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## Expression of the dynamics of the observer in the system coordinates

### Observers for permanent magnet synchronous motors with unkown parameters

#### Quick review of existing normal forms and their observers

**Table of contents :**

**1 Introduction **

**2 Nonlinear observability and observer design problem **

2.1 Observation problem

2.2 Observability and observer design for nonlinear systems

2.3 Organization of the thesis

**I Normal forms and their observers **

**3 Quick review of existing normal forms and their observers **

3.1 State-affine normal forms

3.2 Triangular normal forms

3.3 Conclusion

**4 Observers for the continuous triangular form **

4.1 High gain observer ?

4.2 Homogeneous observer

4.3 Cascade of observers

4.4 Relaxing the assumptions marked with (.)

4.5 Illustrative example

4.6 Conclusion

**II Transformation into a normal form **

**5 Review of existing transformations **

5.1 Transformations into a state-affine normal forms

5.2 Transformations into triangular normal forms

5.3 Conclusion

**6 Transformation into a continuous triangular form **

6.1 Presentation of the problem

6.2 Existence of gi satisfying (6.5)

6.3 Lipschitzness of the triangular form

6.4 Back to Example 4.5 in Chapter

6.5 Conclusion

**7 Transformation into a Hurwitz form **

7.1 Time-varying transformation

7.2 Examples

7.3 Stationary transformation ?

4 Contents

7.4 Conclusion

**III Expression of the dynamics of the observer in the system coordinates**

**8 Motivation and problem statement **

8.1 Example

8.2 Problem statement

9 Around Problem 1 : augmenting an injective immersion into a diffeomorphism

9.1 Submersion case

9.2 The ˜ P[d, dx] problem

9.3 Wazewski’s theorem

**10 Around Problem 2 : image extension of a diffeomorphism **

10.1 A sufficient condition

10.2 Proof of part a) of Theorem 10.1.1

10.3 Application : bioreactor

10.4 Conclusion

**11 Generalizations and applications **

11.1 Modifying and ‘T given by Assumption O

11.2 A global example : Luenberger design for the oscillator

11.3 Generalization to a time-varying

11.4 Conclusion

**IV Observers for permanent magnet synchronous motors with unkown parameters **

**12 Short introduction to PMSMs **

**13 Rotor position estimation with unknown magnet flux **

13.1 Gradient observer

13.2 Alternative path

13.3 Performances

13.4 Tests with real data

13.5 Conclusion

**14 Rotor position estimation with unknown resistance **

14.1 Observability

14.2 Observer design

14.3 Simulations

14.4 Conclusion

**Bibliography **