Model-Based Discrete Linear State Estimator for

R. J. Chang Associate Professor, Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101

Model-Based Discrete Linear State Estimator for Nonlinearizable Systems With State-Dependent Noise A practical technique to derive a discrete-time linear state estimator for estimating the states of a nonlinearizable stochastic system involving both state-dependent and external noises through a linear noisy measurement system is presented. The present technique for synthesizing a discrete-time linear state estimator is first to construct an equivalent reference linear model for the nonlinearizable system such that the equivalent model will provide the same stationary covariance response as that of the nonlinear system. From the linear continuous model, a discrete-time state estimator can be directly derived from the corresponding discrete-time model. The synthesizing technique and filtering performance are illustrated and simulated by selecting linear, linearizable, and nonlinearizable systems with state-dependent noise.

1

Introduction Since the first milestone paper was published by Kalman (1960), the design of a discrete-time linear state estimator has been widely studied in the area of digital control. Numerous researchers have been given considerable effort in deriving and extending the Kalman's results to synthesize a discrete-time state estimator for nonlinear systems under external noise disturbance (Beaman, 1984; Maybeck, 1982). In contrast to the state-estimating problems for the nonlinear systems subjected to external noise disturbance, the estimating problems for nonlinear stochastic systems which consist of statedependent noise have not been given too much investigation (Chang, 1989). Actually, there are many physical systems which can be described and modeled by differential equations with state-noise multiplicative terms. For example, a robotic system with open-chain manipulator attached to a vertically random-fluctuated platform can be described as a nonlinear system with parametric noise entered as an equivalent gravitational term in the system equation (Chang and Young, 1989). However, due to the unsolved complex problems in deriving an appropriate discrete-time model for the nonlinear system with state-dependent noise, the design of discrete-time state estimators for the nonlinear stochastic systems has not been presented. Nonlinear stochastic systems with state-dependent noise may be characterized as a "nonlinearizable" or "linearizable" stochastic system according to their nonlinear characteristics. A nonlinearizable system is usually defined as a dynamic system including nondifferentiable nonlinearities in the system Contributed by the Dynamics Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Manuscript

received by the Dynamic Systems and Control Division July 12, 1988; revised manuscript received February 27, 1989. Associate Editor: A. G. Ulsoy.

equation. This type of nonlinearity is of practical importance in mechanical systems since coulomb friction, backlash, and material hysteretic damping (Badrakhan, 1987) usually appear in these systems. When a stable dynamic system consists of differentiable nonlinearities, the resulting local linearization model may be a stable or an unstable system according to the selection of operating point in the linearization. If the local linearization model is unstable, the linearization method cannot be employed to design a state estimator for the nonlinear system. Hence, a differentiable nonlinear system of this type will be also characterized as a nonlinearizable system in the present sense. For example, a second-order mechanical system with a spring force which is modeled as only a cubic nonlinear displacement term belongs to a nonlinearizable system in this sense if the operating point is selected as zero for the local linearization technique. In this paper, a linear discrete-time state estimator will be synthesized for estimating the states of a nonlinearizable stochastic system corrupted by both state-dependent and external noises through a noisy linear measurement system. Since the local linearization technique cannot be applied to derive the corresponding linearized model for synthesizing a discrete-time linear state estimator, an equivalent linear model will be first derived to emulate the statistical response of the nonlinearizable stochastic system. After an equivalent linear continuous model has been obtained, a discrete-time linear model can be derived and the associated discrete-time state estimator can be directly designed from the Kalman filtering algorithm. The design of a linear discrete-time state estimator by the present technique will be illustrated by selecting three stochastic systems with state-dependent noise. The performance of the state estimator will be simulated and illustrated by employing Monte Carlo techniques. Transactions of the ASME

774/Vol. 112, DECEMBER 1990

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2

Problem Formulation

For an nth-order mean-square stable nonlinear stochastic system with state-dependent noise and under external noise disturbance, a simplified nonlinear dynamic equation in the Ito's sense can be described as (Ibrahim, 1985)

G(X,0 = -x„

X-)

X\

-g,(X)

,(X)

da-,. dx„

Y, (ajXidt + Xjda,)- gd(xl,x2,xi,

x„)dt dW(t) =

Y,8i(Xi,x2,

x„)d$i + du'(t)

da„

(la)

with initial conditions Xj{tQ) = Xj(0),

(lb)

i = \,n

where xu x2, . • • x„ represent n-dimensional states, a-, are some constants, duh d0h du' are mutually independent zeromean Wiener processes with intensities CO,

dP„

i*j

E[da,{.M n + 1

(19)

lay I = \ap\ =

(246) 1, for p = n + 1

For /w = 3, one has and

o.5e„„p22 P\ 1^*33

—

^22

0

0

0

0

M3 ^22

e, =

Q=

0.5Q,raJP.i

(20)

As a result, an wth-order linear time-invariant continuous model for (1) will be derived as (10) with coefficients e,- given by (17) and with the stationary covariance response solved from (1). Unfortunately, the exact stationary covariance response for a nonlinear system with state-dependent noise is usually very difficult to obtain (Young and Chang, 1987) and an approximate approach is usually selected for deriving the stationary covariance response of a nonlinear stochastic system. For a class of nonlinearizable state-noise dependent system with differentiable nonlinearities (Young and Chang, 1988), one practical approach is to employ the Gaussian linearization technique for deriving the stationary covariance response through formulating a statistical linearization model. By employing this technique, a linear system with unknown coefficients, which are the functions of response covariance, and with state-noise dependent terms will be derived. With no loss of generality, the linear stochastic system can be formulated as (1) with gd = gi = 0- For the linear stochastic system, the stationary covariance equation can be directly obtained by replacing m and ek by n and ak, respectively, and substituting an equivalent external noise intensity (Young and Chang, 1987) i + 2«/

skkpkk

°ij+l +-P/+1 j n

Pl+ln-

. .

-0.5S,

-0.5S„,

ptn P=

(24c)

u= 0.56„

Hence, P can be directly obtained by solving (23). Specifically, if « = 4, one has

(21) «1

0

for Quu in (14) to yield =

-0.5S,,

•M1M3 —Pll

"«3

1

«2

0

0

-«1

2 > A = 0

S.i

- s2

3

a

4~

0

fPu

0

P22

-1

Pn

^44

_,

_ ^44

k=\

n

n

0

- 2 £ akPkn + £ SkkPkk + Quu = 0, *=i

0

/t=i

i,j=l,

. .

n-\

(25)

(22)

Now, (22) can be rewritten in a vector form as (A + Q)P = V

^

0 0-5Q„„ _

(23)

where Journal of Dynamic Systems, Measurement, and Control

Furthermore, if the coefficient matrix for P in (25) is independent of P, then Pjj can be solved as DECEMBER 1990, Vol. 112 / 777

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0-5Q M M (g 3 g4-g2)

D

_ o.5e„„a,a4 ••22

P

"

;

0.5Q„„a,a 2 * 0.5Q H B f l l ( g 3 f l 2 -«.«4) A:

(26)

where Ar = — O ^ S n ^ o ^ —o[2) —0.5S22a!iQ;4 -(0.5S 3 3 + 0:2)0:, a2 + a 1 ( a 3 a 2 _ a i a 4 ) ( - 0 . 5 S 4 4 + a 4 )

(27)

For a general nonlinearizable stochastic system or datadependent stochastic system, one simple and direct approach for deriving the stationary covariance response of (1) is to perform the Monte Carlo method for obtaining several sample pathes Xj through numerical integration. Then, the ensemble mean and covariance can be derived by evaluating an unbiased estimation as (Bendat and Piersol, 1971)

divergent estimate X. Hence, a mean-square stable property is guaranteed if the resulting state estimator is employed for a mean-square stable closed-loop system under external noise disturbance. When the nonlinearizable nonlinear systems include state-noise dependent terms, the stability theorem will be valid for the stochastic systems in the mean-square sense if the covariance propagation equation of a closed-loop nonlinear state-noise dependent system can be the same as that of an equivalent time-invariant system subjected to external noise disturbance. Actually, the same covariance propagation equation can be obtained from an equivalent time-invariant system if the state-noise dependent terms in a stochastic system will be interpreted as an equivalent externally excited one in the mean-square sense (Young and Chang, 1987). Based on the above arguments and the mean-square criterion (8) in synthesizing a state estimator, an equivalent linear model constructed from the mean-square sense can be employed effectively for designing a discrete-time state estimator to avoid the difficulties of deriving a local linearization model and discretizing state-noise dependent terms in a continuous dynamic system. The synthesizing of a discretetime state estimator from the equivalent linear time-invariant continuous model will be given in the following section.

4

where TV is a number of Monte Carlo run. An equivalent linear time-invariant continuous model for the nonlinearizable stochastic system (1) has been derived by the present approach. However, the equivalent linear model will have errors in emulating the response of an original nonlinear stochastic system. The modeling error may cause the resulting linear state estimator to be biased or even unstable. For the bias problem, it is mainly due to the fact that the mean response of the nonlinear system and the resulting linear model will be different in spite of the same stationary covariance response obtained in the system and model. The bias problem is a natural result for a nonlinear stochastic system under non-zero mean disturbances since the superposition principle is not valid for a nonlinear system. Hence, the equivalent linear model can be employed to design a linear state estimator for zero-mean regulation problems or for tracking control problems by incorporating an estimator for estimating the mean response. The stability problem of employing the resulting estimator in a nonlinear stochastic control system is very important. Unfortunately, the methodologies for the investigation of the stochastic stability problems of a closed-loop nonlinear control system with statenoise dependent terms are still under development (Ibrahim, 1985). Although the stochastic stability problem is very difficult to be investigated, one may restrict the stability analysis to the mean-square sense (Kozin, 1969) since the present linear time-invariant model constructed is based on the stationary mean-square response of an original nonlinear stochastic system. Now, the stability problem is whether a mean-square stable nonlinear stochastic control system will be destabilized by incorporating the resulting linear state estimator. It is first recalled that the derived equivalent linear model is meansquare stable if the original stochastic control system is stable in the mean-square sense. Thus, the state X estimated by the resulting state estimator is nondivergent in the mean-square sense. For a nonlinear stochastic system without state-noise dependent terms, a stability theorem has been proposed by Safonov and Athans (1978) and is stated as that for a closedloop stable control system, the system will remain bounded if the true state X in a feedback signal is replaced by a non778 / Vol. 112, DECEMBER 1990

Linear Discrete-Time State Estimator

The discrete-time linear state estimator for (1) can be derived through sampling the equivalent linear time-invariant continuous model (10) and employing the discrete-time measurement system (6) for synthesizing a Kalman filter. Several discretization techniques can be employed to derive a discrete-time model from a continuous one. One of the discrete-time models for (10) through a sampler and zeroorder hold can be obtained as (Astrom and Wittenmark, 1984, Maybeck, 1982) m + 1 ) = '(*,) (36) where the corresponding noise intensities are defined in (2). It is desirable to derive a reduced-order discrete-time linear state estimator for the given example. If one assumes that a thirdorder discrete-time linear state estimator is desired, a thirdorder continuous model will be first derived for synthesizing a discrete-time state estimator. By forming a third-order model as

Example 2. A second-order mean-square stable and linearizable Duffing-type system with state-dependent noise and linear measurement system is described as

dyx =y2dt

dxt =x2dt

dy2 =y3dt

dx2 = - a2x2dt - (bdt + dfi)x\ + xldt + du' (r)

(40) (41)

z(t,) = xi(.t,) + v'it,) dyi=~

j}(.e,yi)dt + du'(t)

(37)

then the unknown coefficients e,- can be derived by substituting Pu, P21, and P33 from (26) into (20) to give a4k -axal + {a3a4-a2)a2 04

(a3a4-a2)k ai(-atai + (a3a4-a2)a2)

(38)

where k is given by (27). From (38), the Ee can be expressed as (11) and the reduced-order linear discrete-time state estimator for this example can be directly obtained by (30), (31), and (34) with Ee, D, and //given by 0

1

with E[du'{t)du\t)]=Quu(t)dt E[dmdm]=Tn{t)dt (42)

E[dv'(.t)dv'(t)]=Qm(t)dt

The response characteristics of a second-order linear system with coefficients ex and e2 derived by both the present and local linearization techniques will be first simulated. By selecting h = 0.05, « 2 = 0 . 8 , 6 = 1 . 0 , 7 ^ =0.5, Q„„=0.5, Q„„ = 1.0 and with the zero mean value as an operating point, one has e, = 1.0 and e2 = 0.8 from the local linearization technique. From the present approach, e, and e2 are given by (19) if the stationary response covariance Pn and P22 can be derived from (40). For the nonlinear stochastic system given in this example, the exact Pn and P22 have not been derived in the area of nonlinear random mechanics. Thus, a Monte Carlo simula-

0

E.= -e,

-e

2

-e

3

S\

i=i 3 1=2 ) RILL-ORDER SYSTEM 5 i =3 4 1=2 6.1=3

) REDUCED-ORDER MODEL

D

H=(\

0

0)

(39)

The application of utilizing an equivalent linear model for designing a reduced-order discrete-time state estimator has been addressed in this example. To get a better understanding of the performance of the reduced-order state estimator synthesized by the present technique, one may start to investigate Journal of Dynamic Systems, Measurement, and Control

30

40

Fig. 1 Comparisons of mean-square response of a full-order system and a reduced-order model

DECEMBER 1990, Vol. 112 / 779

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0.52 0.28 0.24 0.2 0.16

H

0.04

:'

2

\

V ' f 1

/

0.03

/"

/— •^,—a^*^

r"*-^

K \

0.02

/

\^ 3

^*

l\

E[7,) E[3ti): BY EQUIVALENT LINEAR MODEL

A