A STUDY ON THE APPLICABILITY OF SYMBOLIC COMPUTATION IN STABILISING CONTROL DESIGN FOR SWITCHED SYSTEMS AT-TASNEEM BINTI MOHD AMIN

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1 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

2 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

3 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

4 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

5 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 Purpose of Study Scope of Study The Significance of Study 4 CHAPTER II STABILITY OF SWITCHED SYSTEM 2.1 Introduction Stability of Switched System Multicontrollers Hurwitz stability State transformation Stabilizing switching control strategy Lie Algebra 2.3 Haris Method Discussion CHAPTER III COMPUTING CAPABILITY TEST FOR METHOD 3.1 Introduction Problem Statement Testing the Computerization Capability cvx 22

6 vi LMI Solver Yalmip Maple Discussion 31 CHAPTER IV THE SYMBOLIC COMPUTATION BASED DESIGN TOOLBOX 4.1 Introduction The Haris Method Hybrid System Controller Design Tool The Transient Response Test 45 CHAPTER V COMPUTING CAPABILITY TEST FOR METHOD 5.1 Introduction Symbolic Computation Numerical Computation LMI Solver cvx Yalmip 5.4 Discussion Case Study CHAPTER VI CONCLUSION 6.1 Summary Suggestions 79 REFERENCES 80

7 vii APPENDICES A Method 85 B Method 96 C Case Study D Case Study E Case Study 4 136

8 viii LIST OF FIGURE No of Figure 1.1 The pathway of a state variable in a hybrid system Switching Compensator Modelling of hyperplanes in DPN framework Representation of a switched system in a DPN framework Circular region for pole location LMI Editor The Method using Maple 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 The toolbox for entering the required information The eigenvalues for the switched system using Maple 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 The plotted range The solution for Method using Maple 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

9 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

10 vii LIST OF TABLE No. of Table 3.1 Feedback controller using the LMI Solver The feedback controller from the Method using Yalmip The feedback controller from the Method using Maple 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 The feedback controller from the Method using the Yalmip 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 Case Study Case Study 4 76