Document
N90-20937
Neural Networks for Aircraft Control
Dennis Linse Department of Mechanical and Aerospace Engineering Princeton University Current research in Artificial Neural Networks indicates that net- works offer some potential advantages in adaptation and fault tolerance.
This research is directed at determining the possible applicability of neural networks to aircraft control. The first application will be to aircraft trim.
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• Introduction to Neural Networks • Neural Networks for Aircraft Trim • Application of Neural Networks to Control \
Introduction to Neural Networks
Artificial neural networks (usually called simply neural networks) are an attempt to model the processing behavior of the nervous system. Neural network research focuses on a much larger group of networks loosely clas- sified by the dense interconnection of simple computational elements.
Since the majority of the processing operations are independent of one another, neural networks can conduct massively parallel computations.
While the human brain is estimated to have 100 billion neurons each with approximately 1000 inputs and outputs, artificial neural networks are interested in the computational capabilities of smaller networks which may or may not be biologically correct.
• Artificial neural networks Abstract simulation of real nervous system Dense interconnection of simple computational elements Massively parallel computation Biologically inspired vs. biologically accurate Estimated 100 billion (1011) neurons in brain Each has 1000 inputs and outputs
Introduction to Neural Networks
Most current applications of neural networks have been in the area of pattern recognition. A smaller number of applications have been devoted to optimization, most notably the famous Traveling Salesman Problem in which the shortest route through selected cities is determined.
The major benefits of neural network are two-fold. First, neural networks have the ability to learn their internal knowledge from presentation of input-output data. This means that the network does not have to be programmed in the traditional sense. It must be trained with examples of the desired input-output relationship. Adaptability arises if the learning process continues while the network is in operation. Neural networks are potentially very fault tolerant due to the massively parallel architecture. The knowledge contained in the network is distributed throughout the network so that the loss of individual computational elements should not seriously degrade the performance of the network.
• Applications Speech/Image/Pattern Recognition, Classification and Restoration Optimization • A few potential benefits Adaptability/Learning Robustness / Fault tolerance
Introduction to Neural Networks
Neural network models are differentiated by the type of computational element (or node), the connection of the nodes, and the learning rule used to update the weighted connections between nodes.
t Neural network models are specified by Node characteristics - linear / nonlinear - analog / discrete (binary) Net topology - node interconnections • layers • feedforward/feedback Learning rule - supervised / unsupervised
Neural Network Node Characteristics
An individual neural network node computes a weighted sum of its inputs and runs the sum through a fixed function. The resulting value is transmitted to all of the following network elements. The fixed function, called the node activation function, is generally a bounded, S-shaped non- linear element. A common activation function is the sigmoid function.
The nonlinearity of the activation function provides the computational power of the network. Networks with linear elements have been tested, but most contain some type of nonlinear function.
• Individual Neural Net Node x 2 w 2
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x n-1 Threshold Node Activation Function Sigmoid fix) = 1.0 / (1.0 + exp(-x)) .
Network Topology and Operation
The multilayer feedforward network is characterized by the distinct layers of network nodes connected only in the forward direction. There are no connections between nodes in a single layer nor are there connections from higher layers to lower layers. A co.nstant unit input is used as a threshold or bias. + The operation of the network is easily characterized by a recursive matrix equation. The output of each layer is simply the weighted sum of its inputs passed through the nonlinear activation function f. The knowledge of the system is contained in the weighted connections, i.e., the weights, W.
As the weights change, the nonlinear input-output relationship function modeled by the network changes.
• Multilayer Feedforward Network w(O) f(') (o) W (_) x I f(o) x(O) 2 (2) X 1 x(O) __ x (o) __ (2) X 2 X(0) x(O) Threshold 1.0 1.0 Forward Operation (Activation, Retrieval) xCk) = f(wCk-1) x(k-1) ), k = 1, ..., N layers x(0) = input x(N) = output
Neural Network Learning
In supervised learning, the neural network is trained to match a par- ticular input-output relationship. First an input is applied to the neural network. Next, the output produced by the network is compared to the de- sired output. Finally, the network weights are adjusted based upon the er- ror.
• Supervised Learning Diagram
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Neural Network Learning
The backpropagation learning algorithm adjusts the weights of the network to minimize the mean square error of all of the outputs. The diffi- culty comes when assessing the importance of individual weights of the in- ternal (or hidden) layers on each ogtput. The backpropagation learning al- gorithm explicitly assigns a portion of the error to each element. The one requirement is that the nonlinear function in each network node must be differentiable.
• Learning by Backpropagation Adjust weights to minimize the mean square error Difficulty of error assignment Differentiability of nonlinearity • Backpropagation Wi+l = Wi + nWi AW i = _ x(k-1) 5-(k) fi is the error term for output layer = ( ud - u) for hidden layers, error is backpropagated
Neural Network for Aircraft Trim
A multilayer feedforward neural network is trained on the input- output relationship for the longitudinal trim of a transport aircraft. Using the backpropagation learning rule, the network is taught to produce the aircraft control positions necessary to maintain a given trim state. Such a trim map could be used as part of a perturbation control law. The power of the neural network would eventually be in the ability to continuously update its knowledge of the aircraft during operation.
• Exploit learning/adaptability features of neural networks Teach neural network the map between trim states and trim control positions Useful for Autotrim system Trim map for perturbation control law Straight and level flight Longitudinal states and controls for learning x = { q - dynamic pressure V - forward velocity T- flight path angle a - angle of attack q - pitch rate h- altitude } u = { 5E - elevator deflection Or - throttle deflection
Aircraft Trim Points
The neural network is trained on aircraft trim points ranging from Mach 0.25 at 1000 feet up to Mach 0.9 at 20000 feet.
Trim Points 25OO0 2OOOO _ 15000 5OOO 0.2 0.4 0.6 0.8 1.0 Mach Example Network Run A simple example is shown in which the network is trained only on the aircraft data at 1000 and 5000 foot altitudes. A three layer network is used with 6 inputs, 6 hidden nodes in one layer, and 2 outputs.
• Train network with 17 trim points, 6 units in hidden layer • 5000 Iterations, _ = 0.5, momentum - 0.9 • Final Total Square Error = 0.0004
Error in Learned Response
The results after 5000 iterations are quite good. The elevator error is less than I percent and the throttle error is less than 3 percent over the en- tire learned range.
0.10 0.05 I I I I I 8OO 2OO 300 400 ,500 600 700 Velocity (fl_) 0.0 -0.5 Error D-Th ......... * ......... Error D-Th 300 400 500 600 700 8OO Velocity (fps)
Neighboring Optimal Control Law
using Neural Network
The neural network trained as a trim map fits quite nicely into a neighboring optimal control scheme. A perturbation control law is used to eliminate the errors between the actual aircraft state and the desired trim value. The resulting perturbation control position is combined with the trim control position from the neural network trim map to produce the con- trol input for the aircraft.
• Full state feedback perturbation control law developed for linearized system • Applied to full nonlinear system using a Neighboring Optimal Control structure • Trim values provided by Neural Network
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Example Histories using Neighboring Optimal
Control with a Neural Net
Using the neighboring optimal control scheme with a neural network as the trim map, a smooth transition from 5000 to 2500 feet is effected for the simulated aircraft. The control positions are smooth and accurate over the entire range.
5o00 3oo0 200o 0 60 120 180 240 300 360 420 480 540 Time (sec) -7.1 -7.2 -7.3 0 60 120 180 240 300 360 420 480 540 Time (sec) 36.8 36.6 36.2 36.4 36.0 35.8 35.6 0 60 120 180 240 300 360 420 480 540 Time (sec)
Neural Networks for Control
In addition to use as a trim map, a neural network could possibly be trained as a feedback control element. If the aircraft is modeled by a discrete time linear system, the weights could be adjusted as to minimize the error between the actual system states and some desired states. The mathematics of such a minimization appears feasible.
• Investigate Neural Network as Feedback Control Element
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• Adjust weights to minimize least squares fit error M I _1
-- -- i_l(Xd(i) --x(i) )T (_xd(i) -_x(i) )
• Steepest descent training based on error gradient
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Neural Networks for Aircraft Control
There are many topics which still must be investigated before any valid conclusions may be made about the usefulness of neural networks in air- craft control. The ability of the network to correctly generalize to trim points between the trained data is an important topic which must be looked into. The fault tolerance of neural networks is based upon massive par- allelism. It is unknown how much of this fault tolerance is retained by smaller networks. There are many more possible applications for neural networks that are, as yet, untapped.
Topics of Interest for Neural Networks • Generalization capabilities of networks What happens between learned points?
• Fault tolerance capabilities of networks What happens if nodes or weights malfunction?
• Other uses of neural networks for aircraft control Model identifier Adaptive control element Conclusions • Neural Networks show some promise for control applications • There is much left to investigate