Suboptimal estimation of systems with large parameter uncertainties

Automatiea, Vol. 10, pp. 413-424. Perllamon Press, 1974. Printed in Great Britain.

Suboptimal Estimation of Systems with Large Parameter Uncertainties' Estimation Suboptimale de Syst6mes avec Ineertitudes de grands Param&res Suboptimale Schtitzung von Systemen mit grol3en Parameterunsieherheiten Cy6onTHMam,Haa oueHxa CHC'reM C 6o mo/i HeonpeeaeHHoeTbIO ]3 3rm,termax napaMeTpax CORNELIUS T. LEONDESt and JACK O. PEARSON +

An estimator that provides open-loop compensation for large uncertainties in plant dynamics and noise covariance matrices may be derived by considering the trade off between minimum estimator sensitivity and minimum estimation error. Summary--The design of a filter that must estimate the

A very appealing approach toward the design of an estimator that must function in an uncertain environment is to estimate simultaneously the uncertain parameters and the state vector. The estimates of the uncertain parameters are used in the state estimator to obtain the absolute minimum estimation error. Estimation of uncertain parameters requires the addition of an on-line identification unit [1]. Since simultaneous identification and state vector estimation transform linear estimarion problems into nonlinear estimation problems, approximations are usually required to obtain pragmatic state vector estimation algorithms. Approximations to adaptive estimators for linear systems with unknown noise mean and covariance matrix parameters have been derived using the maximum likelihood approach [2, 3]. More recently, adaptive estimation has been developed from the point-of-view of statistical hypothesis testing of error residuals [4, 5]. Bounding techniques are primarily open-loop means of preventing "apparent" filter divergence [6] without adding to the on-line estimation algorithm complexity. Obviously, the accuracy potential of bounding techniques is not as great as that of Adaptive estimators. SCHMIDT'S5Method [7] prevents apparent divergence due to data saturation by maintaining the Kalman filter gain above zero. JAZW]NSIa'S limited memory filter [8] prevents filter divergence by not allowing the estimator to form its estimate from stale data. Fading memory filters [9, 10] create stale data rejection by basing the estimator design on process and measurement noise covariance matrices that are the product of the a priori noise covariance

state of a system when large parameter uncertainties in the plant dynamics or plant and measurement noise covariance matrices are present is considered. Filter designs are evaluated relative to the trade off between filter sensitivity to the uncertain parameters and minimum mean-square estimation error. This is accomplished by reformulating the estimation problem as an optimization problem with vector-valued performance index. The dominant design has form similar to the Kalman-Bucy filter, and employ open-loop compensation for the uncertain parameters. 1. INTRODUCTION

THEORETICALLY, the Kalman-Bucy filter gives the unbiased, minimum variance estimate of the state vector of a linear dynamic system disturbed by additive white noise when measurements of the state vector are linear, but disturbed by white noise. Such performance is hardly ever realized in actual practice since the information required to construct the Kahnan-Bucy filter is only approximately known. The noise parameters and models may be based upon only relatively few data points, computer round-off errors may be significant, and the system model may not be adequate. When it is impractical or impossible to arrive at accurate information upon which to base the filter design, adaptive and/or bounding techniques must be considered. * Received 15 January 1973; revised 13 August 1973; revised 19 December 1973. The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by Associate Editor A. Sage. t Present address: University of California, Los Angeles, California. Present address: Hughes Aircraft Company, Culver City, California. § This research was supported under AFOSR Grant 72--7166.

413

414

CORNELIUST. LEONDESand JACK O. PEARSON

matrix and a positive scalar whose magnitude increases with time from the moment of the measurement. Selection of the divergence prevention parameters employed by the various Bounding techniques; such as the fading memory or limited memory time constants, is done primarily by computer simulation. When the system uncertainties are parameter uncertainties, and upper and lower bounds on the parameter uncertainties are known, the divergence prevention parameters can be chosen analytically. D'APPOLITO and HUTCHINSON [11] obtained a procedure for determination of the steady-state estimators, termed "minimax sensitivity filters", that yield the minimax values for Sobral's sensitivity measures when the system noise covariance matrices are uncertain, but bounded with known bounds. Maxinfization is performed over the set of possible uncertain parameter values, while minimization is performed over the set of admissible linear estimator designs. D'APPOLITO and HUTCHINSON [12] derived the steady-state minimax sensitivity filter for a scalar system with uncertain time constant. LEONDESand PEARSON[13] derived a minimax estimator that minimizes the averaged mean-square error when the plant dynamic uncertainties and process noise uncertainties maximize the averaged mean-square error. This article considers a more general approach toward selection of optimizing Bounding technique parameters than the minimax, worst-case, design [13]. Since parameter uncertainties exist, the performance, based on the mean-square error, for any estimator design will be a set of values. The performance sets indicate the sensitivity of the estimator design to the uncertain parameters as well as the relative optimality of the design. Upper and lower bounds of the performance sets are used to form a two-dimensional performance index. The set of non-inferior designs corresponding to the vector-valued performance index is obtained. The "optimum" non-inferior design is then chosen utilizing a suitable secondary performance measure. In the next section it will be shown that the minimax design [13] is a member of the non-inferior design set, and thus a candidate for the "optimum" non-inferior design. However, the minimax design will be the "optimum" non-inferior design only in extreme cases. The estimator to be developed in this article utilizes open-loop compensation for parameter uncertainties. Thus, the Kalman filter gairt matrix is not a function of the measurements or predicted error residual. Therefore, the estimator considered herein requires less on-line computations than "adaptive" estimators. However, its accuracy is

not as great as "adaptive" estimators. This is true of all Bounding estimators. The estimator that will be derived selects optimum Bounding parameter values utilizing knowledge of the upper and lower bounds of the parameter uncertainties. If this additional knowledge is accurate in the sense that the assumed bounds contain the actual parameter perturbations then filter divergence due to data saturation will not occur. However, if the actual parameter values exceed one of the assumed bounds apparent divergence will occur. Thus, one must have accurate bounds or at least "loose" bounds to guarantee divergence prevention. 2. PROBLEM STATEMENT The estimation problem considered assumes the linear continuous system

dx/dt=A(aD t)x(t)+ B(t)u(t); x(to)=xo

(l)

with noisy measurements

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

to
where the dependence of x(t) upon parameter uncertainties at has been suppressed for notational simplicity. The noise processes are zero mean, statistically independent of each other and x o, and have covariance matrices

E(u(t)u(tx)r) = Q(a2, t)f(t- tl) E(v(t)v(q)r)=R(a3, t)6(t- tl). The initial condition for the plant is zero mean and has covariance matrix

E(xSo)=ro. The system uncertainties are known to be such that

A(al, t)e{A=(a~j): (At)~j
(2)

Q(a2, t)e{ Q=(qo): (Q1)ij <-qij

<-(Q2)ij, 2~(Q)>0}

(3)

R(a3, t)e,{R =(rij): (Rt),j < rij _<(R2),j, 2~(R)>0}

(4)

where 21(M) is the ith eigenvalue of M, M = ( m o ) means M is a matrix with

m~j, and (M)~I is the/jth element of M.

ijth element

Suboptimal estimation of systems with large parameter uncertainties The bound matrices A~, Q~, R~; i=1, 2 are derived from the limited knowledge of the system gained from the available data. Since filter divergence due to data saturation can occur if the bound matrices do not encompass all possible parameter perturbation, "loose" bound matrix values should be employed when the available data is not judged sufficient for bound matrix selection.

It is proposed to find an estimate ~(t) of x(t) that is of the form d~/dt=F(t)~(t) + a(t)z(t);

SO(to)=E(xo)

(5)

The estimator matrices F(t) and G(t) are to be chosen such that the performance index (PM) given by equation (6) is minimized in a sense developed in the next section.

J(F, G, a3=

f

T

Trace(D~) dt

(6)

to

where

~(O=E((x- ~)(x- ,~)*) [to, T] is the estimation time interval, and D is a symmetric, positive definite matrix. 3. GENERAL THEORY A general approach toward selection of the optimal estimator design when system parameter uncertainties exist will now be presented. The estimator problem will be reformulated as an optimization problem with vector-valued performance index. The set of non-inferior designs will be identified. Seccndary performance measures will be utilized to obtain a unique dominant estimator design. Consider the problem of optimal estimator design when only the process noise eovariance matrix Q(t) is uncertain. Since the plant matrix A(t) is known, the estimator will be unbiased if one chooses F(t) equal to A(t)-G(t)C(t), and if E(~(to))=E(x(t,)) [14]. In this case, a single matrix differential equation can be written to describe the time behavior of the error oovariance matrix. One has [15]

dy]dt =(.4 - GC)~ + ?(A - GC) r + BQB r + GRG r ?(to)----?o.

(7)

415

The performance index (6) is a function of two variables; the Kalman filter gain matrix G(t) and the uncertain parameter matrix Q(t). Equation (6) can be written as (assuming ~o equals zero)

J(G, Q)=Trace

ff to

Drl~t, ttXBQB r to

+GRGr)~(t, tt) r dtt dt

(8)

where dO(t, tt)/dt=(A-GC)e~(t, tt); O(to, to)-----I. For any Kalman filter gain matrix selection, (8) defines a linear mapping of the space of admissible Q matrix element values into the positive real line. When the Q matrix is uncertain and the admissible uncertainty in Q matrix elements is defined by (3), the mapping range, denoted "the performance set", is an interval of the positive real line for each G(t) selection. Since the admissible Q(t) set is compact and the mapping linear, the performance set has both a maximum and a minimum value. The difference between performance set maximum and minimum values is a measure of the sensitivity of the selected estimator design to parameter uncertainties. The maximum and minimum values taken separately are measures of the design optimality with respect to P M minimization. Thus, the performance set corresponding to an estimator design choice is a measure of both the design choice optimality and the design choice sensitivity to uncertain parameters. Consider the general problem. Let U--(Q, R, A) denote the uncertain parameter variables with admissible set S v defined by (2)-(4), and K---(F, G) denote the estimator design parameters with admissible set S x. Then, a surrogate performance index (PM) that is a measure of both system sensitivity and relative optimality, with respect to the minimum mean-square error, is the twodimensional P M

Jr(K. U)z=(I.(K). IL(K)) where

Ju =max J(K, U) UeS, Jr,=min J(K, U)

UsS~.

(9)

416

CORNELIUST. LEOSDns and JACK O. l~.AgSOtq

The performance index (9) will be employed to obtain an "optimum" design for the uncertain system. The advantages of using this PM in lieu of a more complicated PM are (a) an analytic solution is obtainable, and (b) the "optimum" design's performance set bounds will be evaluated automatically with generation of the optimum design matrices. A unique minimizing design KII=(F 11, G11) exists if for all admissible design choices

K=(F, G)eS~c. J.(K 11)
Consider K/SK and UpeS, such that

J(Kp, Up)=Po=min K

maxJ(K, U) U

Kp is the pessimistic, or minimax, design. Its lower bound is PL m

min J(Kp, U). u

Consider KosSKand UoeSvsuch that

J(Ko, Uo)= OL= min min J(K, U) K

U

Ko is the optimistic, or minimin, design. Its upper bound is

OH= max J(Ko, U). u

ju(K11)J~(K**)

These two estimator designs allow one to determine the noninferior design set.

or

j.(K 1I)> J.(K**) and JL(K 1i)
Theorem 1. The noninferior design set E is contained in the set of design choices K/SK, denoted CE, whose performance set upper and lower bounds, J, and JL, respectively, belong to SN defined by SN ---((Jg(Kj), J£(Kj)): 01.
and Pv ~Ju ~ 0.).

Definition I The subset E of SK, termed "the non-inferior design set", is such that for each Ko=(F., Ge)sE and for all K=(F, G)~$~c

Proof. The optimistic design is a candidate for a unique minimizing design since OL----min minJ(K, U)~gJ(K, U) for all

J.(IO
E

U

KsSK and UsS.. Also, the pessimistic design is

implies

non-inferior to the optimistic design since J.(K)-----J~(K.) and JL(K)=JL(K.). P.ffimin maxJ(K, U)~Ju(K) for all The solution of an optimization problem with vector-valued PM is not as direct as that of an optimization problem with scalar PM since in general a unique minimizing solution cannot be found. Only the non-inferior design set is readily obtainable. To obtain a unique solution, termed "the dominant design", secondary performance measures, termed "evaluators", that compare the performance of each nonlnferior design are required. Evaluators must be such that they are consistent with the original PM in the sense that ff a unique minimizing design exists the evaluator will yield that solution, and that the desired trade off between sensitivity and performance value reduction will occur. Properties that evaluators must satisfy are determined herein.

K

U

KsSK. Finally, all designs Kj with performance bounds belonging to Ss are noninferior to Kp since if Kj has performance bounds in Ss JL(Kj)~ PL while P. ~ J~(Kj).

[]

Sensitivity considerations allow one to find a subset SCE of CE that contains the estimator designs superior to the minimax or m~nimum designs in the sense that averaged over all possible performance values their increase or decrease in sensitivity to parameter uncertainties results primarily in a decrease in performance index value. based on the mean-square error.

Suboptimal estimation of systems with large parameter uncertainties Suppose the minimax design is superior to the minimum design in the sense that averaged over all its possible performance values the minimax design's performance value is smaller. One seeks a design superior to the minimax design. Let KieCE. K~ is more sensitive to uncertain parameters since JL(K3 < Pr and P,
PL- A(K~) > J~(K~)- Pu. Suppose the minimin design is superior to the minimax design. In this case a design K~eCE is less sensitive to parameter uncertainty since OL ~Jr( K3 and J.(K3 < O ,

K i will have performance superior to the minimin design if the decrease in sensitivity occurs primarily in the upper bound of K{s performance set. Thus, one requires

417

and

J.(KJ = v2(&, &)=(J,(KJ 2 + AiK,)2) +.

Property 1 indicates that if a design other than the minimax and minimin designs is chosen, its performance bounds must belong to Sp or S,. An evaluator V~ will guarantee this type of selection if the partial derivative of Jr with respect to Ju along a constant V~ arc satisfies < - 1 in neighborhood of (PL, Pv) (10) > - 1 , but <0 in neighborhood of (Or, Ou) Vt satisfies equation (10). V2 satisfies equation (10) only in a neighborhood of the minimax design bounds. Thus, Vt can be utilized always, while V2 can be utilized only if the minimax design is super;or to the minimin design, dx evaluated for V2 at (Pr, P,) is a large negative value. Thus, V2 penalizes sensitivity more than overall performance value decreases, dK evaluated for VI at (Pr, P~) is minus one. Therefore, V1 penalizes sensitivity less than V2, and has a possible greater reduction in performance value.

Jc(K~)-Or < O.-J.(Kt). These results are summarized in Property 1.

Property 1 The dominant estimator design has performance set bounds that belong either to Sp, defined by

Sp=((J,(KD, JL(K,)): Pr-Jr(K~) ~ J,(K,)-P,) or to So, defined by

So=((J,(K,), &(K,)): O , - & ( K , ) > & ( K , ) - O,3. Obviously, it is impractical and/or impossible to evaluate each and every element of the set CE or any of its subsets, such as Sp and So. However, Theorem 1 and Property 1 can be utilized to select evaluators such that the chosen secondary PM's will yield a suitable dominant design. Secondary performance indices should select a unique design whose performance set has minimum value in some sense. Thus, candidates for an evaluator should penalize excursions of the performance set bounds J~, JL from the origin of the J. x JL plane. Some secondary performance indices that possess this property are

Je(K~)= Vt(J,, Jz)=~(J,(K~) + J~(K,))

4. NOISE COVARIANCE UNCERTAINTIES

Consider the noise covariance matrices Q(t) and R(t) to be uncertain, but constrained by equations (3) and (4). Since the plant matrix A(t) is known, the estimator will be unbiased if one chooses F(t) equal to A(t)-G(t)C(t) and E(~(to))=E(X(to)). Thus, estimator performance is determined by selection of the Kalman filter gain matrix G(t). The dominant estimator design, which is optimum with respect to the evaluator Vl, is obtained from the non-inferior design set by finding the estimator gain matrix G 1~(t) such that

je(G t t) = rain V~Ju(G), Jr(G)). G

Optimization is performed subject to constraints (3), (4), (6) and (7). The dominant design problem is a cascade of two optimization problems; one with respect to Q and R assuming an arbitrary G(t), and a second with respect to G(0 given the solutions of the first problem. First solve for JL(G). One wishes to find Q*(t) and R*(0 such that

JL(G)-- J(G, Q*, R*)-- min f r trace 137 dt. Q, R

dr.

418

CORNELIUST. LEONDESand JACK O. PEARSON

Optimization is performed subject to constraints (3), (4) and (7). Define the Hamiltonian H as

dfl/dt=-OH/Oy= -(A -ac)r~ D

(11)

where OH/O~is a matrix with Uth element OHIO,,U. Since the terms of the Hamiltonian involving Q and R are linear in these variables, the optimizing noise covariance matrices are bang-bang or singular. The bang-bang optimizing matrices have element values as follows. q~ = Sw((Q1)u, (Q2)o; (N1)ij)

r~ = Sw((R1)~j, (R2)u; (N2)u)

(fT trace(Dy') dr,

G

where fl(t) is the adjoint matrix. Along the optimum trajectory the adjo/nt matrix satisfies

#(T)=0

design problem is to find G ~1 such that

j,(G 11)= min Vi

H(y, Q, R, fl)=trace(D~ + (d~?/dt)flr)

- p(a - GC)-

Jz(G) and Ju(G) are incorporated into the dominant

(12)

x,d t o

ft:trace(D,*) dt ) where optimization is performed subject to constraints (11), (12), (13) and (7). The optimization problem posed above is unusual in the sense that the differential constraints are discontinuous with respect to filter gain since the switching matrices defining Q*, Q', R* and R' are functions of G. If B and G are of full rank, the diagonal elements of the Q's and R's will be constants. However, their off-diagonal elements will be time-varying. Changing the time instants when the elements of the switching matrices change sign by varying the elements of G can alter the value of the performance index. The derivation of necessary conditions for optimality will include the influence of the elements of G upon the switching matrix defining Q* and Q' if one adds internal equality constraints of the form

where (Nl(tk))U=0

tke(to,T).

(14)

qi* is the/jth element of Q*, r~ is the/jth element of R*,

Sw(e, p; y) equals e if y is greater than zero, equals p if y is less than zero, and is indeterminate if y equals zero. (Q2)o is the/jth element of Q2. The switching matrix Nx(t) satisfies

N1 =OH/OQ=BrBB while the switching matrix

N2(t) satisfies

N2 =OH[OR=GrflG. The evaluation of Ju(G) is identical to the evaluation of JL(G). The resulting maximizing Q and R matrices have elements as follows.

q[.t=S~((Q2)ij, (Q1)u; (Nt)u)

r[j=Sw((Rz)u, (R1)~j; (N2)u).

(13)

Let 7*(0 denote the error covarianee matrix that results when Q=Q* and R=R*, and ~' denote the error covariance matrix that results when Q=Q' and R=R'. The optimization problem that results when the equations defining

The switching matrix defining R' and R* is an explicit function of the elements of G. Thus, the value of G at any R*, R' element switching may be solved trivially. Satisfaction of these interior constraints may cause discontinuities in elements of G. However, since the time interval over which G is discontinuous is of zero measure, the effect on the system performance is negligible. Thus, the optimal times for N2(t)=0 are arbitrary. Convenient choices for switchings are the time instants when the design based upon the assumption of no switchings experienced switchings. Addition of (14) constrains the solution to have switchings although the optimal solution may not. Thus, one should first solve the problem assuming no switchings occur. If the resulting system trajectories are such that switchings occur, add (14). For simplicity consider a second order system with one switching of the off-diagonal elements of NI(0. First, evaluate the first variation of V~ disregarding constraint (14). Define the Hamiltonian H1 by

Hx(')", "Y*, fl, 2t, 2'1.2,-;I.3, G) Trace (,~.x~. dX,/dt2 r + D(sa(T)~*

+sz(T)~') )

Suboptimal estimation of systems with large parameter uncertainties where Xt =?', Xz=)'* and Xa=fl, and

sdT)=av,/a&; s,(r)=~v,/~J,.

05)

where k~ is a positive constant for all i. Then the performance index is always decreasing unless the stationarity conditions

If the adjoint matrices are chosen to satisfy

dAddt=-OH1/OX:

2~(T)=0

H3(t'()=H3(t +) As(t?) = 2s(t~')+v(N~)#

(16)

and to be continuous at time t=tl, where (Nt)12 =0, then the first variation of Jo is T

trace ((HI)~6G T) dt

6J,=AHx(tl) dq + f

419

to

where the matrix (H1)6 has elements OH2/Ogu, and

are satisfied. The value for v that guarantees dNl(tt)=0 when the stationarity conditions (18)are satisfied can be found by subsituting the chosen forms for 6fl(tt), dtl and 6G into (17). The result is

(f,

AHt (ti)= H1(t'~)- H 1(t~).

If the adjoint matrices are chosen in the previous manner, the first variation of the "performance index" Nt(q) is

Trace (H2)G(H2)~ dt

\dto

)-,

(f,:,,=

Consider constraint (14) as a performance index. Define the Hamiltonian//2 by //2 =Ht(y', ~*, P, ~4, 25, X6, G).

(18)

(H3)G =0

Evaluation of (18) results in the following dominant filter gain matrix.

G11=Pv(t)Cr(R 11)- 1

(19)

where

dNt(tl) = AH2(q) dtt

Pv(t) = sl(T)?*(t) + s2(T)?'(t)- As - ).r

+trace (M(2 6, 1, tl)6flr) trace ((H2)~6Gr) dt (17)

+

(20)

R 11 ~__sI(T)R, + $2(T)R,

to

The minimizing solution will now be found by constructing time histories for the elements of 6G, and selecting values for dh and dfl(tt) that produce M < 0 and satisfy dNl(q)---O. Adjoin the differential of the constraint to the first variation of the performance index. 3J, + v dNt(tl) = AHn(tt) dtl

(21)

QIx =sl(T)Q, +$2(T)Q,"

where M(y, a, t3-- y(t~)- y(ti) +a(N O#.

The matrix Pv(t) satisfies the following Ricatti equation.

dPv/dt = APv + PvAr - PvCr (R 11)- 1CPv + BQllB r Pv(to)=7° Pv(ti') = Pv(t~')- (v)(N,# + N~).

(22)

+trace (M(2s, v, h)3fl T) +

y,

trace ((H_~)~6Gr) dt

to

where H3=HI+vH2, ~.s=~.3+v,~, and ~' is a constant. Choose fir(q), dq and 3G such that the first variation of the performance index augmented with the differential of the constraint becomes

~S, +v dN, --

- -

ks ~AH~(t,)[[2

-NtllM(2 s, v, tt)ll2-ka

I" to

U(H3)aUz dt

Gtl(t) has the same form as the usual Kalman filter gain matrix. However, Pv is not the expected estimation error covarian~ matrix, but, just the matrix that yields the dominant design. Pv(t) evaluation involves solution of a multiple-point, boundary-value problem, which may be solved by stccl~t d e s e r t methods [15]. R tl and Qtl are weighted sums of the maximizing and minimizing noise covariance matrices. The scalar, constant-valued weighting terms, sl(T) and s2(T), are evaluated by solving the fixedpoint problems using 05). Example 1 illustrates the fixed-point problem.

CORNELIUS T. LEONDES and JACK O. PEARSON

420

If it is known that the noise covariance matrix elements change slowly with respect to the estimation interval's duration or are constants, simplification is possible. The optimizing noise covariance matrices are bang-bang, but without switchings. Thus, fl(t) need not be included as a differential constraint. Therefore, the dominant filter gain for constant, but uncertain noise covariance matrices is G~ 1 =PvCT(Rll) -1

(23)

Consider use of the admissible evaluator V~. The dominant estimator design is obtained from the non-inferior set by finding filter matrices F~(t) and Glt(t) such that

Je(F 11, Gll)=min V~(J,(F, G), JL(F, G)). F. G

Optimization is performed subject to constraints (2), (6), (24) and (25). The A(t) values that yield J,(F, G) and JL(F, G) are readily found using the maximum principle. The minimizing A(t) matrix has elements

where

a~=S~,((Al)ll, (A2)0; dPw/dt = A P w + PvcA r _ Pv¢Cr(R 11)- 1CPw

(N3(T)I~=A.),~)

(26)

while the maximizing A(t) matrix has elements

+BQI1B r

a~j=Sw((A2)ij, (A1)ij'~ ( N 3 ( T ) I A = A , ) U ) . (27)

Pv¢(t,) =7°" In this case, there is considerable simplification since only the constant-valued, scalars st(T) and s2(T) remain to be evaluated. To obtain the dominant design time waveform, one must evaluate Pv(t). This is done off-line by solution of a multi-point, boundary-value problem. However, using (20), an alternate means of Pv(t) evaluation is 7', 7" evaluation. Thus, using the alternate approach, one can evaluate the dominant design and its performance set bounds simultaneously off-line.

The switching matrix N3(T) is defined by T

((~2 + fl~)P + [~3ST) dt

Na(T) = to

where the adjoint matrices satisfy

dfll/dt=-Frfll-fllF-D;

]~I(T) = 0

d/32/dt=-Arf12-f12A-f13GC-D; d#3/dt = - a T#3 _ # , r -

fi2(T) =

2(GC)T/~I+ 2D; fl3(r)=o.

5. PLANT MATRIX UNCERTAINTY Consider the plant matrix A(t) to be an uncertain, constant-valued matrix constrained by (2). In this case, the estimator cannot be designed to be unbiased. Define

Let 7', P', P', S' be the matrices 7, P, P, S evaluated at A=A', while 7*, P*, P*, S* are the matrices 7, P, P, S evaluated at A=A*. Then, the dominant estimator matrices F 11, G 11 are such that j,(F 11, G 11) = min Vi

P( t) =E(x( t)x( t) T)

(28)

F, G

Trace (Dy') dt,

\dfo

P(t) =E(:~(O£(t)T) S(0 =E(x(0~(0T). Define the Hamiltonian H as Then, the mean-square error matrix 7(0 satisfies

7 =P + P - S - S T.

(24)

The differential equations defining P, P, S are

dP/dt=AP+PAT + BQBr;

P(to)=7o+So.

dP/dt =FP + PF T + GCS + (GCS) T + GRG T; P(to) =So

dS/dt-~ AS + SF T + PCT GT; S(to)=So=E(x.)E(Xo) r. (25)

H(F, G, P', P', . . . . S*, 21. . . . . 28) =Trace(dP'/dtR~ + . . . +dS*]dtAr). Since F(t) appears linearly in H and is not constrained, a singular optimal control problem must be solved. The general procedure for solving singular optimal control problems is to take time derivatives of OH/OF until F(t) appears explicitly. In this case, two time derivatives are required. The dominant F(t) matrix that results is F il(t)-~Av(t) -- G II(t)C + FG(t)

(29)

Suboptimal estimation of systems with large parameter uncertainties

421

Example I

where

Consider the

linearsystem

Av =(A.$1S, ÷ A,szS,X$1S. ÷ szS, )- 1 dx/dt= { -10

FG=((sI~* + s2~')Cr --(S1 ÷s2)G I;R)(G 11)r(S1S* ÷S2S' )- 1. The stationarity condition for a minimizing

Gft), ~gH/OG=O,yields G 11 (1) =

-

-

-O 2 } x(t)+u(t)

x(t,)=0

z(t) =x(0 + v(0. The measurement noise covariance matrix is the identity matrix, while the plant noise covariance matrix is

(($1 + $2)222) -- 1(21( P , -- S ' )

Q__.{lq0 lq0} 2~q<9.

+2~(e*-s*))c~R -1. 0o) The adjoint matrix 22 is identifical to Pl, the adjoint matrix 23 is identical to sz(T) times P3 evaluated at A(t)=d'(t), and the adjoint matrix 26 is identical to sl(T) times //3 evaluated at

,4(0 =.4*(0. F11(t) has form identical to the unbiased Kalman Filter F(t) matrix with the exceptions that (a) Av(t), a weighted sum of the "bounding" plant matrices, is employed; and (b) there is an additive term FG(t) proportional to the deviation of the dominant estimator gain matrix G 11 from the ideal gain matrix, a weighted sum of "bounding" error covariance matrices. Gl~(t) is a weighted average of the "bounding" matrices P ' - S ' and

P*- S*. Table 1 contains a summary of the estimation algorithms for both noise covariance matrix uncertainty and plant matrix uncertainty.

One wishes to find the estimator design optimum with respect to

v2(G) =((JL(G))2+ (S.(G))2)~

over the time period of 0-1 sex. First, evaluate the dominant design assuming that no q', q* switchings occur. The filter gain matrix optimum with respect to the evaluator I"2 is g" =(:

+ 6 ( T ) / ) / ( l + 6(T))

where ~(T)=$2(T)/$1(T ).

To completethe designone must evaluate6(T). This requiressolutionof the fixed-pointproblem 0 - - f O ) : - ~ T ) + ( T r a c e r ~ ¢(6,

t)dt)

r • /(Tracef,f (6,t)dr)

6. D E S I G N EXAMPLES

Two design examples that illustrate the techniques discussed previously will now be given. The first example involves a second-order system with uncertain plant noise eovariance matrix, while the second example involves a first-order system with uncertain time constant.

First-order gradient algorithmsmay be used to evaluate 6(T); i.e. the (n+ 1)th guess for 6(T) can be found from

a.+ , =a.-(o/ioa)- '/..

TAm.I I. ALoom'nm su~o~Y

Paran~m" uncertainty

On-line algorithm

Off-li~ algorithm

Qft), R(t)

d~/dt=A~+Gl l(z--C.~)

O1=pvCT(R-)-, dPvIdt.. Ah~r÷ PrAT-- PeCT(R11)- I CPv

_a,,(t~-)=~,v0~÷)-(vXNjp+N~p~ Aft)

.~, - t ~ ' ~ " +.¢,'.L~')0~S* +s2S') - ' FG" . : ((*l~" + s 2 -~~" --(SI'+~II~)(~')T(S,S*+S:S')-1 01 I----fiSt +~A2)-I(X3~'(P " - s ' )

+~(e'--S'))C~X-: ii

422

COIU~mLXUS T. LEONDES and JACK O, PEARSON

The filter gain matrix that results from the solution of the fixed-point problem is

In. addition, the maximizing and minimizing q functions are q' =2 and q* =9. Since no switchings of q' or q* occur, g~ given above is indeed the dominant design. The transient response of the dominant performance set is shown in Fig. 1. For comparison purposes the performance sets corresponding to pessimistic (minimax) design and opt.imistie (minimin) design are also shown. Obviously, a significant improvement in performance has been obtained. The dominant performance set's lower bound is 60 per cent of the total possible lower bound reduction, while its upper bound is only 14 per cent of the total possible increase.

4~ IUpper Ibounds g 3.8

/~/~_~" S "-'- "--- --'LLower

o ._E o 3"4

7,/~,

gl 1= _ 0.5(23( p' - s') + 26(p* - s*))/2z f l 1= a = - g l i + (y, + y. _ 2gl t)gl 1/(s* + s')

g~ ~=0.47287'(t) + O.52767'(t).

~ ~ . ~

The dominant estimator design is

/..-------

bounds

a, = - (3s* + s')/(s* + s').

(31)

A two-point, boundary-value problem (TPBVP) must be solved to evaluate (31). The method utilized here is to iterate on the initial conditions for the adjoirtt variables until convergence to the desired terminal values is realized. Define 2 r =(2z, 23, 26). Then, the (n+ I)th guess for the initial condition is ~(to). + ~ = , ~ ( t 0 ) . - ~ w - ' 2 ( T )

where W is the Jacobian matrix with elements d2i(T)/d2fito), and e, 0 < t < l , is chosen to make the algorithm computationally stable. The optimal filter gain, g~l, is proportional to the ratios 26/22 and 23/22. Thus, gtl may be extremely sensitive to the values chosen as the initial conditions for the adjoint variables. This sensitivity aspect of the computation may be avoided by evaluating 22 in terms of the other two adjoint variables using the first order necessary condition Hs=0, e.g. use

o

22 = -- (St23 "4-S'26)/(S' + S*).

c

g / /

2"6

0•

/ I

0.2

I Time (see) t

04

O6

I

0"8

I

lUpper

1.2

to

1=~o. 1. Error bounds with Q(t) uncertainty. g

~o~ E

Example 2

/

Consider the following scalar system with uncertain time constant.

dx/dt=ax+u

g t 804

~.__.=

jbou~,

....

' Pessimistic

//

-3
z ( t ) = x + v.

The measurement noise covaxiance matrix has value 4, while the plant noise covariance matrix has value 1. The initial condition for the plant is zero mean, and has covariance matrix value 0.001. One desires to find the filter design optimal with respect to the evaluator functional

Vt(f,g)=½(JL(f, g)+ J.(f,g)) over the time period of 0-1.0 see.

~2 I

0"2

I

I

|

04" 06 06 Time (sec)

I

I'0

Fro. 2. Error bounds with A(t) uncertainty. The TPBVP solution is shown in Fig. 2. The pessimistic (minimax) and optimistic (minimin) design performance set bounds are included for comparison purposes. The upper bound of the dominant design's performance set is only 3 per cent larger than the minimax design's upper bound,

Suboptimal estimation of systems with large parameter uncertainties while its lower bound is 13 per cent below the minimax design's lower bound. It is interesting to compare the dominant design algorithm with an approximate adaptive estimation algorithm. The extended Kalman filter for this example is defined by d~/dt =dR +Pl l ( z - : f ) / r da/dt =Pl 2(z - ~)/r dpi l/dt = 2dpl I + 2xp12 - p21/r + q dP12/dt=(d + x)p12--pllp12/r

need not be solved. The dominant estimator design is identical to the K a l m a n filter, but with "adjusted" noise eovariance matrices. Systems with constant, but uncertain plant dynamics have the dominant estimator matrices (29) and (30). The estimator algorithm is summarized in Table 1. A weighted sum of plant matrix bounds is utilized to determine the estim a t o r dynamics. The estimator gain matrix is a weighted sum of the "bounding" matrices P ' - S ' and P * - S * . Finally, there is an additive term FGR, whose gain is proportional to the deviation of the dominant estimator gain matrix G tl from an ideal gain matrix, which is a weighted sum of "bounding" error covariance matrices.

where all equations must be solved on-line. The dominant estimator, which is a bounding estimator, must solve d~/dt =,4 eye+ gl 1(z - fc) + FG~

REFERENCES [1] A. P. SAGE and J. L. MELSA: System Identification. Academic Press, New York (1971). [2] J. C. SHELLENBARGER Estimation of Covariance Parameters for an Adaptive Kalman Filter. Proc. National Electronics Conference, pp. 698-702, Chicago, Ill. (1966). [3] A. P. SAGEand G. W. HUSA: Adaptive Filtering with Unknown Prior Statistics. Proc. Joint Automatic Control Conference, pp. 760-765 (1969). [4] R. K. MEHRA: Approaches to adaptive filtering. IEEE Trans. Ant. Control AC-17, 693--698 (1972). [5] N. E. NAm and B. M. SCHAEER: Decision-directed adaptive rtcursive estimators:, divergence prevention. 1EEE Trans. Aut. Control AC-17, 61--69 (1972). [6] R. J. FITZGERALD: Divergence of the Kalman filter. IEEE Trans. Aut. Control AC-16, 736-747 (1971). [71 S. F. SCHMrDret al. : Application of Kaiman filtering to the C-5 guidance and control system, in: Theory and Applications of Kalman Filtering (Edited by C. T. LEor,~). AGARDograph 139 (1970). [8] A. J. JAZWINSKr: Adaptive filtering. Automatica 5, 475--485 (1969). [9] T. J. TARNand J. ZABORSKY:A practical nondiverging filter. AIAA J. 8, 1127-1133 (1970). [10] H. W. SORENSONand J. E. SACKS: Recursive tading memory filtering. Inform. ScL 3, 101-119 (1971). [11] J. A. D'APVOLITOand C. E. HUTCHINSON: A minimax approach to the design of low sensitivity state estimators. Automatica 8, 599-608 (1972). [12] J. A. D'APpoLrro and C. E. HUTCHINSON: LOW sensitivity filters for state estimation in the presence of large parameter uncertainties. IEEE Trans. Aut. Control AC-14, 310--312 (1969). [13] C. T. LEOND~and J. O. PEARSON: A minimax filter for systems with large plant uncertainties. IEEE Trans. Aut. Control AC-17, 266-268 (1972). [14] M. ATHArCSand E. TSE: A direct derivation of the optimal linear filter using the maximum principle. IEEE Trans. Aut. Control AC-12, 690-698 (1967). [15] A. A. BRYSON,JR. and Yu-Cm Ho: Applied Optimal Control. Blaisdell (1969). :

on-line. The constant Av and the time-varying functions gl~ and FG are evaluated off-line. A sequence of piece-wise constant gains approximating the time history of gH and FG are stored in the on-board computer. 7. CONCLUSIONS State vector estimation when large parameter uncertainties in the plant dynamics or plant and measurement noise covarianee matrices of a system are present has been considered. Open-loop compensation of parameter uncertainty was employed in order that the on-line estimation algorithm would not require any more computer storage than the usual Kalman filter. A design theory for uncertain system l:erformance optimization was presented. It was shown that since the performarce of an uncertain system is a set of values, one can view the problem as seeking the minimum of a two-dimensional performance index, whose components are the upper and lower bounds of the performance set. The set of designs non-inferior with respect to the performance index was defined. The dominant estimator design for systems with uncertain noise covariance matrices is given by (19) and summarized in Table 1. Obviously, the estimator form is no more complicated than the usual Kalman filter form. I f the uncertain noise eovariance matrices are time-varying, Pv(t) may have discontinuities. A multi-point, boundaryvalue problem ( M P B V P ) must be solved to evaluate Pv(t). For large dimensional systems the number of discontinuities may make the M P B V P evaluation impractical. However, if the uncertain noise covarianee matrices are constants, a M P B V P

423

APPENDIX Notation E(.) is the expectation operator 7 ( t ) = E ( ( x - ~ ) ( x - 2) r) P(t) = E ( x x r) /~(t) -----E(,¢~ r )

s(t) =E(x~ r)

424

CORNELIUST. LEONDESand JACK O. PEARSON

U=(Q, R, A) are the uncertain parameters with admissible set Sv K=(F, G) are the estimator design parameters with admissible set St J,(.) is the secondary performance measure V~(.,. ) is the evaluator functional (F11, G 11) is the dominant estimator design (Q*, R*, A*) are the minimizing uncertain parameters (Q', R', A') are the maximizing uncertain parameters PL, Pu are the lower and upper bounds of the minimax performance set OL, Ou are the lower and upper bounds of the minirain performance set OH/OAis a matrix with/jth element OH/Oa~j

performance h valeur vectorielle. La conception dominante est de forme semblable au filtre Kalman-Bucy, et fait emploi d'une compensation A boucle ouverte pour les param~tres incertains. Zmmmmmfmmmg--Der Entwurf eines Filters, das den Zustand eines Systems schatzen soil, wenn groBe Unsicherheiten beziiglich der Systemparameter oder der Systemparameter und der Parameter der Kovarianzmatrizen des MeBrauschens auftreten, wird betrachtet. Entwiirfe yon Filtern werden geschatzt und zwar als KompromiB zwischen Filterempfmdlichkeit in Bezug auf die numerischen Parameter und dem kleinsten mittleren quadratischen Schiitzfehler. Dies wird durch Neuformulierung des Sch~itzproblems als Optimierungsproblem mit vektorwertigem Giitekriterium erreicht. Der Hauptenwurf besitzt eine dem Kalman-BucyFilter ~hnliche Form und benutzt f'tir die unsicheren Parameter eine Kompcnsation in oftener Schleife. Pe~oMe---PaccMaTpEmaeTcgpa3pa6oTKa qbH2IbTpa,KOTOpbl~ ~OJ~geH oueHHBaTb C£)CTO~TE[e CHCTeMh,I, K o r e a

EDMCIOT

Me~'I'O 60/l~InHe Heonpc~eHeHHocrH napaM~rpoB

Rtsumt--On considtre la conception d'un filtre qui doit estimer l'ttat d'un systtme lorsque des incertitudes de grands param~tres clans la dynarnique d'installations ou dans les matrices de covariance de bruit d'installations et de mesures sont prtsentes. Les conceptions de filtrcs sont ~valutes en fonction de la rtf~rence entre la sensibilit~ du filtre aux paramttres incertains et rerreur minimum effacace d'estimation. Ceci est achev6 en reformulant le probltmz d'estimation sous forme de probltme d'optimisation avec un indice de

B nHHa~mxe yCTaHOBKa H.rIH ~aHb! MaTpH~I KoBap~auHH IIIyMOB yCTaHOBKH H H3MepeHH1~. Pa3pa6oTza dpan~Tpa oHeHllBaffrcg OTHOCEreJ'I~H0 COOTHOmeHH~I Me)K/Iy qyBCTBHTembHOCTbIO~SHn~Tpa K Heorlp~IeHeHHOCTHnapaMeTpoB H Cp~H-Iela[M~[I~rM./~O~/ ouJ~/~/~I oIIeItKH. ~TOMy conyTCTSyCT nepeabopMa1~sxa n p o 6 n e M ~ o u e m ~ s npo6neMy o~3aIIm~ c BeErOpHOil ouem¢oi~ za~ecTBa. MeTo]I pa3pa6OTXH a~anorwle~ ~b~m~,TpyKanMaHa-Bacx H Hcnon-

b3yeT pa3oMzn~/Ty~ napaMeTpoB.

KOMr/eH~

]I~

Heonpe~e.JieHHl,lX