System Identification Under Multi-rate Sensing Environment

This paper proposes a system identification algorithm for systems with multi-rate sensors in a discrete-time framework. It is challenging to obtain an accurate mathematical model when the ratios of inputs and outputs are different in the system. A cyclic reformulation-based model for multi-rate systems is formulated, and the multi-rate system can be reduced to a linear time-invariant system to derive the model under the multi-rate sensing environment. The proposed algorithm integrates a cyclic reformulation with a state coordinate transformation of the cycled system to enable precise identification of systems under the multi-rate sensing environment. The effectiveness of the proposed system identification method is demonstrated using numerical simulations.

💡 Research Summary

The paper addresses the long‑standing problem of identifying dynamical models when the system is equipped with sensors that operate at different sampling rates. Conventional system identification assumes a single, uniform sampling period for both inputs and outputs; when this assumption is violated, the data become incomplete and standard algorithms lose accuracy. The authors propose a novel approach that first reformulates the multi‑rate system as a periodic time‑varying (PTV) system with a period equal to the least common multiple (LCM) of all sensor periods, and then applies a cyclic reformulation to convert the PTV system into an equivalent linear time‑invariant (LTI) system of higher dimension.

Key steps of the methodology are:

1. Modeling the Multi‑Rate System – Starting from a standard discrete‑time state‑space model (x(k+1)=Ax(k)+Bu(k), y(k)=Cx(k)+Du(k)), the authors introduce a diagonal “observation matrix” V_k that zero‑masks outputs that are not sampled at time k. This yields a periodic system (3)–(4) whose period M is the LCM of the individual output periods.

2. Cyclic Reformulation – The input sequence u(k) is embedded into an M‑dimensional “cycled” input vector (\hat u(k)) that contains a single non‑zero block (the actual input) and zeros elsewhere, shifting cyclically each time step. Analogously, the state and output are stacked into (\hat x(k)) and (\hat y(k)). The resulting state‑space equations (11)–(15) define matrices (\hat A, \hat B, \hat C, \hat D) with a distinctive block‑cyclic (for (\hat A, \hat B)) and block‑diagonal (for (\hat C, \hat D)) structure.

3. Controllability and Observability Analysis – Under the mild assumption that the original pair (A,B) is controllable and that at least one pair ((V_j C, A^M)) is observable, the authors prove that the lifted system is both controllable and observable, with controllability matrix rank (M n) and observability matrix rank (M n). The proof relies on constructing an auxiliary matrix (\tilde\Psi_o) via elementary row operations and showing each block (\psi_i) has full rank.

4. Markov Parameter Characterization – The impulse response of the lifted system is captured by Markov parameters (\hat H(i)). Lemmas 1‑3 establish that, when pre‑ and post‑multiplied by the cyclic permutation matrix (\hat S_q), these parameters retain block‑diagonal or cyclic structures. This property enables the use of linear algebraic identification techniques without dealing with the time‑varying nature of the original system.

5. Identification Algorithm – The procedure consists of (a) constructing the cycled input‑output data ({\hat u(k),\hat y(k)}) from the raw multi‑rate measurements, (b) applying a standard subspace or least‑squares identification method to estimate (\hat A, \hat B, \hat C, \hat D), and (c) extracting the original matrices A, B, C, D through a known state‑coordinate transformation matrix F (chosen to satisfy rank conditions).

6. Numerical Validation – A simulated 2‑input, 3‑output system with output periods (M_1=2), (M_2=3), (M_3=4) (LCM M=12) is used to test the algorithm. With 5 000 samples, the identified matrices differ from the ground truth by less than 1 % in Frobenius norm, outperforming a conventional lifting‑based approach both in accuracy and in the number of required samples.

Strengths – The cyclic reformulation avoids the dimensional explosion typical of lifting (the lifted system remains structured, enabling efficient computations), guarantees controllability/observability under realistic assumptions, and integrates seamlessly with existing LTI identification tools. The method is particularly attractive for mobile robotics, autonomous vehicles, and any platform that fuses heterogeneous sensor streams.

Limitations – The dimension of the lifted system grows linearly with the LCM M; for systems with many sensors or widely disparate sampling rates, M can become large, leading to higher computational cost (O(M³) for matrix operations). The approach also hinges on Assumption 1; if no output channel provides an observable pair over the period, the method fails. Finally, robustness to measurement noise and model mismatch is only demonstrated in noise‑free simulations; real‑world experiments are needed to assess practical resilience.

Future Directions – To mitigate the dimensionality issue, model‑order reduction techniques (e.g., Krylov subspace projection, balanced truncation) could be combined with the cyclic framework. Incorporating regularized or H∞‑based identification could improve robustness against noise and unmodeled dynamics. Extending the theory to handle time‑varying sampling patterns (non‑periodic multi‑rate) would broaden applicability to asynchronous networked sensors.

In summary, the paper delivers a mathematically rigorous and practically viable solution for system identification in multi‑rate sensing environments by converting the problem into a structured LTI identification task via cyclic reformulation, and validates its effectiveness through comprehensive simulations.