Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
Improving Long-Term Learning of Model Reference Adaptive
Controllers for Flight Applications: A Sparse Neural Network Approach
AIAA Guidance, Navigation, and Control Conference January 2017
Scott A. Nivison Pramod P. Khargonekar
Department of Electrical and Computer Engineering University of Florida
Distribution A: Approved for public release; distribution is unlimited.
Outline
Motivation Prior Research MRAC Formulation Unstructured Neural Network (SHL) Structured Neural Network (RBF) Sparse Neural Network Approach (SNN) Simulation Results Future Research Goals
Distribution A: Approved for public release; distribution is unlimited
Motivation
Highly Dynamic Flight Vehicles Trade-offs: Number of Nodes and Learning Rates
Sparse Learning Sparse Auto-encoders Sparse Activation Function: Linear Rectifier Sparse Optimization Techniques: Max-out and
Channel-out
Long-Term Learning Performance: Uncertainty Estimates
Distribution A: Approved for public release; distribution is unlimited.
Prior Research
Develop a MRAC architecture that improves long-term learning and tracking performance of flight vehicles with consistent uncertainties over regions of the flight envelope while utilizing small to moderate learning rates and significant processing constraints.
Research Goals
Enhancements to the MRAC architecture 𝐿1 Adaptive Control Concurrent Learning
Distribution A: Approved for public release; distribution is unlimited.
MRAC Formulation
System Dynamics:
�̇� = 𝐴𝑥 + 𝐵Λ 𝑢 + 𝑓 𝑥 + 𝐵𝑟𝑟𝑟𝑦𝑐𝑐𝑐 𝑦 = 𝐶𝑥
𝐴 ∈ ℝ𝑛×𝑛,𝐵 ∈ ℝ𝑛×𝑐, 𝐵𝑟𝑟𝑟 ∈ ℝ𝑛×𝑐, 𝐶 ∈ ℝ𝑝×𝑛 are constant known matrices
Λ ∈ ℝ𝑐×𝑐 constant unknown diagonal matrix 𝑓(𝑥) ∈ ℝ𝑐 unknown continuous differentiable function
𝑦𝑐𝑐𝑐 ∈ ℝ𝑐 external bounded time-varying command
Reference Model: �̇�𝑟𝑟𝑟 = 𝐴𝑟𝑟𝑟𝑥𝑟𝑟𝑟 + 𝐵𝑟𝑟𝑟𝑦𝑐𝑐𝑐
𝑦𝑟𝑟𝑟 = 𝐶𝑟𝑟𝑟𝑥𝑟𝑟𝑟
𝐴𝑟𝑟𝑟 ∈ ℝ𝑛×𝑛, 𝐶𝑟𝑟𝑟 ∈ ℝ𝑝×𝑛 are constant known matrices 𝐴, 𝐵Λ is controllable
Distribution A: Approved for public release; distribution is unlimited.
Aref= 𝐴 − 𝐵𝐾𝐿𝐿𝐿
MRAC – Adaptive Augmentation
𝑢 = 𝑢𝐵𝐿 + 𝑢𝐴𝐴 Overall Control:
𝑢𝐵𝐿 = −𝐾𝐿𝐿𝐿𝑥, 𝑥 = (𝑒𝐼 , 𝑥𝑝) ∈ ℝ𝑛 𝑒�̇� = 𝑦 − 𝑦𝑐𝑐𝑐
Distribution A: Approved for public release; distribution is unlimited.
MRAC Formulation
State Tracking Error:
𝑒 = 𝑥 − 𝑥𝑟𝑟𝑟
Adaptive Controller:
�̇� = 𝐴𝑟𝑟𝑟𝑒 + 𝐵Λ(𝑢𝐴𝐴 + 𝑓 𝑥 + 𝐼 − Λ−1 𝑢𝐵𝐿)
𝑢𝐴𝐴 = −𝐾�𝐵𝐿(−𝐾𝐿𝐿𝐿𝑥) − Θ�TΦ(𝑥)
Tracking Objective: limt→∞
𝑥 𝑡 − 𝑥𝑟𝑟𝑟 𝑡 ≤ 𝜖
Θ� ∈ ℝ 𝑁+1 ×𝑐, Φ 𝑥 ∈ ℝ 𝑁+1 are matrices of NN weights
Distribution A: Approved for public release; distribution is unlimited.
Unstructured Neural Network (SHL)
State Tracking Error:
𝑓 𝑥 = 𝑊𝑇𝜎 𝑉𝑇𝜇 + 𝜖
𝑊 = 𝑊𝑇 𝑏𝑊 𝑇 ∈ ℝ(𝑁+1)×𝑐
NN Approximation Theorem:
𝑉 = 𝑉𝑇 𝑏𝑉 𝑇 ∈ ℝ(𝑛+1)×𝑁
Adaptive Controller:
𝑢𝐴𝐴 = −𝑊� 𝑇𝜎 𝑉�𝑇𝜇 , 𝜇 ∈ ℝ𝑛+1
Adaptive Update Laws:
𝑊�̇ = 𝑃𝑃𝑃𝑃(2Γ𝑊((𝜎(𝑉�𝑇𝜇)-�̇�(𝑉�𝑇𝜇) 𝑉�𝑇𝜇)𝑒𝑇𝑃𝐵) 𝑉�̇ = 𝑃𝑃𝑃𝑃(2Γ𝑉𝜇𝑒𝑇𝑃𝐵𝑊� 𝑇�̇�(𝑉�𝑇𝜇)) Distribution A: Approved for public release; distribution is unlimited.
Structured Neural Network (RBF)
State Tracking Error:
Adaptive Controller:
𝑢𝐴𝐴 = −𝑊� 𝑇𝜙 𝑥
Adaptive Update Law:
𝑊�̇ = 𝑃𝑃𝑃𝑃 Γ𝑊𝜙 𝑥 𝑒𝑇𝑃𝐵
𝜙 𝑥 = 𝜙1 𝑥 , … ,𝜙𝑁 𝑥 , 1 𝑇 ∈ ℝ𝑁+1 is a vector of 𝑁 RBFs 𝑊� ∈ ℝ(𝑁+1)×𝑐 are the outer layer weights
𝜙𝑖 𝑥 = 𝑒𝑥−𝑥𝑐
2
2𝜎𝐷 , ∀𝑖 = 1, … ,𝑁 𝑥𝑐 is the fixed center 𝜎𝐴 is the RBF width
Distribution A: Approved for public release; distribution is unlimited.
Sparse Neural Network (SNN)
State Tracking Error:
Adaptive Controller:
Distribution A: Approved for public release; distribution is unlimited.
Segment flight
envelope into regions and distribute a specified number of nodes to each region.
Activate only a small percentage of the total number of nodes for control at each point in the operating envelope
Sparse Neural Network (SNN)
State Tracking Error:
Adaptive Controller: 𝑃 = 𝑝1, … ,𝑝𝑇 𝑆 = 𝑠1, … , 𝑠𝑇 𝑇 ∈ ℕ 𝑇 is the total number of segments 𝐼 = {1, … ,𝑇}
𝑠𝑖 = {𝑥𝑜𝑝 ∈ 𝑋: 𝐷 𝑥𝑜𝑝,𝑝𝑖 ≤ 𝐷 𝑥𝑜𝑝,𝑝𝑗 ∀𝑖 ≠ 𝑃}
𝐷:𝑋 × 𝑋 → ℝ 𝑥𝑜𝑝 ∈ ℝ𝑁
SNN Definitions:
Metric Space 𝑋,𝐷 :
Distribution A: Approved for public release; distribution is unlimited.
Sparse Neural Network (SNN)
State Tracking Error:
Adaptive Controller: 𝑁 ∈ ℕ is the total number of nodes 𝑄 ∈ ℕ where 𝑄 = 𝑁
𝑇
𝑄 is the number of nodes per segment
𝐸𝑖∈𝐼 = {𝑒𝐿 𝑖−1 +1, … , 𝑒𝑖𝐿} where 𝑌 =∪𝑖∈𝐼 𝐸𝑖
𝑌 = 𝑒1, … , 𝑒𝑁 𝐵 = {1, … ,𝑁}
SNN Definitions:
Adaptive Controller: Nodes per Segment:
Distribution A: Approved for public release; distribution is unlimited.
Sparse Neural Network (SNN)
State Tracking Error:
Adaptive Controller: 𝐸𝐴𝑖 = 𝐸𝑖 ∀𝑖 ∈ 𝐼
Pure Sparse Approach (R=Q):
Adaptive Controller: Blended Approach (R>Q):
𝑅 ∈ ℕ is the number of active nodes
𝐸𝐴𝑖 ⊆ 𝐸𝑖 ∀𝑖 ∈ 𝐼
Distribution A: Approved for public release; distribution is unlimited.
Each segment activates only nodes that were allocated to that segment
Each segment activates R nearby nodes regardless of node segment assignment
Sparse Neural Network (SNN)
State Tracking Error:
Adaptive Controller:
Adaptive Controller:
Adaptive Update Laws:
𝑢𝐴𝐴 = −𝑊𝑖� 𝑇𝜎 𝑉� 𝑖𝑇𝜇
𝑊𝑖�̇ = 𝑃𝑃𝑃𝑃(2Γ𝑊((𝜎(𝑉𝑖�𝑇𝜇)-�̇�(𝑉𝑖�
𝑇𝜇) 𝑉� 𝑖𝑇𝜇)𝑒𝑇𝑃𝐵)
𝑉𝑖�̇ = 𝑃𝑃𝑃𝑃(2Γ𝑉𝜇𝑒𝑇𝑃𝐵𝑊𝑖� 𝑇�̇�(𝑉𝑖�
𝑇𝜇))
Distribution A: Approved for public release; distribution is unlimited.
Sparse Neural Network (SNN)
Distribution A: Approved for public release; distribution is unlimited.
Simulation
Adaptive Controller: Longitudinal Short-Period Dynamics for High-Speed Flight Vehicle:
Adaptive Controller: Flight Condition:
Distribution A: Approved for public release; distribution is unlimited.
Results - LQR
Distribution A: Approved for public release; distribution is unlimited.
Results - SHL
Distribution A: Approved for public release; distribution is unlimited.
Results - RBF
Distribution A: Approved for public release; distribution is unlimited.
Results – RBF vs SHL
Distribution A: Approved for public release; distribution is unlimited.
Results - SNN
Adaptive Controller: Longitudinal Short-Period Dynamics for High-Speed Flight Vehicle:
Adaptive Controller:
Distribution A: Approved for public release; distribution is unlimited.
Since the disturbance, 𝑓(𝑥) , was designed based on a single input variable, 𝛼 , only the 1-D SNN architecture with T=91 segments was employed for simulation results.
Results – SNN
Distribution A: Approved for public release; distribution is unlimited.
Results and Conclusions
Distribution A: Approved for public release; distribution is unlimited.
Conclusions
Traditional Neural Network schemes typically update adaptive weights based solely on the current state vector which leads to poor long-term learning Sparse Neural Network (SNN) adaptive controllers only update a small portion of neurons at each point in the flight envelope Better memory for uncertainty estimates and weights
from previously visited segments Superior tracking performance and uncertainty
estimates for tasks that have consistent uncertainties and disturbances over regions of the flight envelope
Distribution A: Approved for public release; distribution is unlimited.
Future Work
Develop standard analysis tools to explore trade-offs between variations of neural network adaptive controllers Explore effectiveness of high dimensional sparse
neural network (SNN) adaptive controllers against numerous uncertainties
Investigate structured sparse neural networks (SNN) for adaptive control of flight vehicles
Distribution A: Approved for public release; distribution is unlimited.
Questions?
Distribution A: Approved for public release; distribution is unlimited.