A robust synthesis algorithm is developed for a class of uncertain, linear parameter varying (LPV) systems. The uncertain system is described as an interconnection of a nominal LPV system and a block structured uncertainty. The nominal part is a "gridded" LPV system with state matrices that are arbitrary functions of the parameter. The input/output behavior of the uncertainty is described by integral quadratic constraints (IQCs). The robust synthesis problem leads to a non-convex optimization. The proposed algorithm is a coordinate-wise descent similar to the well-known DK iteration for μ synthesis. It alternates between an LPV synthesis step and an IQC analysis step. Both steps can be efficiently solved as semidefinite programs. The derivation of the synthesis algorithm is less obvious for LPV systems as compared to its LTI counterpart due to the lack of a valid frequency response interpretation. The main contribution is the construction of the iterative synthesis algorithm using time domain dissipation inequalities and a scaled system analogous to that appearing in μ synthesis. It is shown that the proposed algorithm ensures that the robust performance is non-increasing at each iteration step. The effectiveness of the proposed method is demonstrated on a simple numerical example.
Bibliographical noteFunding Information:
This work was supported by the National Science Foundation under Grant #NSF-CMMI-1254129 entitled “CAREER: Probabilistic Tools for High Reliability Monitoring and Control of Wind Farms” and IREE Project RL-0011-13 “Innovating for Sustainable Electricity Systems: Integrating Variable Renewable, Regional Grids, and Distributed Resources”. The material in this paper was partially presented at the 53rd IEEE Conference on Decision and Control, December 15–17, 2014, Los Angeles, CA, USA. This paper was recommended for publication in revised form by Associate Editor Fen Wu under the direction of Editor Richard Middleton.
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- Integral quadratic constraints
- Linear parameter varying systems
- Robust control
- Semidefinite programs