Robust Control Engineering Robust Control Engineering Practical QFT Solutions Mario García-Sanz MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particu- lar use of the MATLAB® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-138-03207-1 (Hardback) This book contains information obtained from authentic and highly regarded sources. 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Isaac Horowitz to Mario García-Sanz (Pamplona, Spain), 2000. “QFT provides one of the most practical robust control design methods available... It is now becoming a much more accessible and valuable method, mainly because the avail- ability of good design software... Prof. Garcia-Sanz has developed a very easy to use design package”. Michael Grimble, founder of the industrial control centre (Glasgow, UK), E-news August 2016. “The most valuable companies in the future won’t ask what problems can be solved with computers alone. Instead, they’ll ask: how can computers help humans solve hard problems”. Peter Thiel, co-fouder of PayPal and Palantir, in his book Zero to One, 2014. “Thanks for making this amazing toolbox!!”. “The QFT Control Tollbox proved to be an excellent tool for robust controller design!!”. “One of the best courses I’ve taken!!”. CWRU students who attended Prof. Garcia-Sanz QFT control courses. “The investment in training will pay off in the long term. 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To Marta, María, Pablo, and Sofía Contents Preface .............................................................................................................................................xv Acknowledgments ......................................................................................................................xix Author ...........................................................................................................................................xxi 1 Introduction .............................................................................................................................1 1.1 The Control Engineer’s Leadership ............................................................................1 1.2 QFT Robust Control Engineering ...............................................................................4 1.3 Book’s Outline ...............................................................................................................7 1.4 Courses and Modules .................................................................................................11 2 QFT Robust Control .............................................................................................................17 2.1 Introduction .................................................................................................................17 2.2 Plant Modeling: Step 1 ...............................................................................................19 2.3 The Nominal Plant: Step 2 .........................................................................................28 2.4 QFT Templates: Step 3 ................................................................................................28 2.5 Stability Speciications: Step 4 ...................................................................................30 2.6 Performance Speciications: Step 5 ...........................................................................31 2.7 QFT Bounds: Steps 6 through 8 ................................................................................34 2.8 Controller Design, G(s)—Loop Shaping: Step 9 ......................................................39 2.9 Preilter Design, F(s): Step 10 ......................................................................................41 2.10 Analysis and Validation: Steps 11 through 13 ........................................................42 2.11 Model Matching ..........................................................................................................54 2.12 Feedforward Control ..................................................................................................59 2.13 PID Control: Design and Tuning with QFT ............................................................63 2.14 Practical Tips ................................................................................................................69 2.14.1 Selection of Speciications .............................................................................69 2.14.1.1 Stability W ......................................................................................69 s 2.14.1.2 Sensitivity δ (ω) ...............................................................................69 3 2.14.1.3 Reference Tracking δ (ω), δ (ω) ................................................71 6-lo 6-up 2.14.2 Loop Shaping—Designing G(s)....................................................................74 2.14.3 Preilter—Designing F(s) ...............................................................................77 2.15 Summary ......................................................................................................................78 2.16 Practice ..........................................................................................................................78 3 Unstable Systems and Control Solutions ........................................................................79 3.1 Introduction .................................................................................................................79 3.2 Understanding Gain and Phase Margins, and W Circles ....................................82 s 3.3 The NSC ........................................................................................................................84 3.4 Nyquist Stability Criterion in the Nichols Chart ....................................................86 3.5 Examples ......................................................................................................................89 3.6 Guidelines to Design Controllers .............................................................................91 3.6.1 Parity Interlacing Property ...........................................................................91 3.6.2 Fundamental Theorem of Feedback Control .............................................92 ix x Contents 3.6.3 Examples .........................................................................................................92 3.7 Analysis of the First Case ..........................................................................................95 3.8 Summary ......................................................................................................................97 3.9 Practice ..........................................................................................................................98 4 Time-Delay and Non-Minimum Phase Systems ...........................................................99 4.1 Time-Delay Systems ...................................................................................................99 4.2 Robust Design of the Smith Predictor ......................................................................103 4.2.1 First Algorithm .............................................................................................107 4.2.2 Second Algorithm ........................................................................................107 4.3 Non-Minimum Phase Systems ...................................................................................114 4.3.1 Analysis .........................................................................................................114 4.3.2 Control Methodology ..................................................................................115 4.4 Summary ....................................................................................................................118 4.5 Practice ........................................................................................................................118 5 Distributed Parameter Systems .......................................................................................119 5.1 Introduction ...............................................................................................................119 5.2 Modeling Approaches for PDE ...............................................................................120 5.3 Generalized DPS Control System Structure .........................................................121 5.4 Extension of QFT to DPS ..........................................................................................124 5.5 Summary ....................................................................................................................134 5.6 Practice ........................................................................................................................134 6 Gain Scheduling/Switching Control Solutions ...........................................................135 6.1 Introduction ...............................................................................................................135 6.2 System Stability Under Switching ..........................................................................135 6.3 Methodology ..............................................................................................................141 6.4 Examples ....................................................................................................................142 6.5 Summary ....................................................................................................................147 6.6 Practice ........................................................................................................................150 7 Nonlinear Dynamic Control ............................................................................................151 7.1 Introduction ...............................................................................................................151 7.2 The Circle Stability Criterion...................................................................................151 7.3 Nonlinear Dynamic Control: One Nonlinearity ..................................................153 7.4 Anti-Windup Solution for PID Controllers ...........................................................156 7.5 Nonlinear Dynamic Control: Several Nonlinearities ..........................................163 7.5.1 Describing Functions ..................................................................................164 7.5.2 Isolines ...........................................................................................................169 7.6 Summary ....................................................................................................................182 7.7 Practice ........................................................................................................................183 8 Multi-Input Multi-Output Systems: Analysis and Control .......................................185 8.1 Introduction ...............................................................................................................185 8.2 Formulation for n × n Systems ................................................................................192 8.3 MIMO Systems—Description and Characteristics ..............................................196 8.3.1 Loop Coupling and Controller Structure .................................................196 8.3.2 Interaction Analysis .....................................................................................197 Contents xi 8.3.3 Multivariable Poles and Zeros ...................................................................199 8.3.4 Directionality................................................................................................201 8.3.5 Gain and Phase ............................................................................................201 8.3.6 Effect of Poles and Zeros .............................................................................202 8.3.7 Disturbance and Noise Signals ..................................................................202 8.3.8 Uncertainty ...................................................................................................202 8.3.9 Stability ..........................................................................................................203 8.4 MIMO QFT Control—Overview ............................................................................204 8.5 Non-Diagonal MIMO QFT—Method 1 ..................................................................207 8.5.1 The Coupling Matrix ...................................................................................208 8.5.2 Tracking .........................................................................................................209 8.5.3 Disturbance Rejection at Plant Input ........................................................211 8.5.4 Disturbance Rejection at Plant Output .....................................................212 8.5.5 The Coupling Elements ...............................................................................213 8.5.6 The Optimal Non-Diagonal Compensator ..............................................214 8.5.6.1 Tracking .........................................................................................215 8.5.6.2 Disturbance Rejection at Plant Input .........................................215 8.5.6.3 Disturbance Rejection at Plant Output .....................................215 8.5.7 The Coupling Effects ...................................................................................215 8.5.7.1 Tracking .........................................................................................216 8.5.7.2 Disturbance Rejection at Plant Input .........................................216 8.5.7.3 Disturbance Rejection at Plant Output .....................................216 8.5.8 Quality Function of the Designed Compensator ....................................217 8.5.9 Design Methodology ...................................................................................218 8.5.10 Some Practical Issues...................................................................................220 8.6 Non-Diagonal MIMO QFT—Method 2 ..................................................................220 8.6.1 Non-Diagonal MIMO QFT Reformulation ...............................................221 8.6.2 Case 1: Reference Tracking and Disturbance Rejection at Plant Output ...........................................................................................................221 8.6.2.1 Methodology .................................................................................222 8.6.3 Case 2: Disturbance Rejection at Plant Input ...........................................226 8.6.4 Stability Conditions and Final Implementation ......................................227 8.6.5 Translating Matrix Performance Speciications ......................................228 8.6.5.1 Case n × n ......................................................................................228 8.6.5.2 Case 2 × 2 .......................................................................................234 8.7 Comparison of Methods 1 and 2 .............................................................................238 8.8 Heat Exchanger, Example 8.1—MIMO QFT Method 1 ........................................240 8.9 Heat Exchanger, Example 8.1—MIMO QFT Method 2 ........................................248 8.10 Summary ....................................................................................................................259 8.11 Practice ........................................................................................................................259 9 Control Topologies .............................................................................................................261 9.1 Introduction ...............................................................................................................261 9.2 Cascade Control Systems .........................................................................................261 9.2.1 Challenge 9.1 .................................................................................................261 9.2.2 Solution 9.1: Cascade Control .....................................................................262 9.3 Feedforward Control Systems .................................................................................263 9.3.1 Challenge 9.2 .................................................................................................263 9.3.2 Solution 9.2a: For Disturbance Rejection ..................................................264 xii Contents 9.3.3 Solution 9.2b: For Reference Tracking. Model Matching ........................266 9.3.4 Solution 9.2c: For Disturbance Rejection and Reference Tracking .......266 9.4 Override Control Systems ........................................................................................267 9.4.1 Challenge 9.3 .................................................................................................267 9.4.2 Solution 9.3: Override Control ....................................................................267 9.5 Ratio Control Systems ..............................................................................................269 9.5.1 Challenge 9.4 .................................................................................................269 9.5.2 Solution 9.4: Ratio Control ..........................................................................269 9.6 Mid-Range Control Systems ....................................................................................270 9.6.1 Challenge 9.5 .................................................................................................270 9.6.2 Solution 9.5: Mid-Range Control ................................................................271 9.7 Load-Sharing Control Systems ...............................................................................272 9.7.1 Challenge 9.6 .................................................................................................272 9.7.2 Solution 9.6: Load-Sharing Control ...........................................................272 9.8 Split-Range Control Systems ...................................................................................273 9.8.1 Challenge 9.7 .................................................................................................273 9.8.2 Solution 9.7 Split-Range Control ................................................................274 9.9 Inferential Control Systems .....................................................................................274 9.9.1 Challenge 9.8 .................................................................................................274 9.9.2 Solution 9.8: Inferential Control .................................................................275 9.10 Auctioneering Control Systems ..............................................................................276 9.10.1 Challenge 9.9 .................................................................................................276 9.10.2 Solution 9.9: Auctioneering Control ..........................................................277 9.11 Summary ....................................................................................................................277 9.12 Practice ........................................................................................................................278 10 Controller Implementation ...............................................................................................279 10.1 Introduction ...............................................................................................................279 10.2 Analog Implementation ...........................................................................................279 10.3 Digital Implementation ............................................................................................284 10.3.1 Sample and Hold ..........................................................................................285 10.3.2 Computer Control Algorithms ..................................................................288 10.3.3 Positional and Velocity Algorithms ..........................................................292 10.3.3.1 Positional Algorithm ....................................................................293 10.3.3.2 Velocity Algorithm .......................................................................293 10.3.4 Switching and Bumpless Algorithms .......................................................294 10.3.5 Pulse Width Modulation .............................................................................294 10.4 Fragility Analysis with QFT ....................................................................................296 10.5 Summary ....................................................................................................................299 10.6 Practice ........................................................................................................................300 Case Study 1: Satellite Control................................................................................................301 Case Study 2: W ind Turbine Control .....................................................................................317 Case Study 3: Wastewater Treatment Plant Control ...........................................................343 Case Study 4: R adio-Telescope Control .................................................................................365 Case Study 5: Attitude and Position Control of Spacecraft Telescopes with Flexible Appendages .........................................................395