A STUDY ON THE APPLICABILITY OF SYMBOLIC COMPUTATION IN STABILISING CONTROL DESIGN FOR SWITCHED SYSTEMS AT-TASNEEM BINTI MOHD AMIN THESIS SUBMITTED IN PARTION FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE FACULTY OF ENGINEERING AND BUILT ENVIRONMENT UNIVERSITI KEBANGSAAN MALAYSIA BANGI 2012
KAJIAN TERHADAP APLIKASI KOMPUTERAN SIMBOLIK DALAM REKA BENTUK KAWALAN KESTABILAN UNTUK SISTEM PERTUKARAN AT-TASNEEM BINTI MOHD AMIN TESIS YANG DIKEMUKAKAN UNTUK MEMENUHI SEBAHAGIAN DARIPADA SYARAT MEMPEROLEH IJAZAH SARJANA SAINS FAKULTI KEJURUTERAAN DAN ALAM BINA UNIVERSITI KEBANGSAAN MALAYSIA BANGI 2012
DECLARATION I hereby declare that the work in this thesis is my own except for quotations and summaries which have been duly acknowledged. 4 October 2012 AT-TASNEEM BINTI MOHD AMIN P50212
x ABBREVIATIONS Z state matrix state matrix in Brunovsky form state matrix state matrix in Brunovsky form output matrix controller gain controller gain in Brunovsky form Lyapunov matrix using Method (LMI Solver) Lyapunov matrix using Method (Yalmip) Lyapunov matrix using Method (cvx, Yalmip, LMI Solver, Maple) switching sequence natural number symmetric positive definite matrix Euclidean Space time input or control vector Lyapunov matrix using Method (theory) state vector derivative of the state vector with respect to time equilibrium point output vector integer number input sequence at time
v TABLE OF CONTENT DECLARATION ABSTRACT ABSTRAK TABLE OF CONTENT LIST OF TABLE LIST OF FIGURE ABBREVIATIONS Page ii iii iv v vii viii x CHAPTER I PREFACE 1.1 Introduction 1 1.2 Purpose of Study 3 1.3 Scope of Study 3 1.4 The Significance of Study 4 CHAPTER II STABILITY OF SWITCHED SYSTEM 2.1 Introduction 5 2.2 Stability of Switched System 6 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 Multicontrollers Hurwitz stability State transformation Stabilizing switching control strategy Lie Algebra 2.3 Haris Method 17 2.4 Discussion 17 9 11 12 15 16 CHAPTER III COMPUTING CAPABILITY TEST FOR METHOD 3.1 Introduction 19 3.2 Problem Statement 20 3.3 Testing the Computerization Capability 21 3.3.1 cvx 22
vi 3.3.2 LMI Solver 3.3.3 Yalmip 3.3.4 Maple 24 27 30 3.4 Discussion 31 CHAPTER IV THE SYMBOLIC COMPUTATION BASED DESIGN TOOLBOX 4.1 Introduction 35 4.2 The Haris Method 36 4.3 Hybrid System Controller Design Tool 40 4.4 The Transient Response Test 45 CHAPTER V COMPUTING CAPABILITY TEST FOR METHOD 5.1 Introduction 51 5.2 Symbolic Computation 52 5.3 Numerical Computation 55 5.3.1 LMI Solver 5.3.2 cvx 5.3.3 Yalmip 5.4 Discussion 64 5.5 Case Study 74 55 58 62 CHAPTER VI CONCLUSION 6.1 Summary 77 6.2 Suggestions 79 REFERENCES 80
vii APPENDICES A Method 85 B Method 96 C Case Study 2 126 D Case Study 3 133 E Case Study 4 136
viii LIST OF FIGURE No of Figure 1.1 The pathway of a state variable in a hybrid system 2 2.1 Switching Compensator 11 2.2 Modelling of hyperplanes in DPN framework 14 2.3 Representation of a switched system in a DPN framework 15 2.4 Circular region for pole location 16 3.1 LMI Editor 24 3.2 The Method using Maple 30 3.3 Location of the pole-zero without the feedback controller and with 33 the feedback controller from the Method using Yalmip 3.4 The transient response for the time intervals of exchange for every 34 (a) 10 seconds, (b) 30 seconds, (c) 50 seconds, and (d) 70 seconds 4.1 Introduction to the hybrid system controller toolbox 41 4.2 The toolbox for entering the required information 41 4.3 The eigenvalues for the switched system using Maple 46 4.4 The location of the pole-zero for the switched system using the 48 Method 4.5 Graph of the transient response for the switched system using the 49 Method 4.6 Graph of the transient response for the switched system for the 50 switching interval occurring every (a) 1 second, (b) 10 seconds, (c) 100 seconds and (d) 500 seconds 5.1 The plotted range 53 5.2 The plotted range 54 5.3 The solution for Method using Maple 54 5.4 The transient response for the switching time intervals for every (a) 1 second, (b) 10 seconds, (c) 100 seconds, and (d) 500 seconds using the LMI Solver for Method 57
ix 5.5 The transient response for the switching time intervals for every (a) 100 seconds, (b) 200 seconds, (c) 300 seconds, and (d) 400 seconds using the cvx for the Method 5.6 The transient response for the switching time intervals for every (a) 10 seconds, (b) 100 seconds, (c) 200 seconds, and (d) 300 seconds using the Yalmip for the Method 5.7 Locations of the pole-zero for the Method and Method with feedback controllers using the Yalmip 61 65 73
vii LIST OF TABLE No. of Table 3.1 Feedback controller using the LMI Solver 26 3.2 The feedback controller from the Method using Yalmip 29 4.1 The feedback controller from the Method using Maple 44 4.2 The steady-state response for the switched system using the 50 Method 5.1 The feedback controller from the Method using the LMI 56 Solver 5.2 The feedback controller from the Method using the cvx 60 5.3 The feedback controller from the Method using the Yalmip 64 5.4 Lyapunov matrix and eigenvalues for the switched system 66 using Method for different types of computing programs 5.5 Lyapunov matrix and eigenvalues for the switched system 67 using Method for different types of computing programs 5.6 Rounding ff the decimal of Lyapunov matrix and the 70 eigenvalues for switched system using the Method for cvx 5.7 The feedback controller for Method using Yalmip (LMILAB 71 and LINPROG) 5.8 Rounding ff the decimal of Lyapunov matrix and the 71 eigenvalues for switched system using the Method for Yalmip 5.9 Case Study 2 74 5.10 Case Study 3 75 5.11 Case Study 4 76