Robustness analysis of Adaptive Model Predictive Control for uncertain non-linear dynamic systems using s-gap metric concepts

Robustness analysis of adaptive control systems, when operating in a certain domain, has been a gulf during the past decades. This problem is more complicated in the case of non-linear dynamic systems including un-modelled dynamics as unstructured uncertainty. To find a clear solution for this famous and interesting problem, limitations and effects of controller operation on performance of on-line model identification procedure (and vice versa) must be determined. In this paper, as the main novelty, we show that it needs some developments and new concepts in robust control theory as the s-gap metric, generalized stability margin (GSM) and modifications on the gain bound calculation. These achievements help us to present an on-line identification method with its convergence proof in sense of the s-gap metric and a relation between GSM and identifier convergence area. Therefore, consideration of GSM in Adaptive Model Predictive Control (AMPC) cost function concludes a systematic solution relating controller robustness and adaptivity, clearly. To this aim, a linear matrix inequality (LMI) representation for GSM constraint is suggested. Also, the stability of AMPC on a certain operating domain is guaranteed in sense of the s-gap metric and GSM. All of these help to determine the attraction area of closed loop system and we show that there exists a trade-off between each two cases of the attraction area size, convergence area size and robustness of closed loop control system. Finally, simulations and experimental results imply on correctness of the proposed method. •Upper bound of the gap metric between two non-linear dynamic systems is calculated.•Generalized stability margin of closed loop control system is calculated in presence of un-modelled dynamics.•A new on-line identification algorithm is introduced to describe a class of non-linear dynamic system as a linear time varying (LTV) model.•Desired robustness of closed loop control system is considered as a constraint in adaptive model predictive control (AMPC) cost function.•A framework is introduced which guarantees stability of AMPC in a defined operating area.