To appear at Conference on Decision and Control (CDC), 2021 Robustness Guarantees for Structured Model Reduction of Dynamical Systems Ayush Pandey and Richard M. Murray Abstract— Model reduction methods usually focus on the error performance analysis; however, in presence of uncer- tainties, it is important to analyze the robustness properties of the error in model reduction as well. In this paper, we give robustness guarantees for structured model reduction of linear and nonlinear dynamical systems under parametric uncertainties. In particular, we consider a model reduction where the states in the reduced model are a strict subset of the states of the full model, and the dynamics for all other states are collapsed to zero (similar to quasi-steady state approximation). We show two approaches to compute a robustness metric for any such model reduction — a direct linear analysis method for linear dynamics and a sensitivity analysis based approach that also works for nonlinear dy- namics. We also prove that for linear systems, both methods give equivalent results. I. I NTRODUCTION For applications of control theory to physical system design, a reduced model is commonly used that describes the dynamics of interest in lower dimensions to simplify the design process. A common way to obtain a reduced model from a detailed model of a system is to make use of different time scales in the full system dynamics. Singular perturbation theory [1] is the formal way of deriving reduced models for system dynamics with time-scale separation. A key feature of singular perturbation theory is that the states of the reduced model are a subset of the states of the full model. In other words, the structure of the model and the meaning of the states and parameters is conserved by construction in any reduced model obtained using singular perturbation theory. This is not automatically the case in other model reduction techniques where transformations are introduced [2] [3] and hence in such techniques, the meanings of the states may not be preserved. We define structured model reduction in this paper as the set of model reduction methods where the states of the reduced models are a strict subset of the states of the full model. The advantage of structured model reduction techniques is that an explicit mapping between the full and the reduced model is readily available [4]. Moreover, since the parameters and the states in the reduced model have the same meaning as in the full model, the design outputs and analysis results obtained using the reduced model can easily be given context and compared with the full The authors are with the Control and Dynamical Systems depart- ment at California Institute of Technology, Pasadena, CA, USA. Email: apandey@caltech.edu model [5]. Due to the strict condition on the possible reduced model states, the structured model reduction methods suffer from the limitation that the choice of reduced models is limited and dependent on the modeling details of the full system. In other words, for a given full model it may not always be possible to analytically derive a reduced model. Other model reduction methods that are projection-based or those which preserve the input-output mapping are better in that respect [6], [7]. In this paper, we focus on the former class of model reduction problems that preserve the modeling structure in the reduced models. The goal with any model reduction problem is to minimize the error in the performance of the reduced model when compared to the full model. This error performance criterion can be general and depend on trajectories of all states and output variables, or specific, such as minimizing a particular metric of interest. Singular perturbation theory for model reduction and its error analysis is a widely studied topic in the literature for different system and control design settings [8]–[10]. A commonly used method for model reduction that is derived from the singular perturbation concept of time- scale separation is the quasi-steady state approximation method (QSSA) [11]–[14]. Here, a subset of states is assumed to be at steady-state and hence their dynamics are collapsed to algebraic relationships. Error analysis for QSSA based model reduction has been studied in [15]– [17]. However, robustness of these model reduction methods is not widely studied in the literature. Robust control design is a well-studied problem in control theory. The extensions of robust control theory to singularly-perturbed systems have been studied in [18] and [19]. Similarly, robust stability analysis of adaptive control problems, linear time-varying systems, and the general parametric uncertainty problems has been of interest as well [20] [21]. A complementary, although not as widely applicable, approach to study the robustness of systems is to use sensitivity analysis of system variables or derived properties under parameter variations. Due to the success of robust control design methods [20] for different applications, the more holistic approach of sensitivity analysis for robustness estimates has not received much attention. In [22], sensitivities of singular values are used to give estimates for robustness properties of a linear feedback system. The advantage of such a method is that it can be used to analyze the effect of