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Quantitative measure of observability for linear stochastic systems
Automatica (
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Brief paper
Quantitative measure of observability for linear stochastic systems✩ Yuksel Subasi a,1 , Mubeccel Demirekler b a
ASELSAN Inc., Ankara, 06750, Turkey
b
Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara, Turkey
article
info
Article history: Received 24 May 2012 Received in revised form 10 February 2014 Accepted 13 March 2014 Available online xxxx Keywords: Stochastic systems Observability Observability measure Subspace observability Mutual information
abstract In this study we define a new observability measure for stochastic systems: the mutual information between the state sequence and the corresponding measurement sequence for a given time horizon. Although the definition is given for a general system representation, the paper focuses on the linear time invariant Gaussian case. Some basic analytical results are derived for this special case. The measure is extended to the observability of a subspace of the state space, specifically an individual state and/or the modes of the system. A single measurement system represented in the observable canonical form is examined in detail. A recursive form of the observability measure for a finite time horizon is derived. The possibility of using this form for designing a sensor selection algorithm is demonstrated by two examples. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction For deterministic systems, observability is acquired and handled by the rank condition of the observability Gramian matrix (Kalman, 1960). The outcome of the procedure is binary; the system is either completely observable or unobservable. The procedure does not provide any information about the degree of observability. The aim of this study is to analyse the degree of observability of linear Gaussian representations and provide an observability measure for them. Quantitative measures for the observability of deterministic systems are proposed by Muller and Weber (1972) and Tarokh (1992). Mode observability is examined for deterministic systems by Choi, Lee, and Zhu (1999), Hamdan and Nayfeh (1989), Lindner, Babendreier, and Hamdan (1989) and Porter and Crossley (1970). Different definitions of the stochastic observability are proposed by Aoki (1967), Bageshwar, Egziabler, Garrard, and Georgiou (2009), Davis and Lasdas (1992), Dragan and Morozan (2006), Han-Fu (1980), Liu and Bitmead (2011), Shen, Sun, and Wu (2013), Ugrinovskii (2003), Van Handel (2009) and Xie, Ugrinovskii, and
✩ The material in this paper was partially presented at the 18th IFAC World Congress, August 28–September 2, 2011, Milano, Italy. This paper was recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (Y. Subasi), [email protected] (M. Demirekler). 1 Tel.: +90 533 3730640; fax: +90 312 8475320.
http://dx.doi.org/10.1016/j.automatica.2014.04.008 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
Petersen (2004). Davis and Lasdas (1992), Van Handel (2009) and Xie et al. (2004) use the probability density functions to define the stochastic observability. Baram and Kailath (1988) provide an estimability definition that is extended to the definition of the stochastic observability by Liu and Bitmead (2011). Baram and Kailath (1988), Han-Fu (1980) and Liu and Bitmead (2011) use the conditional covariance matrices for the definitions. Ugrinovskii (2003) (for linear stochastic uncertain continuous-time systems) and Liu and Bitmead (2011) (for nonlinear systems) define the observability by using information theoretical approaches. In addition to the observability definition, some observability measures are also provided in the literature. Kam, Cheng, and Kalata (1987), Mohler and Hwang (1988) and Sujan and Dubowsky (2003) define observability measures by using the information theoretical approaches. Liu and Bitmead (2011) note that the mutual information can be used as the observability measure. Mohler and Hwang (1988) define the observability measure as the mutual information between the state at the last time and the past measurements. Chen, Hu, Li, and Sun (2007) extend the results of Kam et al. (1987) to continuous state systems by using the quantized versions of continuous variables. Hong, Chun, Kwon, and Lee (2008) define the observability measure by using the observability Gramian. The goal of this paper is to introduce and analyse a new observability measure definition based on the mutual information between the state sequence and the measurement sequence. The definition is given for general nonlinear stochastic systems but is applied to LTI (linear time invariant) discrete time Gaussian
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Y. Subasi, M. Demirekler / Automatica (
case to achieve more specific results. Our approach is different from all the studies presented previously since it addresses a state sequence rather than a state at a given time. Although the information gained about the last state is important in some applications, it is also important how well we know the whole state sequence. As an example, we can mention the applications that do batch processing. A possible application area that may require this definition is the dim target tracking that uses batch data. Another very important application area may be the sensor network design, i.e., decisions made on the number, type and placement of the sensors for the optimal state estimation of the moving targets. Two examples are provided to demonstrate such a case. The application areas are not limited to the ones given above. We believe that the new observability definition offered in this study will fill a gap that exists in this area. An observability measure of ‘‘a subspace of the state space’’ is also defined and analysed in this study. This concept can be expanded to the observability of the modes of the system. In addition, the observability measure based on the state sequence of an individual state is examined in detail for a single measurement system represented in the observable canonical form. A much shorter version of this paper is presented in 18th IFAC World Congress (Subasi & Demirekler, 2011). Compared with that conference paper, this paper provides more concrete definitions and offers proofs for some of the results. We cannot provide all the proofs due to space limitations but the details can be found in Subasi (2012). Furthermore, the observability measure of a subspace of the state space concept is improved by extending the definition to the partial measurement case. The analysis of the single measurement system represented in the observable canonical form is presented here for the first time. In addition, the observability measure based on the state sequence is expressed recursively and two examples are given to demonstrate the use of the recursive form of the observability measure for sensor selection algorithm design. Our main contributions are: definition of an observability measure for an interval; analysis of the relationship between the system matrices and the measure; definition and analysis of the subspace (mode) observability measures; analysis of the system given in the observable canonical form that gives some interesting properties of the information flow in the system and presentation of the observability measure in a recursive way which can be used for sensor selection algorithm designs as demonstrated by the examples. The paper is organized as follows. In Section 2 system under study is defined. Explicit formulae of the observability measure based on the state sequence are obtained in Section 3. The recursive evaluation of the observability measure is also given. In Section 4, the observability measures of a subspace of the state space are given. A single measurement system which is represented in the observable canonical form is examined in detail. In Section 5 two examples are given for sensor selection algorithm design. Section 6 contains the concluding remarks.
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We define x0 , {ωk } and {υk } as the basic random variables. For time k, the state equation of the system can be written in terms of the basic random variables as: k−1
xk = Ak x0 +
Ak−1−i Gωi
(6)
i=0
which yields the measurement equation as: k−1
yk = CAk x0 + C
Ak−1−i Gωi + H υk .
(7)
i=0
Since our goal is to obtain an observability measure for the complete state sequence, the state equations are written in the following form: I 0 x0 A G x1 2 x 2 = A x0 + AG . . . . .. .
.
.
··· ··· ··· .. .
Ak−2 G
···
.. .
Ak−1 G
xk
Ak
Xk
0 0 G
Ak
Gk
0 ω0 0 ω1 0 ω2 .
.. .. . . ωk−1 G
(8)
Wk
Define
Hk = diag H
Ck = diag C
Qk = diag Q
Rk = diag R Y k = y0
V k = υ0
H C
R y2
υ1
υ2
C
···
Q R
H
···
C
Q
y1
···
H
··· ···
Q
yk
(9)
(10)
(11)
(12)
T
(13)
R
υk
···
T
(14)
where (·)T is the transpose operator. Now we can write the state and the measurement equations as: X k = Ak x0 + Gk W k
(15)
Y k = Ck Ak x0 + Ck Gk W k + Hk V k . k
(16)
k
The random vectors X and Y are normal, their mean values are Ak x0 and Ck Ak x0 . Their covariance matrices are given by:
ΣX k = Ak Σ0 Ak T + Gk Qk Gk T
(17)
ΣY k = Ck ΣX k Ck T + Hk Rk Hk T
(18)
ΣX k Y k = ΣX k C k .
(19)
T
3. Observability measure for the state sequence
2. Preliminaries The system that we have analysed is represented by the following equations: xk+1 = Axk + Gωk
(1)
yk = Cxk + H υk
(2)
where xk ϵ ℜn , yk ϵ ℜm and A, G, C , H are constant matrices. It is assumed that {x0 , wk ϵ ℜp , υk ϵ ℜr }∞ k=0 are independent and x0 ∼ N (x0 , Σ0 )
(3)
ωk ∼ N (0, Q ) υk ∼ N (0, R).
(4) (5)
In this section, the definition of the observability measure based on the observability of the complete state sequence is given for LTI discrete-time Gaussian stochastic systems. The mutual information I (X , Y ) between the two continuous random variables with a joint density f (x, y) is defined as (Cover & Thomas, 2006): I (X , Y ) =
f (x, y) log
f (x, y) f (x)f (y)
dxdy.
(20)
Definition 1. The observability measure is the mutual information between the state sequence X k and the measurement sequence Y k .
Y. Subasi, M. Demirekler / Automatica (
Since the variables X k and Y k have Gaussian distributions the mutual information between these variables can be calculated analytically as:
ΣX k ΣY k 1 I (X k , Y k ) = log 2 Σ[X k ,Y k ]
(21)
Ck Ak Σ0 Ak T Ck T + Ck Gk Qk Gk T Ck T + Hk Rk Hk T . (22) I (X , Y ) = log Hk Rk Hk T 2 1
k
Proof. Let Σ[X k ,Y k ] denote the covariance matrix of the joint
variable X
kT
,Y
kT
T
. Determinant of Σ[X k ,Y k ] can be written as:
Σ[X k ,Y k ] = ΣX k ΣY k − ΣY k X k ΣX k −1 ΣX k Y k .
(23)
By substituting (17)–(19) into (23),
Σ[X k ,Y k ] = ΣX k Hk Rk Hk T .
(24)
Note that
Hk Rk Hk T = HRH T k+1 .
(25)
Substitution of (17), (18), (19), and (24) into (21) gives the measure. Next we provide some other useful forms of this expression. Fact 3. The observability measure is: I (X , Y ) = k
k
1
log
k |Σ0 | GQGT
ΣX k − ΣX k Y k ΣY k −1 ΣY k X k 2 ΣX k 1 . = log ΣX k |Y k 2
(27)
In the derivation of the observability measure given in (22), the term ΣX k is cancelled. The singularity of |Σ0 | and/or GQGT creates a problem: the mutual information becomes undefined. For this case, the limit of I (X k , Y k ) as |Σ0 | and/or GQGT approaching zero can be considered as the actual measure. Note that the observability measure is a function of the ratio of the determinants of the conditional and the unconditional covariance matrices of the state sequence. Fact 4. Unobservable states of the pair (C , A) have no effect on the observability measure. Proof. Let (C , A) be written in the following form:
C = 0
A12 A22 C12 .
q22
(30)
(31)
Concentrating on the terms of the mutual information given in (22), we can write: CAr Σ0 CAp
T
T = C12 A22 r Σ22 C12 A22 p r = [0 . . . k] .
(32) (33)
Notice that the term Ck Ak Σ0 Ak T Ck T is independent of the unobservable part. Similarly Ck Gk Qk Gk T Ck T contains only the terms of the form C12 A22 r q22 (C12 A22 p )T . Again, the unobservable part is not involved. The above statement looks like a paradox in the sense that the observability measure is not zero when we have unobservable states. This is obviously not the case because the measurements provide us some information about the state sequence. Later we will discuss the amount of observability of ‘‘unobservable states’’ to make this point clearer. When the observation noise is zero, the observability measure is equal to infinity. Even if there is only one noiseless measurement, i.e., if HRH T is singular, this is true and it can be seen from (22) and (25). This property suggests the following treatments (details of the treatments can be found in Subasi & Demirekler, 2011): (1) Eliminate the perfectly measured part and obtain the observability measure for the remaining part. (2) Add a fixed small noise to all measurements. This approach helps us to differentiate the observability of two perfectly measured modes. After this treatment they may have different observability indices.
(1) The determinant of the covariance matrix of the process noise, Q , increases. (2) The determinant of the initial covariance matrix, Σ0 , increases. (3) The determinant of the state sequence covariance matrix, ΣX k , increases. (4) The determinant of the matrix, C T C , increases. (5) The determinant of the covariance matrix of the measurement noise, R, decreases. Fact 6. When there is no process noise and the initial state uncertainty, the observability measure is zero.
3.1. Discussion on the observability measure
A11 A= 0
Σ11 Σ12 Σ21 Σ22 q q12 . GQGT = 11
Fact 5. The observability measure increases if: (26)
Proof. See Logothetis, Isaksson, and Evans (1997) and Subasi (2012).
3
Σ0 =
p = [0 . . . k] ,
Theorem 2. The mutual information between X k and Y k is:
–
Assume that Σ0 and GQGT are partitioned accordingly:
q21
where |.| is the determinant. A special case of this expression is given in Huang and Chen (2008). Next theorem provides the measure in terms of the basic system parameters.
k
)
The proofs of the above facts can be trivially obtained from (22). For the details see Subasi (2012). The observability measure defined in this paper is a measure for the value of the measurement in the determination of the state vector. If the uncertainty of the state is large, then the measurement is more valuable; thus one gets a larger observability measure value. Fact 6 makes it clear that the measurements are not valuable at all when the initial uncertainty and the process noise are zero since the state sequence is known exactly. In the following theorem the observability measure is expressed recursively.
(28)
Theorem 7. The observability measure based on the mutual information between the state and the measurement sequences can be written recursively as:
(29)
I (X k , Y k ) = I (X k−1 , Y k−1 ) − h(yk |xk ) + h(yk |Y k−1 )
(34)
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Y. Subasi, M. Demirekler / Automatica (
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where h(·) is the entropy. For linear Gaussian systems,
some matrix Lj . That is:
Σy |Y k−1 1 . h(yk |Y k−1 ) − h(yk |xk ) = log k Σy |x 2
I (γ ,
k
(35)
k
Proof. Assuming that the basic random variables are independent, the following relations can be written:
k k Yj
)=
1
log
Σγ k
(42)
Σγk |Y k j ΣΓ k 1 . I (Γ k , Yjk ) = log 2 ΣΓ k |Y k 2
(43)
j
f (X k , Y k ) = f (yk |xk )f (xk |xk−1 )f (X k−1 , Y k−1 )
(36)
f (X k ) = f (xk |xk−1 )f (X k−1 )
(37)
f (Y ) = f (yk |Y
(38)
k
k−1
)f (Y
k−1
)
where f (·) is a probability density function. Substituting (36)–(38) into (20), one can get I (X k , Y k ) =
f (X k , Y k ) log
+
f (X k−1 , Y k−1 ) f (X k−1 )f (Y k−1 )
f (X k , Y k ) log
f (yk , xk ) f (yk |Y k−1 )
dX k dY k
dX k dY k
(39)
which is equal to (34). Details can be found in Subasi (2012). Note that (34) is also valid for nonlinear non-Gaussian systems provided that the basic random variables are independent. (34) indicates that when the information gained from xk about yk is more valuable than the information gained from Y k−1 about yk , the mutual information increases in time. The special form of (34), derived for linear Gaussian systems is given in Logothetis et al. (1997). In some applications, the knowledge about the last state may be important; thus the definition of the observability measure can be changed to concentrate on the last state. This type of definition is given for the first time in Mohler and Hwang (1988). A detailed analysis of this observability measure which lacks in Mohler and Hwang (1988) can be found in Subasi (2012). 4. Observability measures for subspaces of the state space In this section the observability measure of a subspace is defined and analysed both for the state sequence and the final state for linear Gaussian systems. We define the subspace as γk = Mxk . The observability measure of a subspace γk of a stochastic system is defined as the mutual information between γk and the measurement sequence, I (γk , Y k ). From the previous analysis we can compute I (γk , Y k ) as: I (γk , Y ) = k
1 2
Σγ k
. Σγ |Y k k
log
(40)
Note that the formulation allows the computation of the observability measure of a single state. In Mohler and Hwang (1988) only this special case (the observability measure of an individual state) is introduced with no analysis. The definition and the related result given in (40) can be extended to the observability measure of a subspace sequence. Define Γ k = {γi }ki=0 . The observability measure of Γ k is defined as the mutual information between Γ k and the measurement sequence. Similar arguments as given above leads to the result:
ΣΓ k . I (Γ , Y ) = log ΣΓ k |Y k 2 k
k
1
(41)
We can also define the observability measure of γk and Γ by considering only partial measurements defined as Yjk = Lj Y k for k
When the state equation is in the Jordan form, these equations can be used to define the observability measures of the system modes. The observability measures for the subspaces of the state space I (γk , Y k ), I (Γ k , Y k ), I (γk , Yjk ) and I (Γ k , Yjk ) can also be used for general nonlinear systems by using the mutual information definition. In order to examine cases with singular covariances, Liu and Bitmead (2011) mentioned the necessity of studying subspaces of the state space. The following theorem yields a result about the observability measure of the unobservable states. Theorem 8. Let the unobservable representation be in the form given in (28)–(31). Assume that the state vector is represented accordingly as [(X1k )T (X2k )T ]T . The unobservable states of the pair (C , A) of the system have zero observability measure if and only if the off-diagonal blocks of Σ0 , GQGT , and A12 are zero. Proof. The observability measure of X1k can be found from (41). The observability measure is zero if and only if ΣX k X k is zero. This 1 2
requires Σ12 sub block of Σ0 to be zero. Necessity of A12 = 0 is obvious when we consider the cross covariance between the first and the second state blocks at time k = 1, ΣX 1 X 1 = A21 Σ22 A21 T . 1 2
Sufficiency is trivial since all the matrices involved will be block diagonal if the given conditions are satisfied. 4.1. Observability measure of a single measurement system in observable canonical form One interesting question that will be answered in this section is the observability measure of the system represented in the observable canonical form. The problem is analysed for a single output system but the results can be extended to multi output systems as well. The A and C matrices of the observable canonical form are: 0 1 0 A=
0 0 1
0 0 0
.. .
··· ··· ··· .. .
0 0 0
a1 a2 a3
0
0
0
···
1
an
...
C = 0
.. .
0
0
···
.. .
0
.. .
1 .
(44)
(45)
Theorem 9. The observability measure I (X k , Y k ) of the system given in (44) and (45) is equal to the observability measure of the nth state I (Xnk , Y k ), i.e., I (Xnk , Y k ) = I (X k , Y k ). Proof. By using the measurement matrix C given in (45):
ΣY k = ΣXnk + Hk Rk Hk T
(46)
ΣXnk Y k = ΣXnk .
(47)
The observability measure of the nth state I (Xnk , Y k ) can be calculated as:
ΣY k I ( , Y ) = log 2 ΣY k − ΣY k Xnk ΣXnk −1 ΣXnk Y k Xnk
k
1
= I (X k , Y k ).
(48)
(49)
Y. Subasi, M. Demirekler / Automatica (
Theorem 10. The observability measure of the ith state I (Xik , Y k ) (i ̸= n) is smaller than or equal to the observability measure of the nth state I (X k , Y k ), i.e. I (Xik , Y k ) ≤ I (X k , Y k ). Proof. The observability measure of the ith state is: I(
Xik
1
,Y ) = k
2
ΣY k
log
(50)
ΣY k − ΣY k X k ΣX k − 1 ΣX k Y k i
i
i
ΣX k Y k = ΣX k Xnk . i
(51)
i
By substituting (46), (47) and (51) into (50), one can get: I(
Xik
,Y ) = k
1 2
ΣY k
. Hk Rk Hk T + ΣXnk − ΣXnk X k ΣX k −1 ΣX k Xnk
log
i
The term ΣX k −ΣX k X k ΣX k n i
n
−1
i
i
(52)
i
ΣX k Xnk is the covariance matrix of Xnk i
Xik .
|
When this covariance matrix is equal to zero, the observability measure of the ith state I (Xik , Y k ) becomes equal to the observability measure of the system I (X k , Y k ). Otherwise it is smaller than I (X k , Y k ). Theorem 11. If there is no process noise, the observability measure of any state of the system in (44) and (45) becomes equal to the observability measure of the system I (X k , Y k ) when k ≥ n. Proof. Since the system is given in the observable canonical form, if one of the states is known exactly, other states can also be calculated exactly in n steps. When this is the case, the term ΣX k − n
ΣXnk X k ΣX k −1 ΣX k Xnk becomes equal to the zero matrix and I (Xik , Y k ) i
i
i
becomes equal to I (X , Y ). k
k
5. Sensor selection examples The theory developed in the previous parts is applied to two sensor selection algorithm design examples. The availability of the recursive computation of the objective function makes the use of dynamic programming possible. The method is easy to apply when the state vector is quantized to some finite number of levels. Example 12. The observation at time k for this example is either nothing or a noisy version of the single state of the system. Each observation, if it is collected, has a cost of g (ck ). The formulation of the problem is as follows:
max
c0 ,...,cS
I (X S , Y S ) −
S
g (ck )
(53)
i=0
subject to xk+1 = axk + ωk yk = ck xk + υk ,
(54) ck ϵ {0, 1}
(55)
x0 ∼ N (x0 , Σ0 )
(56)
ωk ∼ N (0, Q ) υk ∼ N (0, R).
(57) (58)
In our application we have selected g (ck ) = p log(ck 2 + 1). If p = 0, the optimal sensor selection algorithm obviously measures the state at all times. Different solutions are possible depending on the value of the initial state covariance, p and the interval length S. In this example we assume that a = 0.7, Σ0 = 10, Q = 1, R = 1, S = 10 and p = 0.5. The solution is [1, 1, (0, 1)4 , 1] and the value of the objective function is 1.5769. (0, 1)4 is the concatenation of (0, 1) sequences for 4 times. If p = 0.7, the solution becomes [1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1] and the objective
)
–
5
function takes the value of 0.8253. As the cost of observation increases, the measurement rate decreases as expected. We have conducted an experiment to demonstrate the impact of using the proposed observability measure of the state sequence instead of its final value. The approach that utilizes an instantaneous decision making is applied to the same problem. At time k, the mutual information between the kth state (I (xk , Y k )) and the measurement sequence is computed for the two cases: a measurement is taken and not taken. The difference between them is compared with g (ck ). If the result of the comparison is greater than zero, a measurement is taken. The solution comes out as [(1)1 1] for p = 0.5 with the value of the objective as 1.5355. Repetition of the problem for p = 0.7 yields [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1] with a value for the objective function as 0.7272. Example 13. The second example is a sensor scheduling problem posed by Gupta, Chung, Hassibi, and Murray (2006). The problem is tracking a vehicle that is moving with an approximate constant velocity in a two dimensional space using two sensors. The system matrices are given as:
1 0 A= 0 0
0 1 0 0
h2 /2 0 G= h 0
0 h 0 1
1 0.25
(59)
0
h /2 0 h 2
0.25 1
Q =
h 0 1 0
(60)
(61)
x = [px py vx vy ]T where px and py are the positions and vx and vy are the velocities, h = 0.2. The two sensors measure the positions and are modelled as follows:
1 C = 0
0 1
0 0.4
0.7 0
0 . 1.4
R2 =
0 0
2.4 0
R1 =
0 0
(62)
(63)
(64)
The problem is defined so that only one sensor can be used at a time. The cost is taken as the sum of the traces of the error covariance matrices of the estimates of the two sensors in Gupta et al. (2006). The objective function is defined over a horizon with length 50. The task is to minimize this function. Solution is obtained by an exhaustive search on small sliding windows of length 10. Due to the optimality of Kalman filtering the solution is optimal. We model the problem as: I (X S , Y S )
(65)
xk+1 = Axk + Gωk
(66)
max
υ0 ,...,υS
subject to yk = Cxk + υk
(67)
υk ∼ N
0 , R1 0
x0 ∼ N (x0 , Σ0 )
ωk ∼ N
0 ,Q 0
or N
0 , R2 0
(68) (69)
(70)
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Y. Subasi, M. Demirekler / Automatica (
Fig. 1. Comparison of the two approaches (1: optimal solution of Gupta et al., 2006; 2: optimal solution of the proposed observability measure; 3: only sensor 1 is used; 4: only sensor 2 is used).
where Σ0 is a zero matrix. Vector quantization is used to quantize the covariance matrices to 1000 levels. In 1 we have compared the two different approaches using the cost of Gupta et al. (2006). As seen from the results, the optimal solution of Gupta et al. (2006) that is obtained by an exhaustive search is also achieved by the proposed observability measure.
6. Conclusions We have defined a new observability measure as the information gained about the state sequence from the measurement sequence. Several interesting properties of the observability measure are obtained for linear Gaussian systems. The examples demonstrate possible application areas of the measure. The observability definition is expandable to the non-Gaussian and the nonlinear systems. A Monte Carlo approach seems to be possible for a general nonlinear case.
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Hamdan, A. M. A., & Nayfeh, A. H. (1989). Measures of modal controllability and observability for first- and second-order linear systems. Journal of Guidance, Control, and Dynamics, 12(3), 421–428. Han-Fu, C. (1980). On stochastic observability and controllability. Automatica, 16, 179–190. Hong, S., Chun, H. H., Kwon, S. H., & Lee, M. H. (2008). Observability measures and their application to GPS/INS. IEEE Transactions on Vehicular Technology, 57(1), 97–106. Huang, T., & Chen, B. (2008). Information evolutions in the linear stochastic control systems. In Proceedings of the 7th world congress on intelligent control and automation (pp. 3384–3387). Kalman, R.E. (1960). On the general theory of control systems. In Proc. 1st int. cong. of IFAC (pp. 481–492). Kam, M., Cheng, R., & Kalata, P. (1987). An information theoretic interpretation of stability and observability. In Proceedings of the American control conference (pp. 1957–1962). Lindner, D. K., Babendreier, J., & Hamdan, A. M. A. (1989). Measures of the controllability and observability and residues. IEEE Transactions on Automatic Control, 34(6), 648–650. Liu, A. R., & Bitmead, R. R. (2011). Stochastic observability in network state estimation and control. Automatica, 47, 65–78. Logothetis, A., Isaksson, A., & Evans, R.J. (1997). An information theoretic approach to observer path design for bearings-only tracking. In Proceedings of the 36th conference on decision & control. Mohler, R. R., & Hwang, C. S. (1988). Nonlinear data observability and information. Journal of the Franklin Institute, 325(4), 443–464. Muller, P. C., & Weber, H. I. (1972). Analysis and optimization of certain qualities of controllability and observability for linear dynamical systems. Automatica, 8, 237–246. Porter, B., & Crossley, T. R. (1970). Mode-observability structure of a multivariable linear system. Electronics Letters, 6(21), 664–665. Shen, L., Sun, J., & Wu, Q. (2013). Observability and detectability of discrete-time stochastic systems with Markovian jump. Systems & Control Letters, 62(1), 37–42. Subasi, Y. (2012). Quantitative measures of observability for stochastic systems. Ph.D. Thesis, METU, Ankara, Turkey. Subasi, Y., & Demirekler, M. (2011). Quantitative measure of observability for stochastic systems. In Proceedings of the 18th IFAC world congress (pp. 4244–4249). Sujan, V. A., & Dubowsky, S. (2003). An optimal information method for mobile manipulator dynamic parameter identification. IEEE/ASME Transactions of Mechatronics, 2(2), 215–225. Tarokh, M. (1992). Measures for controllability, observability, and fixed modes. IEEE Transactions on Automatic Control, 37(8), 648–650. Ugrinovskii, V. A. (2003). Observability of linear stochastic uncertain systems. IEEE Transactions on Automatic Control, 48(12), 2264–2269. Van Handel, R. (2009). Observability and nonlinear filtering. Probability Theory and Related Fields, 145, 35–74. Xie, L., Ugrinovskii, V.A., & Petersen, I.R. (2004). Finite horizon robust state estimation for uncertain finite-alphabet hidden Markov models with conditional relative entropy constraints. In Proceedings of the 43th IEEE conference on decision and control (pp. 4497–4502).
References Aoki, M. (1967). Optimization of stochastic systems. New York, USA: Academic Press Inc. Bageshwar, V. L., Egziabler, D. G., Garrard, W. L., & Georgiou, T. T. (2009). Stochastic observability test for discrete-time kalman filters. Journal of Guidance, Control, and Dynamics, 32(4), 1357–1370. Baram, Y., & Kailath, T. (1988). Estimability and regulability of linear systems. IEEE Transactions on Automatic Control, 33(12), 1116–1121. Chen, B., Hu, J., Li, H., & Sun, Z. (2007). Measure observability by the generalized informational correlation. In Proceedings of the 46th IEEE conference on decision and control (pp. 5570–5574). Choi, J.W., Lee, H.C., & Zhu, J.J. (1999). Measures of modal and gross controllability/observability for linear time-varying systems. In Proceedings of the American control conference San Diego. Cover, T. M., & Thomas, J. A. (2006). Elements of information theory (2nd ed.). New Jersey, USA: A John Wiley & Sons, Inc. Davis, M. H. A., & Lasdas, V. (1992). Reducibility and unobservability of Markov processes: the linear system case. IEEE Transactions on Automatic Control, 37(4), 505–508. Dragan, V., & Morozan, T. (2006). Observability and detectability of a class of discrete-time stochastic linear systems. IMA Journal of Mathematical Control and Information, 23(3), 371–394. Gupta, V., Chung, T. H., Hassibi, B., & Murray, R. M. (2006). On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage. Automatica, 42, 251–260.
Yuksel Subasi received his B.S., M.Sc., and Ph.D. degrees in Electrical Engineering from the Middle East Technical University, Ankara/Turkey, in 1996, 1998, and 2012 respectively. He is currently employed in ASELSAN as a research engineer. His main area of interest is estimation theory and stochastic optimal control.
Mubeccel Demirekler received her Ph.D. degree in Electrical Engineering from Middle East Technical University, Turkey in 1979. She is currently a Professor at the same department. Her research interests are tracking systems (on which she conducted several projects), information fusion and stochastic optimization.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Quantitative measure of observability for linear stochastic systems✩ Yuksel Subasi a,1 , Mubeccel Demirekler b a
ASELSAN Inc., Ankara, 06750, Turkey
b
Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara, Turkey
article
info
Article history: Received 24 May 2012 Received in revised form 10 February 2014 Accepted 13 March 2014 Available online xxxx Keywords: Stochastic systems Observability Observability measure Subspace observability Mutual information
abstract In this study we define a new observability measure for stochastic systems: the mutual information between the state sequence and the corresponding measurement sequence for a given time horizon. Although the definition is given for a general system representation, the paper focuses on the linear time invariant Gaussian case. Some basic analytical results are derived for this special case. The measure is extended to the observability of a subspace of the state space, specifically an individual state and/or the modes of the system. A single measurement system represented in the observable canonical form is examined in detail. A recursive form of the observability measure for a finite time horizon is derived. The possibility of using this form for designing a sensor selection algorithm is demonstrated by two examples. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction For deterministic systems, observability is acquired and handled by the rank condition of the observability Gramian matrix (Kalman, 1960). The outcome of the procedure is binary; the system is either completely observable or unobservable. The procedure does not provide any information about the degree of observability. The aim of this study is to analyse the degree of observability of linear Gaussian representations and provide an observability measure for them. Quantitative measures for the observability of deterministic systems are proposed by Muller and Weber (1972) and Tarokh (1992). Mode observability is examined for deterministic systems by Choi, Lee, and Zhu (1999), Hamdan and Nayfeh (1989), Lindner, Babendreier, and Hamdan (1989) and Porter and Crossley (1970). Different definitions of the stochastic observability are proposed by Aoki (1967), Bageshwar, Egziabler, Garrard, and Georgiou (2009), Davis and Lasdas (1992), Dragan and Morozan (2006), Han-Fu (1980), Liu and Bitmead (2011), Shen, Sun, and Wu (2013), Ugrinovskii (2003), Van Handel (2009) and Xie, Ugrinovskii, and
✩ The material in this paper was partially presented at the 18th IFAC World Congress, August 28–September 2, 2011, Milano, Italy. This paper was recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (Y. Subasi), [email protected] (M. Demirekler). 1 Tel.: +90 533 3730640; fax: +90 312 8475320.
http://dx.doi.org/10.1016/j.automatica.2014.04.008 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
Petersen (2004). Davis and Lasdas (1992), Van Handel (2009) and Xie et al. (2004) use the probability density functions to define the stochastic observability. Baram and Kailath (1988) provide an estimability definition that is extended to the definition of the stochastic observability by Liu and Bitmead (2011). Baram and Kailath (1988), Han-Fu (1980) and Liu and Bitmead (2011) use the conditional covariance matrices for the definitions. Ugrinovskii (2003) (for linear stochastic uncertain continuous-time systems) and Liu and Bitmead (2011) (for nonlinear systems) define the observability by using information theoretical approaches. In addition to the observability definition, some observability measures are also provided in the literature. Kam, Cheng, and Kalata (1987), Mohler and Hwang (1988) and Sujan and Dubowsky (2003) define observability measures by using the information theoretical approaches. Liu and Bitmead (2011) note that the mutual information can be used as the observability measure. Mohler and Hwang (1988) define the observability measure as the mutual information between the state at the last time and the past measurements. Chen, Hu, Li, and Sun (2007) extend the results of Kam et al. (1987) to continuous state systems by using the quantized versions of continuous variables. Hong, Chun, Kwon, and Lee (2008) define the observability measure by using the observability Gramian. The goal of this paper is to introduce and analyse a new observability measure definition based on the mutual information between the state sequence and the measurement sequence. The definition is given for general nonlinear stochastic systems but is applied to LTI (linear time invariant) discrete time Gaussian
2
Y. Subasi, M. Demirekler / Automatica (
case to achieve more specific results. Our approach is different from all the studies presented previously since it addresses a state sequence rather than a state at a given time. Although the information gained about the last state is important in some applications, it is also important how well we know the whole state sequence. As an example, we can mention the applications that do batch processing. A possible application area that may require this definition is the dim target tracking that uses batch data. Another very important application area may be the sensor network design, i.e., decisions made on the number, type and placement of the sensors for the optimal state estimation of the moving targets. Two examples are provided to demonstrate such a case. The application areas are not limited to the ones given above. We believe that the new observability definition offered in this study will fill a gap that exists in this area. An observability measure of ‘‘a subspace of the state space’’ is also defined and analysed in this study. This concept can be expanded to the observability of the modes of the system. In addition, the observability measure based on the state sequence of an individual state is examined in detail for a single measurement system represented in the observable canonical form. A much shorter version of this paper is presented in 18th IFAC World Congress (Subasi & Demirekler, 2011). Compared with that conference paper, this paper provides more concrete definitions and offers proofs for some of the results. We cannot provide all the proofs due to space limitations but the details can be found in Subasi (2012). Furthermore, the observability measure of a subspace of the state space concept is improved by extending the definition to the partial measurement case. The analysis of the single measurement system represented in the observable canonical form is presented here for the first time. In addition, the observability measure based on the state sequence is expressed recursively and two examples are given to demonstrate the use of the recursive form of the observability measure for sensor selection algorithm design. Our main contributions are: definition of an observability measure for an interval; analysis of the relationship between the system matrices and the measure; definition and analysis of the subspace (mode) observability measures; analysis of the system given in the observable canonical form that gives some interesting properties of the information flow in the system and presentation of the observability measure in a recursive way which can be used for sensor selection algorithm designs as demonstrated by the examples. The paper is organized as follows. In Section 2 system under study is defined. Explicit formulae of the observability measure based on the state sequence are obtained in Section 3. The recursive evaluation of the observability measure is also given. In Section 4, the observability measures of a subspace of the state space are given. A single measurement system which is represented in the observable canonical form is examined in detail. In Section 5 two examples are given for sensor selection algorithm design. Section 6 contains the concluding remarks.
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We define x0 , {ωk } and {υk } as the basic random variables. For time k, the state equation of the system can be written in terms of the basic random variables as: k−1
xk = Ak x0 +
Ak−1−i Gωi
(6)
i=0
which yields the measurement equation as: k−1
yk = CAk x0 + C
Ak−1−i Gωi + H υk .
(7)
i=0
Since our goal is to obtain an observability measure for the complete state sequence, the state equations are written in the following form: I 0 x0 A G x1 2 x 2 = A x0 + AG . . . . .. .
.
.
··· ··· ··· .. .
Ak−2 G
···
.. .
Ak−1 G
xk
Ak
Xk
0 0 G
Ak
Gk
0 ω0 0 ω1 0 ω2 .
.. .. . . ωk−1 G
(8)
Wk
Define
Hk = diag H
Ck = diag C
Qk = diag Q
Rk = diag R Y k = y0
V k = υ0
H C
R y2
υ1
υ2
C
···
Q R
H
···
C
Q
y1
···
H
··· ···
Q
yk
(9)
(10)
(11)
(12)
T
(13)
R
υk
···
T
(14)
where (·)T is the transpose operator. Now we can write the state and the measurement equations as: X k = Ak x0 + Gk W k
(15)
Y k = Ck Ak x0 + Ck Gk W k + Hk V k . k
(16)
k
The random vectors X and Y are normal, their mean values are Ak x0 and Ck Ak x0 . Their covariance matrices are given by:
ΣX k = Ak Σ0 Ak T + Gk Qk Gk T
(17)
ΣY k = Ck ΣX k Ck T + Hk Rk Hk T
(18)
ΣX k Y k = ΣX k C k .
(19)
T
3. Observability measure for the state sequence
2. Preliminaries The system that we have analysed is represented by the following equations: xk+1 = Axk + Gωk
(1)
yk = Cxk + H υk
(2)
where xk ϵ ℜn , yk ϵ ℜm and A, G, C , H are constant matrices. It is assumed that {x0 , wk ϵ ℜp , υk ϵ ℜr }∞ k=0 are independent and x0 ∼ N (x0 , Σ0 )
(3)
ωk ∼ N (0, Q ) υk ∼ N (0, R).
(4) (5)
In this section, the definition of the observability measure based on the observability of the complete state sequence is given for LTI discrete-time Gaussian stochastic systems. The mutual information I (X , Y ) between the two continuous random variables with a joint density f (x, y) is defined as (Cover & Thomas, 2006): I (X , Y ) =
f (x, y) log
f (x, y) f (x)f (y)
dxdy.
(20)
Definition 1. The observability measure is the mutual information between the state sequence X k and the measurement sequence Y k .
Y. Subasi, M. Demirekler / Automatica (
Since the variables X k and Y k have Gaussian distributions the mutual information between these variables can be calculated analytically as:
ΣX k ΣY k 1 I (X k , Y k ) = log 2 Σ[X k ,Y k ]
(21)
Ck Ak Σ0 Ak T Ck T + Ck Gk Qk Gk T Ck T + Hk Rk Hk T . (22) I (X , Y ) = log Hk Rk Hk T 2 1
k
Proof. Let Σ[X k ,Y k ] denote the covariance matrix of the joint
variable X
kT
,Y
kT
T
. Determinant of Σ[X k ,Y k ] can be written as:
Σ[X k ,Y k ] = ΣX k ΣY k − ΣY k X k ΣX k −1 ΣX k Y k .
(23)
By substituting (17)–(19) into (23),
Σ[X k ,Y k ] = ΣX k Hk Rk Hk T .
(24)
Note that
Hk Rk Hk T = HRH T k+1 .
(25)
Substitution of (17), (18), (19), and (24) into (21) gives the measure. Next we provide some other useful forms of this expression. Fact 3. The observability measure is: I (X , Y ) = k
k
1
log
k |Σ0 | GQGT
ΣX k − ΣX k Y k ΣY k −1 ΣY k X k 2 ΣX k 1 . = log ΣX k |Y k 2
(27)
In the derivation of the observability measure given in (22), the term ΣX k is cancelled. The singularity of |Σ0 | and/or GQGT creates a problem: the mutual information becomes undefined. For this case, the limit of I (X k , Y k ) as |Σ0 | and/or GQGT approaching zero can be considered as the actual measure. Note that the observability measure is a function of the ratio of the determinants of the conditional and the unconditional covariance matrices of the state sequence. Fact 4. Unobservable states of the pair (C , A) have no effect on the observability measure. Proof. Let (C , A) be written in the following form:
C = 0
A12 A22 C12 .
q22
(30)
(31)
Concentrating on the terms of the mutual information given in (22), we can write: CAr Σ0 CAp
T
T = C12 A22 r Σ22 C12 A22 p r = [0 . . . k] .
(32) (33)
Notice that the term Ck Ak Σ0 Ak T Ck T is independent of the unobservable part. Similarly Ck Gk Qk Gk T Ck T contains only the terms of the form C12 A22 r q22 (C12 A22 p )T . Again, the unobservable part is not involved. The above statement looks like a paradox in the sense that the observability measure is not zero when we have unobservable states. This is obviously not the case because the measurements provide us some information about the state sequence. Later we will discuss the amount of observability of ‘‘unobservable states’’ to make this point clearer. When the observation noise is zero, the observability measure is equal to infinity. Even if there is only one noiseless measurement, i.e., if HRH T is singular, this is true and it can be seen from (22) and (25). This property suggests the following treatments (details of the treatments can be found in Subasi & Demirekler, 2011): (1) Eliminate the perfectly measured part and obtain the observability measure for the remaining part. (2) Add a fixed small noise to all measurements. This approach helps us to differentiate the observability of two perfectly measured modes. After this treatment they may have different observability indices.
(1) The determinant of the covariance matrix of the process noise, Q , increases. (2) The determinant of the initial covariance matrix, Σ0 , increases. (3) The determinant of the state sequence covariance matrix, ΣX k , increases. (4) The determinant of the matrix, C T C , increases. (5) The determinant of the covariance matrix of the measurement noise, R, decreases. Fact 6. When there is no process noise and the initial state uncertainty, the observability measure is zero.
3.1. Discussion on the observability measure
A11 A= 0
Σ11 Σ12 Σ21 Σ22 q q12 . GQGT = 11
Fact 5. The observability measure increases if: (26)
Proof. See Logothetis, Isaksson, and Evans (1997) and Subasi (2012).
3
Σ0 =
p = [0 . . . k] ,
Theorem 2. The mutual information between X k and Y k is:
–
Assume that Σ0 and GQGT are partitioned accordingly:
q21
where |.| is the determinant. A special case of this expression is given in Huang and Chen (2008). Next theorem provides the measure in terms of the basic system parameters.
k
)
The proofs of the above facts can be trivially obtained from (22). For the details see Subasi (2012). The observability measure defined in this paper is a measure for the value of the measurement in the determination of the state vector. If the uncertainty of the state is large, then the measurement is more valuable; thus one gets a larger observability measure value. Fact 6 makes it clear that the measurements are not valuable at all when the initial uncertainty and the process noise are zero since the state sequence is known exactly. In the following theorem the observability measure is expressed recursively.
(28)
Theorem 7. The observability measure based on the mutual information between the state and the measurement sequences can be written recursively as:
(29)
I (X k , Y k ) = I (X k−1 , Y k−1 ) − h(yk |xk ) + h(yk |Y k−1 )
(34)
4
Y. Subasi, M. Demirekler / Automatica (
)
–
where h(·) is the entropy. For linear Gaussian systems,
some matrix Lj . That is:
Σy |Y k−1 1 . h(yk |Y k−1 ) − h(yk |xk ) = log k Σy |x 2
I (γ ,
k
(35)
k
Proof. Assuming that the basic random variables are independent, the following relations can be written:
k k Yj
)=
1
log
Σγ k
(42)
Σγk |Y k j ΣΓ k 1 . I (Γ k , Yjk ) = log 2 ΣΓ k |Y k 2
(43)
j
f (X k , Y k ) = f (yk |xk )f (xk |xk−1 )f (X k−1 , Y k−1 )
(36)
f (X k ) = f (xk |xk−1 )f (X k−1 )
(37)
f (Y ) = f (yk |Y
(38)
k
k−1
)f (Y
k−1
)
where f (·) is a probability density function. Substituting (36)–(38) into (20), one can get I (X k , Y k ) =
f (X k , Y k ) log
+
f (X k−1 , Y k−1 ) f (X k−1 )f (Y k−1 )
f (X k , Y k ) log
f (yk , xk ) f (yk |Y k−1 )
dX k dY k
dX k dY k
(39)
which is equal to (34). Details can be found in Subasi (2012). Note that (34) is also valid for nonlinear non-Gaussian systems provided that the basic random variables are independent. (34) indicates that when the information gained from xk about yk is more valuable than the information gained from Y k−1 about yk , the mutual information increases in time. The special form of (34), derived for linear Gaussian systems is given in Logothetis et al. (1997). In some applications, the knowledge about the last state may be important; thus the definition of the observability measure can be changed to concentrate on the last state. This type of definition is given for the first time in Mohler and Hwang (1988). A detailed analysis of this observability measure which lacks in Mohler and Hwang (1988) can be found in Subasi (2012). 4. Observability measures for subspaces of the state space In this section the observability measure of a subspace is defined and analysed both for the state sequence and the final state for linear Gaussian systems. We define the subspace as γk = Mxk . The observability measure of a subspace γk of a stochastic system is defined as the mutual information between γk and the measurement sequence, I (γk , Y k ). From the previous analysis we can compute I (γk , Y k ) as: I (γk , Y ) = k
1 2
Σγ k
. Σγ |Y k k
log
(40)
Note that the formulation allows the computation of the observability measure of a single state. In Mohler and Hwang (1988) only this special case (the observability measure of an individual state) is introduced with no analysis. The definition and the related result given in (40) can be extended to the observability measure of a subspace sequence. Define Γ k = {γi }ki=0 . The observability measure of Γ k is defined as the mutual information between Γ k and the measurement sequence. Similar arguments as given above leads to the result:
ΣΓ k . I (Γ , Y ) = log ΣΓ k |Y k 2 k
k
1
(41)
We can also define the observability measure of γk and Γ by considering only partial measurements defined as Yjk = Lj Y k for k
When the state equation is in the Jordan form, these equations can be used to define the observability measures of the system modes. The observability measures for the subspaces of the state space I (γk , Y k ), I (Γ k , Y k ), I (γk , Yjk ) and I (Γ k , Yjk ) can also be used for general nonlinear systems by using the mutual information definition. In order to examine cases with singular covariances, Liu and Bitmead (2011) mentioned the necessity of studying subspaces of the state space. The following theorem yields a result about the observability measure of the unobservable states. Theorem 8. Let the unobservable representation be in the form given in (28)–(31). Assume that the state vector is represented accordingly as [(X1k )T (X2k )T ]T . The unobservable states of the pair (C , A) of the system have zero observability measure if and only if the off-diagonal blocks of Σ0 , GQGT , and A12 are zero. Proof. The observability measure of X1k can be found from (41). The observability measure is zero if and only if ΣX k X k is zero. This 1 2
requires Σ12 sub block of Σ0 to be zero. Necessity of A12 = 0 is obvious when we consider the cross covariance between the first and the second state blocks at time k = 1, ΣX 1 X 1 = A21 Σ22 A21 T . 1 2
Sufficiency is trivial since all the matrices involved will be block diagonal if the given conditions are satisfied. 4.1. Observability measure of a single measurement system in observable canonical form One interesting question that will be answered in this section is the observability measure of the system represented in the observable canonical form. The problem is analysed for a single output system but the results can be extended to multi output systems as well. The A and C matrices of the observable canonical form are: 0 1 0 A=
0 0 1
0 0 0
.. .
··· ··· ··· .. .
0 0 0
a1 a2 a3
0
0
0
···
1
an
...
C = 0
.. .
0
0
···
.. .
0
.. .
1 .
(44)
(45)
Theorem 9. The observability measure I (X k , Y k ) of the system given in (44) and (45) is equal to the observability measure of the nth state I (Xnk , Y k ), i.e., I (Xnk , Y k ) = I (X k , Y k ). Proof. By using the measurement matrix C given in (45):
ΣY k = ΣXnk + Hk Rk Hk T
(46)
ΣXnk Y k = ΣXnk .
(47)
The observability measure of the nth state I (Xnk , Y k ) can be calculated as:
ΣY k I ( , Y ) = log 2 ΣY k − ΣY k Xnk ΣXnk −1 ΣXnk Y k Xnk
k
1
= I (X k , Y k ).
(48)
(49)
Y. Subasi, M. Demirekler / Automatica (
Theorem 10. The observability measure of the ith state I (Xik , Y k ) (i ̸= n) is smaller than or equal to the observability measure of the nth state I (X k , Y k ), i.e. I (Xik , Y k ) ≤ I (X k , Y k ). Proof. The observability measure of the ith state is: I(
Xik
1
,Y ) = k
2
ΣY k
log
(50)
ΣY k − ΣY k X k ΣX k − 1 ΣX k Y k i
i
i
ΣX k Y k = ΣX k Xnk . i
(51)
i
By substituting (46), (47) and (51) into (50), one can get: I(
Xik
,Y ) = k
1 2
ΣY k
. Hk Rk Hk T + ΣXnk − ΣXnk X k ΣX k −1 ΣX k Xnk
log
i
The term ΣX k −ΣX k X k ΣX k n i
n
−1
i
i
(52)
i
ΣX k Xnk is the covariance matrix of Xnk i
Xik .
|
When this covariance matrix is equal to zero, the observability measure of the ith state I (Xik , Y k ) becomes equal to the observability measure of the system I (X k , Y k ). Otherwise it is smaller than I (X k , Y k ). Theorem 11. If there is no process noise, the observability measure of any state of the system in (44) and (45) becomes equal to the observability measure of the system I (X k , Y k ) when k ≥ n. Proof. Since the system is given in the observable canonical form, if one of the states is known exactly, other states can also be calculated exactly in n steps. When this is the case, the term ΣX k − n
ΣXnk X k ΣX k −1 ΣX k Xnk becomes equal to the zero matrix and I (Xik , Y k ) i
i
i
becomes equal to I (X , Y ). k
k
5. Sensor selection examples The theory developed in the previous parts is applied to two sensor selection algorithm design examples. The availability of the recursive computation of the objective function makes the use of dynamic programming possible. The method is easy to apply when the state vector is quantized to some finite number of levels. Example 12. The observation at time k for this example is either nothing or a noisy version of the single state of the system. Each observation, if it is collected, has a cost of g (ck ). The formulation of the problem is as follows:
max
c0 ,...,cS
I (X S , Y S ) −
S
g (ck )
(53)
i=0
subject to xk+1 = axk + ωk yk = ck xk + υk ,
(54) ck ϵ {0, 1}
(55)
x0 ∼ N (x0 , Σ0 )
(56)
ωk ∼ N (0, Q ) υk ∼ N (0, R).
(57) (58)
In our application we have selected g (ck ) = p log(ck 2 + 1). If p = 0, the optimal sensor selection algorithm obviously measures the state at all times. Different solutions are possible depending on the value of the initial state covariance, p and the interval length S. In this example we assume that a = 0.7, Σ0 = 10, Q = 1, R = 1, S = 10 and p = 0.5. The solution is [1, 1, (0, 1)4 , 1] and the value of the objective function is 1.5769. (0, 1)4 is the concatenation of (0, 1) sequences for 4 times. If p = 0.7, the solution becomes [1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1] and the objective
)
–
5
function takes the value of 0.8253. As the cost of observation increases, the measurement rate decreases as expected. We have conducted an experiment to demonstrate the impact of using the proposed observability measure of the state sequence instead of its final value. The approach that utilizes an instantaneous decision making is applied to the same problem. At time k, the mutual information between the kth state (I (xk , Y k )) and the measurement sequence is computed for the two cases: a measurement is taken and not taken. The difference between them is compared with g (ck ). If the result of the comparison is greater than zero, a measurement is taken. The solution comes out as [(1)1 1] for p = 0.5 with the value of the objective as 1.5355. Repetition of the problem for p = 0.7 yields [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1] with a value for the objective function as 0.7272. Example 13. The second example is a sensor scheduling problem posed by Gupta, Chung, Hassibi, and Murray (2006). The problem is tracking a vehicle that is moving with an approximate constant velocity in a two dimensional space using two sensors. The system matrices are given as:
1 0 A= 0 0
0 1 0 0
h2 /2 0 G= h 0
0 h 0 1
1 0.25
(59)
0
h /2 0 h 2
0.25 1
Q =
h 0 1 0
(60)
(61)
x = [px py vx vy ]T where px and py are the positions and vx and vy are the velocities, h = 0.2. The two sensors measure the positions and are modelled as follows:
1 C = 0
0 1
0 0.4
0.7 0
0 . 1.4
R2 =
0 0
2.4 0
R1 =
0 0
(62)
(63)
(64)
The problem is defined so that only one sensor can be used at a time. The cost is taken as the sum of the traces of the error covariance matrices of the estimates of the two sensors in Gupta et al. (2006). The objective function is defined over a horizon with length 50. The task is to minimize this function. Solution is obtained by an exhaustive search on small sliding windows of length 10. Due to the optimality of Kalman filtering the solution is optimal. We model the problem as: I (X S , Y S )
(65)
xk+1 = Axk + Gωk
(66)
max
υ0 ,...,υS
subject to yk = Cxk + υk
(67)
υk ∼ N
0 , R1 0
x0 ∼ N (x0 , Σ0 )
ωk ∼ N
0 ,Q 0
or N
0 , R2 0
(68) (69)
(70)
6
Y. Subasi, M. Demirekler / Automatica (
Fig. 1. Comparison of the two approaches (1: optimal solution of Gupta et al., 2006; 2: optimal solution of the proposed observability measure; 3: only sensor 1 is used; 4: only sensor 2 is used).
where Σ0 is a zero matrix. Vector quantization is used to quantize the covariance matrices to 1000 levels. In 1 we have compared the two different approaches using the cost of Gupta et al. (2006). As seen from the results, the optimal solution of Gupta et al. (2006) that is obtained by an exhaustive search is also achieved by the proposed observability measure.
6. Conclusions We have defined a new observability measure as the information gained about the state sequence from the measurement sequence. Several interesting properties of the observability measure are obtained for linear Gaussian systems. The examples demonstrate possible application areas of the measure. The observability definition is expandable to the non-Gaussian and the nonlinear systems. A Monte Carlo approach seems to be possible for a general nonlinear case.
)
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Yuksel Subasi received his B.S., M.Sc., and Ph.D. degrees in Electrical Engineering from the Middle East Technical University, Ankara/Turkey, in 1996, 1998, and 2012 respectively. He is currently employed in ASELSAN as a research engineer. His main area of interest is estimation theory and stochastic optimal control.
Mubeccel Demirekler received her Ph.D. degree in Electrical Engineering from Middle East Technical University, Turkey in 1979. She is currently a Professor at the same department. Her research interests are tracking systems (on which she conducted several projects), information fusion and stochastic optimization.