A New Observer for Perspective Vision Systems Under ... - IEEE Xplore

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 2, FEBRUARY 2015

503

A New Observer for Perspective Vision Systems Under Noisy Measurements Ileana Grave and Yu Tang Abstract—A simple design of observers for the range identification problem in perspective vision systems is given based on nonlinear contraction theory and synchronization. Exponential convergence to the object coordinates is achieved. In the presence of significant measurement noise, the performance is improved by synchronization among a group of observers. Index Terms—Contraction synchronization.

theory,

perspective

systems,

I. I NTRODUCTION In vision problems, the dynamics of an object moving in 3-D are described via its image projected in a plane by a perspective dynamical system. For applications such as robot control, surveillance and medical imaging, the unknown depth of the 3-D object must be estimated. Unknown depth estimation has been solved over the years with systems theory and/or with experiment-oriented techniques. Inspired by a parameter identifier, a high-gain observer named IBO (Identifier Based Observer) is given in [1], which concludes exponential convergence of the estimates. Using sliding mode method and adaptation techniques [2] designs a discontinuous observer, which in comparison with the IBO is simpler and has a better performance under noisy measurements. Based on Lyapunov analysis, a fourth-order observer is proposed in [3], whose design is less complex than the discontinuous one. Using the immersion and invariance methodology, a first-order observer with semi-global [4] or global [5] asymptotical convergence is given, however these observers are more sensitive to noise than those in [1]–[3]. For time-invariant linear perspective vision systems other observer forms, easy to handle in the observer design, are obtained in [6]. For practical applications, such as image based visual servoing, two observers built on Lyapunov analysis and nonlinear observer techniques are given in [7] for depth and focal length observation. An autonomous underwater vehicle is implemented in [8] with a global exponential convergent reduced-order observer, which provides an estimate even if the observability condition (in Assumption 2 below) is violated for some time instance. The aforementioned observers assume the dynamics with all the parameters known. The case where the depth and the structure are unknown has been solved with techniques such as stereo vision [9] and homography methods [10]. Recently, an observer which uses part of the linear velocity and acceleration to recover the object coordinates is developed in [11]. In all the cited references, clear vision measurements as well as clear motion parameters are assumed in the observer development. Noise in vision measurements and camera velocities is an important issue in Manuscript received June 11, 2013; revised November 19, 2013 and May 14, 2014; accepted June 2, 2014. Date of publication June 30, 2014; date of current version January 21, 2015. This work was supported by CONACyT under grant 129800 and by PAPIIT 116412. Recommended by Associate Editor P. Pepe. (Corresponding author: Yu Tang.) The authors are with the Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2014.2332692

Fig. 1. Schema of the perspective vision system.

perspective vision systems because if not treated properly, it may affect the performance of the vision system. In this technical note, motivated by metric reduced-order observers theory [12], a simple depth reduced-order observer is first proposed based on contraction theory [13]. Semi-global exponential convergence of the observer is established. Then to cope with the vision measurement noise, a complete-order observer is considered. This observer gives better estimates for a moderate level of noise. To further improve the performance in the presence of strong measurement noise, inspired by the interesting idea of protection from noise using synchronization [14], [15], a group of synchronized observers is derived. The problem of noise in the motion parameters is treated by increasing an order to the proposed observer designs. Contraction theory allows to develop observers for nonlinear systems using “top-down” design methodology [12]. In this methodology, intermediate variables, which may be not measurable, are first introduced giving a simple observer. Analysis is then taken over to guarantee exponential convergence of the observer states. The intermediate variables are substituted by the measurable outputs of the nonlinear system to give the final observer design. The rest of the work is organized as follows. In Section II the problem is formulated and the preliminaries on contraction theory and synchronization concepts are presented. The reduced-order, completeorder and the synchronized observers are derived in Sections III, IV and V, respectively. The modified version of these observers is given in Section VI. Numerical simulations are carried out in Section VII. Concluding remarks are drawn in Section VIII. II. P ROBLEM F ORMULATION AND C ONTRACTION T OOLS A. The Perspective Vision System Problem The dynamics of the perspective vision system are described as

x˙ 1 x˙ 2 x˙ 3

=

a11 a21 a31

a12 a22 a32

a13 a23 a33

x1 x2 x3

0018-9286 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

+

b1 b2 b3

,

(1a)

504


y = [y1 y2 ]T =

x x T 1 2 x3 x3

(1b)

where x ∈ R3 denotes the unmeasurable coordinates of the object in an inertial frame (see Fig. 1) and aij = aij (t), bi = bi (t), ∀i, j = 1, 2, 3, are the known motion parameters. The measurable output y ∈ R2 is the perspective projection of the object in the image plane whereas the focal length of the camera is given by . The objective in the perspective vision system problem is to reconstruct the object coordinates x from the vision measurements y. To achieve this aim, the following assumptions are made. Assumption 1 Perspective Vision System Properties: ([1], [4], [7], [8]). (i) aij , bi ∈ L∞ , ∀i, j = 1, 2, 3 and aij , bi are first-order differentiable. Upper bounds b3 , a3i on bi , a3i are known; (ii) x3 (t) > > 0, without loss of generality, we assume = 1; (iii) y(t) is bounded by known constants y ≤ y(t) ≤ y. Assumption 2 Observability of System (1) ([1]): (b1 − b3 y1 )2 + (b2 − b3 y2 )2 ≥ ω, ω > 0.

The state z = (1/x3 ) < 1/ is unmeasurable. Notice that if an estimate zˆ of z is available, then the object coordinates may be reconstructed through x ˆ1 =

y1 , zˆ

y2 , zˆ

x ˆ2 =

x ˆ3 =

1 . zˆ

(5)

To design the observer, the dynamics of z in (4) is rewritten as v˙ = f3 (y)(v + ζ) − b3 (v + ζ)2 − α(v + ζ) − K(b1 − b3 y1 )f1 (y) − K(b2 − b3 y2 )f2 (y) − γ, z =v + ζ with α, K to be given later. The terms ζ and γ are defined as

b3 b3 y1 y1 + K b2 − y2 y2 , 2 2 ˙b3 ˙b3 γ = γ(y, b) = K b˙ 1 − y1 y1 + K b˙ 2 − y2 y2 . 2 2 ζ = ζ(y, b) = K

b1 −

The following contraction and partial contraction concepts are used for the observers designs. Theorem 1 (Contraction [13]): Consider a nonlinear system

− K(b1 − b3 y1 )f1 (y) − K(b2 − b3 y2 )f2 (y) − γ,

(9b)

being the gain function α = α(y, b)

∂f ∂f + ∂x ∂x

T

(3)

then system (2) is contracting with convergence rate bounded by λ. Contraction implies that any two trajectories starting from different initial conditions converge exponentially to each other. Theorem 2 (Partial contraction [16]): Consider a smooth nonlinear system of the form x˙ = f (x, x, t) and assume that the auxiliary system y˙ = f (y, x, t) is contracting with respect to y. If a particular solution of the auxiliary y system verifies a smooth specific property1 , then all trajectories of the original x system verifies this property exponentially.

α = K (b1 − b3 y1 )2 + (b2 − b3 y2 )2 , K>

≤ −λI, ∀t ≥ 0

(9a)

(2)

where x ∈ Rn is the state vector and f an n × 1 vector function. If the symmetric part of the Jacobian matrix of system (2) is uniformly negative definite, i.e., for some λ > 0 1 2

(8)

vˆ˙ = f3 (y)(ˆ v + ζ) − b3 (ˆ v + ζ)2 − α(ˆ v + ζ)

zˆ = vˆ + ζ(y, b) x˙ = f (x, t)

Js =

(7)

Therefore, the reduced-order observer for system (4) is proposed as

B. Contraction Theory

(6a) (6b)

(10)

c(δz ) 2b3 , c(δz ) = + 2b3 δz + a31 y 1 + a32 y 2 + a33 (11) w

w is the observability condition given in Assumption 2 and δz > 0. The following gives the reduced-order observer convergence properties. Theorem 3 (Reduced-Order Observer): Consider the reduced order observer (9). Under Assumptions 1, 2 and |ˆ z (0) − z(0)| ≤ δz , for some δz > 0, then the estimation error zˆ(t) − z(t) will converge exponentially to zero. Proof: To start with the proof, allow the state z available and take as “virtual” observer z − b3 zˆ2 − α(ˆ z − z). zˆ˙ = f3 (y)ˆ

(12)

It is noted that system (12) has as a particular solution zˆ = z and its Jacobian is given by

III. R EDUCED O RDER O BSERVER

Jr = f3 (y) − 2b3 zˆ − α, ∀t ≥ 0.

System (1) can be rewritten as [1]–[4]

(13)

y˙ 1 = f1 (y) + (b1 − b3 y1 )z,

(4a)

By Assumptions 1 and 2 and condition (11), there exists a λr > 0 such that

y˙ 2 = f2 (y) + (b2 − b3 y2 )z,

(4b)

−Jr = α + 2b3 zˆ + a31 y1 + a32 y2 + a33 ≥ λr

z˙ = f3 (y)z − b3 z

2

(4c)

where f1 (y) = a13 + (a11 − a33 )y1 + a12 y2 − a31 y12 − a32 y1 y2 , f2 (y) = a23 + a21 y1 + (a22 − a33 )y2 − a31 y1 y2 − a32 y22 , f3 (y) = − (a31 y1 + a32 y2 + a33 ). 1 This property may be convergence to a manifold, to a particular trajectory or a relationship between state variables [16].

(14)

because |2b3 zˆ+a31 y1 +a32 y2 +a33 | ≤ |2b3 z+a31 y1 +a32 y2 +a33 | z − z)| + |2b3 (ˆ ≤ c(δz ). The term c(δz ) is given by (11). So, if zˆ(0) is initialized in |ˆ z (0) − z(0)| ≤ δz then Jr ≤ −λr . It follows from Theorem 1 that semiglobal exponential convergence of zˆ to z is guaranteed. Taking into account the equivalent expression of the z dynamics in (6), the virtual observer (12) can be implemented as (9).


Remark 1: The contraction theory provides a simple methodology to design a reduced-order observer, and guarantees semi-global exponential convergence of the solutions for depth estimation in perspective vision systems. Under Assumption 1, one can always choose z (0) − z(0)| ≤ δz in 0 < zˆ(0) ≤ δz − 1/ to fulfill the condition |ˆ Theorem 3. Remark 2: As the one presented in [8], the exponential convergence can be guaranteed if a weaker contraction condition ([17, Section 3.4, p. 16]) is used. This condition is related with the relaxed observability assumption in [8]. In relation to observers developed with the immersion and invariance methodology [4] and [5], the design derived here obtains exponential convergence property with a simpler design procedure instead of asymptotic convergence. In particular, the same observer form of [4] is achieved by setting ζ = β.

505

Therefore, Jc is uniformly negative definite and hence the observer (16) is contracting if J2 > gT J−1 1 g [16]. This condition is satisfied if v + ζ) − f3 (y)] > 1. 4K [α + 2b3 (ˆ

v + ζ) − f3 (y), it follows from (10) and Now, let Λ = α + 2b3 (ˆ Assumption 1 and 2 that

Λ = K (b1 − b3 y1 )2 + (b2 − b3 y2 )2

+ 2b3 (ˆ v + ζ) + a31 y1 + a32 y2 + a33 ≥ Kw + 2b3 (ˆ v + ζ) + a31 y1 + a32 y2 + a33

In the presence of noise in the vision measurements y, the ultimate goal of recovering the object coordinates x is hampered by using the reduced-order observer, because the estimated coordinates are calculated through (5). A complete-order observer will help to achieve ˆ if y ˆ is used in (5) instead of y. In better coordinate estimates x this Section, a complete-order observer based on contraction theory is designed. For this purpose, the dynamic equations of y in (4) are rewritten in terms of v and ζ as y˙ 1 = f1 (y) + (b1 − b3 y1 )(v + ζ),

(15a)

y˙ 2 = f2 (y) + (b2 − b3 y2 )(v + ζ)

(15b)

where ζ and γ are defined in (7) and (8), respectively. The complete-order observer is proposed as

(21)

and v + ζ) + a31 y1 + a32 y2 + a33 | ≤ |2b3 (ˆ v − v)| ≤ c(δz ). |2b3 z + a31 y1 + a32 y2 + a33 | + |2b3 (ˆ

IV. C OMPLETE - ORDER O BSERVER

(20)

(22)

v (0) − The term c(δz ) is given by (11). Since vˆ(0) is initialized in |ˆ v(0)| ≤ δz , observer (16) is contracting. Notice that (20) is implied by 4K[Kw−c(δz )]−1 > 0, whose solution with reference to K is (17). Contraction of the complete-order observer implies that vˆ → v and ˆ → y exponentially. By continuity ζ(ˆ y y, b) → ζ(y, b) and exponential convergence of zˆ to z is obtained, because zˆ − z = (ˆ v − v) + (ζ(ˆ y, b) − ζ(y, b)) → 0. Remark 3: The complete-order observer (18) is built from the ˆ dynamics. This observer reduced-order observer (9) by adding the y structure, corresponding to the choice of an appropriate virtual system [16], facilitates the contraction analysis. For practical implementation y is measured and corrupted by noise, one may initialize the observer ˆ (0) = y(0). It follows from the robustness property of conwith y tracting systems [13], that the distance between the trajectories of the corrupted system and those of (16) are bounded.

v + ζ) − α(ˆ y1 − y1 ), yˆ˙ 1 = f1 (y) + (b1 − b3 y1 )(ˆ

(16a)

V. S YNCHRONIZED O BSERVER

v + ζ) − α(ˆ y2 − y2 ), yˆ˙ 2 = f2 (y) + (b2 − b3 y2 )(ˆ

(16b)

In [14], it is explained analytically how coupled dynamic systems are protected from noise by synchronization. Protection from noise implies that the individual response of a group of identical noisy interconnected systems is close to the noise-free behavior. Based on this idea, a question arises: in the presence of noise can the individual observer behavior be improved by synchronizing a group of interconnected observers? A positive answer to this question is given in Appendix, Theorem 9 based on the results of [14], [15]. Following the last idea, several contracting complete-order observers with a coupling term among them are defined as:

v + ζ) − b3 (ˆ v + ζ)2 − α(ˆ v + ζ) vˆ˙ = f3 (y)(ˆ − K(b1 − b3 y1 )f1 (y) − K(b2 − b3 y2 )f2 (y) − γ (16c) with gain K given by K>

c(δz ) +

c2 (δz ) + w 2w

(17)

vi + ζi ) yˆ˙ 1,i = f1 (yi ) + (b1 − b3 y1,i )(ˆ

c(δz ) is defined in (11), and the estimate of z obtained through zˆ = vˆ + ζ(ˆ y, b).

(18)

Theorem 4 (Complete-order Observer): Consider the observer (16). Under Assumptions 1, 2 and for some δz > 0, the estimation error zˆ(t) − z(t) will converge exponentially to zero provided that y ≤ ˆ (0) ≤ y and |ˆ y v (0) − v(0)| ≤ δz . ˆ = y, vˆ = v and Proof: System (16) has as a particular solution y the negative symmetric part of its Jacobian −Jc is given by

−Jc =

J1 gT

g J2

(ˆ y1,i − yˆ1,j ),

(23a)

j

vi + ζi ) yˆ˙ 2,i = f2 (yi ) + (b2 − b3 y2,i )(ˆ y2,i − y2,i ) − Ks − αi (ˆ

(ˆ y2,i − yˆ2,j ),

(23b)

j

vi + ζi ) − b3 (ˆ vi + ζi )2 − αi (ˆ vi + ζi ) vˆ˙ i = f3 (yi )(ˆ − K(b1 − b3 y1,i )f1 (yi ) − K(b2 − b3 y2,i )f2 (yi )

with J1 = αI ∈ R2×2 , gT = −1/2[(b1 − b3 y1 )(b2 − b3 y2 )], I an identity matrix and v + ζ) + α. J2 = −f3 (y) + 2b3 (ˆ

y1,i − y1,i ) − Ks − αi (ˆ

(19)

− γ − Ks

(ˆ vi − vˆj )

(23c)

j

where j = 1, . . . , N , ζi = ζ(yi , b), αi = α(yi , b) and Ks > 0 is the synchronization gain.

506


Theorem 5 (Synchronized Observer): Under the same assumptions in Theorem 4 the mean value ¯= x

1 î x N

TABLE I RMS E RRORS OF O BSERVERS FOR t ∈ [1.3, 2] (sec)∗

(24)

i

ˆ i = [ˆ yiT vî ]T ∈ R3 , of the synchronized observer recovers the for x estimate of the object coordinates as in the noise free case as N → ∞. Proof: The contraction of the noise-free observer, corresponding the complete-order observer (16), is established in Theorem 4. Also, the assumptions made in Theorem 9 are verified for the perspective dynamics (1). The result follows directly from Theorem 9 by taking into account the leader dynamics (15) and synchronized observer (23). Remark 4: A comment on how to get the image coordinate of a feature point is in order. In perspective vision systems the coordinates of feature points are measured by a CCD camera. Usually, through segmentation and thresholding techniques [18], a set of vision coordinates yi (pixels) of a feature point is obtained, then the center of mass or centroid of this set is calculated and used as the image coordinates y (for reduced and complete-order observers). For the synchronized observer, where different measurements are required, the set of (N ) vision coordinates yi is fed into the group of N synchronized observers directly without the mass center calculation procedure. VI. O BSERVERS U NDER N OISY M EASUREMENTS In the presence of noise in the vision measurements and in the motion parameters, the estimated state zˆ is corrupted directly by both sources of noise in (9b) (reduced-order) or by noise in motion parameters in (18) (complete-order, synchronized). To prevent these measurement noise from propagating directly to the depth estimate, the proposed observers are modified: instead of ζ in (9b) or (18), its filtered version is used. Toward this end, the dynamics of ζ is obtained by deriving (7) and expressing ζ˙ in terms of v˙ and z˙ as ζ˙ = α(v+ζ)+K(b1 −b3 y1 )f1 (y)+K(b2 − b3 y2 )f2 (y) + γ. (25) The modified reduced-order observer follows from (6) and (25) ˆ v + ζ) − b3 (ˆ v + ζ)2 − α(ˆ v + ζ) vˆ˙ = f3 (y)(ˆ

(26a)

− K(b1 − b3 y1 )f1 (y) − K(b2 − b3 y2 )f2 (y) − γ, ˙ ζˆ = α(ˆ v + ζ) + K(b1 − b3 y1 )f1 (y) + K(b2 − b3 y2 )f2 (y) + γ − Kz (ζˆ − ζ)

(26b)

where Kz > 0, α, K are given by (10), (11), respectively, and the depth estimation is given by ˆ zˆ = vˆ + ζ.

(27)

Theorem 6 (Modified Reduced-order Observer): Consider (26) as an observer for system (4) and (25). Under Assumptions 1, 2 and for some δz > 0 the estimation error zˆ(t) − z(t) of the modified reduced-order observer will converge exponentially to zero provided |ˆ v (0) − v(0)| ≤ δz . Proof: System (26) has as a particular solution vˆ = v, ζˆ = ζ and the negative symmetric part of its Jacobian −Jmr is given by a diagonal matrix with terms −Jr , given as (19), and Kz . Jmr is uniformly negative definite under Assumptions 1, 2 and |ˆ v (0) − v(0)| ≤ δz for some δz > 0. Then, the modified reduced-order observer is contracting and semi-global exponential convergence of vˆ to v and ζˆ to ζ is guaranteed. Based on the same idea, the complete-order and synchronized observers are modified to obtain a cleaner estimation of x. These results are given in the following theorems, their proofs are similar to the proof of Theorem 6 and omitted due to space limit.

TABLE II RMS E RRORS OF O BSERVERS FOR t ∈ [1.3, 2] (sec)∗∗

Theorem 7 (Modified Complete-order Observer): Consider (16a), (16b) and (26) with K given by (17) and Kz > 0. Under Assumptions 1, 2 and for some δz > 0, the estimation error zˆ(t) − ˆ (0) ≤ y z(t) will converge exponentially to zero provided that y ≤ y and |ˆ v (0) − v(0)| ≤ δz . Theorem 8 (Modified Synchronized Observer): Consider the modified synchronized observer, consisting of a set of modified completeˆ j ), where ui − u order observers plus the coupling term −Ks j (ˆ ˆ i = [ˆ yi vî ζî ]T , j = 1, . . . , N and ζi = ζ(yi , b). Under the same u assumptions in Theorem 7, the mean value of the synchronized observers recovers the estimate of the object coordinates as in the noise free case as N → ∞. VII. S IMULATION The performance of the observers is tested for the measurement noise case by numerical simulations. Similar to [8], the perspective vision system (1) terms are taken as zero except for a1,3 = −π/30 (rad/s), a3,1 = −a1,3 and b = [0.3 0.4 + 0.1 sin(πt/4) − 0.3] (m/s), the initial conditions are chosen as x(0) = [10 5 2]T (m). For all the observers, White Gaussian noise (WGN) with signalto-noise ratio (SNR) of 20 dB is added to the output y. Aditionally to noise in the vision measurements, the modified versions of the observers in Section VI are implemented with WGN of an intensity 2.5% added in the motion parameters. The initial conditions are taken ˆ ˆ (0) = y(0) and ζ(0) as zˆ(0) = 0.9, y = ζ(y(0), b(0)), where y(0), b(0) are corrupted measurements. The gain was selected as K = 5 for (observers (9a), (16), (23)) and for K = 1.3, Kz = 20 in the modified versions. N = 50 synchronized observers are simulated with Ks = 750. To emulate the real situation where noise in the vision measurement is present (see Remark 4), the input yi and the initial conditions ˆ ˆ i (0) = y ˆ (0), vî (0) = vˆ(0), ζî (0) = ζ(0) y are formed by the signal y corrupted by N different WGN sources. For comparison purposes ¯ (24) is taken as output. with the non-synchronized observers, x To compare the results obtained in this work with the existing designs in literature, the discontinuous observer in [2] was chosen for its better performance under noise measurements than other relevant designs. The reduced (9a), complete (16), synchronized (23) and discontinuous observers root mean square (RMS) estimation error is enlisted in Table I. To emphasize the noise reduction effect, the RMS error is taken over in almost steady-state t ∈ [1.3, 2] (sec). For the presence of noise in the vision measurements and in the motion parameters, the modified version of the observers in Section VI are implemented. The RMS of the estimation error is shown in Table II. The object ˆ3 are depicted in Fig. 2. coordinates x and observation error x3 − x


507

Note that large RMS errors in Table I, for the reduced order observer (9a), are given by the object coordinates calculation in (5) and the dependency of ζ (see (7)) on the observer gain (measurement noise amplification). These errors are significantly reduced in the completeorder observer (16) by shrinking the effect from vision measurement and in the modified reduced-order observer by using ζˆ in (27). Also, notice that further noise reduction is achieved with the modified complete order-observer compared to the modified reduced-order one, though the reduction is less significant as in Table I because the noise in both vision and motion parameters affects the gain α in (10) used in the modified complete-order observer. The estimation error of the discontinuous observer presents a similar performance as the complete-order ones, while the synchronized observers exhibit a further reduction of noise.

VIII. C ONCLUSION This note is a contribution in the design of observers for the perspective vision problem in the presence of measurement noise based on systems theory, in particular contraction theory. This theory allows to derive a set of depth estimators with a simple procedure. The reduced-order observer, as those reported in [4] and [8] was first developed with different tools and served as a building-block to design a complete-order and synchronized observers to cope with noisy vision measurements. When the motion parameters (angular and linear velocity measurements) are also noisy, modified versions of the above observers are designed. Analysis and simulations are provided to show the reduction of noise in the depth estimate. A comparison with the discontinuous observer of [2] shows a good performance of the synchronized observer. An experimental implementation of the observers is necessary to further validate the proposed designs. A PPENDIX Consider a system called “the leader” dy = f1 (y, v)dt, dv = f2 (y, v)dt,

(28) y i = y + σw wi

where yi denotes the i − th measurement of the output y corrupted by a “white” noise wi with intensity σw , i = 1, . . . , N . y ∈ Rm and v ∈ Rp , p ≤ m, represent the measurable and unmeasurable states, respectively. Define an observer, ‘the i-follower’, in the form ˆ i ) − K(t)(ˆ ˆ j )) dt, yi − yi ) − ui (ˆ yi , y dˆ yi = (f1 (yi , v ˆ i ) − ui (ˆ ˆ j )) dt dˆ vi = (f2 (yi , v vi , v

(29)

ˆ j ) = Ks j (ˆ ˆ j ) represents an “all-to-all” couxi , x xi − x where ui (ˆ pling term among the elements in the network, 0 < K(t) ∈ Rm×m a diagonal matrix with positive elements, and Ks > 0. Theorem 9: Assume the noise-free observer ˆ ) − K(t)(ˆ y − y)) dt, dˆ y = (f1 (y, v Fig. 2. The modified complete-order observer is given in green (dashed¯ , in blue (solid) and the discontinuous in dot), the modified synchronized x red (dashed). WGN with SNR of 20 dB is added to y and 2.5% of noise is added to velocity measurements aij , bi ; (a) Coordinate x1 ; (b) Coordinate x2 ; (c) Coordinate x3 (d) Depth estimation error x3 − x ˆ3 .

ˆ )) dt dˆ v = (f2 (y, v

(30)

is contracting, i.e. for some λf > 0, the symmetric part of its Jacobian Js ≤ −λf I uniformly. Let x = [yT vT ]T , where the

508


¯ in (24), and x ˆ , the estimate of x, are similarly average dynamics x defined as x. Then after an exponential transient 2 2 (k2 + l2 )σ 2 + lF2 )σw N (N −1) Hbd (kmax ˆ (t) ≤ + max F w ¯ x(t)− x 4N 2 λf +Ks N N (31) 2

where Hbd ≥ λmax (∂ 2 f /∂x2 ) is a uniform upper bound of ˆ) = (∂ 2 f /∂x2 ), lF > 0 is the Lipschitz constant of f (y, v 2 ˆ ) f2T (y, v ˆ )]T and supt≥0 tr{KT (t)K(t)} ≤ kmax . [f1T (y, v Proof: System (29) can be written as ˆ i ) − K(t)(ˆ ˆ j )) dt + i,1 + wi , yi − yi ) − ui (ˆ yi , y dˆ yi = (f1 (y, v ˆ i ) − ui (ˆ ˆ j )) dt + i,2 vi , v dˆ vi = (f2 (y, v

(32)

ˆ i ) − fn (y, v ˆ i ))dt, n = 1, 2, and with i,n = (fn (yi , v K(t)σw dwi . The derivative of (24), using (29), is given by

wi =

¯ ) − K(t)(¯ d¯ y = (f1 (y, v y − y)) dt + ¯1 + ¯w + δ1 , ¯ )) dt + ¯2 + δ2 d¯ v = (f2 (y, v

(33)

¯ ))dt, ¯n = ((1/N ) i i,n ), where δn = ((1/N ) i fn (y,ˆ vi ))−fn (y, v and ¯w = ((1/N ) i wi ). It is noted that if the terms δn , ¯w and ¯n for n = 1, 2 are made “small”, the behavior of the mean value of the synchronized observers will be close to the noise-free case. The expected sum of ¯w and ¯ = [¯ 1 ¯2 ]T is bounded [14] by

w 2 ≤ E ¯

2 2 ρHbd + lF2 ) σw (kmax , E (¯ ) = N 2N 2

(34)

where ρ represents a bound on the expected sum of the difference xi − between the states of the synchronized elements: E( i