Reachable Set Estimation for Uncertain Linear Systems with Known Inputs
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Reachable Set Estimation for Uncertain Linear Systems with Known Inputs Zhihao Zhang, Xinsheng Wang, Wenbo Xie, Xin Zhang*, and Yi Shen Abstract: This paper studies reachable set estimation for systems with parameter uncertainties and known input. A new method is proposed for estimating real-time reachable sets of systems with parameter uncertainties, known input and non-zero initial condition. In order to reduce the effect of known inputs on the reachable set estimation, we propose a method by separating the uncertain systems into two subsystems: the subsystem with known input and the subsystem with parameter uncertainties. By solving the optimization problem in terms of linear matrix inequalities, real-time reachable sets are obtained. The effectiveness of the proposed method is illustrated by the simulation result of the lateral dynamic system of a vehicle. Keywords: Linear matrix inequality, linear systems, parameter uncertainties, reachable set estimation, system separation.
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INTRODUCTION
State estimation for unmeasurable systems, such as observer design [1–3] and reachable set estimation [4–7], receives wide attention in recent years. Different from the direct estimation of the system state by designing observer, the objective of reachable set estimation is to determine bounding ellipsoids that contained all the reachable states of the considered system. Reachable set estimation appeared in various fields, such as control design [8–10] and safety verification [11–14]. In the past decades, reachable set estimation has been investigated by many researchers, see in [15–17]. Uncertainties are widely existed in pratical systems. Hence, robust design methods have received considerable attention in the recent years [18–23]. However, most existing methods on reachable set estimation do not consider parameter uncertainties. To deal with uncertain parameters, reachable set estimation methods are proposed in [24, 25]. However, the methods proposed in [24, 25] are only applicable to systems with zero initial condition. Note that systems with unknown initial state cannot be mapped directly to systems under zero initial conditions. Therefore, the methods for systems under initial conditions are limited in the application scope. Most of the methods proposed estimate constant reachable sets. However, if we propose a method to update the
reachable set of system state in real time, it will be less conservative than the constant reachable set at this designated time. Thus, real-time reachable set estimation is more useful than ultimate reachable set estimation in practical applications, such as fault detection and safety verification. Althoff and Krogh [26] obtains real-time reachable sets for safety verification of automated driving systems. Tang et al. [27] proposes a method to generate dynamic threshold for fault detection based on the reachable set estimation of error systems. Han et al. [28] studies sensor fault detection of discrete-time descriptor system
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