Post on 08-Apr-2020
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
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Outline
Motivation Prior Research MRAC Formulation Unstructured Neural Network (SHL) Structured Neural Network (RBF) Sparse Neural Network Approach (SNN) Simulation Results Future Research Goals
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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
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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 πΏ1 Adaptive Control Concurrent Learning
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MRAC Formulation
System Dynamics:
οΏ½ΜοΏ½ = π΄π₯ + π΅Ξ π’ + π π₯ + π΅ππππ¦πππ π¦ = πΆπ₯
π΄ β βπΓπ,π΅ β βπΓπ, π΅πππ β βπΓπ, πΆ β βπΓπ are constant known matrices
Ξ β βπΓπ constant unknown diagonal matrix π(π₯) β βπ unknown continuous differentiable function
π¦πππ β βπ external bounded time-varying command
Reference Model: οΏ½ΜοΏ½πππ = π΄ππππ₯πππ + π΅ππππ¦πππ
π¦πππ = πΆππππ₯πππ
π΄πππ β βπΓπ, πΆπππ β βπΓπ are constant known matrices π΄, π΅Ξ is controllable
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Aref= π΄ β π΅πΎπΏπΏπΏ
MRAC β Adaptive Augmentation
π’ = π’π΅πΏ + π’π΄π΄ Overall Control:
π’π΅πΏ = βπΎπΏπΏπΏπ₯, π₯ = (ππΌ , π₯π) β βπ ποΏ½ΜοΏ½ = π¦ β π¦πππ
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MRAC Formulation
State Tracking Error:
π = π₯ β π₯πππ
Adaptive Controller:
οΏ½ΜοΏ½ = π΄ππππ + π΅Ξ(π’π΄π΄ + π π₯ + πΌ β Ξβ1 π’π΅πΏ)
π’π΄π΄ = βπΎοΏ½π΅πΏ(βπΎπΏπΏπΏπ₯) β ΞοΏ½TΞ¦(π₯)
Tracking Objective: limtββ
π₯ π‘ β π₯πππ π‘ β€ π
ΞοΏ½ β β π+1 Γπ, Ξ¦ π₯ β β π+1 are matrices of NN weights
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Unstructured Neural Network (SHL)
State Tracking Error:
π π₯ = πππ πππ + π
π = ππ ππ π β β(π+1)Γπ
NN Approximation Theorem:
π = ππ ππ π β β(π+1)Γπ
Adaptive Controller:
π’π΄π΄ = βποΏ½ ππ ποΏ½ππ , π β βπ+1
Adaptive Update Laws:
ποΏ½Μ = ππππ(2Ξπ((π(ποΏ½ππ)-οΏ½ΜοΏ½(ποΏ½ππ) ποΏ½ππ)ππππ΅) ποΏ½Μ = ππππ(2Ξππππππ΅ποΏ½ ποΏ½ΜοΏ½(ποΏ½ππ)) Distribution A: Approved for public release; distribution is unlimited.
Structured Neural Network (RBF)
State Tracking Error:
Adaptive Controller:
π’π΄π΄ = βποΏ½ ππ π₯
Adaptive Update Law:
ποΏ½Μ = ππππ Ξππ π₯ ππππ΅
π π₯ = π1 π₯ , β¦ ,ππ π₯ , 1 π β βπ+1 is a vector of π RBFs ποΏ½ β β(π+1)Γπ are the outer layer weights
ππ π₯ = ππ₯βπ₯π
2
2ππ· , βπ = 1, β¦ ,π π₯π is the fixed center ππ΄ is the RBF width
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Sparse Neural Network (SNN)
State Tracking Error:
Adaptive Controller:
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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: π = π1, β¦ ,ππ π = π 1, β¦ , π π π β β π is the total number of segments πΌ = {1, β¦ ,π}
π π = {π₯ππ β π: π· π₯ππ,ππ β€ π· π₯ππ,ππ βπ β π}
π·:π Γ π β β π₯ππ β βπ
SNN Definitions:
Metric Space π,π· :
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Sparse Neural Network (SNN)
State Tracking Error:
Adaptive Controller: π β β is the total number of nodes π β β where π = π
π
π is the number of nodes per segment
πΈπβπΌ = {ππΏ πβ1 +1, β¦ , πππΏ} where π =βͺπβπΌ πΈπ
π = π1, β¦ , ππ π΅ = {1, β¦ ,π}
SNN Definitions:
Adaptive Controller: Nodes per Segment:
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Sparse Neural Network (SNN)
State Tracking Error:
Adaptive Controller: πΈπ΄π = πΈπ βπ β πΌ
Pure Sparse Approach (R=Q):
Adaptive Controller: Blended Approach (R>Q):
π β β is the number of active nodes
πΈπ΄π β πΈπ βπ β πΌ
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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:
π’π΄π΄ = βπποΏ½ ππ ποΏ½ πππ
πποΏ½Μ = ππππ(2Ξπ((π(πποΏ½ππ)-οΏ½ΜοΏ½(πποΏ½
ππ) ποΏ½ πππ)ππππ΅)
πποΏ½Μ = ππππ(2Ξππππππ΅πποΏ½ ποΏ½ΜοΏ½(πποΏ½
ππ))
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Sparse Neural Network (SNN)
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Simulation
Adaptive Controller: Longitudinal Short-Period Dynamics for High-Speed Flight Vehicle:
Adaptive Controller: Flight Condition:
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Results - LQR
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Results - SHL
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Results - RBF
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Results β RBF vs SHL
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Results - SNN
Adaptive Controller: Longitudinal Short-Period Dynamics for High-Speed Flight Vehicle:
Adaptive Controller:
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Since the disturbance, π(π₯) , was designed based on a single input variable, πΌ , only the 1-D SNN architecture with T=91 segments was employed for simulation results.
Results β SNN
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Results and Conclusions
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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
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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
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Questions?
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