Design of Diagnostic strategy for Radar System Based on Rollout Algorithm

To solve the problem of how to detect and isolate the fault of radar system with low test cost, a design method of diagnostic strategy for radar system based on rollout algorithm is proposed. This method takes the greedy search method based on information heuristic as the benchmark strategy, and uses the rollout strategy to iterate on the benchmark strategy, so as to overcome the problem of poor optimality of greedy search method. Taking the radar system dependency model as an example, this paper uses rollout algorithm to generate the diagnostic strategy of the radar system, which verifies the effectiveness of the method.


Introduction
With the development and application of high-tech, the structure and function of radar system become more and more complex and refined, and the problems of fault diagnosis and isolation during its use become more and more prominent. To solve this problem, it is necessary to conduct testability design in the design stage of the system [1]. Diagnostic strategy design, as an important part of testability design and analysis, can be used to guide the process of system fault diagnosis and maintenance [2], is the key to reduce the cost of fault diagnosis and improve the efficiency of fault diagnosis.
Diagnostic strategy is a method to optimize system fault diagnosis by combining constraints, objectives and other related factors. Its purpose is to construct a test sequence and quickly isolate the system fault state with less test cost. Pattipati proved that the diagnostic strategy problem is an NPcomplete problem [3]. Although the enumeration method can obtain the optimal solution, the computation time increases exponentially with the complexity of the problem. In order to reduce computational complexity, domestic and foreign scholars apply heuristic search method to diagnostic strategy optimization problem. At present, the most widely used method is the greedy search method based on information heuristic, in which the mutual information of fault state test is the basis for determining the test [4]. However, the greedy algorithm only considers the information obtained from the single-step test when select the test, but does not consider the impact of the selected test on the overall cost and subsequent tests. Therefore, the optimality of the diagnostic strategy obtained by the greedy algorithm is poor.
In view of the problem mentioned above, this paper studies the design method of diagnostic strategy with less test cost based on the radar system dependency model. On the basis of greedy algorithm, this paper proposes the design method of diagnostic strategy for radar system based on rollout algorithm, expounds the implementation process of the algorithm, and constructs the fault diagnostic strategy of radar system. is the set of prior probabilities corresponding to the fault states of the system, where is the prior probability of fault-free state of the system and occupies a large proportion in , and is the prior probability of .

3)
is the test set of the system. It is stipulated that the test can only pass or fail, and the test results are reliable.

4)
is the set of test costs, where represents the cost of executing test . The test cost can refer to the cost measure of different categories, such as test execution time, manpower, material resources, etc. The test cost is set as a dimensionless physical quantity.

5)
is the fault-test dependency matrix, which is a dimensional Boolean matrix, used to represent the logical relationship or dependency between the fault and the test. Elements in the matrix represent the dependency between the test and the fault state.
means that the test can detect the fault , that is, when the fault occurs, the test fails.
indicates cannot detect the fault , that is, when the fault occurs, the test passes.

Inference mechanism of fault diagnosis
According to the order of test execution of the diagnostic strategy, after each test is executed, the fault conclusion of the equipment is inferred according to the results. Assuming that the fault state set of the system is , test is used to detect . There are two outputs of pass and fail , denoted as and respectively. Inference mechanism is the reasoning method of logical inference of system fault state. The inference method for the process from to and is the inference mechanism: , .
is divided into two subsets and by test :

Optimization objective
The objective of the optimal diagnostic strategy is to use tests according to the given logic and sequence when a system failure occurs, to isolate the fault quickly and accurately, and to minimize the expected cost of the test. The calculation equation is[3]: represents the diagnostic strategy with lowest expected cost that isolates all fault without ambiguity; represents the test sequence in the diagnosis tree that isolates fault , represents the length of that sequence; represents the cost of test in the sequence .

Design of diagnostic strategy based on rollout algorithm
3.1. The basic idea of rollout algorithm Rollout algorithm [8] is a one-step look-ahead backtracking algorithm based on greedy search method, which can optimize the benchmark strategy through backtracking iteration of test cost. When constructing the diagnosis tree, the rollout algorithm uses two steps to determine a test. The first step is to extend the diagnosis tree from top to bottom, that is, to construct the diagnosis tree with each alternative test as the vertex by using the benchmark strategy [9]. The second step is test optimization. By calculating and comparing the test cost of each diagnosis tree, the first test corresponding to the diagnosis tree with the lowest test cost is selected as the current optimal test.

Benchmark strategy
According to the knowledge of information theory, if and are discrete random variables, then the information entropy of is: , The conditional entropy of with respect to is: , The amount of information provided by about , that is, the mutual information between and is: , .
In the diagnostic strategy design problem, the more information provided by the test, the more beneficial it is to isolate the fault, and the impact of the test cost should be considered. In the greedy search method based on information heuristic, the ratio of the mutual information of fault and test to the test cost is taken as the basis to select the test, and the heuristic evaluation function is: represents the mutual information between test and the fault state fuzzy set , that is, the information obtained by using test to diagnose the fault state fuzzy set to be isolated. The benchmark strategy selects the test that optimizes the heuristic evaluation function each time, isolated faults according to the test results, classifies fuzzy subsets of faults, and continues to expand further. The detailed steps are as follows: Step 1: Initial failure state set , test set . Step 2: For fault state set , test in test set is selected in turn. According to the test results, fault state set is divided into two subsets and . The total probability of each subset is calculated according to equation (5).
Step 3: Equations (11) and (12) are used to calculate the heuristic function values of each test. The test that minimizes the value of the heuristic function is selected as the current optimal test, denoted as .
Step 4: Test is used to divide the fault state into two subsets and , and the following equation is used to update the probability of the fault state in each subset: .
Step 5: Re-take as each subsets, and is the set after deleting from the original test set. Repeat 2~5 until the number of test subset elements is no more than one.

The specific steps of rollout algorithm
The basic steps of rollout algorithm are as follows: Step 1: Initialize the five-tuple, system failure state set , test set .
Step 2: For fault state set , test in test set is selected in turn. According to the test results, fault state set is divided into two subsets and . The total probability of each subset is calculated according to equation (5). According to equation (13), the failure probability of each subset is updated.
Step 3: The benchmark strategy is used to get the test sequence of each subset. Calculate the expected test cost of each test sequence: . (14) Step 4: Calculate the expected test cost for the diagnosis tree corresponding to test Select the test in the test set that minimizes equation (15), denoted as .
Step 5: Test is used to divide the fault state into two subsets and , and equation (13) is used to update the probability of the fault state in each subset.
Step 6: Re-take as each subsets, and is the set after deleting from the original test set. Repeat 2~6 until the number of test subset elements is no more than one.

Dependency matrix of radar system
Taking radar receiving processing and waveform generation subsystem as an example, the concrete process of diagnostic strategy construction is explained, and the application effect of rollout algorithm and benchmark strategy is analyzed and compared. The multi-signal model of radar receiving processing and waveform generation subsystem is shown in figure 1.  Figure 1. Radar receiving processing and waveform generation subsystem. The system has a total of 11 components and 14 fault modes, among which, and are a group of fuzzy set, and the combination of the two faults is denoted as . Through the selection of test points and tests, 8 test points and 12 tests were finally determined. By analysing the model, the dependency matrix is obtained as shown in table 1. Table 1. Dependency matrix.