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

TL;DR

The paper develops an $LFT$-based framework to model uncertain hybrid continuous/discrete-time multi-rate closed-loop systems as a single-rate discrete-time $LFT$ and to incorporate discretization errors via a structured uncertainty $\Delta$. It introduces $LFT$-preserving ZOH discretization with first- and second-order rational approximations (and higher-order extensions) and an $LFT$-preserving down-sampling procedure, enabling accurate transition from the fast inner loop to slower outer loops. Using $μ$-analysis on the resulting $z_M = F_u(M(z),\Delta)w_M$ model, the method yields robust stability margins $k_r$ and worst-case $\mathcal{H}_\infty$ performance $\gamma_{wc}$, with a branch-and-bound strategy to manage multiple uncertainties. An easily reproducible multi-rate attitude-control-inspired example demonstrates that second-order rational discretization can closely replicate the hybrid model, while exposing scenarios where multi-rate configurations outperform single-rate designs in the presence of uncertainties, underscoring the practical value of the approach.

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 $μ$-based approach. The proposed methodology is illustrated on a realistic and easily reproducible example inspired by the validation of multi-rate attitude control systems.

Figures (8)

• Figure 1: Uncertain hybrid continuous/discrete multi-rate closed-loop system
• Figure 2: LFT-based single-rate full discrete-time approximation
• Figure 3: A graphical illustration of Lemma \ref{['lem1']} and its corollary
• Figure 4: Hybrid closed-loop model with a multi-rate PID controller
• Figure 5: Nominal configuration: continuous-time vs hybrid single & multi-rate implementations

Theorems & Definitions (10)

• Lemma III.1
• proof
• Corollary III.1
• proof
• Remark III.1
• Remark III.2
• Lemma III.2
• proof
• Remark IV.1
• Remark V.1