LFT modelling and $μ$-based robust performance analysis of hybrid multi-rate control systems

Abstract: This paper focuses on robust stability and $H_\infty$ performance analyses of hybrid continuous/discrete time linear multi-rate control systems in the presence of parametric uncertainties. These affect the continuous-time plant in a rational way which is then modeled as a Linear Fractional Transformation (LFT). Based on a zero-order-hold (ZOH) LFT discretization process at the cost of bounded quantifiable approximations, and then using LFT-preserving down-sampling operations, a single-rate discrete-time closed-loop LFT model is derived. Interestingly, for any step inputs, and any admissible values of the uncertain parameters, the outputs of this model cover those of the initial hybrid multi-rate closed-loop system at every sampling time of the slowest control loop. Such an LFT model, which also captures the discretization errors, can then be used to evaluate both robust stability and guaranteed $H_\infty$ performance with a $\mu$-based approach. The proposed methodology is illustrated on a realistic and easily reproducible example inspired by the validation of multi-rate attitude control systems.