Comparative Study of Sliding Mode, Optimal and Extended Kalman-Bucy Filters Performance for Quadratic Stochastic Systems

Comparative Study of Sliding Mode, Optimal and Extended Kalman-Bucy Filters Performance for Quadratic Stochastic Systems

Copyright © IFAC System, Structure and Control ELSEVIER Oaxaca, Mexico, USA, 8-10 December 2004 IFAC PUBLICATIONS www.elsevier.comlloca!elifac CO...

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Copyright © IFAC System, Structure and Control

ELSEVIER

Oaxaca, Mexico, USA, 8-10 December 2004

IFAC PUBLICATIONS

www.elsevier.comlloca!elifac

COMPARATIVE STUDY OF SLIDING MODE, OPTIMAL, AND EXTENDED KALMAN-BUCY FILTERS PERFORMANCE FOR QUADRATIC STOCHASTIC SYSTEMS Michael Basin and Jesus Rodriguez-Gonzaiez·

• Department of Physical and Mathematical Sciences Autonomus University of Nuevo Leon Apdo postal 144-F, C.P. 66450, San Nicolas de Los Garza, Nuevo Leon, Mexiro E-mail: [email protected]@yahoo.com.mx

Abstract This paper presents a filter for nonlinear quadratic stochastic systems over linear observations, based on the sliding mode technique. Performance of the designed filter is compared to those of the optimal filter for quadratic systems and an extended Kalman-llucy filter. Numerical simulation results are obtained and graphically presented. Specifics of the suggested approach are discussed. Copyright © 2004 IFAC

Keywords: Sliding mode, filtering, nonlinear stochastic system

1. INTRODUCTION

designed for linear systems (Utkin, 1992) . In recent. years, the most attention has been paid to linear and nonlinear uncertain systems with bounded disturbances (see, for example, (Slotine and Hedrick, 1987), (Walcott et al., 1987), (Misawa, 1989), (Edwards and Spurgeon, 1994), (Alessandri, 2(00)). The observer design based on higher order sliding modes has been introduced in (Fridman and Levant, 2(02).

One of the most important estimation and control problems is functioning under heavy uncertainty conditions. Although there are a number of sophisticated methods like adaptation based on identification and observation, or absolute stability methods, the most obvious way to withstand the uncertainty is to keep some constraints by brutal force. The most simple way to obtain this is to react immediately react to any deviation from the real system state and apply sufficient energy to suppress a deviation.

Application of the sliding mode technique to design of filters for stochastic system states was initiated in (Drakunov, 1983). The overview of recent results in applying the sliding mode observer design to stochastic systems with bounded disturbances is given in (Poznyak, 2(03). Other original modifications of the sliding mode control technique applicable to disturbance suppression in stochastic systems are suggested in (Shtessel et al., 2003; Poznyak et al., 2(03) .

Sliding modes as a phenomenon present in dynamic systems lead to ordinary differential equations with discontinuities and, therefore, to systems with variable structure. The proper concept of sliding modes appeared in the context of relay-based control systems. It may happen that the control as a function of the system state switches at high, theoretically infinite frequency, and this motion is called sliding mode.

This paper presents a filter for quadratic stochastic systems with white Gaussian noises over linear observations, based on the sliding mode technique. The idea is to assign an appropriate filter gain such that the estimate trajectory reaches a certain sliding mode manifold, specified in view of the observation equation, for a

Application of the variable structure systems and sliding mode technique to design of state observers is actively studied nowadays. The first sliding mode observers were

417

finite time. Under certain conditions, t.his sliding mode motion ensures convergence of the reconstruction error, prm'ided by the filt.er, t.o zero. It is shown that those st.ability conditions are less rest.rictive than the stability condit.ions for t.he optimal filter for quadratic systems (Basin, 2(03) or for an extended Kalman-Bucy filter (Kalman and Bucy, 1961),(Gelb, 1974). Comparison of the sliding mode and optimal and extended KalmanBucy filters reveals certain advantages of the suggested approach from the viewpoint of the absolute difference between values of the real state and its estimate.

the real process x(t) with respect to the obser\'Rtions Yet). Let pet) = E[(x(t} - m(t))(x(t) - m(t)}T I yet)] be the estimate cO\'RTiance (correlation function). The solution to the stated problem is given (Basin. 2003) by the following system of filtering equations, which is closed with respect to the int.roduced \'Rriables, met) and pet}:

dm(t} = (ao(t) + aJ(t)m(t} + a2(t}m(t}m T (t) +a2(t)P(t))dt + P(t)AT(t} (B(t)BT(t))-I[dy(t) - A(t}m(t}dt],

The paper is organized as follows . For reference purposes, Section 2 briefly reviews the optimal filter for quadratic systems over linear observtions and an ext.ended Kalrnan-Bucy filter, and Section 3 discusses the general concepts of sliding modes. Section 4 presents a filt.er for quadratic stochastic systems, based on the sliding mode technique, and compares it to the optimal filter for quadratic systems over linear observtions and an extended Kalman-Bucy filter for stable syst.ems, and Sections 5 does the comparison for unstable ones. N umerical simulation results are graphically presented in Section 6. The obtained results are discussed in Section 7, and Section 8 gives the general conclusions to this paper.

m(to)

+2P(t)(a2(t)m(t})T + b(t}bT(t}}dt - P(t}A T (t)(B(t)BT (t) )-1 A(t}P(t)dt ,

2.2 Extended Kalman-Bucy Filter In the optimal estimate equat.ion (3), the opt.imal filter gain matrh: is equal to Ko(t) = P(t)AT(t)(B(t}BT(t))-1 and can be computed in view of the \'Rriance equation (4). On the other hand, in many engineering applications, t.he filter gain matrix K(t} is assigned according to certain empirical considerations, preserving the general structure of the estimate equation (3) , as follows

dm(t) In this section, the optimal filtering equations for a quadratic state equation over linear observations (obtained in (Basin, 2(03)) are briefly reminded for reference purposes. Let an unobservable random process x(t) satisfy a quadratic equation

+b(t)dW1(t) ,

x(to)

(4)

P(to) =E«x(to) - m(to})(x(to) - m(to))T I Y(to)) .

2.1 Optimal filter for quadratic state equation and linear observations

= (ao(t) + al(t)x(t) + a2(t)x(t)x T (t))dt

= E(x(to) I Y(to}),

dP(t) = (al(t)P(t) + P(t)af(t} + 2(a2(t.)m(t))P(t)

2. FILTERS FOR QUADRATIC STATES AND LINEAR OBSERVATIONS

dx(t)

(3)

= (ao(t) + aJ(t)m(t) + a2(t)m(t)mT (t)}dt +K(t)[dy(t} - A(t)m(t)dt] ,

(5)

m(to} = E(x(to} I Y(to}}· This construction is called an extended Kalman-Bucy filter (Gelb, 1974). A frequently encountered assignment of K(t} is based on using the linearized model, that is, K(t) = Q(t)AT(t)(B(t}BT(t))-I, where the matrbc Q( t} satisfies the Riccati equation corresponding to the linearized system (1)

(1)

= Xo,

and linear observations are given by:

dy(t)

= (Ao(t) + A(t)x(t))dt + B(t)dHl2(t),

dQ(t}

(2)

= (aJ(t}Q(t) + Q(t}af(t) + b(t}bT(t))dt

(6)

-Q(t)A T (t)(B(t)BT (t) )- 1A(t )Q(t)dt Q(to) = E«x(to} - m(to))(x(to) - m(to))T I Y(to))· The extended Kalrnan-Bucy filter given by the equations (5) and (6) , as well as the optimal filter for quadratic systems given by (3) and (4) , will further be used in simulations for the purposes of comparison to the sliding mode filter designed in the following sections.

where x(t) E Rn is the unobservable st.ate vector and y( t) E R!" is the observation process, a 1 (t) is an n dimensional square matrix, a2(t) is a 3D n - dimensional cube tensor, which is syrrunetric over its rightmost indices: (a2)kij(t) = (a2),,:ji(t), k, i,j = 1, ..., n, and nl l (t) and W2 (t) are Wiener processes, whose weak derh'Rtives are Gaussian white noises and which are assumed independent of each other and of the Gaussian initial "'Rlue Xo.

3. SLIDING MODES: GENERAL DEFINITIONS

The filtering problem is to find dynamical equations for the best estimate for the real process x(t) at time t, based in the obsen'Rtions yet) = [yes) I to ~ s ~ t], that is the conditional expectation met) = E[x(t) I yet)] of

Definition 1. (Fridrnan arId Le''Rllt, 2002) Consider the sliding set determined by the equality:

sex, u, t}

418

= 0,

(7)

+a2(t)(x(t)X T (t) - m(t)mT(t»)

where (7) is an m-dimensional condition imposed on the state of the dynamic system (1). Let the sliding set (7) be a non-empty locally int.egral set in Filippov sense (Filippov, 1989), i.e., it consists of Filippov traject.ories (Filippov , 1989) satisfying both the stat.e equation (1) and sliding restrictions (7). Then, the corresponding motion satisfying (7) is called a sliding mode with respect to the constraint functions (7).

+b(t)wJ (t) - K(t)5ign[y(t) - A(t)m(t)], where wl(f) is the state Gaussiall white noise corresponding to the Wiener process W1(t). The sliding mode filter objective is to reach the sliding mode manifold defined as

5{t) = y(t) - A(t)m(t)

Upon reaching the manifold 5{t) error dynamics becomes

x +alx = u + f(t), +cx,

+ b(t)wJ(t).

If the original system (1) is stable (Le., the matrices aJ (t)

and a2(t) are stable), the estimation error converges to zero for any bounded realization of the white noise Wl . For stable systems, the asymptotic convergence of the estimation error to zero also takes place (Jazwins ki , 1970) in the optimal quadratic filter and may occur in the extended Kalman-Ducy filter (described in Section 2). However, it must be noted that, first, the estimation error convergence for those filters is strictly asymptotic, whereas the sliding mode filter estimate (8) reaches a K - vicinity of the real state for a finite time and. second, the stabilizability condition must additionall; be held (Jazwinski, 1970) for convergence of the optimal quadratic and extended Kalman-Ducy filter estimates.

+cx 5. SLIDING MODE FILTER WITH UNSTADLE DYNAMICS

It is important that its solution:

= x{to)e-c(t- tu )

Consider now the case of unstable dynamics in the original system (1). For the observable components of the state 'I.'eCtor x(t), the sliding mode condition 5 = means that

°

depends neither on the plant parameters nor the disturbance. This so-called invariance property looks promising for designing feedback control for dynamic plants operating under uncertainty conditions.

5(t)

The following filter based on the sliding mode technique is proposed for the state of the dynamic system (1)

dm(t)

---;It = ao(t) + al(t)m{t) + a2{t)m(t)m T (t)

(8)

In the optimal quadratic and extended Kalman-Ducy filters, the estimation errors for the observable components are specified by the asymptotic values of the variances P(t) and Q(t), t -+ 00, respecti'l.'ely, which can be quite large if the norms of the unstable matrices (a2(t) or al (t), respectively) are large themseh'eS. This also means that the asymptotic convergence of the optimal

+K(t)sign[y(t) - A(t)m{t)], where m(t) is the designed estimate, and K(t) is the filter gain matrix to be assigned. Then, the reconstruction error dynamics is gi'l.'en by:

= al{t){x{t) -

m{t))

= A(t)[x(t)- m(t)] + W20 = 0,

where W20 are the components of the white noise U'2(t) corresponding to the observable components of x(t). In other words, upon reaching the sliding mode manifold s( t) = 0, the estimates of the observable components of the stat.e 'I.'eCtor converge to a K- vicinity of the their real values for a finite time. Then the difference between them is specified by a realization of the white noise W20(t) .

4. SLIDING MODE FILTER

d dt (x(t)- m (t»

= 0, the reconstruction

+a2{t)(x(t)x T (t) - m(t)mT(t»

°

x(t)

= 0,

d

where .U, al , a2,c are constant parameters and f(t) is a bounded disturbance. The system behavior is analyzed in the state plane (x,x) . The control u undergoes discontinuities at the switching line 5 = 0, and the state trajectories are constituted by two families: the first family corresponds to 5 > 0, u = - AI, and the upper semi plane, and the second family corresponds to 5 < and u = .r..J, and the lower semiplane. Thus, within the n-m - dimensional sector on the switching line, the state velocities are oriented towards the line. This motion with state trajectories on the switching line is called sliding mode. Since, in the course of sliding mode, the state trajectory coincides with the switching line 5 = 0, its equation may be interpreted as the motion equation, that is

°=x

+ W2

dt(x{t) - m(t» = al(t)(x(t) - m(t))

u = -M5ign(5),

s = i:

m{t)]

where W2{t) is the observation Gaussian white noise corresponding to the Wiener process W2{t).

The conwntional example (Utkin et al., 1999) to demonstrate the sliding modes in a state space is a second order time-invariant relay system

i +a2

= A(t)[x(t) -

(9)

419

the observation one, V2- l (t) = 1. In all graphs, Xl and X2 are components of the unobserved real state, and ml and m2 are their estimates, respectively.

quadratic filter estimates for the observable components, which actually takes place (Jazwinski, 1970), can be quite slow. The asymptotic convergence of the extended Kalman-nucy filter estimat.es for the observable components would be even more slow, if it occurs at all.

Exrunple 1. Consider first a linear two-dimensional state with a stable observable component and an unstable unobservable one

For unobservable and unstable components, the dynamics of their estimation errors is not affected by the sliding mode condition set) = yet) - A(t)m(t) = 0 and is given directly by

.z:2

d

y=X2

-(.z:(t)- m (t»),vo = al(t)(x(t)- m (t»),vo dt +a2(t)(x(t)xT(t) - m(t)mT(t»,vo + b(t)U.'l(t),vO,

= mi + lOO x sign [y -

m2] ,

m2 = -m~ + 5 x sign [y -

m2] ,

,i!l

In the optimal quadratic filter, the estimation error for the unobservable and unstable components is given by

dt (x(t)- m (t»,vo = al(t)(x(t) - m(t»).vo

+ W2·

The sliding mode filter equations (8) take the form

where the subindex NO indicates the unobservable components of the state vector. In this case, the reconstruction error obviously diverges from zero.

d

= -X2 ,

where the gain matrix is assigned as

OC = [1~]. The

optimal filter equations for quadratic systems (3)-( 4) are given by

(10)

+a2(t)(.z:(t)X T (t) - m(t)mT(t) - P(t)),vo +b(t)Wl(t) ,vo - P(t)AT(t)(B(t)BT(t))-l

117 1=

111i + Pl l + Pl2 [y -

m2= -m~ - P22

+ P22 [y -

Am2]' Am2] ,

x [dy(t)/dt - A(t)m(t)],vo,

where {Pll , P l 2, P22} are entries of the symmetric positive definite matrix P satisfying the equation (4). The extended Kalman-nucy filter equations (5)-(6) are given by

and its variance pet) satisfies the equation (4), whose solution asymptotically diverges to infinity. In the extended Kalman-nucy filter, the estimation error for the unobservable and unstable components is given by d

dt (x(t)- m (t)),vo

= al(t)(x(t)- m

(t»),vo

m1=

111i + Q12 [y -

m2= -m~

(ll)

+b(t)Wl(t) NO - Q(t)AT(t)(B(t)BT(t))-l

Am2] '

+ Q22 [y - A1112] ,

where {Qll, Ql2 , Q22} are entries of the symmetric positive definite matrix Q satisfying the equation (6).

x [dy(t)/dt - A(t)m(t)].vo ,

The simulation results for the component Xl and its estimates ml corresponding to different filters are shown in Fig. 1. The line marked ST AT E (blue) corresponds to the component Xl itself, the lines marked by F.~1 D (black), FP (red), and F K B E (green) correspond to the estimates given by the sliding mode filter, optimal filter for quadratic systems, and extended Kalman-nucy filter , respectively. The simulation results for the component X2 and its estimates 1112 corresponding to different filters are shown in Fig. 2, with the same line markings.

and its variance Q(t) satisfies the equation (6), whose solution also asymptotically diverges to infinity. Thus, all compared filters give unreliable estimates for the unobservable and unstable components of (1). Nevertheless, the sliding mode estimate looks preferable since its equation contains only one white noise in the right-hand side, whereas both the optimal quadratic and extended Kalman-nucy estimation error equations (10) and (11) contain two of them, taking into account that y( t) .4(t)m(t) is a pure noise in the case of unobservable components.

In accordance with the developed theory, the obtained simulation results demonstrate convergence of the reconstruction errors of all filters to zero for the stable and observable component and their asymptotic divergence to infinity for the unstable and unobservable component. However, it must be noted that the value of the sliding mode filter estimate is less diverged from the unstable real state component at the final time T = 0.12 than the other filters estimates do, and the sliding mode filter estimate reaches the stable real state component faster than the other filters estimates do.

6. SIMULATIONS

In the next two examples, the initial conditions are xl=8, ml = 5, x2=5, m2 = 4, the white Gaussian noise is realized as W2 = sin (5Ot), and the simulation time is equal to T = 0.12. The disturbance is absent in the state equation, Vl (t) = 0, and considered standard in

420

D,--~

_ _

--_-_-~---,

'Ill

'6Il

"" '''' III 6D

./

"'~~~S~T~~'~~~~____ ;=~-~'~~'~"E~-~-~-_-J o~

012

002

004

,0....(1)

0.00

01

0 12

Figure-3. ".----~-------------,

,e

••

. "'.

j

..

~ ;;'2

'0

:J

Ji

;

; 45

, - . -~

'0L ~==7'002:::---:0:-O .0A-:----;:0~"',---;:-0."'~-7'0.,,---,,cO.12 h""

,...

25;;-0-~o.m:;;---;o~ . ",:;---;o:;; .",'----;;o. :;;;",---;: o ;-,--""0.12

Figure-2. EXaIllple 2. Consider now a linear two-dimensional state with an unstable observable component and a stable unobservable one

where {Qu, Q12, Q22 } are entries of the syuunetric positive definite matrix Q satisfying the equation (6). y=Xl

The simulation results for the component X'I and its estimates ml corresponding to different filters are shown in Fig. 3. The line marked ST ATE (blue) corresponds to the component XI itself, the lines marked by F.'II D (black), FP (red) , and F K n E (green) correspond to the estimates given by the sliding mode filter, optimal filter for quadratic systems, and extended Kalman-llucy filter, respectively. The simulation results for t.he component X2 and its estimates m2 corresponding to different filters are shown in Fig. 4, with the same line markings.

+ W2 ·

The sliding mode filter equations (8) take the form

""1 =mi + 1700 x sign[y ""2 = -m~ + 5 x sign [y -

md, md ,

1700 ]. The 5 optimal filter equations for quadratic systems (3)-(4) are given by

where the gain matrix is assigned as IK = [

ml =

mi + P11 + Pll [y -

mF -~ -

P22

+ P I 2 [y -

In accordance with the developed theory, the obtained simulation results demonstrate convergence of the reconstruction errors of the sliding mode filter and optimal filter for quadratic systems to zero for both components, stable and unobservable, as well as unstable and observable. The reconstruction error of the extended Kalrnan-llucy filter fails to converge to zero in both cases. However, again, it must be noted that the values of the sliding mode filter estimates reach the real states much faster than the optimal filter for quadratic systems estimates do, thus demonstrating better behavior of the sliding mode filter estimates from the viewpoint of absolute difference between values of the real state and its estimate.

Amd , Ami] ,

where {PI I, P 12 , P 22 } are entries of the symmetric positive definite matrix P satisfying the equation (4). The extended Kalman-llucy filter equations (5)-(6) are given by

ml=

mi + Q11 [y -

m2= - ~

+ Q12 [y -

Ami]' Amd ,

421

7. DISCUSSION

39th IEEE Conference on Decision and Control. Sydney, Australia. pp. 2593--2598. Dasin, M. V. (2003). Optimal filtering for polynomial system states. ASME Trans. J. Dynamic System.'>, MeasU1Y!ment, and Control 125, 123--125. Drakunov, S. V. (1983). On adaptive quasioptimal filter with discontinuous parameters,. Automation and Remote Control 44, 1167- 1175. Edwards, C. and S. Spurgeon (1994) . On the development of discontinuous observers. Int. J. Control 59, 1211- 1229. Filippov, A. F. (1989). Differential Equations 'With Discontinuous Right-Hand Sides. Kluwer,New York. Fridman, L. and A. Levant (2002). Higher order sliding modes. In: Sliding Mode Control in Engineering. New York. pp. 53- 102. Gelb, A. (1974) . Applip.d Optimal Estimation. The :-'HT Press. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory. Academic Press. Kalman, n.. E. and n.. S. Ducy (1961). New results in linear filtering and prediction theory. ASAfE Transactions. J. Basic Engineering 83, 95-108. Misawa, E. A. (1989) . Nonlinear state estimat.ion using sliding observers. PhD thesis. MIT. Poznyak, A.S. (2003). Stochastic output noise effects in sliding mode state estimation. International J. Control 76, 986-999. Poznyak, A.S., Y.D. Shtessel and C.J. Gallegos (2003). Min-max sliding mode control for multimodel linear time varying systems. IEEE I'rans. Automat. Contr. 48, 2141- 2150. Shtessel, Y. D., A.S.l. Zinober and 1. Shkolnikov (2003). Sliding mode control for nonlinear systems with output delay via method of stable system center. ASME Transactions, J.Dynamic Systems, lI-JeaS1!1Y!ment, and Control 25, 253--257. Slotine, J.J. and J. K. Hedrick (1987) . On sliding observers for nonlinear systems. ASME Transactions, J.Dynamic Systems, lIJeas1!rement, and Controll09, 245-252. Utkin, V. (1992). Sliding Modes in Control and Optimization. Springer. New York. Utkin, V., J. Guldner and J. Shi (1999) . Sliding Afode Control in Electromechanical Systems. Taylor & Francis. London. Walcott, D.L., M. J. Corless and S.H. Zak (1987). Comparative study of nonlinear stat.e observation techniques. Int. J. Control 45, 2109-2132.

As the developed t.heory and simulation results show, the designed sliding mode filter reconstruction error converges to zero for all detectable system states (state components). This means that the reconstruction error asymptotically tends to zero, if the syst.em stat.es are completely observable, or if not all st.ates are completely observable, but. the unobservable states are st.able. Thus, the sliding mode filt.er estimate converges to the real stat.e in t.he same situations where the opt.imal quadrat.ic filter estimates do. The designed sliding mode filter offers better results in the sense of t.he absolute difference between values of t.he real states and their estimates, although loses to the optimal quadratic filt.er if the minimum variance (mean square) criterion is considered. It must be also noted that the sliding mode filter est.imates reach a K-vicinity of the real states for a finite time, whereas the optimal quadratic filter estimates converge asymptotically. Another advantage of the sliding mode filter is that t.he stabilizability condition for the system (1) required for convergence of the optimal quadratic estimate (Jazwinski, 1970) should not be held and can be omitted for the sliding mode filter, since the variance does not participate in formation of the sliding mode filtering equation and there is no need to worry about its rank of values. 8. CONCLUSIONS A filter has been designed for linear stochastic systems with whit.e Gaussian noises, based on application of the sliding mode technique. Performance of the designed filter has been compared to those of the optimal filter for quadratic systems and an extended KalmanDucy filter. It has been demonstrated that the sliding mode filter reconstruction error converges to zero in the same situations where the optimal quadratic filter error converges, however a less number of conditions should be verified for the new filter , whereas the extended Kalman-Ducy filter error converges to zero in an even less number of cases. The simulation results have shown that the designed sliding mode filter offers better results in the sense of the absolute difference between values of the real states and their estimates than the two other filters do, in all considered cases. Thus, it can be concluded that the new filtering algorit.hm is less restrictive, quite efficient, and widely applicable in practical situations. The other area, where application of the designed sliding mode filter should be extensively studied, is nonlinear non polynomial stochastic systems. One can expect even more advantage of the sliding mode filter in that area, since the optimal filter (like the optimal one for quadratic systems) cannot be obtained for most nonlinear non polynomial systems. REFERENCES Alessandri, A. (2000) . Design of sliding mode observers and filters for nonlinear dynamic systems. In: Proc.

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