Robust Multi-loop PI/PID Controller Design for Multivariable Processes

국문 초록 (Abstract)

다중루프 비례-적분/비례-적분-미분(PI/PID) 제어기는 그것이 가진 다양한 장점들 - 산업적 전망, 효율, 생산성 및 품질 향상 – 로 인하여 현재 산업용 제어기로 가장 널리 사용되고 있는 제어기 중의 하나이다. 다중루프 제어시스템은 보다 효율적인 운전전략을 위한 요구사항을 충족하면서 이의 구현을 위한 진보된 도구의 사용을 가능하게 한다. 또한 명확한 이론적 근거를 가지고 있어 이해가 쉬우며 다변수 제어시스템보다 조율을 위한 변수가 적어 경제적 이익을 극대화하고 실제적 응용에 필수적인 요건인 루프 고장에의 강건성 등을 갖추고 있다. 이러한 이유로 다중루프 PI/PID 제어기의 설계를 위한 수많은 방법들이 수년에 걸쳐 공정제어 관련 문헌에 제시되어 왔다. 그 중에서도, 잘 알려져 있는 IMC-PID 접근방법은 설계와 구현 상의 효율성과 강력한 구도로 빈번히 사용되고 있는 방법이다. 이 방법은 강건성과 성능 간의 절충을 한 개의 설계변수, 즉, IMC 필터 상수 또는 닫힌루프 시상수를 조정하여 달성할 수 있기 때문에 제어 엔지니어와 연구자들에게 매우 매력적이다. 이것은 IMC-PID 접근방법이 가지는 매우 유용한 특성으로서 다중루프 제어시스템이 강건 안정성 조건을 만족하도록 조율될 수 있도록 해준다. IMC-PID의 접근법과 비슷한 특징을 가지는 다중루프 PI/PID 제어 시스템을 위한 직접 합성법 또한 많은 저자에 의해 연구되고 있다.
이 논문에서는 다중루프 PI/PID 제어기의 강건한 설계를 위한 새로운 해석적 방법이 일반화된 다중루프 IMC-PIC 접근법과 직접 합성법을 근간으로 제안되었다. 제안된 방법들은 다중시간지연을 가지는 다중변수 공정에서 원하는 닫힌루프 응답을 달성하고자 하는데 목적이 있다. 입력/출력 변수들 간의 상호작용은 다중루프 되먹임 제어 루프에서 공통적으로 접하는 현상으로서 다중루프 시스템에서의 만족스러운 성능 구현에 심각한 장애로 작용할 수 있는 것으로 알려져 있다. 이 장애사항을 극복하기 위한 방안으로서, 가중 합 Mp 기준과 다중루프 Ms 기준 등과 같은 몇 가지 방법들이 상호작용 효과를 완전하게 반영하기 위하여 제안되었다.
강건 안정성과 성능도 본 연구에서 고려된 두 가지 중요한 요소이다. 기본적으로, 실제 시스템과 그 모델 간의 차이점은 닫힌루프의 불안정성을 야기할 수 있다. 제어시스템이 다양한 불확실성 하에서도 만족스럽게 제어될 때 강건하다고 한다. 제안된 다중루프 제어 시스템은 강건 안정성과 성능을 고려하여 설계되었다.
제안된 방법의 우수성을 증명하기 위해 여러가지 시뮬레이션을 수행하였으며 그 결과 제안된 방법에 의하여 설계된 다중루프 PI/PID 제어시스템은 빠르고 균형 잡힌, 그리고 최소의 Integral Absolute Error (IAE) 값을 가지는 강건한 응답을 가져옴을 확인하였다.

다국어 초록 (Multilingual Abstract)

In recent years, multi-loop proportional-integral/proportional-integral-derivative (PI/PID) controllers have become one of the most widely used industrial controllers. The reason for this popularity is due to the industrial perspective, efficiency, improved productivity, and product quality goals since multi-loop control systems satisfy all demands for more effective operational strategies to be integrated in the production line and allows more advanced tools to be implemented. In addition, they are also self-explanatory and easily understood by their sound theoretical foundation, and require fewer parameters to tune than multivariable controllers, which help to maximize economic benefits. Moreover, loop failure tolerance can be easily obtained with the multi-loop control system, which is very important for practical applications. Therefore, various design methods of multi-loop PI/PID controllers have been suggested in the literature over the years. Among them, the well-known IMC-PID approach is often utilized because of its powerful framework and effectiveness for the designing and implementation. It is attracted to control engineers and academics because the tradeoffs between robustness and performance are directly related to one design parameter as the time constant of IMC filter or the closed-loop time constant, equivalently. This is very useful characteristic of the IMC-PID approach, which guarantees multi-loop control systems to be tuned satisfying the robust stability condition. Furthermore, the direct synthesis for the design of multi-loop PI/PID controllers that has the similar features as the IMC-PID approach is also considered by many authors.
In this thesis, the novel analytical methods based on the generalized IMC-PID approach and direct synthesis are proposed for the different states of the robust design of multi-loop PI/PID controllers. The proposed methods are aimed at achieving the desired closed-loop responses for multivariable processes with multiple time delays. It is known that the interactions between input/output variables are a very common phenomenon encountered in control loops, which can be a serious obstacle for achieving a good performance of the multi-loop control system. To overcome this impediment, several techniques are suggested for taking interaction effects fully into account such as weighted sum Mp criterion, multi-loop Ms criterion, etc.
Stability and performance robustness are also two important issues that should be considered in this work. Basically, the gap between the actual system and its model may be major cause of the closed-loop instability. The control system is robust if it performs satisfactorily under parameter variations and various possible uncertainties. Therefore, the proposed multi-loop control system is essentially designed by considering robust stability and performance analysis.
Several simulation studies are carried out to demonstrate the superiority of proposed methods. The simulation results show that multi-loop PI/PID control systems designed by proposed methods achieve fast, well-balanced, and robust responses with minimum integral absolute error (IAE) values.

목차 (Table of Contents)

• Title Page
• Acknowledgements ...................................................................................... i
• Abstract ........................................................................................................ iii
• Contents ....................................................................................................... vi
• List of Tables ............................................................................................... xii
• List of Figures .............................................................................................. xv
• Chapter 1 Introduction
• 1.1. Motivations ...................................................................................... 1
• 1.2. Contributions of the Thesis ............................................................. 6
• 1.3. Thesis Outline .................................................................................. 7
• Chapter 2 Analytical Design of Robust Multi-loop PID Controllers Based
• on the Generalized IMC-PID Method with Mp Criterion
• 2.1. Introduction .................................................................................... 14
• 2.2. Generalized IMC-PID Approach for Multi-loop PID Controller
• Design .................................................................................................... 15
• 2.3. Weighted Sum Mp Criterion for Multi-loop Controller Tuning. ... 19
• 2.4. Integral Absolute Error Index. ........................................................ 22
• vii
• 2.5. Simulation Study. ........................................................................... 23
• 2.6. Conclusions... ................................................................................. 35
• 2.7. References... ................................................................................... 35
• Chapter 3 Analytical Design of Robust Multi-loop PI/PID Controller for
• Enhanced Disturbance Rejection in Multivariable Processes
• 3.1. Introduction .................................................................................... 37
• 3.2. Generalized IMC-PID Method Based on Disturbance Rejection for
• Multi-Loop PI/PID Controller Design ................................................... 39
• 3.3. Design of Multi-loop Proportional Gain ........................................ 42
• 3.4. Design of Multi-loop Integral Time Constant ............................... 43
• 3.5. Ms Criterion for Enhance the Robustness of the Multi-Loop
• Control System ..................................................................................... 44
• 3.6. Robust Performance Analysis ....................................................... 46
• 3.7. Simulation Study ........................................................................... 47
• 3.8. Conclusions .................................................................................... 59
• 3.7. References... ................................................................................... 59
• Chapter 4 Robust Multi-loop PI/PID Controller Design Based on Direct
• Synthesis for Interacting Multivariable Processes
• viii
• 4.1. Introduction .................................................................................... 61
• 4.2. Multi-loop Feedback Controller Design for the Desired
• Closed-loop Responses. ......................................................................... 63
• 4.3. Reduction to the Multi-loop PI/PID Controller .............................. 67
• 4.4. Example of Two-input, Two-output (TITO) Case ........................ 68
• 4.5. Multi-loop Control System Performance and Robustness
• Analysis ................................................................................................ 70
• 4.5.1. Integral Absolute Error Index ............................................... 70
• 4.5.2. Total Variation ...................................................................... 70
• 4.5.3. Robust Stability Analysis ...................................................... 70
• 4.6. Simulation Study ........................................................................... 71
• 4.7. Conclusions .................................................................................... 83
• 4.8. References... ................................................................................... 84
• Chapter 5 Analytical Design of Multi-loop PI Controllers for Interactive
• Multivariable Processes
• 5.1. Introduction .................................................................................... 88
• 5.2. Multi-loop PI Controller Design. ................................................... 89
• 5.3. Analytical Multi-loop PI Controller Tuning Rules ........................ 95
• 5.4. Multi-loop Considerations ............................................................. 98
• ix
• 5.5. Performance and Robustness Measure .......................................... 100
• 5.5.1. Integral Absolute Error Index .............................................. 100
• 5.5.2. Multi-loop Maximum Sensitivity (Ms) to Modeling Error . 100
• 5.6. Simulation Study .......................................................................... 102
• 5.7. Conclusions ................................................................................... 112
• 5.8. References... .................................................................................. 112
• Chapter 6 Effective Open-loop Transfer Function for Multi-loop PI/PID
• Controller Design
• 6.1. Introduction ................................................................................... 115
• 6.2. Preliminaries. ................................................................................. 116
• 6.3. Effective Open-loop Transfer Function and Dynamic Relative
• Gain ...................................................................................................... 118
• 6.4. Reduced EOTF for Controller Design ........................................... 120
• 6.5. Multi-loop PID Controller Design ................................................. 123
• 6.6. Simulation Study .......................................................................... 126
• 6.6.1. Performance Index ............................................................... 126
• 6.6.2. Robustness Index ................................................................. 126
• 6.7. Conclusions ................................................................................... 139
• 6.8. References... .................................................................................. 140
• x
• Chapter 7 Advanced Design of Robust Multi-loop PI Controller for
• Multivariable Processes
• 7.1. Introduction ................................................................................... 144
• 7.2. Multi-loop PI Controller Design via Direct Synthesis and
• IMC-PID Approach .............................................................................. 144
• 7.3. Robust Stability Analysis for Multi-loop System ......................... 145
• 7.4. Simulation Study ........................................................................... 148
• 7.5. Conclusions ................................................................................... 155
• 7.6. References... .................................................................................. 155
• Chapter 8. Conclusions and Further Work
• 8.1. Conclusions ................................................................................... 157
• 8.2. Further Work... .............................................................................. 159
• Notation and Nomenclature ..................................................................... 160
• References ................................................................................................. 163
• Appendix A. Proof of Eq. (6.5) ............................................................... 168
• Appendix B. Matrix Operations and Norms
• B.1. Matrices ........................................................................................ 170
• xi
• B.2. Eigenvalues and Eigenvectors ...................................................... 171
• B.3. Singular Value Decomposition ..................................................... 173
• B.4. Relative Gain Array ...................................................................... 173
• B.5. Vector and Matrix Norms ............................................................ 174
• B.5.1. Vector Norms ........................................................................ 175
• B.5.2. Matrix Norms ........................................................................ 175
• Appendix C. Vectors, Matrices, and Norms in Matlab ........................ 177
• Curriculum Vitae ...................................................................................... 179
• Abstract in Korean ................................................................................... 185