Optimal Nonlinear Neural Network Controllers for Aircraft Joint University Program Meeting October 10, 2001 Nilesh V. Kulkarni Advisors Prof. Minh Q. Phan.

Optimal Nonlinear Neural Network Controllers for

AircraftJoint University Program Meeting

October 10, 2001

Nilesh V. Kulkarni

Advisors

Prof. Minh Q. Phan

Dartmouth College, Hanover, NH

Prof. Robert F. StengelPrinceton University, NJ

Presentation Outline:

Research goals. Definition of the problem. Parametric optimization approach. Modified Approach. Neural Network implementation. Linear System Implementation. Nonlinear System Implementation. Conclusions

Research Goals

To come up with a control approach that is: Optimal or approaching optimality in the

limit Applicable to both linear and non-linear

systems Data-based (no need for an explicit

analytical model of the system) Adaptive to account for slowly time-varying

dynamics and operating conditions. Application to aircraft.

General Problem Statement:

For the system dynamics:

Find a control law:

To maximize a performance index (minimize a cost function)

]),(),(),([)( kkpkukxfkx 1

)]([)( kxuku

1

112

1

i

TT ikRuikuikQxikxJ )]()()()([

)](),([)( kukxhky

Approaches:

Dynamic optimization approaches: Calculus of variations approach.

Euler-Lagrange equations. Dynamic programming.

Hamilton-Jacobi-Bellman equation. Specialization to Adaptive critic designs.

Static optimization approach: Parametric Optimization. Cost-to-go approach.

Direct Parametric Optimization Approach:

Methodology:

Find the unknown coefficients,’G’ , that minimize the cost-to-go function

Disadvantages:

r

i

TT ikRuikuikQxikxGkV1

112

1)]()()()([),(

)( 1kx))(),(( kukxf)),(( Gkxu)(kx

This approach reduces to solving a static optimization problem which is highly nonlinear even for linear systems

Easily gets stuck in spurious local minima even for the case of finding a linear optimal controller for a linear system

Chance of finding a workable optimal controller using such an approach in practice is very limited.

Illustrative Example:For a simple linear time invariant system,

and

we can write,

And so,

As seen the cost-to-go function expressed with a single parameter ‘G’ , is a highly nonlinear function of the parameter and as seen from several test examples was found to contain several minima.

)()()( kBukAxkx 1

)()( kGxku

)()()( kxBGAikx i

)(])()()()[()(

)]()()()([),(

kxBGARGGBGABGAQBGAkx

ikRuikuikQxikxGkV

r

i

iTTiiTiT

r

i

TT

1

11

1

2

1

112

1

]),([)(

...

]),([)(

]),([)(

rr Gkxurku

Gkxuku

Gkxuku

1

1 22

11

Modified Approach: Reformulate the control law:

Set up the cost-to-go function in terms of the

‘G’s’:

r

i

TTr ikRuikuikQxikxGGkV

11 11

2

1)]()()()([),..,,(

‘r’ represents the order of approximation of the cost-to-go function

Modified Approach… Find the G’s by imposing the stationarity conditions:

and

Solving the second set of equations is not as easy and even less implementable in terms of a control architecture.

x(k), the present state of the system appears as a coefficient in the stationarity conditions.

By solving the stationarity conditions for multiple x(k)’s, presents enough equations for solving for the unknown G’s without solving the second set of conditions.

0... 0 021

;;;

rGV

GV

GV

),,,(

...

),,,(

RQGfGG

RQGfGG

rrr 1

122

Illustrative Example:For a simple linear time invariant system,

we can write,

And so,

As seen the cost-to-go function now expressed with the parameters ‘G’s’ , is a quadratic function of the parameter and therefore has a single minimum

)()()( kBukAxkx 1

)()(

...

)()(

)()(

kxGrku

kxGku

kxGku

r

1

1 2

1

)()...()( kxBGABGBGAAikx iiii

111

)(]...)()(...

)...()...[()()(

kxRGGRGGBGAQBGA

BGABGAQBGABGAkxkV

Tr

Tr

T

rrrT

rrrT

1111

11

2

1

Role of Neural Networks

For a nonlinear system, the controller is typically nonlinear.

Cost-to-go function is a nonlinear function. Being universal function approximators, Neural

Networks present themselves as ideal tools for handling nonlinear systems in the proposed Cost-to-go design approach

Neural networks present a straightforward approach for making the design adaptive even in the case of a nonlinear system.

Parameterize the cost-to-go function using a Neural Network (CGA Neural Network)

Inputs to the CGA Network:x(k), u(k),…,u(k+r-1)

Use the analytical model, or a computer simulation or the physical model to generate the future states.

Use the ‘r’ control values and the ‘r’ future states to get the ideal cost-to-go function estimate.

Use this to train the CGA Neural Network

Formulation of the Control Architecture: NN Cost-to-go function Approximator

r

i

TT ikRuikuikQxikxkV1

112

1)]()()()([)(

CGA Neural Network Training

Actual Systemor

Simulation Model

Neural Net Cost-to-go

Approximator

x(k)u(k)

+

V(k)

Vnn(k)

Verr

Neural Network Cost-to-go Approximator Training

u(k+1)u(k+r-1)

Formulation of the Control Architecture: NN Controller

Instead of a single controller structure (G), we need ‘r’ controller structures.

The outputs of the ‘r’ controller structures, generate u(k) through u(k+r-1).

Parameterize the ‘r’ controller structures using an effective Neural Network.

x(k)Neural

Network

Controller

u(k)

u(k+1)…u(k+r-1)

Neural Network Controller Training

Gradient of V(k) with respect to the control inputs u(k) ,…, u(k+r-1) is calculated using back-propagation through the ‘CGA’ Neural Network.

These gradients can be further back-propagated through the Neural Network controller to get, through

Neural Network controller is trained so that1GkV

)(

rGkV

)(

riGkV

i

...,)(

10

x(k)

Neural

Network

Controller

TrainedCost-to-go

Neural Network

u(k)

x(k)

V(k)

1

u(k+1)…u(k+r-1)

iukV

)(

Advantages of the formulation The modified parametric optimization simplifies the

optimization problem. CGA Network training and the controller Network training is

decoupled. Implementation is system independent. So the basic

architecture remains the same for linear or nonlinear systems.

Implementation is data-based. No explicit analytical model needed.

Parameterization using Neural Networks makes the control architecture adaptive.

Order of approximation ‘r’ in the definition of V(k) serves as a tuning parameter.

Implementation for Linear Systems:

Motivation: Linear systems provide an easy way

to see the details of the implementation of the cost-to-go design.

Provides a means for comparison of the results with existing solutions.

Optimal Control of Aircraft Lateral Dynamics

a

r

kφ

kr

kp

kβ

kφ

kr

kp

kβ

0003.00003.0

0179.00024.0

0637.00504.0

0529.00

)(

)(

)(

)(

100098.00004.0

0001.09992.00035.00690.0

002.00075.09665.00848.0

0036.00099.009811.0

)1(

)1(

)1(

)1(

1

))(

)(

10

01)()(

)1(

)1(

)1(

)1(

500000

05000

00100

00010

)1()1()1()1(()(rk

ki r

ara iδ

iδiδiδ

iφ

ir

ip

iβ

iφiripiβkV

)()()()()(

)(kφkrkpkβG

k

k

a

r

Airplane State Variables:

β - Side slip angle

p- Roll rate

r - Yaw rate

φ - Roll angle

Airplane Input Variables:

r - Rudder Deflection

a - Aileron DeflectionPhoenix Hobbico

Hobbistar 60tm model

Order of approximation (‘r’)

Evaluatedcost (‘J’)

Control gain

25 10.1598

35 10.0935

50 10.0925

Gain obtained using the new data-based approach:

4.8498-5.0124-0.3318-7.0398-

19.39011.1945-3.41204.3039-G

1355.55207.44424.05535.6

6319.167872.07868.37907.3G

8351.49646.43411.09927.6

7413.181687.13616.32247.4G

8365.40102.53313.00389.7

3265.191967.14077.33002.4G

Optimal controller gains calculated using LQR optimal solution with perfect knowledge of the system:

Evaluated optimal cost = 10.0925

- State trajectories using the cost-to-go design (r = 35)*** - State trajectories using the cost-to-go design (r = 50) - State trajectories using LQR based optimal control

0 0.5 1 1.5 2 2.5 3-0.2

0

0.2

Sid

e s

lip (

rad)

0 0.5 1 1.5 2 2.5 3-0.2

0

0.2R

oll

rate

(ra

d/s

)

0 0.5 1 1.5 2 2.5 30

0.1

0.2

Yaw

rate

(ra

d/s

)

0 0.5 1 1.5 2 2.5 3-10

-5

0

5x 10

-3

time (sec)

Roll

angle

(ra

d)

Comparison of the state trajectories using the cost-to-go design and the LQR design

0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

Aile

ron

Inpu

t (r

ad)

0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

time (sec)

Rud

der

inpu

t (r

ad)

- Control trajectories using the cost-to-go design (r = 35)*** - Control trajectories using the cost-to-go design (r = 50)

- Control trajectories using LQR based optimal control

Comparison of the control trajectories using the cost-to-go design and the LQR design

Nonlinear Control of Aircraft in an approach configuration

h hglide

xglide

x

V

Aircraft in an approach configuration

e

TTT

Vg

mVL

mVT

gmD

mT

V

Vh

VX

)(

cossin

sincos

sin

cos

.

Equations of motion in the wind-axes system

Vnom (ft/s) nom (deg) nom (deg) Tnom (lb)

235 -3 3.6 16800

Nominal Flight Conditions

M (slugs)

Tmax (lb) Sref (ft2) CL0 CLα CD0 e

4660 42000 1560 1.36

5.04

0.064

4 0.067

Aircraft Parameters

Implementation Details Equations are written with a change of coordinates while maintaining the

nonlinearity.

X and h are transformed through a coordinate transformation,

so that now they represent perturbations along and perpendicular to the approach slope and we can now ignore the dynamics of the perturbation along the approach slope.

Tnomnom

nom

Tnomnomnomnomnom

nom

TtTt

ututu

TtTtVtVththtXtX

txtxtx

)()(

)()(

)()()()()()()(

)()()(

nomnomapproach

nomnomapproach

Xhh

XhX

sincos

cossin

Implementation Details… Equations of motion in terms of the nonlinear perturbation

dynamics:

Equations are discretized with a time step of 0.5 seconds. Specification of the cost function:

e

nomnomnom

nomnom

nom

nomnomnom

nomapproach

TTT

VVg

VVmL

VVmTT

gmD

mTT

V

VVh

)(

)cos()()(

)sin()(

)(

)sin()cos()(

)sin()(

.

t

t

i

TT

f

iuRiuixQixJ1

112

1)]()()()([

Tapproach T ΔΔTVhx ; T

])([

])[(8

824

101

1011010

diagR

diagQ

Control Architecture

Controller hidden layer

(sigmoid)

Controller output layer

(linear)

Controller present and

future outputs CGA hidden

layer (sigmoid)

CGA output layer

(linear)

X(k) V(k)

Combined Neural Network having the CGA Network in front of the Controller Network.

Forming a combined Neural Network Fix the weights of the CGA part of the Network Training inputs to the network: Random values of

x(k) Train the Network so that it gives the output value

of zero for all the input random x(k)

ControllerNetwork

Subnet 1

Subnet 2

Subnet r

x(k)

A LayerWith

QuadraticNeurons

V(k)

x(k+2)

x(k+1)

u(k)

x(k+r)

u(k+1)

u(k+r-1)

u(k)

u(k+1)

u(k+r-1)

A Control Architecture Proposed to Simplify the Neural Network Training Problem

CGA Network

Bringing Structure to the CGA Network

Implementation of the quadratic layer

1x

2x

1q

2q

0

0

211xq

222 xq

)( 222

211 xqxq

Layer withsquare neurons

1

1

Advantages of the new structure:

Guaranteed positive definiteness. Replaced training of a complex

function by the training of several simpler functions.

A good quality control ability. Allow for hybrid architecture

u(k),…,u(k+3)

Subnet 2

Subnet 3

Subnet 4

Subnet 5

Subnet 1

Subnet 3

Subnet 3

Subnet 4

Subnet 4

Subnet 5

Quadratic Layerof Neurons

x(k)

u(k)

u(k),…,u(k+4)

x(k+1)

x(k+3)

x(k+4)

x(k+5)

u(k+5),…,u(k+9)

u(k),…,u(k+9)

V(k)

x(k+6)

x(k+10)

u(k),u(k+1)

u(k),…,u(k+2)

u(k+3),…,u(k+5)

u(k+4),…,u(k+7)

u(k+5),…,u(k+8)

10

1

k

ki

TT RuuQxx )(

Implementation of the Hybrid CGA Network of order ‘r = 10’, using trained subnets of order 1 through 5

Internal Structure of the Neural Network Controller showing the separate Controller Subnets

x(k)

u(k)

u(k+i)

u(k+r-1)

Neural Network Controller training using the trained CGA Network

1

x(k)

ControllerNetwork

(G1,G2,…,Gr)

TrainedCGA

Network

u(k)x(k)

V(k)estu(k+1)

u(k+r-1)

iu

V

Combination Network with the controller Network before the Critic Network

Cumulative value of V(k) getting minimized with training of the Neural Network Controller Weights

. Aircraft response after an Initial perturbation with and without control

. -Open loop dynamics, - Response with the Optimal Nonlinear Neural Network Controller, ‘O’ – Response with the LQR

TX 7000-0.1530-2001 )(

Neural Network Controller Outputs

‘r’ ‘J’

5 410.8852

10 168.2999

15 93.6265

20 92.4814

25 88.5529

A comparison of the cost function, J, as a function of the order of approximation, ‘r’ of

V(k)

Nonlinear Optimization- Global or Local

Conclusions:

New Neural Network Control Architecture for optimal control.

Applicable to both linear and nonlinear systems

Data based. Systematic training procedure. Confirmation on a Nonlinear Aircraft

Model.