Index. A 1-form, 42, 45, 133 Amplitude-frequency tradeoff, 14. Drift field perturbation, 66 Drift vector field, 42

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1 References 1. U. Beis An Introduction to Delta Sigma Converters Internet site Elektronik/ DeltaSigma/ DeltaSigma.html, pp. 1 11, January 21, B. Charlet, J. Lévine and R. Marino Sufficient conditions for dynamic feedback linearization, SIAM J. Control and Optimization, Vol. 29, No. 1, pp , M. D. Di Benedetto and J. Grizzle, Intrinsic notion of regularity for local inversion, output nulling and dynamic extension of non-square systems Control- Theory and Advanced Technology, Vol. 6, No. 3, pp , C. Edwards, S. Spurgeon, Sliding Mode Control: Theory and Applications. Taylor and Francis, London M. Fliess, J. Levine, Ph. Martin and P. Rouchon Sur les systèmes nonlineaires differentiallement plats Comptes Rendus de l Academie des Sciences de Paris, Serie I, Vol. 315, pp , M. Fliess, J. Levine, Ph. Martin and P. Rouchon, Flatness and defect of nonlinear systems: introductory theory and examples International Journal of Control, Vol. 61, No. 6, pp M. Fliess, J. Levine, Ph. Martin and P. Rouchon, A Lie-Bäcklund approach to equivalence and flatness, IEEE Transactions on Automatic Control, Vol. 44, No. 5, pp , May M. Fliess, R. Marquez, H. Mounier An extension of predictive control, PID regulators and Smith predictors to some linear delay systems International Journal of Control Vol. 75, No. 10, pp , M. Fliess, H. Sira-Ramírez and R. Márquez, Regulation of non-minimum phase outputs: A flatness based approach, in Perspectives in Control, D. Normand-Cyrot (Ed.), Springer-Verlag, London B. A. Francis, W. M. Wonham The internal model principle for linear multivariable regulators Applied Mathematics and Optimization, 1975, Volume 2, Issue 2, pp Springer International Publishing Switzerland 2015 H. Sira-Ramírez, Sliding Mode Control, Control Engineering, DOI /

2 254 References 11. L. Fridman, A. Poznyak and F.J. Bejarano Robust LQ Output Control: Integral Sliding Mode Approach, Springer, Berlin, V. Hagenmeyer, E. Delaleau Robustness analysis with respect to exogenous perturbations for flatness-based exact feedforward linearization IEEE Transactions on Automatic Control, Vol. 55, No. 3, pp , A. Isidori, Nonlinear Control Systems, Springer, New York D. Jarman A Brief Introduction to Sigma Delta Conversion, Application Note 9504, Intersil Corporation. May H. G. Kwatny and G. L. Blankenship, Nonlinear control and analytical mechanics with Mathematica, Boston, Birkhauser, J. Levine, Analysis and Control of Nonlinear Systems: A Flatness-based Approach. Springer-Verlag, Berlin, S. Norsworthy, R. Shreier, G. Temes, Delta-Sigma Data Converters: Theory, Design, and Simulation, IEEE Press, Piscataway J. Reiss, Understanding Sigma-Delta modulation: The solved and unsolved issues Journal of the Audio Engineering Society, Vol. 56, No. 1/2, P. Rouchon, Necessary condition and genericity of dynamic feedback linearization, Journal of Mathematical Systems, Estimation and Control, Vol. 4, No. 2, pp , J. Rudolph, Flatness based control of distributed parameter systems, Shaker Verlag, Aachen, J. Rudolph, J. Wnkler, and F. Woittenek, Flatness based control of distributed parameter systems: Examples and computer exercises from various technological domains, Shaker Verlag, Aachen, Y. Shtessel, C. Edwards, L. Fridman and A. Levant, Sliding Mode Control and Observation, Control Engineering Series, Birkhäuser, New York J.J. Slotine, and W. Li, Applied Nonlinear Control, Prentice Hall, New Jersey W. Sluis A necessary condition for dynamic feedback linearization, Systems and Control Letters, Vol. 21, pp , H. Sira-Ramírez, Differential Geometric Methods in Variable Structure Control, International Journal of Control, Vol. 48, No. 4 pp , H. Sira-Ramírez and S.K. Aggrawal, Differentially Flat Systems, Control Engineering Series, Marcel Dekker, Inc. New York H. Sira-Ramírez, M. Fliess, Regulation of nonminimum phase outputs in a PVTOL aircraft in Proc. of the 37th IEEE Conference on Decision and Control, Tampa, Florida, December 13 15, Y. V. Orlov, Discontinuous Systems: Lyapunov Analysis and Robustness under Uncertainty Conditions. Springer-Verlag, London, R. Steele, Delta Modulation Systems, Pentech Press, London Y. Tsypkin Relay Control Systems, Cambridge University Press, Cambridge 1984.

3 References V. I. Utkin, Sliding Modes in the Theory of Variable Structure Systems, MIR, Moscow V. I. Utkin,Sliding Modes in Optimization and Control Problems. Springer, New York V. I. Utkin, J. Guldner, J. Shi, Sliding Mode Control in Electromechanical Systems. Taylor and Francis, London V. Utkin and J. Shi, Integral sliding mode in systems operating under uncertainty conditions, in Proc. 35th IEEE Conference on Decision and Control, Kobe, Japan, December 1996, pp B. Wie and D. Bernstein, A benchmark problem for robust control design Proc. American Control Conference, pp , San Diego, CA, May 1990.

4 Index A 1-form, 42, 45, 133 Amplitude-frequency tradeoff, 14 Bézier polynomial, 20 Bandwidth, 14 Binary valued input, 38 Boost converter, 21, 38, 41, 53 Boost-boost converter, 136 Brunovski s canonical form, 215 Buck-Boost converter, 57 Cascaded buck converters, 162 Chained mass-spring system, 244 Chua s circuit, 87 Co-vector, 42 Computed torque controller, 147 Control and drift field perturbations, 71 Control field perturbation, 70 Control vector field, 42 Cotangent vector, 42 DC motor, 69, 182 Delta modulator, 90 Delta Sigma modulation, 99 Differential, 42 Differential function of the state, 213 Differential independence, 213 Differential parametrization, 214 Differentially flat system, 213 Directional derivative, 42, 130 Double bridge buck converter, 112 Double buck-boost converter, 139 Drift field perturbation, 66 Drift vector field, 42 Equivalent control, 3, 44, 132 Exact linearization, 86 Feedback linearizable system, 215 Finite time reachability, 7 Flat output, 86 Fully actuated rigid body, 145 Generalized Proportional Integral control, 165 Global sliding motions, 29 GPI control, 166 GPI sliding mode control, 209 Higher order Delta modulation, 95 Ideal sliding dynamics, 44 Idealized Delta Modulation, 90 Induction motor, 207 Integral reconstructor, 166 Integral sliding mode, 122 Integral state reconstructors, 165 Internal Model principle, 12 Invariance conditions, 8, 44 Involutive distribution, 79 Lead compensator, 114 Lie bracket, 77 Linearizing output, 86 Springer International Publishing Switzerland 2015 H. Sira-Ramírez, Sliding Mode Control, Control Engineering, DOI /

5 258 Index Local existence of sliding mode, 48 locally decoded signal, 90 Matched perturbation, 67 Matching condition, 135 Matching conditions, 3, 67 Minimum phase output, 26, 37, 82 Multilevel Delta-Sigma modulation, 109 Multivariable nonlinear systems, 128 Non-minimum phase, 84 Non-minimum phase output, 26, 37, 83 Nonholonomic restrictions, 216 Permanent magnet stepper motor, 202 Permanent magnet synchronous motor, 153 Planar rigid body, 223 Plant, 1 Projection operator, 37, 44, 132, 134 Reaching phase, 7 Relative degree, 74 remote decoder, 91 Satellite model, 85 second order Delta modulation, 92 Second order Delta-Sigma modulation, 118 Sensorless control of induction motor, 208 Single axis car, 220 Single link-dc motor system, 248 Sliding manifold, 2 Sliding regime, 8 Sliding surface, 41 Sliding surface coordinate function, 2, 42 Soft landing vehicle model, 84 Sustaining phase, 8 Switched gain, 40 Switched third order integrator, 50 The elements of SMC, 1 The rocket model, 226 The rolling penny, 216 Three tanks model, 236 Transversal condition, 43 Transverse condition, 42 Two degree of freedom robot, 239 Underactuated rigid body, 157 Variable structure system, 38 Vector field, 133 Vector relative degree, 131, 148 Water tank system, 15 Zero dynamics, 26, 37, 83