Part I Fuzzy Modeling |
|
|
|
3 | (18) |
|
1.1 Function Approximation |
|
|
4 | (14) |
|
|
4 | (2) |
|
1.1.2 Approximation Error |
|
|
6 | (2) |
|
1.1.3 Constructing Units in the Fuzzy Models |
|
|
8 | (10) |
|
1.2 Approximation Capabilities of Takagi-Sugeno Fuzzy Models |
|
|
18 | (2) |
|
1.3 Conclusion and Summary |
|
|
20 | (1) |
|
2 Constructing Fuzzy Models from Input-Output Data |
|
|
21 | (38) |
|
2.1 Mosaic or Table Lookup Scheme |
|
|
22 | (4) |
|
2.1.1 Illustrative Example |
|
|
25 | (1) |
|
2.2 Using Gradient Descent |
|
|
26 | (12) |
|
2.2.1 Gradient Updating for Trapezoidal Membership Functions |
|
|
30 | (1) |
|
2.2.2 Gradient Updating for Triangular Membership Functions with Overlap ½ |
|
|
31 | (1) |
|
2.2.3 Gradient Updating for Polynomial Membership Functions |
|
|
32 | (1) |
|
2.2.4 Gradient Updating for Polynomial Membership Functions with Overlap ½ and c i/j = m i/j |
|
|
33 | (1) |
|
2.2.5 Gradient Updating for Gaussian Membership Functions |
|
|
34 | (1) |
|
2.2.6 Illustrative Example |
|
|
34 | (4) |
|
2.3 Using Clustering and Gradient Descent |
|
|
38 | (8) |
|
2.3.1 Algorithm for Mamdani Models |
|
|
38 | (1) |
|
2.3.2 Algorithm for Takagi-Sugeno Models |
|
|
39 | (1) |
|
2.3.3 Illustrative Example |
|
|
39 | (7) |
|
2.4 Using Evolutionary Strategies |
|
|
46 | (4) |
|
2.5 Generalization and Consequences Estimation |
|
|
50 | (5) |
|
2.5.1 Consequence Initialization |
|
|
50 | (1) |
|
2.5.2 Consequence Estimation |
|
|
51 | (4) |
|
2.6 Example of an Industrial Application |
|
|
55 | (3) |
|
|
58 | (1) |
|
3 Fuzzy Modeling with Linguistic Integrity: A Tool for Data Mining |
|
|
59 | (32) |
|
|
59 | (2) |
|
3.2 Structure of the Fuzzy Model |
|
|
61 | (2) |
|
|
63 | (5) |
|
|
68 | (3) |
|
|
71 | (18) |
|
3.5.1 Modeling a Two-Input Nonlinear Function |
|
|
71 | (6) |
|
3.5.2 Modeling of a Three-Input Nonlinear Function |
|
|
77 | (3) |
|
3.5.3 Predicting Chaotic Time Series |
|
|
80 | (5) |
|
3.5.4 Modeling of the Quality Properties on a High-Density Polyethylene (HDPE) Reactor |
|
|
85 | (4) |
|
3.6 Complexity of the AFRELI Algorithm |
|
|
89 | (1) |
|
|
89 | (2) |
|
4 Nonlinear Identification Using Fuzzy Models |
|
|
91 | (32) |
|
4.1 System Identification |
|
|
92 | (1) |
|
4.2 Basic Structure of the Fuzzy System |
|
|
93 | (2) |
|
4.3 Experiment Design for System Identification |
|
|
95 | (3) |
|
4.4 Choosing the Regressors |
|
|
98 | (7) |
|
|
98 | (3) |
|
4.4.2 Regressors Evaluation |
|
|
101 | (4) |
|
4.5 Choosing the Structure |
|
|
105 | (1) |
|
4.6 Calculating the Parameters |
|
|
106 | (2) |
|
|
108 | (1) |
|
4.8 Example Identification of the Box and Jenkins Gas Furnace Data Set |
|
|
109 | (9) |
|
4.9 Identification of Takagi Sugeno Fuzzy Models Using Local Linear Identification |
|
|
118 | (1) |
|
|
119 | (4) |
Part II Fuzzy Control |
|
|
|
123 | (28) |
|
5.1 Model-Free Fuzzy Control |
|
|
124 | (5) |
|
5.1.1 Heuristic Trial-and-Error Design |
|
|
124 | (1) |
|
5.1.2 Design of PID-like Fuzzy Controllers |
|
|
124 | (5) |
|
5.2 Model Based Fuzzy Control |
|
|
129 | (17) |
|
5.2.1 Using Adaptive Methods |
|
|
129 | (5) |
|
5.2.2 Using Direct Synthesis |
|
|
134 | (12) |
|
5.3 Conclusions and Future Perspectives |
|
|
146 | (5) |
|
6 Predictive Control Based on Fuzzy Models |
|
|
151 | (44) |
|
6.1 The Predictive Control Strategy |
|
|
152 | (2) |
|
6.2 Unconstrained Nonlinear Predictive Control |
|
|
154 | (13) |
|
6.2.1 Estimation of the Step Response to Construct G(t) |
|
|
157 | (1) |
|
6.2.2 Example Predictive Control of a CSTR Using a Fuzzy Model |
|
|
158 | (9) |
|
6.3 Constrained Nonlinear Predictive Control |
|
|
167 | (24) |
|
6.3.1 The Constrained Nonlinear Predictive Control Problem |
|
|
168 | (5) |
|
6.3.2 Approach Using Estimated Step Response |
|
|
173 | (4) |
|
6.3.3 Approach Using Takagi-Sugeno Fuzzy Models |
|
|
177 | (1) |
|
6.3.4 Approach Using Takagi-Sugeno Fuzzy Models and Multiple Models in the Predictor |
|
|
178 | (3) |
|
6.3.5 Example Predictive Control of a Steam Generator Using a Fuzzy Model |
|
|
181 | (6) |
|
6.3.6 Example: Nonlinear Predictive Control of a Gas-Phase High-Density Polyethylene (HDPE) Reactor |
|
|
187 | (4) |
|
|
191 | (4) |
|
7 Robust Nonlinear Predictive Control Using Fuzzy Models |
|
|
195 | (12) |
|
|
195 | (1) |
|
7.2 Robust Quadratic Programming |
|
|
196 | (2) |
|
|
198 | (1) |
|
|
199 | (2) |
|
7.5 Formulation of the MPC Problem as a Robust QP |
|
|
201 | (1) |
|
7.6 The Control Algorithm |
|
|
202 | (1) |
|
7.7 Uncertainty Description in Fuzzy Models |
|
|
202 | (3) |
|
7.7.1 Local Uncertainty Described on Each Rule |
|
|
203 | (1) |
|
7.7.2 Using the Active Rules |
|
|
204 | (1) |
|
7.7.3 Using All the Rules |
|
|
204 | (1) |
|
7.7.4 Using the Reachable Set |
|
|
205 | (1) |
|
7.8 Conclusions and Perspectives |
|
|
205 | (2) |
|
8 Conclusions and Future Perspectives |
|
|
207 | (8) |
|
8.1 Conclusions and Summary |
|
|
207 | (3) |
|
8.2 Perspectives and Future Work |
|
|
210 | (5) |
Part III Appendices |
|
|
|
215 | (10) |
|
|
215 | (1) |
|
|
215 | (1) |
|
A.2.1 Some Examples of Membership Functions |
|
|
215 | (1) |
|
A.3 Basic Definitions of Fuzzy Sets |
|
|
216 | (1) |
|
|
216 | (1) |
|
|
216 | (1) |
|
|
217 | (1) |
|
|
217 | (1) |
|
|
217 | (1) |
|
|
217 | (1) |
|
|
217 | (1) |
|
A.4 Operations on Fuzzy Sets |
|
|
217 | (2) |
|
A.4.1 A Is Contained in B |
|
|
217 | (1) |
|
A.4.2 Complement, Negation |
|
|
217 | (1) |
|
|
218 | (1) |
|
|
218 | (1) |
|
|
219 | (1) |
|
A.5.1 Projection of Fuzzy Relations |
|
|
220 | (1) |
|
A.5.2 Composition of Relations |
|
|
220 | (1) |
|
A.6 Approximate Reasoning |
|
|
220 | (1) |
|
|
220 | (1) |
|
A.6.2 Linguistic Variables |
|
|
221 | (1) |
|
A.7 General Structure of a Fuzzy Inference System |
|
|
221 | (4) |
|
A.7.1 Control Rules as a Knowledge Representation |
|
|
221 | (2) |
|
|
223 | (2) |
|
|
225 | (6) |
|
|
225 | (1) |
|
B.2 Using Fuzzy Covariance Matrix: Gustafson and Kessel Algorithm [3] |
|
|
226 | (2) |
|
B.3 Mountain Clustering [4] |
|
|
228 | (3) |
|
C Gradients Used in Identification with Fuzzy Models |
|
|
231 | (12) |
|
C.1 Gradient for the Singleton Consequences |
|
|
231 | (3) |
|
C.1.1 With Trapezoidal Membership Functions |
|
|
232 | (1) |
|
C.1.2 With Polynomial Membership Functions |
|
|
233 | (1) |
|
C.1.3 With Gaussian Membership Functions |
|
|
233 | (1) |
|
C.2 Gradient for the Parameters of the Membership Functions |
|
|
234 | (13) |
|
C.2.1 With Trapezoidal Membership Functions |
|
|
234 | (1) |
|
C.2.2 With Polynomial Membership Functions |
|
|
235 | (2) |
|
C.2.3 With Gaussian Membership Functions |
|
|
237 | (1) |
|
C.2.4 With Triangular Membership Functions with 0.5 Overlap |
|
|
237 | (2) |
|
C.2.5 With Polynomial Membership Functions with 0.5 Overlap |
|
|
239 | (4) |
|
D Discrete Linear Dynamical System Approximation Theorem |
|
|
243 | (4) |
|
E Fuzzy Control for a Continuously Variable Transmission |
|
|
247 | (8) |
|
E.1 Introduction and Process Description |
|
|
247 | (1) |
|
E.2 Performance Specifications |
|
|
248 | (1) |
|
E.3 A Physical Model for the CVT |
|
|
249 | (2) |
|
E.4 Design of the Controller |
|
|
251 | (1) |
|
|
252 | (1) |
|
|
253 | (2) |
References |
|
255 | (6) |
Index |
|
261 | |