# Frequency-Limited Model Reduction for Linear Positive Systems: A Successive Optimization Method
## 1 Introduction  
In various practical fields, such as biology, chemistry, economics and engineering, it is often the case that certain quantities are potentially constrained to be non-negative. The population of a species in an ecosystem [1], the transmission speed that a signal travels in network communication [2], and the distribution of light energy radiated by a light source, to mention a few. This property arises from physical or biological constraints that preclude negative values; the systems that exhibit the positivity are commonly referred to as positive systems whose state variables and output trajectories take non-negative values, provided that the excitation inputs and initial states are non-negative [3; 4; 5]. The study of positive systems can be traced back to the early 1990s when [PERSON] [6] introduced the concept of positive matrices and their applications to linear systems. Later, the theory of linear systems was further developed. It has become a vibrant research area in control theory and systems engineering, including modeling and analysis [7], control and optimization [8; 9; 10; 11], and its applications [12]. In many cases, high-order models are frequently encountered to provide exact characterizations of dynamical behaviors, which poses a significant challenge for the analysis and synthesis of the concerned systems [13]. Therefore, it is crucial to simplify models of dynamical systems by exploring lower-order models that approximate the original high-order ones according to certain criteria [14; 15]. Various model reduction methods, such as balanced truncation [16], Hankel-norm reduction [17], and moment matching [18] have been developed. For a specific positive system, it is reasonably required to preserve the positivity when employing model-reduction methods. Moreover, it suffices to satisfy given approximation specifications within limited frequency ranges [19]; it is especially relevant in engineering applications where practical constraints limit the operating frequency range. There areseveral model reduction approaches available to improve approximation performance over a limited frequency range, including methods such as frequency weighting [20] and frequency-specific balanced truncation [21]. Nevertheless, these methods cannot be directly applied to positive systems as they do not take the essential positivity of the reduced-order model into account. The generalized Kalman-Yakubovich-Popov (KYP) lemma is a powerful tool for characterizing the finite-frequency property, providing an equivalent condition for the solvability of frequency-limited specifications [22]. In the context of model reduction, this condition is expressed in a unified manner as a bilinear matrix inequality (BMI) involving coupled terms between two dependent decision variables [23]. To address this issue, some researchers have conservatively transformed the intractable BMI into a linear matrix inequality (LMI) by performing congruent transformation [24]. However, a potential drawback of this method is that the obtained LMI condition may always be infeasible. By alternately fixing one of the two coupled terms, D-K synthesis has been proposed to solve the BMI problems iteratively [25]; a major challenge of this approach lies in obtaining a feasible initial condition. Note that the introduction of additional positivity constraints and finite-frequency characterizations exacerbates the non-convexity, thereby making it difficult to explore a desired reduced-order model. Motivated by this, the paper aims to overcome this obstacle by developing a novel iterative procedure that can effectively solve the non-convex optimization problem. In this paper, we aim to obtain a model with a lower order that preserves the positivity and captures the most significant behaviors within limited frequency intervals, which requires that the obtained model is positive and the resulting error system is stable with a given finite-frequency performance level.