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TELECOMMUNICATION and CONTROL SYSTEMSLABORATORY

r--4

92 OTIC

ELECRICA ENGNEERNG ETENTr

___E 81 2 2 065vi 7H7 if7

IjI

NONLINEAR MODEL IDENTIFICATIONfFROM OPERATING RECORDS

Bruce K. ColburnAlvin R. Schatte

TCSL- REPQ 80-01

November 1980

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Nonlinear Model Identification /rom Operating Final, 6/79 - 12/79Records, ------

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Bjruce K. .tolburn1-7Alin . SchatteF46 79C09

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IS. SUPPLEMENTARY NOTES

* 19. KEY WORDS (Continue on reverse side /I necessary and identify by block num.ber)

MRAS, Nonlinear Identification, Adaptive Observer, Human Operator

20. ABSTRA.~'oilnuo on reverse side it necessry end Idenify by block number) ietfctinwt-

Two basi c response error (RE) model-reference adaptive system

out dvace nowedg ofthenoninerites.Themethods are developed Msand shown to be applicable for systems with input and output measurementnoise. Design consideration, comparison of linear and nonlinear modelfits to a nonlinear plant are given. The applicability of such anapproach for the human operator model are discussed.

I

i TABLE OF CONTENTS

) page

1. I. NTRODUCTION 1

I

SIII. HYPERSTABLE NONLINEAR RESPONSE 32

m IV. DESIGN REQUIREMENTS 47

V. RESULTS 56

3 VI. SUMMARY AND CONCLUSIONS 75

m APPENDIX 80

REFERENCES 91

.ITI~I

IIII'_ _ _ _ _ _ _ _

I - -. -

III

NONLINEAR MODEL IDENTIFICATION

I FROM OPERATING RECORDS

If ABSTRACT

Practical considerations are given for the use of MRAS identification

techniques for the identification of linear and nonlinear parameters of

dynamic systems. The response error ("parallel MRAS") was used to identify

single and/or multivalued nonlinear parameters whose forms and "transfer

function equivalents" were unknown a priori. All parameter adaptation

occurred simultaneously. This paper develops the problem, discusses dif-

ficulties unique to the nonlinearity approach, and outlines possible solu-

~tions.

:1i

it

' I

NONLINEAR MODEL IDENTIFICATION

FROM OPERATING RECORDS

I I. INTRODUCTION

System identification has become a refined art and science in the

last twenty years [l-5], at least as regards linear system parameter iden-

tification. The area of nonlinear identification, however, has been some-

what neglected. As system operation requirements are tightened, efficien-

cy and energy concerns increase, and more complex systems (large-scale

systems) are contemplated, the need for detailed accurate knowledge of the

full system structure, including nonlinear and time-varying effects, is

increased. Some examples of dynamic systems where nonlinear effects are

significant include power plants, high-performance aircraft, human opera-

tor pilot-models, and transformers.

The objectives of the present work are to develop a model reference

adaptive system (MRAS) identifier for estimating parameters of nonlinear

pilot models to ascertain if information better than traditional (Klelnman

Optimal Control Model, crossover frequency response models) linear models

can result. The application of these results would be to identify human

dynamic models so

(1) stress lists/danger vehicle maneuvers can be investigated safelyby applying the results to the model, not the human

(2) as to use the information in the design of aircraft control sys-tems and for predicting closed-loop flying qualities

(3) as to determine effects of environmental inputs such as tempera-ture, vibration, acceleration, and psychological variables suchas motivation, training, and task difficulty

(4) to parameterize changing pilot characteristics during glide-slopeto flare-up; air to air tracking in unsteady maneuver, and hybrid(loose/tight) control situations

The concept is shown in Figure i-1.

. ...* _ ,,* 1. -2 2 -T 2 Z ... .... .. ..... ... .... .. .. .. .

2

The control of physical systems using modern control theory has become

: increasingly important with the advent of the space age and the energy cri-

sis. The need to operate systems at their peak efficiency and obtain the

utmost in performance from then requires that the control system have ac-

curate knowledge in the form system models of what the system is doing and

what it will do given a set of inputs to it. A special section of control

r theory known as system identification theory is responsible for developing

accurate models of physical systems using the input into the plant and the

output of the plant for use by a control engineer in designing control sys-

tems and to facilitate greater understanding of the physical system.

System identification can be divided into two major catagories: (1)

output tracking, and (2) parametric identification. Output tracking iden-

tification develops models for systems which for a given, fixed input will

approximate the output of the physical system or plant. Since the model is

good only for a particular input, output tracking is of little significance

for identification.

Parametric identification is of greater importance since it yields the

parameters of the plant. Thus, output tracking occurs for any type of in-

put, yielding a much more versatile and accurate model given the least

amount of a priori knowledge of the plant. Parametric system Identifica-

tion consists primarily of three classes: 1) linear, time-invariant, 2)

linear, time-varying, and 3) nonlinear, time-invariant identification.

The linear, time-invariant identification is best understood and most

widely used because such approaches are simple and can be applied easily.

However, in many physical systems, major nonlinearities occur (i.e. hyster-

esis in power systems, saturation in electron devices, etc.) for which

linear, time-invariant identifiers may not supply an accurate description.

Thus, nonlinear parametric system identification has become a subject of

Iincreasing importance as more accurate models become necessary.

Two key approaches exist for systems with one or more nonlinearities.

m These are 1) when the nonlinear form is known a priori [17] and 2)

l when the nonlinear form is unknown a priori [7]. The first approach is

predicated on knowledge of the nonlinear form in advance (e.g. quadratic

j curve, symmetrical saturation, etc.) or where the nonlinearity is one in

which the parameters to be identified enter linearly (e.g. y + ay + cy 2= u).The second approach is based on a series expansion concept (Taylor series,

power series, sine-cosine, etc.) for piecewise continuous curve fittings,

when no a priori knowledge of the nonlinearity is available. This latter

approach is of the greatest interest, especially in regards to its ability

to admit memory nonlinearities (i.e. hysteresis), Figure 1-2.

Several different forms and parameterizations exist for modelling non-

linearities in dynamic systems. These include 1) the Hammerstein model

(8], 2) Volteira series (9], 3) Uryson models [10], 4) memoryless non-

linearity peicewise series fit, 5) memory nonlinearity piecewise series

fit, and 6) known nonlinear model structure. A methodology using forms

4) and 5) which has been developed over the last several years is known as

Model Reference Adaptive System (MRAS) Identification [5]. MRAS identifi-

cation techniques can be broken down into two general catagories: 1) e-

quation error or "series parallel" and 2) response error or "parallel"

-- methods. Block diagrams of these methods are shown in Fig.I-3. The pri-

mary differences between these two methods are the way the error is formed

and the way stability of the composite system is assured. In the equation

error MRAS, the error is formed as a difference between weighted system

output states and input states. In the response error formulation, the

error is formed as the difference between the system output and the ad-

justable model output. The equation error system does not guarantee sta-

K E

I~~ Qo9UCE

LOA OMUE

DYNAMIC MODEL OF

PILOT

Figurer.1.

1 5

N NNLI NE A R ITY CH A RAC T ER IZ A TIO N

S SI NG LE- VA L UE

h 8 hZ h_ h)hkh hfO^x [hlp+ah2p X +a p p +X

I p

M U LTI -VAL UE D

F(w A11l + A12 p p-v +

h hf h(X )X

Figure 1-2. Example of usinga first order Taylor series expansion of f(x)

abutteW-ntx

C66

x

CDC

Il-0

U- 0

a.0

C) 0-a

~~(J%C3 LAC

LAJC-,C c.

It

Noo w

7

bility of the system while the response error system can guarantee a form

of stability. It is this form of stability, known as hyperstability, that

has motivated the recent interest in the response error or parallel MRAS

that this thesis is concerned with.

Hyperstability theory was developed by Popov [21] as an extension of

linear, asymptotic and absolute stability. Its use in linear MRAS iden-

tification assured these algorithms were stable in the sense of hypersta-

bility. Thus, the parametric identification of linear dynamic systems un-

der the hyperstability conditions is assured; a very strong statement which

non of the previous methods could guarantee. The extension of hyperstabil-

ity assurance for nonlinear systems, then, is of interest in order to as-

sure exact, nonlinear, parametric identification.

BACKGROUND

Dynamic system identification had its inception in statistics [1,3],

and numerous such ad-hoc identification algorithms exist. However, with

the resurgence of Lyapanov's stability theory, the field of MRAS was formed

[5]. Lyapanov designs produced several algorithms, but these were of lim-

ited value. In the early 1960's, though, V. M. Popov developed hypersta-

bility theory which brought about a radical change in MRAS identification.

Hyperstable MRAS identification algorithms were develooed by Hang [18],

Landau , and Johnson [12], amongst others, which accomplished linear,

time-invariant identification of dynamic systems. Early systems were con-Ustrained by the requirement of the existence of a strictly, positive realfunction (SPRF), but in 1978, Landau and Johnson removed this condition for

Shyperstability.

Developments of the response error MRAS technique into nonlinear sys-

tem parametric identification, however, have been scarce. In 1977, Tomi-

.-aka [20] applied hyperstability theory successfully to a very special

.1. -'

1 8

I class of nonlinearities. More recently, Colburn and Schatte [22] have

applied hyperstability with limited success to nonlinearitles of a more

I general type.

9

II. MRAS IDENTIFICATION THEORY

The fundamental concepts and theory underlying MRAS identification to

the present will be discussed in this section. The development of nonlin-

ear identification is a straightforward extension of these basic concepts.

Two major MRAS methods will be developed. This is followed by an expose

on hyperstability as it is applied to the MRAS identification process, and,

concluding, with the proof of the hyperstability of a particularly impor-

tant linear, MRAS identification technique.

MRAS IDENTIFICATION TECHNIQUES

There exis two primary MRAS identification techniques: (1) the equa-

tion error (EE) or series-parallel MRAS, and (2) the response error (RE)

or parallel MRAS. The EE method shown in Fig. II-1 was expanded on by

Lion [17], and Kudra and Narendra [6]. It was adapted for use with nonlin-

ear systems by Sprague and Kohr [7] and Sehitoglu and Klein [193. Consider

the nonlinear dynamic system:

anx(n) f (xCe) )x(n-) + .. + fi(xh) )x(i+l) +Fi(x i))

f(x(d) )x( i - l) + + fo(x(S))x u + gl(u(V))u ( 1) + ... +

gm (g)1(m) (2.1)

where an - constant,

(.)(i) dtdt I

u is the input or forcing function, x is the state, and fk' Fi, and g are

nonlinear functions of the indicated argument. Equation (2.l) contains

two types of nonlinear functions, f and F. Functions of the form fk and

gk are assumed to be single-valued and piecewise continuous with fk =,

such as shown in Fig. 11-2. The function, Fi, of which only one may occur

10

'00

4-4-

41

EUL

tm

1 1

I due to uniqueness problems with the constant term associated with the func-

tion is assumed to be piecewise continuous and possible multivalued. Also,

i Fi(0) 0 0 necessarily, see Figure 11-3.

Several assumptions on (2.1) must occur:

I 1) n > m, that is, the order n of x must be greater than the order m

of u

2) the system input, u, and output, x, must be measurable,

3) the input, u, is selectable and must be sufficiently frequencyrich [7] to insure parameter adaptation, and

4) the system may possess at most one multivalued nonlinear function, F.

For the identification process, x and u are filtered by state variable

filters (SVF) which develop "pseudo-states" which are approximations, x and

u, to the system's actual states, x and u. These "pseudo-states" are then

used to form the tracking error or "equation error":

x(n) + (-(e))X(n-1) A (h ) +(Xn)X + ... + -il (h) i +1)

n ^n -i ^ . .. + )

AAl)+ (d)A^(il) A + (s)A A

i(x) + .. + f0(x )x - u +

Ai(u(V u(A) + + g((g u(m) (2.2)

which is formed from (2.1):

(n) (n-l(xCe) )x(n-1) + (x(h) )x ( +

,+ f. 1 (x (d)x (i-l ) + ... + f (x )x - u - gl(u(V))u ( ) -

gm(u (g))u (m ) = 0 (2.3)

of the actual system and where ( ) denotes the approximation to the original

.4 function.

'U Exact identification of the above functions entails the estimation of

an infinite number of points which describe the curve, f vs. x. This is

not practical; therefore, an approximation is sought. Describing each of

.!'[.

I 12

Figure 11-2. Example of fkfunctions.

Figure 11-3. Example of F(') functions.

1 13

the functions fi, gi, and Fi in (2.2) by a Taylor series expansion in in-

tervals over the expected operating domain, piecewise continuous approxi-

I mations to the functions can be obtained. For the single-valued functions,

and g, these expansions take the form:(k) A

)x k l + ak2p(x h p) pfk(x~h ) +ak p " -' +-'

(x(h) r- + (k)

akrp x Xhp) +.. ]x

J where

k denotes the function being expanded,

r denotes the position in the series expansion

(.)r-1 indicates raising the argument to the r-l st power as opposed

to (.)(h) which indicates the hth derivative of (.) with respect totime

p is the interval index

h is the argument index, and

xhp is the expansion point in the pth interval.

For the multivalued function, Fi' the expansion appears as:

A i) A +

Fi (x ) A + A12p(x(i) - Ip) + ... +

A A~) -rlirp x iP + (2.4)

where the same notation as above applies. The nonlinear identification

process has now been reduced to the estimation of the parameters, akrp ,

Airp . and b where the b are the parameters associated with the ex-

pansion of the g (ur)U functions. The error equation now becomes

A n;(n rn- + ^ () ... + +E an + ,l,p +an-l,2,p (^ () e

an~l~r~p~x~e ]A - . .. (n -1) ^

a ep r-l + + [a1+l,lp

A (xh" hp) + + ai+,rp(x(h ) - xhp) + ... (+l)al+l ,2p ...... ~lrp x h

Il

1 14

Ail + A1i2p(x(i) - j)+ +. ip( x(i) x ) +Ca~~~ ~ ^-~l +p ..+a (()-1~ r-Ii)+- ) irp x l

X

+. I + +[al + ao( - +

(xp 02-xS

* Orpx sp Ix u - [bllp + blp ( u - vp +

... +b ir(u(V) -)r-1 A) A

rp Urp + "'"" [b mlp +SA Au(g) A A(g) - r-I (m)

)u + ...+b ( u ) +.Jm2p gp mrp gp ...

(2.6)

The EE method is derived from the minimization of the cost functional

j= 1 2 (2.7)

By taking the partial derivatives of (2.6) with respect to the parameters

y (i.e. akrp), the steepest descent algorithm is obtained:

A T =K AE a (2.8)

where K is a positive constant. This yields the parameter adaptation equa-

tions: ^(n)an =-G xna A A(n ' l)

an-l,lp = "Gn-l,lp E x

t ^( n)r-nGrp n-rp E x ep

ai+lrp -Gi+l)p E x

al~ir p " -Gi+irp E x l(x ) rp r-I

A AA Allp -G il p E

Ii 15

* Ai rp = Gi rp E(x''' - xi -itp

ai-lrp = -iIpE x I~(^X~ (d xdp r

ao = -G O E x'( '

a Orp GOrp ;()^() SP

blp HU E u

Ar 21 HlpEu (v) - Pr-

b =H Eu (u -iE u

it (m) '(g) r1bmrp = mrp Eu W gp

G..r = 0 r p

H.. = 0 t tp (2.9)

The other MRAS approach is known as parallel MRAS or response error

(RE), Figure 11-4. Consider the following linear dynamic system:

x(n) = nE ax (i)X + mr b~u Ci) (2.10)

P0O 1=0

and corresponding model:

xm E a ix m + z b u (.11=0 1-0

Define the tracking error, e, analogous to E in (2.2), as

einx -xm. (2.12)

1 16

.11

II

II

0

+ Ig

CL.

C In

a. L

S. La.

_ I:a,

tL

.eM.

17

This error is then processed thorugh a compensator, C, to yield the response

error, v,: n-I

(n) Mv en) + ?- ciei)(2.13)i=O

Hyperstability theory requires that the transfer functionn-liI1 + E

G(s) i=O (2.14)n-I1 - Z a s1

i=O

be a strictly positive real function (SPRF) for asymptotic hyperstability

of the error system (2.13). An SPRF is defined as:

Re{G(s)} > 0

for all (2.15)

Re{s} > 0.

Hyperstability also yields the following parameter adaptation equations:A (i)[On]

a i = ivx V i c [0,n-l]

(2.16)

it Mbi = Bi V i C € M-1

Because the derivation of this method is governed by exact stability theory

(i.e. hyperstability) parameter identification is guaranteed provided cer-

tain conditions (to be discussed in the next section) are met.DOI3 railLandau and Johnsonrremoved the SPRF requirement by insuring that it

always occurred. They did this by adapting the ci terms in (2.13) by:Y M.

c1 i v e i . (2.17)

In doing this, the transfer function, G(s), became unity and, therefore,

always an SPRF.

Tomiuka [29] applied hyperstability and RE identification to a class

1 i of nonlinear systems of the form:

"1

18

f Dh(m-l) I +". + f0 + fhOx(t) n (u(t)) (2.18)

D n + an-iDnl + ... + aID +h

for m < n and 0 where

nh(u(t)) = nl(u(t)) + n2(u(t)) + ... + n,(u(t)) (2.19)

are known static functions of u(t) and the constants f hi's and ai s must

be identified. Tomi-luka showed that using the parameter adaptation equa-

tions:A

a. = -kai v D x i [O,n-l], kai > 0I 1 a(2.20)

Afhj = kfhj v Di n j e [O,m], h E [1,Z], kfhj > 0

the transfer function (2.14) remained unchanged and that this algorithm

was, therefore, hyperstable.

RE and EE identification methods yield parameter adaptation algorithms

similar in structure. However, there are several fundamental differences

which exist between the two methods. Foremost is that the RE method is

governed by hyperstability theory, while the EE method is not assured to

be convergent. The RE method was developed through hyperstability so that

v - 0 as t - ®. The EE method uses gradient following techniques to min-imize E , with E in equation error analogous to v.

Landau shows that the EE identification yields biased parametric

estimates in the presence of measurement noise, while RE yields unbiased

results. In the nonlinear case, Colburn and Schatte ['(] demonstrated that

the RE state estimate (i.e. xm( )) is much more noise free than the EE

state approximations, (i.e. X (), Figure 11-5. This is because the EE

develops the state information by filtering the noisy input and output

while the RE develops state approximations through its own dynamics, giv-

ing a more noise free approximation.

'I .

-.

Ii 19

U L&J

%.-0

C

06

ci

1 20

I f()

Figure 11-6. EE Identification of saturation plus dead-zone.

'p Figure I1-7. EE Identification of hysteresis.

21

g In the presence of good (i.e. no noise) state information, though,

the EE will identify single-valued and multivalued memory type (e.g. hys-

teresis) nonlinearities very well, see Fig. II_6,7 .This is the chief advan-

5 tage of the EE method over the RE method because the RE method heretofore,

has not been applicable to nonlinear identification.

HYPERSTABILITY CONCEPTS

Hyperstability plays a key note in guaranteeing the stability of the

RE identification technique. It is a generalization of linear asymptotic

and absolute stability. Consider the system shown in Figure 11-8,

Figure 11-8 Basic Hyperstability System

block L is assumed to be composed of linear differential equations. If

block N is a linear feedback, then stability bounds may be derived:

1 < X < X2 (2.21)

where -w(t) = Xv(t). This is the linear asymptotic stability case. If

block N is nonlinear and obeys:

- (t) = 4(v(t)) (2.22)

p(v)v > 0 V v 0 0 (2.23)

(0) = 0 (2.24)

then the limits on block L and (v) can be obtained. This is the absolute

stability problem. Now suppose (2.23) is rewritten

1Ii I t0 -(lv(-) dT 0 V> t (2.25)0 (225

22

this, then, is the hyperstability problem. The problem can be stated as

N find the conditions on block L given a block N such that condition (2.25)

occurs.

To begin the study of hyperstability, suppose that block L may be de-

scribed as:

dx = Ax + bw (2.26)TT

v = cx + dw (2.27)

Several definitions must now be made:

Definition 2.1

Let the hyperstability integral, n, be defined as

n(0,t) = ft - wt!) v(2) dT (2.28)0

Definition 2.2 Minimal Stability

The system (2.26-2.28) is said to posses the property of minimal stabi-

lity if for any initial condition x(0) = x0 there exists a pair of continu-

ous functions x and w, defined for t > 0, and satisfying a) equations (2.26)

and (2.27), b) the restriction

n(0,t) f - (T) V(T) dT < 0 V t > 0, (2.29)

0

and c) the condition

lim x(t) = 0 (2.30)t-tDefinition 2.3 System Transfer Function

The transfer function of system (2.26)-(2.27) is defined as:

g(s) = C(sI-A)- b + d. (2.31)

In order to apply hyperstability to a system (2.26)-(2.28) it is nec-

essary that three conditions for this system (2.26)-(2.28) be satisfied.

I

f 23

f The first condition is that the system (2.26)-(2.28) be minimally stable

in the sense of Def. 2.2. This condition is acceptable since asymptotic

stability is of prime importance and, if the system (2.26)-(2.28) is not

minimally stable, then asymptotic stability in the presence of an inte-

gral restriction of the form (2.28) is impossible. The second condition

is that b # 0. This condition is obviously necessary since the solution

of (2.26)-(2.27) is not a function of u and the problem can be solved

4directly. Finally, the third restriction is that the transfer functionof system (2.26)-(2.27) given in (2.31) be not identically equal to zero.

This is necessary because if (2.31) was identically zero then n(O,t1 )

would be identically zero and (2.28) would be meaningless.

Given, that the above conditions are satisfied, hyperstability may

I now be defined.

Definition 2.4 Hyperstability

System (2.26)-(2.28) is said to be hyperstable if there exist posi-

tive constants (, a, y, and such that the following two properties are

satisfied:

1) Property Hs

For every t e [to,T 0] and every solution of the system (2.26)-

(2.28) in the interval [to,T 0 , if for every constant B0 > 0 for which

r(tot) < 2 V t E [to,T O] (2.32)>4 1 then_

X Hx(t)I I B0 + iIx(t0)I (2.33)

for all t c [to,T 0].

" 2) Property HA

j For every solution of the system (2.26)-(2.28) in the intervalLTTj

1 24

I "yLt~t) > -Yjjx(t 0) 1 - Y I x(t0 )11 SUP aI~X(T)HI (2.34)

Y t E[tOST 01.

Simply put, if

-a < ri(0,t) < 2(2.35)

then the system (2.26)-(2.28) is hyperstable, and if 62=0, then the sys-

I tern is asymptotically hyperstable [21].

The results of Popov show that if, for a linear system (2.26)-(2.28),

the transfer function from w(t) to v(t) given by (2.31) is a strictly pos-

itive real function (SPRF) and the inequality (2.34) holds then the system

(2.26)-(2.28) is asymptotically hyperstable. Willerns [26-27] showed that

the SPRF insures that

n(O,t) = ft- WW'~ V(T) dT < 0 YVt > 0. (2.36)0

RESPONSE ERROR IDENTIFICATION - APPLICATION OF HYPERSTABILITY

Consider the linear dynamic plant:

x(n) n- a (i) + m- bi u(i) (2.37)i=O i=O1

where

x - state variable

u - input or forcing function

ab. - constants

i) dt1

S and the associated model:

x(n) n- z1ax(i) rn+ M1b (i) (2.38)rn = ixr +z 0

1 25

I1 Define the tracking error as:

I e = x - xm (2.39)

and the error dynamics become:

I (n) n-I M n-Ie (n) a e + Z (ai - a +

1 (bi - b)uM i) (2.40)

i=O 1

Let

v(t) = e (n) - n ci e~') (2.41)i=O

thenn-i ^ n-i

v(t) = E (a. - ci)e + z (a1 - ai)Xm i) +i=O i=0

mn-i

(b - b)u(i) (2.42)i=O i

and let

* W(t) = v(t) (2.43)

The hyperstability system now is:

j= a e + E (ai - ai)xm +i=O i=O

rn-i)Em (b -b)u('i:0

(n n-l .b M

v(t) e - Ece2 i (2.44)

W(t) - v(t)

where e is defined in (2.39).

Theorem 2.1

I j If the system (2.44) is minimally stable and the input, u, into the

j 26

plant and the model satisfies a frequency richness criterion, then the

following parameter adaptation equations will yield the hyperstability of

system (2.44) and the parameters a1i ai and bi - bi as t ®:

a. = ),xm V i > 0 i e [0,n-l] (2.45)

iA Bi(i)vb = B. > 0 i C [0,m-l] (2.46)

= y" e(i)v y, >0 le [O,n-l] (2.47)

Proof

Since w(t) = v(t) the transfer function of system (2.44) is

G(s) = 1 (2.48)

which is an SPRF, thus

n(0,t) ft - W(T) v(T) dT < 0 V t > 0 (2.49)0

The other bound on the hyperstability integral must now be determined, that

is

n(0,t) = ft - W(T) v(T) dT > y2 (2.50)00

Substituting (2.42) into (2.50) one obtains

n-i 1 (i)n(0,t) = ft z (a. - ci)e vdT +

0 i=0

n-l (in-ft aM - at )vdT + - (b- vd (2.51)

0 i=0 0 i=0

It is sufficient to look at each of these terms and bound them individu-

ally. Thusn-ift E . (at - cY)e vdT > -y, (2.52)

S0 i=O Y

-tn-l )X(i)v _ft(a - a)x vdT > , and (2.53)

0 i=0

1I

I 27

ft m-1u i vd _,f Z - (bi b)u AT > (2.54)

0 i=O

I The integration and sumation may be interchanged in (2.52)-(2.54)

because they are independent of one another, and since the resultant is a

sum of integrals, it is sufficient that each of these be bounded, yielding:

I f - (ai - ci)e(i)vdr > 2 , i £ [0,n-1) (2.55)0 1 Yi

ft _ (ai - a )Xm(i)vdT > -y2 , E [0,n-l] (2.56)0 -i L,

ft - (b - )u(i) vdT > .y2 , i C [O,m-1] (2.57)0 1 )i

Using (2.55), let

fi(t) = -(ai - c.). (2.58)

Differentiating with respect to t yields

i(t) = ci (2.59)

Define the ci parameter adaptation as

= yi e (i) v , > 0 (2.60)

Substituting (2.60) into (2.59) and solving for fi(t) yields

fi(t) = f ft e ()vdT + fi(O) (2.61)0

where fi(0) = -(a1 - ci(0)). Substituting (2.61) into (2.55) yields

• I ft C fr e(i)vdT,]e(i)vdT + fi(0) ft e(i)vdr >

0 0 0

-Y (2.62)Yj

,I

=~

|228

I Using Leibnitz's rule:

fT f(T'r)dt' fT dT' + f(,T) dd'=T (2.63)

dt 0 0 d " T

I The first term of (2.62) may be integrated. Let

f(T',T) = e(i) (T')V(T') (2.64)

and use Leibnitz's rule yielding:

f 0 and (2.65)T

f(T,T) = e(i) (T)V(T). (2.66)

Thus, if

Fi = fT e(i)vdT' (2.67)0

then from Leibnitz's rule

dFi e(i)(T)V(T) (2.68)

and

dF i = e(i) (T)V(T)dT. (2.69)

This substitution, (2.67), puts (2.62) into the formyi ft Fi(T)dFi(T)d + f (0 ) ft e(i)vdT > _y2 (2.70)

0 0

'I which yields,

'L [ft e(i)vdT]2 + fi(O)fT e(i)vdT > -Y2 (2.71)2 0 0 Yi

Completing the square in (2.71) yields:

Y_ t (i) fi(0) 2 fi2(0) fi2 (0) = 2 . (2.72)

2L0 Y2 Yi

Thus, the boundedness of (2.55) has been proved with the given parameter

'I|

29

I adaptation (2.47). Using similar substitutions:

gi(t) = -(ai - ai ) and (2.73)

hi(t) = -(b i - bi) (2.74)

the boundedness of (2.56) and (2.57) can be proved using the parameter

adaptation (2.45) and (2.46), respectively. Since, all the integrals

(2.55)-(2.57) are bounded the boundedness of (2.51) is assured and is equal

to2 n-l f.2 (0) n- gi2(0) m- 1 h2(0)Y1 = E i + E + E (0 (2.75)

i=0 2yi i=0 2 i= 28i

or, substituting, for fi(0), gi(0), hi(0)

n- A - 2 nl2n-1 [ci(0) - a.] n-I [ai(0) - a.]Y0 = E + +

i=0 2yi izo 1

^ bl]2m-l [bi(0) - b 2(.6

i (2.76)i=0 2i

and the theorem is proved.

Theorem 2.1 was proved under the assumption of minimal stability of

the system (2.44). The following theorem gives the requirements on the

plant (2.37) in order for minimal stability to hold.

Theorem 2.2

If the plant (2.37) is asymptotically stable, then the error system

(2.44) is minimally stable.

Proof

Minimal stability requires that for any initial condition e(O) =e 0

a pair of functions w(t) and e(t) exists such that

1) n(0,t) < 0 and

2) lim e(t) = 0.

Define''I

30

n-I ( m- u'(t) Z Z (aj - ai ) + E (bi - b ) (2.76)

i=O I=0

then (2.40) becomes

(n) n-i (i)e = Z aie + u'(t) (2.77)

Let* n-I ( )

w(t) = (a. - c.)e (2.78)i=O

(this can be done by setting ai = ai, i E [O,n-1] and bi = bi, I E (0,m-l]

in (2.42)-(2.43), and is certainly an allowable w(t)). By using this W(t),

u'(t) = 0 and (2.77) becomese(n) n-I l( i )e = ni e (2.79)

1=0

which is asymptotically stable only if the plant is asymptotically stable

for any initial condition x(O) = xO. Since the w(t) was chosen from a set

for which

-y2 < n(Ot) < 0 (2.80)-Y

then the first condition of minimal stability is maintained and the theorem

is proved.

The frequency richness criterion in Theorem 2.1 was discussed by

Sprague and Kohr [7]. They showed that for parametric identification to

occur when the error was identically equal to zero the number of distinct

frequency components of the input, u, to the plant and model is given byn+m+l

N= 2 ' (2.81)

when the parameters to be identified were the ai i e [O,n-l] and bi,

i [O,m-l], and the plant was linear. They showed this mathematically

for the linear plant and no zero case and inferred it experimentally in

the general linear case. For the nonlinear case it was found to be atI

31

m least as large as (2.81). For the case in which there are also adaptive

ci terms the number of distinct frequencies was found to be equal to

N. 2n + m + 1 (2.82)N= 2I

using experimental results and empirical inferences.

The development of nonlinear RE identification algorithms is based on

the linear algorithm presented here. Its proof is very similar to the

I proof just presented.

I!)

i

32

III. HYPERSTABLE NONLINEAR RESPONSE ERROR IDENTIFICATION (HNREI)

g The development of linear MRAS identifiers and hyperstability prin-

ciples along with the nonlinear concepts of the EE method lead to the de-

velopment of nonlinear RE identification techniques. Heretofore, the ex-

tension to nonlinear identification using the RE method has been thwarted

by the requirement of the SPRF which is not defined for general nonlinear

systems. However, with Landau's [11] and Johnson's [12] papers elimina-

ting the SPRF difficulty through problem reformulation, an approach for

HNREI algorithms now exists. In this chapter two different nonlinear

identifier formulations will be presented and two nonlinear identification

algorithm proved to be hyperstable are given.

Nonlinear Formulations

For the development of HNREI algorithms, two nonlinear formulations

were considered: 1) piecewise linear expansions in intervals of the

state space and 2) power series expansions over the entire state space.

Both of these have distinct advantages and disadvantages. The piecewise

linear expansion, Figure 111-1, allows the nonlinear function to be divided

into a series of intervals in the state space and a linear approximation to

the nonlinearity formed. This method has a serious disadvantage, though,

when perfect state information is lacking, Figure 111-2. It is possible

that the identification algorithm could be adapting a parameter in the wrong

interval yielding erroneous results. Thus, the degree of accuracy obtain-

able by the piecewise linear formulation decreases with the loss of per-

fect state information.

The power series expansion technique, Figure 111-3, uses a continuous

approach and, hence, does not suffer from the intervalization problem as-

sociated with the piecewise linear method. The power series method attempts

to approximate the actual nonlinear function as a finite power series:I''

IFI

I ok + al k (x(h) x kh)

X(h)

7kh

Figure III-1. Piecewise Linear Expansion

noisy state

stateintervals

time

Figure 111-2. Interval Selection Fromi States.

I 34

I~ f(x)

do..

.00,

Fiur 11-. PwrSrisEpnin

INf(x) Z N aix'. (3.1)

i= l *

The principal disadvantage of this method is that for some nonlinearities

which can be approximated with a small number of intervals using the piece-

wise linear expansion may require many terms using a power series expansion.

Noisy state measurements also present problems because of the possible

high degree of the polynomial and noise biasing.

Nonlinear Identification Algorithms

The nonlinear expansion techniques just discussed were used to de-

velop HNREI. The piecewise linear HNREI required a special interval con-

dition to insure hyperstability while the power series expansion technique

required no special conditions.

Piecewise Linear HNREI

Consider the following nonlinear plant:

x(n = nE1 fi(x(i)) + g(x (Z)) + Z hi(u(i)) (3.2)i=O i=O

with a corresponding model:

x(n) n-I 0i)) + ̂ (xm(o) + ME, NO (3.3)

i m 9z m i=O

where (.(i) , and Z 6 [O,n-l]. Leti dti

fi(x (i )) = ailX(i)(34

.igt(x ( ) = a a ( (91) - p) (3.5))) ato p U ~p(

hi(u(i)) = bilpU(i) (3.6)

which have corresponding model nonlinearities

I ! I

36

fi(xm () = ilpX(i) (3.7)

1t~ m (Z il= tpm+aZP( Z z 38(x ( )() - m2 (3.8)g (x )= a 0p + a~lp xm - mp

t h~) ()lp (1(.g

where p denotes the pth interval of the state space x i) or u( ). Thus,

the plant and model may be rewritten, respectively, as:

(n) = n-I ailX(i) + a + a i(X(Z) ) +

m-I bilpZIPi)

E il (3.10)i=O

=m(n ailp mi) + a Op + a .lp ( Z xi z

ii=0

i=0ini (i)

where the parameters bilp' ailp' atop, and a~1p must be identified in each

interval p. Define the tracking error as

e = x - xm (3.12)

The error dynamics which are to be made hyperstable are:

n-i-A

) e(n) n-I p1 e (i) -a _ .i~ _m~ .+2 0 ^a

~i=0

A W,, nE )x (i)+ (a - aP )(Xm -XmZp) = (aiip -a )

i=0

+ - (b b )u(i) (3.13)

i=0 P UP

Let/ . e( n ) n -Iv(t) = Z c( le (3.14)

I i lp• I

I.

. 9

37

thenn-1 A p()

v(t) E (ai - )e -a p(X -X ) +i=O ip lp ZIP XP mp

atop " op + (a ip a zlp)(Xm C Xmzp +

mE (a - a )x (i) + m (bilp b ip)Ui (3.15)

i=0 lp lP m i=0 i

Define

S= v(t) (3.16)

The hyperstability system is given by (3.13), (3.14) and (3.16).

Theorem 3.1

If the system (3.13)-(3.16) is minimally stable, the input, u, into

the plant and the model is sufficiently frequency rich, and the condition

X =p XMIp are satisfied, then the following parameter adaptation equations

guarantee the hyperstability of system (3.13)-(3.16) and the parametric

identification: a t0p - a top, a ip and b lpo b as t

Aa Lip a zip v a mp > 0 (3.17)

" V(pm( p

a Ip a IP Vx - Xm~p)' aLIp>U (3.18)

a lp = lp V m i [0,n-l], i L UP > 0 (3.19)

bA (i) ,b = i UP vu U > 0, i E [0,m-l] (3.20)

Clip = yx ve i) i > 0, i e [0,n-lJ (3.21)

Proof

Since w(t) = v(t),

/, , (0,t) - -t.W(T)V(T)d < 0 (3.22)

0

"I

- I I • I " " 'mm- " n . . . . . .. . .. I l m m i l

38

and, thus, one limit on n(0,t) is established. To establish hyperstability

the other limit must be satisfied:

n(0,t) = ft - W(T)V(T)dT > -Y (3.23)0Y(

Substituting (3.15) into (3.23) for w(T) using (3.16) yields

n(O,t) ft n-I l p e +i

E z - (all - ci )e vdT + ft0 i=0 P 0

(-X )vd + ft (a O- a Lp)vd + ft

. -a a op)vd,]+ Z - (rn- i)u vdT > -d 0 (3.24)

i It is sufficient to bound each of the terms in (3.24) in order to insure

the boundedness of (3.24) and by exchanging the integration and summation

i of terms 1, 5, and 6 and using the same sufficiency argument (3.24) yields:

f t - ci )e(i)vdT > Y2 i 6 [O,n-l] (3.25)

0 (ail cp Yilp

l ft - (a - a 0op)vdT > y2 (3.26)

0 Lop

(a,1 - ip)(Xm(4) - mp)VdT > C (3.27)

I A Ift - (all - a p)Xm ivdT >-Y i C [0,n-l], is2. (3.28)

S0ip p -- P

tMf- (b 11p ~b 1 p)u 'vdT > - a i I [0,m-i] (3.29]

Using (3.25), let

I Afp (t) -(ailp cilp). (3.30)

I Differentiating (3.30) with respect to time yields:

I fuip(t) = cilp (3.31)

1. ..

39

Define the cil p parameter adaptation as:

Aleiv, 0 (3.32)Cilp= Yi p 0

Substituting (3.32) into (3.31) and solving for filp(t) yields

f (t ~ t f~ i vdt + p()(3.33)Sfilp ( t ) Yilp 0e filp(0)

0

where f ilp(O) =-(ailp -Cilp(O)). (3.34)

Subs tituting (3.33) into (3.25) yields

ft [f T (i)vd']e(i)vd + f11l 1t e(i)vd (3.35)

0 0 l 0

By using Leibnitz's rule and completing the square, (3.35) yields:

1. f 2 (0)il[ft e(i)vdr + >( - fip 0 = 2 (3.36)2 1l 0- - 2yilp Yilp

Thus, on the first integral has been obtained using the parameter adapta-

tion (3.21).

Similarly, using (3.26), let

g9op(t) = -(ato p - atop). (3.37)

Following a development similar to (3.30)-(3.36), (3.26) can be bounded

using the parameter adaptation (3.17) obtaining:

g g(0) 2 . (0)l g2 (0)njZO 2 Eft vd £ ODj 2 tl tp0 p r atOpp

(3.38)

The other integrals (3.27)-(3.29) can be bounded using (3.8)-(3.20), re-

spectively, and obtaining, respectively,

IiI

ii 40(0

!L-'R [ft (xm ('1) )vdT + 2..1DLID -

p0 1 az"p 1 p

> - = _ Y2 (3.39)- i1p i p

Ili [ft Xm(i)vdT + g~ip(Oi ]2 >

+il(0) 2.2al

2a _ a (3.40)ip ip

8i_ [ft u(i)vd h2il(0) ]2 h211p(O)2 ip 2 lip

h2ilpO .Y 2 (3.41)" 2iip = ilp

where

hilp(t) = -(bil p - bilp). (3.42)

Thus, all of the integrals (3.25)-(3.29) have been bounded, and,

therefore, the integral (3.24) is bounded by -y' where

n-if1 n-Ig 2 i(0) m()___ 2iZ + li +O 0 iO ilp 10 2ailp i=0 20ilp

g2o(0) g2 (0)t + 11 (3.43)

b or, substituting for filp(0) gilp(O), hilp(O)

n-I (a. (O)-a n-I (a -a ' 1Y.. 2yiO i=O 2aiilp 0 =0

(b lp'-blIp + (aOD-alop) + (a.p-a,,I) 2 (3.44)2 il p 2a top 2aLIP

Thus,

1 41

I 0 < n(ot) < 0 (3.45)

Iand hyperstability is proved.The conditions for minimal stability of the error system (3.13)-(3.16)

Iare given in the following theorem.Theorem 3.2

If the plant (3.10) is asymptotically stable, then the error system

(3.13)-(3.16) is minimally stable.

I Proof

The proof of Theorem 3.2 parallels the proof of Theorem 2.2. Let

n-lw(t) = E (a -c )e(i )+ a -a i -7ip ) (3.46)

i=0 ilp ilp top Lip zP

By using this input, the error system becomes:

e (n) = n-l a1 1 pe(i) + a + a ZPX p.

(Xm Xmip)] (3.47)

by setting

atop -0 (3.48)

at, = a11p i c [0,n-l]

bilp = bil p i E [O,m-l]

and uM() 0 1 C El'm-l)

Thsxm()(0) = 0 1 e [O,n-lJ.

Thus, (3.47) becomes

1 x(n) = Z 1 apx(i) + azop + a.l p(x(t) 'x (3.49)

• , Ijot

S 42

I which must be asymptotically stable in order to insure the minimal stabil-

ity of (3.13)-(3.16). The final condition of Theorem 3.1,

X = xmip (3.50)

requires substantive a priori knowledge of the plant. This type of know-

j ledge is not readily available and condition (3.50) limits the piecewise

linear HNREI.

I Power Series HNREI

Consider the following nonlinear plant:

n = n E j= aix (1 iJ] + - (i)j -- . )J] (3.51)i=0 j1l 1=0 O

1 where k, Zi is a finite integer and (.)(l)j indicates d L-m raised to the1 dt

jth power. This plant has a corresponding model:

I (n) = n-I k i -I ziXm (n [ E ajmij + ME [ Z b )J( ] (3.52)

i=0 j=o i_10 j=l

Define the tracking error as:

1 • - x - xm (3.53)

I and the error dynamics become

n-l ki n-I kia e (n) = n E a i j (x( i ) J ' x m(i ) ) ] + niO E (aij-aij)xm()]

* 1=0 j=l i=0 Jul ij1 mI

1 b (b- )u(i)J (3.54)1-0 j l

I Let the response error be defined,

n-I kv(t) z e(n) - E [ cij(x(i)i-x (i)j (3.55)

iJ=O Jul

I mand define

W(t) v(t) (3.56)

II 43

I Thus, the error system to be made hyperstable is (3.54)-(3.56).

i Theorem 3.3

If the system (3.54)-(3.56) is minimally stable, and the input, u,

Iinto the plant (3.51) and model (3.52) is sufficiently frequency rich,

then the following parameter adaptation equations insure the hyperstability

I of system (3.54)-(3.56) and the parameter convergence

a.. - a1j, i .-" b.. as t- :I A bM3

ct. iJx v le[0,n-1], jcel,ki), a1j)>O (3.57)

b. z 81.U~)V le[0m-l], jE:[l,2. 1) 8 >0 (3.58)

=i ''~x _xM (I)Jv ic[0,n-l1, je(l,k1], Yij>0 (3.59)

Proof

Since (3.56) holds the upper limit of ri(0,t) defined as

n(0,t) - ft W(Tr) V(T) dT < 0. (3.60)0

Thus, the other limit for (3.60) must be obtained to prove hypersLabillty.

Substituting (3.55) and (3.56) into (3.60) yields:

nE~)-f [ E - (a jcjj)x _x )d0 i=O J-1

k

'0=0- i li )xm ('JvT+ ft

.2(b 1i'-bij)u(1 )Jvd-r 0 - (3.61)

By using a sufficiency argument similar to that used in Theorem 2.1 and

I Theorem 3.1, (3.61) can be written:

1 44-t (ij ()j) d >_2

f (a. (-c))vdt > y2 iE[O,n-l],

Saij -ij m Yij

I je[l,k i] (3.62)^ - i >"y 1

t - (ai -aij)x m vdT > CL iE[O,n-l], jE[lk i] (3.63)

101J - 8ij'

(bi -bi )u()JvdT > -Y 2 ic[Om-l], jecl,k iJ (3.64)

Also, following an argument similar to Theorem 2.1 and Theorem 3.1, and

defining

f. (t) = -(a ij-c.ij) (3.65)

gij(t) = -(aij-aij) (3.66)

hij(t) = -(bij-b.i) (3.67)

the inequalities (3.62)-(3.64) are found to be, respectively,

2 ft(x(i)JXm ()j)vd + fij(O) 2 f2 i(O)2 0 Yij 2yif2 i .(0)

=-2

=2Y y2 (3.68)

2_ Yl j yl j 2

jLioXm JvdT +-a.. . 2al > 2ai..02 13j --

"Y 2 (3.69)

[ft () h(0)2 h2i(0) h21 (0)vd[ + u(1Jvd0 +2] U > . j =

2 0 d i+ 2 8ij - 28ij

"Y 2 (3.70)

which bounds (3.61) by -y0 where11'. . ...I. . . .. . . I Il II i . . l I I I IlI l .. .

I, 45

0)k2 (0) 2n-li Cc. -a' a a.) rn-i* j -2 + ] +

0 i=0 jzl 3 1 2i0

| i (bi (O )-b .)2 ]

1[ 2 (3.71)

j=l 2aii

where the initial conditions

fij(0) = c i(0) - aij (3.72)

gij(0) = aij(0) - aij (3.73)

and h. (0) = bi(0) - b.. (3.74)13 ii 1

have been substituted in (3.68)-(3.70) to obtain (3.71), and hyperstability

is proved.

The conditions on the plant (3.51) to insure minimal stability of the

error system (3.54)-(3.56) are obtained in the following theorem:

Theorem 3.4

If the plant (3.51) is asymptotically stable, then the error system

(3.54)-(3.56) is minimally stable.

Proof

The proof of this theorem parallels that of Theorems 2.2 and 3.2 and

consists of finding a w(t) such that

1) n(O,t) < 0 and

2) llm e(t) = 0

for any initial condition e(0) = e0. Letn-l ki

S w(t) = [ z (ajj-cij)(x ''' -X j)) (375)iO J=l I

by letting

aij , aij (3.76)

bij = bij (3.77)

1 46

I The error equation (3.54) then becomes:

k.e(n) . n-i i ij (1)

a. (x x ] (3.78)i=0 j=l 1

i

which has lim e(t) = 0 only if the plant (3.51) is asymptotically stable.t-cn

Thus, minimal stability is proved.

Thus, two HNREI algorithms have been developed which yield parameter

adaptation. This parameter convergence is guaranteed if the plant is asymp-

U totically stable and the input, u, to the plant and model is sufficiently

i frequency rich. The next section discusses methods of implementing these

algorithms and design criteria associated with them.

l

l

l

l

4 'SI

i I

1447jIV. DESIGN REQUIREMENTS

In the design of nonlinear, parametric identifiers, several design

considerations must be made. These considerations can be roughly grouped

into two catagories: 1) what type of identifier is to be chosen, and 2)

specific nonlinearity considerations. The answer to the first depends

on the a priori knowledge of the nonlinear system and the maount of noise

in the system. If the functional form of the nonlinearities is known,

then a linear, time-invariant identifier might be advised. Similarly, if

the nonlinearity forms are unknown and only small (<1%) amounts of noise

exist, then the series parallel (equation error) MRAS approach might be

I best. The criteria for choosing between identifiers is largely a func-

tion of experience. However, with unknown structure nonlinearities in

the presence of noise, parallel MRAS yields best results.

I e Fig. IV-I shows a general block diagram of the nonlinear response

error parametric identification algorithm. The algorithm may be broken

I into two main loops. The inner loop consists of state variable filters

(SVF) adjustable model, dynamic compensator (N(s)), and the parameter

I adaptation. This inner loop presents several considerations for the de-

£ signer:

1 1) SVF design

i 2) Parameter gains93) PRF and the dynamic compensator design

Ii The SVF's function is to provide "pseudo-derivatives" for use in develop-d

ing the response error. If p d then L(p) denotes the SVF and (1) be-

comes

L• l(pnx P) + ... + L[F i(P ix P) + ... + L[f°O(p k xp)x p ]

U L[go(p u p)u p] + ... + L[gm(pWu p)u p] (4.1)

!I

44

( -i

LU .

2: LU_

CD)

tnr

k44

49

where xp and up are the plant output and input respectively. Sprague and

Kohr [7] have shown that if L(.) is a transport operator then L(.) commutes

4with the nonlinearity to giveSL(pnx p) + ... + [Fi L(pJxp) + k+ fo[L(pkxp)] Xp

g0[L(pu p)]up + ... + gm[L(pwu p)]up (4.2)

allowing the SVF to be moved from the output of the model to its input.

This gives compatability between the model states and the corresponding

plant "pseudo-states".

Parameter gains should be chosen in such a manner as to give rapid

convergence and decrease asymptotically to zero to prevent noise bias.

Several algorithms exist [11,12] for the decreasing adaptive gains'

The PRF requirement was addressed earlier. If a priori knowledge of

the range of plant parameters exists then a fixed ci compensator can be

developed, saving much computation time and implementation hardware. How-

ever, in general, the nonlinear adaptive compensator must be used to ob-

tain the PRF requirement.

The outer loop of the nonlinear identification algorithm consists of

the pattern recognition, model structure, and nonlinear interval blocks.

Important design consideration occurring in the outer loop are:

1) Pattern recognition block

2) Model structure blocka) Model order (numerator, denominator degree)b) Nonlinearity location

c) State information

3) Nonlinearity interval blocka) Interval sizingb) Number of intervals

4) Identification time frame

I1

50

The pattern recognition block must determine if a "possible" nonlin-

earity has been identified. This block decides when a nonlinearity des-

1cribed by a set of points is Class C , i > o, to qualify as an allowable

4 nonlinearity, Fig. IV-2. Define:

AIfl = P2 - Pl

A1 f2 = P3 " P2 (4.3)

Alfn-l= Pm - Pn-l

the "first difference" fo the set of points.

Then A2fi A l AI

and AfA fi+l f Ai-I f (4.5)and Aif = Ai-lj+l f.

represent higher differences of the set of points. Now the performance

criteria: iN A -Z.

< E e is greater than zero (4.6)Nj=l TZ1

i

defines the class of functions C i- where only the A ftj that differ in

sign from the previous, A i fgj-l are used to define the class. N is the

number of sign changes, and T.i is the width of the interval between the

points.

The results of the pattern recognition block feed into the model

structure block. In this block almost all of the decisions concerning the

model structiire and nonlinearity placement are made. This block is nearly

always "filled" by the designer himself.

The first piece of information necessary from the model structure

block is the model order and preliminary structure. Several model order

algorithms exist [13,14,15] to provide possible model structures and in-

formation on the number of poles and zeros for the model and any time de-

1 51

P ATT E RN R E COG NI TIO N

P4

N AI)PEJ

J=L TJ

FO0R CL A SS C

Figure IV-2.

52

lays.

The placing of any nonlinearities in the afore found model structure

is largely a matter of trial and error or intuition as to what states

4will be affected by the nonlinearity. By minimizing J in (4.6) of the

nonlinear pattern recognition block with respect to i an E can be picked

which fixes the class of functions allowable for that particular model

state. If the expected plant nonlinearity was not of this class then

another model state might be chosen in which to locate a model nonlinea-

rity.

Since both plant and model state information and used to identify

possible nonlinearities, it is important that the proper state information

is obtained to get the correct answer. A decision must be made whether to

use the noisy plant states, themodel states, or a combination of the two.

The problem is illustrated in Fig. (IV-3). As can be seen, at t1 the ac-

tual plant state (unmeasurable in general) is in interval 2 while the model

state is in interval 3 and the noisy plant state is in interval 4 of the

intervalized state space. This problem can be posed as an optimal com-

munications problem where a decision function:

S (X) + x) (4.7)

is defined and the functions f and q are chosen to minimize the probabil-

ity of selecting the incorrect interval. Q, then, is the independent var-

iable used to determine the interval number of the model's state space.

The nonlinearity interval block decides how the model's state spaces

will be broken up in order to identify the nonlinearities (if any) exis-

ting in the corresponding plant's state spaces. This problem can be bro-

ken into two interrelated parts: 1) interval size and 2) number of in-

tervals. Statistical measurements can be made for a particular input to

determine how the Q's of (4.7) range. The interval size can then be cho-

I!

53

I LU

CD,

3fnVA 31V-

54

I sen to allow for uniform (if any) convergence of the parameters.

The number of intervals presents other problems. As the number of

intervals increase, the amount of state space in these intervals decrease.

As teh SNR of the independent variables decreases, the probability of an

internal misselect increases. Let

Yu==o output with noise present and no input

Yuso = output with noise present and nonzero input drivingfunction

NZ YuMo(it)2

SNR = N- (4.8)

SYu=o(it)2

i=l

As the SNR decreases, the number of intervals the nonlinearity can be

broken into and still get an "allowable" (in the sense of (4.6)) fit de-

creases, (Number of Intervals) c(SNR (independent variable), where the pro-

portionality constant is designer-selected. As offline test for the proper

number of intervals is given using the Theil criterion [16].

Nl 2

K i(4.9)N N p

As shown in Fig. (IV-4), for a particular example, an optimum number of

intervals was found.

The identification time frame is also designer-selected. In general,

the longer the time for identification, the more accurate the final answer.

However, in practice the amount of time must be weighed against the speed

of parameter adaptation and measurement noise levels in order to give an

optimum adaptation time.i |

I

* 55

'-I

CD,

C) L-%

1 56

V. RESULTS

The response error MRAS approach was tested on numerous models,

I since no human pilot operating records were available from FDL at Wright-

Patterson AFB, Ohio. The examples illustrate the good parameter track-

I ing possible. It should be emphasized that since no optimal control

pilot model formulation was explicitly a part of this study, but instead

only the feasibility of MRAS identification for plants with correlated

noise, results and their interpretation should be limited to the scope

of MRAS nonlinear function and linear parameter estimation.

Example 1 (No Noise)

Plant: xp + f(x)x + x = u

where f(x)x is the saturation function shown in Figure V-i

Model: xm + ailqxm + xm a bou

bo = constant

allq = supposed intervalized nonlinearity

Using the interval select nonlinear MRAS approach, with ci's constant.

With SVF poles at -20 and -30, the results are shown in Figure V-i,

with noisy measurements

Xavail X +

E{n} - 0

a a -= .05xp

Example 2 (Noise)

Plant:

Spx -f(Xp) - xp + uP p

y(output) x p + nx'7

I!

'~1

57

IL

ILa

N

LLLLI

! --

!58

The available (measured) u is u' - u + nu

u Z Aisin witIi=l

I ( {.2, .4, .6, .8, 1, 1.2, 1.4, 1.6}

First, a linear model will be fit to the system and then a nonlinear

I plant, to compare the improvement in tracking accuracy of the nonlinear

version over that of the linear version. The linear version is

Ym blm(S) =u-u-(ss2 + a21s + a,,

and the nonlinear version

xm = u fl(xm) f2(Xm)

Ym = f3(Xm) + f4 (Xm)

Each case used ci values varying. The nonlinear case used 5 intervals

for each nonlinearity.

The results for the linear case are shown in Figures V-2 through

V-13. The nonlinear case is shown in Figures V-14 through V-16.

Results show that the linear model has a poor fit to the (actual)

nonlinear plant, because of the forced fit, or lack of degrees of freedom,

of the linear model. The nonlinear results, on the other hand, show

) good model-plant correlation, as can be seen in Figure V-16(b), where a

phase plane of yp vs ym is plotted for the test input. If the model were

a perfect replication, there would be a single line at 450; only small

variations about this line occur.

A complete computer program listing is given in the Appendix of the

identifier system for practical nonlinear identification by the interval

approach. A flowchart is shown in Figure V-17.

/

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75

IVI. SUMMARY AND CONCLUSIONS

The response error MRAS identification can result in meaningful

nonlinear identification if set correctly. Chief problems include ci

varying by interval, noise biasing effects. This is shown in Figure VI-l

and 2. If one uses fixed ci terms, the nonlinear intervalized response

acts like time varying ai(t) terms for the plant. Even if one uses varying

ci terms, ci , there is the sharp time-varying tracking problem as different

intervals are entered (Figure VI-2). As shown, in Figure VI-3, there must

be vector arrays of ci terms, the vector entries being for different in-

tervals. This means that if there are two ci terms, c1 and c2 , whereAT A lI Cl2

T, . ., c ] where there are p intervals to the nonlin-

earity.

The effect of the number of intervals in the presence of noise has

been shown to be a parabola, Figure VI-4. Too few intervals yields an

"almost linear" fit, and too many, given noise, yields nonsense due to the

interval search problem discussed previously.

To perform total human operator identification, a complete Optimal

Control Model (0CM) computer package, plus operator remnant construction,

would be needed. Such work was beyond the scope of this project, but

could be extended from the present results.

The methods developed here do provide for good nonlinear model iden-

tification, if the noise levels are reasonable. Work is underway to mini-

mize the effects of the large correlated noise levels.

elA

t.

76

PRF PROBLEM

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REFERENCES

[1] Astrom, Eykhoff, "System Identification - A Survey," Automatica,vol. 12, 1971, pp. 123-162.

[2] Rajbman, N.S., "The Application of Identification Methods in theU.S.S.R. - A Survey," Automatica, vol. 12, 1976, pp. 73-95.

[3] Sage, Melsa, System Identification, Academic Press, 1971.

[4] Bekey, Beneken, "Identification of Biological Systems: A Survey,"Automatica, vol. 14, no.1, 1978, pp. 41-47.

[5] Landau, "A Survey of Model-Reference Adaptive Techniques - Theoryand Applications," Automatica, vol. 10, 1974, pp. 353-379.

[6] Kudva, Narendra, "An Identification Procedure for Discrete MultivariableSystems," IEEE Trans. Auto. Control, vol. AC-19, no. 5, Oct. 1974

[7] Sprague, Kohr, "The Use of Piecewise Continuous Expansions in theIdentification of Nonlinear Systems," ASME J. Basic Engineering,June 1969, pp. 179-184.

[8] Gallman P., "A Comparison of Two Hammerstein Model IdentificationAlgorithms," IEEE Trans. Auto. Control, vol. AC-21, February 1974,pp. 124-126.

[9] Landau, M. and Leondes, C.T., "Volterra Series Synthesis of Non-linear Stochastic Tracking System," IEEE Trans. Aero ElectronicSystems, vol. AES-lI, no. 2, March 1975, pp. 245-265.

[10] Gallman, P., "An Interactive Method for the Identification of Non-linear Systems Using aUryson Model," IEEE Trans. Auto. Control,vol. AC-20, no. 6, December 1976, pp. 771-775.

[11] Landau, I.D. "Elimination of the Real Positive Condition in theDesign of Parallel MRAS," IEEE Trans. Auto Control, vol. AC-23no. 6, December 1978, pp. 1015-1020.

[12] Johnson, C.R., "SCHARF", Submitted July 1979 to Proc. IEEE.

[13] Wellstead, P., "Model Order Identification Using an AuxillarySystem," Proc. IEEE, vol. 123, No. 12, December 1976, pp. 1373-1379.

[14] Van den Boom, A. and Van den Eden, A., "The Determination of theOrders of Process and Noise Dynamics," Automatica, vol. 10, 1974,PP. 245-256.

[15] Unbehauen, H. and Gohring, B., "Tests for Determinating Model Orderin Parameter Estimation," Automatica, vol. 10, 1974, pp. 233-244.

I

7 A-A094 411 TEXAS A AND M UNIV COLLEGE STATION TELECOMMUNICATION--ETC F/6 12/2NONLINEAR MODEL IDENTIFICATION FROM OPERATING RECOROS. (U)NOV A0 B K COLGURN, A R SCHATTE F'49620-79-C-0091

UNCLASSIFIED TCSL-80-1 AFOSR-TR-81-0076 NL2EF

I [16) Kheir, N.A. and Holmes, W.M., "On Validating Simulation Modelsof Missile Systems," Simulation, April 1978, pp. 117-128.

m [17] Lion, "Rapid Identification of Linear and Nonlinear Systems" AJournal, October 1967, pp. 1835-1842.

[18] Hang, "The Design of Model Reference Parameter Estimation Syst,Using Hyperstability Theories," Proc. 3rd IFAC Sym. Sys. IdentJune 1973, pp. 741-744.

[19] Sehitoglu, Klein, "Identification of Multitalued and Meibory Noilinearities in Dynamic Processes," Simulation, Sept. 1975, pp.

[20] Tomizuka, "Series-Parallel and Parallel Identification Schemesa Class of Continuous Nonlinear Systems," ASME J. Dyn. Sys. MeCon., June 1977.

l [21) Colburn, B.K. and Schatte, A.R., "Nonlinear Dynamic System Idefication Using the MRAS Approach," Proc. 13th Asilomar Conf. Cand Systems, Nov. 1979.

II

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