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Much progress has been made in text summarization, fueled by neural architectures using large-scale training corpora. However, in the news domain, neural models easily overfit by leveraging position-related features due to the prevalence of the inverted pyramid writing style. In addition, there is an unmet need to generate a variety of summaries for different users. In this paper, we propose a neural framework that can flexibly control summary generation by introducing a set of sub-aspect functions (i.e. importance, diversity, position). These sub-aspect functions are regulated by a set of control codes to decide which sub-aspect to focus on during summary generation. We demonstrate that extracted summaries with minimal position bias is comparable with those generated by standard models that take advantage of position preference. We also show that news summaries generated with a focus on diversity can be more preferred by human raters. These results suggest that a more flexible neural summarization framework providing more control options could be desirable in tailoring to different user preferences, which is useful since it is often impractical to articulate such preferences for different applications a priori.