Root locus

Abstract: This paper presents a small-signal analysis for parallel-connected inverters in stand-alone AC supply systems. The control technique of the inverters is based on frequency and voltage droops, which depends on the local variable measurements and does not need control interconnections. Simulation and experimental results show that the system is well represented by the small-signal model. Some root locus plots for the system are provided, which make the stability studies and design easier.

Abstract: 1. Introduction. 2. Mathematical Tools for Control Systems Analysis. 3. First-Order Dynamic Systems. 4. Higher-Order Dynamic Systems. 5. Basic Components of Control Systems. 6. Design of Single Loop Feedback Control Systems. 7. Tuning of Feedback Controllers. 8. Root Locus and Frequency Response Techniques. 9. Cascade Control. 10. Ratio, Override and Selective Control. 11. Feedforward Control. 12. Multivariable Process Control. 13. Dynamic Simulation of Control Systems. Appendix A: Instrumentation Symbols and Labels. Appendix B: Design Case Studies. Appendix C: Sensors, Transmitters, and Control Valves. Appendix D: Tuning Case Studies. Appendxi E: Operating Case Studies. Index.

Abstract: Preface to the third edition- Preface to the second edition- Preface to the first edition- 1 Introduction- 11 Active versus passive- 12 Vibration suppression- 13 Smart materials and structures- 14 Control strategies- 141 Feedback- 142 Feedforward- 15 The various steps of the design- 16 Plant description, error and control budget- 17 Readership and Organization of the book- 18 References- 19 Problems- 2 Some concepts in structural dynamics- 21 Introduction- 22 Equation of motion of a discrete system- 23 Vibration modes- 24 Modal decomposition- 241 Structure without rigid body modes- 242 Dynamic flexibility matrix- 243 Structure with rigid body modes- 244 Example- 25 Collocated control system- 251 Transmission zeros and constrained system- 26 Continuous structures- 27 Guyan reduction- 28 Craig-Bampton reduction- 29 References- 210 Problems- 3 Electromagnetic and piezoelectric transducers- 31 Introduction- 32 Voice coil transducer- 321 Proof-mass actuator- 322 Geophone- 33 General electromechanical transducer- 331 Constitutive equations- 332 Self-sensing- 34 Reaction wheels and gyrostabilizers- 35 Smart materials- 36 Piezoelectric transducer- 361 Constitutive relations of a discrete transducer- 362 Interpretation of k2- 363 Admittance of the piezoelectric transducer- 37 References- 38 Problems- 4 Piezoelectric beam, plate and truss- 41 Piezoelectric material- 411 Constitutive relations- 412 Coenergy density function- 42 Hamilton's principle- 43 Piezoelectric beam actuator- 431 Hamilton's principle- 432 Piezoelectric loads- 44 Laminar sensor- 441 Current and charge amplifiers- 442 Distributed sensor output- 443 Charge amplifier dynamics- 45 Spatial modalfilters- 451 Modal actuator- 452 Modal sensor- 46 Active beam with collocated actuator-sensor- 461 Frequency response function- 462 Pole-zero pattern- 463 Modal truncation- 47 Admittance of a beam with a piezoelectric patch- 48 Piezoelectric laminate- 481 Two dimensional constitutive equations- 482 Kirchhoff theory- 483 Stiffness matrix of a multi-layer elastic laminate- 484 Multi-layer laminate with a piezoelectric layer- 485 Equivalent piezoelectric loads- 486 Sensor output- 487 Beam model vs plate model- 488 Additional remarks- 49 Active truss- 491 Open-loop transfer function- 492 Admittance function- 410 Finite element formulation- 411 References- 412 Problems- 5 Passive damping with piezoelectric transducers- 51 Introduction- 52 Resistive shunting- 53 Inductive shunting- 54 Switched shunt- 541 Equivalent damping ratio- 55 References- 56 Problems- 6 Collocated versus non-collocated control- 61 Introduction- 62 Pole-zero flipping- 63 The two-mass problem- 631 Collocated control- 632 Non-collocated control- 64 Notch filter- 65 Effect of pole-zero flipping on the Bode plots- 66 Nearly collocated control system- 67 Non-collocated control systems- 68 The role of damping- 69 References- 610 Problems - 7 Active damping with collocated system- 71 Introduction- 72 Lead control- 73 Direct velocity feedback (DVF)- 74 Positive Position Feedback (PPF)- 75 Integral Force Feedback(IFF)- 76 Duality between the Lead and the IFF controllers- 761 Root-locus of a single mode- 762 Open-loop poles and zeros- 77 Actuator and sensor dynamics- 78 Decentralized control with collocated pairs- 781 Cross talk- 782 Force actuator and displacement sensor- 783 Displacement actuator and force sensor- 79 References- 710 Problems- 8 Vibration isolation- 81 Introduction- 82 Relaxation isolator- 821 Electromagnetic realization- 83 Active isolation- 831 Sky-hook damper- 832 Integral Force Feedback- 84 Flexible body- 841 Free-free beam with isolator- 85 Payload isolation in spacecraft- 851 Interaction isolator/attitude control- 852 Gough-Stewart platform- 86 Six-axis isolator- 861 Relaxation isolator- 862 Integral Force Feedback- 863 Spherical joints, modal spread- 87 Active vs passive- 88 Car suspension- 89 References- 810 Problems- 9 State space approach- 91 Introduction- 92 State space description- 921 Single degree of freedom oscillator- 922 Flexible structure- 923 Inverted pendulum- 93 System transfer function- 931 Poles and zeros- 94 Pole placement by state feedback- 941 Example: oscillator- 95 Linear Quadratic Regulator- 951 Symmetric root locus- 952 Inverted pendulum- 96 Observer design- 97 Kalman Filter- 971 Inverted pendulum- 98 Reduced order observer- 981 Oscillator- 982 Inverted pendulum- 99 Separation principle- 910 Transfer function of the compensator- 9101 The two-mass problem- 911 References- 912 Problems- 10 Analysis and synthesis in the frequency domain- 101 Gain and phase margins- 102 Nyquist criterion- 1021 Cauchy's principle- 1022 Nyquist stability criterion- 103 Nichols chart- 104 Feedback specification for SISO systems- 1041 Sensitivity- 1042 Tracking error- 1043 Performance specification- 1044 Unstructured uncertainty- 1045 Robust performance and robust stability- 105 Bode gain-phase relationships- 106 The Bode Ideal Cutoff- 107 Non-minimum phase systems- 108 Usual compensators- 1081 System type- 1082 Lead compensator- 1083 PI compensator- 1084 Lag compensator- 1085 PID compensator- 109 Multivariable systems- 1091 Performance specification- 1092 Small gain theorem- 1093 Stability robustness tests- 1094 Residual dynamics- 1010References- 1011Problems- 11 Optimal control- 111 Introduction- 112 Quadratic integral- 113 Deterministic LQR- 114 Stochastic response to a white noise- 1141 Remark- 115 Stochastic LQR- 116 Asymptotic behavior of the closed-loop- 117 Prescribed degree of stability- 118 Gain and phase margins of the LQR- 119 Full state observer- 1191 Covariance of the reconstruction error- 1110Kalman-Bucy Filter (KBF)- 1111Linear Quadratic Gaussian (LQG)- 1112Duality- 1113Spillover- 11131Spillover reduction- 1114Loop Transfer Recovery (LTR)- 1115Integral control with state feedback- 1116Frequency shaping- 11161Frequency-shaped cost functionals- 11162Noise model - 1117References- 1118Problems- 12 Controllability and Observability- 121 Introduction- 1211 Definitions- 122 Controllability and observability matrices- 123 Examples- 1231 Cart with two inverted pendulums- 1232 Double inverted pendulum- 1233 Two dof oscillator- 124 State transformation- 1241 Control canonical form- 1242 Left and right eigenvectors- 1243 Diagonal form- 125 PBH test- 126 Residues- 127 Example- 128 Sensitivity- 129 Controllability and observability Gramians- 1210Internally balanced coordinates- 1211Model reduction- 12111Transfer equivalent realization- 12112Internally balanced realization- 12113Example- 1212References- 1213Problems- 13 Stability- 131 Introduction- 1311 Phase portrait- 132 Linear systems- 1321 Routh-Hurwitz criterion- 133 Lyapunov's direct method- 1331 Introductory example- 1332 Stability theorem- 1333 Asymptotic stability theorem- 1334 Lasalle's theorem- 1335 Geometric interpretation- 1336 Instability theorem- 134 Lyapunov functions for linear systems- 135 Lyapunov's indirect method - 136 An application to controller design- 137 Energy absorbing controls- 138 References- 139 Problems- 14 Applications- 141 Digital implementation- 1411 Sampling, aliasing and prefiltering- 1412 Zero-order hold, computational delay- 1413 Quantization- 1414 Discretization of a continuous controller- 142 Active damping of a truss structure- 1421 Actuator placement- 1422 Implementation, experimental results- 143 Active damping generic interface- 1431 Active damping- 1432 Experiment- 1433 Pointing and position control- 144 Active damping of a plate- 1441 Control design- 145 Active damping of a stiff beam- 1451 System design- 146 The HAC/LAC strategy- 1461 Wide-band position control- 1462 Compensator design- 1463 Results- 147 Vibroacoustics: Volume displacement sensors- 1471 QWSIS sensor- 1472 Discrete array sensor- 1473 Spatial aliasing- 1474 Distributed sensor- 148 References- 149 Problems- 5 Tendon Control of Cable Structures- 151 Introduction- 152 Tendon control of strings and cables- 153 Active damping strategy- 154 Basic Experiment- 155 Linear theory of decentralized active damping- 156 Guyed truss experiment- 157 Micro Precision Interferometer testbed- 158 Free floating truss experiment- 159 Application to cable-stayed bridges- 1510Laboratory experiment- 1511Control of parametric resonance- 1512Large scale experiment- 1513 References- 16 Active Control of Large Telescopes- 161 Introduction- 162 Adaptive optics- 163 Active optics- 1631 Monolithic primary mirror- 1632 Segmented primary mirror- 164 SVD controller- 1641 Loop shaping of the SVD controller- 165 Dynamics of a segmented mirror- 166 Control-structure interaction- 1661 Multiplicative uncertainty- 1662 Additive uncertainty- 1663 Discussion- 167 References- 17 Semi-active control- 171 Introduction- 172 Magneto-rheological fluids- 173 MR devices- 174 Semi-active suspension- 1741 Semi-active devices- 175 Narrow-band disturbance- 1751 Quarter-car semi-active suspension- 176 References- 177 Problems- Bibliography- Index

Abstract: Passive damping is the most adopted method to guarantee the stability of LCL-filter-based grid converters. The method is simple and, if the switching and sampling frequencies are sufficiently high, the damping losses are negligible. This letter proposes the tuning of different passive damping methods and an analytical estimation of the damping losses allowing the choice of the minimum resistor value resulting in a stable current control and not compromising the LCL-filter effectiveness. Stability, including variations in the grid inductance, is studied through root locus analysis in the z-plane. The analysis is validated both with simulation and with experiments.