Robust Fault-Tolerant Control for Satellite Attitude Stabilization Based on Active Disturbance Rejection Approach with Artificial Bee Colony Algorithm

Algorithm 2

Robust fault-tolerant control algorithm for satellite stabilization based on ADRC with bandwidth parameter optimization by ABC.

Input 𝑋_𝑙(0),𝑈_𝑙(0),𝑈_𝑟(0),𝑢_𝑓,𝑣₀₁,𝑣₀₂,𝑣₀₃,𝜔_best

Initialize 𝐴_𝑙,𝐵_𝑙,𝑁_𝑡,ℎ₁,𝑟₁,ℎ₂,𝑟₂

For 𝑘=1 to 𝑁_𝑡

The model of the attitude control system is

𝑋_𝑙(𝑘+1)=𝐴_𝑙𝑋_𝑙(𝑘)+𝐵_𝑙𝑈_𝑙(𝑘)+𝐺_𝑙𝑇_𝑙𝑌(𝑘)_𝑙(𝑘)=𝐶_𝑙𝑋_𝑙(𝑘)

𝑣₀₁,𝑣₀₂,𝑣₀₃ are input to TDs.

 𝑣_𝑙1(𝑘+1)=𝑣_𝑙1(𝑘)+ℎ₁𝑣_𝑙2(𝑘);

 𝑣_𝑙2(𝑘+1)=𝑣_𝑙2(𝑘)+ℎ₁fhan(𝑣_𝑙1(𝑘)−𝑣₀₁(𝑘),𝑣_𝑙2(𝑘),𝑟₁,ℎ₁);

 𝑣_𝑙3(𝑘+1)=𝑣_𝑙3(𝑘)+ℎ₁𝑣_𝑙4(𝑘);

 𝑣_𝑙4(𝑘+1)=𝑣_𝑙4(𝑘)+ℎ₁fhan(𝑣_𝑙3(𝑘)−𝑣₀₂(𝑘),𝑣_𝑙4(𝑘),𝑟₁,ℎ₁);

 𝑣_𝑙5(𝑘+1)=𝑣_𝑙5(𝑘)+ℎ₁𝑣_𝑙6(𝑘);

 𝑣_𝑙6(𝑘+1)=𝑣_𝑙6(𝑘)+ℎ₁fhan(𝑣_𝑙5(𝑘)−𝑣₀₃(𝑘),𝑣_𝑙6(𝑘),𝑟₁,ℎ₁).

ESO is established:

 𝑍_𝑙(𝑘+1)=𝐴_𝑙𝑍_𝑙(𝑘)+𝐵_𝑙𝑈_𝑙(𝑘)+𝐿_𝑙𝐸_𝑙(𝑘).

Feedback state errors:

 𝑒_𝑙1(𝑘)=𝑣_𝑙1(𝑘)−𝑧_𝑙4(𝑘);

 𝑒_𝑙2(𝑘)=𝑣_𝑙2(𝑘)−𝑧_𝑙1(𝑘);

 𝑢_0𝑥(𝑘)=−fhan(𝑒_𝑙1(𝑘),𝑒_𝑙2(𝑘),𝑟₂,ℎ₂);

 𝑒_𝑙3(𝑘)=𝑣_𝑙3(𝑘)−𝑧_𝑙5(𝑘);

 𝑒_𝑙4(𝑘)=𝑣_𝑙4(𝑘)−𝑧_𝑙2(𝑘);

 𝑢_0𝑦(𝑘)=−fhan(𝑒_𝑙3(𝑘),𝑒_𝑙4(𝑘),𝑟₂,ℎ₂);

 𝑒_𝑙5(𝑘)=𝑣_𝑙5(𝑘)−𝑧_𝑙6(𝑘);

 𝑒_𝑙6(𝑘)=𝑣_𝑙6(𝑘)−𝑧_𝑙3(𝑘);

 𝑢_0𝑧(𝑘)=−fhan(𝑒_𝑙5(𝑘),𝑒_𝑙6(𝑘),𝑟₂,ℎ₂).

Control laws are

 𝑢_𝑥𝑢(𝑘)=^0𝑥(𝑘)−𝑧_𝑙7(𝑘)𝑏_𝑥;

 𝑢_𝑦𝑢(𝑘)=^0𝑦(𝑘)−𝑧_𝑙8(𝑘)𝑏_𝑦;

 𝑢_𝑧𝑢(𝑘)=^0𝑧(𝑘)−𝑧_𝑙9(𝑘)𝑏_𝑧.

Under fault-free condition, control torques of flywheels are

𝑢_𝑟𝑥(𝑘)=𝑢_𝑥(𝑘);

𝑢_𝑟𝑦(𝑘)=𝑢_𝑦(𝑘);

𝑢_𝑟𝑧(𝑘)=𝑢_𝑧(𝑘);

𝑢_𝑟𝑠(𝑘)=0.

When a fault is detected, isolate the faulty flywheel and activate the spare flywheel.

If 𝑢_𝑥(𝑘)−𝑢_𝑟𝑥(𝑘)≥𝑢_𝑓, then

 𝑢_𝑟𝑠(𝑘+1)=√3𝑢_𝑥(𝑘+1),

 𝑢_𝑟𝑥(𝑘+1)=0,

 𝑢_𝑟𝑦(𝑘+1)=𝑢_𝑦(𝑘+1)−𝑢_𝑥(𝑘+1),

 𝑢_𝑟𝑧(𝑘+1)=𝑢_𝑧(𝑘+1)−𝑢_𝑥(𝑘+1);

If 𝑢_𝑦(𝑘)−𝑢_𝑟𝑦(𝑘)≥𝑢_𝑓, then

 𝑢_𝑟𝑠(𝑘+1)=√3𝑢_𝑦(𝑘+1),

 𝑢_𝑟𝑦(𝑘+1)=0,

 𝑢_𝑟𝑥(𝑘+1)=𝑢_𝑥(𝑘+1)−𝑢_𝑦(𝑘+1),

 𝑢_𝑟𝑧(𝑘+1)=𝑢_𝑧(𝑘+1)−𝑢_𝑦(𝑘+1);

If 𝑢_𝑧(𝑘)−𝑢_𝑟𝑧(𝑘)≥𝑢_𝑓, then

 𝑢_𝑟𝑠(𝑘+1)=√3𝑢_𝑧(𝑘+1),

 𝑢_𝑟𝑧(𝑘+1)=0,

 𝑢_𝑟𝑥(𝑘+1)=𝑢_𝑥(𝑘+1)−𝑢_𝑧(𝑘+1),

 𝑢_𝑟𝑦(𝑘+1)=𝑢_𝑦(𝑘+1)−𝑢_𝑧(𝑘+1).

End.

Output 𝑌_𝑙