Reial-time identification of missile aerodynamics using a linearised Kalman filter aided by an artificial neural network

M.P. Horton

Indexing terms: Missile uerudynamics, Lineurised Kulman filter, Artijkiul neural network

- Abstract: The paper investigates the problem of real-time identification of aerodynamic derivatives in a guided missile application. This application provides a severe test for any parameter estimator, since it has to identify the linearised parameters of a multivariable, nonlinear, time variant, noisy plant, which is initially unstable and then becomes lightly damped. Initially, two radically different approaches are taken by designing both a linearised Kalman filter (LKF) estimator and an artificial neural network (ANN) based estimator. A lhybrid estimator is then formed by an LKF, whch is aided by the ANN. This produces a new estimator which has superior performance to those from which it is derived. The performance of these estimators is assessed with a nonlinear single plane model against eight types of engagements.

1 Introduction

I. 1 Adaptive control In a tactical missile, the problem of controlling the air- frame throughout flight is handled by a lateral autopi- lot which responds to lateral acceleration demands generated by the guidance loop [I]. The autopilot must maintain acceptable performance over varying altitude, velocity and general engagement conditions, and this requirement can be solved by adaptive control. Usu- ally, autopilots have a simple adaptive nature and fall into the class known as gain scheduled systems. With this solution, the autopilot is designed offline at a number of operating conditions and the required gains are prestored against related conditions which can either be measured directly (e.g. time, pitot pressure) or deriveid in an inertial navigation system (INS) (e.g. velocity, altitude).

An alternative solution is to adapt the gains in response to knowledge acquired directly online about the performance during the engagement. This system is a self-adaptive autopilot and offers the advantages that

0 IEE, 1997 IEE Proceedings online no. 19971 125 Paper received 5th September 1996 The author is with the Guidance and Control Group, Flight Dynamics Department, British Aerospace Defence Ltd., Dynamics Division, Bristol, UK

aerodynamic performance need not be known to the same degree of accuracy and this could reduce the amount of wind tunnel testing and reduce the required build accuracy of the airframe.

A schematic autopilot for a homing application, based on a self-tuning regulator (STR) [l, 21, is shown in Fig. 1. This solution contains a real-time plant parameter identification routine which is used with a real-time gain design law to tune the autopilot to the required response. This response requirement is also derived in real-time by monitoring the seeker signal-to- noise level, closing velocity and range-to-go.

This paper concentrates on the identification aspects of this autopilot and will initially describe two estima- tors based on two quite different approaches: a line- arised Kalman filter (LKF) and an artificial neural network (ANN). These two methods are then brought together to form a hybrid estimator which is an LKF aided by an ANN. This is demonstrated to have supe- rior performance over the individual estimators from which it is derived.

I. 2 Linearised equations of motion After linearisation, the missile airframe equations of motion for a roll-stabilised airframe [l] reduce to the state and output equations in the yaw plane:

L A L J L

where U is the missile longitudinal velocity, v and r are lateral velocity and body rate, s is the rudder fin deflec- tion, y , and y s are aeronormalised force derivatives due to lateral velocity and fin deflection, n,,, n, and ns are aeronormalised moment derivatives due to lateral velocity, body rate and fin deflection, aym is the meas- ured acceleration and 1, is the accelerometer moment arm from the centre of gravity.

The response of this second-order plant (called the weathercock mode) is given by the eigenvalues of the plant matrix. With typical aeronormalised derivatives (hereafter referred to as ‘derivatives’), the mode is found to be lightly damped with a weathercock fre- quency which is dominated by the product Un,. All these derivatives vary with velocity, altitude, Mach number and incidence. In addition, the moment deriva- tives also vary with the centre of gravity movement which occurs with propellant burn. In particular, n, will vary in both sign and magnitude and, when it is nega- tive, the airframe is statically unstable.

299 IEE Proc.-Control Theory Appl , Vol. 144, No. 4, July 1557

, 1::;:Fa targetlmi ssi l e propertes - geometry bandwtth

I seeker 1-4 requirement I t I

t i t

demand latax

fin demand

Fig. 1 Self-tuning regulator (STR) autopilot stwcture

tion of the iden cation routines A detailed single model of a tactical missile has been used to assess the capabilities of each of the three parameter estimators. This model represents a generic ground launched tactical missile in an anti-air role [3]. It has a range of about 8km, which is achieved by a

propellant motor which boosts for about 5.5 sec- to give a peak velocity of Mach 3.5. After burn-

out, the missile coasts to achieve a range of 8km at about 10 seconds. The airframe is initially unstable but, as the propellant burns, the missile centre of gravity moves forward and the airframe becomes stable. Against a high diving target, the missile achieves an altitude of 7km. The model therefore, enables the parameter estimators to experience a time varying envi- ronment over th conditions. The s (i) single plane nonlin ties;

fin servo, sensor and body bending dynamics and cessing delays;

noise, sensor self

(v) atmospheric disturban was also assumed to be

(i) a high quality measurement of missile velocity was assumed since the missile would use an inertial naviga- tion system of good quality. In this evaluation the ‘real world’ value was used; (ii) a simple mass properties model was incorporated so that time-varying data on mass, inertia and accelerome- ter moment arm was available. This model assumed nominal values of mass, tre of gravity and inertia and high quality (‘real WO values for the accelerom- eter position and fin centre of pressure. At this stage, the ad ve system is based on ‘real world’ derivatives enabling the parameter estimators to be assessed without the complication which could result by using these estimated parameter values in the gain design law. In this way, there was enough detail to draw realistic conclusions about the dominant charac- teristics of the parameter estimators.

All initial parameter estimates are set to the ‘best knowledge’ given by the nominal, noise free values. Some evaluation work was also done with uncertain

aerodynamics set at 90% confidence limits and were arbitrary in the choice of extreme, except that the uncertainties in centre of pressure and centre of gravity were arranged to give the most unstable aerodynamic configuration. This was done in anticipation that it will give the most testing identification conditions. This combination represents an extreme condition with a very low probability of occurrence.

In addition, the parameter estimators were evaluated using trajectory data from engagements against both radial and crossing targets. This allows the perform- ance to be assessed against eight cases derived from: (i) the nominal and uncertain aerodynamics; (ii) the sea level and high diving trajectories; (iii) the radial and crossing target engagements. The results for these eight cases appear in summary tables using the measures of quality described in Sec- tion 1.5. In addition, representative time responses are shown for the identification of the derivative n, in the engagement against a 60” diving, radial target with aer- odynamics set at the extremes of uncertainty.

1.4 identification accuracy requirements The requirements for parameter identification accuracy have been found from sensitivity analysis of the autopi- lot. They are expressed as a percentage, except for n, which is expressed as an absolute value since it changes from negative to positive as the fuel burns (Table 1).

Table 1: Derivative accuracy requirements

Yv 10%

nV 0.25

YS 15%

n5 11%

n, 580%

The derivative n, is thus relatively unimportant since it is augmented by damping from the autopilot. Other- wise, these requirements provide a severe test of the parameter estimators which have to identify the line- arised parameters of a multivariable, nonlinear, time variant, noisy plant which is initially unstable and then lightly damped.

1.5 Measures of q A number of criteria have been used to assess the qual- ity of the parameter identifiers.

IEE Proc -Control Theory Appl , Vol 144, NO 4, July 1997 300

(i) Over a complete trajectory of T time steps, the qual- ity of each of the five derivative estimates can be assessled by deriving the RMS error from the estima- tion error Sl, at each time step.

(ii) Over a complete trajectory the quality of all the identified derivatives can be assessed by a performance index (PI) which is the root of the sum of all the squares of the RMS errors as a ratio of their require- ments, Blrry,, given in Table 1

This 1’1 has the merit of placing a disproportionate emphasis on errors which exceed the requirement. (iii) Over a number of trajectories the PIS can be RMSd to give an overall assessment of the parameter estimator.

1.6 Design of a probing signal For successful identification, it is essential that the airframe is excited to give a sequence of fin and meas- urement responses which contain the necessary infor- mation. Although the lateral acceleration response of the airframe can be highly contaminated by both motor vibration noise and seeker thermal noise, this natural excitation has been found to be either unrelia- ble or inadequate for identification purposes, and an explicit probing signal is needed.

This probing signal must have spectral characteristics which excite the airframe modal response and have as large an amplitude as possible subject to control and operational considerations. The best position to apply this probing signal is at the demand to the fin servo rather than as an autopilot acceleration demand because this point provides less attenuation of the high frequency components by the controller.

A suitable probing signal can be designed from a maximal length pseudorandom binary sequence [4]. For simplicity, the probe frequency should be a submultiple of the autopilot sample frequency and a six-stage proc- ess with a bit frequency of 50Hz, gives a sequence period of 63 bits and a sequence frequency of 0.79Hz to provide the necessary spectral content.

Over the majority of the engagement, the probing amplitude is found to be limited by the fin rate capabil- ity and in the latter stages there is a need to reduce the probe amplitude so that it does not increase the miss distance. From investigations with the simulation mod- elling, this sets a probe amplitude of 0.02rad for the majority of the engagement which is reduced linearly, from a range-to-go of 1 km, down to zero at intercep- tion.

2 Liinearised Kalman filter estimator

2. I The K.alman filter is a recursive technique which was originally designed to provide the best estimates of the states of a linear time invariant dynamic system when provided with noisy measurements [5, 61. It is also a versatile technique that can be adapted to far more

Kalman filter for linear systems

IEE Proc -Control Theory Appl , Vol 144, No 4, July 1997

complex situations than those for which it was origi- nally intended. It is one of these adaptations which will be used here.

In the situation where a system has nonlinear dynam- ics or measurements, or when knowledge of the system dynamics is incomplete, it may be desirable to estimate not only the time varying states of the system but also some of the system parameters which govern the dynamics. The Kalman filter can be modified to cope with this situation, thereby making the filter applicable to a wide range of situations that are not of a simple linear form. The modified filters required for nonlinear systems are the extended Kalman filter (EKF) and the linearised Kalman filter (LKF). The EKF and LKF essentially Kalman filters embedded in a process of lin- earisation, and their derivation therefore relies heavily on the theory of the linear Kalman filter which can be found in a number of references [7, 81. In parameter identification, the system input is also required, and this must be incorporated into the equations.

For a linear system with states x, inputs y and system noise w the system dynamics can be described by the normal way in the and discrete form as

XtS1 = Qxt + ru t + T W ~ The system noise is assumed to be a zero mean Gaus- sian white noise sequence of covariance matrix Q.

The measurements provided at the plant output at discrete intervals can be represented in the normal way by the linear equation:

The measurement noise v is assumed to be a zero mean discrete Gaussian white noise process of covariance matrix R.

The essence of the Kalman filter is that it contains a model of the ‘real world’ system in which filter states are taken to be the expectation or mean states:

yt = cx t + Dut + Vt

E { X k } = x so that the error between the ‘real world’ and the esti- mated states 2 = x - ri has covariance given by

and The filter model is a deterministic state estimator with dynamics driven by the mean states such that

E{&,:} = P,t = n E{xtxT} = 0 , t # n

x t + l = axt + ru t and f t = Cxt + Dut The filter seeks to bring the mean filter states into alignment with the ’real world’ states by comparing the values of the actual and predicted measurements, 9 . Any differences between these two quantities are fed back in an optimal way to correct the state estimates by multiplying the difference by the Kalman gains Kt.

x;t+ = x; + Kt(yt - y t )

where the + superscript is the best estimate after incor- porating the measurement and the - superscript is the best estimate before incorporating the measurements.

The Kalman gains are derived by minimising the var- iance of estimation errors. Thus, when a measurement becomes available, it is incorporated as an unbiased, minimum covariance estimate using

Kt = P[ CT (CPL CT + R)-’ to give the covariance as

P,s = ( I - KtC)P,-

301

P- predict I P- I r

I

system , I

Fig. 2 Linear Kalmun filter structure

Kolmon filter

I I P+

I I

I

I

I

I I I I

' C I

I I : - I I I I I system

Fig. 3 Linearised Kalman filter (LKF) structure

Immediately before the next measurement, the states and covariance are predicted to the next sample instant using the embedded model as

X ~ + ~ = (ail$ + r u t and PGl = (aP;@ + TQTT

This Kalman filter is shown in Fig. 2.

2.2 Development of the linearised Kalman filter (LKF) The formulation above is applicable to linear systems only. To use the Kalman filter for identification, the fil- ter needs to be developed for systems that have nonlin- earities inherent in their dynamics. The continuous dynamics of a nonlinear system can be described by

and yt = g(xt, ut) + vt The principle is that a nominal nonlinear model is cho- sen whose states are close to the 'real world' value of the system states, so that the errors from the nominal values are therefore small. In this way, error equations can be formed and the state and measurement equa- tions can be linearised about their nominal values. A linear Kalman filter can then be applied to these equa- tions to obtain the error states. A filter developed about some a priori nominal trajectory is a linearised Kalman filter whilst a filter developed about its current best estimate is an extended Kalman filter. Neither the

A -

X = f(x, U) + Sw

r 1 1 . I ' I ' I

EKF nor the LKF is linear or optimal, but the per- formance is well behaved and good results can be obtained in practice.

A nominal trajectory of states and measurements, X and y can be propagated according to the equations:

x = f(Si,u) and yt = g ( x t , u t ) There will be differences between nominal and esti- mated values:

- J x = k - ~ and S y = y - j i If these differences are small and the input U is known, the nonlinear equations can be expanded as a first- order Taylor series about the nominal values. f (X , U) = f(%, U) + At&X and g ( x , U) = g(x, U) + CthX where A , = df/dxl,=, and C, = dg/dx/,,,

The LKF structure is shown in Fig. 3.

2.3 Linearised Kalman filter for parameter identification The LKF can be used to identify the parameters, 8, of a system by formulating nonlinear equations which give the unknown parameters as pseudostates. These parameter states now augment the natural system states to form a new state vector, so that a filter can be designed to estimate both the system states and the sys-

IEE Proc.-Control Theory Appl., Vol. 144. No. 4, July 1997 302

tem parameters. The dynamics of the parameter states are treated as constants with a random variation. Thus,

The original nonlinear system dynamics and output equations can be recast to emphasise the dependence on the unknown parameters and to introduce the sys- tem input into the state estimation. With this aug- menteld system, the filter is no longer linear and an LKF is now required to provide the best estimates of the augmented state vector. Defining the state and parameter state vectors as

x = [U TI' and e = [ y v n, y, n, nTIT Then, with 'real world' fin deflection angle available from the fin servo, velocity from the INS and a nomi- nal estimate of accelerometer moment arm the error dynamics and measurement matrices are

-U v 0 q 0 0 1 A ~ = ~ T L , f i r 0 .ij 0 5 ~1

1 1, ct = [ g u +i,fi, ranT v I,v s LC

0 1 0 0 0 0 YIt A useful method of accounting for time-varying param- eters without increasing the number of state variables, is to consider the variation as random walk noise. Thus,

The standard deviation of this random walk noise should correspond roughly with the rate of change of the aerodynamic derivatives:

&e E

- 0 c? we

and the system noise covariance is derived as

TQTT = diag[0 0 I h2Qe] Nominal trajectories of the derivatives can be specified as simple time-dependent functions. In the numerical work, these were derived by using the single plane model to investigate the nominal parameters over the engagement envelope. From the set of trajectories for each (derivative, a simple first-order curve was fitted which defined an initial value, a constant slope over the boost period and a constant slope over the coast period.

After initialisation, the LKF parameter estimator for airframe identification is as shown in Fig. 4.

2.4 LKF identification experiments The 1,KF estimator was assessed against the eight engagements using the nonlinear single plane model discussed in Section 1.3. When the parameter estimator was defined with nominal values of = 0 and 7 = 0, the estimates of derivatives yv, nu and n, did not update from their nominal values. Closer inspection showed that the system is not completely observable with the system linearised at these values of lateral velocity and body rate. In particular, the observability matrix:

[ C 1 C A I . . . I CA']' gives ;a 14 x 7 matrix with null columns in the positions of the derivatives y,, yz, and n,. Because of this, the Kalman gains for these derivatives remain at zero and the nominal values never update as new measurements are incorporated.

IEE Puoc -Control Theoiy Appl , Vol 144, No 4, July 1997

At each measurement instant. 1 , measure the airframe inputsand outputs ;, and

[aym 11: and derive the velocity. U,. and the estimated accelerometer moment ann. 1 , Intermgate the motor burn condition to determine \\hether to use the boost or coast rates of change of aerodynamic derivatives

Predict the nominal >illlie ofthe deri\ati\e from

g , = Bc., + h a , Calculate ths Jacobian measurement matrix

0 I 0 0 0 0 0 , Predict the airframe measurements usine.

Calculate the optimal (Kalman) gains as

Incorporate the new measurements by updating the Sfdle and covariance estimates using K , = P,-C,T(C,P,-C,T + R).'

Calculate the transitiw matrix based on the Jacobian

Form a predictinn ofsystem state5 and error covariance to the neht sample instant

.. _. 7 e,., = e , + h e , P i , =O,P,*O: + Diag[O 0 I h'Q,]

Repeat the cycle with each iiew measurement ~ ~ -

Fig.4 LKF algorithm

Pass the scaled input "ector though ths AhV to produce the response 6 , Derive the nominal derivative estimate from the scaled A N N output vector using S,

a, = [ e , ii, ?< E< fir]: =so ?, . Calculate the Jacobian measuicmciit matr,Y

[';I, =['" + I a ; " . " '."];[9]; +['; +:"'l;, 0 1 . Calculate the optimal (Kalman) gains

K, = P;C,'(C,P;C,' +RI- ' . Incorporate the new nieasurements by updating the state and covariance e s t n " unng I r ; i * r s i -

* Calculate the transition matrix based on the Jacobian

- Form a prediction ofsystem states and error covanancz to the next sample instant

-~ ^ + - - e,., =e , + e , - e,.b

Fig. 5 Hybrid algorithm

The system can be made observable if it is linearis about non-zero values of lateral velocity and body rate, and this can be done by linearising about their esti- mated values. This parameter estimator is thus a mix- ture of the principles of the EKF and the LKF.

The nominal derivative model is defined by

&,o = [-2.4 -0.5 +300 -700 -1.61T

- = [-0.8 +0.3 +210 -370 -0.8IT in boost

- e '= [+0.5 -0.2 -90 +170 +0.4IT in coast

303

The simulation model was run to set up the estimator with reasonable initial values of the covariances and system and measurement noises. They were determined as Pt=o = diag[25 0.1 0.02 0.03 4E3 2E4 0.41 R = diag [ 400 QS = diag[0.04 0.02 2E3 6E3 0.031 The iteration frequency was set at 200Hz, which is a sufficiently high frequency to cope with the dynamics of the probing signal. The initial experimental results were poor, and closer examination showed that the covariances converged to small values to produce low Kalman gains, so that little weight was attached to the new measurements. Thus the estimator had placed high confidence in the embedded model. However, this model was chosen to meet the conflicting requirements of accuracy and tractability and is likely to have many inadequacies. A number of methods have been devel- oped to compensate for general inadequacies [7] and the simplest one, which does not require any change in the estimator structure, is to formulate the filter with pseudonoise. This allows the estimator to decrease the confidence in its estimates and can be done either by increasing the value of the system noise Q or by decreasing the value of the measurement noise R. This is essentially an ad hoc process although a number of automated methods of performing this tuning have been developed 191. The approach is somewhat problem specific, although by tuning over the eight trajectories a best overall result can be achieved.

After manual retuning, the best results were judged to be achieved with

0.011

R = diag [ 100 0.0051 &e = diag[0.1 0.03 2E3 2E4 0.031

The results are summarised in Table 2 giving an overall PI of 2.8 and the response of the representative deriva- tive is shown in Fig. 9 (and includes a 2-second delay for the launch phase). The most difficult derivative to identify was found to be yv.

3 ANN estimator

3. I Introduction Over the last 10 years or so, one branch of artificial intelligence (AI) to have flourished has been the study of artificial neural networks (ANNs). Work on the sys- tem’s identification which seeks to replicate the response of a nonlinear system without trying to uncover its internal structure has been reported widely

[lo]. However, in the parameter identification applica- tion, the pattern matching characteristics of a network will be used to estimate the derivatives.

In essence, the role of the ANN is to perform pattern matching by mapping an input vector into an output vector. In this sense then, the ANN is ideally suited to identification where the input vector consists of param- eters which influence the derivatives and the output vector consists of the derivatives themselves. It has been shown that an ANN architecture with at least one hidden layer is capable of mapping any nonlinear char- acteristic [ l l ] . In principle then, a network can be trained offline with representative sets of trajectory parameters, which have been derived by modelling. This trained network can then be installed in the mis- sile and used in flight. Any deviations from the training sets, either through new trajectories or through aerody- namic variations, should then be accommodated by the interpolation properties of the network.

3.2 Application to the identification of aerodynamic derivatives The input vectors to the network at each time step are the main missile environmental factors of velocity, alti- tude and mass. Thus,

T u t = [ U H h], The derivatives are nonlinear with missile incidence, but this is considered to be a secondary factor and it is not included. The output of the network are the five derivatives:

In recent years a profusion of networks have been developed (some with several associated training algo- rithms) [ 10, 121. However, a multilayered perceptron (MLP) will be used here to establish the basic princi- ples, whilst alternative networks and training algo- rithms may provide useful refinements in future work.

The experimental architecture of the network con- tains the three inputs, which are taken to a hidden layer of neurons. The output of this layer is then taken to the network output of five neurons. The basic con- siderations in the size of the hidden layer are that too few neurons produces underfitting of data with a poor match against training vector pairs, whereas, too many neurons produces overfitting with a good fit against training vector pairs but with poor interpolation and extrapolation properties. A few simple experiments have established that a hidden layer of five neurons provides about the right compromise for this applica-

Table 2: RMS accuracies of the LKF parameter estimator

304

Yv nv Y; ns nr PI

10 0.25 15 11 580 -

Radial sea level nominal aero 21 0.26 13 12 137 2.8

- - uncertain aero 20 0.30 15 10 44 2.7

high diver nominal aero 22 0.17 16 5.5 272 2.6

- - uncertain aero 15 0.22 35 11 144 3.1

Crossing sea level nominal aero 22 0.30 5.8 11 144 2.7

- - uncertain aero 25 0.22 21 6.5 54 3.1 - high diver nominal aero 15 0.14 19 3.8 171 2.1

- - uncertain aero 16 0.14 43 9.0 91 3.4

RMS 20 0.23 24 9.8 149 2.8

Required accuracies % % % %

-

IEE Proc.-Control Theory Appl., Vol. 144, No. 4, July 1997

tion. Sigmoidal activation functions [lo] are used in the hidden layer but linear activation functions are used in the output layer.

Back error propagation (BEP) [ 131 is well established and was used to train this network. The method adjusts the network weights over a number of presentations P to minimise a cost function J , which is the sum squared error between the achieved output vector !2 and the target output vector e*.

P

p=l

There is an advantage in scaling the network input and outputs to lie within the useful range of the activation functilons of 21.0. This avoids large weight transients which could slow or even prevent convergence and allows a single pair of learning rate and momentum terms to be chosen for use at all points in the network to provide good convergence. The maximum values of the inputs and output elements were be found by run- ning the simulation model to set the input and output scaling vectors S, and So as

S, = diag [ 1000 500 1001 and So = diag[7 2 1300 2200 61

so that the network input and output vectors are now modified to

ut =S,’[U H f i ] T T and 4, = S o [ $ , 6 , 6, fi, % I t

3.3 Sequencing the training sets Training data was produced by running the nonlinear missile model through chosen flight profiles to produce the selquences of three inputs and five outputs. The air- frame was excited using the pseudorandom sequence described in Section 1.6.

Init ial experiments presented the sequenced training sets at 200Hz from a single trajectory of about 10 sec- onds flight time. However, when the data was sequenced in chronological order, the results proved to be very unsatisfactory. The ANN was found to be tem- porally unstable by forgetting training sets presented earlier in the sequence.

chronological training sequence

skipped training sequence Fig. 6 Skipped sequence for training

Good results were, however, produced by presenting the training sets in a nonchronological sequence by skipping at 0.5s intervals, as illustrated in Fig. 6. With the basic format for the presentation of data estab- lished. the ANN was trained with a mixture of data

from nominal sea level and high diving radial engage- ments. This was done by producing skipped sequences for the two trajectories and then interleaving these to produce a single data sequence. The interleaving was done by alternately taking a training set from each skipped sequence as shown in Fig. 7.

w Ie lPlPlel I etc

0 50 100 150 200 250 300 epochs

Fig. 8 Network convergence

The interleaved sequence of noise free training sets was presented to the ANN a number of times and the cost function over each epoch was used as a measure of convergence. Fig. 8 shows that, from random initial weights, the convergence is well established after only a few epochs and was judged to be adequate after about 150 epochs when the cost function converged to 0.82. The results were obtained with a learning rate of 0.1 and a momentum term of 0.5 which gave a good com- promise between speed and stability.

-2000 0 T 2 L 6 0 10 12

time of flight, s

Fig. 9 ~ estimated, ~ ~ ~ ~ true

Representative LKF identijkation response

IEE Proc-Control Theory Appl., Vol. 144, No. 4, July 1997 305

Table 3: RMS accuracies of the ANN parameter estimator

Required accuracies

Radial - -

-

Crossing -

-

RMS

sea level -

high diver -

sea level -

high diver -

nominal aero

uncertain aero

nominal aero

uncertain aero

nominal aero

uncertain aero

nominal aero

uncertain aero

4.1 0.06

12 0.48

3.6 0.04

11 0.32

23 0.19

27 0.52

27 0.22

30 0.36

20 0.31

1.3

42

1.5

42

2.5

40

3.4

39

28

0.7 1.4 0.48

40 34 5.1

1.2 1.5 0.42

40 34 4.9

2.3 3.0 2.5

38 33 5.5

2.9 4.1 2.8

37 33 5.4

27 24 3.7

3.4 Experimental results The resulting network was evaluated against the eight trajectories and summarised in Table 3 with the response of the representative derivative in Fig. 10. The network reproduced derivatives for the two nominal trajectories against which it was trained with a very high accuracy and gave PIS of 0.48 and 0.42, respec- tively. In a departure from the routine assessment pro- cedure, the network was also found to produce high accuracy results for an intermediate elevation trajectory thereby demonstrating interpolation between the sea level and high diving training sets.

For the remaining trajectories, the interpolation properties were not found to encompass the aerody- namic uncertainty or crossing engagements and, as Fig. 10 shows, poor results were obtained. Overall, a PI of 3.7 was found which is not as good as the LKF estimator. The derivatives yv and n, were found to be the most difficult derivatives to identify.

-2ow 2 0 2 L 5 8 10 12

time of flight,s Representative ANN identification response Fig. 10

~ estimated, - - - ~ true

4 Hybrid estimator

4. I ANNs compared with recursive estimators Like the ANN, the LKF estimator also 'learns' or can be 'taught', although these terms are rarely used to describe its operation. However, there are fundamental differences in the capability of the two estimators. The recursive estimator used a priori knowledge of structure of the system model and initial estimates of states to propagate better estimates of system states in response to a changing environment. As the estimator is used, the system continues to adapt and 'learn' based on the experiences gained online. In general, the estimator does not have to be preprogrammed to respond to any specific environment just so long as the structure of the system model does not change. The quality of the esti- mates is determined, in the main, by the basic capabili-

ties of the algorithm and by a priori knowledge of the system model and initial estimates.

By contrast, the simple ANN structure has to be trained offline over the range of environmental and tol- erance conditions that it will experience in use. By this means the system model is, effectively, trained into the network and can contain changes in the structure of the system model though this can be invisible to the outside world. In use, the ANN does not need any ini- tial estimates of the system parameters and it does not use any additional online experiences to continue its training. The quality of parameter estimates is deter- mined by the basic capabilities of the chosen network and the range of conditions experienced in the offline training. Some robustness is achieved by interpolation within the network. There are, however, limitations to this simple, passive, concept which are produced by the accuracy and generality of an ANN to perform under plant uncertainties and environmental conditions which were not contained in the training sets. These limita- tions can be overcome either by providing online retraining or by using the ANN as a basis for a self- adaptive estimator. It is the latter of these two approaches which will be used here.

4.2 Hybrid LKF/ANN structure In this concept, the simple nominal model of the deriv- atives contained in the LKF is replaced by the output from the trained ANN. In this way, the derivatives provided by the ANN provide a nominal value around which the filter can be linearised. Even with the simple nominal model, the estimation accuracy was fairly good and it can be expected that, with the better qual- ity nominal model provided by an ANN, these results could be improved.

4.3 Application to the identification of missile aerodynamics It was found in Section 2.4 that, to make some of the pseudostates observable, the LKF should be linearised around the current estimates of v and r and this enables some simplification to be made to the estimator pre- sented in Section 2.3. Thus, after initialisation, the hybrid parameter estimator for airframe identification is as in Fig. 5 .

4.4 Experimental results The parameter estimator was judged to work well with the same tuning as the LKF and the results are summa- rised in Table 4 under the name Hybrid-1. The overall PI is 2.3 with the derivatives yv and y , now showing the

306 IEE Proc.-Control Theory Appl., Vol. 144, No. 4, July 1997

Table 4: RMS accuracies of the hybrid-I parameter estimator

Yv nv Required accuracies Yo

10 0.25

Radial sea level nominal aero 8.1 0.10 - - uncertain aero 9.5 0.35 - high diver nominal aero 12 0.06 - - uncertain aero 7.6 0.23

Crossing sea level nominal aero 18 0.14

ys n5 Yo % 15 11

3.1 5.1

25 7.1

16 4.0

29 7.6

3.4 4.4

nr PI % 580 -

128 1.1

96 2.5

145 1.7

142 2.4

132 1.9

- - uncertain aero 19 0.20 30 8.1 46 3.0

- high diver nominal aero 20 0.13 13 4.4 205 2.3

- - uncertain aero 21 0.18 26 8.8 80 2.9

RMS 15 0.19 21 6.4 129 2.3

Table 5: RMS accuracies of the hybrid-2 parameter estimator

Required accuracies

Radial sea level nominal aero 8.1 0.10 5.1 5.1 1.0 1.1

- - uncertain aero 9.5 0.35 7.3 7.1 35 1.9

- high diver nominal aero 12 0.06 4.0 4.0 2.4 1.3

- - uncertain aero 7.6 0.23 7.7 7.6 36 1.5

Crossing sea level nominal aero 18 0.14 4.4 4.4 2.9 1.9

- - uncertain aero 19 0.20 7.8 8.1 34 2.3

- high diver nominal aero 20 0.13 4.4 4.4 3.9 2.1

- - uncertain aero 21 0.18 8.7 8.8 35 2.5

RMS 15 0.19 6.4 6.4 25 1.8

greatest difficulty in meeting the requirements. This performance shows improvements over the two estima- tors from which it is derived.

Fro" observation, a further improvement can be made by ignoring the rather poor estimate of y , given by the filter, and using the better estimate of ns to form a new estimate of y , from the relationship between fin force and moment:

IO,",

where m is missile mass, xCg and xf are centre of gravity and fin centre of pressure positions from the nose and lyy is the lateral moment of inertia.

A further improvement can be made by ignoring the filter estimate of n, and using the nominal value from the ANN in its place. With these minor modifications the overall PI is reduced to 1.8 and the results are sum- marised in Table 5 as Hybrid-2. Fig. 11 shows the response for the representative derivative. The most difficult derivative to identify is again found to be y,.

O'

-2000 1 0 2 4 6 8 10 12

time of f\ight,s

Fig. 1 1 ~ estimated. ~ ~ ~ ~ true

Representutive hybrid-2 identificution response

5 Conclusions

The problem has been motivated by the design of a self tuning regulator (STR) autopilot for a tactical missile. This requires aeronormalised derivatives to be identi- fied in an airframe which is a multivariable, nonlinear, time variant and noisy plant which is initially unstable and then lightly damped. The qualities of each estima- tor have been assessed in a single plane nonlinear model containing the principles features of missile dynamics using eight trajectories designed to cover the flight envelope. A probing signal has been designed, which has spectral and amplitude characteristics which meet the conflicting requirements of missile guidance and of parameter identification. The requirements for identification accuracy have been set by sensitivity analysis of the autopilot and these have led to criteria to assess the performance of each estimator.

Initially two quite different real-time parameter esti- mators have been investigated. A linearised Kalman fil- ter (LKF) estimator has been designed which is linearised about a simple nominal model of derivatives and this performs fairly well with an overall perform- ance index (PI) of 2.8. An artificial neural network (ANN) estimator has also been designed but this does not perform quite so well with an overall PI of 3.7.

A hybrid estimator was then formed from an LKF which is aided by the ANN which produced an overall PI of 2.3. Further improvements were then achieved from observation of these results and a PI of 1.8 was finally attained. These hybrid estimators therefore dem- onstrate superior performance over the original two estimators.

Throughout the assessment, the normal force deriva- tive y , was the most difficult to identify because it is

IEE Proc.-Control Theory Appl., Vol. 144, No. 4, July 1997 307

the most nonlinear of the derivatives. Even with Hybrid-2, it still exceeded the original requirement. However, since the accuracy of the other derivatives with Hybrid-2 is comfortably within the requirements, the performance of this estimator is judged to be satis- factory.

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