Multi-Criteria Optimization for Filter Design of L1 Adaptive Control

This paper considers the problem of optimal trade-off between robustness, in terms of time-delay margin, and performance of the $\mathcal {L}_{1}$ adaptive controller. Although the architectures of the $\mathcal {L}_{1}$ adaptive control theory can be systematically tuned to trade-off performance for robustness, there is no practical methodology as of today for the design of the underlying filter toward obtaining the optimal performance. The only available results in this direction were given in the form of linear matrix inequalities (LMIs), which are numerically tractable but tend to yield conservative results similar to what one would obtain by applying methods from robust control theory. This paper presents design schemes that are based on search for the design parameters of the filter. We consider filters of certain structures to satisfy the design specifications and investigate their properties over the space of the design parameters. To handle uncertain variables in the design specifications, greedy randomized algorithms are adopted to analyze and synthesize the system performance and robustness in the presence of uncertainties. In addition, we compute approximate sets of allowable design parameters in both analytical and numerical ways. Illustrative examples are provided to demonstrate that these methods can achieve the desired performance and robustness specifications with reasonable degree of conservatism.