In this section, we provide a brief empirical analysis of the validity of the myopic approximation of sensor utility used in our DIF implementations. In our DIF imple- mentations, an estimator approximates sensor utility by evaluating the most recent sensor observation received. The value of the observation is computed as the reduc- tion in entropy realised by fusing the observation into the estimator. This approach is advantageous due to the simplicity of implementation. Entropy reduction can be computed e ciently without additional data storage. The disadvantage of this approximation is that it is myopic. Myopic approximations only re ect the instan- taneous eect of a sensor observation on the information gathering performance of a single robot. Myopic approximations are commonly used since the long-term value of a sensor observation can be di cult to compute in the general case [100].
Inspired by the recent success in exploiting the submodularity property of mutual information for the sensor selection problem [38], one may assume that such property would prove bene cial for providing a sensor utility estimate. However, submodularity does not readily extend to information gathering tasks with dynamic environments which is the case of interest. For further details, Appendix A provides an analysis of the submodularity of linear-Gaussian systems and gives a simple counterexample. Further insight into the error introduced by a myopic approximation for linear- Gaussian systems can be obtained by the comparison with the upper bound provided in [4]. De ne Pkas the covariance matrix of the estimate at time k and de ne φkas the
propagation function of the covariance from time 0 to time k such that Pk= φk(P0).
Then, due to the concavity of the discrete Riccati equation [92] we have the upper bound given by Equation 4.47 for the error introduced by an deviation of P0 in the
direction of Q ∈ S+. If we assume that αI Pk βI, then we have the bound
4.5 Summary 77 by Equation 4.49. log |φk(P0+ Q)| − log |φk(P0)| ≤ d dlog |φk(P0+ Q)| = tr (φk(P0)) −1 d dφk(P + Q) ≤ 1 λmin(φk(P0)) tr d dφk(P0+ Q) (4.47) tr d dφk(P0+ Q) ≤ β β β + α k tr(P0−1Q) (4.48) log |φk(P0+ Q)| − log |φk(P0)| ≤ β α β β + α k tr(P0−1Q) (4.49) To evaluate the myopic approximation for xed receding-horizon planning relative to the upper bound, a multi-robot mapping simulation was performed. The simulation scenario included two mobile robots mapping a spatio-temporal varying eld. To permit the use of the performance bound, open-loop control was assumed. At the end of each time horizon, the rst robot computed the expected information content of its estimate at the end of the second horizon with and without fusing the observation from the second robot. The resulting information content during the simulation is shown alongside the utility bound in Figure 4.3. As shown in the gure, the myopic approximation is much closer to the multiple time-step utility when compared with the conservative upper-bound. Although the upper bound is easy to compute, for many practical applications, the myopic approximation provides a better estimate.
4.5 Summary
In this chapter, we proposed an e cient decentralised solution for both the min-cost- DIF and threshold-DIF problems de ned in Sections 3.2 and 3.3. Our solution to min- cost-DIF was adapted from recent results in multicast routing, which we extended to allow for negative link costs that represent sensor utility. In threshold-DIF, ow
0 5 10 15 20 25 30 0 5 10 15 20 25
Utility Approximation for a 3−Time−Step Planning Horizon
Time Steps
Sensor Utility
Exact Utility
Concavity−Based Approximation Myopic Approximation
Figure 4.3 Comparison of xed-horizon sensor utility approximations. The solid line shows the exact sensor utility computed over a xed time horizon. The dashed line shows the myopic approximation while the dotted line shows the concavity-based approximation.
rates are optimised based on the value of information while obeying local computation limits and global communication limits. Our solution to threshold-DIF is based on a distributed version of ADMM that requires neighbour-to-neighbour communication only. Finally, we proved that the convergence time of our solution is polynomial in the size of the network. In the following chapter, we present a solution to the third problem considered in this thesis, negotiation-DIF.
Chapter 5
Negotiation-DIF
In this chapter, we present a solution to the negotiation-DIF problem. Negotiation- DIF addresses communication e ciency at both the data fusion and decision making layers concurrently.
First, we begin with a brief introduction of background material in Section 5.1. This introduction will aid our presentation of linear-quadratic information structure op- timisation (LQISO), a solution algorithm for the comms-LQ problem. Presented in Section 5.2, LQISO provides a communication-e cient decision making solution for LQ problems. Then, in Section 5.3, we extend LQISO to provide a communication- e ciency solution for decision making in decentralised information gathering. Finally, in Section 5.4, we combine the solution of min-cost-DIF with the extended version of LQISO to present a solution to negotiation-DIF.
5.1 LMIs in LQ Optimal Control
In this section, we introduce background material on the use of linear matrix in- equalities (LMIs) in LQ optimal control. This background information is necessary for the presentation of the algorithm described in Section 5.2. A detailed discussion on LMIs and optimal control is outside the scope of this thesis and can be found in
[12, 81]. LMI formulations of the LQ optimal control problem allow the addition of extra criteria such as communication constraints.
To this end, consider a system with linear dynamics given by Equation 5.1 and quadratic cost function given by Equation 5.2. Assume that the control vector is set to the feedback law given by Equation 5.3. When the matrix K is the solution of the algebraic Riccati equation, then Equation 5.3 is the optimal feedback control policy. ˙x = f (x, u) = Ax + Bu (5.1) g(x, u) = 1 2 x TQx + uTRu (5.2) u = −R−1BTKx (5.3)
If we consider the quadratic function given by Equation 5.4 where K positive de nite, then the dissipation inequality [98] is given by Equation 5.5 and in dierential form in Equation 5.6. V (x) = 1 2x TKx (5.4) t1 Z t0 g(x, u)dt + V (x1) ≥ V (x0) (5.5) ∂V ∂x T f (x, u) ≥ −g(x, u) (5.6)
If V satis es the dissipation inequality for the choice of control action, then it is a lower bound on the value function of the system. If we substitute the de nitions of V, f and g into the dissipation inequality we obtain Inequality 5.7. Consequently, we obtain the inequalities (5.8-5.11). Inequality 5.9 is obtained since a scalar is equal to its transpose. Inequality 5.10 results from the arbitrary choice of x and Inequality 5.11 is a result of the Schur complement lemma.
xTKAx − xTKBR−1BTKx + 1 2x TQx +1 2x TKBR−1 BTKx ≥ 0 (5.7)
5.2 LQISO 81