system, illustrates the parameters related to the damping characteristic, designer arbitrarily set. In particular, σ indicates the time until the output of the control system reaches about 60% of the step target value. δ is an attenuation characteristic, and is designed to be about 0 to 1.0. Refer to Non-Patent Document 2 for details of CMAC-FRIT.

≪Construction of nonlinear PID controller using GMDH≫
<Nonlinear PID controller>
As described above, the nonlinear PID controller according to the present embodiment is generated by describing each PID gain constructed by CMAC-PID as a nonlinear model given by equation (10). That is, the PID gain at each time is described as a nonlinear function of r (t), e (t), and Δe (t). At that time, GMDH (Group Method of Data Handling) which is one of network models is used.

<GMDH>
GMDH is one of network models used to construct a nonlinear model using system input / output data. The GMDH network has a feature that it is relatively easy to express a model as a polynomial. Therefore, the CMAC-PID controller 10 of FIG. 1 which is a non-linear controller can be expressed approximately.

FIG. 5 shows a block diagram of a GMDH-PID controller which is a nonlinear PID controller according to the present embodiment. The GMDH-PID controller 10A includes a deviation calculator 12 that calculates e (t) by subtracting y (t) from r (t), and a differential calculation that calculates a differential value Δe (t) of e (t). Based on the GMDH theory, K _P (t), K _I (t), K _D (t) from the part 14 (corresponding to the above difference operator) and r (t), e (t), Δe (t) GMDH-PID adjustment unit 16A that calculates the PID control unit that performs PID control of the nonlinear system 100 with the calculated K _P (t), K _I (t), and K _D (t) based on e (t) 18.

The GMDH network is a hierarchical network configured by an appropriate combination of adderline operators (N-Adaline) 164 as shown in FIG. As shown in FIG. 7, the adderline operator 164 is a unit having two inputs and one output, and the input variables are x ₁ (t) and x ₂ (t), and the output variable is z (t). z (t) is expressed as the following equation.

However, w _i (i = 0,..., 5) represents a weighting coefficient. In addition, Sq. Indicates square calculation.

  The least square method of the following formula can be used for calculating the weight. Of course, the weight may be calculated by other methods.

  The GMDH network has three types of combinations as shown in FIG. Among them, the one having the highest contribution rate defined by the following equation (for example, a network surrounded by a broken line in FIG. 6) is determined as the GMDH network 165 to be mounted on the GMHD-PID adjustment unit 16A.

In the above equation, N _s represents the number of input / output data, tilde z (t) represents an estimated output value, and bar z represents an average value of output data.

<Controller design>
Using the above-described GMDH, a non-linear model of PID gain is designed according to the following procedure.
step 1) Input r ₀ (t), e ₀ (t), Δe ₀ (t) to the learned CMAC load table, and tilde K _P (t), tilde K _I (t), tilde K _D (t ) Is calculated.
(Step 2) The weight w is determined by the equation (12). In Expression (11), x ₁ (t) and x ₂ (t) are _two of r ₀ (t), e ₀ (t), and Δe ₀ (t), and z (t) is a tilde K. _P (t), a Childer _K I (t), Childer _K D (t). The weight w is a different weight for each PID gain.
step 3) The weight w of the second layer is determined by the GMDH of the first layer and the least square method using the estimated values of the PID gains.
(Step 4) The estimated value in each combination is calculated, and the contribution rate is calculated by the equation (13). The one with the highest contribution rate is defined as a non-linear model of PID gain.

  With this method, each PID gain calculated using CMAC can be expressed by a GMDH network, and the memory required for CMAC can be greatly reduced.

In the CMAC-PID controller 10, three CMACs 162 for calculating K _P (t), K _I (t), and K _D (t) are respectively converted into a three-dimensional input space (the size is 11 ³ , for example). There are 11979 (= 11 ³ × 3 × 3) parameters to be stored in the memory. For this reason, when the CMAC-PID controller 10 is realized by a microcomputer or the like, for example, one parameter is set as 64-bit data, and the microcomputer requires a memory capacity of about 93 Kbytes. On the other hand, in the GMHD-PID controller 10 </ b> A, the GMHD-PID adjustment unit 16 </ b> A determines K _P (t), K _I (in each of the three GMDH networks 165 from r (t), e (t), and Δe (t). t) and K _D (t) are calculated. Therefore, the parameters to be stored in the memory need only be six weighting factors of each adderline operator 164 in each GMDH network 165, that is, 54 (= 3 × 3 × 6) parameters. For this reason, when the GMDH-PID controller 10A is realized by a microcomputer or the like, for example, one parameter is set to 64-bit data, and the memory capacity required for the microcomputer is only 432 bytes. Thus, in the GMDH-PID controller 10A according to the present embodiment, the necessary memory capacity can be significantly reduced.

≪Control result≫
In order to verify the effectiveness of the present embodiment, the GMDH-PID controller 10A is applied to a Billierar model given by the following equation.

Here, ξ (t) represents Gaussian white noise having an average of 0 and a variance of 1.0 × 10 ⁻³ . In addition, the static characteristics of the Billilinear model are shown in FIG. Furthermore, the target value in each step was set as shown in equation (15).

<Offline learning of CMAC>
First, in order to obtain input / output data necessary for offline learning of CMAC, control by a fixed PID gain was performed in consideration of the stability of the closed loop system. At this time, the PID gain was determined by using the CHR (Chien, Hrones, Reswick) method using data in the vicinity of y = 5. The PID gain is as follows:

  The control result by the fixed PID controller with this parameter is shown in FIG. As a result, the target value followability is not necessarily good for a low target value.

  Next, offline learning of CMAC is performed by CMAC-FRIT based on experimental data by fixed PID control. The reference model at that time was designed as shown in equation (17).

In addition, the learning coefficients were designed to be 10 ⁻³ for η _P , η _I , and η _D. 10 and 11 show the transition of the control result and PID gain in CMAC-PID. It can be seen that the CMAC is sufficiently learned off-line and good control can be performed even at a low target value.

<Controller design by GMDH>
The offline learned CMAC was configured as a network by GMDH, and a simulation was performed on the target value given by equation (15).

  The results of simulation and the transition of PID gain are shown in FIGS. From the results, it can be seen that GMDH-PID has almost the same control performance as CMAC-PID.

<Verification of generalization ability>
The generalization ability is considered by controlling the unlearned target value shown in Equation (18). FIG. 14 shows the control result with the fixed PID, FIG. 15 shows the control result with the CMAC-PID, and FIG. 16 shows the control result with the GMDH-PID according to the present embodiment. FIG. 17 shows the transition of the PID gain in CMAC-PID, and FIG. 18 shows the transition of the PID gain in GMDH-PID according to this embodiment.

  From these control results, it is considered that the GMDH-PID according to the present embodiment has almost the same control performance as the CMAC-PID. In this simulation, the amount of memory required for the load table of CMAC was 264 Kbytes as a double precision floating point number, whereas the weight of GMDH in this embodiment was 432 bytes, and the memory was reduced by 99.8%. Further, since the CMAC memory may increase depending on the control target, this embodiment can be said to be useful for reducing the memory capacity.

10A GMDH-PID controller (nonlinear PID controller)
12 Deviation calculation unit 14 Differential calculation unit 16A GMDH-PID adjustment unit 162 CMAC
164 Adaline Operator 165 GMDH Network 18 PID Control Unit 100 Nonlinear System

Claims (5)

A deviation calculator for calculating a deviation between the output value of the nonlinear system to be controlled and the target value;
A differential operation unit for calculating a differential value of the deviation;
GMHD-PID adjustment for calculating a proportional gain, an integral gain, and a differential gain according to PID control of the nonlinear system based on a GMDH (Group Method of Data Handling) theory from the three values of the target value, the deviation, and the differential value And
A non-linear PID controller comprising: a PID control unit that PID-controls the non-linear system with the calculated proportional gain, integral gain, and differential gain based on the deviation. The nonlinear PID controller of claim 1,
The GMHD-PID adjustment unit calculates each gain individually from the three values in each GMDH network;
A non-linear PID controller, wherein each GMDH network is formed by hierarchically connecting adderline operators. The non-linear PID controller according to claim 2,
Each GMDH network receives first and second adderline operators to which any two of the three values are input, an output of the first adderline operator, and an output of the second adderline operator. A non-linear PID controller comprising: a third adderline operator. In the method of determining the weighting factor of GMDH in the nonlinear PID controller as described in any one of Claim 1 to 3,
Each CMAC (Cerebellar Model Articulation Controller) that individually calculates proportional gain, integral gain, and differential gain related to PID control of the nonlinear system from the three values is constructed on a computer,
Causing each CMAC to learn about PID control of the nonlinear system;
A GMDH weighting factor determination method, wherein a weighting factor of each GMDH network for calculating each gain from the three values is determined based on learned input / output sample data of each CMAC. In the GMDH weighting coefficient determination method according to claim 4,
Two adderline operators to which any two of the three values are input are arranged in the first hierarchy, and the adderline operator to which the output of the two adderline operators in the first hierarchy is input is in the second hierarchy. For each of the three arranged GMDH networks, based on the input / output sample data, determine the weighting factors of the two adderline operators in the first hierarchy,
After determining the weighting factors of the two adderline operators in the first layer, again determine the weighting factor of the adderline operator in the second layer based on the input / output sample data;
After determining the weighting factors of all the Adaline operators, calculate the contribution rate of each of the three GMDH networks,
One of the three GMDH networks having a high contribution rate is selected, and the weighting factor of the selected GMDH network is determined as the weighting factor of each GMDH network that calculates each gain from the three values. A GMDH weighting factor determination method characterized by the above.