An adaptive learning control system for aircraft

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STATUS REPORT For Period May 1, 1975 - January 31, 1976 AN ADAPTIVE LEARNING CONTROL SYSTEM FOR AIRCRAFT by Dr. Ralph Mekel and Mr. Solomon Nachmias i^ N76-18140 (NASA-CE-146283) AN ADAPTIVE LEARNING CONTROL SYSTEM FOR AIFCRAFT Status Report, •

1 May 1975 - 31 Jan. 1976 (City Coll. of the

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City Univ. of New York.) 11 p HC $3.50

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•^ Prepared for Flight Dynamics and Control Division NATIOPdAL A RONAUTICS AND SPACE ADMI7,IISTRATION Grant No. N,SG-1169 I ! _, v iS y

THE CITY COLLEGE

O F THE CITY UNIVERSITY of NEW YORK _ff

THE CITY COLLEGE RESEARCH FOUNDATION THE CITY COLLEGE OF THE CITY UNIVERSITY OF NEW YORK NEW YORK, N. Y. 10031 S T AT US REPORT For Period May 1, 1975 - January 31, 1976 AN ADAPTIVE LEARNING CONTROL SYSTEM FOR AIRCRAFT by Dr. Ralph Mekel and Mr. Solomon Nachmias Prepared for Flight Dynamics and Control Division NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Grant No. NSG-1169 February 12, 7.976

1. Introduction This research is concerned with adaptive learning control systems for aircraft. The research to date, has led to the development of a learning control system which blends the gain scheduling and adaptive control into a single learning system that has the Advantages of both. An important feature of the developed learning control system is its capability to adjust the gain schedule in a prescribed manner to account for changing aircraft operating characteristics. Furthermore, if tests performed by the criteria of the learn- ing system preclude any possible change in the gain schedule, then the overall system becomes an ordinary gain scheduling system.

The research accomplished to date is presented in the following sections.

First, a statement of the problem is given and then the development of the learning control system is described. Two examples are also discussed. The results of the first example were presented at Langley on July 31, 1975.

2. Statement of Problem Let the aircraft motion be described by (1) where x is the state vector and r is the input vector. The aircraft is con- trolled using a feedforward gain matrix G and a feedback gain matrix K. The augmented system is therefore characterized by x A )+ Bp( • K xp

y) + [Bp ( X) • GI r ( 2)

p = [ L The net of linear equations in Eg. (1) is obtained by linearizing the nonlinear equations of motion in the neighborhood of the operating point y.

After this is done one obtains data points for the elements of the matrices Ap(y) and B (y)for descrete values of y. Lut 6 Y, where Y is the vector Y p

behavior of the aircraft through the continuum of the subspace Y, we inter- polate the data points and therefore the elements of the matrices A p(y) and B (y)become functions of the vector ye Our next step is to obtain a better p approximation of the parameters than this initial interpoation has given and also be able to detect changes in the elements of matrices A (y) and B (y) p p so as to control the aircraft effectively by updating the matrices K and G.

For this purpose we developed an adaptive learning control system or briefly a learning control system which is described in the next sections.

3.• The Learning Control System A block diagram illustrating the functional organization of the learn- ing control system is depicted in Fig. 1. One of the features of this system is its capability to adjust the feedforward &nd feedback gains in a prescribed and learned manner to account for changing aircraft operating characteristics.

As shovm in Fig. 1., the leaning system consists of, three basic subsystems: 1. The information acquisition subsystem, 2. The learning algorithm sub- system and 3. The memory and control process subsystem. The tasks of each subsystem and their mathematical development are described in the next three sections.

3.1. The Information- Acquisistion Subsystem The information acquisition subsystem identifies the values of the elements of Ap (y) and B Y) matrices. Several techniques have been formulated - p ( all were based on the second method of Liapunov to insure convergence of the identification process. The technique that produced the best results is de- scribed in this section. In this technique the plant was represented by a model of the form x -Fx +(A -F)x + B r (3) -m -m m -p M_ where F is a stable matrix and the model has the same dimensionality as the

sr ^k 4AL_ 3.

plant. The adaptation error is defined as e • • x m •xx (4) and the error differential equation describing the adaptation error is obtained f as R ti ere+b i ui)x +(^d wi)r i (5) where ti biui - Am - Ap(y) • B (6) (y) • K p ti diwi . m _ p(Y) • G (7) L=I Vectors b i and d i are constant for all i, and ui , wi are vectors whose compo- nents are the misallignments of the parameters of the i•-th row.

An appropriate Liapunov function for Eq. (5) is N ti V . e i u i (8) T Me +^ uiN + wiQiwi tat X81 where M, N i and Q are symmetric positive definite matrices with constant elements. Applying Liapunov's stability criterion to V and its time deriva- tive V, one derives a set of controller equations which when related to the model matrices A m and m yields Am w Ap(Y) + Bp(Y)•K - (bi-(biMe )tlildt (9) L:i of t n.

B i a By(E) • G ALT(-210Qildt (10) ^:^ 0

a convergence criterion. This criterion is defined presently by • V(t)/V(0) (11)

#1 min

"Z

where is prescribed by the designer according to his desired accuracy

I

min of identification for a prescribed length of time while the system is sub- jected to sufficient excitation. Other performance criteria are also being studied.

3.2. The Learning Al -.aorithm Subsystem The identified instantaneous values of the parameters affecting the motion of the aircraft are fitted to predetermined analytical expressions describing the behavior of the parameters over the subspace Y. The predeter- mined analytical expressions were obtained by interpolating the apriori avail- able data and representing it by polynomials. In general, any set of linearly independent functions can be used for this purpose as long as they span the space of the parameter functions.

The elements of the model matrices A and B are formed into an nll m m dimensional vector P (y)given by I (12) PI (Y) - H(Y) • c + v Vector c is to be updated sequentially upon receiving new information (PI, Y)• This type of learning falls into the category of stochastic approximation method. There are a few ways of updating c; a survey is given in reference ( 1) . Examining the dufcrcnt elgorithris for updating vector c one observes that there is a tradeoff between computation complexity and rate of conver- gence. In our case, we used the least square error algorithm which converges

-1 (14) 14) X P k+l ' Pk - ' kA+l(Rk+l + Hk+1PkN+1) where Rk is the covariance matrix of the error vector (15) Vk - PI,k - HkCk For CO we use the coefficients of the interpolation polynomials over the apriori available data and P O can be chosen any positive definite matrix.

The choice of PO influences considerably the rate of convergence of the algorithm.

A necessary test for the learning algorithm must be to continually evaluate the validity of the information in vector C k . This test is per- formed by the confidence criterion (16) !C- k +l) E ( Ok +l K11 a a where vivi or iteratively Ok+l Ok + vk+1Vk+1' 0k +1 c., The quantity d is fixed by the designer, and is the maximum tolerable mean square error.

3.3. The Memory and Control Process Subsystem After passing the confidence criterion, vector C is used to compute the elements of matrices Am and Bm . The stored values of this vector (in a Ap( Y ) and B (Y).

dynamic memory) are used to compute the elements of matrices The gain matrices K, G are then computed using the equations (6) and (7) by a a letting u i 0, w i 0. We therefore have

n G • p l (y)B (18) After the gains are computed, vector C updates the previous value in the dynamic memory.

Examples 4.

Two examples were studied in order to verify the validity of the learn- ing control system and be able to check out the programs used in the process.

The first example illustrates the learning of three parameter curves for a second order representation of the longitudj.nal dynamics of an aircraft.

The sta g : variables for this system are pitch rate q and pitch attitude 8. The independent variable for the parameter curves is e. The three parameter curves to be learned are Mp(9), Dp(0) and Cp(e) -- (moment, damping and control ef- fectiveness). The results were presented at.NASA, Langley Research Center on July 31, 1975.

The second example illustrates the learning of 10 parameter curves for a fourth order representation of the longitudinal dynamics of the F-8 aircraft.

The state variables of the system are pitch rate q, variational speed V, angle of attack a C, and pitch attitude e. The parameter curves are functions of the mach number M and of the altitude. The equations of motion are linearized at selected flight conditions and can be written as q q V V °^ • A B

(19)

long °< + long ' `fie e

7.

where all a12 a13 a22 a?.3 (20) A long ^ a32 a33 0 0 0 0 bbl b12 (21) Blong b13 and cfe is a linear combination of the states and the pilot's input. The a ij 's and bij t s are given in reference (2). In this example we used data from refereri-e (2) and considered the four wing-down (CO) configurations at sea level; hence the 10 unknown elements . of matrices and were Along Blong functions of the mach number M. We did not use the data point at M = 1 since it was discontinuous with the rest of the data and it would amount for non valid predetermined functional representations.

Our first step was to interpolate polynomials through the data in the least square sense and determine a functional representation of the system parameters with respect to M. After this was done we had a vector of coeffi- cients (CIS) for the model. Then we chose a different set of coefficients (Cp) according to which we computed the plant's unlmown parameters for a given mach number. We simulated a flight at sea level at three different mach num- bers and the learning system reproduced the parameter curves (a ij & bij curves).

An interestinq observation is that the parameter curves that multiply the state var.t3ble V are learned in an easier fashion than the other parar.:eters

8.

since the response of V to a given input is sensitive to these parameters.

To overcome the difficulty that the state variables q and @C are not sensitive to their corresponding parameters we had to give a number of inputs at each operating condition so as to achieve approximate identification and therefore good learning. Note that the a ij I s and bij f s discussed in this example are related to the stability derivatives for the F-8 aircraft as shown on pg. 3 in reference (2).

5. References 1, Gera, I. A., "An Algebraic Solution of the State Estimation Problem," ,AIAA Journal, Vol. 7 9 No. 7, July 1969, pp. 1242-1247.

2. Gera, J., "Linear Equations of Motion for F-8 DFS-1 Airplane at Selected Flight Conditior.s," NASA, Langley Research Center, F-8 Digital Fly-By- Wire Internal Document, Report No. 010-74.

9.

Information acquisition subsystem.

Convergence criterion.

PI Learning Aircraft algorithm (plant) subsystem.

x Confidence criterion.- Control input 'fe c ^ I Lookup Table -----x Control Laws + and Memory (Memory) (Gain scheduling) control process Pilot input ; subsystem I ( ^ c Figure 1. A block diagram illustrating the functional organization of a learning control system.