Presentation is loading. Please wait.

Presentation is loading. Please wait.

Pariz-Karimpour Feb 2011 1 Chapter 3 Reference: Switched linear systems control and design Zhendong Sun, Shuzhi S. Ge.

Published by Modified over 9 years ago

Similar presentations

Presentation on theme: "Pariz-Karimpour Feb 2011 1 Chapter 3 Reference: Switched linear systems control and design Zhendong Sun, Shuzhi S. Ge."— Presentation transcript:

1 Pariz-Karimpour Feb 2011 1 Chapter 3 Reference: Switched linear systems control and design Zhendong Sun, Shuzhi S. Ge

2 Pariz-Karimpour Feb 2011 2 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

3 Pariz-Karimpour Feb 2011 3 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

4 Pariz-Karimpour Feb 2011 4

5 5 Introduction

6 6 Introduction

7 7 ?Introduction

8 8 Introduction Example

9 Pariz-Karimpour Feb 2011 9 LetIntroduction

10 Pariz-Karimpour Feb 2011 10 Introduction

11 Pariz-Karimpour Feb 2011 11 This lecture provide: Basic observation on the ability and limitation of switching design Analyze and design of some switching for Stability and robustness Introduction

12 Pariz-Karimpour Feb 2011 12 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

13 Pariz-Karimpour Feb 2011 13

14 Pariz-Karimpour Feb 2011 14 3.2.1. Algebraic Criteria 3.2.2. Equivalence of the Stabilization Notions 3.2.3. Periodic and Synchronous Switchings 3.2.4. Special Systems 3.2.5. Robustness Issues General Results

15 Pariz-Karimpour Feb 2011 15 Algebraic Criteria

16 Pariz-Karimpour Feb 2011 16 Algebraic Criteria

17 Pariz-Karimpour Feb 2011 17 Algebraic Criteria

18 Pariz-Karimpour Feb 2011 18 Example Algebraic Criteria

19 Pariz-Karimpour Feb 2011 19 Algebraic Criteria

20 Pariz-Karimpour Feb 2011 20 Example Algebraic Criteria

21 Pariz-Karimpour Feb 2011 21 Algebraic Criteria

22 Pariz-Karimpour Feb 2011 22 3.2.1. Algebraic Criteria 3.2.2. Equivalence of the Stabilization Notions 3.2.3. Periodic and Synchronous Switchings 3.2.4. Special Systems 3.2.5. Robustness Issues General Results

23 Pariz-Karimpour Feb 2011 23 Does this equivalence still hold for switched linear systems To establish the equivalence, we need the concept of switched convergence Equivalence of the Stabilization Notions

24 Pariz-Karimpour Feb 2011 24 Equivalence of the Stabilization Notions

25 Pariz-Karimpour Feb 2011 25 Equivalence of the Stabilization Notions

26 Pariz-Karimpour Feb 2011 26 Equivalence of the Stabilization Notions

27 Pariz-Karimpour Feb 2011 27 Equivalence of the Stabilization Notions

28 Pariz-Karimpour Feb 2011 28 R2R2... RlRl R1R1 RiRi Equivalence of the Stabilization Notions

29 Pariz-Karimpour Feb 2011 29 R2R2... RlRl R1R1 RiRi Equivalence of the Stabilization Notions Since

30 Pariz-Karimpour Feb 2011 30 Equivalence of the Stabilization Notions

31 Pariz-Karimpour Feb 2011 31 Equivalence of the Stabilization Notions

32 Pariz-Karimpour Feb 2011 32 Equivalence of the Stabilization Notions

33 Pariz-Karimpour Feb 2011 33 Equivalence of the Stabilization Notions

34 Pariz-Karimpour Feb 2011 34 3.2.1. Algebraic Criteria 3.2.2. Equivalence of the Stabilization Notions 3.2.3. Periodic and Synchronous Switchings 3.2.4. Special Systems 3.2.5. Robustness Issues General Results

35 Pariz-Karimpour Feb 2011 35 Periodic and Synchronous Switchings

36 Pariz-Karimpour Feb 2011 36 Periodic and Synchronous Switchings

37 Pariz-Karimpour Feb 2011 37 Periodic and Synchronous Switchings

38 Pariz-Karimpour Feb 2011 38 Periodic and Synchronous Switchings

39 Pariz-Karimpour Feb 2011 39 3.2.1. Algebraic Criteria 3.2.2. Equivalence of the Stabilization Notions 3.2.3. Periodic and Synchronous Switchings 3.2.4. Special Systems 3.2.5. Robustness Issues General Results

40 Pariz-Karimpour Feb 2011 40 Special Systems

41 Pariz-Karimpour Feb 2011 41 Special Systems

42 Pariz-Karimpour Feb 2011 42 Special Systems

43 Pariz-Karimpour Feb 2011 43 3.2.1. Algebraic Criteria 3.2.2. Equivalence of the Stabilization Notions 3.2.3. Periodic and Synchronous Switchings 3.2.4. Special Systems 3.2.5. Robustness Issues General Results

44 Pariz-Karimpour Feb 2011 44 Robustness Issues

45 Pariz-Karimpour Feb 2011 45 Robustness Issues

46 Pariz-Karimpour Feb 2011 46 Robustness Issues Proof of theorem 3.15 (continue)

47 Pariz-Karimpour Feb 2011 47 Robustness Issues Proof of theorem 3.15 (continue)

48 Pariz-Karimpour Feb 2011 48 Robustness Issues

49 Pariz-Karimpour Feb 2011 49 Robustness Issues

50 Pariz-Karimpour Feb 2011 50 Robustness Issues

51 Pariz-Karimpour Feb 2011 51 Proof of theorem 3.19 (continue) Robustness Issues

52 Pariz-Karimpour Feb 2011 52 Proof: By theorem 3.19 we have Robustness Issues

53 Pariz-Karimpour Feb 2011 53 Robustness Issues

54 Pariz-Karimpour Feb 2011 54 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

55 Pariz-Karimpour Feb 2011 55

56 Pariz-Karimpour Feb 2011 56 Periodic Switching

57 Pariz-Karimpour Feb 2011 57 0 12m …… 12m 12m Periodic Switching

58 Pariz-Karimpour Feb 2011 58 Define the fundamental matrix as: Periodic Switching

59 Pariz-Karimpour Feb 2011 59 Periodic Switching

60 Pariz-Karimpour Feb 2011 60 0 12m …… 12m 12m = 1 = 2 Periodic Switching

61 Pariz-Karimpour Feb 2011 61 0 12m …… 12m 12m = 1 = 2 Periodic Switching

62 Pariz-Karimpour Feb 2011 62 Periodic Switching

63 Pariz-Karimpour Feb 2011 63 i) The system state is bounded if the perturbation is bounded ii) The system state is bounded and convergent if the perturbation is bounded and convergent iii) The system state is exponentially convergent if the perturbation is exponentially convergent Periodic Switching

64 Pariz-Karimpour Feb 2011 64 i) Periodic Switching

65 Pariz-Karimpour Feb 2011 65 ii) Periodic Switching

66 Pariz-Karimpour Feb 2011 66 iii) Periodic Switching

67 Pariz-Karimpour Feb 2011 67 Periodic Switching

68 Pariz-Karimpour Feb 2011 68 Periodic Switching

69 Pariz-Karimpour Feb 2011 69 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

70 Pariz-Karimpour Feb 2011 70

71 Pariz-Karimpour Feb 2011 71 3.4.1. State-space-partition-based Switching 3.4.2. A Modified Switching Law 3.4.3. Observer-based Switching State-feedback Switching

72 Pariz-Karimpour Feb 2011 72 State-space-partition-based Switching

73 Pariz-Karimpour Feb 2011 73 Switching strategy State-space-partition-based Switching

74 Pariz-Karimpour Feb 2011 74 State-space-partition-based Switching

75 Pariz-Karimpour Feb 2011 75 State-space-partition-based Switching

76 Pariz-Karimpour Feb 2011 76 State-space-partition-based Switching

77 Pariz-Karimpour Feb 2011 77 State-space-partition-based Switching

78 Pariz-Karimpour Feb 2011 78 State-space-partition-based Switching

79 Pariz-Karimpour Feb 2011 79 State-space-partition-based Switching

80 Pariz-Karimpour Feb 2011 80 State-space-partition-based Switching

81 Pariz-Karimpour Feb 2011 81 State-space-partition-based Switching

82 Pariz-Karimpour Feb 2011 82

83 Pariz-Karimpour Feb 2011 83 function y=myfun2(x) if x(1)~=x(2);y=1; else y=0; end function y=myfun1(w) if w==1; y=[1;0];end if w==2; y=[0;1];end end function y=myfun(w) x=w(1:2);sigk=w(3); A1=[-2 0;0 1];A2=[1 0;0 -2];x0=[1;-1]; P=0.5*eye(2); Q(1).s=A1'*P+P*A1;Q(2).s=A2'*P+P*A2;r(1)=0.4;r(2)=0.4; if (x'*Q(sigk).s*x) > (-r(sigk)*x'*x) [c,y]=min([x'*Q(1).s*x, x'*Q(2).s*x]); else y=sigk; end State-space-partition-based Switching

84 Pariz-Karimpour Feb 2011 84 State-space-partition-based Switching

85 Pariz-Karimpour Feb 2011 85 State-space-partition-based Switching

86 Pariz-Karimpour Feb 2011 86 State-space-partition-based Switching

87 Pariz-Karimpour Feb 2011 87 State-space-partition-based Switching

88 Pariz-Karimpour Feb 2011 88 3.4.1. State-space-partition-based Switching 3.4.2. A Modified Switching Law 3.4.3. Observer-based Switching State-feedback Switching

89 Pariz-Karimpour Feb 2011 89 A Modified Switching Law

90 Pariz-Karimpour Feb 2011 90 Modified Switching strategy A Modified Switching Law

91 Pariz-Karimpour Feb 2011 91 A Modified Switching Law

92 Pariz-Karimpour Feb 2011 92 A Modified Switching Law

93 Pariz-Karimpour Feb 2011 93 A Modified Switching Law

94 Pariz-Karimpour Feb 2011 94 A Modified Switching Law

95 Pariz-Karimpour Feb 2011 95 A Modified Switching Law

96 Pariz-Karimpour Feb 2011 96 A Modified Switching Law

97 Pariz-Karimpour Feb 2011 97 A Modified Switching Law

98 Pariz-Karimpour Feb 2011 98 A Modified Switching Law

99 Pariz-Karimpour Feb 2011 99 A Modified Switching Law

100 Pariz-Karimpour Feb 2011 100 A Modified Switching Law

101 Pariz-Karimpour Feb 2011 101 A Modified Switching Law

102 Pariz-Karimpour Feb 2011 102 A Modified Switching Law

103 Pariz-Karimpour Feb 2011 103 A Modified Switching Law

104 Pariz-Karimpour Feb 2011 104 3.4.1. State-space-partition-based Switching 3.4.2. A Modified Switching Law 3.4.3. Observer-based Switching State-feedback Switching

105 Pariz-Karimpour Feb 2011 105 Observer-based Switching

106 Pariz-Karimpour Feb 2011 106 Observer-based Switching

107 Pariz-Karimpour Feb 2011 107 Observer-based Switching

108 Pariz-Karimpour Feb 2011 108 Observer-based Switching

109 Pariz-Karimpour Feb 2011 109 Observer-based Switching

110 Pariz-Karimpour Feb 2011 110

111 Pariz-Karimpour Feb 2011 111

112 Pariz-Karimpour Feb 2011 112 Observer-based Switching

113 Pariz-Karimpour Feb 2011 113

114 Pariz-Karimpour Feb 2011 114 Observer-based Switching

115 Pariz-Karimpour Feb 2011 115 Observer-based Switching

116 Pariz-Karimpour Feb 2011 116 1- Check the assumption 3.2 for the system 2- Repeat the system simulatrion by 3- Choose suitable L 1 and L 2 and repeat the simulation. 4- Examine the system for y=x 1 for the first system and y=x 2 for the second one. Exercises: 5- According to exercise 4 derive another condition for observer base switching.

117 Pariz-Karimpour Feb 2011 117 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

118 Pariz-Karimpour Feb 2011 118

119 Pariz-Karimpour Feb 2011 119 Combined Switching

120 Pariz-Karimpour Feb 2011 120 Periodic switching 0 12m …… 12m 12m Combined Switching

121 Pariz-Karimpour Feb 2011 121 State feedback switching Combined Switching

122 Pariz-Karimpour Feb 2011 122 Combined Switching

123 Pariz-Karimpour Feb 2011 123 3.5.1. Switching Strategy Description 3.5.2. Robustness Properties 3.5.3. Observer-based Switching 3.5.4. Extensions Combined Switching

124 Pariz-Karimpour Feb 2011 124 Switching Strategy Description

125 Pariz-Karimpour Feb 2011 125 Switching Strategy Description

126 Pariz-Karimpour Feb 2011 126 tktk t k+2 t k+1 Switching Strategy Description

127 Pariz-Karimpour Feb 2011 127 Proof: Switching Strategy Description

128 Pariz-Karimpour Feb 2011 128 Switching Strategy Description

129 Pariz-Karimpour Feb 2011 129 Switching Strategy Description

130 Pariz-Karimpour Feb 2011 130 Switching Strategy Description

131 Pariz-Karimpour Feb 2011 131 Switching Strategy Description

132 Pariz-Karimpour Feb 2011 132 Switching Strategy Description

133 Pariz-Karimpour Feb 2011 133 Switching Strategy Description

134 Pariz-Karimpour Feb 2011 134 Switching Strategy Description

135 Pariz-Karimpour Feb 2011 135

136 Pariz-Karimpour Feb 2011 136 By student (#2) Switching Strategy Description

137 Pariz-Karimpour Feb 2011 137 3.5.1. Switching Strategy Description 3.5.2. Robustness Properties 3.5.3. Observer-based Switching 3.5.4. Extensions Combined Switching

138 Pariz-Karimpour Feb 2011 138 Proof: By one of the student (#3) Robustness Properties

139 Pariz-Karimpour Feb 2011 139 By student (#3) Robustness Properties

140 Pariz-Karimpour Feb 2011 140 3.5.1. Switching Strategy Description 3.5.2. Robustness Properties 3.5.3. Observer-based Switching 3.5.4. Extensions Combined Switching

141 Pariz-Karimpour Feb 2011 141 Observer-based Switching

142 Pariz-Karimpour Feb 2011 142 Observer-based Switching

143 Pariz-Karimpour Feb 2011 143 Proof: By one of the student (#4) Observer-based Switching

144 Pariz-Karimpour Feb 2011 144 3.5.1. Switching Strategy Description 3.5.2. Robustness Properties 3.5.3. Observer-based Switching 3.5.4. Extensions Combined Switching

145 Pariz-Karimpour Feb 2011 145 Extensions

146 Pariz-Karimpour Feb 2011 146 Extensions

147 Pariz-Karimpour Feb 2011 147 Extensions

148 Pariz-Karimpour Feb 2011 148 Extensions

149 Pariz-Karimpour Feb 2011 149 Let

150 Pariz-Karimpour Feb 2011 150 Let t0t0 t4t4 t1t1 t2t2 t3t3 Extensions

151 Pariz-Karimpour Feb 2011 151 Extensions

152 Pariz-Karimpour Feb 2011 152 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

153 Pariz-Karimpour Feb 2011 153

154 Pariz-Karimpour Feb 2011 154 Numerical Examples

155 Pariz-Karimpour Feb 2011 155 Numerical Examples

156 Pariz-Karimpour Feb 2011 156 Numerical Examples

157 Pariz-Karimpour Feb 2011 157 Numerical Examples

158 Pariz-Karimpour Feb 2011 158 Numerical Examples

159 Pariz-Karimpour Feb 2011 159 Numerical Examples

160 Pariz-Karimpour Feb 2011 160 Numerical Examples

161 Pariz-Karimpour Feb 2011 161 Numerical Examples

162 Pariz-Karimpour Feb 2011 162 Numerical Examples

163 Pariz-Karimpour Feb 2011 163 Numerical Examples

164 Pariz-Karimpour Feb 2011 164 Numerical Examples

165 Pariz-Karimpour Feb 2011 165 Numerical Examples

166 Pariz-Karimpour Feb 2011 166 Numerical Examples

167 Pariz-Karimpour Feb 2011 167 Numerical Examples

168 Pariz-Karimpour Feb 2011 168 Numerical Examples

169 Pariz-Karimpour Feb 2011 169 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems Summary

170 Pariz-Karimpour Feb 2011 170 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

171 Pariz-Karimpour Feb 2011 171 Stabilizing Switching for Autonomous Systems Summary 3.1. Introduction

172 Pariz-Karimpour Feb 2011 172 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

173 Pariz-Karimpour Feb 2011 173 Stabilizing Switching for Autonomous Systems Summary 3.2. General Results 3.2.1. Algebraic Criteria

174 Pariz-Karimpour Feb 2011 174 Stabilizing Switching for Autonomous Systems Summary 3.2. General Results 3.2.1. Algebraic Criteria 3.2.2. Equivalence of the Stabilization Notions

175 Pariz-Karimpour Feb 2011 175 Stabilizing Switching for Autonomous Systems Summary 3.2. General Results 3.2.1. Algebraic Criteria 3.2.2. Equivalence of the Stabilization Notions 3.2.3. Periodic and Synchronous Switching

176 Pariz-Karimpour Feb 2011 176 Stabilizing Switching for Autonomous Systems Summary 3.2. General Results 3.2.1. Algebraic Criteria 3.2.2. Equivalence of the Stabilization Notions 3.2.3. Periodic and Synchronous Switching 3.2.4. Special Systems 3.2.5. Robustness Issues

177 Pariz-Karimpour Feb 2011 177 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

178 Pariz-Karimpour Feb 2011 178 Stabilizing Switching for Autonomous Systems Summary 3.3. Periodic Switching 0 12m …… 12m 12m

179 Pariz-Karimpour Feb 2011 179 Stabilizing Switching for Autonomous Systems Summary 3.3. Periodic Switching

180 Pariz-Karimpour Feb 2011 180 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

181 Pariz-Karimpour Feb 2011 181 Stabilizing Switching for Autonomous Systems Summary 3.4. State Feedback Switching Switching strategy

182 Pariz-Karimpour Feb 2011 182 Stabilizing Switching for Autonomous Systems Summary 3.4. State Feedback Switching Modified Switching strategy

183 Pariz-Karimpour Feb 2011 183 Stabilizing Switching for Autonomous Systems Summary 3.4. State Feedback Switching Observer Based Switching strategy

184 Pariz-Karimpour Feb 2011 184 3.1. Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

185 Pariz-Karimpour Feb 2011 185 Stabilizing Switching for Autonomous Systems Summary 3.5. Combined Switching

186 Pariz-Karimpour Feb 2011 186 Stabilizing Switching for Autonomous Systems Summary 3.5. Combined Switching (Robustness Property)

187 Pariz-Karimpour Feb 2011 187 Stabilizing Switching for Autonomous Systems Summary 3.5. Combined Switching (Extension)

188 Pariz-Karimpour Feb 2011 188

Download ppt "Pariz-Karimpour Feb 2011 1 Chapter 3 Reference: Switched linear systems control and design Zhendong Sun, Shuzhi S. Ge."

Similar presentations

The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Coordination of Multi-Agent Systems Mark W. Spong Donald Biggar Willett Professor Department of Electrical and Computer Engineering and The Coordinated.

Coordination of Multi-Agent Systems Mark W. Spong Donald Biggar Willett Professor Department of Electrical and Computer Engineering and The Coordinated.
Lecture 7: Basis Functions & Fourier Series

Lecture 7: Basis Functions & Fourier Series
INTRODUCTION to SWITCHED SYSTEMS ; STABILITY under ARBITRARY SWITCHING

INTRODUCTION to SWITCHED SYSTEMS ; STABILITY under ARBITRARY SWITCHING
Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory.

Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory.
Chapter 2: Second-Order Differential Equations

Chapter 2: Second-Order Differential Equations
A New Eigenstructure Fault Isolation Filter Zhenhai Li Supervised by Dr. Imad Jaimoukha Internal Meeting Imperial College, London 4 Aug 2005.

A New Eigenstructure Fault Isolation Filter Zhenhai Li Supervised by Dr. Imad Jaimoukha Internal Meeting Imperial College, London 4 Aug 2005.
Investment Science D.G. Luenberger

Investment Science D.G. Luenberger
ECE 8443 – Pattern Recognition ECE 8527 – Introduction to Machine Learning and Pattern Recognition Objectives: Jensen’s Inequality (Special Case) EM Theorem.

ECE 8443 – Pattern Recognition ECE 8527 – Introduction to Machine Learning and Pattern Recognition Objectives: Jensen’s Inequality (Special Case) EM Theorem.
Probability theory 2010 Main topics in the course on probability theory  Multivariate random variables  Conditional distributions  Transforms  Order.

Probability theory 2010 Main topics in the course on probability theory  Multivariate random variables  Conditional distributions  Transforms  Order.
4.I. Definition 4.II. Geometry of Determinants 4.III. Other Formulas Topics: Cramer’s Rule Speed of Calculating Determinants Projective Geometry Chapter.

4.I. Definition 4.II. Geometry of Determinants 4.III. Other Formulas Topics: Cramer’s Rule Speed of Calculating Determinants Projective Geometry Chapter.
Universidad de La Habana Lectures 5 & 6 : Difference Equations Kurt Helmes 22 nd September - 2nd October, 2008.

Universidad de La Habana Lectures 5 & 6 : Difference Equations Kurt Helmes 22 nd September - 2nd October, 2008.
Lecture 2 Linear Variational Problems (Part II). Conjugate Gradient Algorithms for Linear Variational Problems in Hilbert Spaces 1.Introduction. Synopsis.

Lecture 2 Linear Variational Problems (Part II). Conjugate Gradient Algorithms for Linear Variational Problems in Hilbert Spaces 1.Introduction. Synopsis.
Pariz-Karimpour Apr 2010 1 Chapter 2 Reference: Switched linear systems control and design Zhendong Sun, Shuzhi S. Ge.

Pariz-Karimpour Apr Chapter 2 Reference: Switched linear systems control and design Zhendong Sun, Shuzhi S. Ge.
SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart.

SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart.
Probability theory 2011 Main topics in the course on probability theory  The concept of probability – Repetition of basic skills  Multivariate random.

Probability theory 2011 Main topics in the course on probability theory  The concept of probability – Repetition of basic skills  Multivariate random.
Some Fundamentals of Stability Theory

Some Fundamentals of Stability Theory
4. Convergence of random variables  Convergence in probability  Convergence in distribution  Convergence in quadratic mean  Properties  The law of.

4. Convergence of random variables  Convergence in probability  Convergence in distribution  Convergence in quadratic mean  Properties  The law of.
1 A Lyapunov Approach to Frequency Analysis Tingshu Hu, Andy Teel UC Santa Barbara Zongli Lin University of Virginia.

1 A Lyapunov Approach to Frequency Analysis Tingshu Hu, Andy Teel UC Santa Barbara Zongli Lin University of Virginia.

Similar presentations

Ads by Google