State Observability and Observers of Linear-Time-Invariant Systems Under Irregular Sampling and Sensor Limitations

Abstract

Abstract-State observability and observer designs are investigated for linear-time-invariant systems in continuous time when the outputs are measured only at a set of irregular sampling time sequences. The problem is primarily motivated by systems with limited sensor information in which sensor switching generates irregular sampling sequences. State observability may be lost and the traditional observers may fail in general, even if the system has a full-rank observability matrix. It demonstrates that if the original system is observable, the irregularly sampled system will be observable if the sampling density is higher than some critical frequency, independent of the actual time sequences. This result extends Shannon's sampling theorem for signal reconstruction under periodic sampling to system observability under arbitrary sampling sequences. State observers and recursive algorithms are developed whose convergence properties are derived under potentially dependent measurement noises. Persistent excitation conditions are validated by designing sampling time sequences. By generating suitable switching time sequences, the designed state observers are shown to be convergent in mean square, with probability one, and with exponential convergence rates. Schemes for generating desired sampling sequences are summarized.

Target Tracking of a Linear Time Invariant System under Irregular Sampling

“…[10,11] discuss some related issues of non-uniformly sampled systems, including model derivation, controllability and observability, computation of single-rate models with different sampling periods, reconstruction of continuoustime systems, and parameter identification of nonuniformly sampled discrete-time systems. [12] introduces a method of designing observers for linear time invariant systems with irregular sampling time sequences, which can be generated either passively in event-triggered sampling or by actively controlling the system input or sensor threshold when the sensor is binary-valued. Studies on Approach 2) remain an active area of research, see [13][14][15][16] and the references therein for some recent work in this area.…”

“…We note that the results about estimation methods of the discrete Kalman filter for irregular sampling [12] cannot be used for the target tracking problem directly. For example, the matrix exponential of A , i.e., A e is obtained by a polynomial function of A based on the Cayley-Hamilton theorem in [12], which requires A to be a full-rank matrix. But we know this condition does not hold for a target tracking system.…”

State and event estimation for regime-switching systems under irregular and random sampling schemes

“…This is a special case of threshold control in generating sampling sequences [29,30,33,34]. This scheme allows a one-bit observation sequence that can drastically reduce communication resources.…”

“…We start with a review of certain basic relationships from [29,30,33] that characterize sampling complexity for state and event estimation under irregular sampling. In this section, all subsystems are observable.…”

“…What is the probabilistic description of observability? Our exploration of these topics utilizes some fundamental results from our recent work on minimum sampling density for state observability [33,34]. Our recent work [29,30] deals with deterministic regime-switching processes.…”