Multi-rate and Periodically Time-Varying Systems
Overview
In practical control systems, sensors and actuators often operate at different sampling rates. For example, a position sensor may provide measurements at a high rate, while a force sensor or a vision system operates at a slower rate. Similarly, the actuation period may differ from the sensing periods. Such systems, where multiple components operate at distinct sampling rates with mutually rational ratios, are called multi-rate systems.
Multi-rate systems are naturally described as linear periodically time-varying (LPTV) systems, whose internal parameters repeat with a known period. This periodic structure is a defining feature rather than an obstacle: by properly exploiting it, one can design observers, feedback controllers, and system identification algorithms that are tailored to the multi-rate environment.
Our research employs cyclic reformulation as a unifying framework for the analysis, control design, and identification of multi-rate and LPTV systems. The cyclic reformulation converts an LPTV system into an equivalent higher-dimensional linear time-invariant (LTI) system while preserving the original sampling rate — unlike the classical lifting approach, which reduces the sampling rate by a factor equal to the system period. This rate-preserving property makes the cyclic reformulation particularly well suited for multi-rate problems, as the relationship between the reformulated system parameters and the original LPTV parameters remains transparent and structurally exploitable.
Based on this framework, our work addresses the following topics:
Multi-rate state observer design: A periodically time-varying observer for systems with multiple sensors operating at different rates. The observer gains are designed via linear matrix inequality (LMI) conditions to minimize the l2-induced norm of the estimation error.
Observer-based feedback controller design: Extension of the observer design to observer-based feedback control for multi-rate sensing and actuating environments, where both observation and control periods may vary across sensors and actuators.
System identification for LPTV systems: A subspace identification algorithm that recovers the LPTV state-space parameters from input–output data through the cyclic reformulation. A key feature is that it does not require specific periodic input signals — general excitation signals suffice. The algorithm identifies the cycled system using standard subspace methods and then applies a state coordinate transformation, designed by exploiting the sparsity structure of the Markov parameters, to extract the original LPTV parameters.
Multi-rate system identification: Extension of the LPTV identification approach to systems with different sensor and actuator sampling rates.
For a detailed explanation of the cyclic reformulation-based identification algorithm, see the blog article: Cyclic Reformulation-Based System Identification for Periodically Time-Varying Systems (Qiita)