Closed-Loop Identification of Periodically Time-Varying Systems via Cyclic Reformulation

Technical Summary: Closed-Loop Identification of Periodically Time-Varying Systems via Cyclic Reformulation

Problem Statement
This paper addresses the system identification of Linear Periodically Time-Varying (LPTV) plants, specifically targeting scenarios where the plant is open-loop unstable. In such cases, open-loop experiments are not practically feasible because the plant output diverges without feedback. Existing LPTV identification methods, including subspace approaches and lifting-based techniques, are primarily formulated for open-loop data or assume open-loop stability. While closed-loop identification for Linear Time-Invariant (LTI) unstable plants is well-established, a realization-theoretic framework that explicitly recovers a finite-dimensional state-space LPTV plant model directly from closed-loop data has remained underdeveloped.

Methodology
The proposed framework integrates the joint input-output approach with cyclic reformulation to bridge the gap between closed-loop data and LPTV model recovery. The methodology proceeds through the following stages:

1. System Formulation: The paper considers a square, strictly proper LPTV plant controlled by an LPTV controller with a relative degree of one. The closed-loop system is modeled as an augmented LPTV system driven by an external reference signal $r(k)$, producing joint outputs of the plant output $y(k)$ and control input $u(k)$.
2. Cyclic Reformulation: The periodic closed-loop system is converted into an equivalent Linear Time-Invariant (LTI) system using cyclic reformulation. This involves constructing "cycled" signals where the input and output vectors are expanded to dimension $M$ times the original (where $M$ is the period), effectively stacking the periodic variations into a single LTI state-space model.
3. Subspace Identification: Standard subspace identification algorithms (e.g., N4SID) are applied to the cycled reference signal $\check{r}(k)$ and the stacked cycled joint output $\check{z}(k) = [\check{y}(k)^\top, \check{u}(k)^\top]^\top$. This yields a shared state-space realization $(A, B, C)$ for the closed-loop maps from $\check{r}$ to $\check{y}$ and $\check{r}$ to $\check{u}$. Crucially, both transfer functions share the same state dynamics $(A, B)$.
4. Algebraic Plant Extraction: The core of the method is an exact algebraic theorem that recovers the cycled plant transfer matrix $\check{P}(z)$ from the identified closed-loop maps. Since $\check{P}(z) = \check{T}_{yr}(z) \check{T}_{ur}(z)^{-1}$, the paper derives a state-space realization of $\check{P}(z)$ directly from the shared matrices $(A, B, C_y, C_u)$ without state augmentation. The extraction relies on the invertibility of the controller path's direct feedthrough term ($C_u B$).
5. Parameter Recovery: The extracted cycled realization is reduced in order (removing controller dynamics via balanced truncation) and transformed back into the original periodic state-space parameters $(A_{p,k}, B_{p,k}, C_{p,k}, D_{p,k})$ using a coordinate transformation that exploits the shifted block-diagonal sparsity of the Markov parameters.

Key Contributions
The paper presents four primary contributions:

1. Exact Algebraic Theorem: It establishes a theorem proving that the cycled plant transfer matrix can be recovered from a shared-$A$ realization of the closed-loop maps without increasing the state dimension, even if the recovered plant realization is unstable.
2. Stability Shift: The framework shifts the stability requirement for data generation from the open-loop plant to the internally stable closed-loop system, enabling the identification of open-loop unstable LPTV plants.
3. Explicit Assumption Separation: The work explicitly separates the assumptions required for data generation (closed-loop stability), identification (minimality of the cycled closed-loop system), and algebraic inversion (nonsingularity of the controller path's direct term).
4. Algorithmic Implementation: It provides a complete identification algorithm (Algorithm 1) that combines subspace identification with the derived algebraic extraction and coordinate transformation steps.

Numerical Results
The method is validated through three numerical examples:

• Stable SISO Plant: Demonstrates that the method recovers parameters with machine precision in noise-free conditions and high accuracy (Frobenius norm errors $\approx 10^{-6}$) under 40 dB signal-to-noise ratio (SNR).
• Open-Loop Unstable SISO Plant: Confirms the method's ability to recover an unstable plant realization (with eigenvalues outside the unit circle) from stable closed-loop data. The extracted realization correctly reflects the unstable monodromy eigenvalues, and the algebraic extraction remains valid despite the instability.
• MIMO Plant: Validates the approach for a Multi-Input Multi-Output system, showing that the framework scales to higher dimensions and maintains coordinate-invariant accuracy via Markov parameter comparisons.

Significance and Claims
The paper claims that this work fills a specific gap in system identification theory: the lack of a realization-theoretic route to recover finite-dimensional state-space LPTV models from closed-loop data, particularly for unstable plants. The significance lies in the exact algebraic nature of the plant extraction, which does not rely on iterative optimization or stability assumptions on the plant itself. The method distinguishes itself from:

• LTI closed-loop identification: By exploiting the periodic structure via cyclic reformulation.
• LPV identification: By addressing time-periodicity rather than external scheduling signals.
• Open-loop LPTV identification: By removing the prerequisite of open-loop stability.
• Lifted frequency-domain methods: By targeting finite-dimensional state-space models rather than nonparametric frequency responses.

The authors note that the reliability of the extraction step is governed by the conditioning of the inverse controller path ($C_u B$) and that the method assumes a square, strictly proper plant and a controller with relative degree one. Future work is suggested to address non-square plants and non-strictly-proper cases.