KR19990059514A - Nonlinear system follow-up control device - Google Patents

Abstract

The present invention relates to a tracking control apparatus for a nonlinear system using a CMAC neural network.

The nonlinear system tracking control apparatus includes a neural network unit that can simulate a nonlinear function with respect to an input signal, an amplifying unit that amplifies an input signal with a predetermined gain value, and a controller that is commonly connected to the neural network unit and the amplifying unit. And an adaptive control means connected to the error generating means for varying the mixed follow-up error including the inter-interval gain value having an arbitrary value according to the condition between the slits, Respectively.

In the nonlinear system follow-up control apparatus of the present invention, the size of the sand dune can be automatically adjusted.

Description

Nonlinear system follow-up control device

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a nonlinear system, and more particularly, to a nonlinear system follow-up control apparatus using a CMAC neural network.

In recent years, as the need for precision instruments or precision systems increases, controllers with improved control performance over existing controllers are required. For this purpose, researches on control techniques using neural networks have been actively carried out. As a kind of these neural networks, neural networks such as Mulilayer Perceptron (RBFN) and Radial Basis Function Network (RBFN) have a problem in that they are difficult to be applied to actual industrial fields because of a large amount of calculation and a complicated structure. The Cerebellar Model Articulation Controller (CMAC) proposed by Albus in 1975, in contrast to other neural networks, has a simple look-up table structure, And it is possible to implement it in actual industrial field because the amount of calculation is small. Since the price of memory was high at the time when it was proposed, it has not been applied in many industrial fields. However, as the proportion of memory in the entire system becomes smaller due to the development of semiconductor technology, it is applied to various fields. The industrial field where CMAC is used has a significant performance improvement effect, which has been proved in the industrial field where this neural network is applied. However, According to "Practical stability issues in CMAC neural network control systems" published by Chen and C. -H Chang, the controller using CMAC works well with greatly reducing the tracking error initially, but after a long time it becomes unstable .

In order to prevent such a phenomenon, we have proved that a stable controller can be obtained by learning CMAC only when the error is larger than the size of the sphere by using the interval of learning rule. However, this method has been found to be problematic in two respects, one of which is that the size of the sand dune is determined through trial and error through experimental methods, and the other is that the size of the sand dune is not certain to be.

SUMMARY OF THE INVENTION Accordingly, it is an object of the present invention to provide a tracking control apparatus for a non-linear system in which the size of a sand dune is automatically controlled.

It is another object of the present invention to provide a nonlinear system follow-up control apparatus which automatically adjusts the size of a sand dune, thereby mathematically proving the stability of the system so as to clarify the relationship between the parameters of the controller and the control characteristics, .

1 is a block diagram showing a tracking control apparatus for a nonlinear system according to an embodiment of the present invention;

Fig. 2 is a diagram showing the structure of a CMAC neural network in Fig. 1; Fig.

DESCRIPTION OF THE REFERENCE NUMERALS

2, 8, 10, 14, 18: adders 4, 12, 16: amplifiers

6: Adaptive Amplifier 20: CMAC Neural Network

22: Controlled system

In order to achieve the above object, a tracking control apparatus for a nonlinear system of the present invention includes: neural network means capable of simulating a nonlinear function with respect to an input signal; amplifying means for amplifying an input signal with a predetermined gain value; Means for generating a mixed follow-up error, which is commonly connected to the error generating means, for generating a mixed follow-up error including a blank interval and a mixed follow-up error including a blank interval, And an adaptive control means for varying the adaptive control means.

Other objects and advantages of the present invention will become apparent from the following description of embodiments with reference to the accompanying drawings.

Hereinafter, embodiments of the present invention will be described in detail with reference to FIGS. 1 and 2 attached hereto.

1 shows a tracking control apparatus for a nonlinear system according to an embodiment of the present invention.

In the configuration of Fig. 1, the tracking control apparatus of the nonlinear system of the present invention is configured to control A first adder 2 supplied with a desired speed A third adder 18 to which a desired trajectory x _d is supplied and a first amplifier 4 commonly connected to the first adder 2 and the second adder 8, A fourth adder 10 commonly connected to the first adder 2 and the first amplifier 4 via a first node n1 and a fourth adder 10 commonly connected to the third adder 18 and the fourth adder 10, An adaptive amplifier 6 connected to the output terminal of the first adder 2 and a third amplifier 12 connected to the output terminal of the fourth adder 10, The CMAC 20 connected to the second adder 8 and the third adder 18 via the second node n2 and the third node n3 respectively and the CMAC 20 connected to the adaptive amplifier 6, The fifth adder 14 and the fifth adder 14 to which the output signals of the adder 12 and the CMAC 20 are supplied and the fourth adder 14 and the fifth adder 14 via the fourth node n4 and the fifth node n5, A second adder 8, a third adder 18 and a control target system 22 commonly connected to the CMAC 20. [

The first adder 2 adds the desired acceleration < RTI ID = 0.0 > And the inverted terminal is supplied with the output of the first amplifier 4 And outputs an output signal To the adaptive amplifier (6). The gain value d of the adaptive amplifier 6 is adaptively variable according to the conditions of the dune. The first amplifier 4 amplifies the speed error supplied via the first node n1, To its own gain value? And supplies it to the first adder 2. The second adder 8 receives the current speed of the control target system 22 via the fifth node n5 and the second node n2, Inverted terminal and the desired speed Are supplied to the inverting terminals to sum these signals. The output signal of the second adder 8 is a signal Is commonly supplied to the first amplifier 4 and the fourth adder 10 via the first node n1. The current position x of the control target system 22 is supplied to the non-inverting terminal of the controlled system 22 via the fourth node n4 and the third node n2 to the third adder 18 and the desired trajectory x _d is applied to the inverting terminal . The third adder 18 sums these signals and outputs a tracking error And the following error To the second amplifier (16). The fourth adder 10 multiplies the speed error inputted via the first node n4 And the output of the second amplifier 16 To generate a mixed follow-up error s, and supplies this signal to the third amplifier 12. [ The CMAC 20 may be applied to other neural networks and adaptive control systems than CMAC, but a local neural network is preferred for smooth operation of the tracking control of the nonlinear system of the present invention. The fifth adder 14 serves to supply the control input u (t) to the control target system 22, and the non-inverting terminal of its own is connected to the output signal of the adaptive amplifier 6 And the output signal u _c of the neural network are supplied to the inverting terminal and the output signal ks of the third amplifier 12 is inputted to the inverting terminal to add the signals to generate the control input u (t).

The operation of the tracking control apparatus of the nonlinear system of the present invention will be described in detail with reference to the following equations (1) to (20).

The controlled system 22 appears as a nonlinear system represented by equation 1 representing most of the mechanical devices.

[Equation 1]

Where m> 0, b <0, u (t) is the control input and z (t) is the noise (nosie). The current position x and the current speed Which represents the nonlinearity of the system. The control objective of the present invention is to control the tracking error &lt; _{RTI ID} = 0.0 _&gt; . Equation 1 can be expressed as Equation 2 below.

[Equation 2]

In the above equation 2, the nonlinear function The CMAC 22 is used. An example of a CMAC when the input is two-dimensional is shown in Fig. In FIG. 2, the interval of each input variable is divided into 17 steps, and four storage locations correspond to one input element. The number of storage locations corresponding to one input is referred to as a generalization size and denoted by N _g . Adjacent input elements share N _g -1 memory locations and only one memory location is different. H (9, 7) = {Cb, Hg, Pp, Ww} for the input elements 9 and 7 is defined as a mapping that connects N _g storage locations in the input element H .

The output of the CMAC 20 can be expressed as: &lt; EMI ID = 3.0 &gt;

[Equation 3]

Here, n is the total number of storage locations and is 100 in the case of Fig. Is a value stored in each memory location and is called a weight. a _j represents the address of the jth storage location, and Cb, Gh, and so on represent it. χ (a _j ) is defined as follows.

The control input u (t), that is, the output signal of the fifth adder 14 is obtained as shown in Equation (4).

[Equation 4]

Here, the mixed follow-up error s from the fourth adder 10 is the speed error from the second adder 8 And the gain value? Of the second amplifier 16 and the positional error from the third adder 18 As shown in FIG. That is, the mixed follow-up error s , And the output signal of the first adder 2 . In Equation 4, k > 0 is the feedback gain Respectively. The estimation law of the controller parameters is given by Equation 5, Equation 6 and Equation 7 below.

[Equation 5]

[Equation 6]

[Equation 7]

Here, the error including the interval As shown in Equation 8 below, . &Lt; / RTI &gt;

[Equation 8]

[Equation 9]

[Equation 10]

Size of sand dunes Is expressed by Equation 11 below.

[Equation 11]

The stability of the above control law can be proved through the following process. When the control input expression 4 is substituted into the expression (2)

[Equation 12]

. Here, the estimation error of the gain value d to be. Can be simulated by the CMAC as shown in Equation (13).

[Equation 13]

here, Is a finite function due to the CMAC error, and W _j Is an ideal weight that minimizes Respectively. Substituting Equation 13 into Equation 12 results in Equation 14 below.

[Equation 14]

Here, to be. Coarse error And the noise z (t).

[Equation 15]

Here, ε is defined as the minimum value of the quantity satisfying the above inequality and is assumed to be unknown in advance. epsilon Respectively.

The following Lyapunov function V is defined by Equation 16 below.

[Equation 16]

here, to be. The time derivative of the above equation (16) is obtained by the following equation (17).

[Equation 17]

when, The equation (17) is expressed by the following equation (18).

[Equation 18]

Using Equation 14 and Equation 15, Equation 19 is expressed as Equation 19 below.

[Expression 19]

, Equation (5), Equation (6), and Equation (7) are substituted into Equation (19).

[Equation 20]

. When Because of Converge to zero and the parameters of the controller are finite. Converges to zero Is finite Can be understood as finite. Therefore, the tracking error of position and velocity Can be verified.

Thus, although only the system represented by Equation 1 has been considered, the nonlinear system follow-up control apparatus of the present invention can be applied to a general nonlinear system (including a linear system) in addition to the system represented by Equation 1 through a similar inductive process.

As described above, in the nonlinear system follow-up control apparatus of the present invention, the size of the sand dune can be automatically adjusted.

The tracking control apparatus of a nonlinear system according to the present invention can implement an effective controller by mathematically proving the stability of the nonlinear system by automatically adjusting the size of the dune so as to clarify the relationship between the parameters of the controller and the control characteristics.

It will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit or scope of the invention. Therefore, the technical scope of the present invention should not be limited to the contents described in the detailed description of the specification, but should be defined by the claims.

Claims (6)

A neural network means capable of simulating a nonlinear function with respect to an input signal, Amplifying means for amplifying an input signal with a predetermined gain value, An error generating means connected in common to the neural network means and the amplifying means for generating a mixed follow-up error including the inter- And an adaptive control unit connected to the error generating unit and adapted to vary a mixed tracking error including the inter-interval gain value of the inter-interval gain having an arbitrary value according to the condition of the inter- Device. The method according to claim 1, The first adaptive gain is η ₁ , the mixed follow-up error including the interval , a _j is the address of the jth memory location, and H (x) is the mapping of N _g memory locations in the input element, then it has a value of "1" when a _j ∈ H (x) When there is χ (a _j ) having a value of "0" in other conditions, The jth estimated weight of the neural network The Of the nonlinear system is calculated by the following equation. The method according to claim 1, The second adaptive gain is η ₂ , the mixed follow-up error including the interval , Desired acceleration And the velocity error amplified by the gain value? The difference signal of In other words, Estimated gain value The Of the nonlinear system is calculated by the following equation. The method according to claim 1, The third adaptation gain is η ₃ , z (t) is the noise, x The simulation error of the neural network as a function of Ε is the minimum value of the quantity satisfying this inequality, Estimate of ε Quot; Of the nonlinear system is calculated by the following equation. The method according to claim 1, The mixed follow-up error is s, In other words, The mixed follow-up error including the above- Quot; Of the nonlinear system is calculated by the following equation. The method according to claim 1, z (t) is the noise, position x and velocity The simulation error of the neural network as a function of Ε is the minimum value of the quantity satisfying this inequality in the inequality of ε, , And an arbitrary gain value is k, Size of sand dunes The Of the nonlinear system is calculated by the following equation.