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Adaptive Output Regulation of Nonlinear Systems with Unknown High-Frequency Gains

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Adaptive Output Regulation of Nonlinear Systems with Unknown High-Frequency Gains ADAPTIVE OUTPUT REGULATION OF NONLINEAR SYSTEMS WITH UNKNOWN HIGH-FREQUENCY GAINS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Sciences and Engineering 2020 Jingyu Chen Department of Electrical and Electronic Engineering School of Engineering Contents List of Tables 6 List of Figures 7 Symbols 8 Abbreviations 9 Abstract 10 Declaration 12 Copyright Statement 13 Publications 14 Acknowledgements 15 1 Introduction 17 1.1 Background . 17 1.2 Research Problems . 19 2 1.3 Contributions and Organization . 20 2 Literature Review 23 2.1 Output Regulation Problem for SISO system . 23 2.1.1 Development of Output Regulation . 24 2.1.2 The Effects of the Knowledge of the High-Frequency Gain Sign . 28 2.2 Consensus Output Regulation of MASs . 30 2.2.1 Development of Consensus Control . 30 2.2.2 Control Directions in Consensus Output Regulation . 32 2.3 Summary . 34 3 Preliminary Studies and Problem Descriptions 35 3.1 Fundamentals of Nonlinear Systems . 35 3.1.1 Nonlinear Systems . 35 3.1.2 Stability Concepts for Nonlinear Systems . 37 3.1.3 Tools for Control Design . 39 3.2 Centre Manifold Theory . 40 3.3 The Output Regulation Theory . 41 3.4 Steady-State Generator . 45 3.4.1 Linear Immersion Assumption . 47 3.4.2 Nonlinear Immersion Assumption . 48 3.5 The Internal Model Principle . 49 3 3.6 Nussbaum Gain Method . 52 3.7 Graph Theory Basics . 53 3.8 State Feedback Consensus Protocols . 56 4 Adaptive Output Regulation of Output Feedback Nonlinear Systems with Unknown Nonlinear Exosystems and Unknown High-Frequency Gain Sign 59 4.1 Problem Formulation . 60 4.2 State Transformation . 62 4.3 The Internal Model Design . 64 4.4 Controller Design . 68 4.5 Stability Analysis . 71 4.6 Numerical Example . 73 4.7 Summary . 77 5 Adaptive Consensus Output Regulation of a Class of Heterogeneous Nonlinear Systems with Unknown Control Directions 78 5.1 Problem Formulation . 79 5.2 State Transformation and The Internal Model . 81 5.3 A Novel Nussbaum-Type Function for Consensus Control . 85 5.4 Consensus Output Regulation Design, ρ =1 ............... 87 5.5 Consensus Output Regulation Design, ρ > 1............... 90 5.6 Example . 93 4 5.7 Summary . 95 6 Distributed Adaptive Consensus Output Tracking of Nonlinear Sys- tems on Directed Graphs with Unknown High-Frequency Gains 96 6.1 Problem Statement . 97 6.2 Preliminary Results . 99 6.3 Control Design . 100 6.4 Simulation Example . 105 6.5 Summary . 106 7 Conclusions and Future Works 110 7.1 Final Conclusions . 110 7.2 Future Works . 112 Bibliography 114 5 List of Tables 3.1 Steady-state generators and internal models under different assumptions. 52 6 List of Figures 2.1 The control process of output regulation problem for a single system. 24 4.1 The system output e and control input u for b = 1. 75 4.2 Feedforward control η1 and its estimationη ^1 for b = 1. 75 4.3 The system output e and control input u for b = −1. 76 4.4 Feedforward control η1 and its estimationη ^1 for b = −1. 76 5.1 The subsystem outputs. 94 5.2 The subsystem control inputs. 95 6.1 The subsystem outputs yi; i = 0;:::; 4. 107 6.2 The subsystem states xi;2; i = 0;:::; 4. 107 6.3 The adaptive gains ki; i = 1;:::; 4. .................... 108 6.4 The tracking errors ζi; i = 1;:::; 4. .................... 108 6.5 The control inputs ui; i = 1;:::; 4...................... 109 6.6 The control inputs ui; i = 1;:::; 4 when t between 0-2s. 109 7 Symbols R set of real numbers Rn n-dimensional Euclidean space Rn×m set of n × m real matrices =: defined as 8 for all 2 belongs to ⊂ subsets of ! tends to P summation Π product ⊗ the Kronecker product of matrices kxk the Euclidean norm of the vector x AT transpose of the matrix A AH conjugate transpose of the matrix A A−1 inverse of the matrix A diag(Ai) a block-diagonal matrix with Ai; i = 1;:::;N, on the diagonal kAk the induced 2-norm of the matrix A λmin(A)(λmax(A)) the minimum (maximum) eigenvalue of the matrix A sup supremum, the least upper bound inf infimum, the greatest lower bound L1 L1 space Lf h the lie derivative of h with respect to the vector field f f : S1 ! S2 a function f mapping a set S1 into a set S2 N(·) Nussbaum function 8 Abbreviations SISO Single-Input-Single-Output MASs Multi-Agent Systems HST Hubble Space Telescope GRORP Global Robust Output Regulation Problem ARE Algebraic Riccati Equation 9 Abstract The University of Manchester Jingyu Chen Doctor of Philosophy Adaptive Output Regulation of Nonlinear Systems with Unknown High- Frequency Gains October 31, 2020 Disturbance rejection and output regulation problem is an important topic in control design since disturbances are inevitable in practical systems. This thesis is concerned with this problem for a class of single-input-single-output (SISO) nonlinear systems and multi-agent nonlinear systems, especially when the high-frequency gains are un- known. Output regulation aims to design a control law to achieve asymptotic tracking in the presence of a class of disturbances while maintaining the stability of the closed- loop system. \Multi-agent systems (MASs)" is referred as a group of agents which are connected together by a communication network. In many applications, the subsys- tems or agents are required to reach an agreement, which is regarded as "consensus control". The main contribution of this thesis is to provide feasible methods to deal with the output regulation of SISO system and consensus output regulation for multi- agent systems. The adaptive output regulation for SISO nonlinear output feedback system with un- known exosystem and unknown high frequency gain is first investigated. By transform- ing the system with ρ relative degree to 1, the augmented system is obtained. Then, using the invariant manifold theory, an adaptive internal model is proposed to tackle 10 the unknown feedforward input. In terms of the unknown high-frequency gain, a type of Nussbaum gain is presented, the control law is then obtained by adopting this gain and the adaptive internal model. To establish the closed-loop stability, the Lyapunov function is invoked, and this control scheme is finally validated by simulation example. The adaptive consensus output regulation of multi-agent nonlinear systems with un- known control directions is then investigated. A new type Nussbaum gain with a potentially faster rate is proposed such that the boundedness of the system parame- ters can be established by an argument of contradiction even if the Nussbaum gain parameter for only one of the subsystems goes unbounded. The adaptive laws and control inputs with the information available from the subsystems and their neigh- bourhood are proposed, and therefore the adaptive laws and inputs are viewed as decentralized. The proposed control can deal with the subsystems with different dy- namics as long as the subsystems with the same relative degree. An example is finally included to demonstrate the proposed control design. The consensus tracking problem for multi-agent nonlinear systems with unknown high- frequency gains is finally investigated for the subsystems connected over directed graph. An integral-Lyapunov function is proposed to tackle the asymmetry of the Laplacian matrices, then the new Nussbaum gain and the adaptive internal model are combined to design the controller. It can be shown that the control scheme and the adaptive laws are fully distributed, and the simulation result illustrates that the proposed control scheme guarantee the convergence of errors to zero asymptotically and the boundedness of the state variables. 11 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 12 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the \Copyright") and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intel- lectual property (the \Intellectual Property") and any reproductions of copyright works in the thesis, for example graphs and tables (\Reproductions"), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any rele- vant Thesis restriction declarations deposited in the University Library, The Univer- sity Library's regulations (see http://www.manchester.ac.uk/library/aboutus/regul- ations) and in The University's Policy on Presentation of Theses. 13 Publications 1).
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