1 research outputs found
μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ ννμ νμ₯λ λΉμ ν κ΄μΈ‘κΈ° μ μ€νμ ν΅ν λΉμ ν κ΄μΈ‘κΈ° μ€κ³
νμλ
Όλ¬Έ (λ°μ¬)-- μμΈλνκ΅ λνμ : μ κΈ°Β·μ»΄ν¨ν°κ³΅νλΆ, 2014. 8. μμ§ν.λ³Έ λ
Όλ¬Έμ λΉμ ν μμ€ν
μ λν κ΄μΈ‘κΈ° μ€κ³ λ¬Έμ λ₯Ό λ€λ£¨κ³ μλ€. κ΄μΈ‘κΈ° μ€κ³ λ¬Έμ λ μ£Όμ΄μ§ μμ€ν
μ μ
λ ₯κ³Ό μΆλ ₯ μ 보λ§μ νμ©νμ¬ λμ μμ€ν
μ μν λ³μλ₯Ό μΆμ ν μ μλ μμ€ν
μ μ€κ³νλ κ²μ΄λ€. μ ν μμ€ν
μ κ²½μ°μλ 루μλ²κ±° κ΄μΈ‘κΈ°(Luenberger observer)λ‘ μλ €μ§ μΌλ°μ μΈ ν΄λ²μ΄ μ‘΄μ¬νλ λ°λ©΄, μΌλ°μ μΈ λΉμ ν μμ€ν
μ λν΄ κ΄μΈ‘κΈ°λ₯Ό μ€κ³νλ λ°©λ²μ λν μ°κ΅¬ κ²°κ³Όλ νμ¬κΉμ§ λ³΄κ³ λ λ°κ° μλ€. λ€λ§, νΉμ ν ννμ λΉμ ν μμ€ν
μ λν΄ κ΄μΈ‘κΈ°λ₯Ό μ€κ³νλ λ¬Έμ μ λν μ°κ΅¬λ νλ°νκ² μ§νλμ΄ μ€κ³ μλ€. κ΄μΈ‘κΈ° μ€μ°¨ μ νν(observer error linearization) κΈ°λ²μ μ΄ λ¬Έμ μ λν κ°μ₯ μ μλ €μ§ λ°©λ²λ‘ μ€μ νλλ‘μ, μ£Όμ΄μ§ λΉμ ν μμ€ν
μ μ’ν λ³νμ ν΅ν΄ κ΄μΈ‘ κ°λ₯ν μ ν μμ€ν
κ³Ό μΆλ ₯μ£Όμ
(output injection) λΆλΆλ€λ‘ ꡬμ±λ λΉμ ν κ΄μΈ‘κΈ° μ μ€ν(nonlinear observer canonical form)μΌλ‘ λ³νμν€λ λ¬Έμ μ΄λ€. λΉμ ν κ΄μΈ‘κΈ° μ μ€νμΌλ‘ λ³ν κ°λ₯ν μ’νκ³μμλ μμ€ν
μ λͺ¨λ λΉμ νμ±μ΄ μμ€ν
μ μ
λ ₯κ³Ό μΆλ ₯μ ν¨μλ‘ μ΄λ£¨μ΄μ§ μΆλ ₯ μ£Όμ
λΆλΆμ μ νλλ―λ‘, μ΄λ₯Ό μμμν΄μΌλ‘μ¨ μ ν μμ€ν
μ κ²½μ°μ λΉμ·ν ννμ 루μλ²κ±°νμ κ΄μΈ‘κΈ°(Luenberger-type observer)λ₯Ό μ€κ³νλ κ²μ΄ κ°λ₯νκ³ , μ΄μ λ°λΌ μ ννλ κ΄μΈ‘κΈ° μ€μ°¨ λμν(observer error dynamics)μ μ»μ μ μλ€. κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ μΆν μ΄λλ‘, μ΄λ₯Ό μ μ©ν μ μλ μμ€ν
μ λ²μλ₯Ό νμ₯μν€κΈ° μν μ¬λ¬ μ°κ΅¬κ° μ§νλμ΄ μλ€. κ·Έ μ€ νλλ μ£Όμ΄μ§ μμ€ν
μ λ³΄λ€ λμ μ°¨μμ λΉμ ν κ΄μΈ‘κΈ° μ μ€νμΌλ‘ λ³νμν€λ λ°©λ²μ΄λ€. μ΄λ¬ν λ°©μμλ μμ€ν
μ΄λ¨Έμ Ό κΈ°λ²κ³Ό λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν(dynamic observer error linearization) κΈ°λ²μ΄ μλλ°, κ·Έ μ€μμλ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ νΉμ§μ λ€μκ³Ό κ°μ΄ ν¬κ² λ κ°μ§λ‘ μμ½λ μ μλ€. 첫째λ λμ μμ€ν
μ μΆλ ₯μ μ
λ ₯μΌλ‘ νλ 보쑰 λμν(auxiliary dynamics)μ μ€κ³νλ κ²μ΄κ³ , λμ§Έλ 보쑰 λμνμ ν¬ν¨νλ νμ₯λ μμ€ν
μ λμ μμ€ν
λ³΄λ€ λμ μ°¨μμ μΌλ°νλ λΉμ ν κ΄μΈ‘κΈ° μ μ€ν(generalized nonlinear observer canonical form)μΌλ‘ λ³ννλ κ²μ΄λ€. λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μμ μ μλ μΌλ°νλ λΉμ ν κ΄μΈ‘κΈ° μ μ€νμ κ΄μΈ‘ κ°λ₯ν μ ν μμ€ν
κ³Ό μΌλ°νλ μΆλ ₯ μ£Όμ
(generalized output injection)μΌλ‘ ꡬμ±λμ΄ μκ³ , μΌλ°νλ μΆλ ₯ μ£Όμ
μ λμ μμ€ν
μ μΆλ ₯ λΏλ§ μλλΌ λ³΄μ‘°λμνμ μν λ³μμ λν ν¨μλ‘ μ΄λ£¨μ΄μ Έ μλ€λ μ°¨μ΄μ μ΄ μλ€. νμ§λ§, μ΄ λ°©λ²λ‘ μ κ΄μΈ‘κΈ°μ μ°¨μκ° λμ μμ€ν
μ μ°¨μλ³΄λ€ ν¬λ€λ λ¨μ μ κ°μ§κ³ μλ€. μ΄λ¬ν λ¬Έμ λ₯Ό ν΄κ²°νκΈ° μν΄, μ΅κ·Όμλ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ ννμ λ³νλ κΈ°λ²μΌλ‘μ μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν(reduced-order dynamic observer error linearization)λ κΈ°λ²μ΄ λ¨μΌ μΆλ ₯ μμ€ν
μ λν΄ μλ‘κ² μ μλμλ€. μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ² μμ 보쑰 λμνμ μ€κ³νμ¬ νμ₯λ μμ€ν
μ μΌλ°νλ λΉμ ν κ΄μΈ‘κΈ° μ μ€νμΌλ‘ λ³νμν¨λ€λ μ μμ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²κ³Ό 곡ν΅μ μ κ°μ§λ§, λ³νλ μΌλ°νλ λΉμ ν κ΄μΈ‘κΈ° μ μ€νμ μ°¨μκ° λμ μμ€ν
μ μ°¨μμ κ°λ€λ μ°¨μ΄μ μ΄ μλ€. λΉλ‘ μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ΄ μ μ© κ°λ₯ν μμ€ν
μ λ²μ£Όλ λμ κ΄μΈ‘κΈ° μ νν κΈ°λ²μ΄ μ μ© κ°λ₯ν μμ€ν
λ²μ£Όλ₯Ό λ²μ΄λ μλ μμ§λ§, μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ λμ κ΄μΈ‘κΈ° μ νν κΈ°λ²μ λΉν΄ λ μμ μ°¨μμ κ΄μΈ‘κΈ°λ₯Ό μ€κ³ν μ μλ€λ μ΄μ μ΄ μκ³ , 보쑰 λμνμ κ°λ
μ λμ
ν¨μΌλ‘μ¨ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ λΉν΄ λ λμ λ²μ£Όμ μμ€ν
μ μ μ© κ°λ₯νλ€λ μ₯μ μ μ§λλ€. λΏλ§ μλλΌ, μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ κ°λ
μμ²΄κ° κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ κ°λ
κ³Ό λ§€μ° ν‘μ¬νκΈ° λλ¬Έμ (보쑰 λμνμ κ³ λ €νμ§ μμ μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν λ¬Έμ λ κ΄μΈ‘κΈ° μ€μ°¨ μ νν λ¬Έμ μ μΌμΉνλ€.) μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ λν μ°κ΅¬λ₯Ό ν΅ν΄ κΈ°μ‘΄μ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ ν΄μν μλ μλ€.
μ΄μ λ°λΌ, λ³Έ λ
Όλ¬Έμμλ μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ νν κΈ°λ²μ λ€μ€ μΆλ ₯ μμ€ν
μ λν΄ νμ₯μν€κ³ , μ΄μ λν μ°κ΅¬λ₯Ό μννμ¬ κΆκ·Ήμ μΌλ‘λ μ£Όμ΄μ§ λ€μ€ μΆλ ₯ μμ€ν
μ΄ μ΄ κΈ°λ²μ μν΄ μΌλ°νλ λΉμ ν κ΄μΈ‘κΈ° μ μ€νμΌλ‘ λ³νλ μ μλ νμμΆ©λΆ μ‘°κ±΄μ μ μνλ€. μ΄ κ²°κ³Όλ νμ¬κΉμ§ ν립λμ§ μμλ μΌλ°μ μΈ ννμ μΆλ ₯ λ³νκΉμ§ κ³ λ €νμμ κ²½μ°μ λ€μ€ μΆλ ₯ μμ€ν
μ λν κ΄μΈ‘κΈ° μ€μ°¨ μ νν λ¬Έμ μ νμμΆ©λΆ μ‘°κ±΄μ λ΄ν¬νκ³ μλ€.
λν, λ³Έ λ
Όλ¬Έμμλ λΉμ ν κ΄μΈ‘κΈ° μ μ€νμ μ ν λΆλΆ λν μμ€ν
μ μΆλ ₯κ³Ό 보쑰 λμνμ μν λ³μμ μν΄ κ²°μ λλ νμ₯λ λΉμ ν κ΄μΈ‘κΈ° μ μ€ν(extended nonlinear observer canonical form)μ μ μνκ³ , μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ ννμ νμ₯λ κΈ°λ²μΌλ‘μ μ£Όμ΄μ§ λ¨μΌ μΆλ ₯ μμ€ν
μ 보쑰 λμνμ μ€κ³νμ¬ νμ₯λ λΉμ ν κ΄μΈ‘κΈ° μ μ€νμΌλ‘ λ³ννλ λ¬Έμ λ₯Ό μ μνκ³ μ΄μ λν νμμΆ©λΆ μ‘°κ±΄μ μ μνλ€. λν μ΄ κ²°κ³Όλ₯Ό λ’°μ¬λ¬ μμ€ν
(Rossler system)μ μ μ©μμΌλ΄μΌλ‘μ¨ μλ‘κ² μ μλ λ°©λ²λ‘ μ΄ μΆμ μ°¨μ λμ κ΄μΈ‘κΈ° μ€μ°¨ μ ννμ λΉν΄ λ λμ λ²μ£Όμ μμ€ν
μ μ μ©λ μ μμμ μμ¦νλ€.This dissertation contributes to the observer design problem for some classes of nonlinear systems. The observer design problem is to construct a dynamic system (called observer) that can estimate the state of a given dynamic system by using available signals which are commonly the input and the output of the given system. While a standard solution (called Luenberger observer) to the problem was solved for linear systems, there has not been a unified solution for general nonlinear systems. However, there have been significant research efforts on the problem of designing observers for special classes of nonlinear systems. Observer error linearization (OEL) is one of the well-known methods, and it is the problem of transforming a nonlinear system into a nonlinear observer canonical form (NOCF) that is an observable linear system modulo output injection. If a nonlinear system can be transformed into the NOCF, then all the nonlinearities of the system are restricted to the output injection term which is a vector-valued function of the system input and the system output. As a result, we can design a Luenberger-type observer that cancels out the output injection and thus has a linear observer error dynamics in the transformed coordinates. In order to extend the class of systems to which the OEL approach is applicable, a lot of attempts have been made in the past three decades. One of them is to transform a nonlinear system into a higher-dimensional NOCF: system immersion and dynamic observer error linearization (DOEL). In particular, the main idea of DOEL is twofold: the first is to introduce an auxiliary dynamics whose input is system output, and the second is to transform the extended system into a generalized nonlinear observer canonical form (GNOCF) that is an observable linear system modulo generalized output injection depending not only on the system output but also on the state of auxiliary dynamics. By introducing such an auxiliary dynamics, the DOEL problem can be solved for a larger class of systems compared with the (conventional) OEL problem. However, it has a drawback on the dimension of observer. That is, the dimension of observer designed by the DOEL approach is larger than that of the given system, because the dimension of GNOCF equals to the sum of dimensions of the given system and the auxiliary dynamics. Recently, inspired by this fact, a new approach called reduced-order dynamic observer error linearization (RDOEL) was proposed for single output nonlinear systems. In the framework of RDOEL, we also introduce an auxiliary dynamics and transform the extended system into GNOCF in a similar fashion to DOEL, but the coordinate transformation preserves the coordinates corresponding to the state of auxiliary dynamics so that the dimension of GNOCF equals to that of the given system. Although RDOEL is a special case of DOEL (that is, the class of systems to which the RDOEL approach can be applied is a subset of that of DOEL), the RDOEL approach offers a lower-dimensional observer compared to the DOEL approach, and it is also applicable to a larger class of systems compared to the (conventional) OEL approach. In addition, since the framework of RDOEL is coterminous with that of OEL (in fact, the OEL problem is identical to the RDOEL problem with no auxiliary dynamics), most of results for the RDOEL problem can be also used to analyze the OEL problem by slight modification.
In this respect, one of the topics of this dissertation is to deal with the RDOEL problem for multi-output systems. We first formulate the framework of RDOEL for multi-output nonlinear systems and provide three necessary conditions. And then, by means of the necessary conditions, we derive a geometric necessary and sufficient condition in terms of Lie algebras of vector fields. Since the proposed RDOEL problem is a natural extension of the (conventional) OEL problem, the result can be easily translated into a geometric necessary and sufficient condition for the OEL problem, which has not yet been completely established in the case where an output transformation of general form is considered.
The other topic of the dissertation is to introduce an extended nonlinear observer canonical form (ENOCF) whose linear part also depends on the system output and the state of auxiliary dynamics, and to deal with the problem of transforming a single output nonlinear system with an auxiliary dynamics into the ENOCF as an extension of the RDOEL problem. Since the proposed ENOCF admits a kind of high-gain observers, the solvability of the problem allows us to design observers for a class of single output nonlinear systems. We also first present two necessary conditions, and then derive a geometric necessary and sufficient condition for the problem. Furthermore, as a case study, we apply the results to the Rӧssler system in order to show that the proposed method enlarges the class of applicable systems compared with the RDOEL approach.ABSTRACT i
List of Figures ix
Notation and Acronyms x
1 Introduction 1
1.1 Research Background 1
1.2 Organization and Contributions of the Dissertation 5
2 Mathematical Preliminaries 7
2.1 Manifolds and Differentiable Structures 7
2.2 Vector Fields and Covector Fields 10
2.3 Lie Derivatives and Lie Brackets 13
2.4 Distributions and Codistributions 16
3 Review of Related Previous Works 21
3.1 Observability of Multi-Output Nonlinear Systems 21
3.2 Observer Error Linearization (OEL) 23
3.3 System Immersion 28
3.4 Dynamic Observer Error Linearization (DOEL) 30
3.5 Reduced-Order Dynamic Observer Error Linearization (RDOEL) for Single Output Systems 36
3.6 Inclusion Relation among OEL, System Immersion, DOEL, and RDOEL 39
4 Reduced-Order Dynamic Observer Error Linearization (RDOEL) for Multi-Output Systems 43
4.1 Problem Statement 43
4.2 Necessary Conditions 47
4.2.1 Observability 47
4.2.2 Inverse Output Transformation 52
4.2.3 System Dynamics 61
4.3 Necessary and Sufficient Conditions 65
4.3.1 Necessary and Sufficient Condition for RDOEL 65
4.3.2 Necessary and Sufficient Condition for OEL 80
4.3.3 Procedure to Solve OEL and RDOEL 81
4.4 Illustrative Examples 85
5 Extension of RDOEL: System into Extended Nonlinear Observer Canonical Form (ENOCF) 97
5.1 Problem Statement 99
5.2 Necessary Conditions 102
5.2.1 Output Transformation and Observability 102
5.2.2 System Dynamics 105
5.3 Necessary and Sufficient Condition 109
5.4 Case Study: Rӧssler System into ENOCF 117
6 Conclusions 125
BIBLIOGRAPHY 129
κ΅λ¬Έμ΄λ‘ 139
κ°μ¬μ κΈ 143Docto