Method and control unit for controlling an overdetermined system, system and aircraft
A method for controlling an overdetermined system with multiple actuators, for example an aircraft ( 1 ) with multiple propulsion units ( 3 ). The actuators perform at least one primary task and at least one non-primary task, including: a) determining a pseudo-control command u p ∈ p′ based on a physical model of the system, which 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, where u 1 =D −1 u p u 1 ∈ k represents a control command for the actuators to perform the primary task, c) projecting the non-primary task into the null space N(D) of the primary task, so that Du 2 =0 if u 2 u 2 ∈ k represents a control command for the actuators to perform the non-primary task, and d) providing the control commands from b) and c) to the actuators. In this way, the solution of the primary task is not adversely affected by the non-primary task or its solution.
1. A method for controlling an overdetermined system with multiple actuators, wherein the actuators perform a primary task and a non-primary task, the method comprising:
a) determining a pseudo-control command u p ∈ p′ based on a physical model of the overdetermined system,
b) determining a control matrix D, D∈ ∈ p′×k according to u p =Du, so that u 1 =D −1 u p u 1 ∈ k represents a control command for the actuators to perform the primary task,
c) projecting the non-primary task into a null space N(D) of the primary task, so that Du 2 =0 if u 2 u 2 ∈ k represents a control command for the actuators to perform the non-primary task, and
d) controlling the actuator using the control commands provided from b) and c).
2. The method as claimed in claim 1 , further comprising calculating the null space N(D) according to
N ( D )= I−D T ( D −1 ) T ∈ k×k
with u 1 =D −1 u p , wherein u 1 ∈ k is a vector or control command which contains actuator commands which solve the primary task and generate necessary physical control commands u p for the actuators, with
u 1 ∈U:={u∈ k |u min ≤u≤u max }.
3. The method as claimed in claim 2 , wherein u 2 org ∈ k is first calculated to solve the non-primary task, and then u 2 org is projected onto the null space N(D) of the control matrix D according to u 2 =Nu 2 org .
4. The method as claimed in claim 3 , further comprising subsequently calculating a complete control command u as follows:
u=u 1 +u 2 ,
wherein u and u 1 generate same physical control commands u p .
5. The method as claimed in claim 4 , further comprising specifying for u 2 org that
u 2 org ∈u 2 :={u 2 org ∈ k |u min −u 1 ≤N 2 u 2 org ≤u max −u 1 }.
6. The method as claimed in claim 1 , wherein the overdetermined system is for an aircraft having multiple, k, propulsion units, and the propulsion units form at least some of the actuators of the overdetermined system.
7. The method as claimed in claim 1 , further comprising calculating an inverse of the control matrix D according to u 1 =D −1 u p for the primary task such that an expression u 1 T Wu 1 is minimized, where u=D −1 (W,u min , u max )u p , with weighting matrix W i ∈ k×k and physical load limits u min ∈ k and u max ∈ k with ∀i=1, . . . , k:u k min ≤u i ≤u k max , for each of the actuators.
8. The method as claimed in claim 7 , wherein the expression u 1 Wu 1 is minimized using the Moore-Penrose inversion method.
9. The method as claimed in claim 7 , further comprising calculating the null space N(D) according to
N ( D )= I−D T ( D −1 ) T ∈ k×k
with u 1 =D −1 u p , wherein u 1 ∈ k is a vector or control command which contains actuator commands which solve the primary task and generate necessary physical control commands up for the actuators, with
u 1 ∈U:={u∈ k |u min ≤u≤u max },
wherein u 2 org ∈ k is first calculated to solve the non-primary task, and then u 2 org is projected onto the null space N(D) of the control matrix D according to u 2 =Nu 2 org , and
the control commands for solving the non-primary task, u 2 =Nu 2 org , are calculated such that components of the control commands for solving the primary task, u 1 , approximate to an average by:
(i) calculating an average of the control commands for solving the primary task, u 1 :
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(ii) determining a difference between the control commands for solving the primary task, u 1 , and the average:
u d =u 1 −u 1,m , and
(iii) calculating the control commands for solving the non-primary task in order to reduce u d :
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10. The method as claimed in claim 1 , further comprising, in addition to the primary task and the non-primary task, defining at least one subordinate task which is to be carried out by the actuators, with control commands u 3 =Nu 3 org , so that
u 3 org ( j )=−(1−μ j )( u 1 ( j )+ u 2 ( j )),
wherein a complete control command is then calculated as follows:
u= 1 +u 2 +u 3 .
11. The method as claimed in claim 10 , wherein the at least one subordinate task is a load reduction μ j for at least one of the actuators as defined by j, μ j ∈[0,1].
12. The method as claimed in claim 1 , further comprising, in addition to the primary task and the non-primary task, defining at least one subordinate task which is to be carried out by the actuators, with control commands u 3 =Nu 3 org , so that
u 3 org ( j )=−(1−μ j ) u max ,
wherein a complete control command is then calculated as follows:
u=u 1 +u 2 +u 3 .
13. The method claim 12 , wherein the at least one subordinate task is a load reduction μ j for at least one of the actuators as defined by j, μ j ∈[0,1].
14. The method as claimed in claim 1 , wherein the actuators are multiple propulsion units for an aircraft, and said pseudo-control command represent torques and a total thrust force acting on the system.
15. A control unit comprising a processor for controlling an overdetermined system with multiple actuators, wherein the actuators perform a primary task and a non-primary task, the control unit being configured for:
a) determining a pseudo-control command u p ∈ p′ based on a physical model of the system,
b) determining a control matrix D, D∈ p′×k according to u p =Du, so that u 1 =D −1 u p u 1 ∈ k represents a control command for the actuators to perform the primary task,
c) projecting the non-primary task into a null space N(D) of the primary task so that Du 2 =0 if u 2 ,u 2 ∈ k represents a control command for the actuators to perform the non-primary task, and
d) controlling the actuators using the control commands provided from b) and c).
16. The control unit as claimed in claim 15 , wherein the actuators are multiple propulsion units for an aircraft, and said pseudo-control command represent torques and a total thrust force acting on the system.
17. The control unit as claimed in claim 16 , further comprising sensors for at least one of measuring or determining parameters and states of at least one of the system or the actuators, which are required for a determination of the pseudo-control command according to step a).
18. The control unit as claimed in claim 15 , further comprising an actuator observation device configured to determine, based on measured or certain parameters and states of at least one of the system or the actuators, the actuators defined by j for which a subordinate task is to be carried out and to provide an associated value μ j ∈[0,1] if necessary.
19. An overdetermined system comprising a control unit and multiple actuators actively connected to the control unit, and operable to perform a primary task and a non-primary task, wherein the control unit is configured to:
a) determine a pseudo-control command u p ∈ p′ based on a physical model of the system,
b) determine a control matrix D, D∈ p′×k according to u p =Du, so that u 1 =D −1 u p u 1 ∈ k represents a control command for the actuators to perform the prima task,
c) project the non-primary task into a null space N(D) of the primary task so that Du 2 =0 if u 2 ,u 2 ∈ k represents a control command for the actuators to perform the non-primary task, and
d) control the actuators using the control commands provided from b) and c).
20. The overdetermined system of claim 19 , wherein the overdetermined system comprises a multicopter aircraft comprising the control unit and the actuators, wherein the actuators comprise multiple propulsion units of the multicopter aircraft including a number of electrically driven rotor units.