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9780817642662

PID Controllers for Time-Delay Systems

by ; ; ;
  • ISBN13:

    9780817642662

  • ISBN10:

    0817642668

  • Format: Hardcover
  • Copyright: 2005-01-01
  • Publisher: Birkhauser

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Summary

The Proportional-Integral-Derivative (PID) controller operates the majority of modern control systems and has applications in many industries; thus any improvement in its design methodology has the potential to have a significant engineering and economic impact. Despite the existence of numerous methods for setting the parameters of PID controllers, the stability analysis of time-delay systems that use PID controllers remains extremely difficult and unclear, and there are very few existing results on PID controller synthesis. Filling a gap in the literature, this book is a presentation of recent results in the field of PID controllers, including their design, analysis, and synthesis. The focus is on linear time-invariant plants, which may contain a time-delay in the feedback loop---a setting that captures many real-world practical and industrial situations. Emphasis is placed on the efficient computation of the entire set of PID controllers achieving stability and various performance specifications---both classical (gain and phase margin) and modern (H-infinity norms of closed-loop transfer functions)---enabling realistic design with several different criteria. Efficiency is important for the development of future software design packages, as well as further capabilities such as adaptive PID design and online implementation. Additional topics and features include: * generalization and use of results---due to Pontryagin and others---to analyze time-delay systems * treatment of robust and nonfragile designs that tolerate perturbations * examination of optimum design, allowing practitioners to find optimal PID controllers with respect to an index * study and comparison of tuning techniques with resepct to their resilience to controller parameter perturbation * a final chapter summarizing the main results and their corresponding algorithms The results presented here are timely given the resurgence of interest in PID controllers and will find widespread application, specifically in the development of computationally efficient tools for PID controller design and analysis. Serving as a catalyst to bridge the theory--practice gap in the control field as well as the classical--modern gap, this monograph is an excellent resource for control, electrical, chemical, and mechanical engineers, as well as researchers in the field of PID controllers.

Table of Contents

Preface xi
1 Introduction 1(20)
1.1 Introduction to Control
1(2)
1.2 The Magic of Integral Control
3(3)
1.3 PID Controllers
6(1)
1.4 Some Current Techniques for PID Controller Design
7(9)
1.4.1 The Ziegler-Nichols Step Response Method
7(2)
1.4.2 The Ziegler-Nichols Frequency Response Method
9(2)
1.4.3 PID Settings using the Internal Model Controller Design Technique
11(2)
1.4.4 Dominant Pole Design: The Cohen-Coon Method
13(1)
1.4.5 New Tuning Approaches
14(2)
1.5 Integrator Windup
16(2)
1.5.1 Setpoint Limitation
16(1)
1.5.2 Back-Calculation and Tracking
17(1)
1.5.3 Conditional Integration
17(1)
1.6 Contribution of this Book
18(1)
1.7 Notes and References
18(3)
2 The Hermite-Biehler Theorem and its Generalization 21(18)
2.1 Introduction
21(1)
2.2 The Hermite-Biehler Theorem for Hurwitz Polynomials
22(5)
2.3 Generalizations of the Hermite-Biehler Theorem
27(10)
2.3.1 No Imaginary Axis Roots
29(2)
2.3.2 Roots Allowed on the Imaginary Axis Except at the Origin
31(4)
2.3.3 No Restriction on Root Locations
35(2)
2.4 Notes and References
37(2)
3 PI Stabilization of Delay-Free Linear Time-Invariant Systems 39(18)
3.1 Introduction
39(1)
3.2 A Characterization of All Stabilizing Feedback Gains
40(11)
3.3 Computation of All Stabilizing PI Controllers
51(5)
3.4 Notes and References
56(1)
4 PID Stabilization of Delay-Free Linear Time-Invariant Systems 57(20)
4.1 Introduction
57(1)
4.2 A Characterization of All Stabilizing PID Controllers
58(9)
4.3 PID Stabilization of Discrete-Time Plants
67(8)
4.4 Notes and References
75(2)
5 Preliminary Results for Analyzing Systems with Time Delay 77(32)
5.1 Introduction
77(1)
5.2 Characteristic Equations for Delay Systems
78(4)
5.3 Limitations of the Pade Approximation
82(7)
5.3.1 Using a First-Order Pade Approximation
83(2)
5.3.2 Using Higher-Order Pade Approximations
85(4)
5.4 The Hermite-Biehler Theorem for Quasi-Polynomials
89(3)
5.5 Applications to Control Theory
92(7)
5.6 Stability of Time-Delay Systems with a Single Delay
99(7)
5.7 Notes and References
106(3)
6 Stabilization of Time-Delay Systems using a Constant Gain Feedback Controller 109(26)
6.1 Introduction
109(1)
6.2 First-Order Systems with Time Delay
110(12)
6.2.1 Open-Loop Stable Plant
112(4)
6.2.2 Open-Loop Unstable Plant
116(6)
6.3 Second-Order Systems with Time Delay
122(12)
6.3.1 Open-Loop Stable Plant
125(4)
6.3.2 Open-Loop Unstable Plant
129(5)
6.4 Notes and References
134(1)
7 PI Stabilization of First-Order Systems with Time Delay 135(26)
7.1 Introduction
135(1)
7.2 The PI Stabilization Problem
136(1)
7.3 Open-Loop Stable Plant
137(13)
7.4 Open-Loop Unstable Plant
150(9)
7.5 Notes and References
159(2)
8 PID Stabilization of First-Order Systems with Time Delay 161(30)
8.1 Introduction
161(1)
8.2 The PID Stabilization Problem
162(2)
8.3 Open-Loop Stable Plant
164(15)
8.4 Open-Loop Unstable Plant
179(10)
8.5 Notes and References
189(2)
9 Control System Design Using the PID Controller 191(32)
9.1 Introduction
191(1)
9.2 Robust Controller Design: Delay-Free Case
192(11)
9.2.1 Robust Stabilization Using a Constant Gain
194(2)
9.2.2 Robust Stabilization Using a PI Controller
196(3)
9.2.3 Robust Stabilization Using a PID Controller
199(4)
9.3 Robust Controller Design: Time-Delay Case
203(10)
9.3.1 Robust Stabilization Using a Constant Gain
204(1)
9.3.2 Robust Stabilization Using a PI Controller
205(3)
9.3.3 Robust Stabilization Using a PID Controller
208(5)
9.4 Resilient Controller Design
213(4)
9.4.1 Determining kappa, T, and L from Experimental Data
213(1)
9.4.2 Algorithm for Computing the Largest Ball Inscribed Inside the PID Stabilizing Region
214(3)
9.5 Time Domain Performance Specifications
217(5)
9.6 Notes and References
222(1)
10 Analysis of Some PID Tuning Techniques 223(20)
10.1 Introduction
223(1)
10.2 The Ziegler-Nichols Step Response Method
224(5)
10.3 The CHR Method
229(4)
10.4 The Cohen-Coon Method
233(4)
10.5 The IMC Design Technique
237(4)
10.6 Summary
241(1)
10.7 Notes and References
241(2)
11 PID Stabilization of Arbitrary Linear Time-Invariant Systems with Time Delay 243(22)
11.1 Introduction
243(1)
11.2 A Study of the Generalized Nyquist Criterion
244(4)
11.3 Problem Formulation and Solution Approach
248(2)
11.4 Stabilization Using a Constant Gain Controller
250(3)
11.5 Stabilization Using a PI Controller
253(3)
11.6 Stabilization Using a PID Controller
256(7)
11.7 Notes and References
263(2)
12 Algorithms for Real and Complex PID Stabilization 265(32)
12.1 Introduction
265(1)
12.2 Algorithm for Linear Time-Invariant Continuous-Time Systems
266(10)
12.3 Discrete-Time Systems
276(1)
12.4 Algorithm for Continuous-Time First-Order Systems with Time Delay
277(7)
12.4.1 Open-Loop Stable Plant
279(1)
12.4.2 Open-Loop Unstable Plant
280(4)
12.5 Algorithms for PID Controller Design
284(11)
12.5.1 Complex PID Stabilization Algorithm
285(2)
12.5.2 Synthesis of Hoc PID Controllers
287(4)
12.5.3 PID Controller Design for Robust Performance
291(2)
12.5.4 PID Controller Design with Guaranteed Gain and Phase Margins
293(2)
12.6 Notes and References
295(2)
A Proof of Lemmas 8.3, 8.4, and 8.5 297(10)
A.1 Preliminary Results
297(4)
A.2 Proof of Lemma 8.3
301(1)
A.3 Proof of Lemma 8.4
302(1)
A.4 Proof of Lemma 8.5
303(4)
B Proof of Lemmas 8.7 and 8.9 307(6)
B.1 Proof of Lemma 8.7
307(1)
B.2 Proof of Lemma 8.9
308(5)
C Detailed Analysis of Example 11.4 313(10)
References 323(6)
Index 329

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