KR101818133B1 - Control apparatus and method using adaptive tracking control for uncertain switched nonlinear systems in nonstrict-feedback form - Google Patents

Abstract

Disclosed is a control device. The control device of the present invention, which follows and controls an unknown nonlinear switched system having a non-rigorous feedback form, comprises: a signal processing part converting a coordinate for a dynamic surface design; and a control performing part having a following control device which performs a follow-up control based on the dynamic surface design by using an error surface value obtained from the signal processing part.

Description

BACKGROUND OF THE INVENTION 1. Field of the Invention [0001] The present invention relates to a control apparatus and a control method using an adaptive tracking control for an unknown nonlinearly switched system having an unstrict feedback form,

The present invention relates to a control apparatus and a control method for performing follow-up control.

Stability analysis and control methods for switched systems such as network control systems, robotic systems or switching power systems are being developed. In particular, the Lyapunov function technique and the back stepping technique have been applied to address the stability problem for randomly switched nonlinearly switched systems. However, conventional control techniques and systems have not considered control over unknown nonlinearly switched systems due to the complexity of the control. Recently, an attempt has been made to perform adaptive tracking control on an unknown nonlinearly switched system having a strict-feedback form, but there is a problem in that an unknown (non-strictly feedback form) No study has been done on the control of nonlinearly switched systems.

SUMMARY OF THE INVENTION The present invention provides a control apparatus and control method for performing tracking control on an unknown nonlinearly switched system having an unusual feedback form.

According to an aspect of the present invention, there is provided an apparatus for tracking an unknown nonlinearly switched system having an irreversible feedback form, the apparatus comprising: a signal processing unit for transforming coordinates for dynamic surface design; There is provided a control apparatus including a control performing unit including a tracking controller that performs tracking control based on a dynamic surface design using an error surface value.

According to another aspect of the present invention, there is also provided a method of tracking and controlling an unknown nonlinear switched system having a non-strict feedback form, the method comprising: transforming and signal processing coordinates for dynamic surface design; There is provided a control method including a step of approximating a function using a fuzzy logic system and a step of controlling the tracking based on a dynamic surface design using an error surface value and a function approximation value obtained in signal processing .

According to the embodiment of the present invention, it is possible to perform control that converges at a finite time through tracking control for an unknown nonlinearly switched system having a non-strict feedback form.

1 shows a control device according to an embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Embodiments of a control apparatus and a control method according to the present invention will be described in detail with reference to the accompanying drawings. In the following description with reference to the accompanying drawings, the same or corresponding components are denoted by the same reference numerals, A description thereof will be omitted.

It is also to be understood that the terms first, second, etc. used hereinafter are merely reference numerals for distinguishing between identical or corresponding components, and the same or corresponding components are defined by terms such as first, second, no.

1 is a block diagram illustrating a control apparatus according to an embodiment of the present invention.

1, the target system in the present embodiment is an unknown nonlinear switched system having a non-strict feedback form, and the control apparatus 100 of the present embodiment includes a signal processing unit 110 and a control performing unit 120 do. The signal processing unit 110 transforms coordinates for dynamic surface design and the control execution unit 120 performs an operation based on the dynamic surface design using the error surface value obtained from the signal processing unit 110 And performs tracking control.

To begin with, the target system of the present embodiment may be an unknown nonlinear system that is arbitrarily switched and has a non-strict feedback form. For example, a switched system, such as a network control system, a robotic system, or a switching power system, may be the target system. Such a target system can be represented illustratively by the following equation.

[Unknown switched nonlinear systems with non-strict feedback form]

Equation 1

은 각각 상태 변수들 및 출력이다.

는 외부 교란(external disturbances)이고,

는 스위칭 신호이다.

은 k번째 서브 시스템의 제어입력이고,

는

인 알려지지 않은

비선형 함수이다. 엄격 피드백 양식

에서

항은, 반복적인 제어 디자인(recursive control design)을 위한 비엄격 피드백 항

에 의해 제거되지 않는다고 가정된다.

Are state variables and outputs, respectively.

Is the external disturbances,

Is a switching signal.

Is the control input of the k < th > subsystem,

The

Unknown

It is a nonlinear function. Strict Feedback Form

in

The term is defined as a non-strict feedback term for a recursive control design

Lt; / RTI >

The tracking control of the present embodiment is such that all signals of a closed-loop system are bounded at a finite time, so that the output of the target system can be traced to a desired trajectory r (t) even in any switching state Track it.

및 미분값

은 연속적이고 유계 된다. 즉,

인 상수

가 존재한다.Assumption 1: Target signal

And derivative values

Lt; / RTI > In other words,

In constant

Lt; / RTI >

는 알려지지 않았으나, 그 신호들은 알려져 있다. 그리고,

,

인 상수

가 존재한다. 일반성을 잃지 않고,

의 신호들은 양수로 가정된다.Assumption 2: Nonlinear function

Are not known, but their signals are known. And,

,

In constant

Lt; / RTI > Without losing generality,

Are assumed to be positive numbers.

는

를 만족시키는 함수들이고,

는 미지의 class-K 함수들이다.Assumption 3:

The

, ≪ / RTI >

Are unknown class-K functions.

(

는 미지의 상수)인 동안 유계 된다.Assumption 4: External disturbance

(

Is an unknown constant).

On the other hand, to simplify the above equations, the following definitions and lemmas are needed.

Definition 1:

,

는 표준적인 시그넘(signum) 함수를 지칭한다.

,

Refers to the standard signum function.

(

이고

)을 가정한다.

(

는 상수)인 양의 스칼라 함수(positive definite scalar function)

가 존재한다면, 시스템은 한정된 시간에서 안정적(finite-time stable)이다. 더불어, 안정 시간(settling time)

는

를 만족시킨다.Subdivision 1: Dynamic system (dynamicsystem)

(

ego

).

(

Is a positive definite scalar function,

The system is stable (finite-time stable) at a limited time. In addition, settling time,

The

.

(

)Subproject 2:

(

)

In the present embodiment, a function approximator 130 is used for the control and stability analysis, and the function approximation unit 130 can process the nonlinear term. As the function approximation device 130, a fuzzy logic system may be used. The configuration of the fuzzy logic system includes, for example, a fuzzy rule, several fuzzy IF-THEN rules, a fuzzy inference engine and a non-fuzzy rule, Lt; / RTI >

로 부터 출력 y로의 매핑(mapping)을 수행하는 퍼지 IF-THEN 규칙을 채택할 수 있다. i번째 퍼지 규칙은 다음과 같이 정의된다.Here, the fuzzy inference apparatus includes an input linguistic vector,

THEN rule that performs a mapping from the output IF to the output y. The i-th fuzzy rule is defined as follows.

는 퍼지 변수들이고,

는 싱글톤 수(singleton number)이다.)(

Are fuzzy variables,

Is a singleton number.)

[Fuzzy logic system]

Illustratively, a fuzzy logic system with a singleton fuzzifier, a product inference, and a center average defuzzifier can be expressed as:

Equation 2

는 퍼지 변수

의 소속 함수 값(membership function value)이고, r은 퍼지 규칙들의 번호이다.

는 조정 가능한 파라미터 벡터(adjustable parameter vector)이고,here,

Is a fuzzy variable

, And r is the number of the fuzzy rules.

Is an adjustable parameter vector,

는 퍼지 기초 함수(fuzzy basis function)이다. 기초 함수 벡터

는

를 만족시킨다.

Is a fuzzy basis function. Basic function vector

The

.

를 컴팩트 세트

에서 연속 함수라고 하면,Proposition 3:

Compact set

In this case,

(상수

)인 수학식 2의 형태의 퍼지 로직 시스템

이 존재한다.

(a constant

Fuzzy logic system of the form < RTI ID = 0.0 > (2)

Lt; / RTI >

In this embodiment, tracking control in finite time is presented through dynamic surface design. The following coordinate transformations can be used for dynamic surface design. Coordinate transformation can be performed in the signal processing unit.

[Coordinate transformation for dynamic surface design]

Equation 3

(

)는 에러 표면(error surface)이고,

는 경계층(boundary layer)에러이고,

는 공통 가상 제어 규칙(common virtual control law)이고,

는 다음의 제1차 필터에 의해 얻어진 신호들이다.here,

(

) Is an error surface,

Is a boundary layer error,

Is a common virtual control law,

Are the signals obtained by the following first-order filter.

,

,

The tracking controller 140 performs tracking control using an error surface value obtained from the signal processor 110. [ The tracking controller 140 uses a common tracking control system, which will be described later, based on dynamic surface design.

The common adaptive finite-time control scheme at finite time can be expressed as:

[Common adaptive finite-time control scheme]

Equation 4

는 양의 설계 파라미터들이고,

는

의 추정치이다.

는 퍼지 근사기

의 가중 파라미터이다. here,

Are positive design parameters,

The

Respectively.

Is a purge-

.

를 위한 추종 규칙은 다음과 같이 정의될 수 있다.And,

Can be defined as follows.

Equation 5

는 양의 설계 파라미터들이다.

과 수학식 5는

를 이끌어 낸다.here,

Are positive design parameters.

And Equation (5)

.

)에 대한 정보 없이도 제시될 수 있다. 또한, 수학식 4에서

항은 제안된 제어 시스템에서 유한한 시간에서의 안정화를 위하여 포함된다. 이러한 항의 미분은

로 인하여 에러 표면의 특이성 문제(singularity problem)를 일으킬 수도 있다. 따라서, 가상 제어 규칙의 도함수 값을 얻기 위해, 미분에 의한 문제를 피할 수 있는 동적 표면 설계가 백스텝핑(backstepping) 기법을 대신하여 사용된다.In the present embodiment, the tracking control system represented by Equation (4) includes control coefficient functions

) Can be presented without any information. In Equation 4,

Are included for stabilization in the finite time in the proposed control system. The derivative of these terms

May cause a singularity problem of the error surface. Therefore, in order to obtain the derivative value of the virtual control rule, a dynamic surface design that avoids the problem by differential is used instead of the backstepping technique.

Proposition 4: For Equation 3, the following inequality is established.

here,

는 다음의 부등식을 만족한다.Subproject 5: Nonlinear function

Satisfies the following inequality.

이고, here,

ego,

,

,

,

이고,

는

인 상수이다.

,

ego,

The

Is a constant.

은 에러 변수들(error variables) 및 추종 변수들(adaptive parameters)과 연관되는 비선형성(nonlinearities)으로 변환된다.By the above-described method, non-strict feedback nonlinear functions associated with all state variables

Is transformed into nonlinearities associated with error variables and adaptive parameters.

[Theorem 1]

(

는 상수,

는 리아프노프 함수)를 만족시키는 어떠한 초기 조건들에 대해서도, 폐회로(closed-loop) 시스템의 모든 신호들은 준 전역적(semi-globally)이고 균일하게 궁극적으로 유계(uniformly ultimately bounded) 되고, 추적 에러(tracking error)는 다음의 조정 가능한 원점 이웃한 영역(adjustable neighborhood of the origin)으로 수렴된다.If the unknown switched nonlinear system is tracked controlled by the proposed equations 4 and 5,

(

Is a constant,

All the signals of the closed-loop system are semi-globally uniformly ultimately bounded for any initial conditions that satisfy the Lyapunov function, the tracking error converges to the next adjustable neighborhood of the origin.

Equation 6

에서,

in,

는 양의 상수,

는 유한한 시간이고here,

Is a positive constant,

Is a finite time

(

)이다.

(

)to be.

Thus, the tracking error and the convergence time can be kept small by adjusting a and b.

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 of the invention as set forth in the appended claims. The present invention can be variously modified and changed by those skilled in the art, and it is also within the scope of the present invention.

100: Control device
110: Signal processor
120:
130: function approximation device
140: tracking controller

Claims (11)

An apparatus for tracking and controlling an unknown nonlinearly switched system having a non-strict feedback form,
A signal processor for converting coordinates for dynamic surface design; And
And a controller for performing tracking control based on the dynamic surface design using an error surface value obtained from the signal processing unit.
The method according to claim 1,
Wherein the control performing unit comprises:
A controller comprising a function approximator using a fuzzy logic system.
3. The method of claim 2,
Wherein the fuzzy logic system is expressed by the following equation.

(here,

Is a fuzzy variable

, And r is the number of the fuzzy rules.

Is an adjustable parameter vector,

Is a fuzzy basis function. Basic function vector

The

Lt; / RTI >
The method of claim 3,
And the signal processing unit performs the following conversion.

(here,

(

) Is an error surface,

Is a boundary layer error,

Is a common virtual control law,

Are the signals obtained by the following first-order filter.

,

)
5. The method of claim 4,
In the control performing unit,
Wherein the tracking controller is expressed by the following equation.

(here,

Are positive design parameters,

The

Respectively.

Is a purge-

Lt; / RTI >
And,

Can be defined as follows.

(here,

Are positive design parameters.

and

The

.)
6. The method of claim 5,
Wherein the tracking error and the convergence time in the tracking controller are expressed by the following equations.

in,

(here,

Is a positive constant,

Is a finite time

(

)to be.
Also,

(

Is a constant,

Has some initial conditions that satisfy the Lyapunov function).
CLAIMS 1. A method of tracking and controlling an unknown nonlinear switched system having an unstable feedback form,
Transforming coordinates and performing signal processing for dynamic surface design;
Approximating a function using a fuzzy logic system; And
Controlling the tracking based on the dynamic surface design using an error surface value and a function approximation value obtained in the signal processing.
8. The method of claim 7,
Wherein the function approximation step comprises:
Wherein the fuzzy logic system is expressed by the following equation:

(here,

Is a fuzzy variable

, And r is the number of the fuzzy rules.

Is an adjustable parameter vector,

Is a fuzzy basis function. Basic function vector

The

Lt; / RTI >
9. The method of claim 8,
Wherein the signal processing step performs the following conversion.

(here,

(

) Is an error surface,

Is a boundary layer error,

Is a common virtual control law,

Are the signals obtained by the following first-order filter.

,

)
10. The method of claim 9,
Wherein the tracking control step is expressed by the following equation.

(here,

Are positive design parameters,

The

Respectively.

Is a purge-

Lt; / RTI >
And,

Can be defined as follows.

(here,

Are positive design parameters.

and

The

.)
11. The method of claim 10,
Wherein the tracking error and the convergence time in the tracking control step are expressed by the following equations.

in,

(here,

Is a positive constant,

Is a finite time

(

)to be.
Also,

(

Is a constant,

Has some initial conditions that satisfy the Lyapunov function).

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