Discrete-Time Estimators with Guaranteed Peak-To-Peak Performance

Copyright © 1996 IFAC 13th Triennial World Congress. San Francisco. USA

3a-17 2

DISCRETE-TIME ESTIMATORS WITH GUARANTEED PEAK-TO-PEAK PERFORMANCE! T. Vincent

, .2

J. Abedor" K. N agpal' P. P. Khargoneka.·'

• EECS Department, University of Michi9an, Ann Arbor, MI, 48109- 2122 ,- ilml,ex Corporation, Redwood City, CA, 94063-3199

Abstract. In this pa.per is considered discrete--time es timation ,,,ith v.'orst-case peakto-p eak gain a."i the performance measure. Both linear and a. H~~tricted cla,ss of nonlinear estimat.ion prohlems arc t reated. In the linear case estimators are synthesized tha.t minimize t.he *-uorm , the best upper bound on the induced loo norm that onc eau outrun by bounding the reachable set with incscapa 1I]e e1lipsoids. In tlw non linear ca."ie cOllditioll~ are pr~ellted that are sufficient to ensure a. given level or peak-to- peak performance. K eywords. Maximum Peak to Peak Gain, Linear Estimation, Konlinear Estimation

I. JNTR.ODCC'TIO'l

The I t optimal control problt-'m was forrnulaterl by Vidyasagar (1986). Tllt~ problem is to synthesize a controller that minimi zes the worst ca.'3c amplification from disturbancp signa.ls to regulated sip;nals, where the signal siZf! (norm) is taken to he the signal's peak value. This problem was first ~ tu(hcrl hy Dahleh and Pearson (1987}) where the authors using interpolation ideas proposed a solut.ioIl to t.he problem ba."ied on linea.r programming. Subsequen tly, this problem has been st udied ~xt.r.n~ivdy and gen eralized by various researchers {~r.e for exampl" (Dahleh «lid Diaz-Bobillo, 1995; Shamma, 1993; Staffans, 1991; VOIlIgaris, 1993) and the references therein ). A gf!nerai con~ept ual framework for st udying worst-case estimat.ion problems s ll~h as I filtering is discussed in the work of Milanf'~~e and Tempo (1985). The problem of f l filtering ove r it tiniw horizon has been addressed in a iim e-domain seltillg by Tempo (1988). Most of these approaches suffer froIll complexity prob-

e

I Support€d in pan. by AFOSR Cont.l'ad Ko F -49620-93-10246DEF alld AltO bra.llt .'1 0 DAAHn4-9.1- C:-OOI2 " SU I)ported by Nati o nal Science Foundat.ion FI!llowsh ip

lems. The optimal fil ters a nd controllers lend 1.0 be of very high order and th e computational requirements for obtaining t heIIl (Ia.rge linear programs) are also very str in~eut.

Here a \'cry different approach is taken. In the linear case the problem considered is ~Yllthesiz ing estimators tho..t minimi7:c the *-llorm, which is the best upper bound on the indur.ed lno norm that one can obtain by bounding the reachable .~et with inescapable ellip.'wids (t he precise definitions are given in the Seetiol1 3). This approach was firsl introduced by Schweppe (1973). Our work is motivated Ly the recent book by Boyd et al. on linear matrix inequalities ill syst.ems theor:; (1993) and the work of Yakubovich (1971) on applications of S-procedurc ideas to the areas of systems a.nd control.

Designing ohservers Or estimators for nOli-linear systr.ms with good guaranteed performance is very difficult. Extended Kalrnan filtering (EKF) offers one popular approach of d(~signing estimators for non-linear sy~tems. Some performance issues of EKF are investigated in (Bara. et "I.. 1988; Scala, 1994; Song and Griz,le, 1995). Mo~t o f tlwst' result s are. local iu nature and ~how t.hat,

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the EKF approach performs well if the disturbances, initial error and the growth rate of nonlinearities are sufficiently t'smaIP. Here is examined the estimation problem for a special class of Ilonlinear systems where the nonlinearities satis(y a Lipschit:;, type condition. Based on S-procedure ideas it is shown that. if a c.ertain Riccati equation admits a solut.ion, the corr0.sponding nonlinear filter is globally stabk and ensures a g:iven level of peakto-peak performance.

In this preliminary version of the pa.per, all the systems arc assumed. to be linear time invariant (LTI) and the results arc presented without proofs.

2. :.rOTATIO:.r The notation is standard. Z+ is the set of nonnegative integers, R is the set of real numbers, and Rk is the set of all k-tuplr:s of real numb(!rs. If the type of norm is not explicitly indicated, it is taken to be a 2-Ilorm. Thus for x E Rk, IIxll:=.,;;c;, aIllI for Q E Rmx", IIQII is the largest singular value of Q. Let p(Q) denote the spectral radius of Q. Let v be a k-channel signal. That is, v: Z+ ---I' RI... The loo norm (peak norm) ofv is defined to he the ::iUp over all time of the 2-norm of v at each time instant (in contrast. t.o the llslla.l definition, where the DO-norm is taken at each tim(~ instant). Thus Ilvll oo := esssuPk>O Ilv(k)lI· The space of signals l~ is defined to be the set of all k-channel signals with finite loo norm. Given a linr:ar operator H: 1::::: --+ l~), the induced Ice norm of H is defined to be IIHII ,~,:=

sup II'!I.':I-,,-~J

IIHwll oo .

Thus IIHllioc' iS 1 roughlYl the maximum peak-to-peak gain of H. Finite-dimensional linear shift-invariant systems, that is) systems that admit the usual realization x(k + 1) = A:r(k) + B1I'(1:), z(k) = C"(k) + DW(k), will often be described using system matrix notation; that is, such syst.ems will often be given as

l~w=

[i¥v],

where T zw is the transfer function mapping input w to output z.

3. PREDICTORS AND FILTERS FOR LINEAR

SYSTEMS In this section the problems of linear single step ahead prediction and filtering will be considered. First some analysis questions are addressed.

3.1 Analysis Let us consider the stable) minimal system H realized by x(k

+ 1) = z(k) =

+ BW(k) Cx(k) + Dw(k) Ax(k)

.0:(0)

= 0,

(1 )

(2)

with w(k) E Rm, l,(k) ER", and z(k) E RP. The induced loe norm (the worst-case peak-to-peak amplification) of H is defined to be

IIHllioo:=

sup

Ilu'II,~l

IIHwll"".

This norm may be described in terms of the reachable set X c Rn which is the set of all states reachable from the origin for som~ to \vith IIwll(X) '$ 1, i.e., N

X:=

:1' E Rn :

X

=

{

L AiBw(i) where i=Q

N

EZ+ and Ilwlloo <: I}.

It is easy to see t.hat the induced l:::o norm of H can be described in terms of the reachable set as follows:

IIHllioo

=

sup

Ilex + Dwll·

l'EX

w'w-~l

The following definition describes when a set is mescapable.

Definition 1: A set FeRn is said to be an inescapable set ifAx + Bw E F whenever x E F and IIwll <: l. Any set t.ha.t is inescapable must contain the reachable set and in fact the reachable set is the smallest inescapable set.. Any inescapable set (say, F) gives rise to an upper bound on the induced loo norm of H:

11H11,oo <: max IICx + Dwll. xEF w'w·~l

The idea of this paper, a."i in (Abcdor et al., IU1.) , irs to consider upper bounds on the induced lex) norm obtained from a certain class of inescapable sets, namely, the cla.":is of all inescapable ellipsoids. The least (;onservativc such upper bound is what is definE'cl to be the *-llorm of the system H. The key to the *-llorm approach is the next theorem. It. provides a simple c.haracterization of inescapable ellipsoids in terms of a single rcalIJarameter and a Lyapunov inequality. It. should be emphasized that thifl if.; a characterization of all inescapabk el1ipsoius. The sufficiency

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part of this result goes back to Sehweppe (1973) and can be derived cleanly using the S-procedure as in Boyd et al. (1993). "iecessity follows from the losslessness theorem of Yakubovitch (1971). Theorem 1. Let p be the spectral radius of A for the system H, a.nd let Q be positivp definite. The ellipsoid E = {;r : :r.'Q-I X :oS I} is iIlf~scapable if and only if there exists a real number (l' C (0, 1 ~ p2) such that -r-AQA' - Q + "-BB' < O. 1--0' Cl-

Now fix Q E (0,1 - p'). It is not hard to show that t.he smallest inescapable ellipsoid corresponding to this particular n satisfies the Lyapnnov equation 1 r --.4Q"A - Q" 1-0'

+

1 I -BB = O. n

Define Na(H) to be the corresponding upper bound IIHllioo. that is,

max

IIG.,· + Dwll·

(

3

)

ea,ily show that N"(H) = IICQ"C'1I 1 /', so computing Na(H) requires only the sollLtion of the discrete Lyapunov equation (3). It follows that for stric.tly proper systems

where It is the uIlit pulse response of H, that is, h(k) = CA' B. Computing the '-norm of H requires that N.(H) be minimil';ed over 0: E (0, 1- (2). This is straightforward because the square of No(H) (regarded as a fUllction of a) is convex. Computing Na(H) and IIHII. when H is not strictly proper reduces to the solution to an LMI eigenvalllc problem, as in th~ continuous-time case (Abedor et al., n.d.). Specifically,

OIl

(4)

N. (IJ) = inf {

-r : 3"

E

R "Heh that

c'Q;-:l r:Sl tI,1 1.O

,::;1

"Q'

Note that N(,>;(H) is the least conservative upper bound on IIHllioo detennined by inesca.pahle ellipsoids corresponding to a particular Cl' hetween zero and 1 _ p2. The *-norm of H is then defined to be be the minimum NoJ H) over all 0: in this interval:

IIHII.

:=

inf

nUO,l

iV,,(H).

[

0" C

1

0 C' ] (1 - ,,)1 D' D

,'J

}

>0 ,

where Qa it) the solution to the Lyapunov equation (3). Moreover, it can be shown that N~(H) is quasicoIlvex (as a fUllction of (t) on this interval, hence computing the *-llorm of H is again straightforward.

p2)

The *-llorm is thus the least cOllservative upper bound IIHlliex; determined by the snt of all inescapable ellipsoids.

3.2 One Step Ahead Pn.:didiun

OIl

The following analysis theorem provides an LMI (linear matrix inequalities) test: of whether a system's No;-norm is less than some p;iven numbC!r "':' or not. This is used to prove the linear pn'!dic.tion and filtering results of sections 3.2 and 3.3. Theorem 2. Let (1) and (2) be" minimal realization of H, let p be the spectral radius of H, fix Q E (0,1- p'), and let , he a positivi" real number. Then H is stable and Nu(H) < " if and only if there ('xists a symmetric matrix Q > 0 and a real number (T > 0 such that both 1/(1 - o)AQA' - q + l/oBB' <~ ()
,,'2- 1

o

[ C

0 C' ] (1 - rr)J D' > O.

D

,ll

Next consider the problem of how to compute No.(H) and IIHII •. In the strictly proper case (D = 0), one can

The one step ahead prediction problem is defined as follows. Given the system x(k

+ 1) = Ax(k) + Bw(k)

(5)

z(k) = C 1 x(k)

(6)

1/(k) = C,x(k)

+ DW(k),

(7)

it is desired to estimate the signal z( k) using the noisecorrupted measnrement Y Hp to time k ~ 1. That is, it is desired to synthesize a predictor P such that the estimate i(k) given by

2(k)

= P(y(O).··· ,y(k -1»

is dose to z(k) in the sense that the worst-case peak-topeak amplification from noise w to the estimation error e := z - is small. This can he assured by designing P to make IITe!~·II" small, where Tew denotes the linear opC!rator mapping u} to c. Hereafter the dependence of the map from lIJ to (; on the c.hoiC'e of predictor is explicitly

z

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exhibited by writing TC'IJI(P) and the following standard a.'lsnmptions are marlp.:

(PI) (C".4) is detectable and (A, B) is controllable. (P2) BD' = 0; DD' = I

IIT,,,,(P,,)II. < I

AS8umptiolL (PI) is required to insnre that a certain Riccati equation (given in the t.heorem statement) has a posit.ive definitel:)olution, and (P2) means that the process noise Bw is entirely uncoupled from the measurement noise D11I and that all measurements are corrupted by noise. The result that follows can be used to solve this prediction problem. For a fixed 0' E (0,1), this result gives necessary and sufficient conditions for the solution of the suboptimal .lV~(t-llorm problem as well as a predictor that achieves the desire level of performance when these conditions are met. Theor·em 3. Fix any 0: E (0,1) and 1 > 0, and consider the system given by (5-7) under assumptions (PI) and (P2). The following statements a.rc equivalent:

(1) There exists a. finite-dimensional, linear shiftinvariant predictor P that renders N" (T,w(I')) < -y (2) IIC, Y"C~ 11 < ,', where Ye< is the stabilizing solution to the discrete algebraic Riccati equation () =

In light of the previous result it is apparent that minimi7.ing the *-norm from w to e amounts to minimi:6ing Nupt(n) over the interval (0,1).

Corollary 5. infp(p IIT,w(P)II. infoE(o,l) Nupt(a). .Moreover, if N opt has a minimum at 0: = O'opt., then the predictor p",.", (given by (9) and (10) with a = a op .) achieves this innmlllli. In summary, solving the *-Ilorru prediction problem has been reduced to minimizing l\'op~, a continuous function of a single real variable, over the interval (0,1).

3.3 Filtering

Consider the following linear shift-in:rariallt system:

x(k

+ !..BB'

(8)

0:

R:= C:.lYClC~

+

+ 1) = 2(k)

A±(k)

l--;/~I.

+ Al:,C;W' [:~(k)

= C,f(k)

+ B-w(k + 1)

- C,x(k)] (9) (10)

Define the real-valued function of a real variable N op ': (0, 1) ~ R by N opt (<» := IIC,YoC;1I 1 /', where Ya is the stabilizing solution to the Riccati eqnation (8). 2,From the previous theorem it. follows that the best achievable No-norm from IV to e is given by Nopt(a) , and a predictor that achieves this minimum is Pa • The "'-norm suboptimal problf!m thus reduces to a. feasibility problem defined in terms of the function N opt ' Corollary 4. Let P be the seL of candidate predictors (linear shift-invariant, finite-dimensional, strictly proper). There exists aPE P ,uch that 111~w(P)II. <

(11)

(12)

+ lJw(k).

(13)

The filtering problem is essentially the same as the one step ahead prediction problem, except that we wish to e::;tirnate z(k) using al1 measurements up to and including time k. Th us for a filter }',

Moreover, if either (hence both) of these statements hold, then the predictor Pc. given hy

:i;(k

.4x(k)

y(k) = C,x(k)

1-0' 1-0:

+ 1) =

z(k) = C,x(k)

_1_ Ay".4' _ 1:, _1_ AY"C;W'C,l:,A'

and

, if and only if there exists an a E (0,1) such that Nopt(a) < I'. Moreover, for any 0: E (0,1) that satisfies Nopt(a) < " the filter P" givf'll by (9) and (10) achieves

i(k) = F(y(O),···, y(k)). The objective is design F, a linear shift-invariant system, to make the estimation error small relative to the noise in the sense of making IITewll. (hence IIT,wll,,,,) small. The result that follow, can be used to solve this problem. For a fixed <> E (0,1), this result gives necessary and sufficient conditions for the solution of the suboptimal N(},-lIorm problem as well as a filter that aehicv(~s the desi[(~ level of performance when these conditions are met.

Theorem 6. Fix any n E (0,1) and , > 0, and consider the setup given by (11-13) under assumptions (PI) and (P2). The following statements are equivalent: (1) There exists a finite-dimensional, linear shiftinvariant filter F that renders

N.(Tm(F)) < ,. (2)

IIC1P"C: 1 < " satisfy'

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where the matrices poJ'o

> 0

1";;1

= _1_.4P;;lA' 1 - 0:

+ ~BB' 0

Po = PC( + UC~ C2

(14)

0, f> > 0, " E (0, I), fi > 0, and p > 0 sLl ch that the following are satisfied

(15)

1>-1

Moreover, if either (hence hoth) of these statements hold 1 then the filter F.}. given by .f. (k

+ I ) = Ax(k)

= A«I - al P - pJ)A' p

+ ~BB' + ~K1K: a

p

= 1" - ,11 + o(JC'(aK; K, + {3J) - 1C p C: [ Cl ')' I

(16)

1 >- 0

.f (k + I )=x(k+ 1)+ nP':; 'C;(y(k

il k)

+ I)

- C,i(k + I»

=C,i(kl

(17)

then the following non linear estimator

(18) x(k

achieves No(T",, (F,,» < 1· Remark: By changing the notation to 00. := ~ p and Q 0 := ~ P : it c..:an lie readily verified that as et - f 0 , the above filter reduces to the celebrated Kalman filt.er . A~

+ I) =

Ar(k)

+

f(x(k»

x (k+I)=i:(k + l) +

in the case o f o nc sl.cp ahca.cl prediction, this theorem

o fJ P lC~(aK; K2 i (k

+ J) =

where e(k achieves

can be used to reduce the problem of synthesizing the optimal *-norm filter t.o minimizing a continuous fune-

Clx(k

+ 1) =

y(k

+ fJ I) - l e(k + 1)

+ I) + I)

- C 2 i(k

+ 1) -

q(i (k

+ 1»

II z(k) - i(k)1I ::;

tiOll of a sin gle v(l.ria.hh"! over t.he nn it interval.

(1 ~ a) ~ IIPII t lI·e(O) - x(O) 1I + 11Iwll ~·

4. FILTERS FOR NO!\LINEAR SYSTEMS --VVe now consider the filtering problem for a nonlinear system ciesnibed by the follov,;iug equations: x (k

+ I)= z( k )

.4x( k ) + f (x(k»

y (k l = C 2T(k) We

will

+ B1V(k + I)

=C ,x( l:)

5. ACKNOWLEDGMENTS The authors wish to thank Jessy Gri •• le anu S.R. Venkatesh for ma.ny helpful discussions.

(J9) (20)

+ g(x (k » + 1J(k )

assume that the functions

f

6. REFEHJ·;N CES

(21)

and 9 are known

and satisfy I,hp. following Lipschit7. eondil.ions with som ~ ku own matrices K, and. K2 (hr allowing Kl and K 2

to be matrices instead of scalars, one c.an less conserva.tively incorporate the nature of nonlinearities) :

f(y)ll ::; IIK,(x -/llll V :c,y E Rn (22) Ilg(xl - g(y)lI::; IIK,(J:-/l)11 V x,y E Rn (23)

Ilf( x ) -

As in the linear t:a.':!e the problem is to design a system F that estimates z(k) based on the noise-corrupted measurelllents :V(O) through :V(k). L"t " := z - i and w := lw' v' }T. The following result gives a sufficient condition for cxi stenf.e of an estimator with a. ~uaranteed bouud on the peak to peak noise gain plus a geometrically decreasing bOllnd on th p (,ffor dup. to initial condi tions. Th e01'em 7. Let the system reali?..ed by (19 ) through (20 ) satisfy a..'isumptions (22 ) and (23). If there eXists a P >

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tl

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Op-

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