Abstract-We discuss some multiple sampling problems that arise in finite horizon real-time estimation when there is an upper limit on the number of allowable samples. Measuring estimation quality by the aggregate squared error, we compare the performances of the best deterministic, level-triggered and the optimal sampling schemes. We restrict the signal to be either a Wiener or an Ornstein-Uhlenbeck process. For the Wiener process, we provide closed form expressions and series expansions, whereas for the Ornstein Uhlenbeck process, procedures for numerical computation. Our results indicate that the best level-triggered sampling is almost optimal when the signal is stable. In many decision and control problems, we can impose a certainty-equivalence type separation into a signal estimation part and a control or decision design part. For example, an optimal control or a signal detection problem can be solved by certainty-equivalence policies which treat a leastsquares estimate of the signal waveform as the true state. In these situations, the processing of available measurements should be geared towards obtaining the best quality signal estimate. In Networked Control Systems [1] where the sensors have only a limited number of packets (samples) to transmit to the supervisor, the sampling design affects the quality of the signal estimate. Kushner [2] has treated a problem of picking a fixed number of deterministic sampling times for a finite horizon linear optimal control problem. He establishes the validity of the separation principle and obtains closed form expressions for the minimum cost in the scalar case. The collection [3] treats some randomized as well as deterministic but irregular sampling schemes for smoothing and control. Cambanis and Masry [4] have treated the problem of picking the best deterministic and random sampling schedules for hypothesis testing based on a smoothed estimate. Their random sampling schedules are however not adapted to the signal observed by the sensor. For the problems treated in this paper, we seek to characterize the performance gains provided by event-trigerred sampling policies. Event-triggered sampling has been referred to as 'Lebesgue-type' sampling in the control literature tailored to a signal filtering task. The design objective is to minimize, over a finite horizon, the distortion of a filter (real-time estimator) of the signal based upon the stream of samples. This minimization is performed with a fixed constraint on the maximum number of samples used. In [6] a related problem in discrete-time is treated. For the signal to be estimated (the state process) x t , t ∈ [0, T ], we will assume that a sensor has perfect observation of the state and transmits at times it chooses, current samples of the state process. The sensor is allowed to generate at most N samples to be transmitted to a supervisor. The sampling times S = {τ 1 ,... ,τ N } have to be an increasing sequence of stopping times with respect to the x-process. They also have to lie within the interval [0, T ]. Based on the samples and the sampling times, the least-squares estimate for the supervisorx t is given by [7] : (1) The quality of this estimate is measured by the aggregate squared error distortion: We will consider three sampling strategies and characterize their performance. The strategies are: Deterministic sampling: The sampling sequence S is chosen apriori and hence independent of the signal trajectory. It is chosen to minimize the expected distortion J. In this scheme, the supervisor too knows in advance when the sensor will generate and transmit samples. The sensor chooses the sampling times based on times the error signal x t −x t crosses chosen thresholds. The actual sampling times are the lesser of these threshold times and the end time T . The sampling times are dependent on the actual sensor observations. Optimal sampling: To choose sampling times, the sensor solves a sequence of optimal stopping time problems and applies the resulting stopping rule. Here too, these times are dependent on the actual sensor observations. Let us now start with a detail on the single sampling instance, i.e. N = 1, since this is going to serve as a basis for solving the general case.
In REF , a multiple sampling method for Wiener and Ornstein-Uhlenbeck processes was investigated, respectively, and it is shown that the proposed method is almost optimal when the signal is stable.
Multiple Sampling for Estimation on a Finite Horizon
Proceedings of the 45th IEEE Conference on Decision and Control
Proceedings of the 45th IEEE Conference on Decision and Control
Mathematics