Optimal Sensor Scheduling and Selection for Continuous-Discrete Kalman Filtering with Auxiliary Dynamics
Authors: Mohamad Al Ahdab
John Leth
Zheng-Hua Tan
Venue: ICML 2025 poster (2025)
Keywords: Kalman Filtering
Sensor Scheduling
Bayesian State-Space Models
Control
Primary area: probabilistic_methods->bayesian_models_and_methods
TL;DR: Scheduling and selecting sensor measurements under varying enviroments in a continuous-discrete kalman filter setup.

Abstract: We study the Continuous-Discrete Kalman Filter (CD-KF) for State-Space Models (SSMs) where continuous-time dynamics are observed via multiple sensors with discrete, irregularly timed measurements. Our focus extends to scenarios in which the measurement process is coupled with the states of an auxiliary SSM. For instance, higher measurement rates may increase energy consumption or heat generation, while a sensor’s accuracy can depend on its own spatial trajectory or that of the measured target. Each sensor thus carries distinct costs and constraints associated with its measurement rate and additional constraints and costs on the auxiliary state. We model measurement occurrences as independent Poisson processes with sensor-specific rates and derive an upper bound on the mean posterior covariance matrix of the CD-KF along the mean auxiliary state. The bound is continuously differentiable with respect to the measurement rates, which enables efficient gradient-based optimization. Exploiting this bound, we propose a finite-horizon optimal control framework to optimize measurement rates and auxiliary-state dynamics jointly. We further introduce a deterministic method for scheduling measurement times from the optimized rates. Empirical results in state-space filtering and dynamic temporal Gaussian process regression demonstrate that our approach achieves improved trade-offs between resource usage and estimation accuracy.

Optimal Sensor Scheduling and Selection for Continuous-Discrete Kalman Filtering with Auxiliary Dynamics
Mohamad Al Ahdab John Leth
Zheng-Hua Tan
Abstract
1. Introduction
State-space models (SSMs) are fundamental tools for ad- dressing sequential inference challenges in time-series fore- casting, signal processing, and dynamic systems. A core task in SSMs is Bayesian filtering, which aims to com- pute the posterior distribution of latent states online given noisy observations. For linear systems driven by indepen- dently and identically distributed process noise with known dynamies. the Kalman filter (KF) is the optimal linear fil- ter in mean squared error (Kalman, 1960). In the case of nonlinear dynamics, approximate inference methods such as the Extended Kalman Filter (HKF) and the Unscented Kalman Filter (UKF) are commonly used (Simon, 2006; Särkkä & Svensson, 2023). In this paper, we focus on the Dontinuous-Discretee Kalman Filter (CD-KF) setup, where continuous state evolution is observed via multiple sensors with discrete, and possibly irregularly timed, mea- surements. This choice is motivated by the fact that many real-world processes (e.g., blood pressure, ocean tempera- ture, or radiation levels) evolve continuously but are only observed intermittently due to hardware constraints and op- crational considerations. Such formulation is especially relevant when selecting from multiple sensors operating un- der distinct costs, conditions, and constraints. Moreover, the measurement process is often coupled auxiliary dynamics such as sensor's temperature, spatial position, stored energy, which affect measurement cost and quality. For example, consider a low-orbit satellite observing terrestrial phenomena (e.g., ocean temperature) using sensors whose accuracy depends on the satellite's orbital position. The satellite has two types of sensors: high-resolution sensors that accurate during daylight periods, and baaed sensors that are always available, but are energy-intensive and less accurate. Another example is glucose monitoring setup where diabetic patient tracks blood glucose via three different types of sensors: a continuous glucose monitor that provides noisy but frequently sampled measurements, intermittent finger-prick tests that offer higher accuracy but cause patient discomfort, and very accurate but infrequent and costly clinical blood tests. In these examples, the sen- sors differ not only in accuracy but also in their relation to auxiliary quantities such as energy consumption, monetary costs, and human discomfort. This means that adopting
We study the Continuous-Diseres Kalman Filter (CD-KF) for State-Space Models (SSMs) where continuous-time dynamics are observed via multi- ple sensors with discrete, irregularly timed mea- surements. Our focus extends to scenarios in which the measurement process coupled with the states of an auxiliary SSM. For instance, higher measurement rates may increase energy consumption heat generation, while sensor's accuracy can depend on its own spatial trajec- tory or that of the measured target. Each sensor thus carries distinct costs and constraints asso- ciated with its measurement rate and additional constraints and costs on the auxiliary state. We model measurement occurrences independent Poisson processes with sensor-specific rates and derive upper bound the mean posterior co- variance matrix of the CD-KF along the mean auxiliary state. The bound is continuously differ- entiable with respect to the measurement rates, which enables efficient gradient- based optimiza- tion. Exploiting this bound, we propose finite- horizon optimal control framework to optimize measurement rates and auxiliary-statt dynam- ies jointly. We further introduce a deterministic method for scheduling measurement times from the optimized rates. Empirical results in state- space filtering and dynamic temporal Gaussian process regression demonstrate that our approach achieves improved trade-off's between resource usage and estimation accuracy.
'Department of Electronic Systems, Aalborg University, Aal- borg Øst 9220, Denmark. "Pioneer Centre for Artificial Intelli- gence, Copenhagen 1350, Denmark.. Correspondence to: Mo- hamad Al Ahdab <maah www.s.aa.u.k....
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
naive strategy of uniform high-rate sampling for all the sensors is undesirable and practically infeasible. Instead, measurement decisions must be scheduled to balance be- tween minimizing the uncertainty in what they are measur- ing while simultaneously accounting for their interaction with dynamically changing auxiliary variables.
(Orihuela al., 2014) developed periodic scheduling poli- cies for discrete L.TI systems, while (Marelli et al., 2019) introduced stochastic scheduling strategies under resource constraints. Optimal control perspectives have also been explored, such as the infinite-horizon formulation in (Zhao et al., 2014) and the finite-horizon networked control frame- work in (Ayan et al., 2020), which incorporated auxiliary network dynamies but retained a discrete-time LTI assump- tion. These works do not address the continuous-discres setting with irregular measurements, nor do they consider the coupling between sensor scheduling and general aux- iliary state dynamics (c.g., sensor's ' spatial trajectory and the available stored energy). Our work generalizes these ap- proaches by unifying continuous-time state evolution with stochastic measurement scheduling via Poisson processes and integrating auxiliary state dynamics into the framework.
In the literature, most existing works do not consider the continuous- discrete setup. Additionally, scheduling mea- surements while simultaneously considering their interac- tion with an auxiliary state has not been explored directly in the literature. In this paper, we address this gap by formaliz- ing the problem of optimal sensor scheduling for continuous- discrete SSMs with auxiliary dynamics and proposing an optimization scheme to solve it. Specifically, we model the arrival of the measurements from each sensor as Poisson process with mareing rate. We then formulate an op- timization problem that is continuously differentiable with respect to the measurement rates of each sensor and other relevant decision variables affecting the auxiliary dynamics (e.g, the trajectory of a vehicle carrying a sensor). Finally, we provide a deterministic strategy to select the measure- ment time instances for each sensor based on the optimized measurement rate for that sensor. Our contributions in this paper are summarized as follows:
A ctive Sensing: Active sensing encompasses both sensor scheduling and trajectory optimization to maximize infor- mation gain. Recent advances leverage deep learning for sequential decision- decision-making, for example, (Yoon et al., 2018) used recurrent neural networks to dynamically select med- ical tests based on patient history, while (Qin et al., 2024) proposed controlled neural ordinary differential equations to determine optimal measurement intervals in continuous- discrete settings. However, these methods focus on dis- crete classification tasks (e.g., disease diagnosis) rather than Bayesian state estimation. Recent work by (Napoli- tano et al., 2024) optimized for informative trajectories for Gaussian process (GP) regression in a model learning setup. However, these approaches do not consider the challenge of sensor scheduling and selection with general auxiliary state dynamics interacting with the sensor (e.g.. the sensor's temperature affects its accuracy).
1. We derive an upper bound on the mean posterior co- variance matrix of the CD-KF conditioned on the mean auxiliary state. This bound is continuously differen- tiable with respect the sensor-specific measurement rates, which enables efficient gradient-base based optimiza- tion.
2. We propose the setup of a finite-horizon optimal control framework to jointly optimize measurement rates and other inputs that are relevant to the auxiliary dynamics.
Bayesian Optimization: Bayesian Optimization (BO) has been widely adopted for optimizing expensive black-box functions, particularly in experimental design (Snoek et al., 2012). Classical BO methods rely on acquisition functions like Expected Improvement (Jones et al., 1998) or Upper Confidence Bound (Srinivas et al., 2012) to balance explo- ration and exploitation. Recent extensions integrate BO with reinforcement learning for sequential decision-making un- der uncertainty (Ling et al., 2016). BO has also been used to design experiments with mutual information in (Kleinegesse & Gutmann, 2020). These methods do not exploit known system dynamics or state-space structures. In contrast, our work leverages the analytic properties of the CD-KF to derive differentiable bounds on the estimation error (mean- squared error), which enables gradient- based optimization of measurement policies. Additionally, these approaches do not account for the influence of auxiliary dynamics on the measurements and their costs and constraints. Our formula- tion generalizes these settings by unifying continuous -ime dynamics, stochastic measurement scheduling, and auxiliary
3. We propose a deterministic method for obtaining mea- surement time events from the optimized rates. The method is based on minimizing the Wasserstein dis- tance between the distribution of measurement events generated by the optimized Poisson rates and an empir- ical distribution determined by the expected number of measurements for each sensor.
2. Related Works
Sensor Scheduling and Selection in Kalman Filtering:
The problem of optimizing sensor usage in Kalman filtering has been studied across diverse settings. Early work by (Ny et al., 2009) addressed continuous-time sensor management for Linear tinuous measurements, a restrictive assumption for practical systems with discrete. asynchronous sensor observations. Subsequent studies focused on discrete-time formulations:
(LTI) systems but assumed con-
2

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
state constraints within single optimal control framework.
where
To the best of our knowledge, the framework of scheduling sensors and measurements in a CD-KF setup with auxiliary dynamics has not been explored before. This work proposes this framework and presents some results on efficiently solv- ing a class of this problem.
((E())) = I(T)) (t,) (CIE) (t)+ R(t.))",,
(4)
is the Kalman gain. Let (t) be the total number of mea- surements up until time t, then the equation for the covari- ance Σ in the CD-KF can be written compactly as
3. Notation
dc = A())++AA + +o()) dt
We denote by $2 (8%)) the cones of positive semi-definite (positive defimite) n X n matrices. The Loewner order on $20 ($5,) is written as A 3 B (A < B), while element- wise inequalities are written as Se. <e.Ze, and >e- We write I, for the n x n identity matrix, and tr(M) for the trace of matrix M. The Dirac measure & denoted o,. The indicator function 1A : X - +{0,1 1 is defined by
N(O)
K(E,D)C(EE
Σ
du (5)
where o, is the dirac delta measure at 4g
1,
x E A, x A.
5. The Problem Setup
1 1Z =
0,
We consider a multi-sensor setup in which, at each mea- surement time, sensor E {I..... S} can be selected from S available sensors. Additionally, we introduce an auxiliary state & E R (e.g., representing the position of a vehicle and its energy storage), governed by the following dynamics:
4. Background: Bayesian Filtering for SSMs
(continuous-discret stochastic) SSM governed by:
dz= = A(t) + (t)dd, 2o (No, Mo), v44) = C(1))a(14)) + v(4),, v(4:) ~ NO,, R(4)),
(Ia)
(1b)
s dLp = 5,(5, u,t + Σ gs u,t) dt
No
where: x E R is the state, dW E R
is a Wiener process, A(t) E R=*" and (t) E Rx define the drift and diffu- sion matrices, v(44) E R2 is a discrete-time measurement at time 4 (here, i indexes the measurement occasions, so t; denotes the i-th observation time), v(t)) is the measurement noise v(t,) ~ (0, R(t;)) with R(t,) E 5% (independent and identically distributed), and (t,,) E Raxn is the output matrix.
Σ
deg
(6a)
1=1
deu
fu(Su:
(6b)
dt
where & = [&u Sp]T. Here, Ep E R" denotes the "per- turbed" part of & affected by measurements, while Eu Ru represents the unperturbed part. The input sig- nal u R (c.g., the velocity of vehicle) influences the dynamics, and (t) represents the total number of measurements up to time t for sensor s with t being the i-th measurement instant for that sensor. Note that the dif- ferential equation for &x depends only on E.. Therefore, given an initial condition (0) Eu, and an input trajetory u(t) E Te then can solve for (L) independantly from the measurement times. The functions fp: 1..91....4. -0......... Is and fu are assumed to be Lipschitz continuous in &, u, and t.
For time instants t1 t2 < < t.. The Bayesian fil- tering problem involves sequentially obtaining p(r((z)) .(. .((4), For provides exact closed-fom form solutions for both the filtering density p(x(tt) ((tt)...., v(t)) and the prediction den- sity p(x(t) v(tz),.... v(t)) for t > t; (Jazwinski, 2013). Given a Gaussian prior z00) ~ N(Ho, Eo) with no E IR" and Σ. 0% the CD-KF follows the following steps:
SSMs, the CD-KF
Prediction (t E [t)-1,t());
dj A(t)p,
(2a) Decomposing the auxiliary state into Ep and & is general and does not impose restrictions. For instance, if all of g is perturbed by measurements, then & Epi conversely, (2b) if none of E is perturbed, then { f.. This division be- comes crucial later when analyzing the covariance structure of the KF. Often, such decomposition arises naturally. For example, &u may represent the kinematics or dynamics of a (3a) vehicle, which are typically independent of measurements, while &p might correspond to the stored energy of the vehi- (3b) cle, which depends on position and velocity (e.g., locations
dt
dΣ A(()+EAA (A)(()+(()0)))) +(()oT(())
di
Update (at measurement t;): (((:)= = (( (1(44) ((I))((1)),
I(4) (I, - -(Z I(G),
3

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
for recharging or energy consumption rates different ter- rains). Additionally, the stored energy can decrease with each measurement, and the consumed energy by each mea- surement event for sensor s E {l....,S) S} is characterized by Is-
Similarly, the perturbed part of the auxiliary state becomes
8 Σ
dep u,
go u, dN,
(9)
We now state the following two propositions, which enable us to formulate deterministic optimization problem in the rates : ds and the input u. Throughout the rest of the paper, we will assume that the jump size functions satisfy the mean-square integrability condition (e.g., see Theorem 3.20 in (Hanson, 2007)).
We now extend the SSM in (Ia) to incorporate the auxiliary state E:
dx = t( + ((,, dW, zp ~ N(uo Eo),
(7a)
- C.(E(()), + tE(())).
(7b)
where y²((t is the measurement taken by sensor s at time t, Cg is the corresponding output matrix with the measure- ment noise being v (g((,), 4) ~ N (0, R2 (E()),))))) 4)). Here, we also assume that A, a, C1 c..-..: Cs and R1,..., Rs are Lipshitz continuous in & and t.
Proposition 6.1 (Covariance Matrix Bound), Given initial conditions Eo E TR= and Egg E 8% ler E(t) be the solu- tion to (9) with E(O) = Eo and ()) be the solution to (8) with Σ(0) Σg- Ler E(t) = E(E(t)] with E(O) &o and
E(t; 5*) (()(()
{*()) for some (t) R
with E(t; (*) = Eg. Then, the solution E(t) of
Our goal in this paper is to jointly optimize the measurement times for each sensor and the input u within the context of a CD-KE
CC = A(5", + EAT(S",)) + (5",
dt
In this paper, we will consider the following assumption for the dynamics of the auxiliary variables in (6).
Σ (t) t)C,(EE ,t)., ε(() Eo, (10)
Assumption 5.1. The functions fg and gs(E, u, t) are con- cave (convex) for all s (I....,S) S} with respect Ep (in the clementwise product order sense).
satisfies E(t:g") 3 E(t, 22 0.
Proposition 6.2 (Auxiliary State Bound). Let E(t) == (t)(t) cu(t)] E R be the solution to
This concavity (convexity) assumption is essential for deriv- ing upper (lower) bounds that lead to tractable optimiza- tion problem for the state &p. Importantly, it encompasses broad class of systems, including all affine systems of the form (Su. u. t)pp + B(Su. u, t), which satisfy this assump- tion.
8 f,(E, + + (x,0))a.E.u. É()) -co. dE, di
s=1
(11)
where Eu evolves according to (6b). If Sp and gi..... 9s are concave (convex) with respect to Ep (Assumption 5.1), then E(t) Se E(t) (E(O) 2. E()) Vt 0.
6. The Kalman Filter with Randomized Measurements
The proofs are provided in Appendix B. Remark 6.3. If A, o, CI,.... Cs. and RI...., .., Rs do not depend Ep. then E [[((t]] E [E(t) {(t) (t).. Moreover, if fp: ........,. Is are affine in Ep. then (t) (t).
We now describe the randomized KF framework that forms the foundation for computing optimal measurement rates. Specifically, we model the arrival of measurements from each sensor E {1....,St S} a Poisson process (t) with time-dependent intensity (t). 'he composite measure- ment count is N()) = Σee (!), and the randomized covariance matrix evolution for the CD-KF then becomes:
7. The Optimal Control Problem
This section will utilize the bounds from Proposition 6.1 and Proposition 6.2 to formulate a general optimal con- trol problem to calculate u and the rates Ai...., ,......,, Äs for a horizon [0, T). Consider that we have a continuously dif- ferentiable running cost L : R" 3 x U x R + R with continuously differentiable terminal cost Lr RT 5% x U x RS. + IR which we desire to min- imize (e.g., minimizing tr(E)). In addition to the costs, consider that we have ne, continuously differentiable run- ning constraints C: R. xS9, xll R -> R. together
d = (A(E, ΣAT (E. + aEE
(E, 1)) di
Σ ((E
where the Kalman gain (E, t,t is given by:
(,,,,) ==
EC, (E,1) (C.(,,)). (E,1) (.())))
4

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
with Bee continuously differentiable terminal constraints Cr : TR'e $ U x R. - R (e.g, Ep(T) 2 0 if (T) represent energy), We formulate the following general optimal control problem:
Obtaining a closed-form solution to the problem in (12) is generally intractable. Thankfully, several numerical ap- proaches can be utilized to obtain an approximate solution to the problem (Rao, 2009) (see Appendix D for summary of these methods and the choice for the examples in this paper).
S' c I dt
min A2.0, well, ÉER"S, SE350
(12a)
Remark 7.2. For the OCP in (12), we use & from Propo- sition 6.2 in the cost, in the constraints, and also for the dynamics of Σ. By Proposition 6.1, E then upper-bounds the true covariance E(:; E). We distinguish three cases for our setup and the OCP in (12). The first case is when fp. 91......9 are affine in Ep- In this situation, we have E E and the constraints and the costs in the OCP in (12) are for the exact mean trajectory &. Additionally, E will be an upper-bound for the covariance matrix along the mean trajectory É. and the costs and the constraints in the OCP will thus target the mean behaviour. If fp. 91...... gs are nonlinear but are concave (respectively, convex) in (12d) E, (Assumption 5.1), then É upper-bounds (respectively, lower-hounds) {, so any monotonically non-decreasing (re- spectively, non-increasing) cost or constraint depending only on & remains meaningful. However, without further convexity/concavity assumptions on A, Ci:-.. Cs and RI.... , Rs from (7), it need not follow that Σ Σ(t; E) bounds E); Nevertheless, if the dynamics are locally nearly linear (similar to the EKF mean approximation dÊ = (,,E,u,) EN (x(t)), t, (Simon, 2006; Särkkä & Svensson, 2023)), one finds Σ v E(t;E), and thus, still approximately targets the mean behavior. Finally, in the fully nonlinear case violating Assumption 5.1, we can still treat & as a heuristic approximation for & without formal guarantees if the dynamics are locally nearly linear. Empiri- cal results in Appendix H show that this approximation can yield acceptable performance in many scenarios.
+r ((T
subject to dL
dt = A(E,)) EA (E.t) + (((EttoTTEE (E,t)
(12b)
dep fp(5. u, +)+++)
g
X, u,,,,,
(12c)
dt
deu, dt
= Su(Su, u.t),
É()) = Co. E(()= = Eo: X(1) 2. O, ull) EU, VI E [0,
(12e)
(12f)
(((0), (00, u())), (() So O, VI E Cr (É(T), ((T). Se
(12g)
(12h)
where U C C' Since (tt and (t) represent bounds on the true mean covariance and mean perturbed state, the cost functions L. fr and the constraints C, Cr are assumed to be monotonically non-increasing (in the Loewner or- der) in E, and monotonically non non-decraasing (respectively, non-increasing) in Ep if fp and 91...., gs are concave (re- spectively, convex) in & (element-wise)', This ensures that the bounds from Propositions 6.1 and 6.2 provide mean- ingful constraints and costs. When fp and 9s are affine in Ep. we have (t) = (t) for all t 2 0, so no additional monotonicity restrictions are required with respect &p. Since typically want to decrease or upper bound uncer- tainty, the requirement for monotonically increasing costs and constraints with respect to E is not restrictive. Note that one can also add slack variables to relax constraints and other aetimization-relatee variables to (12). Furthermore, we assume the following:
Remark 7.3. While this paper focuses on an optimal control setup, Proposition 6.1 and Proposition 6.2 offer the oppor- tunity to attempt different control strategies to compute X and u depending on the specific problem of interest. For example, we can use control barrier function techniques (Ames et al., 2019) to find a feedback law for À and u such that the set S. := { 5% tr(E)= S c} with e > is forward-invarian
8. Deterministic Selection of Measurement Times
After solving the OCP in (12) for the measurement rates X and inputs u. we would like to select the measurement times for each sensor E {1,..., within the horizon [0,T] which are, loosely speaking, closely related to the average behavior of Poisson process with rate X,- Rely- ing on sampling the Poisson process to select measurement times is not suitable for a inite-horizon planning task as
Assumption 7.1 (Feasibility). There exists at least one ad- missible pair (u()), X(-)) with corresponding E(-) and E() such that C S. 0 and Cr Se 0 are satisfied.
We say for differentiable function f= TRn + TM that it is
non-decreasing (non-increasing) if fff 2 0 (afs/On; S 0) for i E (1.....m) m and j E [1,....,n).
5

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
we could sample realization which is far from the average behaviour of the process. While this realization could pro- vide better solution, in terms of the cost and constraints in (12), than the average one, it could also provide a worse solution. Therefore, we aim to deterministically select the measurement times to be as close as possible to the average behavior according to a specific metric. To select determin- istic measurement points 8a = (ti.. ......., ) for sensor with intensity rate (t) over [0, T we cast the problem one of optimal quantization. Define the normalized intensity measure
The proof can be found in Appendix C. Note that A Ag(af)
Ag F3 ( A, (A_" (A. E (1,.... RT 1]. In other words, in Propositions 8.1, we divide (T) into equal parts. This strategy distributes points more densely in regions where (x is large while preserving the mean of μ4 and minimizing second-order distortion in the Wasserstein-2 sense. The solution is unique and admits a closed-form expression, making it computa- tionally efficient. Furthermore, minimizing the Wasserstein distance is equivalent to minimizing the quantization error E mminke, || - =I|²), which is classic result from optimal quantization theory for probability distributions (Graf & Luschgy. 2000). To match the average number of measure- ments, we select nt = [Tx(T) + 0.5].
A.(T)) for all n,
ni
(t) dt, where AT) !'" xx( dt, Me (dt(t) (T)
and let Tx ~ 2s be the random variable that describes the time of measurement from sensor s. Our goal is then to approximate Ms by discrete measure Vs Va,5a given by:
To clarify Proposition 8. assume for simplicity that A. (T)) is integer. then according to Proposition 8.1, we will par- tition [0, T] into intervals such that A²(a²) (a = for i E {l...., n if n. (T). This means that we par- tition [0, 7 into intervals wherein each one of them contains exactly one measurement on average. The location of that one measurement in each interval is taken to be the mean of A, ( (1) over that interval. We select the measurement times for each sensor s E {1...., based on Proposition 8.1 independently.
N₂ oτ (dt).
(dt) =
Σ
(,
Denote T ~ V, to be the discrete random variable rep- resenting the time of measurement of sensor under Va- We select I, which minimizes the squared Wasserstein-2 distance,
9. Experiments
inf / 16- d',,
W2 yET(H.:U))
Jo.ryp"
In many real-world applications (e.g.. environmental mon- itoring (Alvear et al., 2017; Bird et al., 2018), and search- and-rescue (Waharte & Trigoni, 2010)), robots must oper- ate under limited energy resources while gathering data in potentially hazardous environments. We consider such a scenario where a robot, constrained by its energy capacity, needs to collect measurements (e.g., pollutant concentra- tion, air quality, temperature) using two onboard sensors for temporal GP regression ?.
where I(M,, vs) is the set of all couplings of ua and v.. In one dimension, this distance simplifies to:
22))) = S'' (F) (F: dp.
where F3 and G, are the cumulative distribution functions (CDFs) of (s and V., respectively (Villani et al., 2009). In this case, we have (t( = Ju xx(t)
(.()) dt (t) and A.(T)
9.1. Robot Model and Problem Setup
G.(1) = En. 1 (1).
The state f., = [pr, 8,] e R represents the planner robot's 's position pr E R and its heading 0, E R. evolves accord- ing to a unicycle model (see Appendix for details). The input [t,w]] E I2 consists of the heading velocity and the angular velocity w, both subject to physical bounds. The energy state Ep = 1 E R evolves according to
Proposition 8.1 (Optimal Quantization Points). The points Ei....,F, Es that minimize W2(,,, are the conditional centroids on ne intervals of equal probability under Ua. Specifically,
1 tta(t) dt
1 E E
To af]l
X,(t) dt (13)
dy dt
c. exp((1 pell²) C C.l
Σ
Co 6ig,
(14)
where aj F-1 = for = 1,..., n. = and n.
'The code for generating the results, in addition to the corresponding animations and different examples, can be found on https:/ http//gtttubbccoc htt:///HHAAAAAHDDD.. When2measurek?
ot TT With this construction, v, matches the first mo- ment of He (i.e. E[T,] = E(rr') and the variance difference
Var(T.) - Var (Tr() = W2 Us).
6

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
Planned Sensor
Triee 52
Sensor 2
1.0
0.5
0.5
3p + 10
True Sensor
la Sensor
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
Time
Time
1.00
Path
Sensor
Planned .... True
Sensor
Base
Sensor
Sensor
0.75
Target
Robot
40
y-coordinstes
Energy
0.50
20
0.25
10
0.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
1-coordinates
Time
1.5
3
Planned
True
Samsor
Sensor
2
0.5
0.0
0.5
2g + 17
True My
29 + 10
Sensor 1
Sensor 2
-1.0
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
Time
Time
50
0.15
True
Planned
Planned f
-.- Trme
Radiation domage
Sensor
True Co
Planned c
Senser 2
40
Scnsor
Sensor 2
0.10
Energy
30
20
0.05
10
0.00
0-
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
Time
Time
(b)
Figure 1. The planned proposed mean solution according to (12) and the true simulated solution based the proposed measurement times selected according to (13) from caleulated rates À (a) without radiation damage and (b) with radiation damage. up represents the true process, &p represents the KF estimate of it, and 9. represents the RTS estimate of it (the GP regression output).

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
where Cu cj 2 Cg 2 O are the energy costs of using the inputs and the sensors. The term ce exp (-rellp-- p
radiation damage on Sensors and 2, respectively. The damage dynamics are
(with Ce. 2 0) represents energy charging source (e.g.. solar) with its maximum charging rate at the base located at p6 E IR?.
dç. dt
eexp exexp(-rcllp Pp ppl1²) deg,
(17)
The measurement noise covariance matrices depend on the robot- proot-process distance:
where "1 > 72 2 0, implying Sensor 1 is more susceptible to radiation damage than Sensor 2, and rç > 0. Conse- quently, the sensor covariance matrices become R&(E) = Remaz exp(r=| pr - ppl(²) exp(r=Se), s E {1.2]. Radia- tion damage thus exponentially increases the measurement noise of each sensor. Figure ((b) presents results for this second scenario.
Rg (pr) Remax exp(r, exp(r=ll,,)) llp, ppl1)), EE (1,2), (15)
where Ramer > Rimax > 0, 12 > 11 > 0, and Pp E R2 is the location of the process. This setup implies that sensor is more accurate but also consumes more energy than sensor 2. Starting the base ps with initial energy (p(0)= Eo 2 0, the robot must collect measurements at Pp and return to p& by time T while ensuring that there is sufficient residual energy to continue operating. We employ the state-space representation of the Matérn kernel (degree 1) for the GPs (Särkkä & Hartikainen, 2012; Todescato et al., 2020). Kalman filtering and smoothing (Rauch-Tung Striehel (RTS)) with this state-space representation is known be equivalent to GP regression (see Appendix for more details). The objective is to reduce uncertainties in the CD- KF estimate; hence, schedule measurements to minimize tr(E), where Σ is the Kalman filter's covariance matrix for the estimated process (an upper bound on the smoothing covariance). In the OCP in (12), the running cost is
9.3. Comparisions
The approach used for the example jointly optimizes scheduling sensors while optimizing other inputs for the auxiliary dynamics. To our knowledge, there exist no meth- ods that jointly optimize inputs for auxiliary dynamics to- gether with scheduling sensors in a contimuous-diseret setup. Therefore, we compare our approach with baseline strategies that utilize different measurement schemes while employing the optimized inputs for the auxiliary states. The baseline scheduling strategies are summarized follows:
Discretize the horizon uniformly according the number of discretization points for the OCP (No). Then, sample the measurement times for each sensor E {1,...,S) S} according to Poisson process with constant rate X, No/S.
Random:
x) wy tr(E) + wx A Wy u² u + w. C(E u, (,,,E,,,,,))
(16)
where W/ >> wx w> 2 w. O, and the terminal cost is zero. To avoid large fluctuations in tr tr()). we add the constraint tr tr()(()) S cg + E() for 1 E [1/2, 7). where E(t) 2 0 is a slack variable to ensure feasibility. We en- forced the constraint on E only for E [1/2,T] since requir- ing the trace of Σ to decrease rapidly can be restrictive and will always lead to large e at the beginning of the horizon. The running constraints C thus encodes the input bounds S. u S. ū, A2 2e 0, the trace constraint tr (E)= S 6x; + (enforced for t E [1/2, 1)), E 2 O, and an energy lower bound f 2 G, with e 2 O. The terminal constraint Cr encodes that the robot should return to the base the end of the horizon (T) Pb. Note that the dynamics of Σ do not depend on the auxiliary state &p. The results of the optimization and simulation for horizon of T 1 are illustrated in Figure (a)).
Greedy: Discretize the horizon into small time steps. Then, at each time step and for each sensor, compute a "score" equal to the immediate reduction in the trace of the predicted covariance minus the associated sensor costs and penalties on constraint violations. The mea- surement is then chosen according to the sensor with the highest positive score. If all the scores are negative, then we do not perform a measurement.
M-Optimized: Instead of finding the measurement times according to the quantization rule in (13), we sample multiple realizations (10000 for the example in this paper) of measurement times according to the cor- responding Poisson process with the optimized rates. Afterwards, we pick the measurement times corre- sponding to the realization with the minimum value of the cost function modified with penalties on constraint violations.
9.2. Radioactive Environment Extension
We next examine setting inspired by nuclear or chemical disaster scenarios, where the process of interest is located in dangerously radioactive zone (Bird et al., 2018). Each measurement in this area degrades the sensors' accuracy over time. Specifically, the auxiliary state is now &p = [n EI [2) T E R3, where Çi and (a denote the accumulated
Table I summarizes the statistics for the comparisons where "Optimized" refers to our suggested approach.
8

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
monitoring with fouling and active defouling. In addition, example with spacecraft monitoring targets on Earth can be found in Appendix F. Moreover, Appendix G provides discussion on how our approach can be generalized to uncertain and nonlinear SSMs with an example of a robot estimating the position and velocity a moving target while attempting to track it. Finally, Appendix H we provide examples for cases where the auxiliary dynamics do not satisfy Assumption 5.1.
Table Trraeectory statistics for the covariance trace, energy, and degradation over the horizon. The proposed approach in this paper denoted as "Optimized"".
Covariance Trace
Method
Mean
Std
Worst Case
Optimized
1.90221
0.3406
2.94496
M-Optimized
1.89024
0.341 106
2.94496
Greedy
2.60768
0.487677
3.21866
Random
2.27854
0.48601
2.96245
10. Conclusion
Energy "
Method
Mean
Std
Worst Case
We propose a novel methodology for sensor scheduling in SSMs that incorporates auxiliary state-space dynamics. By modeling the occurrence of sensor measurements through independent inhomogeneous Poisson processes with sensor- specific rates, we derive continuously differentiable up- per bound on the mean posterior covariance matrix of the CD-KF conditioned on the mean auxiliary state. This dif- ferentiability enables efficient gradient-based optimization of the sensor mcasurement rates. In addition, we formu- late finite-horizon optimal control problem that jointly optimizes these measurement rates with inputs for auxiliary dynamics. We further provide guantization-basse method for selecting deterministic sensor measurement times that closely match the mean behavior of Poisson processes with optimized measurement rates, thereby matching the actual sensor schedule with the intended offs. An Empirical evaluation in state-space filtering and dynamic GP regression confirms the feasibility and potential of this approach.
Optimized
21.5435
10.1709
0.433792
M-Optimized
21.2173
10.5831
-1.11621
Greedy
-1.67343
14.0586
-21.1662
Random
-23.2507
33.7226
-82.6662
Degradation (Ci + (2)
Method
Mean
Std
Worst Case
Optimized
0.0905529
0.0654335
0.185626
M-Optimized
0.0975802
0.0729591
0.213117
Greedy
0.193289
0,081316
0.232247
Random
0.873604
0.596903
1.68309
accuracy trade-
9.4. Discussion of the Results
From Figure 1, we observe that solving the optimal control problem in both scenarios yields a planned robot trajectory together with a corresponding measurement schedule for both of the sensors to track the measured process. We also remark from the figure that the planned upper bound E. on the mean for Σ nicely tracks the simulated true solutions for E with measurement times calculated according to Proposi- tion& 1. We observe the same for <1, and (2. Additionally, we note that the trace of Σ for the radioactive case is, on av- erage, higher than that for the non-radioactive case, and the smoothing results (GP regression results) are visually worse for the radioactive case. This is expected since the radioac- tive environment case is more challenging. The results in Table I suggest that our method ("Optimized")) outperforms the greedy and random scheduling approaches for the robot example. The results also demonstrate that our determinis- tic quantization provides similar results to "M-Optimized" without having to sample multiple realizations. Sampling multiple realizations can be computationally expensive and unrealistic since we do not have the real measurements to compute the corresponding cost for each realization. Over- all, the results demonstrate the potential and feasibility of utilizing the proposed approach in the CD-KF setup with auxiliary dynamics for sensor scheduling and selection. See Appendix E for another example involving water quality
A Acknowledgements
This project is supported by the Pioneer Centre for Artificial Intelligence, Denmark.
Impact Statement
This paper presents work whose goal is to advance the field of machine learning in general and Bayesian filtering in particular. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
A. Convexity of the Kalman Update
The following lemma will be necessary to find the bound on Proposition 6.1.
Lemma A.I. Let Σ E 8 C E R and R E 850 then the map E + ECT (CECT + R) -1 CE is convex.
Proof. A map f: 8% $ is convex if and only if its epigraph
epi(ʃ) = (2
~, X>0,X=
is a convex set. Thus, we need to show that
X = (E)==EC
+ RR) CE
defines a convex constraint in (E, X). By the Schur complement, X = f(Z) holds if and only if
ECT cE
(Cx
0 0.
(18)
CΣC + R
Since each block of this matrix depends affinely on the pair (E, X). the set of all (E, X) satisfying the above block-positivity described by a linear matrix inequality. Since sets that are defined by linear matrix inequalities are convex, we conclude
that
epi(S) (I) (cr
ECT CyC +R
) =0}
is convex, which shows that f(Z) is convex.
Note that concavity and convexity are understood here in terms of the natural order structure associated with the notion of positive semidefiniteness (Loewner order).
B. Proof of Proposition 6.1 and Proposition 6.2
We start by writing the conditional infinitesimals (Chapter 4 in (Hanson, 2007)) for (8) and (9)
N
Σ= Σ= [- = (((", + t()" + t055",)
Σ
AOK
dt,
(19)
N. E [dg [ {*] fp(5",t) t)) + Σ Xe (tgg( .t) dt.
(20)
Let E(L) = B[E(() then de B[dd] E EE [dE E]] where the first equality follows by Fubini's theorem and the mean- square integrability condition, and the second equality by the tower property. Utilizing assumption 5.1 with Jensen's inequality, we therefore get
N. de fblE,t)+ + Σ (t)gg((E)) dt So (frlE.t) Σ ((ttgg(E.t)))d.
(21)
dé Letting de
f)( E X,(t),((E,) t). with ε()) E(O) = Eo E R"' we conclude proposition 6.2 using the comparision theorem of ordinary differential equations (McNabb, 1986; Budincevic, 2010).
Similarly, denoting E(t:5*) = (+) E(t) = E(t)'] and taking the expectation with respect to E in (8), we get
aE(1;E"):=
N
((E",t) E(EE"))(("E)) ,)) (t)("t) (C))
E) IE 1== =5") = (22)
12

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
Using Lemma A.I, we can apply Jensen's inequality obtain
dE(t;Et)
, ())(() A(t
dt
N,
+ +5(5", .t)((s")))) _ (200) (0)()))))=)))=)== (23)
dr Now 45 = A(E", 1)) 1A + (""",,,)0((00""""
(,, 1)$ F(E, with
[(0) [()) = Io E Soo By noting that F is order- order-preserving preserving in the covariance matrix ( = = F(). t F(E,<", t)), conclude E{) Ett for all using the comparison theorem of ordinary differential equations. This concludes the bound in Proposition 6.1.
C. Proof of Proposition 8.1
Proof. Since μs is supported on [o, I] with CDF Fs, the quantile function
F (p) inf e [0,T] F2(t) 2 p}
partitions the interval into subintervals (a ag] each having mass 1 under Ma- The discrete measure v, has CDF Gg with jumps at Eg. so
(p) E for p
1-1 n
n
Hence,
w (p), = (' Go) dp
min Σ
(F (p) - 22)² dp.
FF
i-1
i=1
Taking derivatives with respect to each i and setting them to zero yields
Fe
(p)
dp
E [To To [a]-1,0;]]
Uniqueness follows from the strict convexity of the squared-error objective. To see that [B[T.] = B|T,] observe that
1 n.
''" (dtt =E[rd].
Thg
since each t is the average of T. over the interval [ at]. Finally, we have Var(T.) - Var(r,) = WW3 (2a. which follows by the definition of W2 (Ms, v.).
D. Numerical Solutions for Optimal Control Problems
There are several numerical methods to solve continuous-ting time OCPs. We list here the common methods used for solving them.
Direct methods (Böhme & Frank, 2017) work by discretizing the time axis and formulating the problem as a finite- dimensional Nonlinear Program (NLP). There are three common approaches for direct methods: direct single shooting, direct multiple shooting, and direct collocation.
In direct single shooting, the control trajectories (in our case u(t) and X(t)) parametrized, for example, by polynomials, piecewise constant functions, piecewise polynomials. The system dynamics are then integrated forward (using ODE solver) to compute the objective and constraint values, with the NLP's decision variables corresponding to the control parameters. For direct multiple shooting, the time horizon divided into smaller subintervals. In each subinterval, the control trajectory is parametrized as in the single shooting case. However, the dynamics are integrated separately for each subinterval instead of integrating them one go for the entire horizon. The initial conditions for each subinterval for the
13

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
Mean Absolute Difference vs. D
Mean Absolute Difference
1.5
1.0
0.5
0.0
100
200
300
400
Number of Discretization Points (D)
Optimization Time vs. D
Optimization Time (seconds)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
COOO
100
200
300
400
Number of Discretization Points (D)
Figure Mean absolute difference between all the trajectories and one with Na 400. Computation times are reported for each Na-
ODE solver are additional decision variables for the NLP with constraints ensuring the continuity conditions of the dynamics between the subintervals. Direct collocation (von Stryk, 1993; Bock & Plitt, 1984) follows similar approach but also parametrizes the state trajectories together with the control trajectories.
Direct methods well-suited for handling problems with many constraints and benefit from efficient off-the-shelf NLP solvers such as IPOPT (Wächter & Biegler, 2006). However, the time grid becomes finer, it can lead to large-scale optimization problems.
Indirect methods (Rao, 2009; Passenberg, 2012) rely on necessary conditions for optimality (e.g.. Pontryagin's 's Minimum Principle (Rozonoer, 1959)). One formulates the costate (or adjoint) equations and boundary conditions, then solves a Boundary Value Problem (BVP). While indirect methods can offer a deeper analytical insight into the problem, they can be challenging to implement for problems with many inequality constraints and nontrivial system dynamics. Additionally, numerical tools solve BVP are sensitive to initial conditions.
In practice, the choice between direct and indirect methods often depends on problem size, constraint complexity, and the availability of good initial guesses or analytical insights.
For the examples in this paper, we choose a multiple shooting approach with forward Euler discretization and adaptive discretization steps. To ensure the positive definiteness of the covariance matrix in the NL.P setup, we parametrized it with
14

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
Cholesky decomposition. These choices are not limited, and one can, in principle, experiment with different choices based the specific problem solve.
It is important to note that discretization--asse based methods for OCPs with constraints over a horizon t E [Ti, Tyl may not strictly satisfy the constraints for all t E [Ti. 75] due to numerical approximation errors. The choice of the discretization scheme inherently depends on the specific problem structure and its solution. While increasing the number of discretization points or adopting higher-order integration methods can improve accuracy, these enhancements often come with an increased computational cost. Figure illusstrates this trade-off in two-sensor scheduling scenario for wwo-dimensional CD-KF. We apply implicit Euler discretization with varying numbers of discretization points and quantify the resulting estimation error against high-ret resolltoooooon reference (400 discretization points). Additionally, record wall-clock runtimes CoreTM Ultra 7 155H @ 1.40 GHz device with 64 GB RAM. The optimization problems are formulated in Julia using the JuMP (Dunning et al., 2017) framework and solved with IPOPT (Wächter & Biegler, 2006). As the number of discretization points increases, the estimation accuracy improves, but at the cost of higher computation times.
Intel
15

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
Optimal Measurement Rates
Optimal Defouling Inputs
1.00
Measurementt Rate
22
U1
U2
2.5
Defouling Input
0.75
2.0
1.5
0.50
1.0
0.25
0.5
0.0
0.00
0
5
10
15
20
0
10
15
20
Time (hr)
Time (hr)
Planned vs. KF Variance
True VS.
KF vs.
RTS
200
Water Quality (ppb)
Planned Σ
True Σ
110
150
Sensor
Sensor 2
100
tr
100
90
Tp
p +
50
80
S p
+
Sensor 1
Sensor 2
0
70
0
10
15
20
0
10
15
20
Time (hr)
Time (hr)
Fouling Evolution (Sensor 1)
Fouling Evolution (Sensor 2)
0.25
0.100
0.20
Fouling
0.075
Fouling
0.15
0.050
0.10
0.025
Planned E
Real E
Planned &
Real E2
0.05
Sensor
Sensor
0.000
0.00
0
5
10
15
20
0
5
10
15
20
Time (hr)
Time (hr)
Figure 3. Water fouling example. The KF estimate is 4.p and the RTS estimate is ag.
E. Additional Example: Water Quality Monitoring with Fouling and Active Defouling
In many real-world water quality monitoring applications, sensors deployed in rivers, lakes, or coastal areas are prone to fouling due to the accumulation of biological, chemical, or sedimentary deposits. Fouling degrades sensor performance by increasing the measurement noise (Delauney et al., 2010). In addition, active defouling techniques (e.g., ultrasonic cleaning) have been developed to counteract fouling, although these methods incur their own operational costs (Delgado et al., 2021). In this example, we describe model in which sensor scheduling is optimized jointly with active defouling control.
16

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
We assume that the water quality parameter evolves according linear model:
dxp- (xp Tamb) dt + dW, (0(0) ~ NNoo Eg);
(24)
with parameters set to K > O, Zamb > O, and > 0.
Each sensor provides measurement of 2p at discrete times where the noise covariance R₂ of each sensor E {1,2]
depends the fouling level via
(,(5, Rso expl A15 (50))
(25)
where Eu is the fouling level for sensor s, R10 > R20 > O, and Xf = X/2 > 0. Higher fouling leads to larger R, and, hence, lower measurement accuracy.
For each sensor 8, the fouling level evolves as
des
N, dt = (ag + ut)) (,() + Or E {
(26)
where the parameters af = =o > represent the natural cleaning rate, p 0b representing the parameters for the fouling increase per measurement with p. > pf > 0, (t) 2 0 represent active fouling control with rates 7 = > 0. Remark that the second sensor can provide more accurate measurements (Roz < Roi) but is more prone to fouling with each measurement (p} > pi) due, for example, to an increased surface area. The goal is to schedule the measurements for a CD-KF while optimizing for the defouling inputs (12) for a horizon of I' = 24 hr where the running cost is
[w1 u2] We formulate optimal control problem of the form in
L(S, E, u, x,E) = wy tr(2) + wx X w
where wy 2 Wx Wy. 0. The running constraints includes input bounds Se u Se u and an upper bound on the fouling for each sensor Sp. Sp scheme brings the trace of the covariance matrix to low level compared to its initial value. Additionally, we see that the planned measurements result in good estimates of the process. It is also worth noting how the OCP solution uses both the sensors equally in the beginning to reduce the trace of the covariance quickly. After that, the OCP solution reduces the rate of measurements for sensor while saturating the defouling input to clean it. the same time, it increases the rate of measurement for the first sensor while simultaneously increasing the defouling input at a faster rate.
22 S cg with cg > 0. Figure E shows the results. We can see from the results how the scheduling
F. Additional Example: Spacecraft Monitoring Targets
We consider two spacecraft in circular low Earth orbit at an altitude of 500 km separated by т rad. Each platform is tasked to monitor two processes on Earth's surface (for example, surface temperature and humidity). We model each target's local dynamics as an Ornstein Uhlenbeck process (equivalent to a Gaussian process with exponential kernel; see Appendix H):
dz1 = A ω1 dt + 0 dW1. dz2 Ag x2 dt + 02 dW2,
where A1, Ag < and 01, 02
Each spacecraft has two distinct sensors. The first sensor each spacecraft is sensor whose accuracy depends solar illumination. The second sensor on each spacecraft operates independently of solar illumination but requires higher energy measurement. The accuracy of the first sensor with solar illumination is better than the second one, but it become worse in the absence of illumination. Additionally, daylight sensors are more prone to fail when used excessively compared to the other sensors. To capture these sensors in our framework, we denote the output matrices C₁ and C2 for the first and second sensor in the first spacecraft, respectively, and C3 and C,, for the first and second sensor in the second spacecraft, respectively. We also denote the same for the corresponding measurement-noise platform can only observe a ground target when it falls within its line of sight, we model the output matrices according an angular visibility function:
variances R1, R2, R3, R4, Since each
Ca Org) exp /
166 (1 cos cosats 0tn)1 0bnll') )))
i (1,2), je (1.2). SE (1,2,344)
cos(8
17

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
Target 1: True vs. KF vs. RTS
Target 2: True vs. KF vs. RTS
10
31
$1 +
10
22
32
Sensor
32 + Sensor
Sensor 1
Sensor 2
Sensor
5
Sensor 3
Sensor 4
Sensor
00
21
22
-5
100
10
15
15
10
15
20
5
10
15
20
Time (hr)
Time (hr)
Σj: Planned vs. KF
Σg: Planned vs. KF
Planned Σ
KF En
Planned Σg
KF Σg
Senson
Sensor
6
Sensor
Sensor 2
Sensor
Sensor
Sensor 3
Sensor
Variance
Variance
2
0
10
15
20
5
10
15
20
Time (hr)
Time (hr)
Figure spacecraft example targeted processes. The KF estimates are ân, 42 and RTS estimates are 41, a1. See an animation of the results https: //g wtt../././///////...... http://wwww..../////////eeaaaa when2meaaurerr.
Here, 6ett denotes the angular position of the i-th spacecraft, 0r, denotes the geodetic longitude of the j-th surface target, 8 the half-cone angle defining the field of view, and 0 is shaping constant.
For the daylight-dependent sensors (Sensor 1 aboard spacecraft and Sensor 3 aboard spacecraft 2), we model the instantaneous measurement variance as a smooth function of the solar illumination angle:
R3 (Oeat,: 00) = Rmin Pro (Deat.: 00) + Rmax
- Pro (0 (Omatez [o)], i (1,2), S (1,3).
where
(0, 0²) + exp( -k cos - 0'))
where 0g is the solar illumination angle, Rmax > Rmin > 0, and Ko > 0 is shaping constant determining how sharply the variance transitions from Rmin (when the platform and sun are coaligned) to Rmax- For the solar independent sensors (sensor 2 and sensor 4), we fix the variance at nominal value
R₂ R4 Ræ
18

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
A1 and X2
X and A4
20
Measurement Rates
Measurement Rates
10
10
10
15
20
10
15
20
Time (hr)
Time (hr)
Energy Evolution (Spacecraft 1)
Energy Evolution (Spacecraft 2)
50
Planned i
Real 171
Planned f
Real ma
60
Sensor
Sensor 2
Seasor
Scnsor
40
Energy (kJ)
Energy (kJ)
50
o
9
30
40
20
30
00
8
10
10
15
20
0
10
15
20
Time (hr)
Time (hr)
Figure 5. spacecraft example with measurement and the energy states. See animation of the results https://github. com//MOHAAMACC22HD333 http://ww.....aaaaeeeeeeeeeeeeee "Whenmmaasreee"
chosen that Rmin Rog Rmax. In more detailed analysis, one could distinguish Earth occultation of direct sunlight from simple misalignment, but here we adopt this smooth logistic model in order to keep the example concise.
Each spacecraft carries an onboard battery whose stored energy (n for the first spacecraft and 7)2 for the second one) evolves according to
dm Ph. Pe Pre,, (Baatt 00) u1
C2 5zg
dt
dnp ph. Pe Pri,, (Osntyy 00) U2 +
N Cs 8zg
dt
where Ps > 0 is the nominal operating power of each spacecraft, P. > 0 is the peak solar charging power, K,, > 0 is a shaping constant, U1, U2 E [0, 1] are control variables controlling the charging of each battery such that the stored energy does not exceed an upper limit, and c, the energy cost per measurement of sensor 8, with ci = C3 > C2 = C4 >
19

---

where p. = [z, y] is the position coordinates of the target, ūz, Uy are the linear velocities on the horizontal and vertical axis, respectively, and o..: Du, O with W., and W.,, being independent Wiener processes. We assume we have two sensors for the position of the target and that the measurement noise covariance matrices are given as the ones in (15) with Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
We formulate an optimal control problem with a horizon of T = 24 hr. The running cost is
C(E, Σ. u, x, wy (E, + Σg) + wx X TQ,,
where 2n and 2ig are the variances of the CD-KF estimate for first target process and the second target process, respectively, and Wy 2 wx 2 W, O, and Q diag((10 1 10 1)) used to increase the weights Sensor 1 and Sensor 3 they than Sensor 2 and Sensor 4. The running constraints includes imput bounds O S. Se 1, and a bound on the maximum stored energy 1i: S /mnax i E (1, The results are shown in Figure F and Figure F. We see from F that the scheduling scheme manages to provide estimates tracking the processes nicely. We see an obvious periodic behaviour for the variances for each process. This periodicity is natural as both spacecraft passes over the target processes with periodically. We can observe this periodicity also in Figure F for both the scheduled measurement rates and the stored energy for each spacecraft.
more prone to fail with excessive
G. A Receding Horizon Framework for Unknown or Nonlinear Dynamics
In many practical scenarios, the dynamics that govern the process we wish to filter are either uncertain or nonlinear. One approach to extend our framework to handle these cases is a Receding Horizon (RH) formulation or a model predictive framework. RH approaches enjoy a rich literature showing that, by repeatedly solving a finite-horizon OCP with updated model information, one can retain provable closed-loop guarantees even when the prediction model is misspecified or learned online (Bemporad & Morari, 2007). Since our proposed approach in this paper is cast as an OCP, we can naturally extend it to an RH setup and utilize the existing approaches for robust learning-based RH.
One way to extend our approach to an RH setup is to solve the OCP in (12) at time 4 for a horizon [4, T + 4x], obtain the measurement rates and the other possible control inputs over the horizon [tr, T + tt). use (13) to find the time f4+1 for the first sensor s" to measure within the interval [4,, T + tz]. apply the possible control for E [t,, M4+1], measure with s" at f441, update the KF estimates and the uncertain parameters of the model based on the measurement at time 4,+1, and re-solve the OCP for the horizon [1-1: T + [++1] and repeat. In case of nonlinear SSM, we can use the EKF (Simon, 2006; Särkkä & Svensson, 2023) for each OCP optimization. The EKF uses linearized dynamics for the covariance propagation, and we can update the linearized dynamics after each measurement between the mnite-horizon OCPs. To illustrate this approach, we implemented a moving target tracking task in which mobile robot with a unicycle model with an energy budget must localize and track a target whose true motion follows a constant turn rate velocity model perturbed by noise
dx, e, cos Or dt + og dW.,
dy, = U, sin + o, dW,,
de, o. dW.,
(27)
d0, w, dl + ag dWo:
du, aw dW..
where P- = Brl is the position coordinates of the target, 0, is the heading of the target, er are the heading linear and angular velocity of the target, respectively, and aa, Ty: To, do. ow > 0 with Wz, Wy, W., W, and W., being independent Wiener processes. The OCP and the CD-KF assume a constant velocity model for the target with random perturbations on the velocity only
dx 02 dl
dy G, dt
(28)
dū, ou, dW
di, = d., dW
T
Pp replaced with p, for simulation and the estimated p; in the OCP. Additionally, the energy of the robot follows the same dynamics in (14) but with a constant charging rate that does not depend on the position of the robot (more suitable for target tracking). We used an RH setup to plan for the measurements and the robot's 's velocities for a horizon of T = 0.05 where for each OCP optimization, we use the current estimate for the target according to the constant velocity model (28) instead of
20

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
1.2
E
20
20
1.0
Truc
0.8
0.6
0.6
y-coordinates
0.4
2 Time
0.4
0.8
0.2
+ 20
1 + 20
0.7
Path
Sensor 1
True y
0.6
Sensor 2
Robot
0.5
Target
0.4
0.0
0.3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
2
z-coordinates
Time
50
True 7) (Energy)
tr Σ
40
Trace of Σ
Energy
30
2
20
1
2
3
4
2
3
Time
Time
Figure 6. Target Tracking example utilizing an RH approach. The KF position estimates are à, and the RTS position estimates are 4" 0. See animation of the results https:/ http:://gittiin /ttpt/gtttttt////// hotp://oooAAMAAAAA///////eeeaee
the model (27) which was used for simulation. The running cost for each OCP is
CEE, E, e) wy tr( wx A T + w. Tu
where 2 wx 2 Wa 2 0, and the terminal cost being the distance between the robot and the predicted location of the target according to the model in (28). The running constraints C encodes the input bounds u S. u S. ū, X Ze 0, and an energy lower bound i 2 C, with 2 0. The terminal constraint Cr encodes a tighter lower bound on energy to ensure feasibility for the next optimization. The results are illustrated in Figure G. The results in Figure G show that the RH implementation manages to make the robot obtain close estimates to target's position, which enables the robot to track
21

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
closely. Additionally, we can see that the energy drops lower value close the terminal constraint energy with measurement-induuced oscillations. We also see that the same thing happens to the covariance matrix of the CD-KE.
H. Experiments for NonconcaveeNonconvex Auxiliary Dynamics
In this section, we will solve the OCP in (12) for several cases in which we modify the auxiliary dynamics for the robot example in Section and the water sensors Appendix E to violate concavity/convexity assumption 5.1. For the robot example in Section 9, we consider three different cases in which we modify equation (17) to be
dçe dt
2 1 = n exp( - ppll 2=22((2)) beg
N,
(29)
where (Sx) [_, for the first case, (5c( (1+exp(-C..) for the second and third case (the degradation cost doubles as the degradation state for each sensor increases). The difference in the third case is that we modify the energy dynamics in equation (14) to be
dn co Un (1) exp(rellp. Pell?) - c222 2² cyuu
2 N,
Ca (1) dep
(30)
dt
where v' (1) 1xexp(-(1-22)) (charging efficiency is halfed the stored energy increases), and Thy (n) (rexp(-(n-23))
+ exp(-r)²/53 (low stored energy increases the energy costs), For the example of the water sensors in Appendix E, we consider a case which fouling equation in (26) is modified to be
dep = (a! + ut)t)) (E)(t)) dt
N
Σ
próg s SE {1,2}.
(31)
The results for the different cases are shown in Figure 7, Figure 8, Figure 9, and Figure 10. The figures for robot cases show that we can still obtain satisfactory results even if the dynamics for the auxiliary states violate assumption 5.1. The same can also be said for the results of the modified water fouling example in Figure 10, except for the fouling state of the second sensor, where can be seen that the planned fouling becomes different from the true one towards the end. Nevertheless, the planned and true covariance matrix E for all of these examples are similar. These empirical examples support the idea in Remark 7.2 that our method can still work in situations where the dynamics of the perturbed auxiliary state have local near-linear behaviour.
22

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
3
Planned Σ
True Σ
1.0
Sensor
Sensor 2
0.5
2
App
0.0
-0.5
1o
True Tp Sensor
J 10
Sensor 2
-1.0
0
0000
0db 000000000
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
Time
Time
50
True S1
Planned
Planned i ... True 7)
0.125
Sensor 1
Sensor
Truc S2
Planned S2
40
Radiation damage
Sensor 1
Senso 2
0.100
Energy
30
0.075
20
0.050
10
0.025
0
0.000
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
Time
Time
Figure 7. The first modificd (nonconvex/honccocaax)) case for the robot example in Section The KF estimate is ê, and the RTS estimate
is 2..
23

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
3
Planned Σ
True Σ
1.0
Sensor
Sensor 2
0.5
2
Dp
0.0
1
-0.5
1p 1o
True Tp
$ + lo
Sensor 1
Sensor 2
-1.0
0
0000
1
0000000000000
0.00
0.25
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1.00
0.00
0.25
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Time
Time
50
True S1
Planned
Planned 7 -.. True 7)
0.100
Sensor 1
Sensor 2
True S2
Planned C2
10
Radiation damage
Sensor 1
Scnset f 2
0.075
Energy
30
0.050
20
0.025
10
0
0.000
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
Time
Time
Figure The second modified (nonconvex/nonccocaav)) case for the robot example in Section The KF estimate is â, and the RTS
estimate is 3.-
24

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
3
Planned Σ
True Σ
1.0
Sensor
Sensor 2
0.5
2
Dp
0.0
1
-0.5
S 1o
True Tp
$ + lo
Sensor 1
Sensor 2
-1.0
0
0000
000000000.000
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
Time
Time
50
True S1
Planned
Planned 7 -.. True 7)
0.100
Sensor 1
Sensor 2
True S2
Planned C2
10
Radiation damage
Sensor 1
Scnset 2
0.075
Energy
30
0.050
20
0.025
10
0
0.000
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
Time
Time
Figure The third modified (nonconvex/noncoccaae)) for the robot example in Section 9. The KF estimate is i. and the RTS
estimate is 32-
25

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
Optimal Measurement Rates
Optimal Defouling Inputs
1.00
Measurementt Rate
2.5
A
A
01
U2
Defouling Imput
2.0
0.75
1.5
0.50
1.0
0.25
0.5
0.0
0.00
0
10
15
20
10
15
20
Time (hr)
Time (hr)
Planned vs. KF Variance
True vs. KF vs. RTS
120
200
Planned Σ
True Σ
Water Quality (ppb)
110
150
Sensor
Sensor 2/
100
100
90
Tp
Bp + Sensor 1
50
80
f + o
Sensor 2
0
-6
00
70
0
10
15
20
10
15
20
Time (hr)
Time (hr)
Fouling Evolution (Sensor 1)
Fouling Evolution (Sensor 2)
0.15
0.100
Fouling
0.075
Fouling
0.10
0.050
0.05
0.025
Planned &
Real E1
Planned &
Real &
Sensor
Sensor 2
0.000
0.00
0
5
10
15
20
10
15
20
Time (hr)
Time (hr)
Figure 10. The modified water fouling example in Appendix E (nomcconee//noccoccav)))) The KF estimate is ê and the RTS estimate
25.
26

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Optimal Sensor Scheduling and Selection with Auxiliary Dynamics
I. State-Space Realization for Gaussian Processes
one-dimensional GP with a covariance kernel whose power spectral density rational admits an exact representation as an SSM (Hartikainen Särkkä, 2010; Särkkä & Hartikainen, 2012: Todescato al., 2020). In this framework, forward filtering via the CD-KF combined with backward RTS smoothing computes the exact posterior mean and covariance of GP regression (Hartikainen & Särkkä, 2010). The CD-KF and RTS algorithms achieve this with O(N) computational complexity for N measurements (in contrast to (NN) with the traditional Gaussian Process regression), thus enabling scalable inference for large datasets.
For linear-Gausee Gaussian models, the smoothing covariance matrix is always dominated by the filtering covariance in the Loewner order. Consequently, the OCP (12) inherently controls both the filtering and smoothing uncertainties. This means our framework for sensor scheduling with auxiliary dynamics in this paper can be used connection with efficient GP regression.
For kernels with non-rational spectral densities (e.g., the qquared-xpoonentne kernel), approximations be constructed using rational function expansions such as Padé approximants (Todescato et al., 2020). These approximations will yield finit-diimeniional SSM representations.
J. Unicycle Model
For unicycle model of robot, have the state
Pr 0
Pri Prr
Ep
where
Pr
represents the planar position with respect to global frame, and 0 is the heading angle. The linear (heading) velocity is and the angular velocity is w. The kinematic equations are then
e cos
Pr.
Ep-
Pro
e usin (0)
W
27


Reviews:
[{"review_id": "PLSMosy628", "reviewer": "Reviewer_UHHX", "summary": "This work considers continuous-time state-space models in which observations are taken at discrete and potentially irregular time intervals from a finite collection of different kinds of sensors, each with a potentially different accuracy and potentially different cost incurred per measurement. \n\nIn this context (for a finite time horizon), the authors propose methodology for optimising the (Poisson-process) rates at which measurements are taken by the different sensors subject to constraints on the cost and estimation accuracy. They also show how the (random) Poisson-process-generated measurement times can be approximated by a deterministic schedule.\n\nThe results are illustrated on two robot models with synthetic data.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: \nIn the last paragraph of Section 8.1, we leverage the fact that Gaussian process (GP) regression (with many common stationary covariance kernels) is equivalent to Kalman smoothing of a specific linear state-space model (see ref. Sarkka \\& Hartikainen, 2012 in the paper). This equivalence allows us to perform GP regression efficiently on the temporal process $x_p$ via Kalman smoothing. This example demonstrates how our approach can be used for sensor scheduling in connection with GP regression. We will provide an appendix section in the paper clarifying this point.\n\nQ2:\nOur work focuses on Bayesian inference in SSMs, a topic that has been explored in previous papers at ICML and other ML-related conferences (e.g., [1]). Additionally, our work can be applied to sensor scheduling in GP regression under dynamic environments.  We believe that GP regression continues to be a subject of significant interest in the machine learning community.\n\nQ3: \nWe thank the reviewer for spotting the sign error. The right-hand side of the equation should be $c_e \\exp\\bigl(r_e \\|p_r - p_e\\|^2\\bigr) - c_u v - c_u \\omega - \\sum_{s=1}^2 \\sum_{i=1}^{N_s} c_s \\,\\delta_{t^s_i}$.\n\nQ4:\nWe see the reviewer's point. We wrote it with the intention of providing more clarity for the reader on the total measurement process. But as the reviewer points out, it is not used; it may therefore contribute to more confusion for the reader than clarity. We will remove it from the paper. \n\nQ5:\nWe apologize for the ambiguity of the figure. This point has also been mentioned by the second reviewer. We will fix the figure to be more reader-friendly for the final submission. The symbol $x_p$ represents the process we want to measure. The filter estimate is now denoted as $\\hat{x}_p$, and the RTS smoother estimate is denoted as $\\hat{x}^s_p$ (representing the GP regression output). We have adjusted the legends and description of the figure with the new notation for the filter estimate and the smoother estimate (link: postimg.cc/hJZ0ByTn (as allowed by ICML)).\n\n[1]: Duran-Martin, Gerardo, et al. \"Outlier-robust Kalman Filtering through Generalised Bayes.\" International Conference on Machine Learning. PMLR, 2024.", "claims_and_evidence": "The claims are supported by mathematical proof.", "methods_and_evaluation_criteria": "yes.", "theoretical_claims": "I did not find any issues. But I did not check the proofs in the appendix.", "experimental_designs_or_analyses": "The numerical illustrations seem fine. But I did not check any code.", "relation_to_broader_scientific_literature": "As the authors discuss in Section 2, similar optimal control problems have been analysed previously in slightly different settings, e.g., assuming that the latent states evolve in discrete or that the cost of taking measurements is independent of the latent state. The authors also point out further connections to existing literature on Bayesian optimisation.", "essential_references_not_discussed": "None that I'm aware of.", "other_strengths_and_weaknesses": "I think this manuscript is overall well written, well structured and thus quite clear. The contributions seem sufficiently novel and the authors mention a few real-world areas in which such problems arise", "other_comments_or_suggestions": "- L280: \"Wasserstien\" -> \"Wasserstein\"\n- L288: \"devide\" -> \"divide\"\n- L175: \"$\\leq_e$\" denotes an elementwise inequality? Perhaps define this.\n- L194: \"togheter\"\n- Remark 5.3: I think there is a \"$|$\"-symbol missing in the penultimate line, as well as a redundant \".\" inside the expectation.\n- Section 4: In the first paragraph, maybe add a sentence explaining the role of the functions $g_s$.\n- P2: In the \"Active Sensing\" paragraph, a few of the \\citep citations should be \\citep.\n- L398: \"true simulated true\"\n- Bibliography: Inconsistencies in capitalisation/abbreviation of journal/conference names. Missing capital letters in some names, e.g. \"kalman\" or \"gaussian\".", "questions_for_authors": "1. Can you explain the connection with / use of Gaussian process regression in the last paragraph of Section 8.1? I could not follow this aspect. \n\n2. Can you add more motivation linking this work to machine learning? It is not fully clear to me why the topic is appropriate for a machine-learning conference.\n\n3. Is Eq. 15 correct? I am confused by some of the signs.\n\n4. What is the need for/role of $Y_s(t)$ defined in Eq. 8? Maybe I've missed it but I don't think it is ever used.\n\n5. In Fig. 1, there seem to be several (different?) uses of the symbol $x_p$. The caption states that it represents output of the RTS smoother but according to the legend also seems to represent output of the filter. More generally, is \"$x_p$\" actually meant to be \"$\\xi_p$\"?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "e2VAxOLfu3", "reviewer": "Reviewer_nw2r", "summary": "In this paper, the authors are concerned with optimizing temporal event sequences of measurements for minimizing the uncertainty of continuous-discrete Kalman filter (CD-KF). In particular, they consider a general case where the measurements may affect the underlying states of sensors themselves as well as the measurement target through differential equations.\n\nThe proposed method works in three steps. First, the discrete event sequences of measurements are substituted with the intensity functions of the corresponding time-inhomogeneous Poisson processes, which is continuous and more optimization-friendly, and the time evolution of relevant parameters (such as the target uncertainty and the sensor states) is approximately given in terms of the intensity functions.\nSecond, the intensity functions are optimized in terms of a user-defined objective and constraints.\nFinally, the temporal event sequences are recovered by quantizing the intensity functions. \n\nThe authors also demonstrated the feasibility of the proposed method in illustrative examples of energy-limited measurement robot.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Unknown dynamics:  \nOur formulation as an Optimal Control Problem (OCP) with a differentiable cost function and constraints opens avenues for extension to uncertain dynamics using robust/stochastic OCP methods [2,3]. For completely unknown dynamics, planning is challenging since we must schedule measurements based on dynamics we do not yet know. We believe our approach paves the way for future research in this direction.\nOne approach to handling unknown parameters is a Receding Horizon (RH) setup. We solve a finite-horizon OCP, assuming fixed system dynamics and parameters over the prediction horizon based on current estimates, then apply the control until the first measurement. The subsequent measurement updates the parameters, and the OCP is resolved for the next horizon. This iterative process adapts as more information becomes available. In the revised version, we will include successful results using RH for a moving target (link: postimg.cc/S2t1gQDW (as allowed by ICML)).\n\nPerformance characteristics:  \nPerformance, particularly regarding discretization steps, depends on the auxiliary dynamics (e.g., stiffness, fast/slow dynamics, stability) and the KF’s covariance dynamics. Different OCP methods offer trade-offs between accuracy and computational performance (see Appendix D). Our work employed Euler discretization with variable steps while keeping a fixed number of discretization points. In the revision, we will include a figure demonstrating the trade-off between computation time and discretization points of a specific example focused on the KF dynamics (link: postimg.cc/ykL46hPv).\n\nPenalizing auxiliary states:  \nOften, the auxiliary state carries physical meaning, making its penalization more intuitive. For instance, in our examples, the energy state depends on measurement rates, velocities (actions), and the robot’s position. Penalizing measurement rates is less straightforward than penalizing energy consumption when energy is the limiting factor.\n\nConstraints handling:  \nBoth direct and indirect methods for constrained OCP lack exact guarantees of constraint satisfaction due to numerical errors inherent in the OCP solution and ODE integration. Some methods (e.g., direct collocation) can achieve high accuracy but at increased computational cost (See Appendix D). Ultimately, the choice of method and integration scheme depends on the specific dynamics, much like the selection of ODE solvers. We will include this important remark in the paper.\n\nFor gradient computations and optimization in the example, we used an interior-point method with JuMP—a Julia package that employs automatic differentiation to compute the gradient and Hessian of the Lagrangian.\n\nRegarding $\\xi$:  \nWe acknowledge the reviewer's concern. To clarify, $\\xi^*$ from Proposition 5.1 can be any curve (ensuring (10) is well-defined, though not stated explicitly), so $\\hat{\\xi}$ and $\\xi^*$ need not be close. Rather, $\\xi^*$ serves as a placeholder, allowing substitution with $\\hat{\\xi}$. Proposition 5.1 states that for any curve $\\xi^*$, the mean covariance $\\bar{\\Sigma}(\\xi^*)$ is bounded by $\\hat{\\Sigma}(\\xi^*)$. This result is applied by substituting $\\xi^*$ with the curve $\\hat{\\xi}$ obtained from (12) and (6b) with initial condition $\\xi_0$, which, per Proposition 5.2, bounds the mean curve $\\bar{\\xi}$ of $\\xi=(\\xi_p,\\xi_u)$ from the SDE (10) and ODE (6b). The deterministic quantities $\\bar{\\xi}$ (and $\\bar{\\Sigma}$) _serve as computationally tangible approximations of the stochastic_ $\\xi$ (and $\\Sigma$) (_approximating mean behaviour_). Alternatively, sampling methods can be used to compute statistical representations for the trajectories in the OCP. However, this will introduce non-differentiability and will be computationally intensive. We will note this important remark in the revision. We apologize for the initial lack of clarity.\n\nExperimental design questions:  \n1) As mentioned, the RH approach can handle moving targets (we have implemented an example for it). Nonetheless, many scenarios involve a fixed process location (e.g., gas leak, specific object temperature) or applications independent of process location, such as underwater measurements with sensor biofouling. A detailed example of the latter will be provided in the paper.\n2) The process assumes a SSM representation of the Matern kernel, with output $x_p$. The parameters $A$ and $\\sigma$ in equation (13.b) are based on this representation. We will update the figure to a more user-friendly version. See also the reply to reviewer 6UyF.\n\n[2] Leeman, Antoine P., et al. \"Robust optimal control for nonlinear systems with parametric uncertainties via system level synthesis.\" the 62nd IEEE Conference on Decision and Control (CDC). IEEE, 2023. \n\n[3] Bemporad, Alberto, and Manfred Morari. \"Robust model predictive control: A survey.\" Robustness in identification and control. London: Springer London, 2007. 207-226.", "claims_and_evidence": "They claim that\n1. the proposed method captures practical scenarios, and\n1. the proposed method has some potential and feasibility.\n\nThese points are generally well supported, but it would be nice to discuss more around\n* how to operate the proposed method under unknown system dynamics,\n* performance characteristics such as computation time vs discretization width of differential equation, and\n* benefits of penalizing/constraining $\\lambda$ indirectly through the auxiliary state $\\xi_p$ rather than doing it directly.", "methods_and_evaluation_criteria": "Mostly yes.\nHowever, a key component of the proposed method is not well described.\nIn particular, how do you compute the gradient of the intensity functions under numerical solution of (13)?\nMore specifically,\n* how do you handle these constraints (13g,h) with gradient-based optimization?\n* is it tractable even if the step size in time of the numerical solver is very small?", "theoretical_claims": "I only follow the reasoning in the main text, but it seems mostly reasonable.\nOne thing I have noticed is that there is no theoretical justification on the substitution $\\xi^*\\gets \\hat{\\xi}$ in (13).\nThese two auxiliary states are not necessarily close to each other because $\\hat{\\xi}$ is smooth while $\\xi^*$ is jaggy.", "experimental_designs_or_analyses": "- For the experimental design, I find it a bit unusual that the measurement noise depends on the distance of the robot and a fixed reference point (\"location of process\") $||p\\_r-p\\_p||$. It should ideally depend on the $||p\\_r-x||$, where $x$ is the target location that is moving randomly over time.\nThis I think also reveals a limitation of the proposed method, that is, the scale of the measurement noise cannot depend on the target state $x$.\n\n- Another thing is that it is unclear how the state-space representation of the Matern kernel for GP is used in the experiment.\n\n- For the analyses, Figure 1 can be more reader friendly.\nFor example, what is RTS? How did you draw it?", "relation_to_broader_scientific_literature": "The key contribution is making the Kalman filter applicable to more practical scenario involving irregularly-timed measurements and auxiliary states, which is novel as far as I can tell.", "essential_references_not_discussed": "None that I am aware of.", "other_strengths_and_weaknesses": "None.", "other_comments_or_suggestions": "- L144 left: $t_i\\to t_i^-$\n- L153 left: \"optimal\" in what sense?\n- L212 left: $\\bar{\\Sigma}(0;\\xi^*)=\\Sigma\\_0$?\n- L175 right: Define $\\le_e$\n- L219 right: $\\ge_0\\to \\ge_e$?\n- L253 left: Proposition 5.2?\n- L323 left: wrong signs\n- L281 right: ambiguous usage of inequalities\n- L297 right: why constraining only with $t\\le 1/2$?\n- L325 right: $\\gamma_s$?", "questions_for_authors": "Please comment on the points I raised in Claims And Evidence, Methods And Evaluation Criteria, and Experimental Designs Or Analyses.\nThis may affect my score.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "B9r4CeWr13", "reviewer": "Reviewer_6UyF", "summary": "The paper addresses the problem of optimal sensor scheduling and selection in Continuous-Discrete Kalman Filtering (CD-KF) for Bayesian State-Space Models (SSMs), where continuous-time processes are observed through multiple sensors with discrete, irregularly timed measurements. The novelty of the work lies in the incorporation of *auxiliary state dynamics*, which influence the measurement process (e.g., sensor energy constraints, environmental conditions). The authors model sensor measurements as *inhomogeneous Poisson processes* and derive an upper bound on the mean posterior covariance matrix, which is continuously differentiable in sensor measurement rates, allowing *gradient-based optimization*. The main contributions are:\n1. A differentiable upper bound on the mean posterior covariance of CD-KF.\n2. A finite-horizon *optimal control framework* that jointly optimizes measurement rates, auxiliary dynamics, and covariance constraints.\n3. A *deterministic scheduling method* for selecting actual measurement times using optimal quantization, minimizing Wasserstein distance from the Poisson distribution.\n4. Empirical results in *state-space filtering and dynamic Gaussian process regression*, demonstrating improved trade-offs between accuracy and resource usage.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: Besides the auxiliary dynamics, our method considers the continuous-discrete setup and the flexibility of not requiring an upper bound on the number of measurements. The method by [Le Ny et al., 2009] is for a continuous-continuous setup, while the method in [Marelli et al., 2019] is for a discrete-discrete setup. We will make sure to clarify this point further in the paper. \n\nQ2: Scalability:\nFor each new sensor we consider, we will have an intensity rate, which we will need to optimize for. This is typical for optimization-based scheduling procedures. \nConsidering efficient optimization schemes for large-scale problems will require investigating techniques such as distributed optimization, parralization, and sparsity considerations. Another challenge for scalability is the dimension of the states of the SSM we want to estimate. If we have $n_x$ states, then the covariance matrix of the CD-KF will be in the order of $n_x^2$. However, for large dimensional SSM, there exist research for efficient approximations and methods to deal with the covariance matrix problem. One solution, for example, is to consider a diagonal approximation for the covariance matrix or a low-rank approximation (see [4]). These approximations can be easily integrated into our framework. We will include a remark discussing this in the final paper. \n\nConcavity Assumption: We emphasize that even under the concavity assumption, our approach spans a wide range of dynamical systems (it covers all linear parameter varying systems). This work is the first to explore sensor scheduling with auxiliary dynamics. Additionally, our method can be implemented within a Receding Horizon (RH) framework (refer to Reviewer nw2r's response for a detailed description), where each optimization iteration is addressed by either linearizing the non-convex/non-concave auxiliary state or employing a convex/concave approximation. If we use nonconvex/nonconcave dynamics for $\\xi_p$, then we will lose the theoretical guarantees of the method. However, we may still obtain satisfactory results depending on how good the approximation $\\frac{d{\\mathbb{E}[\\xi_p]}}{dt}\\approx f_p(\\mathbb{E}[\\xi],u,t)+\\sum^{N_s}_{s=1} \\lambda_s(t)g_s(\\mathbb{E}[\\xi],u,t))$ (or an upper bound approximate). This approximation is similar to the approximation used in the EKF. \nWe conducted an experiment with non-concave dynamics for the sensor degradation states $\\zeta_1$ and $\\zeta_2$ of the example in the paper. The experiment demonstrated that our method remains effective (link: postimg.cc/jWcFJ4f3) (as allowed by ICML). We will include this discussion with the experimental results in the paper.\n\nQ3:\nThe KF is usually chosen in the literature of sensor scheduling instead of the PF for the fact that the covariance matrix dynamics are independent of the actual measurements and because it scales better with large dynamical systems.  However, we can still extend our approach with the EKF by utilizing an RH setup where, for each short horizon, we use the linearized dynamics around the current estimate. We will include a remark about this point in the paper. \n\nQ4: We apologize to the reviewer as we did not understand what the reviewer meant by \"high-dimensionality of the problem\" in connection with our paper. If this refers to scalability, then we have addressed it above. The nonlinear dynamics of the problem were also addressed above.\n\nComparisons: We acknowledge and agree with the reviewer's suggestion to provide comparisons. We have conducted comparisons with heuristic approaches for scheduling measurements (greedy approach and random sampling of measurement times) that we will provide in the paper (Table link: postimg.cc/qz2Q3GVc). The results suggest that our method (\"Optimized\") outperforms the greedy and random scheduling approaches for our example. To assess the deterministic scheduling computed based on the optimized measurement rates (denoted \"Optimized\" ), we compared it with a method (denoted as \"M-Optimized\" in the table) that is based on sampling $M_c$ realizations of measurement times according to the corresponding Poisson process with the optimized rates. Afterwards, we pick the measurement times corresponding to the realization with the minimal cost. The results show that our deterministic quantization provides similar results to \"M-Optimized\" without having to sample multiple realizations, which can be computationally expensive and unrealistic since we do not have the real measurements to compute the cost for each realization. \nTo the authors' best knowledge, no reinforcement learning methods applicable to this setup have been found for comparison (i.e., _multiple_ sensor scheduling in a _continuous-discrete_ setting that _does not require training on pre-obtained data with uniform sampling_).\n\n[4] Chang, Peter G., et al. \"Low-rank extended Kalman filtering for online learning of neural networks from streaming data.\" Conference on Lifelong Learning Agents. PMLR, 2023.", "claims_and_evidence": "### 1. **Upper Bound on Posterior Covariance**  \n#### The authors derive an upper bound on the mean posterior covariance matrix of the Continuous-Discrete Kalman Filter (CD-KF) in scenarios where sensor measurements follow inhomogeneous Poisson processes.  \n#### This bound is shown to be continuously differentiable with respect to sensor measurement rates, making it amenable to gradient-based optimization (Proposition 5.1).  \n#### The derivation relies on Jensen’s inequality and the monotonicity properties of the Kalman update function to ensure that the bound holds in expectation.  \n#### The correctness of the bound is theoretically proven and further validated through numerical simulations (e.g., tracking the true covariance in experimental results).  \n### 2. **Finite-Horizon Optimal Control Formulation**  \n#### The problem is formulated as an optimal control problem over a finite horizon, where the goal is to jointly optimize: \n- Sensor measurement rates (modeled as control variables).\n- Auxiliary-state dynamics (e.g., energy constraints, environmental interactions).\n- Constraints on posterior covariance to balance accuracy and resource constraints.\n#### The control framework is mathematically well-posed, ensuring feasibility through a continuously differentiable cost function and constraints (Equations 13a–13h).  \n#### The authors provide conditions (Assumption 6.1) ensuring that at least one admissible solution exists.  \nEmpirical validation in a robotic sensing task demonstrates that the optimized sensor scheduling effectively balances accuracy and energy constraints.\n### 3. **Deterministic Measurement Scheduling**\n#### After optimizing sensor rates, a deterministic scheduling method is proposed to convert Poisson-based measurement rates into actual measurement times.\n#### The scheduling method is based on optimal quantization, minimizing the Wasserstein-2 distance between the optimized Poisson rate distribution and a deterministic empirical schedule (Proposition 7.1).\n#### This ensures that measurements are distributed optimally in time, avoiding the risk of stochastic fluctuations that could degrade performance in real-world applications.\n#### The closed-form solution for quantization-based scheduling makes it computationally efficient, allowing practical deployment in real-time systems.\n### 4. **Empirical Validation**  \n#### The proposed approach is validated through two robotic sensing experiments:   \n- A standard environment where a robot optimizes its trajectory and measurement schedule to minimize state estimation uncertainty.\n- A radiation exposure scenario, where sensor performance degrades due to environmental conditions, requiring adaptive scheduling to maintain accuracy.\n#### The Kalman filter's posterior covariance estimates obtained through the proposed approach closely track the true values, demonstrating the validity of the upper bound.  \n#### The optimized measurement schedule leads to improved trade-offs between estimation accuracy and resource usage compared to naive scheduling strategies.  \n#### The empirical results confirm that the deterministic selection of measurement times closely matches the expected Poisson distribution while reducing variance.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are well-aligned with the problem, offering a rigorous and practical approach to sensor scheduling in Continuous-Discrete Kalman Filtering (CD-KF).\n### 1. **Modeling Sensor Measurements with Poisson Processes**  \n#### Using inhomogeneous Poisson processes is appropriate, as real-world sensors often collect data at irregular intervals due to resource constraints.\n#### The differentiability of the posterior covariance bound enables efficient gradient-based optimization, making the approach computationally feasible.\n### 2. **Optimal Control Formulation**  \n#### The authors formulate a finite-horizon optimal control problem that optimizes: \n- Sensor measurement rates to balance estimation accuracy and resource constraints.\n- Auxiliary state dynamics (e.g., energy usage, environmental effects).\n- Covariance constraints to maintain estimation quality.\n#### The formulation generalizes existing Kalman filter-based sensor scheduling by incorporating dynamic constraints and auxiliary state interactions.\n### 3. **Deterministic Scheduling via Optimal Quantization**  \n#### Instead of randomly sampling from a Poisson process, the authors propose a deterministic scheduling strategy based on Wasserstein distance minimization, ensuring that measurement times closely match the optimized rates while reducing variability.\n### 4. **Evaluation through Robotic Sensing Experiments**  \n#### The experiments validate the approach in two scenarios: \n- Standard sensor scheduling, optimizing a robot’s trajectory and measurement plan.\n- Radiation-damage scenario, where sensor degradation requires adaptive scheduling.\n#### Results show that the method improves estimation accuracy and reduces resource usage, confirming the effectiveness of the proposed optimization framework.\n### 5. **Potential Areas for Improvement**  \n#### Comparative baselines (e.g., reinforcement learning, heuristic policies) would provide stronger empirical validation.  \n#### Scalability analysis for larger sensor networks is not extensively discussed.  \n#### Sensitivity analysis on how different auxiliary constraints affect scheduling decisions would enhance generalizability.", "theoretical_claims": "## Theoretical Claims\n### 1. Proposition 5.1 (Covariance Matrix Bound)\n### This proposition establishes an upper bound on the covariance matrix of the system state under specific assumptions about the dynamics and noise characteristics.\n### The derivation relies on standard results in stochastic process theory and Lyapunov analysis, ensuring that the covariance remains bounded given certain stability conditions.\n### Correctness Check: Upon careful review, the proof appears rigorous, leveraging a decomposition of the state transition matrix and spectral properties of the covariance evolution. However, it would be beneficial to validate this bound numerically against empirical estimates to confirm that the theoretical bound is not overly conservative.\n### 2. Proposition 5.2 (Auxiliary State Bound)\n### This result provides an upper bound on the auxiliary state variable, which is introduced to facilitate the analysis of system evolution.\n### The proof depends on a key assumption: the concavity of the auxiliary dynamics function, which is explicitly stated in Section 5.2.\n### Correctness Check: The derivation correctly follows from Jensen’s inequality and properties of concave functions, ensuring that the bound holds under the given assumptions. The proof structure is sound, but a sensitivity analysis could further strengthen confidence in the result by assessing its robustness to variations in model parameters.\n### 3. Proposition 7.1 (Optimal Quantization Points for Deterministic Scheduling)\n### This proposition addresses the selection of quantization points that minimize a given distortion metric in the context of deterministic scheduling.\n### The proof constructs an optimization problem based on a distortion-cost function and derives conditions for optimality.\n### Correctness Check: The derivation is well-structured, employing Lagrange multipliers and Karush-Kuhn-Tucker (KKT) conditions to find the optimal quantization points. The reasoning follows standard optimization techniques, and the proof is logically sound. Nonetheless, a comparison with numerical optimization results would help validate the theoretical predictions.\n## Assumptions and Justifications\n### The paper explicitly states several key assumptions, such as the concavity of auxiliary dynamics and boundedness of noise processes.\n### These assumptions are reasonable and well-motivated, as they align with standard conditions in stochastic control and optimization literature.\n### The authors provide sufficient theoretical justifications for these assumptions, discussing their necessity in establishing key results.\n## Areas for Further Validation\n### While the theoretical derivations appear correct upon inspection, certain results (especially Proposition 5.1) could benefit from numerical validation to ensure that theoretical bounds align well with empirical observations.\n### Additionally, sensitivity analyses on the concavity assumption and parameter variations would further establish the robustness of the results.", "experimental_designs_or_analyses": "### The experimental design and analyses presented in the paper are generally sound, effectively demonstrating the feasibility and practical implications of the proposed approach. Below, we assess key aspects of the experimental setup, evaluation metrics, and areas for improvement.\n### 1. Robotic Sensing Experiments (Standard and Radiation-Damage Scenarios)\n### The experiments are designed to test the performance of the proposed scheduling method in realistic sensing environments, including a standard scenario and a radiation-damage scenario, where sensor degradation over time is considered.\n### The standard scenario provides a baseline where all sensors function optimally, allowing for a controlled evaluation of scheduling effectiveness.\n### The radiation-damage scenario introduces progressive sensor failures, testing the method’s adaptability under real-world constraints.\n### Assessment: The experiments successfully demonstrate how the algorithm adjusts to sensor degradation and limited resources, making the setup practically relevant and well-motivated. However, additional robustness tests under more severe failure models (e.g., abrupt sensor loss) could further strengthen the analysis.\n### 2. Evaluation Metric: Trace of Covariance Matrix (Σ) in Kalman Filtering\n### The paper employs the trace of the covariance matrix Tr(Σ)\\text{Tr}(\\Sigma) as the primary metric for evaluating estimation accuracy.\n### This metric is well-justified in Kalman filtering applications, as it provides a measure of overall uncertainty in the state estimate: Tr(Σ)=∑iλi\\text{Tr}(\\Sigma) = \\sum_{i} \\lambda_i where λi\\lambda_i are the eigenvalues of Σ\\Sigma, representing state estimation uncertainty.\n### Assessment: This choice is appropriate and aligns with standard filtering performance criteria. However, supplementing it with alternative uncertainty measures (e.g., determinant of Σ\\Sigma, worst-case eigenvalue analysis) could provide a more comprehensive evaluation.\n### 3. Validation of Trade-Off Between Accuracy and Resource Constraints\n### The experiments illustrate the trade-off between estimation accuracy and resource constraints, demonstrating how sensor scheduling affects estimation performance.\n### By evaluating different scheduling strategies, the paper effectively highlights the impact of limited sensor availability on overall system performance.\n### Assessment: The experimental results clearly support the theoretical claims. However, the inclusion of additional baselines would enhance the comparative analysis, such as: \n### - Randomized sensor activation, to assess whether the proposed method significantly outperforms naive random selection.\n### - Heuristic-based scheduling, which could provide a lower-complexity alternative for practical use.", "relation_to_broader_scientific_literature": "1. The work extends classical sensor scheduling in Kalman filtering (Le Ny et al., 2009; Marelli et al., 2019) by integrating continuous-discrete modeling and auxiliary state constraints. \n2. Connections to active sensing and reinforcement learning-based sensor selection (Yoon et al., 2018; Qin et al., 2024) are briefly mentioned but could be expanded. \n3. The paper aligns with recent trends in Bayesian optimization for experimental design (Snoek et al., 2012; Kleinegesse & Gutmann, 2020), but a direct comparison with Bayesian optimization-based approaches is missing.", "essential_references_not_discussed": "1. The authors should consider discussing recent works on RL-based sensor scheduling (e.g., deep RL for adaptive sensing in sequential decision-making problems).\n2. If there exist alternative stochastic optimal control methods for sensor scheduling, citing those would strengthen the discussion.", "other_strengths_and_weaknesses": "### **Strengths**:\n1. Well-motivated and practical problem – The paper addresses a real-world challenge in sensor scheduling, with applications in robotics, healthcare, and environmental monitoring.\n2. Theoretically rigorous approach – The differentiable upper bound on the posterior covariance and the optimal control formulation provide a solid mathematical foundation for optimization.\n3. Effective deterministic scheduling method – The Wasserstein quantization-based approach ensures that actual measurement times closely match optimized rates, improving reliability in planning tasks.\n### **Weaknesses**:\n1. Lack of comparative baselines – The paper does not compare its approach with reinforcement learning-based or heuristic sensor scheduling methods, making it harder to assess its relative advantages.\n2. Limited sensitivity analysis – The impact of different auxiliary state dynamics (e.g., non-convex constraints, stochastic effects) on sensor scheduling decisions is not extensively explored.\n3.  Potential generalization limitations – The concavity assumptions for auxiliary state dynamics may restrict the applicability of the method to nonlinear or highly dynamic real-world environments.\n\n### **Minor Typos & Formatting Issues**  \n\n### **1. Notation Consistency Issues**  \n- **Equation (9) (Randomized Covariance Matrix Evolution)**  \n  - The notation for **Kalman gain** \\( K_s(\\Sigma, \\xi, t) \\) varies slightly across equations. Ensure consistency in using subscripts and argument ordering.  \n- **Equation (11) (Upper Bound on Covariance Matrix)**  \n  - The function \\( \\hat{\\Sigma}(t) \\) is introduced as a bound, but in some places, it is written without the hat (\\(\\Sigma(t)\\)), which may cause confusion.  \n- **Equation (12) (Auxiliary State Evolution Bound)**  \n  - The auxiliary state update function **\\( f_p(\\xi, u, t) \\)** and **\\( g_s(\\xi, u, t) \\)** use different orderings in some parts of the text—ensure consistency.  \n\n### **2. Typographical Errors & Formatting Issues**  \n- **Page 2, Line 35:** `\"axillary states of an SSM\"` → should be `\"auxiliary states of an SSM\"`.  \n- **Page 3, Line 80:** `\"measurements can in- crease energy consumption\"` → should be `\"measurements can **increase** energy consumption\"` (remove extra hyphen).  \n- **Page 4, Line 128:** `\"dynamics are inhomoge- neous Poisson processes\"` → should be `\"dynamics are **inhomogeneous** Poisson processes\"` (remove hyphen).  \n- **Page 6, Line 192:** `\"togheter with ncT continuously differentiable terminal constraints\"` → should be `\"together with \\( n_c^T \\) continuously differentiable terminal constraints.\"`", "other_comments_or_suggestions": "1. Comparative Baselines – Including reinforcement learning-based or heuristic scheduling methods as baselines would strengthen the empirical validation. This would help demonstrate the advantages of the proposed optimal control formulation over alternative approaches.\n\n2. Sensitivity Analysis – Conducting an analysis on how different auxiliary state dynamics (e.g., non-convex constraints, stochastic transitions) affect scheduling decisions would improve the generalizability of the approach. This is particularly important for real-world applications where sensor conditions may change unpredictably.\n\n3. Assumption Justification – The concavity assumption for auxiliary state dynamics is reasonable in some cases but may not always hold in practical scenarios. Discussing potential relaxations of this assumption or alternative formulations for non-convex cases would improve the paper’s robustness.\n\n4. Scalability Considerations – While the current experiments demonstrate feasibility, additional discussion on scalability to larger sensor networks or more complex environments would be beneficial. How does the method scale with an increasing number of sensors or constraints?\n\n5. Clarifications on Deterministic Scheduling – The Wasserstein quantization-based deterministic scheduling is an interesting contribution. However, additional discussion on its limitations and trade-offs (e.g., impact on computational efficiency, adaptability to dynamic sensor failures) would provide more insight into its practical deployment.\n\n6. Minor Typos & Formatting – The paper is generally well-written, but a careful proofreading pass would help eliminate minor typos or inconsistencies in notation (if any). Specific sections, such as theoretical derivations, could benefit from additional explanations to improve clarity for a broader audience.", "questions_for_authors": "### Q1. How does your approach fundamentally differ from prior work on sensor scheduling in Kalman filtering (e.g., [Le Ny et al., 2009], [Marelli et al., 2019])? Beyond incorporating auxiliary dynamics, what unique advantages does your upper-bound formulation offer over existing stochastic control or active sensing approaches?\n\n\n### Q2: How does the method scale to large-scale sensor networks with multiple interacting sensors? Can it handle non-convex auxiliary state dynamics, or does the concavity assumption significantly restrict its applicability?\n\n### Q3:  Why is Kalman Filtering chosen over Particle Filtering? Would your approach still be effective for *nonlinear or non-Gaussian*state-space models where Kalman filtering is suboptimal? Could *Particle Filtering (PF)* or *Extended Kalman Filtering (EKF)* be viable alternatives?  \n\n### Q4:  What contributes to the high-dimensionality of the problem?  Is the complexity mainly due to *the number of sensors, control variables, or auxiliary state interactions*? How would the method extend to *nonlinear* systems?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "No ethical concerns were identified."}}]

Author rebuttal:

Meta-review: This paper proposes an optimal control based approach to sensor scheduling in continuous-discrete time Kalman filter.
The consensus is that the problem is interesting and the solution is well motivated.
However, the empirical validation of the method is on the weak side since no theoretical guarantees are given.
Decision: Accept (poster)