Analysis of a distributed estimation and control scheme for formation flying spacecraft

Aerospace Science and Technology 73 (2018) 232–238

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Analysis of a distributed estimation and control scheme for formation flying spacecraft Thanh Vu a , Amir Rahmani b a b

University of Miami, Coral Gables, FL 33146, USA Jet Propulsion Laboratory, Pasadena, CA 91109, USA

a r t i c l e

i n f o

Article history: Received 24 October 2016 Received in revised form 4 September 2017 Accepted 18 October 2017 Available online 31 October 2017 Keywords: Distributed estimation Distributed control Formation flying spacecraft

a b s t r a c t The stability characteristics of a distributed consensus-based Kalman filter estimation and control scheme are studied through analytic and numerical means. This estimation scheme seeks to minimally reduce the necessary bandwidth for communication while maintaining overall stability. A weaker form of the separation principle is proven to hold whereby control could be designed independently but not estimation. However, actuation limitations still provide the possibility for a semi-independent design of estimators. Numerical simulations confirm that the stability depends very heavily on consensus on estimation and ultimately the amount of information available to the system as it evolves. © 2017 Elsevier Masson SAS. All rights reserved.

1. Introduction Formation flying satellites are becoming a major new development in space operations. The commercial, scientific, and military sectors all wish to expand the mission capabilities of their satellite fleets such as increased communication volume for information transfers, increased field of surveillance, and improved navigational accuracy for military and civilian aircraft [1]. While current distributed missions, such as GPS, are conducted in satellite constellations, these constellations do not have a coupled control law that takes into account the states of other satellites. Therefore, formation flying would be able to perform missions with more stringent requirements on formation positions. With limited funds, implementing formation flying into existing satellite technology is a cost-effective way to extract more utility. Formation flying capabilities increases not only the scope of satellite missions but also the reliability. In addition to performing synchronous measurements, formations have redundancies in operation, which means failure of one spacecraft would not endanger the integrity of the mission [2]. A notable but novel use of formation flying can be seen in the distributed aperture telescope system [3]. In light of current restrictions on the cost and logistics of sending large telescopes for scientific observations, formation flying spacecraft could be used to circumvent this problem. Each of the satellites within the formation would act as a section of a larger reflecting telescope, and the formation, as a whole, would become a “virtual telescope” with

E-mail address: [email protected] (T. Vu). https://doi.org/10.1016/j.ast.2017.10.028 1270-9638/© 2017 Elsevier Masson SAS. All rights reserved.

an aperture several times larger than their conventional reflecting counterparts [4]. This would give astronomers access to better clarity and resolution compared to individual telescopes. Lastly, the robustness of architecture would avert a incident similar to the Hubble telescope, which had a manufacturing defect in its lens and had to be repaired in space. Smaller mirrors would be more costefficient to manufacture, and satellites with defective instruments can be replaced and substituted fairly easily. This application is notable for its tight requirement on the shape of the formation in order to achieve satisfactory resolution. Thus, accurate estimation of the global formation structure is paramount to the success of such a mission. Distributed space systems are seen to be the successor to current monolithic systems that are too cumbersome to organize. Through the comprehensive survey on guidance and control techniques for formation flying spacecraft by Scharf et al., one can see that a central-control framework lacks robustness to changes within the formations [5,6]. Distributed systems also allow for additional autonomy in conducting missions as they can reduce the reliance on receiving instructions from a ground station [2]. However, the fundamental challenge to the implementation of distributed systems is achieving a desired global outcome from isolated, local interactions. Tillerson et al. [7] investigates the effectiveness of an LP controller with regards to different methods of localizing a formation of satellites; however in order to implement distributed control, the control input to each spacecraft would have to be published through a fully-connected network. From an estimation perspective, Olfati-Saber proposed using a distributed

T. Vu, A. Rahmani / Aerospace Science and Technology 73 (2018) 232–238

233

Nomenclature Note that units for all variables depends on designer’s choice of state and control variables n number of agents state of agent i xi control input to agent i ui ˜ A dynamics matrix of environment control matrix of agent i Bi process noise of agent i vi X global state vector U global control input V global process noise measurement vector of agent i zi measurement matrix of agent i Hi measurement noise of agent i Wi A global dynamics matrix

Kalman filter scheme that incorporates a consensus algorithm so that a group of observers can, collectively, estimate and agree on the states of a process [8]. Olfati-Saber and Jalalkamali furthered these results with moving observers who estimate a moving target [9]. Ranzer provided insight into using multiple controllers, accessing different measurements to control a distributed system and proved a separation principle for these cases [10]. Smith and Hadaegh sketched a formation controller that uses parallel estimators, however the full formation states have to be observable by every estimator [11]. Building on all of this, Rahmani et al. presented a distributed estimation and control architecture in which each spacecraft would generate its own estimates of both the states and controls of the entire formation. These estimates are derived from both its sensors and the information transmitted by neighboring spacecraft [12]. Thus through local interactions, each spacecraft can build its own image of what the entire formation is doing and respond as required. From a design perspective, one of the most powerful theorems from linear control has been the separation principle, which states that it is possible to combine independently designed controller and estimators together to form a stable system. For distributed systems, with each spacecraft performing local communication, the fundamental issue is: Is there a generalization for the separation principle for distributed systems? If so, what form will it take? Unfortunately it will be shown that the separation principle strictly does not apply to these distributed systems. However, by relying on engineering constraints and a weaker formulation of the separation principle, designers would still be able to design local estimators while ensuring stability.

B H K xˆ i Ki Pi Ri Q

γ Ni i Uˆ i

ηi η

global input matrix global measurement matrix state feedback gain global state estimate from agent i Kalman gain of agent i covariance of state error perceived by agent i covariance of measurement noise perceived by agent i covariance of process noise consensus coefficient neighborhood set of agent i control selection matrix of agent i global control input perceived by agent i global estimate error of agent i concatenation of all estimate error

Fig. 1. Sketch of distributed estimation and control.

A sketch of the control scheme, along with its inter-dependencies, is summarized in Fig. 1. The major concept is that each spacecraft is not only estimating its own states, but also the states, controls, and formation assignment of the other spacecraft. Using its estimate of the formation control and orientation, it would then implement its own control and select its own goal, respectively. Communication of information enables estimation of states even if no spacecraft can observe the entire formation. With this framework in mind, the following sections will implement the structure with respect to the linear dynamics on the spacecraft formation. 2.1. Dynamics Consider first, n spacecraft, each under linear dynamics influenced by an exogenous, zero mean white Gaussian noise, v i .

2. Formulation

x˙ i = A i xi + B i u i + v i

The problem considered is to determine how would the separation principle hold in the context of distributed systems. First, a framework is needed to consider the states of such a system. For this analysis, a spacecraft flying formation is considered, but the framework would be valid for any linear system. The dynamics of a distributed system can be formed by first considering the dynamics of a single spacecraft and aggregating their respective states to form the state of the entire formation. Again, we will only consider linear dynamics for each spacecraft. While at first glance this might look restrictive, in practice dynamics of most planned formation flying missions can be represented by linearization around an operation point of interest, like the Clohessy–Wiltshire–Hills equations for relative orbital dynamics [13]. Since spacecraft formation operate in relatively close proximity, the use of linearized dynamics is justified.

The states of each spacecraft, xi , can be concatenated to form an aggregated state vector, X = [x1T · · · xnT ] T . Similarly, the control input and noise disturbances can also be aggregated as U = [u 1T · · · unT ] T and V = [ v 1T · · · v nT ] T respectively. Under this formulation, the state and control matrix can be written in a block diagonal form: A = diag( A 1 , · · · , A n ) and B = diag( B 1 , · · · , B n ) respectively. These aggregated dynamics can now be written in a form analogous to equation (1). Note that while the use of different individual state matrices A i keeps this analysis general for diverse, heterogeneous systems, for homogeneous systems within the same environment, the state matrix would be identical.

X˙ = A X + BU + V

(1)

(2)

We assume, each spacecraft i can measure a subset of these aggregated states zi , usually their own and a few select states of their

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neighbors, or a linear combination of them via a measurement matrix H i , with an additive W i , a zero mean Gaussian measurement noise. An important case of this would be measurement models based on inter-spacecraft distances, such as those employed by Vetrisano and Vasile [14]. Thus, an aggregated system of measurements can be found.

zi = H i X + W i

(3)

It should be noted that it has not been required that ( A , H i ) be observable, which allows the possibility of the whole system not being observable by one spacecraft. Instead, a weaker condition is assumed where each state could be constructed by at least one spacecraft. In technical terms, this means that ( A , H ) is observable where H = [ H 1T · · · H nT ] T .

2.3. Distributed control Since each spacecraft would have to obtain its own controls from its respective estimate, we can define i as the selection function of its estimate inputs. This selection matrix is a block diagonal matrix whose blocks consist of zero matrices and one identity matrix. These block matrices are square matrices whose length is the size of the control input to each spacecraft, and the identity matrix is situated on the position corresponding to spacecraft i. Now it is possible to write out actual control input to the entire formation.

i = diag (0, 0, . . . I . . . 0)   U (t ) = i Uˆ i = i K Xˆ i i

2.2. Distributed estimation We would like to employ state feedback to ensure global stability of the network, either by LQR or pole placement. However, both methodologies require that K , the state feedback gain, is calculated by assuming full knowledge of the entire formation states.

U ideal = K X

(4)

Since complete knowledge of entire system states in a distributed setting is not possible, we assume each spacecraft can measure some of its own and neighboring spacecraft’s states represented by zi = H i X . Provided the spacecraft can locally communicate and share their estimate of the entire formation state Xˆ i with their immediate neighbors (sensing and communication neighborhoods do not necessarily need to be the same), each spacecraft can run its local version of a distributed consensus-based Kalman filter to estimate the state of the entire network. Olfati-Saber show that such a filter will converge to the same estimate in the absence of any external control. However, the problem is that the dynamics of the entire formation depends on knowledge of actual control U implemented by all spacecraft. Since each spacecraft implements its own control locally and does not communicate this value with the entire formation in order to save bandwidth, we use the following distributed estimation and control proposed by [12]. Therefore, it is only necessary to perform consensus on the state estimates and does not need to implement consensus again on the control input for the entire formation. The distributed Kalman filter in presence of locally constructed controls is as follows:

 ˙ Xˆ i = ( A + B K ) Xˆ i + K i ( zi − H i Xˆ i ) + γ P i ( Xˆ j − Xˆ i ),

(5)

j ∈Ni

Ki =

P i H iT

−1

Ri ,

P˙ i = A P i + P i A T + Q − K i R i K iT

(6)

(8) (9)

i

Essentially, each spacecraft i holds a local estimate of what every spacecraft control action should be, i.e. Uˆ i = K Xˆ i . The selection function i only selects the control pertaining to spacecraft i and sets everything else to zero. This is the actual control implemented by spacecraft i. Hence, the actual control implemented by the entire formation is the concatenation of these locally constructed controls. This is achieved via the summation sign since the selection matrix, i , returns values of spacecraft i’s control at its corresponding locations and zero at every other location. To show the convergence of each spacecraft’s estimation to the global state, one can define a spacecraft’s error vector. Subsequently the system dynamics can be derived in terms  of state variables and the state error by exploiting the fact that i = I . i

ηi = X − Xˆ i

(10)

X˙ = A X + BU X˙ = A X + B



i K Xˆ i

i

X˙ = A X + B



i K ( X − ηi )

i

X˙ = ( A + B K ) X −



B i K ηi

(11)

i

Now it is possible to write the error dynamics in terms of the state error and simplify the consensus term since X j − X i = ηi − η j .

η˙ i = ( A + B K − K i H i )ηi + γ P i −





(η j − ηi )

j ∈Ni

B j K η j

(12)

j

(7)

where P i , Q , and R i are the symmetric covariance matrices of the error, process and measurement noise, respectively. Every term except for the third term in equation (5) is the standard local Kalman filter, with the addition of the extra term, which is the consensus algorithm. This term is weighted by both a scalar constant, γ , and the error covariance, P i . The consensus, γ , is the relative weighting factor for the entire consensus contribution to the estimate, while P i is weighing each state by its uncertainty. This means each spacecraft would put more “trust” in incoming data if its own estimate is more uncertain. Naturally, for states that it is not observing, then it would need to rely on incoming data to achieve an accurate estimate. Therefore, it is not necessary for any single spacecraft to observe the entire formation, and it is only required that ( A , H ) be a fully observable pair, i.e. any state is observable by at least one spacecraft.

where Ni is the communication neighborhood set of spacecraft i. 3. Convergence conditions Since we wish to consider the validity of the separation principle to distributed systems, then it is necessary both for the estimates to converge onto the global formation and for the formation itself to converge onto the desired formation simultaneously. With this in mind, we consider a Lyapunov function of the form:

V (t ) =



ηiT P i−1 ηi + X T X

(13)

i

By construction, this Lyapunov function is positive definite and can be zero if and only if the error vector of every spacecraft and the state vector are identically zero.

T. Vu, A. Rahmani / Aerospace Science and Technology 73 (2018) 232–238

V˙ =

 (η˙ iT P i−1 ηi + ηi P i−1 P˙ P i−1 ηi + ηiT P i−1 η˙ i ) + 2 X T X˙ i



V˙ = −

i

+2



⎡

⎛

⎣η T P −1 ⎝ B K η i − i

i



η

T i

η j − ηi

B  j K η j ⎠⎦





+ 2X T (A + B K )X − 2X T

B i K ηi

(15)

i

Next we define the following relations for concise notation:



η = η1T η2T . . . ηnT

T

η¯ = [ X T η T ]T 1 i = H iT R − H i + P i−1 Q P i−1 i

M1 = diag(1 , 2 , . . . , i ) Using this notation, then it is possible to assemble block matrices for an even simpler representation:

−







1 −1 −1 ηiT H iT R − ηi + 2 X T ( A + B K ) X i Hi + Pi Q Pi

i

⎡

⎢ ⎢ ⎢ = −η¯ T ⎢ ⎢ ⎣  = −η¯ T

−2( A + B K )

0

1 2 .. 0

−2( A + B K ) 0

By definition, 2γ

0

i

j ∈N i

.



ηiT P i−1

i

⎡



... ...

P n−1 B 1 K

P n−1 B 2 K

...

= −2η T ⎢ ⎣

.. .



2

 i

⎢ ⎢ ⎣

ηiT P i−1 B K ηi = 2η T ⎢

T

⎤

P 1−1 B n K P 2−1 B n K ⎥ ⎥

.. .

⎥η ⎦

P n−1 B n K

B n K

...



η (18)

⎤ ⎥ ⎥ ⎥η ⎦

P 2−1 B K

..

. P n−1 B K

= 2η T M3 η

By assuming that ( A , B ) is controllable, K was constructed to ensure that the closed loop system is stable, then A + B K ≺ 0, and since every i ≺ 0, then M1  0. Since the eigenvalues of each P i are bounded from below by the Cramer–Rao bounds [15], which are nonzero because of the process noise Q , then P i−1 is guaranteed to have bounded eigenvalues. Since L is assumed to represent a connected graph, it is positive semi-definite, and its nullspace is within span{1}. Therefore, one can select a sufficiently large γ such that S  0. Intuitively this means that consensus must be reached first so that each agent will have a reliable estimate of total control input. Once this occurs, then each agent would be running an estimation scheme as if there is a singular global Kalman filter. Thus, the system is stable if the following assumptions are satisfied:

4. Design considerations

= −2 X T B¯ η P 1−1 B K

(23)

(27)

i

⎡

S = M1 + 2M2 − 2M3 + 2γ L  0

∃γ : M1 + 2M2 − 2M3 + 2γ L  0

= −2γ η Lη where L

B 2

(22)

(26)

(17)

 i K ηi = −2 X T B 1 K

−2( A + B K )  0

nul L = span{1}

= −2η T M2 η −2 X T B

(21)

(25)

BKηj P 1−1 B 2 K P 2−1 B 2 K

(20)

( A , H ) is observable

j

P 1−1 B 1 K ⎢ P −1 B 1 K ⎢ 2

η¯

⎥ ⎥ ⎥ ⎥ η¯ ⎥ ⎦

(16)





(24)

is the Laplacian of the graph. Now let us consider the following terms:

−2

0

2B¯ M1 + 2M2 − 2M3 + 2γ L

( A , B ) is controllable

η¯

ηi η j − ηi

−2( A + B K )

⎤

n



M1    T



For this system to be stable, then H  0. Since H is block triagonal, this must imply that blocks on the diagonal are positive definite:

⎞⎤

j

j ∈Ni

i



V˙ = −η¯ T

= −η¯T Hη¯

1 ηiT ( H iT R − H i + P i−1 Q P i−1 )ηi i

i

+ 2γ

(14)

235

(19)

We can combine the previous equations to show that the time derivative of the candidate Lyapunov function be characterize by a single block matrix:

Since the design of γ is influenced by the values of K , strictly speaking the separation principle cannot be applied. Fortunately, there is only a unidirectional dependence, as the controller can to be designed first without considering the estimator. Only the estimator depends on the controller gains. Mathematically, this can be seen in the derivation of the differential Riccati equation by Kalman and Bucy, [16] which assumes that the control input is explicitly known at all times, and has no effect in the evolution of the error. Therefore, once there is a consensus in estimation, each agent knows exactly what the global input is to the system. However, practically speaking, in most applications, there are physical or design constraints on the gain values, therefore initial design consideration could assume the worst case bound on the gain values and then optimize as the design progresses. In terms of selecting an appropriate γ , the last stability condition is rather cumbersome to use in designing a system because S has an implicit time dependence due to P i−1 . Nevertheless, one

¯ 2 and M ¯ 3 whose eigenvalues would be can construct matrices M a worst case upper bound on the their respective counterparts. Using these matrices, γ could be designed offline and also ensure stability for all time. As mentioned earlier, the eigenvalues of P i are bounded from below by the Cramer–Rao bounds [15], therefore the eigenvalues from the Fisher information matrix, J , are the ¯ 2 and M ¯3 upper bound for the eigenvalues of P i−1 . Therefore, M can be constructed by replacing each P i−1 by J :

⎡

J B 1 K ⎢ J B 1 K ⎢

¯ 2=⎢ M ⎣

.. . J B 1 K

J B 2 K J B 2 K

... ...

J B 2 K

...

⎤

J B n K J B n K ⎥ ⎥

⎥ .. ⎦ . J B n K

(28)

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T. Vu, A. Rahmani / Aerospace Science and Technology 73 (2018) 232–238

Fig. 2. Graph representation of communication topology among spacecraft.

⎡ ⎢ ¯ 3=⎢ M ⎢ ⎣

⎤

J BK

⎥ ⎥ ⎥ ⎦

J BK

..

.

(29)

J BK Similarly, a time-invariant version of the stability criterion can ¯ 2 and M ¯ 3 and neglecting M1 : be evaluated using M

¯ 2−M ¯ 3 +γL0 S¯ = M

(30)

Fig. 3. Convergence of estimate and goal.

Since M1 is always positive definite, it was neglected in order to add a factor of safety to the stability of the system. One might also note the implicit assumption in stating that the eigenvalues of J are bounded. For this assumption to be valid, the designer will need to make sure that the covariance of the process noise, Q , is positive definite, which is standard practice. This prevents the eigenvalues of P i from going to zero which allows P i−1 to exist and J is bounded. 5. Simulations To illustrate the implementation of this distributed control scheme on a system of satellites, the Hill–Clohessy–Wiltshire equations [13] are considered as the natural dynamics of the environment, which are inherently unstable under most initial conditions [17]. Therefore, maintaining positions in the radial, along track and cross track direction would indicate a validation of the control and estimation algorithm. The spacecraft would be orbiting at an attitude of 415 km above the surface of the Earth, which corresponds to the orbit of the International Space Station. To fully illustrate the effectiveness of this estimation scheme, each spacecraft will be able to measure the states of its neighbors. Neighbors are defined by the graph representation, given in Fig. 2. Since each state is observable by at least one spacecraft, it ensures that ( A , H ) is fully observable. As for controllability, it was assumed that these spacecraft have control over all three acceleration components, therefore the formation state vector X contains both position and velocity components of the Hills coordinate system. The control gain was chosen by an LQR regulator which gives more weight to the states in order to show convergence of the system to the desired configuration. Though the initial states and estimated states were both randomly drawn from uniform distributions, the estimated states were given a wider range so that they would be more dispersed from the actual states. To simply assure positive-definite covariance matrices, positive scalar multiples of the identity matrix were chosen for the P , Q , and R matrices. Since the control scheme was given for the continuous Kalman-filter, a Runge–Kutta (4, 5) scheme with a time step of 0.1 seconds was utilized for a high-fidelity simulation. This means that every 0.1 s, a Runge–Kutta (4, 5) scheme was integrated until the next 0.1 s. It was noted that changing the time step would surprisingly yet noticeably affect the trajectories because the time step would have a physical significance. During every iteration of the Runge–Kutta integration, each spacecraft would place a zero order hold on the states communicated from its neighbors. Therefore, the time-step would represent the time lag in communication signals between the spacecraft. The two graphs in Fig. 3 are illustrating the two mode convergence of the spacecraft formation. The first mode, given left

Fig. 4. Illustration of formation estimation and control in a 415 km circular orbit under CWH equations of motion.

axis, is the convergence of the entire swarm to their desired global goal within the desired formation. The second mode, given by the left axis, is the convergence of the Kalman estimates of each spacecraft. They are plotted together to illustrate the simultaneous convergence of both modes. Because the formation state vector X contains both position and velocity components, then units are not shown as only the 2-norm of each respective vector is plotted. Estimates remain bounded close to their true position, with numerical errors and noise disturbances keeping from going to zero completely. An visual illustration of these results can be seen in Fig. 4. In this figure, the solid satellites represents the true locations and each color-corresponding transparent satellites represents the

T. Vu, A. Rahmani / Aerospace Science and Technology 73 (2018) 232–238

Fig. 5. Convergence of norm of state error and estimation errors for various

237

γ values: (a) γ = 0.01; (b) γ = 0.1; (c) γ = 1; (d) γ = 10.

Fig. 6. Effect of increasing time lag in the consensus algorithm (γ = 3): (a) t = 0.5 s; (b) t = 0.1 s.

current self-estimate of their position. Finally the colored circles represents the desire goal of each satellite. The initial configuration shows that the formation has a really bad initial estimate of itself and the combined estimation and control scheme allows the formation to correctly localize itself and converge the desired goals. To explicitly characterize the convergence of the states and estimates, the time history of the state and estimate errors are shown in Fig. 5, over multiple values of the consensus coefficient, γ . As one can see there is strong correlation between the convergence of the estimations to the convergence of the states, which is expected in an unstable environment. The convergence of state and estimates is shown to be very quick as the errors dampen out within 16 seconds. One might also note that the continued oscillations in the estimates after convergence. This is due to a combination of numerical errors from the zero order hold in received information and noise. Figs. 5(a) through 5(d) seems to illustrate that 1/γ seems to act as a dampening coefficient for the convergence rate. As γ starts to increase, notably at γ = 10, the estimation error converges much slower, thereby greatly increasing the state convergence time, which seems to contradict the Lyapunov-based convergence criterion. On the other hand, reducing the gain values does seem to decrease convergence time for the estimates. This reversal is possibly due to the zero order hold implementation. Since the estimation is trying to converge onto a formation the previous time step, then this causes a time lag in the control input to the system. Therefore, having high gain values for γ gives more weight

to discrete jumps in the estimates, which is a consequence of the discrete implementation of the consensus part of the estimation filter. However, since each satellite can only measure its neighbor states and not its own, then γ = 0 would definitely be unstable. So, there must be some optimal value of γ that would lead to stability and desired convergence properties. In order to support this claim that the discrete implementation is responsible for the instability, Fig. 6 illustrates the effect of increasing the time delay of the zero order hold from 0.1 s to 0.5 s, with γ = 3 chosen as an intermediate value because there was such as significant change between γ = 1 and γ = 10. As expected, there are slights oscillations with a longer time delay in consensus communication, and the estimate error is not even converging. A possible solution in realtime implementation might be to reduce the control gain initially to give time for the consensus algorithm to converge. Another is to simply have a faster update rate for the incoming estimates relative to the measurement updates. Either way, this phenomenon shows the necessity to have faster convergence of estimates, as in the case of a single controller and estimator. In order to investigate the effect of initial conditions on the convergence rate, changes in the initial estimation and the initial covariance are shown in Figs. 7 and 8. Fig. 7 changes the initial estimate from uniformly distributed in a 4 km by 4 km by 4 km region at the origin to a normal distribution around the true initial state with varying spreads, while in Fig. 8 the initial uncertainty

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Fig. 7. Convergence of norm of state error and estimation errors (normally distributed initial distribution): (a) With covariance Q ; (b) With covariance Q /50.

Fig. 8. Convergence of norm of state error and estimation errors: (a) Reduced initial covariance trace; (b) Increased initial covariance trace.

was changed by both increasing and reducing the trace of initial covariance matrix. As one expects, the convergence of the estimate error is only affected by the initial covariance. While it might appear strange that higher initial uncertainty would lead to better estimate convergence, it just means that the higher initial guess was probably closer to true uncertainty in the system. However, the state convergence is dependent on both the initial covariance and initial state estimate, therefore having more information on the true state would improve the transient response. 6. Conclusion The stability criterion for distributed estimation and control of a collection of agents is derived in a linear continuous system. It is also shown that a fully generalized version of the separation principle is not achievable with the estimation scheme, and a weak version with one way dependence could be used for combined estimation and control. This combined scheme ultimately reduces communication bandwidth constraints because it only requires transmission of state estimates. Simulations were performed for satellites operating under linear orbital dynamics which validated results expected from the model. They illustrated the necessity for estimation to converge faster than state convergence. Since estimates are communicated in discrete intervals, these results would need to be extended into the discrete domain. It would also be desirable to obtain a mathematical relation for the convergence rate of the estimation in terms of given parameters. Conflict of interest statement None declared. References [1] H. Helvajian, Microengineering Aerospace Systems, 1999, pp. 29–36.

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