Adaptive Feedforward Control for Uncertain Linear System

Yang WangSupervisor: Prof. T. Parisini

Imperial College London

UKACC PhD Presentation Showcase

Problem Formulation

UKACC PhD Presentation Showcase Slide 2

PlantInput Output

Reference

• LTI• Stable• Uncertain

Problem Formulation

UKACC PhD Presentation Showcase Slide 2

Plant

Disturbances

Input Output

• LTI• Stable• Uncertain

Reference

Problem Formulation

UKACC PhD Presentation Showcase Slide 2

Plant

Disturbances

Input Output

• LTI• Stable• Uncertain

• Periodic• Source unknown• Not measurable

Reference

Problem Formulation

UKACC PhD Presentation Showcase Slide 2

Plant

Disturbances

Input Output

• LTI• Stable• Uncertain

• Periodic• Source unknown• Not measurable

Reference

Problem Formulation

UKACC PhD Presentation Showcase Slide 2

Plant

Disturbances

Input Output

• LTI• Stable• Uncertain

• Periodic• Source unknown• Not measurable

Reference

1. Engine Noise

2. Wind Noise

3. Tire (Break) Noise

Problem Formulation

UKACC PhD Presentation Showcase Slide 2

Plant

Disturbances

Input Output

• LTI• Stable• Uncertain

• Periodic• Source unknown• Not measurable

Reference

Goal: Minimize the influence of the disturbances

Problem Formulation

UKACC PhD Presentation Showcase Slide 3

Plant

Disturbances

Input OutputReference

Goal: Minimize the influence of the disturbances

lim𝑡𝑡→∞

(𝑦𝑦 𝑡𝑡 − 𝑟𝑟 𝑡𝑡 ) = 0

𝑑𝑑(𝑡𝑡)

𝑟𝑟(𝑡𝑡)

𝑑𝑑(𝑡𝑡) = 𝑎𝑎 sin(𝜔𝜔∗𝑡𝑡 + 𝜙𝜙)

�̇�𝑥 = 𝐴𝐴 𝜇𝜇 𝑥𝑥 + 𝐵𝐵 𝜇𝜇 (𝑑𝑑 + 𝑟𝑟)

𝑦𝑦(𝑡𝑡) = 𝐶𝐶 𝜇𝜇 𝑥𝑥(𝑡𝑡)

Solution: Adaptive Forward Control

UKACC PhD Presentation Showcase Slide 4

Plant

Disturbances

Reference 𝑦𝑦(𝑡𝑡)

𝑑𝑑(𝑡𝑡)

Solution: Adaptive Forward Control

UKACC PhD Presentation Showcase Slide 4

Plant

Disturbances

𝑦𝑦(𝑡𝑡)

𝑑𝑑(𝑡𝑡)

Solution: Adaptive Forward Control

UKACC PhD Presentation Showcase Slide 4

Plant

Disturbances

𝑦𝑦(𝑡𝑡)

DisturbancesCopy

�̂�𝑑(𝑡𝑡)

𝑑𝑑(𝑡𝑡)

FeedForward Control

Solution: Adaptive Forward Control

UKACC PhD Presentation Showcase Slide 4

Plant

Disturbances

𝑦𝑦(𝑡𝑡)

DisturbancesCopy

�̂�𝑑(𝑡𝑡)

𝑑𝑑(𝑡𝑡)

FeedForward Control

𝜔𝜔𝑎𝑎𝜙𝜙

Solution: Adaptive Forward Control

UKACC PhD Presentation Showcase Slide 4

Plant

Disturbances

𝑦𝑦(𝑡𝑡)

DisturbancesCopy

�̂�𝑑(𝑡𝑡)

𝑑𝑑(𝑡𝑡)

FeedForward Control

𝜔𝜔∗

𝑎𝑎𝜙𝜙

Key Point: 𝑑𝑑(𝑡𝑡) not available

𝑦𝑦 𝑡𝑡 = 𝑎𝑎 𝑊𝑊 𝑗𝑗𝜔𝜔∗ sin 𝜔𝜔∗𝑡𝑡 + 𝜙𝜙 + ∠𝑊𝑊 𝑗𝑗𝜔𝜔∗

unknown unknown

Solution: Adaptive Forward Control

UKACC PhD Presentation Showcase Slide 4

Plant

Disturbances

𝑦𝑦(𝑡𝑡)

DisturbancesCopy

�̂�𝑑(𝑡𝑡)

𝑑𝑑(𝑡𝑡)

FeedForward Control

𝜔𝜔∗

�𝑎𝑎(𝑡𝑡)�Φ(𝑡𝑡)

Key Point: 𝑑𝑑(𝑡𝑡) not available

𝑦𝑦 𝑡𝑡 = 𝑎𝑎 𝑊𝑊 𝑗𝑗𝜔𝜔∗ sin 𝜔𝜔∗𝑡𝑡 + 𝜙𝜙 + ∠𝑊𝑊 𝑗𝑗𝜔𝜔∗

unknown unknown

Estimator

Adaptive Observer

Estimator

UKACC PhD Presentation Showcase Slide 5

�𝜃𝜃(1)𝑅𝑅𝑅𝑅[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

𝐼𝐼𝐼𝐼[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

Estimator

UKACC PhD Presentation Showcase Slide 5

�𝜃𝜃(1)𝑅𝑅𝑅𝑅[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

𝐼𝐼𝐼𝐼[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

Estimator

UKACC PhD Presentation Showcase Slide 5

�𝜃𝜃(1)𝑅𝑅𝑅𝑅[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

𝐼𝐼𝐼𝐼[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

Estimator

UKACC PhD Presentation Showcase Slide 5

�𝜃𝜃(1)𝑅𝑅𝑅𝑅[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

𝐼𝐼𝐼𝐼[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]Price:

• Computation burden

• Slower convergence

Estimator

UKACC PhD Presentation Showcase Slide 5

�𝜃𝜃(1)𝑅𝑅𝑅𝑅[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

𝐼𝐼𝐼𝐼[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]Price:

• Computation burden

• Slower convergence

Achievements:

• Time-varying system

• Time-varying disturbance

Simulation Results

UKACC PhD Presentation Showcase Slide 6

Plant :𝑊𝑊 𝑠𝑠 = 2(𝑠𝑠−1)𝑠𝑠2+2𝑠𝑠+5

1) 𝜔𝜔1∗ = 1 , 𝑑𝑑(𝑡𝑡) = 2 sin(𝑡𝑡)

𝑅𝑅𝑅𝑅[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

𝐼𝐼𝐼𝐼[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

𝑊𝑊 𝑗𝑗𝜔𝜔1∗

Simulation Results

UKACC PhD Presentation Showcase Slide 6

Plant :𝑊𝑊 𝑠𝑠 = 2(𝑠𝑠−1)𝑠𝑠2+2𝑠𝑠+5

1) 𝜔𝜔1∗ = 1 , 𝑑𝑑(𝑡𝑡) = 2 sin(𝑡𝑡) 2) 𝜔𝜔2∗ = 3 , 𝑑𝑑(𝑡𝑡) = 2 sin(3𝑡𝑡)

𝑅𝑅𝑅𝑅[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

𝐼𝐼𝐼𝐼[𝑊𝑊 𝑗𝑗𝜔𝜔∗ ]

𝑊𝑊 𝑗𝑗𝜔𝜔1∗

𝑊𝑊 𝑗𝑗𝜔𝜔2∗

Results

UKACC PhD Presentation Showcase Slide 7

Plant :𝑊𝑊 𝑠𝑠 = 2(𝑠𝑠−1)𝑠𝑠2+2𝑠𝑠+5

0 50 100 150

-3

-2

-1

0

1

2

0 50 100 150

-3

-2

-1

0

1

2

1) 𝜔𝜔1∗ = 1 , 𝑑𝑑(𝑡𝑡) = 2 sin(𝑡𝑡) 2) 𝜔𝜔2∗ = 3 , 𝑑𝑑(𝑡𝑡) = 2 sin(3𝑡𝑡)

Intermediate Results and Further Plans

Simplify controller

Extend to MIMO system

Combine controller with frequency estimator

UKACC PhD Presentation Showcase Slide 8

Remove the SPR-like conditions in AFC of uncertain system

Current results:

Plans: