In this section, we provide a general framework for the (discrete-time) sensor scheduling problem and briefly review the corresponding literature.
Consider the discrete-time linear system
x[k + 1] = Ax[k] + w[k], (4.1)
where x[k] ∈ Rn is the system state, w[k] ∈ Rn is a zero-mean white Gaussian noise process with Ew[k](w[k])T = W for all k ∈ N, and A ∈ Rn×n is the system dynamics matrix. We assume throughout this chapter that the pair (A, W12) is stabilizable.
The set of sensors to be chosen must come from a given set Q consisting of q sensors. Each sensor i ∈ Q provides a measurement of the form
yi[k] = Cix[k] + vi[k], (4.2)
where Ci ∈ Rsi×nis the state measurement matrix for that sensor, and vi[k] ∈ Rsiis a zero-mean white Gaussian noise process. For convenience, we define
y[k] , y1[k] .. . yq[k] , C , C1 .. . Cq , v[k] , v1[k] .. . vq[k] .
Then the measurement equation corresponding to the output of all sensors is
y[k] = Cx[k] + v[k]. (4.3)
We denote Ev[k](v[k])T = V and take E v[k](w[j])T = 0 for all j, k ∈ N.
Let zi[k] be the indicator variable of sensor i at time-step k, i.e., zi[k] ∈ {0, 1} and zi[k] = 1 if and only if the measurement of sensor i is chosen at time-step k. Based on the requirement of the specific application, one can specify certain constraints on the selection of sensors (e.g., impose an upper bound on the number of sensors that can be chosen at each time-step). Let z[k] ∈ {0, 1}q be the indicator vector of the set of chosen sensors at time-step k, and denote the set of feasible indicator vectors (which satisfy the specified constraints) at time-step k to be Z[k]. Given an estimation strategy for the state x, the sensor scheduling problem is to design a scheduling policy P = {z[k]|z[k] ∈ Z[k], ∀k} such that the average error covariance of the cor- responding estimator ˆxP is minimized. Specifically, for the finite-horizon version of the problem over time interval [0, T ], the objective is to solve the following optimization problem:
min P,ˆxP JT , 1 T T X k=1 E[(x[k] − ˆxP[k])TQk(x[k] − ˆxP[k])],
where {Qk} is a sequence of semidefinite weighting matrices.
For the infinite-horizon version, the goal is to solve the following problem: min
P,ˆxP
lim sup T →∞
JT.
Note that the specific form of the estimator ˆxP depends on the estimation strategy adopted and its performance depends on the scheduling policy P. A common choice of the estimation
strategy is Kalman filtering and the corresponding sensor scheduling problem has been studied extensively, especially the finite-horizon version of the problem.
In [40], the Kalman filter is used for state estimation after data fusion and the authors pro- posed a stochastic sensor scheduling algorithm to minimize an upper bound on the expected steady state estimation error covariance. In [58], the authors gave a convex relaxation based ap- proach for parameter estimation, which provided a general framework for various cost functions (e.g., performance criteria or energy and topology constraints); however, [58] assumes that the sensor measurements are uncorrelated, which may not be true in practice. Thus, in [95], an- other framework is proposed to handle correlated measurements. Some other interesting works include [144], where optimal and suboptimal sensor scheduling algorithms based on tree prun- ing techniques are given, and [74], where the optimization problem is decomposed into coupled small convex optimization problems which can be solved in a distributed fashion. However, so far, the solutions proposed for the finite-horizon problem either are computationally inefficient or consist of heuristics with no guaranteed performance (except for the greedy policies relying on submodularity of the corresponding objective functions) [55].
Recently, the infinite-horizon sensor scheduling problem has received increasing attentions, e.g., see [56,71,74,99,150]. In [71], the authors considered the continuous-time sensor schedul- ing problem; leveraged the results of the classical Restless Bandit Problem, they provided an- alytical expressions for a simplified scalar version of the problem and proposed a family of periodically switching policies for the multi-dimensional systems. In [74], the authors studied a discrete-time version of the problem. While the original problem is deterministic, they pro- posed a stochastic strategy with performance bounds; moreover, they prove the monotonicity and trace-convexity properties of the underlying discrete-time modified ARE (MARE) and provided a closed-form solution for a special class of MARE. In [56], the authors considered the discrete- time sensor scheduling problem and characterized the conditions under which there exists a schedule with uniformly bounded estimation error covariance. When such conditions are sat- isfied, they proposed a scheduling algorithm that guarantees bounded error covariance. In [150], some interesting properties of the solutions of the discrete-time infinite-horizon scheduling prob- lem are demonstrated; specifically, the authors showed that both the optimal infinite-horizon average-per-stage cost and the corresponding optimal schedules are independent of the covari- ance matrix of the initial state, and the optimal estimation cost can be approximated arbitrarily closely by some periodic schedule.
Finally, we note that the actuator scheduling problem (i.e., the problem of scheduling actu- ators to minimize control efforts) can be regarded as a dual problem of the sensor scheduling problem; see [26] for more details.