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  Contents  

6.1  Example  18

6.2  Numerical Inversion  19

6.3  Analytical Inversion  20

6.4  Propering Filter  21

6.5  Proper Inversion  23

6.5.1  Analytical Proper Inversion  24

6.5.2  Inverse via Limit  25

6.5.3  Root Locus  25

6.5.4  Implementation of the Proper Inverse  26

44  A Matlab Function sym2tf  

List of Figures

  List of Figures  

1  Turbulence model extraction  5

2  General inversion scheme  7

3  Transfer function inversion block diagram  8

4  Transfer function inversion result  8

5  State-space inversion block diagram  12

6  Improper system inversion block diagram  14

7  Instability resulting from inadequate solver  15

8  Impact of propering poles  15

9  Proper inversion block diagram  16

10  Proper inverse result  17

11  Pole zero map of MIMO example  19

12  State-space proper inversion block diagram  23

13  Extended root locus of proper inverse  26

14  Instability  28

15  Stabilized system block diagram  29

16  Stabilized system result  30

17  Mirrored transmissions zero block diagram  32

18  Time delay resulting from all-pass  33

19  Phase shift of system mirrored system and mirrored system with time  

  delay  34

20  Nonlinear system inversion block diagram  35

21  Nonlinear system inversion result  36

22  Input and output signals of EC 135 model  37

23  EC 135 inversion block diagram  41

24  EC 135 inversion result  42

25  Zoom of EC 135 inversion result  43

4

1 Introduction

- is defined:

”The overall objective of this task is to improve the modeling accuracy and understanding of helicopter dynamics and control. Improved modeling and understanding of the important issues can be used to increase the fidelity of ground-based simulations, thus allowing early pilot evaluation during the de- velopment of new control systems, compatibility checks for improved safety, decreases in experimental flight testing, and hence a reduction in costs and risks.”

One of the recent subtasks under Task IX has been a disturbance rejection study [1], resulting in a UH-60 Black Hawk control equivalent turbulence simulation model [2].

) and the reaction (rates, velocities, . . . ) of the helicopter (x).

remains.

This equivalent turbulence input can then directly be used as an additional control input in any kind of simulator; without the need to gain and implement more complex turbulence models.

During the actual model extraction approach [1] it became clear that

1 Introduction

used.”

This paper will therefore present and discuss different approaches to invert dynamical

systems.

6

2 Transfer Function Inversion

The inversion of a general dynamic system is depicted in Figure 2

y

u.

0

of the system respectively.

The inversion of a transfer function is a very straightforward exchange of numerator and denominator

1

poles

>> G_tf = tf ([0.1 1] , [1 1])

Transfer function :

0.1 s + 1 --------- s + 1

2 Transfer Function Inversion

>> G_star_tf = inv ( G_tf )

Transfer function :

s + 1 --------- 0.1 s + 1

Simulink can then simulate the identity system (Figure 3) according to Figure 2.

Figure 4 demonstrates that, even though the doublet with its sharp edges (and the corresponding high frequency spectrum portion) poses quite a challenge for the inverter, it is regained almost perfectly.

1.5

8

Also, the influence of the transfer function zero with the time constant of 0.1 sec can be

found in the immediate ”jumping” of the system output at every doublet edge.