A New Efficient Algorithm for Analyzing and Optimizing the System of Sensors

Abstract

In this paper, we propose a new algorithmic approach for design of a sensor system that maximizes the diagnosability of the system and minimizes the cost of the sensor placement. This approach also allows us to analyze extend of the coverage of the sensor system and to determining its diagnosability degree. The quality and efficiency of a diagnosis system depends on the availability and relevance of the information it can retrieve from the diagnosed plant. The quality of the measurements is expressed by the diagnosability degree, i.e., given a set of sensors, which faults can be discriminated? There is no straightforward relation between the number of sensors and diagnosability of the systems; increasing the number of sensors alone does not guarantee a higher level of diagnosability. The relevance of information provided by an additional sensor and its correlation with information provided by other sensors must also be taken into account. Besides the issue of diagnosability, we also consider economics issues. We must provide a sensor system that achieves a desired degree of diagnosability at the lowest possible cost. Our new algorithmic approach allows us to find efficient solutions for these two important problems regarding the system of sensors. The first problem is to analyze and certify the sensor system. Here the main problem is to determine the diagnosability degree of the system, i.e., characterize the set of the faults that can be discriminated. Our method also determines an optimal set of new sensors that needs to be added to the system to enhance its diagnosability. The second issue is the optimization problem of sensor placement. In this regard, the main questions are as follows. (1) Minimal sensor set: finding a minimal additional sensor set that guarantees a specific degree of diagnosability. (2) Minimal cost sensors: in the case that different sensors are assigned with different costs, finding the minimal cost additional sensors which achieve a specific degree of diagnosability. The primary application of this method is at the design level where the number and position of sensors are not known. Our new approach for solving these problems is motivated by our successful method for solving the diagnosis problem. For sensor placement, we start by observing that it can be mapped onto a 0-1 IP problem. The objective function, in the most general case, is not linear, but the constraints are linear and defined by a 0-1 matrix. For the sensor placement problem, we start with the structural model of the system. The structure analysis of the system and the potential information carried by each sensor provide a set of relations usually called the analytical redundant relations (ARRs). We can also consider the additional sensors (the potential sensors that will provide the desired degree of diagnosability) and their corresponding ARRs. The information of all these ARRs can be summarized in a signature matrix. Then the above sensor optimization problems can be formulated as combinatorial problem regarding the signature matrix, or as integer programming problem involving this matrix. The existing methods for solving these combinatorial problems usually boil down to exhaustive search methods. For solving the diagnosis problem, we have found a new branch-and-bound technique which has achieved order of magnitude speedup over the standard algorithms. Combination of this new algorithm and the ARR approach would provide a powerful efficient technique for solving the difficult problem of sensor placement optimization. We have developed a Mathematica code for validating and benchmarking this algorithm.