# A Hybrid Tolerance Design Method for the Active Phased-Array Antenna
## 1 Introduction  
The phased-array antenna plays an increasingly important role in the communications and military fields, and is basically composed of two main kinds of parts: (1) a large number of discrete elements, (2) continuous reflection plates, as shown in Figure 1. The geometric errors of the elements seriously affect the electrical performance [1; 2; 3], where the position errors cause the dominant effect while the effects of rotational errors are negligible [4]. The position error of the element is composed of two parts: the discrete error and the continuous error. The discrete error results from the assembly of the element, and the continuous error is caused by the manufacture of the reflection plate. The discrete errors are independent of each other, while the continuous errors are dependent variables due to the continuous deformation of the plate [5]. The tolerance design of the elements and the reflection plate is very important. The tolerance design of the antenna mainly went through three stages: (1) the empirical design stage, (2) the feedback design stage and (3) the precise design stage. (1) The empirical design stage: the tolerance design mainly relies on empirical equations. Based on a series of experiments, [PERSON] [6] summarized the empirical formula that describes the empirical relation between the root mean square (RMS) of the array element errors and the gain degradation. Following [PERSON]'s work, the empirical function between the RMS and the sidelobes degradation was determined [7; 8; 9]. The empirical tolerance design method is simple, and it has been widely used. However, it is not precise. (2) The feedback design stage: the tolerance design mainly relies on the performance prediction model and the feedback process, as illustrated in Figure 2.  
The Monte Carlo method was applied to the feedback design by [PERSON] [10], and the performance of the realistic array with geometric errors was calculated based on the commercial software. To improve the simulation precision, [PERSON] [11] established the multifield coupling model between the performance and the geometric errors. To reduce the computational cost, several performance prediction methods were proposed. The worst-case method [12] was proposed based on the Cauchy-Schwarz inequality, it can predict the worst boundary of the performance interval. To improve the interval accuracy, [PERSON] [13; 14] proposed the interval method, including the rectangular interval method and the circular interval method. These prediction models can calculate the performance rapidly. The feedback tolerance design method is time-consuming, and what is worse is that recalculation is required for different array antennae. Another significant disadvantage of the above methods is  
Figure 1: The computer-aided design (CAD) model of a planar phased array antenna. Figure 2: The process of the feedback tolerance design. that only the discrete errors of the elements were considered. The feedback tolerance design methods ignored the form error of the reflection plate, which results in inaccurate tolerance design. (3) The precise design stage: the tolerance design relies on the precise and direct design model. Based on the linear programming support vector regression (LPSVR) algorithm, [PERSON] [15] established the surrogate model between the power pattern and the continuous form errors and random discrete errors. This method has very high calculation accuracy, however, it requires a large number of samples. Moreover, it needs to re-model for each different antenna. In our previous work [16], the precise position tolerance design of the discrete elements has been developed. To optimize position tolerance of the discrete elements and the flatness of the continuous reflection plates simultaneously, the hybrid tolerance design method was proposed in this paper.