Selecting Sensors in Biological Fractional-Order Systems

In this paper, we focus on sensor selection, that is, determine the minimum number of state variables that need to be measured, to monitor the evolution of the entire biological system, that is, all state variables, when modeled by discrete-time fractional-order systems (DTFOS) that are subject to modeling errors, process, and measurement noise. These systems are particularly relevant while modeling the spatiotemporal dynamics of processes in which the impact of long-range memory cannot be properly modeled by multivariate autoregressive integrative moving-average models. Therefore, the DTFOS enable a unified state-space framework to model the evolution of several biological (e.g., stem cell growth and bacteria evolution) and physiological signals (e.g., electroencephalogram and electromyogram). Therefore, in this paper, we focus on the solution to four different (yet related) problems of sensor selection for DTFOS, which are motivated by constraints on the data acquisition that are enforced by the detrimental impact of the sensing mechanisms to the biological system, the cost of performing the measurements with the current sensing technology, or spatial constraints that limit the number of sensors that can be deployed. Toward determining the solution to these problems that we show to be NP-hard, we leverage the representation of the DTFOS to derive new objectives and conditions that, ultimately, enable us to efficiently approximate a solution to the different problems by exploiting the submodularity structure, which enables us to establish suboptimality guarantees.