Observer design for a class of kinematic systems

Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007

WeB13.5

Observer design for a class of kinematic systems Pedro Batista, Carlos Silvestre, Paulo Oliveira

Abstract— An observer design methodology is introduced for a class of kinematic systems that often arise in the development of Navigation Systems for vehicular applications. At the core of the proposed methodology there is a time varying orthogonal coordinate transformation that renders the observer error dynamics linear time invariant (LTI). The problem is then formulated as a virtual control problem which is solved by resorting to the standard H∞ output feedback control synthesis technique, thus minimizing the L2 induced norm from a generalized disturbance input to a performance variable. The resulting observer error dynamics are globally exponentially stable (GES) and several input-to-state stability (ISS) properties are derived. A relevant example is provided that demonstrates the potential and usefulness of the proposed design methodology and simulation results are offered to illustrate the filter achievable performance in the presence of extreme environmental disturbances and realistic sensors’ noise.

I. I NTRODUCTION This paper addresses the design of observers for a class of dynamic systems with direct application to the estimation of linear quantities in Integrated Navigation Systems. The observer design technique here proposed is motivated by previous work that can be found in [1], where resorting to a time varying orthogonal coordinate transformation the resulting observer error dynamics become linear time invariant (LTI). Examples of application of the proposed observer can be foreseen in the design of accurate navigation and positioning systems for a great variety of mobile platforms. To tackle this class of problems several different approaches have been proposed in the literature. In [2] a passive globally exponentially stable (GES) observer for ships (in two-dimensions) that includes features such as wave filtering and bias estimation is presented and in [3] an extension to this result with adaptive wave filtering is available. An alternative filter was proposed in [4] where the problem of estimating the velocity and position of an autonomous vehicle in three-dimensions was solved by resorting to special bilinear time-varying complementary filters. A passivity based controller-observer This work was partially supported by Fundac¸a˜ o para a Ciˆencia e a Tecnologia (ISR/IST plurianual funding) through the POS Conhecimento Program that includes FEDER funds and by the projects PDCT/MAR/55609/2004 - RUMOS of the FCT and MEDIRES of the AdI. The work of P. Batista was supported by a PhD Student Scholarship from the POCTI Programme of FCT, SFRH/BD/24862/2005. The authors are with the Institute for Systems and Robotics, Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal.

{pbatista,cjs,pjcro}@isr.ist.utl.pt

1-4244-1498-9/07/$25.00 ©2007 IEEE.

design for n degrees of freedom robots is proposed in [5] and a sliding mode observer for robotic manipulators is reported in [6]. The development of nonlinear observers for EulerLagrange systems has been addressed in [7] and [8]. In these problems it is often common to have available to the design only a subset of the desired physical quantities. Moreover, these measurements are usually corrupted by sensors’noise and environmental disturbances such as the wind, sea waves, etc., that must be taken into account in the design phase. The particular problem of estimation of the linear quantities falls in the general framework presented in the paper, with direct application to the design of Integrated Navigation Systems. The devised observer error dynamics are GES and several input-to-state stability (ISS) properties are derived that demonstrate the robustness of the solution relative to disturbances on the key physical quantities. The proposed design technique minimizes the L2 induced norm from a generalized disturbance input to a performance variable, whereas the augmented observer error dynamics may include frequency weights to shape both the exogenous and the internal signals. The present paper is organized as follows: Section II introduces the class of dynamic systems and the estimation problem addressed in this work. In Section III the proposed observer design technique is presented and some properties are derived in Section IV. A motivating example that demonstrates the potential and usefulness of the proposed design methodology is offered in Section V and simulation results are included to illustrate the filter achievable performance. Finally, Section VI summarizes the main contributions of the paper. II. P ROBLEM S TATEMENT Consider the class of dynamic systems 8 η˙ 1 = f1 (t, ξ, ω, η 1 ) + γ1 η 2 > > > > > η˙ 2 = f2 (t, ξ, ω, η 1 ) + γ2 η 3 − S (ω) η 2 > > < ... , (1) > η˙ N −1 = fN −1 (t, ξ, ω, η 1 ) + γN −1 η N − S (ω) η N −1 > > > > η˙ = fN (t, ξ, ω, η 1 ) − S (ω) η N > > : N ψ = η1

where η i = η i (t) ∈ Xi ⊆ R3 , i = 1, . . . , N are the system states, ψ = ψ(t) is the system output, fi (.), i = 1, . . . , N are smooth functions of their arguments, ξ and ω are parameterizing vectors, possible time varying, i.e., ξ = ξ(t) and ω = ω(t), γi , i = 1, . . . , N − 1, represents nonzero scalar constants, and S (ω) is a skew-symmetric matrix that verifies S (a) b = a × b, with × denoting the cross product, and that satisfies R˙ = RS(ω), where R is a

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 rotation matrix. The time dependence of η, ψ, ξ, ω, and R will be omitted in the sequel for the sake of simplicity. The following assumption is made: Assumption 1: The values of R(t), ξ(t), and ω(t) are available to be used in the observer. Moreover, ω(t) is bounded for all t, i.e. ∃0 ˙ > ˆ N − S (ω) η ˆ N −1 − τ N −1 η ˆ = f N −1 (t, ξ, ω, ψ) + γN −1 η > N −1 > : ˙ ˆN − τ N η ˆ N = fN (t, ξ, ω, ψ) − S (ω) η (2) 8 > > > > > <

ˆ ) , i = 1, 2, . . . , N, are virtual where τ i = τ i (t, ξ, ω, ψ, η control variables that will be h used to stabilize iT the observer ˆ = η ˆ T1 η ˆ T2 . . . η ˆ TN . Notice that, error dynamics, with η ˆ 1 ), this apart from the output injection term S (ω) (ψ − η structure is an exact copy of the nominal system. The reasoning behind the introduction of this term will become clear in the paper. ˜ i = ηi − η ˆ i , i = 1, . . . , N , denote the state Let η estimation errors. Hence, from (1) and (2), it follows that the observer error dynamics can be written as 8 > ˜ 2 − S (ω) η ˜1 + τ 1 η ˜˙ 1 = γ1 η > > < ... , > ˜ N − S (ω) η ˜ N −1 + τ N −1 η ˜˙ N −1 = γN −1 η > > : η ˜N + τ N ˜˙ N = −S (ω) η

WeB13.5 where 2

6 6 6 6 Ap = 6 6 6 6 4

x˙ p = Ap xp + Bp T(t)τ ,

0 .. . .. .

...

...

... .. . .. . .. . ...

0 .. .

3

7 7 7 7 7, 7 0 7 7 γN −1 I 5 0

which is a linear time invariant system. Thus, with the coordinate transformation (3) the observer error dynamics are ˆ 1) rendered LTI. The introduction of the term S (ω) (ψ − η is now evident. Naturally, not all the error states are available for feedback. In fact, only x1 is accessible. Thus, to complete the observer error dynamics, define as output yp := Cp xp ,

(6)

where Cp = [I 0 . . . 0]. Notice now that the LTI system (5)-(6) is both controllable and observable. Therefore, any control design methodology for linear time invariant systems can be employed to stabilize the observer error dynamics, in particular the H∞ output feedback control synthesis. The employment of this design technique allows for the natural use of frequency weights to shape both the exogenous and the internal signals. To that purpose, consider Fig. 1, where the linear observer error dynamics are shown together with weight matrix transfer functions Wi (s), i = 1, . . . , 4. In  T  T the figure, w = w1T w2T and z = zT1 zT2 represent the generalized disturbance and performance vectors, respectively. Notice that the models for the disturbance inputs and sensor noise are not exact as they live in the transformed space. The same applies to the performance weights. w2

(4)

γ1 I .. .

 T Bp = I, and τ = τ T1 τ T2 . . . τ TN . Applying the same coordinate transformation to the virtual control input of the observer error dynamics, i.e., u = T(t)τ , allows to rewrite (4) as x˙ p = Ap xp + Bp u, (5)

which are inherently time varying. Next, this extra complexity is overcome through the use of an appropriate orthogonal time varying coordinate transformation. To that purpose,  T T T T define xp = x1 x2 . . . xN as

xp := T(t)˜ η, (3)   T T T ˜ TN ˜2 . . . η ˜ = η ˜1 η and T(t) is the coordinate where η transformation matrix defined as T(t) := diag (R, . . . , R) . Notice that (3) is a Lyapunov transformation [9] as • T(t) is continuous differentiable for all t; ˙ • Under Assumption 1 both T(t) and T(t) are bounded ˙ for all t, where T(t) = T(t)MS (ω), with MS (ω) := diag (S(ω), . . . , S(ω)) ; • det [T(t)] = 1. In the new coordinate space the observer error dynamics can be written as

0 .. . .. . .. . 0

w1

W2 (s)

W1 (s)

u

Bp

+

1 s

xp

yp Cp

+ W3 (s)

y z1

Ap W4 (s)

Fig. 1.

z2

Generalized Linear Observer Error Dynamics

 T Define x = xTp xTW 1 xTW 2 xTW 3 xTW 4 , where xW i , i = 1, . . . , 4, denote the states of the state space realizations of the frequency weights Wi , i = 1, . . . , 4. Then, the augmented plant can be written, in a compact form, as   x˙ = Ax + B1 w + B2 u z = C1 x + D12 u , (7)  y =C x+D w 2 21

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 where the definition of the various matrices is omitted as they are evident from the context. The standard design setup and nomenclature in [10] is adopted and it is assumed that the H∞ control problem is well-posed. Let Tzw (s) denote the closed-loop operator from the generalized disturbance vector w to the generalized performance vector z. Then, the solution of the H∞ control problem for the augmented plant (7) yields a stabilizing compensator  x˙ K = AK xK + BK y , (8) u = CK xK that minimizes kTzw (s)k∞ . Combining (2) with (8) finally yields the observer in the original coordinates 8 > ˆ˙ < η

> : x˙ K

= =

ˆ − MS (ω)ˆ f (t, ξ, ω, ψ) + Ap η η , +CTp S(ω)ψ − [T(t)]T CK xK ˆ1) AK xK + BK R (ψ − η

(9)

WeB13.5 Assumption 2: The function f (t, ξ, ω, ψ) is globally Lipschitz in (ξ, ψ), i.e., ∃0 > > :

x˙ K

=

=

“ ” ˜ ω, ψ − ψ ˜ f (t, ξ, ω, ψ) − f t, ξ − ξ, ˜ ˜ − MS (ω)˜ +Ap η η + CTp S(ω)ψ + [T(t)]T Bp CK xK ˜ AK xK + BK R˜ η 1 − BK Rψ

. (11)

Consider the Lyapunov-type function (10). Its time deriva˜ satisfies tive, under the presence of disturbances ξ˜ and ψ, ‚" #‚2 ‚ ‚ ˜ η ‚ ‚ ˙ V ≤ −‚ ‚ + LV ‚ xK ‚

‚" #‚ #‚ ‚" ‚ ‚ ξ˜ ‚ ‚ ˜ η ‚ ‚‚ ‚ ‚‚ ˜ ‚, ‚ ‚ ‚ xK ‚ ‚ ψ

(12)

with LV = 2 kP0 k [L1 + W + σmax (BK )], where σmax (.) denotes the maximum singular value of a matrix. Let 0 < θ < 1. Then, it is easy to show that V˙ verifies ‚" #‚ #‚2 ‚" ‚ ‚ ‚ ‚ ˜ ˜ η η LV ‚ ‚ ‚ ‚ V˙ ≤ − (1 − θ) ‚ ‚≥ ‚ ∀‚ ‚ xK ‚ ‚ xK ‚ θ

‚" ‚ ‚ ‚ ‚

ξ˜ ˜ ψ

#‚ ‚ ‚ ‚. ‚ (13)

Since, in addition to (13), it can be shown that V satisfies ‚" ‚" #‚2 #‚2 ‚ ‚ ‚ ‚ ˜ ˜ η η ‚ ‚ ‚ ‚ λmin (P0 ) ‚ ‚ , ‚ ≤ V ≤ λmax (P0 ) ‚ ‚ xK ‚ ‚ xK ‚

h T T iT ˜ it follows that the observer error is ISS from input ξ˜ ψ [12]. The additional presence of disturbances in ω and R is addressed in the following theorem. Assumption 2 must be strengthened, as to include the Lipschtiz condition in ω too. The new assumption is: Assumption 3: The function f (t, ξ, ω, ψ) is globally Lipschitz in (ξ, ψ, ω), uniformly in t. Theorem 3: Suppose that ξ, ψ, ω and R in (9) are ˜ ˜ replaced by disturbed variables ξhm = ξ− ξ, i ψ m = ψ − ψ, ˜ , where ξ, ˜ ψ, ˜ ˜ and Rm = R I − S λ ω m = ω − ω, ˜ are the disturbances, respectively. Then, under the ˜ and λ ω, conditions of Theorem 1 and Assumption 3, the observer h T T iT ˜ ,ω ˜T ˜T, λ error is locally ISS, with ξ˜ , ψ as input.

Proof: The proof follows the same steps of Theorem 2 and is therefore omitted. The difference resides in the fact that only local ISS is now achieved.

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Last, the optimality of the proposed observer design is addressed. To that purpose, consider the generalized observer error dynamics depicted in Fig. 2. The main difference between this generalized plant and the one of Fig. 1 is that: i) the generalized disturbances go through the transformation TT (t) and; ii) the generalized performance vector takes into account the system states and the control signal after the transformation T(t). In spite of these transformations, the magnitude of the signals is preserved - only the directionality  T T T ˜ xW 1 . . . xTW 4 . Notice is affected over time. Let χ = η that χ = TTc (t)x, with dynamics given by χ˙ = A(t)χ + B1 (t)w + B2 (t)τ . where A(t) = −



MS (ω) 0 0 0



+ TTc (t)ATc (t),

B1 (t) = TTc (t)B1 , and B2 (t) = TTc (t)B2 T(t). The performance vector ζ can be written as ζ = C 1 (t)χ + D 12 (t)τ , where C 1 (t) = C1 Tc (t) and D 12 (t) = D12 T(t). The generalized output is given by ψ = C 2 (t)χ + D 21 (t)w, where C 2 (t) = RT (t)C2 Tc (t) and D 21 (t) = RT (t)D21 . The following theorem addresses the optimality of the proposed solution. w2 w1

W2 W1

n

d

RT(t)

TT(t)

+

τ

Bp

+ +

− +

R

˜ η

Cp

ψp

ψ

+ +

Ap T (t)

W3

T (t)

W4

ζ1

S (ω)

Fig. 2.

ζ2

Generalized Observer Error Dynamics

Theorem 4: Under the conditions of Theorem 1, the proposed observer minimizes the L2 induced norm from w to ζ, assuming that w is a finite energy signal, i.e., w is square integrable. Proof: Suppose that w ∈ L2 , where L2 denotes the set of real-valued finite energy signals, and consider the closed-loop systems from w to z and from w to ζ. Let γ ∗ , associated to the control input τ ∗ (t), be the minimum γ that satisfies Z Z T 2 ζ (t)ζ(t)dt ≤ γ wT (t)w(t)dt, and γl∗ , associated with the control input u∗ (t), be the minimum γl that satisfies Z Z zT (t)z(t)dt ≤ γl2 wT (t)w(t)dt.

WeB13.5 Notice that u∗ (t) is the control law resulting from the H∞ synthesis. Choosing τ (t) = TT (t)u∗ (t), it is easy to show that Z Z T T ∗ ζ (t)ζ(t)dt ≤ γl w (t)w(t)dt . τ (t)=TT (t)u∗ (t) from which one concludes that γ ∗ ≤ γl∗ . On the other hand, choosing u(t) = T(t)τ ∗ (t), it is easy to show that Z Z T ∗ T z (t)z(t)dt ≤ γ w (t)w(t)dt . u(t)=T(t)τ ∗ (t) from which one concludes that γl∗ ≤ γ ∗ . Since γ ∗ ≤ γl∗ and γl∗ ≤ γ ∗ it must be γ ∗ = γl∗ with τ ∗ (t) = TT (t)u∗ (t). Thus, the proposed observer minimizes the L2 induced norm from w to ζ. It is important to remark that the observer structure was previously imposed and did not arise naturally from the solution of an optimization problem. Nevertheless, good performance can be achieved with the minimization of the L2 induced norm from w to ζ in the augmented error dynamics depicted in Fig. 2, as it will be clearly demonstrated in the next section. V. S IMULATION R ESULTS This section presents a case study of practical interest in marine applications that demonstrates the potential and usefulness of the proposed observer design methodology. This problem was first posed in [1]. Consider an Underwater Vehicle equipped with an acoustic positioning system like an Ultra Short Base Line (USBL) and suppose that there is a moored buoy in the mission scenario where an acoustic transponder is installed. The linear velocity kinematics of the vehicle can be written as p˙ = Rv, where p is the position of the origin of the body-fixed coordinate system {B} described in the inertial coordinate system {I}, R is the rotation matrix from {B} to {I}, that verifies R˙ = RS(ω), v is the linear velocity of the vehicle relative to {I}, expressed in body-fixed coordinates, and ω is the angular velocity, also expressed in body-fixed coordinates. Assume that the buoy where the transponder is installed is subject to wave action of known power spectral density that affects its position over time, and suppose that the position of the vehicle with respect to the transponder is available, in body-fixed coordinates as measured by an USBL sensor installed onboard. Suppose also that the body angular velocity ω and the rotation matrix R are available from an Attitude and Heading Reference System (AHRS). Finally, suppose that the vehicle is moving in deep waters (far from the wave action), in the presence of an ocean current of constant velocity, which expressed in body-fixed coordinates is represented by vc . The problem considered here is that of estimate the velocity of the current and the position of the vehicle with respect to the transponder. Further consider that the velocity of the vehicle relative to the water is available from the measures of an onboard Doppler velocity log. In shallow waters, this sensor can be employed to measure both the velocity of the vehicle relative to the inertial frame and relative to the water. However, when the vehicle is far from the bottom the inertial velocity is usually not available.

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007

the inertial frame, signal n in Fig. 1, to the position and current velocity estimate errors in the inertial frame, R˜ e and R˜ vc , respectively. The diagram shows that the performance requirements are met by the resultant closed loop system, which is evident from the band rejection characteristics of the notch filter present in both singular value diagrams. The 20 From noise to position error From noise to velocity error

0 −20

(dB)

−40 −60 −80 −100 −120 −3 10

−2

−1

10

Fig. 3.

0

10

1

10 Frequency (rad/s)

2

10

3

10

10

Singular values of the closed loop system

structure of the resulting observer is depicted in Fig. 4, where the H∞ output feedback compensator is of order 18. +

e vr

− −− − − −

+

ˆ e

R

−

S (ω)

R

ˆc v

S (ω)

Ck

TT (t)

R

+

Bk

+

R (t)

Ak H∞ Compensator

Fig. 4.

Position and current velocity observer structure

To illustrate the performance of the proposed observer a simulation was carried out with a simplified model of the SIRENE underwater vehicle [14]. In addition to the disturbances induced by ocean waves, which were confined to intervals of about 10 m of amplitude, and the disturbances of the USBL positioning sensor, in the simulation the measurements of the velocity of the vehicle relative to the water and the angular velocity were also assumed to be corrupted by Gaussian noise, with standard deviations of 0.01 m/s and 0.02 °/s, respectively. The time evolution of the observer estimates is presented in Fig. 5. The position of the buoy if there were no ocean waves is also shown, as well as the actual velocity of the fluid, all expressed in body-fixed coordinates. From these plots the performance of the observer is already evident - only the initial transients are noticeable. The evolution 800

2.5

x y z

x y z

2 Current Velocity (m/s)

600 Buoy Position (m)

By estimating the ocean current velocity, an estimate of the velocity of the vehicle relative to the inertial frame is immediately obtained. Let e denote the position of the transponder and vr denote the velocity of the vehicle relative to the fluid, both expressed in body-fixed coordinates. Since the position of the transponder is assumed constant (in the absence of environmental disturbances) in the inertial frame, the time derivative of e is given by e˙ = −vr − vc − S (ω) e. On the other hand, as the velocity of the fluid is assumed to be constant in the inertial frame, the time derivative of this quantity expressed in body-fixed coordinates is simply given by v˙ c = −S(ω)vc . Notice that the vehicle velocity relative to the inertial frame satisfies v = vr + vc . Clearly, the problem of estimating the velocity of the fluid, vc , falls into the class of problems addressed in the paper. To make it explicit, just consider the system (1) with η 1 = e, η 2 = vc , ξ = vr , f1 (t, ξ, ω, η 1 ) = −ξ − S(ω)η 1 , γ1 = −1, f2 (t, ξ, ω) = 0, and N = 2. Thus, it is possible to design an observer as detailed in Section III. Note that, in this case, the position of the transponder changes with time as the latter is assumed to be mounted in a buoy moored close to the sea surface, subject to strong wave action. The buoy wave induced random motion can be modeled as errors in the USBL positioning system expressed in the inertial frame, and their description embedded in the frequency weights presented in Section III. As closed loop design objective consider the rejection of the wave induced disturbances from the position measurements to the position and current velocity estimates. The disturbances induced by the three-dimensional wave random fields in the position of the buoy are modeled using three second-order harmonic oscillators representing the disturbance models along the x, y and z directions, [2], [13] σi s Hwi (s) = 2 2 , i = 1, 2, 3, s + 2ξi ω0i s + ω0i where ω0i is the dominating wave frequency along each axis, ξi is the relative damping ratio, and σi is a parameter related to the wave intensity. In the simulation the dominating wave frequency was set to ω0i = 0.8975rad/s and the relative damping ratio to ξi = 0.1. Thus, the sensor frequency weight matrix transfer function W2 (s) was chosen as   σi s I3 . W2 (s) = 5 1 + 2 2 s + 2ξi ω0i s + ω0i Notice that a direct term was added, not only to satisfy the requirements of the H∞ design but also to model the errors of the position sensor, which were assumed Gaussian with standard deviation of 1 m. As the observer nominal model, that corresponds to the kinematics of the linear motion, is exact and there is neither model uncertainty nor state disturbances the weight W1 (s) was set to W1 (s) = 0.01I6 . Using the fact that this is a pure disturbance rejection control problem, the performance weights were selected as W3 (s) = I6 . Finally, the virtual control input weights were chosen as W4 (s) = 2 (s + 1) / (s + 10) I6 to properly tune the input-output behavior of the closed loop system. Fig. 3 shows the singular values of the linear closed loop transfer functions from the position error measurements in

WeB13.5

400 200 0 −200

1.5 1 0.5 0 −0.5 −1 −1.5

−400

−2 −600 0

100

200 t (s)

Fig. 5.

300

−2.5 0

100

200

300

t (s)

Actual (dash-dot lines) and estimated (solid lines) variables

of the observer error is shown in Fig. 6. As the initial

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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 transients are of no interest - they arise due to the mismatch of the initial conditions of the states of the observer and can be considered as a warming up time of 180 s of the corresponding Integrated Navigation System - the observer error is shown in more detail in Fig. 7. From the various plots it can be seen that the disturbances induced by the waves, as well as the sensors’ noise, are highly attenuated by the observer, producing very accurate estimates of the velocity of the current and the position of the buoy. To conclude 20

1.5

Position Observer Error (m)

Current Velocity Observer Error (m/s)

x y z

15 10 5 0 −5 −10 −15 −20 0

50

100

Fig. 6.

150 t (s)

200

250

x y z

1 0.5 0 −0.5 −1 −1.5 −2 −2.5 0

300

50

100

150 t (s)

200

250

300

Time evolution of the observer error

WeB13.5 Integrated Navigation Systems. At the core of the proposed design technique there is a time varying orthogonal coordinate transformation that renders the observer error dynamics linear time invariant (LTI). The problem was then cast into a virtual control problem that was solved resorting to the standard H∞ output feedback controller design technique. The resulting observer error dynamics were shown to be globally exponentially stable (GES) and several input-to-state stability (ISS) properties were derived. The proposed design technique minimizes the L2 induced norm from a generalized disturbance input to a performance variable, whereas the augmented observer error dynamics may include frequency weights to shape both the exogenous and the internal signals. A case study of practical interest in marine applications was presented that demonstrates the potential and usefulness of the proposed observer design methodology. Simulation results were offered that illustrate the filter achievable performance in the presence of extreme environmental disturbances and realistic sensors’ noise. R EFERENCES

0.4

0.03

Position Observer Error (m)

0.3

Current Velocity Observer Error (m/s)

x y z

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 180

200

220

Fig. 7.

240 t (s)

260

280

300

x y z

0.02 0.01 0 −0.01 −0.02 −0.03 180

200

220

240 t (s)

260

280

300

Detailed evolution of the observer error

this case study, it should be noticed that, if the position of the transponder in the inertial frame, at rest, is known to the vehicle, then, an estimate of the actual position of the vehicle in the inertial frame is simply given by p = I (e) − Re, where I (e) is the position of the transponder expressed in the inertial frame. Fig. 8(a) depicts the actual and the estimated vehicle trajectory. For comparison purposes, the non-filtered position of the vehicle is plotted on Fig. 8(b). It is clear how accurate the observer estimates the trajectory described by the vehicle, even in the presence of severe wave action affecting the position of the buoy and realistic sensors’ noise.

Actual Trajectory Non−filtered Trajectory

300

300

200

200 z (m)

z (m)

Actual Trajectory Filtered Trajectory

100 0 −100 −100

200 100 0

0 −100 100 200 300 −200 y (m) x (m)

(a) Estimated trajectory Fig. 8.

100 0 −100 −100

0

200 100 0 −100 100 200 300 −200 y (m) x (m)

(b) Non-filtered trajectory

Vehicle trajectory

VI. C ONCLUSIONS This paper presented an observer design methodology for a class of kinematic systems with application to the design of

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