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PID Controllers for Time Delay Systems
This is the results on the proportional-integralderivative (PID) controller and its design, analysis, and synthesis. The author focuses on linear time-invariant plants that may contain a time delay in the feedback loop. The results of the book published in Structure and Synthesis of PID Controllers by Datta, Ho, and Bhattacharyya. This is described by the author the efficient computation of the entire set of PID controllers achieving stability and various performance specifications. Finding the entire set is the key enabling step to realistic design with several design criteria. It has efficient computation that reduces most often to linear programming with a sweeping parameter, which is typically the proportional gain. It is achieved by developing some preliminary results on root counting, which generalize the classical Hermite-Biehler Theorem, and by exploiting some fundamental results of Pontryagin on quasi-polynomials to extract useful information for controller synthesis. The is important for developing software design packages such as adaptive PID design and online implementation and creating a realistic interactive design environment where multiple performance specifications that are appropriately prioritized can be overlaid and intersected to telescope down to a small and satisfactory controller set. The PID controller is very important in control engineering applications. Classical control theory approaches generally produce a single controller based on ad hoc loop-shaping techniques and are also inadequate for the kind of computer-aided multiple performance specifications design applications advocated in this book. Below are the contents of the book:
1 Introduction 1
1.1 Introduction to Control 1
1.2 The Magic of Integral Control 3
1.3 PID Controllers 6
1.4 Some Current Techniques for PID Controller Design . . . . 7
1.4.1 The Ziegler-Nichols Step Response Method 7
1.4.2 The Ziegler-Nichols Frequency Response Method . . 9
1.4.3 PID Settings using the Internal Model Controller
Design Technique 11
1.4.4 Dominant Pole Design: The Cohen-Coon Method . . 13
1.4.5 New Tuning Approaches 14
1.5 Integrator Windup 16
1.5.1 Setpoint Limitation 16
1.5.2 Back-Calculation and Tracking 17
1.5.3 Conditional Integration 17
1.6 Contribution of this Book 18
1.7 Notes and References 18
2 The Hermite-Biehler Theorem and its Generalization 21
2.1 Introduction 21
2.2 The Hermite-Biehler Theorem for Hurwitz Polynomials . . 22
2.3 Generalizations of the Hermite-Biehler Theorem 27
2.3.1 No Imaginary Axis Roots 29
2.3.2 Roots Allowed on the Imaginary Axis Except at the
2.3.3 No Restriction on Root Locations . 35
2.4 Notes and References 37
PI Stabilization of Delay-Free Linear Time-Invariant
Systems 39
3.1 Introduction 39
3.2 A Characterization of All Stabilizing Feedback Gains . . . . 40
3.3 Computation of All Stabilizing PI Controllers 51
3.4 Notes and References 56
PID Stabilization of Delay-Free Linear Time-Invariant
Systems 57
4.1 Introduction 57
4.2 A Characterization of All Stabilizing PID Controllers . . . . 58
4.3 PID Stabilization of Discrete-Time Plants 67
4.4 Notes and References 75
Preliminary Results for Analyzing Systems with Time
Delay 77
5.1 Introduction 77
5.2 Characteristic Equations for Delay Systems 78
5.3 Limitations of the Pade Approximation 82
5.3.1 Using a First-Order Pade Approximation 83
5.3.2 Using Higher-Order Pade Approximations 85
5.4 The Hermite-Biehler Theorem for Quasi-Polynomials . . . . 89
5.5 Applications to Control Theory 92
5.6 Stability of Time-Delay Systems with a Single Delay . . . . 99
5.7 Notes and References 106
Stabilization of Time-Delay Systems using a Constant Gain
Feedback Controller 109
6.1 Introduction 109
6.2 First-Order Systems with Time Delay 110
6.2.1 Open-Loop Stable Plant 112
6.2.2 Open-Loop Unstable Plant 116
6.3 Second-Order Systems with Time Delay 122
6.3.1 Open-Loop Stable Plant 125
6.3.2 Open-Loop Unstable Plant 129
6.4 Notes and References 134
PI Stabilization of First-Order Systems with Time Delay 135
7.1 Introduction 135
7.2 The PI Stabilization Problem 136
7.3 Open-Loop Stable Plant 137
7.4 Open-Loop Unstable Plant 150
7.5 Notes and References 159
8 PID Stabilization of First-Order Systems with Time Delay 161
8.1 Introduction 161
8.2 The PID Stabilization Problem 162
8.3 Open-Loop Stable Plant 164
8.4 Open-Loop Unstable Plant 179
8.5 Notes and References 189
9 Control System Design Using the PID Controller 191
9.1 Introduction 191
9.2 Robust Controller Design: Delay-Free Case 192
9.2.1 Robust Stabilization Using a Constant Gain 194
9.2.2 Robust Stabilization Using a PI Controller 196
9.2.3 Robust Stabilization Using a PID Controller . . . . 199
9.3 Robust Controller Design: Time-Delay Case 203
9.3.1 Robust Stabilization Using a Constant Gain 204
9.3.2 Robust Stabilization Using a PI Controller 205
9.3.3 Robust Stabilization Using a PID Controller . . . . 208
9.4 Resilient Controller Design 213
9.4.1 Determining fc, T, and L from Experimental Data . 213
9.4.2 Algorithm for Computing the Largest Ball Inscribed
Inside the PID Stabilizing Region 214
9.5 Time Domain Performance Specifications 217
9.6 Notes and References 222
10 Analysis of Some PID Tuning Techniques 223
10.1 Introduction . 223
10.2 The Ziegler-Nichols Step Response Method 224
10.3 The CHR Method 229
10.4 The Cohen-Coon Method 233
10.5 The IMC Design Technique 237
10.6 Summary 241
10.7 Notes and References 241
11 PID Stabilization of Arbitrary Linear Time-Invariant
Systems with Time Delay 243
11.1 Introduction 243
11.2 A Study of the Generalized Nyquist Criterion 244
11.3 Problem Formulation and Solution Approach 248
11.4 Stabilization Using a Constant Gain Controller 250
11.5 Stabilization Using a PI Controller 253
11.6 Stabilization Using a PID Controller 256
11.7 Notes and References 263
12 Algorithms for Real and Complex PID Stabilization 265
12.1 Introduction 265
12.2 Algorithm for Linear Time-Invariant Continuous-Time
Systems 266
12.3 Discrete-Time Systems 276
12.4 Algorithm for Continuous-Time First-Order Systems with
Time Delay 277
12.4.1 Open-Loop Stable Plant 279
12.4.2 Open-Loop Unstable Plant 280
12.5 Algorithms for PID Controller Design 284
12.5.1 Complex PID Stabilization Algorithm 285
12.5.2 Synthesis of Hoc PID Controllers 287
12.5.3 PID Controller Design for Robust Performance . . . 291
12.5.4 PID Controller Design with Guaranteed Gain and
Phase Margins 293
12.6 Notes and References 295
A Proof of Lemmas 8.3, 8.4, and 8.5 297
A.l Preliminary Results 297
A.2 Proof of Lemma 8.3 301
A.3 Proof of Lemma 8.4 302
A.4 Proof of Lemma 8.5 303
B Proof of Lemmas 8.7 and 8.9 307
B.l Proof of Lemma 8.7 307
B.2 Proof of Lemma 8.9 308
C Detailed Analysis of Example 11.4 313
The book is a dingle source of information designed to complement the codes and standards concerned with the installation of electrical equipment in hazardous locations. It begins with thorough explanation what hazardous locations are. This book covers among the hundred important subjects.