Stabilization and control of unstable time delay systems

Lu [5] investigated stabilization of severalpopular unstable including integral delay processes by simple controllers PID or itsspecial cases, established explicit and complete stabiliza

STABILIZATION AND CONTROL OF

UNSTABLE TIME DELAY SYSTEMS

LEE SEE CHEK

NATIONAL UNIVERSITY OF SINGAPORE

2012

STABILIZATION AND CONTROL OF

UNSTABLE TIME DELAY SYSTEMS

LEE SEE CHEK

(B.Eng (Hons., 1st Class) UTM, M.Sc NUS)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2012

Abstract

Control theories and designs for stable delay-free systems have been well developed inresearch society and widely adopted in industry Study of time delay systems remains ahot research topic while the unstable systems are gaining great attention from researchersrecently Control of unstable delay systems is the most challenging and difficult case andbecomes a research frontier in process control, and its progress is yet at a preliminarystage Unlike stable systems, simply detuning the controller is not a trivial solution toachieve stability of the closed loop

PID and lead-lag controllers are the two most popular type of controllers used in dustrial control (often in single loop configuration) In this thesis, the Nyquist stabilitycriterion, combined with some algebraic analysis, is used to perform frequency domainanalysis which then leads to the establishment of stabilizabilty conditions and controllerdesign parameterization Particularly, for all-pole process, and first order processes withzero dynamics, both necessary and sufficient stabilizability conditions are derived andpresented Stabilizability conditions (necessary and/or sufficient) for more complex pro-cesses with zero dynamics are also derived

in-As seen from the PID stabilizability results in the literature, whether a first-orderunstable time delay process can be stabilized or not, depends on the time delay magni-tude When the normalized time delay exceeds 2, a PID controller has no stabilizationsolution In this thesis, a controller of higher order form is developed and stabilization

is achieved for the time delay beyond such bound The method used to derive such astabilizer is either internal model control (IMC) principle or genetic algorithm

Performance of a control system is also as important as stabilization A stabilizedunstable process may exhibit large overshoot, prolonged settling time, poor disturbanceresponse, etc In this thesis, an IMC-like scheme is proposed for better performance andstabilization The scheme can suit a wide range of processes with an arbitrary high-order

of stable lags and permits a larger time delay bound Simulation results show a betterperformance than other comparable schemes from literature

Unstable multivariable (MIMO) systems exists and pose a more difficult control lem than that of a single variable (SISO) case due to the interactions from other loops

In this thesis, a design scheme for multiloop P/PI/PD/PID control has been developedfor a MIMO system that contains a combination of stable and unstable loop The stabi-lizability and controller design for SISO case developed in the earlier part of the thesis isused in MIMO multiloop controller design Gershgorin band principle is used to ensurethe interactions of other loops are within the range such that the stability achieved foreach individual closed loop is still maintained

The schemes and results presented in this thesis have both practical values and retical contributions to the newly emerged research interest in control research of unstablesystem and dynamics

Acknowledgments

I would like to express my thanks to all the tutors, colleagues, friends, and family fortheir support of my research and life During the period of my PhD program, I benefitedand learned much from them, especially when I met obstacles

First of all, I want to thank my supervisor Prof Wang Qing-Guo for his patient ance and advice on my research, writing and presentation throughout my PhD studies.His uncompromising research attitude and stimulating advice helped me in overcomingobstacles in my research Without him, I would not be able to finish the work here Ialso wish to take this opportunity to thank Prof Lee Tong Heng, Prof Ben Chen,Assoc Prof Xiang Cheng and Prof Xu Jianxin for their courses which built up myfundamentals on the theory of control Besides, I am grateful to my colleagues for theirconstant support and encourage

guid-Finally, I would like to express my gratitude to my mother and my family for their sistent support Without their encouragement and love, I may not complete my researchduring the period at the university

1.1 Motivation 1

1.2 Contributions 5

1.3 Organization of the thesis 7

2 PID Stabilization for Unstable All-Pole Time Delay Processes 9 2.1 Introduction 9

2.2 Problem formulation and preliminaries 10

2.3 P/PI controller 15

2.4 PD/PID controller 20

2.5 Conclusion 25

3 PID/lead-lag Stabilization for Unstable Processes with A Zero 26 3.1 Introduction 26

3.2 Problem Formulation and Preliminaries 28

3.3 First-order processes 30

IV

Contents V

3.3.1 P controller 30

3.3.2 PI controller 32

3.3.3 PD/PID controller 38

3.3.4 Lead-lag controller 40

3.4 Second-Order Processes 45

3.5 Higher-Order Processes 47

3.5.1 P/PI controller 47

3.5.2 PD/PID controller 52

3.6 Conclusion 57

4 High-Order Stabilizer for First-Order Unstable Processes with Large Time Delay 58 4.1 Introduction 58

4.2 Problem Formulation and Preliminaries 60

4.2.1 Approach 61

4.3 The IMC Principle 64

4.4 Pole Placement via Genetic Algorithm 68

4.5 Conclusion 74

5 An IMC-like Compensation Scheme for Better Stabilization and Per-formance 76 5.1 Introduction 76

5.2 Control Scheme 79

5.3 Controller Design 83

5.3.1 Inner Loop Controller K 83

Contents VI

5.3.2 Outer Loop Controller C 86

5.4 Internal Stability 88

5.5 Examples 89

5.6 Conclusion 99

6 Multiloop PID Controller Design for Unstable Delay Processes 102 6.1 Introduction 102

6.2 Problem Formulation and Preliminaries 104

6.3 Approach 105

6.3.1 Stable gll(s) [1] 107

6.3.2 Unstable gll(s) 110

6.4 P/PD controller 110

6.4.1 Stability 111

6.4.2 Design 111

6.4.3 Examples 113

6.5 PI/PID controller 115

6.5.1 Stability 115

6.5.2 Design 116

6.5.3 Examples 117

6.6 Conclusion 120

7 Conclusions 121 7.1 Main Findings 121

7.2 Suggestions for further work 123

Contents VII

List of Figures

2.1 Unity output feedback 12

2.2 Stabilization for G(s) of (2.16), ¯L = 0.5 19

2.3 Stabilization for G(s) of (2.16), ¯L = 5.5 24

3.1 P controller 33

3.2 Stabilizable delay bound L for α ≥ 1 37

3.3 PI controller 39

3.4 Lead-lag controller 45

3.5 P/PI controller for G(s) of (3.43) 51

3.6 PD/PID controller for G(s) of (3.43) 56

4.1 C stabilizing ˆG(s) 62

4.2 Multiplicative uncertainty model 63

4.3 Stabilization under the proposed IMC principle 65

4.4 e−jω ¯L− np (jω) d p (jω)  TGCˆ (jω)

plotted over ω 68

4.5 Step response from IMC design 69

4.6 Step response of perturbed G(s) stabilized by C(s) 69

4.7

 e−jω ¯ L− np (jω) d p (jω)  TGCˆ (jω) ... involved and onlyfirst-order delay system is addressed Huang and Chen [2] proved upper bounds on delayfor stabilization by P and PD control X Lu [5] investigated stabilization of severalpopular unstable. .. largetime delay< /p>

A common type of unstable process is in a first order form with time delay Forstabilization by a conventional PID controller, there is a limit in which beyond sometime delay. .. integral) delay processes by simple controllers (PID or itsspecial cases), established explicit and complete stabilizability results in terms of the up-per limit of time delay size, and developed