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E-grāmata: Analysis and Identification of Time-Invariant Systems, Time-Varying Systems, and Multi-Delay Systems using Orthogonal Hybrid Functions: Theory and Algorithms with MATLAB(R)

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Analysis and Identification of Time-Invariant Systems, Time-Varying Systems, and Multi-Delay Systems using Orthogonal Hybrid Functions: Theory and Algorithms with MATLAB(R)Palielināt
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This book introduces a new set of orthogonal hybrid functions (HF) which approximates time functions in a piecewise linear manner which is very suitable for practical applications.





The book presents an analysis of different systems namely, time-invariant system, time-varying system, multi-delay systems---both homogeneous and non-homogeneous type- and the solutions are obtained in the form of discrete samples. The book also investigates system identification problems for many of the above systems. The book is spread over 15 chapters and contains 180 black and white figures, 18 colour figures, 85 tables and 56 illustrative examples. MATLAB codes for many such examples are included at the end of the book.
1 Non-sinusoidal Orthogonal Functions in Systems and Control 1(24)
1.1 Orthogonal Functions and Their Properties
2(1)
1.2 Different Types of Non-sinusoidal Orthogonal Functions
3(11)
1.2.1 Haar Functions
3(2)
1.2.2 Rademacher Functions
5(1)
1.2.3 Walsh Functions
6(1)
1.2.4 Block Pulse Functions (BPF)
7(1)
1.2.5 Slant Functions
7(2)
1.2.6 Delayed Unit Step Functions (DUSF)
9(1)
1.2.7 General Hybrid Orthogonal Functions (GHOF)
10(1)
1.2.8 Variants of Block Pulse Functions
11(1)
1.2.9 Sample-and-Hold Functions (SHF)
11(1)
1.2.10 Triangular Functions (TF)
12(1)
1.2.11 Non-optimal Block Pulse Functions (NOBPF)
13(1)
1.3 Walsh Functions in Systems and Control
14(3)
1.4 Block Pulse Functions in Systems and Control
17(1)
1.5 Triangular Functions (TF) in Systems and Control
18(1)
1.6 A New Set of Orthogonal Hybrid Functions (HF): A Combination of Sample-and-Hold Functions (SHF) and Triangular Functions (TF)
19(1)
References
19(6)
2 The Hybrid Function (HF) and Its Properties 25(24)
2.1 Brief Review of Block Pulse Functions (BPF)
25(1)
2.2 Brief Review of Sample-and-Hold Functions (SHF)
26(1)
2.3 Brief Review of Triangular Functions (TF)
27(1)
2.4 Hybrid Function (HF): A Combination of SHF and TF
28(2)
2.5 Elementary Properties of Hybrid Functions
30(3)
2.5.1 Disjointedness
30(1)
2.5.2 Orthogonality
31(1)
2.5.3 Completeness
32(1)
2.6 Elementary Operational Rules
33(14)
2.6.1 Addition of Two Functions
33(4)
2.6.2 Subtraction of Two Functions
37(2)
2.6.3 Multiplication of Two Functions
39(5)
2.6.4 Division of Two Functions
44(3)
2.7 Qualitative Comparison of BPF, SHF, TF and HF
47(1)
2.8 Conclusion
47(1)
References
48(1)
3 Function Approximation via Hybrid Functions 49(38)
3.1 Function Approximation via Block Pulse Functions (BPF)
49(2)
3.1.1 Numerical Examples
50(1)
3.2 Function Approximation via Hybrid Functions (HF)
51(1)
3.3 Algorithm of Function Approximation via HF
52(2)
3.3.1 Numerical Examples
52(2)
3.4 Comparison Between BPF and HF Domain Approximations
54(2)
3.5 Approximation of Discontinuous Functions
56(11)
3.5.1 Modified HF Domain Approach for Approximating Functions with Jump Discontinuities
58(4)
3.5.2 Numerical Examples
62(5)
3.6 Function Approximation: HF Versus Other Methods
67(7)
3.7 Mean Integral Square Error (MISE) for HF Domain Approximations
74(5)
3.7.1 Error Estimate for Sample-and-Hold Function Domain Approximation
75(1)
3.7.2 Error Estimate for Triangular Function Domain Approximation
76(3)
3.8 Comparison of Mean Integral Square Error (MISE) for Function Approximation via HFc and Hm Approaches
79(5)
3.9 Conclusion
84(2)
References
86(1)
4 Integration and Differentiation Using HF Domain Operational Matrices 87(28)
4.1 Operational Matrices for Integration
87(9)
4.1.1 Integration of Sample-and-Hold Functions
88(4)
4.1.2 Integration of Triangular Functions
92(4)
4.2 Integration of Functions Using Operational Matrices
96(4)
4.2.1 Numerical Examples
97(3)
4.3 Operational Matrices for Differentiation
100(6)
4.3.1 Differentiation of Time Functions Using Operational Matrices
100(3)
4.3.2 Numerical Examples
103(3)
4.4 Accumulation of Error for Subsequent Integration-Differentiation (I-D) Operation in HF Domain
106(4)
4.5 Conclusion
110(3)
References
113(2)
5 One-Shot Operational Matrices for Integration 115(26)
5.1 Integration Using First Order HF Domain Integration Matrices
116(1)
5.2 Repeated Integration Using First Order HF Domain Integration Matrices
117(1)
5.3 One-Shot Integration Operational Matrices for Repeated Integration
118(9)
5.3.1 One-Shot Operational Matrices for Sample-and-Hold Functions
119(3)
5.3.2 One-Shot Operational Matrices for Triangular Functions
122(4)
5.3.3 One-Shot Integration Operational Matrices in HF Domain: A Combination of SHF Domain and TF Domain One-Shot Operational Matrices
126(1)
5.4 Two Theorems
127(2)
5.5 Numerical Examples
129(9)
5.5.1 Repeated Integration Using First Order Integration Matrices
130(2)
5.5.2 Higher Order Integration Using One-Shot Operational Matrices
132(4)
5.5.3 Comparison of Two Integration Methods Involving First, Second and Third Order Integrations
136(2)
5.6 Conclusion
138(2)
References
140(1)
6 Linear Differential Equations 141(26)
6.1 Solution of Linear Differential Equations Using HF Domain Differentiation Operational Matrices
142(3)
6.1.1 Numerical Examples
143(2)
6.2 Solution of Linear Differential Equations Using HF Domain Integration Operational Matrices
145(7)
6.2.1 Numerical Examples
148(4)
6.3 Solution of Second Order Linear Differential Equations
152(7)
6.3.1 Using HF Domain First Order Integration Operational Matrices
152(2)
6.3.2 Using HF Domain One-Shot Integration Operational Matrices
154(1)
6.3.3 Numerical Examples
155(4)
6.4 Solution of Third Order Linear Differential Equations
159(6)
6.4.1 Using HF Domain First Order Integration Operational Matrices
160(2)
6.4.2 Using HF Domain One-Shot Integration Operational Matrices
162(2)
6.4.3 Numerical Examples
164(1)
6.5 Conclusion
165(1)
References
166(1)
7 Convolution of Time Functions 167(18)
7.1 The Convolution Integral
168(1)
7.2 Convolution of Basic Components of Hybrid Functions
169(7)
7.2.1 Convolution of Two Elementary Sample-and-Hold Functions
170(1)
7.2.2 Convolution of Two Sample-and-Hold Function Trains
171(1)
7.2.3 Convolution of an Elementary Sample-and-Hold Function and an Elementary Triangular Function
172(1)
7.2.4 Convolution of a Triangular Function Train and a Sample-and-Hold Function Train
173(1)
7.2.5 Convolution of Two Elementary Triangular Functions
174(1)
7.2.6 Convolution of Two Triangular Function Trains
174(2)
7.3 Convolution of Two Time Functions in HF Domain
176(5)
7.4 Numerical Example
181(2)
7.5 Conclusion
183(1)
References
184(1)
8 Time Invariant System Analysis: State Space Approach 185(36)
8.1 Analysis of Non-homogeneous State Equations
186(11)
8.1.1 Solution from Sample-and-Hold Function Vectors
188(7)
8.1.2 Solution from Triangular Function Vectors
195(2)
8.1.3 Numerical Examples
197(1)
8.2 Determination of Output of a Non-homogeneous System
197(5)
8.2.1 Numerical Examples
201(1)
8.3 Analysis of Homogeneous State Equation
202(6)
8.3.1 Numerical Examples
202(6)
8.4 Determination of Output of a Homogeneous System
208(4)
8.4.1 Numerical Examples
212(1)
8.5 Analysis of a Non-homogeneous System with Jump Discontinuity at Input
212(7)
8.5.1 Numerical Example
215(4)
8.6 Conclusion
219(1)
References
220(1)
9 Time Varying System Analysis: State Space Approach 221(20)
9.1 Analysis of Non-homogeneous Time Varying State Equation
222(10)
9.1.1 Numerical Examples
230(2)
9.2 Determination of Output of a Non-homogeneous Time Varying System
232(1)
9.3 Analysis of Homogeneous Time Varying State Equation
233(5)
9.3.1 Numerical Examples
234(4)
9.4 Determination of Output of a Homogeneous Time Varying System
238(1)
9.5 Conclusion
238(1)
References
239(2)
10 Multi-delay System Analysis: State Space Approach 241(30)
10.1 HF Domain Approximation of Function with Time Delay
241(5)
10.1.1 Numerical Examples
245(1)
10.2 Integration of Functions with Time Delay
246(2)
10.2.1 Numerical Examples
247(1)
10.3 Analysis of Non-homogeneous State Equations with Delay
248(18)
10.3.1 Numerical Examples
260(6)
10.4 Analysis of Homogeneous State Equations with Delay
266(3)
10.4.1 Numerical Examples
267(2)
10.5 Conclusion
269(1)
References
270(1)
11 Time Invariant System Analysis: Method of Convolution 271(18)
11.1 Analysis of an Open Loop System
271(5)
11.1.1 Numerical Examples
272(4)
11.2 Analysis of a Closed Loop System
276(10)
11.2.1 Numerical Examples
283(3)
11.3 Conclusion
286(1)
References
287(2)
12 System Identification Using State Space Approach: Time Invariant Systems 289(18)
12.1 Identification of a Non-homogeneous System
289(5)
12.1.1 Numerical Examples
291(3)
12.2 Identification of Output Matrix of a Non-homogeneous System
294(3)
12.2.1 Numerical Examples
295(2)
12.3 Identification of a Homogeneous System
297(1)
12.4 Identification of Output Matrix of a Homogeneous System
297(1)
12.5 Identification of a Non-homogeneous System with Jump Discontinuity at Input
297(7)
12.5.1 Numerical Examples
299(5)
12.6 Conclusion
304(1)
References
305(2)
13 System Identification Using State Space Approach: Time Varying Systems 307(12)
13.1 Identification of a Non-homogeneous System
307(4)
13.1.1 Numerical Examples
309(2)
13.2 Identification of a Homogeneous System
311(5)
13.2.1 Numerical Examples
311(5)
13.3 Conclusion
316(1)
References
317(2)
14 Time Invariant System Identification: Via 'Deconvolution' 319(12)
14.1 Control System Identification Via 'Deconvolution'
319(10)
14.1.1 Open Loop Control System Identification
320(3)
14.1.2 Closed Loop Control System Identification
323(6)
14.2 Conclusion
329(1)
References
330(1)
15 System Identification: Parameter Estimation of Transfer Function 331(26)
15.1 Transfer Function Identifications
331(1)
15.2 Pade Approximation
332(2)
15.3 Parameter Estimation of the Transfer Function of a Linear System
334(16)
15.3.1 Using Block Pulse Functions
336(4)
15.3.2 Using Non-optimal Block Pulse Functions (NOBPF)
340(2)
15.3.3 Using Triangular Functions (TF)
342(3)
15.3.4 Using Hybrid Functions (HF)
345(4)
15.3.5 Solution in SHF Domain from the HF Domain Solution
349(1)
15.4 Comparative Study of the Parameters of the Transfer Function Identified via Different Approaches
350(2)
15.5 Comparison of Errors for BPF, NOBPF, TF, HF and SHF Domain Approaches
352(2)
15.6 Conclusion
354(1)
References
355(2)
Appendix A: Introduction to Linear Algebra 357(10)
Appendix B: Selected MATLAB Programs 367(58)
Index 425
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