Optimal and Suboptimal Estimation of Quadratic

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Optimal and Suboptimal Estimation of Quadratic Functionals of the State Vector in Linear Stochastic Systems Il Young Song, Vladimir Shin & Won Choi To cite this article: Il Young Song, Vladimir Shin & Won Choi (2017): Optimal and Suboptimal Estimation of Quadratic Functionals of the State Vector in Linear Stochastic Systems, IETE Journal of Research, DOI: 10.1080/03772063.2016.1274238 To link to this article: http://dx.doi.org/10.1080/03772063.2016.1274238

Published online: 15 Feb 2017.

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Date: 15 February 2017, At: 14:54

IETE JOURNAL OF RESEARCH, 2017 http://dx.doi.org/10.1080/03772063.2016.1274238

Optimal and Suboptimal Estimation of Quadratic Functionals of the State Vector in Linear Stochastic Systems Il Young Song1, Vladimir Shin2 and Won Choi3 1 Department of Sensor Systems, Hanwha Corporation R&D Center, Daejeon, Republic of Korea; 2Department of Information and Statistics, Research Institute of Natural Science, Gyeongsang National University, Jinju, Republic of Korea; 3Department of Mathematics, Incheon National University, Incheon, Republic of Korea

ABSTRACT

KEYWORDS

This paper focuses on estimation of a quadratic functional (QF) of a random signal in dynamic systems described by a linear stochastic differential equations. The QF represents a quadratic form of state variables, which can indicate useful information of a target system for control. The optimal (in mean square sense) and suboptimal estimators of a QF represent a function of the Kalman estimate and error covariance. The proposed estimation algorithms have a closed-form estimation procedure. The quadratic estimators are studied in detail, including derivation of the exact formulas for mean square errors. The obtained results we demonstrate on practical example, and comparison analysis between optimal and suboptimal estimators is presented.

Kalman filtering; Quadratic functional; Random signal; State vector; Stochastic system

Research highlights

I An optimal mean square estimator for an arbitrary QF in linear stochastic systems is derived. I The proposed estimator is a comprehensively investigated, including derivation of matrix equation for its mean square error.

I Performance of the optimal and suboptimal estimators is illustrated on theoretical and practical examples for real QFs.

1. INTRODUCTION The Kalman filtering and its variations are well-known signal estimation techniques in wide use in a variety of applications such as navigation, target tracking, vehicle state estimation, communications engineering, air traffic control, biomedical and chemical processing, and many other areas [1–11]. However, some applications require the estimation of not only a signal but also a quadratic functional (QF), which expresses practical and worthwhile information for control systems. For instance, a QF of a state vector can be interpreted as a current square distance between targets [1] or in a mechanical application, such QFs include energy or work of an object [12]. Many researchers have studied various statistical approaches for estimation of a QF in different application areas (see for example, [13–18] and references therein). In [13,14], the optimal matrix of QF is searching based on cumulant criterions. Statistical properties of an empirical correlation matrix representing a QF are studied in [15]. The minimax results for estimation of a square norm of unknown signal belong to given sets © 2017 IETE

proposed in [16,17]. Optimal extraction of a diffusion coefficient from Brownian trajectories is described in [18]. However, most authors have not focused on estimation of a general QF for vector signals in dynamical models defined by stochastic equations. To the best of our knowledge, there are no references for mean square estimation of an arbitrary QF in dynamical systems in the literature. Therefore, the aim of this paper is to develop an optimal mean square estimator for an arbitrary QF in linear stochastic differential systems. We propose an optimal (in the mean square error sense) and suboptimal estimators, and investigate their theoretical and practical efficiency. This paper is organized as follows. Section 2 presents a statement of the estimation problem for QFs within the continuous-time Kalman filtering framework. In Section 3, the optimal and suboptimal estimators are derived. In Section 4, we theoretically study an estimation accuracy of the proposed quadratic estimators and derive the exact formulas for their mean square errors (MSEs).

2

I. Y. SONG ET AL.: OPTIMAL AND SUBOPTIMAL ESTIMATION OF QUADRATIC FUNCTIONALS OF THE STATE VECTOR IN LINEAR STOCHASTIC SYSTEMS

Section 5 presents several examples of mechanical models with specific QFs. In Section 6, an efficiency of the proposed estimators is studied for a scalar random signal. Numerical simulations of the MSEs confirm theoretical results presented in Section 4. In Section 7, the quadratic estimators are simulated on real model of the wind tunnel system. Finally, we conclude the paper in Section 8.

2. PROBLEM STATEMENT Let us consider a linear system described by the stochastic differential equation: x_ t ¼ Ft xt þ Gt vt ; t
0;

(1)

where xt 2 Rn is a state vector (unobservable signal), and vt 2 Rr is a zero-mean Gaussian white system noise with intensity Qt, i.e. E ðvt vTs Þ ¼ Qt dts ; dt is the Dirac delta-function, and Ft 2 Rnn ; Gt 2 Rnr ; and Qt 2 Rrr : Suppose that an observable signal is determined by the model yt ¼ Ht xt þ wt ;

(2)

where yt 2 Rm is an observation vector, and wt 2 Rm represents a zero-mean Gaussian white noise with intensity Rt , i.e. E ðwt wTs Þ ¼ Rt dts ; and Ht 2 Rmn . We assume that the initial state x0 » Nðx 0 ; P0 Þ; and system and observation noises vt ; wt are mutually uncorrelated. A problem associated with such system (1) and (2) is that of estimation of the QF of state vector, zt ¼ fðxt Þ ¼ xTt Vt xt þ aTt xt ; VTt ¼ Vt ;

(3)

from the overall noisy observations ¼ fys : 0stg. Here, Vt
0 and at are an arbitrary matrix and vector, respectively. yt0

3. OPTIMAL AND SUBOPTIMAL ESTIMATION ALGORITHMS In this section, the new optimal estimation algorithm for a QF of the state vector is derived. The algorithm gives a minimum MSE estimate for an arbitrary QF. Also, we consider a simple suboptimal estimation algorithm of the QF. The both algorithms include two stages: the optimal Kalman estimate of the state vector ^x t computed at the first stage is used at the second stage for the optimal and suboptimal estimation of the QF. First step (“Calculation of Kalman estimate of the state vector”). The optimal mean square estimate ^x t ¼ Eðxt j yt0 Þ of the state xt based on the overall observations yt0 and its error covariance Pt ¼ Eðet eTt Þ; et ¼ xt  ^x t are given by the continuous Kalman filter (KF) equations [8–11]: ^x_ t ¼ Ft ^x t þ Kt ðyt  Ht^x t Þ; t
0; ^x t¼0 ¼ x 0 ; Kt ¼ Pt HTt Rt 1 ; P_ t ¼ Ft Pt þ FTt Pt  Pt HTt Rt 1 Ht Pt þ Gt Qt GTt ; Pt¼0 ¼ P0 : (4) Second step (“Calculation of estimate of the QF”). Next, the optimal mean square estimate of the QF zt D f(xt) based on the overall sensor observations yt0 also represents a conditional mean: t ^z opt t ¼ Eðzt j y0 Þ:

(5)

The conditional mean (5) can be explicitly calculated in terms of the state estimate ^x t and its error covariance Pt. We have opt

Theorem 3.1: The optimal mean square estimate ^z t Eðzt j yt0 Þ is given by ^z opt x t ^x Tt Þ þ aTt ^x t ; t ¼ tr½Vt ðPt þ ^

¼

(6)

where tr(A) is the trace of a matrix A, and the Kalman estimate ^x t and error covariance Pt satisfy Equation (4). Proof: Using the formula for a second-order vector moment E ðxT xÞ ¼ mT m þ trðCÞ; where m D E(x), C D Cov(x,x) D E[(x-m)(x-m)T], it is easy to derive that

Typical examples of such QF may be an arbitrary quadratic form f ðxt Þ ¼ xTt t xt ; representing an energy-like function of an object or Euclidean square distance f ðxt Þ ¼ d2 ðxt ; ~x t Þ between two vector signals xt and ~x t .

EðxT VxÞ ¼ tr½VðC þ mmT Þ:

We propose optimal and suboptimal estimation algorithms for the QF and study their performance in the subsequent sections.

Taking into account that a conditional pdf p ðxt j yt0 Þ ¼ Nð^x t ; Pt Þ is a normal with conditional mean ^x t ¼ Eðxt j yt0 Þ and covariance Cov ðxt j yt0 Þ ¼ Pt [8–11] and using

(7)

I. Y. SONG ET AL.: OPTIMAL AND SUBOPTIMAL ESTIMATION OF QUADRATIC FUNCTIONALS OF THE STATE VECTOR IN LINEAR STOCHASTIC SYSTEMS

Lemma 4.1: Let X 2 R3n be a composite multivariate normal vector:

the fact (7) we obtain optimal estimate (5) for the QF: T T t ^z opt t ¼ Eðxt Vt xt þ at xt j y 0 Þ T t T ¼ Eðxt Vt xt j y0 Þ þ at Eðxt j yt0 Þ ¼ trfVt ½Pt þ Eðxt j yt0 ÞEðxTt j yt0 Þg þ aTt ^x t ¼ tr½Vt ðPt þ ^x t ^x Tt Þ þ aTt ^x t :

(8)

X » NðmX ; SX Þ; XT ¼ ½UT VT WT ; U; V; W 2 Rn ; 2 3 2 Suu Suv mu 6 7 6 mX ¼ EðXÞ ¼ 4 mv 5; SX ¼ CovðX; XÞ ¼ 6 4 Svu Svv mw

This completes the derivation of Equation (6). Remark 3.1: Note that the deterministic version of the formula (7) derived in [19]. It takes the form, xT Vx ¼ tr ðVxxT Þ; where x 2 Rn is a non-random vector. In parallel with the best optimal estimate (6) consider a simple suboptimal estimate of the QF, such as ^z sub ¼ ^x Tt Vt ^x t þ aTt ^x t : t

(9)

4. MEAN SQUARE ERRORS FOR OPTIMAL AND SUBOPTIMAL QUADRATIC ESTIMATORS Let us compare estimation accuracy of the optimal (6) and suboptimal (9) estimators. The following result completely defines the precise MSEs opt

opt

sub 2 Pz;t ¼ E½ðez;t Þ2 ; Psub z;t ¼ E½ðez;t Þ ; opt

opt

ez;t ¼ zt  ^z t ; esub z sub z;t ¼ zt  ^ t ;

opt

Theorem 4.1: The optimal and suboptimal MSEs Pz;t and Psub z;t are given by opt

Psub z;t

7 Svw 7 5:

Sww

(14) Then the third- and fourth-order vector moments of the composite random vector X are given by ðiÞ EðUT VWT Þ ¼ mTu mv mTw þ trðSuv ÞmTw þ mTv Suw þ mTu Svw ; ðiiÞ EðUT UVT VÞ ¼ mTu mu mTv mv þ 2trðSuv Svu Þ þ trðSuu ÞtrðSvv Þ þ trðSuu ÞmTv mv þ trðSvv ÞmTu mu þ 4mTu Suv mv ; ðiiiÞ EðUT VVT UÞ ¼ mTu mv mTv mu þ trðSuu Svv Þ þ trðSuv ÞtrðSvu Þ þ trðS2uv Þ þ mTv Suu mv þ mTu Svv mu þ 2trðSuv ÞmTu mv þ mTv Suv mu þ mTu Svu mv ; ðivÞ EðUT VWT UÞ ¼ mTu mv mTw mu þ trðSuv ÞtrðSuw Þ þ trðSuu Swv Þ þ trðSuw Suv Þ þ trðSuv ÞmTu mw þ trðCuw ÞmTu mv þ mTv Suu mw þ mTv Suw mu þ mTu Svu mw þ mTu Svw mTu :

(15) The derivation of the vector formulas (15) for calculating the high-order moments is based on their scalar versions [20,21] Eðxi xj xk Þ ¼ mi mj mk þ mi Sjk þ mj Sik þ mk Sij ; Eðxi xj xk xl Þ ¼ mi mj mk ml þ Sij Skl þ Sik Slj þ Sil Sjk þ mi mj Skl þ mi mk Sjl þ mi ml Sjk þ mj mk Sil þ mj ml Sik þ mk ml Sij ; where mh ¼ Eðxh Þ; Spq ¼ E½ðxp  mq Þðxq  mq Þ;

(16)

and standard matrix manipulations.

(12)

respectively. Here, the unconditional mean mt ¼ Eðxt Þ and covariance Ct ¼ Covðxt ; xt Þ of the state vector xt are determined by the Lyapunov equations [9–11]: m_ t ¼ Ft mt ; t
0; m0 ¼ x 0 ; C_ t ¼ Ft Ct þ Ct FTt þ Gt Qt Gt ; C0 ¼ P0 :

Swv

3

(11)

¼ 4trðVt Pt Vt Ct Þ  2trðVt Pt Vt Pt Þ

þ tr2 ðVt Pt Þ þ 4mt Vt Pt Vt mt þ aTt Pt at þ 4mTt Vt Pt at ;

Swu

Suw

(10)

for the quadratic estimators (6) and (9), respectively.

Pz;t ¼ 4trðVt Pt Vt Ct Þ  2trðVt Pt Vt Pt Þ þ 4mt Vt Pt Vt mt þ aTt Pt at þ 4mTt Vt Pt at ; and

3

(13)

The derivation of the MSEs (11) and (12) is based on the following Lemma.

Proof of Theorem 4.1: We are now in a position to derive the first MSE (11). For simplicity we omit time index, i.e. xt ! x; ^x t ! ^x ; Pt ! P; . . . Then using Equations (3) and (6), the estimation error can be written as z opt ¼ xT Vx þ aT x  tr½VðPþ^x ^x T Þ  aT ^x eopt z ¼ z^ ¼ xT Vx^x T V^x  trðVPÞ þ aT e ¼ ðeþ^x ÞT Vðeþ^x Þ  ^x T V^x  trðVPÞ þ aT e ¼ eT Ve þ 2eT V^x þ aT e  trðVPÞ; where e ¼ x^x ; trðV^x ^x T Þ ¼ ^x T V^x ; ^x T Ve ¼ eT V^x : Next, using the unbiased and orthogonality properties of the Kalman estimate E ðeÞ ¼ 0; E ðe^x T Þ ¼ 0, and P

4

I. Y. SONG ET AL.: OPTIMAL AND SUBOPTIMAL ESTIMATION OF QUADRATIC FUNCTIONALS OF THE STATE VECTOR IN LINEAR STOCHASTIC SYSTEMS

¼ EðeeT Þ we obtain the optimal MSE:

and MSEs take the form:

T opt 2 T T x ^x T VeÞ Popt z ¼ E½ðez Þ  ¼ Eðe Vee VeÞ þ 4Eðe V^ þ aT Pa þ tr2 ðVPÞ þ 4EðeT VeeT V^x Þ þ 2EðeT VeeT Þa  2tr2 ðVPÞ þ 4EðeT V^x eT Þa: (17)

Using Lemma, we can calculate high-order moments in Equation (17). We have T

^z opt x t ^x Tt Þ ¼ k^x t k2 þ trðPt Þ; t ¼ trðPt þ ^ ^z sub x t k2 ; t ¼ k^ opt Pz;t ¼ 4trðPt Ct Þ  2trðP2t Þ þ 4mTt Pt mt ;

(20)

2 T 2 Psub z;t ¼ 4trðPt Ct Þ  2trðPt Þ þ 4mt Pt mt þ tr ðPt Þ:

In next section, we present typically used examples of QF for mechanical models.

ðaÞ Eðe Vee VeÞ ¼ 2trðVPVPÞ þ tr ðVPÞ; U ¼ e; V ¼ Ve; ðbÞ EðeT V^x ^x T VeÞ ¼ trðPVP^x ^x VÞ þ mT VPVm ¼ trðPVCPÞ  trðVPVPÞ þ mT VPVm; U ¼ e; V ¼ V^x ; ðcÞ EðeT VeeT V^x Þ ¼ EðeT Ve^x T VeÞ ¼ 0; U ¼ e; V ¼ Ve; W ¼ V^x ; ðdÞ EðeT VeeT Þ ¼ 0; U ¼ e; V ¼ Ve; W ¼ e; ðeÞ EðeT V^x eT Þ ¼ mT VP; U ¼ e; V ¼ V^x ; W ¼ e; T

zt ¼ f ðxt Þ ¼ kxt k2 ¼ xTt xt ;

2

5. EXAMPLES OF QUADRATIC FUNCTIONALS AS ENERGY AND WORK

(18) where m ¼EðxÞ ¼ Eð^x Þ; C ¼ Covðx; xÞ; P ¼ Covðe; eÞ; EðeÞ ¼ 0; EðV^x Þ ¼ Vm; P^x ^x ¼ Covð^x ; ^x Þ ¼ C  P; Covðe; VeÞ ¼ PV; CovðVe; VeÞ ¼ VPV: Substituting Equation (18) into Equation (17), and after some manipulations, we get the optimal MSE (11). The unknown mean m D E(xt) and covariance C D Cov (xt,xt) of the state vector of system (1) satisfy the Lyapunov equations (13).

Example 5.1 (Work done from moving particle): Consider a particle with mass m and speed yt, which is moving along the x-axis. Let the particle has an original velocity y1 at displacement x1, and a final velocity y2 at displacement x2. Then, a work done represents a QF such as 1 1 W ¼ my22  my21 ¼ xT Vx; 2 2 where " # " # y1 1 m 0 : x¼ ; V¼ 2 y2 0 m Example 5.2 (Work done from forced pump): A pump forces up water at a speed of yt from a well into a reservoir at a rate of M. Let the water level be raised to a vertical height of ht. Then, the work done at current time will represent a QF:

This completes the derivation (11). In the case of the suboptimal estimate ^z sub t , the derivation of the MSE (12) is similar. Thus, Equations (11) and (12) completely define the true MSEs of the optimal and suboptimal quadratic estimators (6) and (9), respectively. opt

Corollary 4.1: Comparison of the MSEs Pz;t and Psub z;t shows that the difference between them is equal: opt

2 et ¼ Psub z;t  Pz;t ¼ tr ðVt Pt Þ;

(19)

1 2

Wt ¼ Mght  My2t ¼ xTt Vxt þ aT xt ; g ¼ 9:81; where " xt ¼

ht yt

"

#

;V¼

#

0

0

0

 M=2

" ; a¼

Mg 0

# :

Example 5.3 (Energy of a harmonic oscillator): The equation of motion of a harmonic oscillator is m x€t ¼  kxt or "

#

"

where Pt is the error covariance of the state vector determined by the KF equations (4).

X_ t ¼

Corollary 4.2: In the particular case with Vt ¼ 1 and at ¼ 0; the QF zt ; the optimal and suboptimal estimates

yt ¼ x:t ; v2 ¼ k=m;

0

1

 v2

0

Xt ; Xt ¼

xt yt

# ;

I. Y. SONG ET AL.: OPTIMAL AND SUBOPTIMAL ESTIMATION OF QUADRATIC FUNCTIONALS OF THE STATE VECTOR IN LINEAR STOCHASTIC SYSTEMS

where m is the mass and k is the force constant of a spring. Then, energy is conserved as 1 1 E ¼ my2t þ kx2t ¼ XTt VXt ; 2 2 where " # 1 k 0 : V¼ 2 0 m

   vt u:t ; ¼  v20 sinðut Þ v:t

The KF equation (4) gives the following: 1 Pt ðyt  ^x t Þ ; ^x 0 ¼ x 0; r 1 P_ t ¼ 2aPt  P2t þ q ; P0 ¼ s 20 : r

^x:t ¼ a^x t þ

vt ¼ u:t ;

v20 ¼

g ; l

where vt is the angular velocity, and v20 ¼ g=l is the natural frequency of small oscillations. The conserved energy is

k1 þ k2 ; Pt ¼ k2 þ 2 ½ðs 0 þ k1 Þ=ðs 20  k2 Þe2bt  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ a2 þ q=r; k1 ¼ rðb  aÞ; k2 ¼ rðb þ aÞ: (23) Further, we consider a power of the signal xt which is proportional to its square. In this case one can take zt ¼ x2t : Using Equations (6) and (9) we obtain the best optimal and suboptimal estimates of power of the signal: ^z opt x 2t þ Pt ; ^z sub x 2t ; t ¼^ t ¼^

Small oscillations: If we assume the angle ut is small, we may write cosðut Þ  1; in which case the energy represents a QF, such as 1 2

Let us compare estimation accuracy of the optimal and suboptimal estimates (24). Using Theorem 4.1 we obtain precise formulas for the opt MSEs of these estimates, Pz;t and Psub z;t ; respectively,

E ¼ ml2 v2t  mgl ¼ XTt VXt þ a0 ; a0 ¼ mgl;

Pz;t ¼ E½ðx2t  ^x 2t  Pt Þ2  ¼ 4Pt Ct  2P2t þ 4m2t Pt ;

where"

2 Psub x 2t Þ2  ¼ 4Pt Ct  P2t þ 4m2t Pt ; z;t ¼ E½ðxt  ^

Xt ¼

ut vt

"

; V¼

1 0 2 0

0 ml2

opt

# :

(25)

Further, we consider theoretical example of application of the quadratic estimators (6) and (9).

6. THEORETICAL EXAMPLE. ESTIMATION OF POWER OF SCALAR SIGNAL

where the mean mt and covariance Ct of the signal xt are determined by Lyapunov equations (13) m:t ¼ amt ; m0 ¼ x 0; _C t ¼ 2aCt þ q; C0 ¼ s 20 :

(26)

with solutions

If xt is a scalar random signal measured in additive white noise then the system and measurement models are x_ t ¼ axt þ vt ; x0 » Nðx 0 ; s 20 Þ; yt ¼ xt þ wt ; t
0;

(24)

where ^x t and Pt are determined by Equations (22) and (23), respectively.

E ¼12 ml2 u:t 2 þ Uðut Þ ¼ 12 ml2 v2t  mglcosðut Þ:

#

(22)

Analytical solution of the Riccati equation takes the form

Example 5.4 (Energy of a pendulum): Next, consider the simple pendulum, composed of a mass point m affixed to a massless rigid rod of length l: The potential is U ðut Þ ¼ mglcosðut Þ: Hence, ml u€t ¼ mglsinðut Þ. This is equivalent to 

5

(21)

where vt and wt are the uncorrelated zero-mean white Gaussian noises with intensities q and r, respectively, a D const. Assume that signals, noises, and parameters in Equation (21) are physically dimensionless.

at mt ¼ x 0e ; Ct ¼ ðs 20 þ q=2aÞe2at  q=2a:

(27)

Thus, the analytical solutions (23) and (27) with formulas (25) completely establish the exact MSEs for the optimal and suboptimal estimates (24). According to Corollary 4.1, the difference between MSEs opt 2 is equal et ¼ Psub z;t  Pz;t ¼ Pt : Figures 1 and 2 show the numerical values of the MSEs and relative error Dt ð%Þ ¼

6

I. Y. SONG ET AL.: OPTIMAL AND SUBOPTIMAL ESTIMATION OF QUADRATIC FUNCTIONALS OF THE STATE VECTOR IN LINEAR STOCHASTIC SYSTEMS

from a chosen equilibrium point of the following quantities: x1 ¼ Mach number, x2 ¼ actuator position guide vane angle in a driving fan, and x3 ¼ actuator rate. Then, the system model is given by 2

0:4032 4 0 x_ t ¼ 0

Figure 1: Optimal and suboptimal MSEs

0 0:1245  2:5247

3 0 0:0701 5xt þ vt ;  0:5488

(28)

where a wind tunnel system with the initial mean and covariance are x 0 ¼ ½3 28 10T and P0 ¼ diag½1 1 0: The intensity matrix of the system white Gaussian noise vt 2 R3 is subjected to Q ¼ diag½0:01 0:01 0:01: Two sensory measurement models are given by 

1 yt ¼ 0

Figure 2: Relative error between optimal and suboptimal MSEs

    sub opt opt   Pz;t  Pz;t =Pz;t  100%; respectively, for the values a ¼ 1; q ¼ 0:5; x 0 ¼ 0; s 20 ¼ 4; and r ¼ 0:1: From Figure 2 we observe that the relative error Dt varies from 3% to 6% within the time zone t 2 ½0:1; 1:1; and then it increases. In steady-state regime t > 4 the relative error is reached the value D 1 ¼ 20:4%; and at the same time zone the absolute values of the MSEs are sub equal Popt 1 ¼ 0:1029 and P 1 ¼ 0:1239 (Figure 1). Thus, the numerical results show that the suboptimal estimate ^z sub ¼ ^x 2t may be seriously worse than the optimal one t opt ^z t ¼ ^x 2t þ Pt :

7. PRACTICAL EXAMPLE. WIND TUNNEL SYSTEM Here, a comparative experimental analysis of the optimal and suboptimal quadratic estimators is considered on example of a total kinetic energy of the wind tunnel system. The state differential equations and the design based on a continuous-time model have been applied to the highspeed closed-air unit wind tunnel model by Armstrong and Tripp [22]. The state vector xt 2 R3 consists of the state variables x1t ; x2t ; and x3t ; representing derivatives

0 0

 0 xt þ wt ; 1

R ¼ diag½0:05 0:05:

(29)

The total kinetic energy of an actuator representing the QF, zt ¼ xTt Vxt ; can be expressed as the sum of the translational kinetic energy of the centre of mass, Etr ¼ my2t =2; and the rotational kinetic energy about the centre of mass Er ¼ Iv2t =2; where I is rotational inertia, vt ¼ x_ 2t is angular velocity, m is mass, and yt ¼ x3t is linear velocity. The total kinetic energy can be expressed in the following QF (see Section 5): 1 2 1 I_x 2;t þ mx23;t ¼ XTt VXt ; 2 2 Xt ¼ ½ x1;t x2;t x3;t x_ 2;t T ; V ¼ diag½ 0 0 m=2 I=2 ;

zt ¼ Er þ Etr ¼

(30)

where Xt 2 R4 is extended state vector, and I D 0.136 kgm2, m D 7.39 kg. Using the obtained result (30) an optimal and suboptimal quadratic estimators take the form: T t ^ ^T ^z opt t ¼ EðXt VXt j y0 Þ ¼ tr½VðPt þ X t X t Þ; ^ Tt VX ^ t; ^z sub ¼X t

(31)

^ t 2 R4 and where the optimal estimate of the state X 44 are determined by the KF error covariance Pt 2 R equations. Our point of interest is the behaviour of the MSEs, opt opt 2 z sub Pz;t ¼ E½ðzt  ^z z;t Þ2  and Psub z;t Þ : Based z;t ¼ E½ðzt  ^

I. Y. SONG ET AL.: OPTIMAL AND SUBOPTIMAL ESTIMATION OF QUADRATIC FUNCTIONALS OF THE STATE VECTOR IN LINEAR STOCHASTIC SYSTEMS

7

In a view of importance of QFs for practice, the proposed estimation algorithms are illustrated on theoretical and practical examples. The examples show that the optimal quadratic estimator yields reasonably good estimation accuracy.

DISCLOSURE STATEMENT No potential conflict of interest was reported by the authors.

Figure 3: Comparison of MSEs of the total kinetic energy zt using optimal and suboptimal estimators

FUNDING This work was supported by the Incheon National University Research Grant in 2014–2015.

on Theorem 4.1, formulas for the MSEs are given by

REFERENCES

opt

Pz;t ¼ 4trðVPt VCt Þ  2trðVPt VPt Þ þ 4mt VPt Vmt ; Psub z;t

¼

opt Pz;t

þ tr ðVPt Þ; 2

(32) respectively. Here, the mean mt and covariance Ct of the state vector satisfy Lyapunov equation (13). opt We observe in Figure 3 that the optimal estimator ^z t has the best performance in contrast to the suboptimal opt sub one ^z sub t , Pz;t < Pz;t : The relative error Dt varies from 6.2% to 10% within the initial zone, t 2 ½0:02; 0:07; of operation of the system, and then it decreases. In time zone, t > 0:07; the values of the MSEs and relative error opt are equal Pz;t ¼ 0:89; Psub z;t ¼ 0:94; and Dt ¼ 5:6%; respectively. As a result, we confirm that the proposed optimal quadratic estimator for a QF is more suitable for data processing in practice.

8. CONCLUSION In many application problems, a QF of signal brings useful information of the target systems for control. In order to estimate an arbitrary QF, optimal and suboptimal algorithms are proposed. The quadratic estimators are comprehensively investigated, including derivation of compact matrix forms for optimal and suboptimal estimates and their MSEs. The main advantages of the paper include the following: (1). The optimal quadratic estimator (6) gives the minimum MSE. (2). The matrix formulas (11), (12), and (19) completely define the MSEs of the optimal and suboptimal estimators, and show the difference between them.

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Authors Il Young Song received the BS degree from the Department of Control and Instrumentation Engineering, Changwon National University, Korea, in 2007, and the MS degree from the School of Information and Mechatronics, Gwangju Institute of Science and Technology (GIST), Gwangju, in 2008. He received the PhD degree from the Department of Information and Communications, GIST in 2012. He is currently a senior researcher in Hanwha Corporation R&D Center, Pangyo, Korea. E-mail: [email protected] Vladimir Shin received the BSc and MSc degrees in applied mathematics from Moscow State Aviation Institute, Moscow, Russia, in 1977 and 1979, respectively. In 1985, he received the PhD degree in mathematics from the Institute of Control Science, Russian Academy of Sciences, Moscow. From 1984 to 1999, he was the Head of the Statistical Methods Laboratory with the Institute of Informatics Problems,

Russian Academy of Sciences. During 1995, he was a visiting professor at the Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, Korea. From 1999 to 2002, he was a member of the technical staff at the Samsung Institute of Technology. He is currently a professor with the Department of Information and Statistics, Gyeongsang National University, Jinju, Korea. His research interests include estimation, filtering, tracking, data fusion, stochastic control, identification, and other multidimensional data processing methods. E-mail: [email protected] Won Choi received his PhD degree at Sung Kyun Kwan University under the direction of Yeong Don Kim. Since 1993 he has been at the University of Incheon. In 1996 and 2003, he was in Steklov Mathematical Institute of Russia and University of Iowa, respectively. His research interests include stochastic processes and bio-mathematics. E-mail: [email protected]