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    μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™”μ™€ ν™•μž₯된 λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•μ„ ν†΅ν•œ λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° 섀계

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    ν•™μœ„λ…Όλ¬Έ (박사)-- μ„œμšΈλŒ€ν•™κ΅ λŒ€ν•™μ› : 전기·컴퓨터곡학뢀, 2014. 8. μ„œμ§„ν—Œ.λ³Έ 논문은 λΉ„μ„ ν˜• μ‹œμŠ€ν…œμ— λŒ€ν•œ κ΄€μΈ‘κΈ° 섀계 문제λ₯Ό 닀루고 μžˆλ‹€. κ΄€μΈ‘κΈ° 섀계 λ¬Έμ œλž€ 주어진 μ‹œμŠ€ν…œμ˜ μž…λ ₯κ³Ό 좜λ ₯ μ •λ³΄λ§Œμ„ ν™œμš©ν•˜μ—¬ λŒ€μƒ μ‹œμŠ€ν…œμ˜ μƒνƒœ λ³€μˆ˜λ₯Ό μΆ”μ •ν•  수 μžˆλŠ” μ‹œμŠ€ν…œμ„ μ„€κ³„ν•˜λŠ” 것이닀. μ„ ν˜• μ‹œμŠ€ν…œμ˜ κ²½μš°μ—λŠ” 루엔버거 κ΄€μΈ‘κΈ°(Luenberger observer)둜 μ•Œλ €μ§„ 일반적인 해법이 μ‘΄μž¬ν•˜λŠ” 반면, 일반적인 λΉ„μ„ ν˜• μ‹œμŠ€ν…œμ— λŒ€ν•΄ κ΄€μΈ‘κΈ°λ₯Ό μ„€κ³„ν•˜λŠ” 방법에 λŒ€ν•œ 연ꡬ κ²°κ³ΌλŠ” ν˜„μž¬κΉŒμ§€ 보고된 λ°”κ°€ μ—†λ‹€. λ‹€λ§Œ, νŠΉμ •ν•œ ν˜•νƒœμ˜ λΉ„μ„ ν˜• μ‹œμŠ€ν…œμ— λŒ€ν•΄ κ΄€μΈ‘κΈ°λ₯Ό μ„€κ³„ν•˜λŠ” λ¬Έμ œμ— λŒ€ν•œ μ—°κ΅¬λŠ” ν™œλ°œν•˜κ²Œ μ§„ν–‰λ˜μ–΄ 였고 μžˆλ‹€. κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™”(observer error linearization) 기법은 이 λ¬Έμ œμ— λŒ€ν•œ κ°€μž₯ 잘 μ•Œλ €μ§„ 방법둠 μ€‘μ˜ ν•˜λ‚˜λ‘œμ„œ, 주어진 λΉ„μ„ ν˜• μ‹œμŠ€ν…œμ„ μ’Œν‘œ λ³€ν™˜μ„ 톡해 κ΄€μΈ‘ κ°€λŠ₯ν•œ μ„ ν˜• μ‹œμŠ€ν…œκ³Ό 좜λ ₯μ£Όμž…(output injection) λΆ€λΆ„λ“€λ‘œ κ΅¬μ„±λœ λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•(nonlinear observer canonical form)으둜 λ³€ν™˜μ‹œν‚€λŠ” λ¬Έμ œμ΄λ‹€. λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•μœΌλ‘œ λ³€ν™˜ κ°€λŠ₯ν•œ μ’Œν‘œκ³„μ—μ„œλŠ” μ‹œμŠ€ν…œμ˜ λͺ¨λ“  λΉ„μ„ ν˜•μ„±μ΄ μ‹œμŠ€ν…œμ˜ μž…λ ₯κ³Ό 좜λ ₯의 ν•¨μˆ˜λ‘œ 이루어진 좜λ ₯ μ£Όμž… 뢀뢄에 μ œν•œλ˜λ―€λ‘œ, 이λ₯Ό μƒμ‡„μ‹œν‚΄μœΌλ‘œμ¨ μ„ ν˜• μ‹œμŠ€ν…œμ˜ κ²½μš°μ™€ λΉ„μŠ·ν•œ ν˜•νƒœμ˜ λ£¨μ—”λ²„κ±°ν˜•μ˜ κ΄€μΈ‘κΈ°(Luenberger-type observer)λ₯Ό μ„€κ³„ν•˜λŠ” 것이 κ°€λŠ₯ν•˜κ³ , 이에 따라 μ„ ν˜•ν™”λœ κ΄€μΈ‘κΈ° 였차 동역학(observer error dynamics)을 얻을 수 μžˆλ‹€. κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” κΈ°λ²•μ˜ μΆœν˜„ 이래둜, 이λ₯Ό μ μš©ν•  수 μžˆλŠ” μ‹œμŠ€ν…œμ˜ λ²”μœ„λ₯Ό ν™•μž₯μ‹œν‚€κΈ° μœ„ν•œ μ—¬λŸ¬ 연ꡬ가 μ§„ν–‰λ˜μ–΄ μ™”λ‹€. κ·Έ 쀑 ν•˜λ‚˜λŠ” 주어진 μ‹œμŠ€ν…œμ„ 보닀 높은 차수의 λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•μœΌλ‘œ λ³€ν™˜μ‹œν‚€λŠ” 방법이닀. μ΄λŸ¬ν•œ λ°©μ‹μ—λŠ” μ‹œμŠ€ν…œ 이머젼 기법과 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™”(dynamic observer error linearization) 기법이 μžˆλŠ”λ°, κ·Έ μ€‘μ—μ„œλ„ 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” κΈ°λ²•μ˜ νŠΉμ§•μ€ λ‹€μŒκ³Ό 같이 크게 두 κ°€μ§€λ‘œ μš”μ•½λ  수 μžˆλ‹€. μ²«μ§ΈλŠ” λŒ€μƒ μ‹œμŠ€ν…œμ˜ 좜λ ₯을 μž…λ ₯으둜 ν•˜λŠ” 보쑰 동역학(auxiliary dynamics)을 μ„€κ³„ν•˜λŠ” 것이고, λ‘˜μ§ΈλŠ” 보쑰 동역학을 ν¬ν•¨ν•˜λŠ” ν™•μž₯된 μ‹œμŠ€ν…œμ„ λŒ€μƒ μ‹œμŠ€ν…œλ³΄λ‹€ 높은 차수의 μΌλ°˜ν™”λœ λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•(generalized nonlinear observer canonical form)으둜 λ³€ν™˜ν•˜λŠ” 것이닀. 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” κΈ°λ²•μ—μ„œ μ œμ•ˆλœ μΌλ°˜ν™”λœ λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•μ€ κ΄€μΈ‘ κ°€λŠ₯ν•œ μ„ ν˜• μ‹œμŠ€ν…œκ³Ό μΌλ°˜ν™”λœ 좜λ ₯ μ£Όμž…(generalized output injection)으둜 κ΅¬μ„±λ˜μ–΄ 있고, μΌλ°˜ν™”λœ 좜λ ₯ μ£Όμž…μ€ λŒ€μƒ μ‹œμŠ€ν…œμ˜ 좜λ ₯ 뿐만 μ•„λ‹ˆλΌ λ³΄μ‘°λ™μ—­ν•™μ˜ μƒνƒœ λ³€μˆ˜μ— λŒ€ν•œ ν•¨μˆ˜λ‘œ 이루어져 μžˆλ‹€λŠ” 차이점이 μžˆλ‹€. ν•˜μ§€λ§Œ, 이 방법둠은 κ΄€μΈ‘κΈ°μ˜ μ°¨μˆ˜κ°€ λŒ€μƒ μ‹œμŠ€ν…œμ˜ μ°¨μˆ˜λ³΄λ‹€ ν¬λ‹€λŠ” 단점을 가지고 μžˆλ‹€. μ΄λŸ¬ν•œ 문제λ₯Ό ν•΄κ²°ν•˜κΈ° μœ„ν•΄, μ΅œκ·Όμ—λŠ” 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™”μ˜ λ³€ν˜•λœ κΈ°λ²•μœΌλ‘œμ„œ μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™”(reduced-order dynamic observer error linearization)λž€ 기법이 단일 좜λ ₯ μ‹œμŠ€ν…œμ— λŒ€ν•΄ μƒˆλ‘­κ²Œ μ œμ•ˆλ˜μ—ˆλ‹€. μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” 기법 μ—­μ‹œ 보쑰 동역학을 μ„€κ³„ν•˜μ—¬ ν™•μž₯된 μ‹œμŠ€ν…œμ„ μΌλ°˜ν™”λœ λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•μœΌλ‘œ λ³€ν™˜μ‹œν‚¨λ‹€λŠ” μ μ—μ„œ 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” 기법과 곡톡점을 κ°–μ§€λ§Œ, λ³€ν™˜λœ μΌλ°˜ν™”λœ λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•μ˜ μ°¨μˆ˜κ°€ λŒ€μƒ μ‹œμŠ€ν…œμ˜ μ°¨μˆ˜μ™€ κ°™λ‹€λŠ” 차이점이 μžˆλ‹€. 비둝 μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” 기법이 적용 κ°€λŠ₯ν•œ μ‹œμŠ€ν…œμ˜ λ²”μ£ΌλŠ” 동적 κ΄€μΈ‘κΈ° μ„ ν˜•ν™” 기법이 적용 κ°€λŠ₯ν•œ μ‹œμŠ€ν…œ λ²”μ£Όλ₯Ό λ²—μ–΄λ‚  μˆ˜λŠ” μ—†μ§€λ§Œ, μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” 기법은 동적 κ΄€μΈ‘κΈ° μ„ ν˜•ν™” 기법에 λΉ„ν•΄ 더 μž‘μ€ 차수의 κ΄€μΈ‘κΈ°λ₯Ό 섀계할 수 μžˆλ‹€λŠ” 이점이 있고, 보쑰 λ™μ—­ν•™μ˜ κ°œλ…μ„ λ„μž…ν•¨μœΌλ‘œμ¨ κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” 기법에 λΉ„ν•΄ 더 넓은 λ²”μ£Όμ˜ μ‹œμŠ€ν…œμ— 적용 κ°€λŠ₯ν•˜λ‹€λŠ” μž₯점을 μ§€λ‹Œλ‹€. 뿐만 μ•„λ‹ˆλΌ, μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” κΈ°λ²•μ˜ κ°œλ… μžμ²΄κ°€ κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” κΈ°λ²•μ˜ κ°œλ…κ³Ό 맀우 ν‘μ‚¬ν•˜κΈ° λ•Œλ¬Έμ— (보쑰 동역학을 κ³ λ €ν•˜μ§€ μ•Šμ€ μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” λ¬Έμ œλŠ” κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” λ¬Έμ œμ™€ μΌμΉ˜ν•œλ‹€.) μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” 기법에 λŒ€ν•œ 연ꡬλ₯Ό 톡해 기쑴의 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” 기법을 해석할 μˆ˜λ„ μžˆλ‹€. 이에 따라, λ³Έ λ…Όλ¬Έμ—μ„œλŠ” μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” 기법을 닀쀑 좜λ ₯ μ‹œμŠ€ν…œμ— λŒ€ν•΄ ν™•μž₯μ‹œν‚€κ³ , 이에 λŒ€ν•œ 연ꡬλ₯Ό μˆ˜ν–‰ν•˜μ—¬ κΆκ·Ήμ μœΌλ‘œλŠ” 주어진 닀쀑 좜λ ₯ μ‹œμŠ€ν…œμ΄ 이 기법에 μ˜ν•΄ μΌλ°˜ν™”λœ λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•μœΌλ‘œ λ³€ν™˜λ  수 μžˆλŠ” ν•„μš”μΆ©λΆ„ 쑰건을 μ œμ‹œν•œλ‹€. 이 κ²°κ³ΌλŠ” ν˜„μž¬κΉŒμ§€ ν™•λ¦½λ˜μ§€ μ•Šμ•˜λ˜ 일반적인 ν˜•νƒœμ˜ 좜λ ₯ λ³€ν™˜κΉŒμ§€ κ³ λ €ν•˜μ˜€μ„ 경우의 닀쀑 좜λ ₯ μ‹œμŠ€ν…œμ— λŒ€ν•œ κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™” 문제의 ν•„μš”μΆ©λΆ„ 쑰건을 λ‚΄ν¬ν•˜κ³  μžˆλ‹€. λ˜ν•œ, λ³Έ λ…Όλ¬Έμ—μ„œλŠ” λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•μ˜ μ„ ν˜• λΆ€λΆ„ λ˜ν•œ μ‹œμŠ€ν…œμ˜ 좜λ ₯κ³Ό 보쑰 λ™μ—­ν•™μ˜ μƒνƒœ λ³€μˆ˜μ— μ˜ν•΄ κ²°μ •λ˜λŠ” ν™•μž₯된 λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•(extended nonlinear observer canonical form)을 μ œμ•ˆν•˜κ³ , μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™”μ˜ ν™•μž₯된 κΈ°λ²•μœΌλ‘œμ„œ 주어진 단일 좜λ ₯ μ‹œμŠ€ν…œμ„ 보쑰 동역학을 μ„€κ³„ν•˜μ—¬ ν™•μž₯된 λΉ„μ„ ν˜• κ΄€μΈ‘κΈ° μ •μ€€ν˜•μœΌλ‘œ λ³€ν™˜ν•˜λŠ” 문제λ₯Ό μ œμ•ˆν•˜κ³  이에 λŒ€ν•œ ν•„μš”μΆ©λΆ„ 쑰건을 μ œμ‹œν•œλ‹€. λ˜ν•œ 이 κ²°κ³Όλ₯Ό 뒰슬러 μ‹œμŠ€ν…œ(Rossler system)에 μ μš©μ‹œμΌœλ΄„μœΌλ‘œμ¨ μƒˆλ‘­κ²Œ μ œμ•ˆλœ 방법둠이 μΆ•μ†Œ 차원 동적 κ΄€μΈ‘κΈ° 였차 μ„ ν˜•ν™”μ— λΉ„ν•΄ 더 넓은 λ²”μ£Όμ˜ μ‹œμŠ€ν…œμ— 적용될 수 μžˆμŒμ„ μ˜ˆμ¦ν•œλ‹€.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 RΣ§ssler 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: RΣ§ssler System into ENOCF 117 6 Conclusions 125 BIBLIOGRAPHY 129 ꡭ문초둝 139 κ°μ‚¬μ˜ κΈ€ 143Docto
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