Diversity and Non-Integer Differentiation for System Dynamics.

Cover -- Title Page -- Copyright -- Contents -- Acknowledgments -- Preface -- Introduction -- Chapter 1: From Diversity to Unexpected Dynamic Performances -- 1.1. Introduction -- 1.2. An issue raising a technological bottle-neck -- 1.3. An aim liable to answer to the issue -- 1.4. A strategy idea liable to reach the aim -- 1.4.1. Why diversity? -- 1.4.2. What does diversity imply? -- 1.5. On the strategy itself -- 1.5.1. The study object -- 1.5.2. A pore: its model and its technological equivalent -- 1.5.2.1. The model -- 1.5.2.2. The technological equivalent -- 1.5.3. Case of identical pores -- 1.5.4. Case of different pores -- 1.5.4.1. On differences coming from regional heritage -- 1.5.4.1.1 Differences of technological origin -- 1.5.4.1.2. A difference of natural origin -- 1.5.4.1.3. How is difference expressed? -- 1.5.4.2. Transposition to the study object -- 1.6. From physics to mathematics -- 1.6.1. An unusual model of the porous face -- 1.6.1.1. A smoothing remarkable of simplicity: the one of crenels -- 1.6.1.2. A non-integer derivative as a smoothing result -- 1.6.1.3. An original heuristic verification of differentiation non-integer order -- 1.6.2. A just as unusual model governing water relaxation -- 1.6.3. What about a non-integer derivative which singles out these unusual models? -- 1.6.3.1. On the sinusoidal state of the operator of order n E [0, 2] -- 1.6.3.1.1. 0 ≤ n ≤1 -- 1.6.3.1.2. 1≤ n ≤ 2 -- 1.6.3.2. On the impulse state of the operator of order n E ]0,1[ -- 1.6.3.3. An original heuristic verification of time non-integer power -- 1.7. From the unusual to the unexpected -- 1.7.1. Unexpected damping properties -- 1.7.1.1. Relaxation damping insensitivity to the mass -- 1.7.1.2. Frequency verification of the insensitivity to the mass -- 1.7.2. Just as unexpected memory properties -- 1.7.2.1. Taking into account the past.

1.7.2.2. Memory notion -- 1.7.2.3. A diversion through an aspect of human memory -- 1.7.2.3.1. The serial position effect -- 1.7.2.3.2. A model of the primacy effect -- 1.8. On the nature of diversity -- 1.8.1. An action level to be defined -- 1.8.2. One or several forms of diversity? -- 1.8.2.1. Forms based on the invariance of the elements -- 1.8.2.2. A singular form based on the time variability of an element -- 1.9. From the porous dyke to the CRONE suspension -- 1.10. Conclusion -- 1.11. Bibliography -- Chapter 2: Damping Robustness -- 2.1. Introduction -- 2.2. From ladder network to a non-integer derivative as a water-dyke interface model -- 2.2.1. On the admittance factorizing -- 2.2.2. On the asymptotic diagrams at stake -- 2.2.3. On the asymptotic diagram exploiting -- 2.2.3.1. Step smoothing -- 2.2.3.2. Crenel smoothing -- 2.2.3.3. A non-integer differentiator as a smoothing result -- 2.2.3.4. A non-integer derivative as a water-dyke interface model -- 2.3. From a non-integer derivative to a non-integer differential equation as a model governing water relaxation -- 2.3.1. Flow-pressure differential equation -- 2.3.2. A non-integer differential equation as a model governing relaxation -- 2.3.2.1. Pressure as a variable of the differential equation -- 2.3.2.2. Flow as a variable of the differential equation -- 2.3.3. Electrical analogy -- 2.4. Relaxation expression -- 2.5. From a non-integer differential equation to relaxation damping robustness -- 2.5.1. Operational approach -- 2.5.1.1. A multiform characteristic equation -- 2.5.1.2. A pair of conjugate complex roots -- 2.5.2. Frequency approach -- 2.5.3. On the representation of robustness in a symbolic domain -- 2.5.3.1. Continuous representation in the plane s -- 2.5.3.2. Discrete representation in the plane z -- 2.6. Validation by an experimental simulation in analog electronics.

2.6.1. Simulation functional diagram -- 2.6.2. Simulation electronic circuit -- 2.6.2.1. Design -- 2.6.2.2. Achievement -- 2.6.2.3. A greatly simplifying approximation -- 2.6.2.4. Modeling -- 2.6.2.4.1. Order 1 integrator -- 2.6.2.4.2. Order 0.5 integrator -- 2.6.2.4.3. Open loop and closed loop -- 2.6.3. On the simulation itself: damping robustness tests -- 2.7. Bibliography -- Chapter 3: Non-Integer Differentiation, its Memory and its Synthesis -- 3.1. Introduction -- 3.2. From integer differentiation to non-integer differentiation -- 3.2.1. Toward non-integer differentiation: the adopted approach -- 3.2.2. Generic form of the order 1 and 2 derivatives -- 3.2.3. Generalization to the integer and non-integer case -- 3.3. From repeated integer integration to non-integer differentiation through non-integer integration -- 3.3.1. Repeated integer integration -- 3.3.2. Non-integer integration -- 3.3.3. Non-integer differentiation -- 3.3.3.1. Riemann-Liouville and Caputo definitions -- 3.3.3.2. Impulse response of the non-integer differentiation operator -- 3.4. Non-integer differentiation in sinusoidal steady state -- 3.4.1. A definition of the Fresnel vector -- 3.4.2. A direct application to kinematic magnitudes -- 3.4.3. Non-integer derivative of position -- 3.4.3.1. First case: 0 ≤ n ≤1 -- 3.4.3.2. Second case: 1≤ n ≤ 2 -- 3.4.4. Verification of the decomposition of -- 3.5. On memory associated with non-integer differentiation -- 3.5.1. From the local to the global by taking into account the past -- 3.5.2. On memory notion -- 3.5.3. On an aspect of human memory (an investigation trail) -- 3.5.3.1. Serial position effect -- 3.5.3.2. A primacy effect model -- 3.5.3.3. Optimal integration non-integer order proof -- 3.6. On the synthesis of non-integer differentiation -- 3.6.1. Synthesis of a frequency-bounded real non-integer differentiator.

3.6.2. Synthesis of a frequency-bounded complex non-integer differentiator -- 3.6.2.1. Definition of the space ĉ -- 3.6.2.2. Demonstration of the validity of the synthesis transmittance for a complex non-integer differentiation order -- 3.6.3. Stability of the synthesis transmittance -- 3.6.4. Distribution of synthesis zeros and poles: real and imaginary orders of differentiation -- 3.6.5. Determination of the number of synthesis zeros and poles -- 3.6.5.1. Impulse response energy -- 3.6.5.2. Relative energy difference -- 3.6.6. Validation in time domain -- 3.6.6.1. Impulse response of the differentiator to be synthesized -- 3.6.6.2. Impulse response of the synthesized differentiator -- 3.6.6.3. Comparative performances -- 3.6.6.4. Achievement of the synthesis transmittance -- 3.7. Bibliography -- Chapter 4: On the CRONE Suspension -- 4.1. Introduction -- 4.2. From the porous dyke to the hydropneumatic version of the CRONE suspension -- 4.2.1. Concept -- 4.2.2. From concept to achievement -- 4.2.3. Vehicle implementation -- 4.3. Metallic version of the CRONE suspension -- 4.3.1. A technological difference in terms of suspensions -- 4.3.2. Performance and robustness objective -- 4.3.3. Strategy -- 4.3.4. Contract collaboration -- 4.3.5. Principle of the CRONE suspension -- 4.3.6. Transfers of the usual and CRONE suspensions -- 4.3.7. Initial behavior: no initial acceleration for the CRONE suspension -- 4.3.8. Stability degree robustness -- 4.3.8.1. Robustness tests -- 4.3.8.2. Stability degree measurement -- 4.3.8.3. Frequency and step responses: robustness of the resonance ratio and thefirst overshoot for the CRONE suspension -- 4.3.8.4. Characteristic equation roots: robustness of the damping ratio for the CRONE suspension -- 4.3.9. Idea of the synthesis of a non-integer order dashpot -- 4.3.10. Active character of the CRONE suspension.

4.3.11. Piloted passive CRONE suspension -- 4.4. Bibliography -- Chapter 5: On the CRONE Control -- 5.1. Introduction -- 5.2. From the porous dyke to the CRONE control of first and second generations -- 5.2.1. First interpretation of the relaxation model: first generation CRONE strategy -- 5.2.1.1. Dynamic behavior of the water mass -- 5.2.1.2. Dynamic behavior of the water-dyke interface -- 5.2.1.3. Functional representation highlighting a constant phase CRONE controller -- 5.2.1.4. Idea of the first generation CRONE strategy -- 5.2.2. Second interpretation of the relaxation model: second generation CRONE strategy -- 5.2.2.1. Non-integer order differential equation as a relaxation model -- 5.2.2.2. Functional representation leading to an open-loop frequency template -- 5.2.2.3. Idea of the second generation CRONE strategy -- 5.3. Second generation CRONE control and uncertainty domains -- 5.3.1. Uncertainty domains -- 5.3.2. Particular open-loop uncertainty domains -- 5.3.3. Adequacy of the second generation CRONE control template to the particular uncertainty domains -- 5.4. Generalization of the vertical template through the third generation CRONE control -- 5.4.1. First level of generalization -- 5.4.1.1. Generalized template -- 5.4.1.2. Generalized template and complex non-integer integration -- 5.4.2. Second level of generalization -- 5.4.2.1. Curvilinear template -- 5.4.2.2. Operational description of the curvilinear template -- 5.4.3. Open-loop transfer integrating the curvilinear template -- 5.4.4. Optimization of the open-loop behavior -- 5.4.4.1. Optimal template -- 5.4.4.2. Criterion to be minimized by the optimal template -- 5.4.5. Structure and parametric estimation of the controller -- 5.4.6. Application -- 5.5. Bibliography -- Chapter 6: Recursivity and Non-Integer Differentiation -- 6.1. Introduction.

6.2. Indefinite recursive parallel arrangement of series RC cells.

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