PID controllers for time-delay systems

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 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 respect to their resilience to controller parameter perturbation \* a final chapter summarizing the main results and their corresponding proposed 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. Cover......Page 1 Control Engineering Series......Page 3 Title Page ......Page 4 Publication Data ......Page 5 Contents......Page 7 1.1 Introduction to Control......Page 14 1.2 The Magic of Integral Control ......Page 16 1.3 PID Controllers......Page 19 1.4.1 The Ziegler-Nichols Step Response Method......Page 20 1.4.2 The Ziegler-Nichols Frequency Response Method......Page 22 1.4.3 PID Settings using the Internal Model Controller Design Technique......Page 24 1.4.4 Dominant Pole Design: The Cohen-Coon Method......Page 26 1.4.5 New Tuning Approaches......Page 27 1.5.1 Setpoint Limitation......Page 29 1.5.3 Conditional Integration......Page 30 1.7 Notes and References......Page 31 2.1 Introduction......Page 33 2.2 The Hermite-Biehler Theorem for Hurwitz Polynomials......Page 34 2.3 Generalizations of the Hermite-Biehler Theorem......Page 39 2.3.1 No Imaginary Axis Roots......Page 41 2.3.2 Roots Allowed on the Imaginary Axis Except at the Origin......Page 43 2.3.3 No Restriction on Root Locations......Page 47 2.4 Notes and References......Page 49 3.1 Introduction......Page 50 3.2 A Characterization of All Stabilizing Feedback Gains......Page 51 3.3 Computation of All Stabilizing PI Controllers......Page 62 3.4 Notes and References......Page 67 4.1 Introduction......Page 68 4.2 A Characterization of All Stabilizing PID Controllers......Page 69 4.3 PID Stabilization of Discrete-Time Plants......Page 78 4.4 Notes and References......Page 86 5.1 Introduction......Page 87 5.2 Characteristic Equations for Delay Systems......Page 88 5.3 Limitations of the Pade Approximation......Page 92 5.3.1 Using a First-Order Pade Approximation......Page 93 5.3.2 Using Higher-Order Pade Approximations......Page 95 5.4 The Hermite-Biehler Theorem for Quasi-Polynomials......Page 99 5.5 Applications to Control Theory......Page 102 5.6 Stability of Time-Delay Systems with a Single Delay......Page 109 5.7 Notes and References......Page 116 6.1 Introduction......Page 118 6.2 First-Order Systems with Time Delay......Page 119 6.2.1 Open-Loop Stable Plant......Page 121 6.2.2 Open-Loop Unstable Plant......Page 125 6.3 Second-Order Systems with Time Delay......Page 131 6.3.1 Open-Loop Stable Plant......Page 134 6.3.2 Open-Loop Unstable Plant......Page 138 6.4 Notes and References......Page 143 7.1 Introduction......Page 144 7.2 The PI Stabilization Problem......Page 145 7.3 Open-Loop Stable Plant......Page 146 7.4 Open-Loop Unstable Plant......Page 159 7.5 Notes and References......Page 168 8.1 Introduction......Page 169 8.2 The PID Stabilization Problem......Page 170 8.3 Open-Loop Stable Plant......Page 172 8.4 Open-Loop Unstable Plant......Page 187 8.5 Notes and References......Page 197 9.1 Introduction......Page 199 9.2 Robust Controller Design: Delay-Free Case......Page 200 9.2.1 Robust Stabilization Using a Constant Gain......Page 202 9.2.2 Robust Stabilization Using a PI Controller......Page 204 9.2.3 Robust Stabilization Using a PID Controller......Page 207 9.3 Robust Controller Design: Time-Delay Case......Page 211 9.3.1 Robust Stabilization Using a Constant Gain......Page 212 9.3.2 Robust Stabilization Using a PI Controller......Page 213 9.3.3 Robust Stabilization Using a PID Controller......Page 216 9.4.1 Determining k, T, and L from Experimental Data......Page 221 9.4.2 Algorithm for Computing the Largest Ball Inscribed Inside the PID Stabilizing Region......Page 222 9.5 Time Domain Performance Specifications......Page 225 9.6 Notes and References......Page 230 10.1 Introduction......Page 231 10.2 The Ziegler-Nichols Step Response Method......Page 232 10.3 The CHR Method......Page 237 10.4 The Cohen-Coon Method......Page 241 10.5 The IMC Design Technique......Page 245 10.7 Notes and References......Page 249 11.1 Introduction......Page 250 11.2 A Study of the Generalized Nyquist Criterion......Page 251 11.3 Problem Formulation and Solution Approach......Page 255 11.4 Stabilization Using a Constant Gain Controller......Page 257 11.5 Stabilization Using a PI Controller......Page 260 11.6 Stabilization Using a PID Controller......Page 263 11.7 Notes and References......Page 270 12.1 Introduction......Page 271 12.2 Algorithm for Linear Time-Invariant Continuous-Time Systems......Page 272 12.3 Discrete-Time Systems......Page 282 12.4 Algorithm for Continuous-Time First-Order Systems with Time Delay......Page 283 12.4.1 Open-Loop Stable Plant......Page 285 12.4.2 Open-Loop Unstable Plant......Page 286 12.5 Algorithms for PID Controller Design......Page 290 12.5.1 Complex PID Stabilization Algorithm......Page 291 12.5.2 Synthesis of Hoc PID Controllers......Page 293 12.5.3 PID Controller Design for Robust Performance......Page 297 12.5.4 PID Controller Design with Guaranteed Gain and Phase Margins......Page 299 12.6 Notes and References......Page 301 A.1 Preliminary Results......Page 302 A.2 Proof of Lemma 8.3......Page 306 A.3 Proof of Lemma 8.4......Page 307 A.4 Proof of Lemma 8.5......Page 308 B.1 Proof of Lemma 8.7......Page 311 B.2 Proof of Lemma 8.9......Page 312 C Detailed Analysis of Example 11.4......Page 316 References......Page 326 Index......Page 331 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 that 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, which is important for the development of future software design packages, as well as further capabilities such as adaptive PID design and online implementation. TOC:Preface.- Introduction.- The Hermite-Biehler Theorem and Its Generalization.- PI Stabilization of Delay-Free Linear Time-Invariant Systems.- PID Stabilization of Delay-Free Linear Time-Invariant Systems.- Preliminary Results for Analyzing Systems with Time Delay.- Stabilization of Time-Delay Systems Using a Constant Gain Feedback Controller.- PI Stabilization of First-Order Systems with Time Delay.- PID Stabilization of First-Order Systems with Time Delay.- Control System Design Using the PID Controller.- Analysis of Some PID Tuning Techniques.- PID Stabilization of Arbitrary Linear Time-Invariant Systems with Time Delay.- Algorithms for Real and Complex PID Stabilization.- A Proof of Lemmas 8.3, 8.4, and 8.5.- B Proof of Lemmas 8.7 and 8.9.- C Detailed Analysis of Example 11.4.- References.- Index This monograph presents our recent results on the proportional-integr- derivative (PID) controller and its design, analysis, and synthesis. The fo­ cus is on linear time-invariant plants that may contain a time delay in the feedback loop. This setting captures many real-world practical and in­ dustrial situations. The results given here include and complement those published in Structure and Synthesis of PID Controllers by Datta, Ho, and Bhattacharyya [10]. In [10] we mainly dealt with the delay-free case. The main contribution described here is the efficient computation of the entire set of PID controllers achieving stability and various performance specifications. The performance specifications that can be handled within our machinery are classical ones such as gain and phase margin as well as modern ones such as Hoo norms of closed-loop transfer functions. Finding the entire set is the key enabling step to realistic design with several design criteria. The computation is efficient because it reduces most often to lin­ ear programming with a sweeping parameter, which is typically the propor­ tional gain. This is achieved by developing some preliminary results on root counting, which generalize the classical Hermite-Biehler Theorem, and also by exploiting some fundamental results of Pontryagin on quasi-polynomials to extract useful information for controller synthesis. The efficiency is im­ portant for developing software design packages, which we are sure will be forthcoming in the near future, as well as the development of further capabilities such as adaptive PID design and online implementation. 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. Emphasis is placed on the efficient computation of the entire set of PID controllers achieving stability and various performance specifications, which is important for the development of future software design packages, as well as further capabilities such as adaptive PID design and online implementation. 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. "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."--BOOK JACKET In this chapter we give a quick overview of control theory, explaining why integral feedback control works, describing PID controllers, and summarizing some of the currently available techniques for PID controller design.