Design of a reduced-order adaptiveobserver

Prof. N. Minamide, Dr.Eng., F.S.I.C.E., Prof. P.N. Nikiforuk, Ph.D., Dr.Sc,C.Eng., F.I.E.E., F.lnst.P., F.E.I.C., and Prof. M.M. Gupta, Ph.D., Fel.I.E.E.E.

Indexing terms: Adaptive control, Matrix algebra, Mathematical techniques

Abstract: A reduced-order implicit adaptive observer for a single-input single-output time-invariant nth-orderlinear system is considered. A state vector described in observable canonical form is explicitly parameterised interms of the unknown system parameters and the filtered input and output vectors. For parameter estimation, aleast-square-type adaptation scheme that can afford an arbitrarily fast exponential convergence is proposed.

1 Introduction

Various design methods for reconstructing the state of anunknown system using only the input-output measure-ments have been reported [1, 2, 5, 6]. Among these, animplicit-type adaptive observer has attracted considerableattention. The advantage of this type of observer is thatthe state reconstruction and parameter identification canbe considered separately. A structure of this type of obser-ver was first studied by Kreisselmeier [2] in observablecanonical form and later by Nuyan and Carroll [6], whoincluded reduced-order as well as full-order adaptiveobservers. More recently, the authors [5] gave an explicitcharacterisation of the full-order adaptive observer inobservable canonical form.

This paper presents the derivation of a parametric rep-resentation of a reduced-order implicit adaptive observer,and a least-square-type adaptation scheme that can assurean arbitrarily fast exponential rate of convergence.

In Section 2, the parametric representation of a statevector of a continuous-time system described in observablecanonical form is given in terms of filtered input andoutput vectors generated by (n — l)th-order dynamics. InSection 3, a least-square-type adaptation scheme is pro-posed. An arbitrarily fast exponential convergence of thescheme is obtained using the Lyapunov method. Theresults are a natural extension of those obtained byLandau and Silveira [4] for discrete-time systems.

Parameterisation of a reduced-order adaptiveobserver

Consider a time-invariant observable linear systemdescribed by

xp(t) = Ap xp(t) + bp u(t), xp(0) = x0

y(t) = c'pxp(t)

where xp(t) is a state vector of known dimension n, u(t) andy(t) are scalar input and output, respectively, and Ap, bpand cp are in observable canonical form

— a ,— a

Paper 4547D (C8, C9), received 25th March 1985

Prof. Minamide is with the Department of Electrical Engineering, Faculty of Engin-eering, Mie University, 1515 Kamihama-cho, Tsu-City 514, Japan. Prof. Nikiforukis Dean of the College of Engineering and Prof. Gupta is with the Department ofMechanical Engineering, University of Saskatchewan, Saskatoon, Saskatchewan,Canada

IEE PROCEEDINGS, Vol. 133, Pt. D, No. 3, MAY 1986

0

In eqn. 1, system parameters a, b and an initial state x0 areunknown.

An adaptive observer must adaptively identify theunknown system parameters a, b and asymptoticallyreconstruct the state of the unknown system eqn. 1 frominput-output measurements.

Let the components of the state vector xp(t) be dividedas

where z(t) e R"'1 is an unknown component of the statexp{t). From eqn. 1, y(t) and z(t) obey the following differen-tial equations:

y(t) = -aiy(t) + Zl(t) + MW (2)

z(t) = Sz(t) - ay(t) + j]u(t)

where a, /? £ R" 1 and S e 1tively, by

a' = [a2 a3 • • • a j , /?' =

and F € R("~l)x{n i] is an asymptotically stable matrix ofthe form

Pu(t) (3)

are defined, respec-

(1) F = \-fr-A=-fc'o 0 0]' e R'

Taking the Laplace transforms of eqns. 2 and 3 yields,respectively,

(s + ai)Y(s) = Z^s) + b.Uis) + y0 (4)

Z(s) = (si - n-'ifZM - *Y(s) + flU(s) + zo) (5)

where [^o, z'oT = *o is the initial state of xp(t). SubstitutingZ^s) from eqn. 4 into eqn. 5 and using the identity

(si - F)-l{fsY(s)} =fY(s) + (si - F)~1{FfY(s)}

yields

Z(s) =/T(s) + (si - F)~ll(Ff+ « i / " a)Y{s)

(6)

133

Let H(f, h) with/, h e R"'1 be an (n - 1) x (n - 1) matrixdefined by

H(f,

where

L(h)

h)

=

= L(h){I 4

'h h2

h '

• U(f)} -

•• K

0

L(f)U(h)

-1

5

U(f) =0

Jn - 1

• o

By direct calculation, it can be verified that H(f, h) is sym-metric and satisfies

(7)Ftf (/, h) = H(f, h)F' = H(f, Fh)

H(f, h)c0 = h

Thus, by the identityn-2

(si - F)~l = {s"-2I + X (Fk +flFk~l

k=l

+ ---+fkI)s"-k-2}/dQt(sI-F)

and eqn. 7,

(si — F)~lh = (si — F)~1H(f, h)c0

= H(f,h)(sI-FTic0 (8)

is valid. By eqns. 6 and 8, a parametric representation ofz(t) is obtained:

<t) =fy(t) + H(f, axf-ai + Ff)<f>a(t)

) + £o(t) (9)

where eo(0 = eFt(z0 — y0 / ) , and (f)a(t) and (f)b(t) are definedby

(10)

The reduced-order implicit observer is thus obtained fromeqn. 9 by replacing the system parameters a and b by theirestimates a(t) and 8(i) at time t, respectively,

z(t) =fy(t) + H{f, ax(t)f- <x(t) + Ff}4>a(t)

+ H{f, f{t) - £x(t)f}mFor parameter estimation, an output equation (eqn. 2) isnow expressed in terms of system parameters and filteredfunctions <f)a(t) and <f>b(t). Substituting eqn. 9 into zY(t) =c'o z(t) of eqn. 2 and using eqn. 7 gives

y(t) + tort) = (/i + A0 - ai){y(t) -rut)}

+ & i M 0 -f'<t>b(t)} + P'4>b(t) + c'O£O(t)

where Ao( > 0) is a design parameter to be determined later,£i(t) = c'oeo(t)and

1*' = [/i + ^o - fli,{(/i + ^o)/+ Ff- a}', blt n

-f'tJLt), 4>'JA u(t) -f'Ut), Um

Let D denote a differential operator. Then, y(t) is expressedas

134

y(t) =

+An adaptation scheme described in the following Sectionrequires that H^s) — k/2 be strictly positive real for somepositive constant X. This condition is easily satisfied byrewriting eqn. 11 as

(12)

where e(t) = 8^/(0 + ?n) and Ax is another design param-eter to be determined later. In eqn. 12, however, a newfiltered vector {ij/(t)/(D + Xt)} that has to be generated forparameter estimation is introduced. To reduce the numberof such new variables to be generated to the minimum, thefollowing argument is used.

Let {9%, k = 1,2,..., 2n} be recursively defined by

2 — r\l —

)* = n*(13)

' n + 2 — f / n + 2 ~ ^ l ^ n + l * • • • > ^ 2 ^ ~ ^ 2 n ~ A1^2n-l

Solving for {rj*, k = 1, 2, . . . , 2n} in eqn. 13, and substitut-ing the resulting equations into eqn. 11 and using eqn. 10,gives

y(t) =l

+ + fii(

- <#)] (I4)

where <j)a k(t) and (/>b k(t) are the /cth components of (f>a(t)and 0fc(t), respectively, and

Let the estimate of the output y(t) be

This gives the following error equation:

e(t) = y(t) - y(t) = H(D){ - 6'(t)<l>(t) + e(t)} (15)

where 9(t) = 6(t) — 9*, H(s) is a proper transfer functionand s(t) is an exponentially decaying time function.

Remark: An extension of the results to discrete-timesystems described in an observable canonical form

xt + l = Apxt + bput

is almost immediate. In this case, take / = 0 in eqn. 9 andthe parameterised representation of unknown componentszt of the state xt is given then by

• - a n b 2 b 3 ••• b— a-

— a-.

—a-.

IEE PROCEEDINGS, Vol. 133, Pt. D, No. 3, MAY 1986

3 Adaptation scheme

In this Section, a least-square-type adaptation scheme thatcan guarantee the global asymptotic stability of the errorsystem of eqn. 15 is considered.

In the discussion that follows, H(s) appearing in theerror system eqn. 15 can be any transfer function such thatH(s) — A/2 is strictly positive real. In particular, when H(s)is given by eqn. 14, H(s) — A/2 is strictly positive real, ifand only if A, Ao and At are chosen such that

0 < A < 2 0 < A o A < 2 / 1

Let x(t) be the state of the error system eqn. 15 corre-sponding to any minimal realisation of H(s) = c(sl

Aylb di

Remark 1: Boundedness of K(t) is always guaranteed ifeither qx(t) is set to zero or

x(t) = Ax{t) + b{u(t) + e(t)}

e(t) = cx(t) + d{u(t) + s(t)}(16)

where u(t) = — 0'(O</>(O- Then, consider the following adap-tation scheme:

§(t) = K(t)cf>(t)e(t) 0(0) = 0O (17a)

K(t) = qi(t)K(t) - q2(t)K(tWt)4>'(t)K(t) K(0) = gl

(lib)

where K(t)eR2"*2" is a symmetric gain matrix, qt{t)(i = 1, 2) are non-negative scalar functions and g is a largepositive constant. It is to be noted here that K(t) is positivedefinite as K~l(t) satisfies the differential equation

l(t) + q2(tMt)<t>'(t)

Theorem 3.1: Suppose that H(s) in the error system eqn. 15is a transfer function such that H(s) — A/2 is strictly posi-tive real for some positive number A, and that q2(t) in theadaptation scheme of eqn. lib is chosen to satisfy 0 ^q2(t) ^ q2i < A with q21 as a positive constant. Then,

(i) x(t) e L2, u(t) G L2 and e(t) e L2

(ii) l i m ^ ^ r ) = 0 and limf^oo0'(O^~1(O0W exists(iii) 0(r) is uniformly bounded if K(t) is uniformly

bounded.

Proof: Because it is assumed that H(s) — A/2 is strictlypositive real, there exist positive definite symmetric matrixP, a positive constant \i and vectors W and L such that

A'P + PA = -\i2P - LL

b'P + WE = c,

W'W = d + d' - A

Consider the functional V(t) defined by

V(t) = x'(t)Px(t) + d\t)K-\t)6{t)

d2 b'Pb

(18)

^ - , 2 w - ^ * <19)

Then,

v(t) = -11^(0 - £ ( 0 ^ I I P - 11^(0 + wu(t)\\2

-qi(t)0'(t)K-l(t)9(t) - [{A - q2(t)Y'2u(t)

+ &{t)d/{X - q2(t)}1/2] 2 (20)

whence V(t) converges and condition (i) follows. As A ineqn. 16 is asymptotically stable and u(t), e(t) e L2, it followsthat lim,^^ x(0 = 0. Because V(t) has a limit and lim,^^x(t) = 0, a limit l im^^ 9'(t)K~l(t)9(t) thus exists. Condi-

tion (iii) is a direct consequence of condition (ii).

In the latter case, tr{K(t)}, the trace of K(t), becomes con-stant.

Remark 2: Theorem 3.1 can be extended as the followingstability theorem of a class of feedback systems (cf. Refer-ence 4). A linear time invariant system described by

x(r) = Ax(t) + Bu(t)

y(t) = Cx(t) + Du(t)

in feedback connection with a time-varying system

z(t) = A(t)z(t) + B(t)v{t)

w(t) = C(t)z(t) + D(t)v(t)

with y(t) = v(t) and u(t) = — w(t) is globally asymptoticallystable if the following conditions (a)-(c) are satisfied:

(a) C(sl - A)''B + D - A/2 = H(s) - A/2 is strictlypositive real

(b) There exist non-negative definite matrix functionsP(t), Q(t) and R(t), a matrix function S(t), and a symmetricmatrix function F(t), such that the following system ofequations is satisfied:

P(t) + A'(t)P(t) + P(t)A(t) = -Q(t) + C(t)F(t)C(t)

B'(t)P(t) + S'(t) = C(t) + D'(t)F(t)C(t)

R(t) = D(t) + D'(t) + D'(t)T(t)D(t)

and

r<2(o s(t)i\_S'(t) R(t)j

M(t)

(c) A - r(o > o.The proof parallels that of theorem 3.1 by taking, as aLyapunov function, V(t) = x'(t)Px(t) + z'(t)P(t)z(t), whereP is defined by eqn. 18. Note that in the_ case of thetheorem 3.1, letting P(t) = K-\t), z(t) = 0(0, ^4(0 = 0,

S(t) = 0, R(t) = 0, T(0 = q2(t) and A = A verifies conditions(aHc).

The following theorem shows that the proposed schemecan provide an arbitrarily fast exponential convergenceunder the persistently exciting condition of eqn. 21.

Theorem 3.2: Suppose that qx{t) and q2(t) in the adaptationscheme eqn. lib are chosen to satisfy q^t) = qi0 and<?2o ̂ <?2W ̂ <?2i < K respectively, with q10 and q2i

(i = 0, 1) as positive constants. If there exists a positiveconstant v such that

(f){xW{r) dx ^ v/ for all t ^ T (21)i - r

then K(t) is uniformly bounded and 0(0 converges to 0*, inan exponential fashion with degree not less than <5 = min{qi0, n1, 0>}/2, where \i is a constant defined in eqn. 18,

and Of is the negative of the maximum real part of eigen-values of the characteristic equation s"'1 +fis"~2 + • ••+ /„_ i = 0 determined by the observer's dynamics.

Proof: Let

V0(t) = x'(t)Px(t) + O'(t)K-l(t)e(t)

Then, by eqns. 19 and 20 and the uniform boundedness of||3c(0ll ^ Mo with Mo equal to a constant depending only

1EE PROCEEDINGS, Vol. 133, Pt. D, No. 3, MAY 1986 135

on the initial conditions, V0(t) obeys the following differen-tial inequality:

where <50 = min {ql0, fi2} and c^i = 1, 2) are positive con-stants depending only on the initial conditions. Thus, V0(t)decays exponentially with degree not less than 25. On theother hand, by eqn. lib, K~l{t) is bounded below as

,-qiot

dx

Hence, by the chain of inequalities,

qoe'(t)e(t) ^ d\t)K-\t)e(t) ^ vo(t)

6'(t)6(t) converges to zero exponentially with degree notless than 26.

4 Simulation

To test the usefulness of the proposed scheme of eqns. 17, asimulation study was performed. A system chosen forsimulation was the second-order system described by

with an input function

u(t) = 5{sin 3t + sin At]

The free parameters used in eqns. 9 and 17 were chosen as

Xo=\.2, Xx = 1.5, F=-fl = -1.0

g = 1000, qi(t) = 0.5, q2(t) = 1.0

Fig. 1 shows the parameter estimates @i(t)/6f (i = 1, 2, 3, 4)in ratio, where 6* = [—1, —0.5, 2, —6]'. Fig. 2 shows thecomparison of the trajectories z(t) = xp2(t) and their esti-mates z(t). These results show a considerably rapid con-vergence in parameter estimation and state reconstruction.

5 Conclusion

A parameterised representation of a reduced-order adapt-ive observer has been explicitly developed, and a least-square-type adaptation scheme has been shown to assurean arbitrarily fast exponential convergence by the Lyapu-nov method. This, together with the parameterised rep-resentation of the unknown components of the state

vector, implies the global asymptotic convergence of thereduced-order adaptive observer.

V 1

<£- _!

- 3

-50.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

time

Fig. 1 Parameter estimates in ratio

9

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-21

Fig. 2 Comparison of trajectories z(t) and z(t)

6 References

1 CARROLL, R.L., and LINDORFF, D.P.: 'An adaptive observer forsingle-input single-output linear systems', IEEE Trans., 1973, AC-18,pp. 428-435

2 KREISSELMEIER, G.: 'Adaptive observers with arbitrary exponentialrate of convergence', ibid., 1977, AC-22, pp. 2-8

3 LANDAU, I.D.: 'An extension of a stability theorem applicable toadaptive control', ibid., 1980, AC-25, pp. 814-817

4 LANDAU, I.D., and SILVEIRA, H.M.: 'A stability theorem withapplications to adaptive control', ibid., 1979, AC-24, pp. 305-312

5 MINAMIDE, N., NIKIFORUK, P.N., and GUPTA, M.M.: 'Design ofan adaptive observer and its application to an adaptive pole placementcontroller', Int. J. Control, 1983, 37, pp. 349-366

6 NUYAN, S., and CARROLL, R.L.: 'Minimal order arbitrarily fastadaptive observers and identifiers', IEEE Trans., 1979, AC-24, pp. 289-297

136 IEE PROCEEDINGS, Vol. 133, Pt. D, No. 3, MAY 1986