Method and control unit for controlling an overdetermined system, system and aircraft

1. A method for controlling a system, which is overdetermined, with multiple power-restricted actuators, wherein the actuators perform at least one primary task and a number of non-primary tasks, the method comprising:

a) determining a pseudo-control command u p ∈ p′ based on a physical model of a system, said pseudo-control command represents the torques (L, M, N) and a total thrust force (F) acting on the system,

b) determining a control matrix D, D∈ p′×k according to u p =Du, wherein u denotes actuator control commands,

c) dissociating the control matrix D into sub control matrices D i according to

D

=

[

D

1

⋮

D

q

]

,

 q≤p′, so that

u

p

=

[

u

p

,

1

⋮

u

p

,

q

]

 and u p,i =D i u,

wherein the sub control matrices D i and corresponding sub pseudo-control commands u p,i for i=1 correspond to the primary task and for i>1 correspond to the at least one non-primary task and wherein a priority of the non-primary tasks decreases with increasing index i,

d) determining actuator control commands for solving the primary task according to u 1 =D 1 # u p,1 , where #denotes a matrix inversion and u 1 ∈ k comprises said actuator control commands which solve the primary task, with u 1 ∈U:={u∈ k |u min ≤u≤u max }

e) projecting the non-primary tasks, i>1, into a null space of the primary task, i=1, and into respective null space of all of the non-primary tasks of higher priority, if present, so that D i u j =0, with u j ∈ k , j>i, comprises actuator control commands for performing a non-primary task, and

f) providing the actuator control commands from d) and e) at the actuators.

2. The method as claimed in claim 1 , further comprising

first calculating u 2 org ∈ k , in order to solve a first one of the non-primary tasks, according to u 2 org =D 2 # (u p,2 −D 2 u 1 ), and subsequently projecting u 2 org onto the null space N 2 (D 1 ) of the sub control matrix D 1 according to u 2 =N 2 u 2 org , wherein N 2 =I−D 1 T (D 1 # ) T .

3. The method as claimed in claim 2 , further comprising

subsequently calculating a complete actuator control command u as follows:

u=u 1 +u 2 .

4. The method as claimed in claim 3 , wherein

for u 2 org it is specified that u 2 org ∈U 2 :={u 2 org ∈ k |u min −u 1 ≤N 2 u 2 org ≤u max −u 1 }.

5. The method as claimed in claim 2 , further comprising

subsequently calculating u 3 org ∈ k in order to solve a second one of the non-primary tasks, according to u 3 org =D 3 # (u p,3 −D 3 (u 1 −u 2 )), and subsequently projecting u 3 org onto the null space N 3 (D 1 ,D 2 ) of the sub control matrices D 1 ,D 2 according to u 3 =N 3 u 2 org , wherein

N

3

=

(

I

-

[

D

1

D

2

]

T

⁢

(

[

D

1

D

2

]

#

)

T

)

.

6. The method as claimed in claim 5 , further comprising

subsequently calculating a complete actuator control command u as follows: u=u 1 +u 2 +u 3 .

7. The method as claimed in claim 6 , wherein

for u 3 org it is specified that u 3 org ∈U 3 :={u 3 org ∈ k |u min −u 1 −u 2 ≤N 3 u 3 org ≤u max −u 1 −u 2 }.

8. The method as claimed in claim 1 , further comprising

calculating the null space N i (D), i>1 according to

N

i

(

D

)

=

(

I

-

[

D

1

⋮

D

i

-

1

]

T

⁢

(

[

D

1

⋮

D

i

-

1

]

#

)

T

)

∈

k

×

k

.

9. The method as claimed in claim 8 , further comprising

for solving a non-primary task, i>1, calculating:

u i org =D i # ( u p,i −D i Σ j=1 i−1 u j )

u i =N i u i org ;

u=Σ i=1 n u i ; with

u i ∈U i :={u i ∈ k |u min −u Σ ≤N i u i ≤u max −u Σ } and with

u Σ =Σ j=1 i−1 u j .

10. The method as claimed in claim 1 , wherein the method is applied to an aircraft ( 1 ) with multiple propulsion units ( 3 ), and said propulsion units form at least some of the actuators of the system.

11. The method as claimed in claim 1 , further comprising

prioritizing attitude control concerning rolling and pitching, over directional control concerning yawing and vertical control concerning a flying altitude.

12. The method as claimed in claim 11 , further comprising

selecting D 1 ∈ 2×k such that it represents a mapping of the actuator control commands u onto a rolling and pitching moment, with u p,1 ∈ 2 , and selecting D 2 ∈ 2×k such that it represents a mapping of the actuator control commands u onto a thrust and yawing moment, with u p,2 ∈ 2 .

13. The method as claimed in claim 10 , further comprising

prioritizing attitude control concerning rolling and pitching over directional control concerning yawing and a total thrust.

14. The method as claimed in claim 13 , further comprising

selecting D 1 ∈ 2×k such that it represents a mapping of the actuator control commands u onto a rolling and pitching moment, with u p,1 ∈ 2 ,

selecting D 2 ∈ 1×k such that it represents a mapping of the actuator control commands u onto a yawing moment, with u p,2 ∈ , and

selecting D 3 ∈ 1×k such that it represents a mapping of the actuator control commands u onto the total thrust, with u p,3 ∈ .

15. The method as claimed in claim 10 , further comprising

prioritizing a total thrust over attitude control concerning rolling and pitching, and over directional control concerning yawing.

16. The method as claimed in claim 15 , further comprising

selecting D 1 ∈ 1×k such that it represents a mapping of the actuator control commands u onto the total thrust, with u p,1 ∈ 2 ,

selecting D 2 ∈ 1×k such that it represents a mapping of the actuator control commands u onto a pitching moment, with u p,2 ∈ ,

selecting D 3 ∈ 1×k such that it represents a mapping of the actuator control commands u onto a rolling moment, with u p,3 ∈ , and

selecting D 4 ∈ 1×k such that it represents a mapping of the actuator control commands u onto a yawing moment, with u p,4 ∈ .

17. A control unit ( 2 a ) for controlling a system, which is overdetermined, with multiple power-restricted actuators, wherein the actuators perform at least one primary task and at least one non-primary task, comprising a controller configured to

a) determining a pseudo-control command u p ∈ p′ based on a physical model of a system, said pseudo-control command represents torques (L, M, N) and a total thrust force (F) acting on the system,

b) determining a control matrix D, D∈ p′×k , according to u p =Du, wherein u denotes actuator control commands,

c) dissociating the control matrix into sub control matrices

D

=

[

D

1

⋮

D

q

]

,

 q≤p′, so that

u

p

=

[

u

p

,

1

⋮

u

p

,

q

]

 and u p,i =D i u,

wherein the sub control matrices D i and corresponding sub pseudo-control commands u p,i for i=1 correspond to the primary task and for i>1 correspond to the at least one non-primary task and wherein a priority of the non-primary tasks decreases with increasing index i,

d) determining the actuator control commands for solving the primary task according to u 1 =D 1 # u p,1 , where #denotes a matrix inversion and u 1 ∈ k comprises said actuator control commands which solve the primary task, with u 1 ∈U:={u∈ k |u min ≤u≤u max },

e) projecting the non-primary tasks, i>1, into a null space of the primary task, i=1, and into respective null space of all of the non-primary tasks of higher priority, if present, so that D i u j =0, if u j ∈ k , j>i, comprises the actuator control commands for performing a non-primary task, and

f) providing the actuator control commands from d) and e) at the actuators.

18. The control unit ( 2 a ) as claimed in claim 17 , wherein the controller is further configured such that it first calculates u 2 org ∈ k , in order to solve a first one of the non-primary tasks, according to u 2 org =D 2 # (u p,2 −D 2 u 1 ), and subsequently projects u 2 org onto the null space N 2 (D 1 ) of the sub control matrix D 1 according to u 2 =N 2 u 2 org , wherein N 2 =I−D 1 T (D 1 # ) T .

19. The control unit ( 2 a ) as claimed in claim 17 , further comprising at least one of devices ( 4 ) for at least one of measuring or determining parameters and states of the system or the actuators which are required for the determination of the pseudo-control command according to step a) connected to the controller.

20. An overdetermined system comprising multiple actuators actively connected to the control unit ( 2 a ) as claimed in claim 17 .

21. An aircraft ( 1 ) comprising the overdetermined system as claimed in claim 20 , wherein the actuators multiple propulsion units ( 3 ).