This article explains a new system identification method that turns measurement noise — usually treated as a nuisance to be averaged out — into a structured uncertainty description suitable for robust control design. The key idea is to deliberately apply cyclic reformulation with period $N$ to a linear time-invariant plant, exploit the resulting noise-induced parameter variations as polytope vertices, and design a robust $H_{\infty}$ controller from a single identification experiment. Related articles and MATLAB code links are placed at the bottom.

Author: Hiroshi Okajima, Associate Professor, Kumamoto University, Japan — 20 years of control engineering research

This article is based on the following paper.

Okajima, S. Shirahama, T. Hayashi and N. Matsunaga, From Noise to Knowledge: System Identification with Systematic Polytope Construction via Cyclic Reformulation, IEEE Access (2026), Vol. 14 (Open Access)

This paper is co-work with Shun Shirahama, Tatsunori Hayashi and Prof. Nobutomo Matsunaga, all at Kumamoto University.

Why Systematic Polytope Construction Matters
Plant Setup and Polytope Representation
Intentional Periodicity Induction
Algorithm and Computational Complexity
Numerical Examples
Robust H∞ Control Using the Constructed Polytope
Connections to Related Research
MATLAB Code
Related Articles and Videos
- Blog Articles (blog.control-theory.com)
- Research Web Page
Paper Information
SystemIdentification #PolytopicUncertainty #CyclicReformulation #RobustControl #HInfinityControl #LMI #SubspaceIdentification #N4SID #ControlEngineering #IEEEAccess
- Working Hypothesis and the Best In-Polytope Point

Why Systematic Polytope Construction Matters

Robust model-based control depends on two things: an accurate nominal model and a faithful description of how that model is wrong. A widely used uncertainty description is the matrix polytope, in which the system matrices are written as a convex combination of vertex matrices. Once a polytope is available, the rich machinery of LMI-based synthesis — robust $H_{\infty}$ control, parameter-dependent Lyapunov functions, gain-scheduled controllers, robust filtering — becomes directly applicable.

The bottleneck has not been the synthesis side. It has been how to construct the polytope from data in the first place. Most existing methods either

require multiple independent identification experiments,
assume the uncertainty structure (which parameters vary, and in what range) is known a priori, or
rely on bounded-noise assumptions of set-membership identification, which place the problem in a deterministic worst-case setting rather than a stochastic one.

The paper proposes a different route: given a single identification experiment under stochastic process and observation noise, generate a polytope automatically by applying cyclic reformulation with period $N$ to the plant. The construction has a single design parameter ( $N$ ) and yields exactly $N$ polytope vertices. The vertices are then used directly for LMI-based robust $H_{\infty}$ synthesis.

Plant Setup and Polytope Representation

The plant is an $n$ -th order discrete-time LTI system with process and observation noise:

$\displaystyle x(k+1) = Ax(k) + Bu(k) + d_u(k)$

$\displaystyle y(k) = Cx(k) + Du(k) + d_y(k)$

where $x(k) \in \mathbb{R}^{n}$ is the state, $u(k) \in \mathbb{R}^{m}$ is the input, $y(k) \in \mathbb{R}^{q}$ is the output, $d_{u}(k)$ is the process noise, and $d_{y}(k)$ is the observation noise. The pair $(C, A)$ is observable and $(A, B)$ is controllable.

A matrix polytope is a parametrized family of LTI systems

$\displaystyle A(\lambda) = \sum_{i=0}^{N-1} \lambda_i A_i, \quad B(\lambda) = \sum_{i=0}^{N-1} \lambda_i B_i$

$\displaystyle C(\lambda) = \sum_{i=0}^{N-1} \lambda_i C_i, \quad D(\lambda) = \sum_{i=0}^{N-1} \lambda_i D_i$

with $\lambda$ on the standard simplex

$\displaystyle \varepsilon = \left\{ \lambda \in \mathbb{R}^{N} : \lambda_i \geq 0, \sum_{i=0}^{N-1} \lambda_i = 1 \right\}$

The matrices $A_{i}, B_{i}, C_{i}, D_{i}$ are the vertex matrices. The exact value of $\lambda$ is unknown — only the information that $\lambda \in \varepsilon$ is available — and this is what makes the polytope a set-valued uncertainty model rather than a single point estimate.

The question the paper addresses is: given input-output data from a noisy LTI plant, how can the vertex matrices ${ (A_{mi}, B_{mi}, C_{mi}, D_{mi}) }_{i=0}^{N-1}$ be obtained systematically from a single experiment?

Intentional Periodicity Induction

Cyclic reformulation is a classical technique for converting an $N$ -periodic linear time-varying system into an equivalent time-invariant system on a lifted state space. The author has previously used this technique to identify periodically time-varying systems and multi-rate systems (see the Connections section below).

In this paper, the same machinery is applied in the opposite direction: to a system that is already linear time-invariant. The authors call this intentional periodicity induction.

The mechanism is the following. When cyclic reformulation with period $N$ is applied to an LTI plant, the augmented system has a block-cyclic structure. After subspace identification followed by a coordinate transformation that recovers this cyclic structure, $N$ parameter sets

$\displaystyle \{ (A_{mi}, B_{mi}, C_{mi}, D_{mi}) \mid i = 0, 1, \ldots, N-1 \}$

are extracted from the augmented model.

In the ideal noise-free case, all $N$ sets would coincide:

$\displaystyle (A_{mi}, B_{mi}, C_{mi}, D_{mi}) = (A, B, C, D), \quad i = 0, \ldots, N-1$

This is a direct consequence of the time-invariance of the underlying plant.

In the practical noisy case, each of the $N$ sets corresponds to a different phase position within the cycling period, and is therefore subject to a different realization of the noise. The sets scatter around the true parameters:

$\displaystyle (A_{mi}, B_{mi}, C_{mi}, D_{mi}) \approx (A, B, C, D) + \delta_i$

where $\delta_{i}$ is a phase-dependent perturbation.

Rather than averaging the $\delta_{i}$ out, the proposed method uses the scattered parameter sets as polytope vertices. The period $N$ is the sole design parameter, and the number of polytope vertices is determined by this choice.

Algorithm and Computational Complexity

The procedure takes input-output data ${ u(k), y(k) }_{k=1}^{N_{\mathrm{data}}}$ and a period $N$ as inputs, and returns an $N$ -vertex polytope. The steps are:

Signal transformation: Convert input-output data to cyclic signals ${ \check{u}(k), \check{y}(k) }$ .
Subspace identification: Apply N4SID to the cyclic signals to obtain $(\check{A}_{\ast}, \check{B}_{\ast}, \check{C}_{\ast}, \check{D}_{\ast})$ .
Coordinate transformation: Recover the block-cyclic structure using a transformation matrix $T$ .
Parameter extraction: Extract $N$ parameter sets ${ (A_{mi}, B_{mi}, C_{mi}, D_{mi}) }$ from the cyclic matrices.
Polytope construction: Use the $N$ sets as polytope vertices.

The dominant cost is [tex: O*1].

Two theoretical observations frame the interpretation of $\lambda^{\ast}$ :

(i) FIT( $\lambda^{\ast}$ ) does not reach 100 in general. Each vertex carries a finite-data subspace estimation error, and even if the true plant lay inside the convex hull, the convex combination minimizing the validation error need not exactly reproduce it.

(ii) FIT( $\lambda^{\ast}$ ) can still exceed the conventional single-model FIT. If the vertices are distributed around the true plant rather than concentrated on one side of it, the convex combination selected by $\lambda^{\ast}$ partially cancels their individual errors. This is analogous to variance reduction in ensemble methods such as bagging.

Two important points are emphasized in the paper:

$\lambda^{\ast}$ requires knowledge of the noise-free output $y_{\mathrm{val}}$ , which is unavailable in deployment. It is therefore used only as an offline diagnostic of polytope quality.
The deployable output of the method is the polytope itself, which serves as the uncertainty description for the robust $H_{\infty}$ design demonstrated below.

$\lambda^{\ast}$ is computed numerically via Particle Swarm Optimization (PSO) over the simplex $\varepsilon$ .

Numerical Examples

The benchmark is a 3rd-order discrete LTI system with 1 input and 2 outputs (SIMO):

$\displaystyle A = \begin{pmatrix} 0 & 0 & -0.3025 \cr 1 & 0 & 0.5800 \cr 0 & 1 & 0.7000 \end{pmatrix}, \quad B = \begin{pmatrix} 1 \cr 0 \cr 0 \end{pmatrix}$

$\displaystyle C = \begin{pmatrix} 1 & 1.9000 & 2.2450 \cr 3 & 1.4000 & 1.7100 \end{pmatrix}, \quad D = \begin{pmatrix} 0 \cr 0 \end{pmatrix}$

This is the controllable canonical form realization of the original system in the paper. All errors are evaluated under this common realization so that Frobenius-norm comparisons between models are well-defined.

Core Result: Polytope Quality across Noise Levels

With $N_{\mathrm{data}} = 3000$ , $N = 6$ , $N_{\mathrm{val}} = 1000$ , five noise configurations are tested. For each configuration, the conventional single-model FIT (worst-channel) and the best in-polytope FIT obtained by PSO are compared, together with the in-polytope parameter error $E_{\lambda^{\ast}}$ .

Noise Level	Conv. FIT_min [%]	PSO FIT_min [%]	E_{λ*}
Low	99.81	99.98	0.000009
Medium	99.01	99.89	0.000314
High	98.02	98.98	0.021872
Process only	98.07	99.30	0.012323
Observation only	97.40	99.82	0.000911

The improvements range from 0.17 to 2.42 percentage points, and the gap widens with the noise level. The in-polytope parameter error is several orders of magnitude smaller than the conventional identification error. In other words, the polytope contains a convex combination whose Frobenius-norm parameter error and FIT both improve upon the conventional single-model estimate, under every noise configuration tested.

A notable feature of the high-noise case is that one vertex was a severe outlier (Frobenius error $E_{0} \approx 12.45$ , roughly 356 times larger than the smallest vertex error). Nevertheless, the optimal weight distribution placed essentially zero mass on this corrupted vertex (98.1% of the weight went to the two best vertices), so the convex hull still enclosed a point near the truth. The polytope tolerates a poorly identified vertex as long as the other vertices distribute around the true plant.

Comparison with Bootstrap-Inspired Resampling

A natural alternative to cyclic reformulation is to partition the dataset and identify each partition independently. The paper evaluates two such bootstrap-inspired variants:

Bootstrap A (non-overlapping): Split $N_{\mathrm{data}}$ into $N$ non-overlapping consecutive subsets of length $\lfloor N_{\mathrm{data}} / N \rfloor$ ; apply N4SID to each.
Bootstrap B (overlapping): Extract $N$ consecutive subsets of length $\lfloor N_{\mathrm{data}} / 3 \rfloor$ with a systematic shift; apply N4SID to each.

For $N = 4$ , $N_{\mathrm{data}} = 3000$ , high-noise conditions, 10 independent trials, the results are:

Method	FIT_{y1} [%] mean (std)	FIT_{y2} [%] mean (std)
Cyclic (proposed)	98.92 (0.93)	98.92 (0.81)
Bootstrap A (750, non-ovlp)	97.96 (1.21)	98.18 (1.07)
Bootstrap B (1000, overlap)	98.16 (1.14)	98.05 (0.90)
Conventional (single)	95.69 (1.73)	96.06 (1.47)

The proposed method achieves the highest mean best in-polytope FIT with the smallest standard deviation, and it attains the best per-trial FIT_min in 9 out of 10 trials. The structural reason is data utilization: cyclic reformulation exploits all 3000 samples simultaneously through the algebraic structure of the lifted system, whereas Bootstrap A/B identify each subset (750 or 1000 samples) independently. The vertices of the proposed polytope therefore have higher individual estimation quality.

Robustness and Generalization

Two additional experiments support the method beyond the core setting:

Noise distribution: With variance-matched uniform noise (in place of Gaussian), the constructed polytope still contains in-polytope points whose FIT exceeds the conventional estimate, and all vertices remain stable.
MIMO: A 4th-order MIMO system ( $n = 4, m = 2, q = 2$ ) with $N = 4$ (augmented dimension $Nn = 16$ ) is evaluated. The in-polytope FIT matches or exceeds the conventional estimate for both output channels at $N_{\mathrm{data}} \in {3000, 6000, 9000}$ , and all four vertices remain stable.

Robust H∞ Control Using the Constructed Polytope

The polytope is then used as-is as the uncertainty description for robust $H_{\infty}$ state-feedback design. For each vertex $(A_{mi}, B_{mi})$ , a generalized plant is formed with disturbance weighting $Q_{w}^{1/2}$ and performance output $z(k) = C_{e} x(k) + D_{e} u(k)$ . A single gain $K$ is sought that simultaneously stabilizes all vertex models and minimizes the worst-case closed-loop $H_{\infty}$ norm over the polytope. The synthesis uses the discrete-time state-feedback LMI formulation of de Oliveira et al. with a common Lyapunov matrix at all vertices; for the explicit LMI condition, see Eq. (48) and the surrounding discussion in the paper.

The key point is that this synthesis uses only the noisy identification data (through the polytope) — no knowledge of the true plant is assumed. Since the true plant is not guaranteed to lie inside the convex hull, stability of the resulting closed-loop on the true system is not an a priori consequence of the design.

To assess whether the polytope nevertheless captures sufficient uncertainty information for control synthesis, the full pipeline (identification → polytope → LMI design → closed-loop evaluation on the true plant) is repeated over 10 independent noise realizations under the medium-noise setting with $N = 6, N_{\mathrm{data}} = 3000$ .

Quantity	Mean	Std	Min	Max
γ_opt (polytope design bound)	4.344	0.127	4.223	4.618
γ_{P-fix} (true sys, P̂, K̂ fixed)	4.191	0.106	4.023	4.348
‖T_zw‖_∞ (true closed-loop)	4.060	0.088	3.893	4.156
γopt / γ{P-fix} (conservatism)	1.036	0.017	1.013	1.064

Across all 10 trials, the polytope-based LMI was feasible, the designed controller stabilized the true system, and both the bounded-real LMI and the common Lyapunov condition held on the true plant. The mean conservatism ratio $\gamma_{\mathrm{opt}} / \gamma_{P\text{-fix}}$ was 1.036 (maximum 1.064), meaning the polytope-based design was only marginally more conservative than a controller designed with full knowledge of the true plant.

This is the central practical claim of the paper: the polytope constructed from noise-induced parameter variations carries enough uncertainty information for robust $H_{\infty}$ synthesis to stabilize the true plant with low conservatism, despite the absence of any formal containment guarantee.

The proposed method extends two earlier works by the author group, both of which use cyclic reformulation in its conventional direction (handling time-varying or multi-rate systems). The present paper applies cyclic reformulation in reverse — to LTI systems — for uncertainty quantification.

Okajima, Y. Fujimoto, H. Oku and H. Kondo, Cyclic Reformulation Based System Identification for Periodically Time-Varying Systems, IEEE Access, Vol. 13, pp. 26483–26493 (2025) (Open Access) — The foundational paper establishing cyclic reformulation as an identification framework for LPTV systems.

Okajima, R. Furukawa and N. Matsunaga, System Identification Under Multirate Sensing Environments, Journal of Robotics and Mechatronics, Vol. 37, No. 5, pp. 1102–1112 (2025) (Open Access) — Extension of cyclic reformulation to multi-rate sensing.

MATLAB Code

MATLAB code for the proposed method is not publicly available at this time. Future releases will be announced on the author's research website and GitHub repository.

Blog Articles (blog.control-theory.com)

System Identification: From Data to Dynamical Models — A Comprehensive Guide — The hub article covering the full system identification landscape; the present method sits in the uncertainty quantification branch.
Cyclic Reformulation-Based System Identification for Periodically Time-Varying Systems — Companion article on the LPTV identification framework that this paper builds on.
System Identification Under Multirate Sensing Environments — Companion article on the multi-rate extension.

Research Web Page

www.control-theory.com/en/ — Research overview.

Paper Information

Citation: H. Okajima, S. Shirahama, T. Hayashi and N. Matsunaga, "From Noise to Knowledge: System Identification with Systematic Polytope Construction via Cyclic Reformulation", IEEE Access (2026), DOI: 10.1109/ACCESS.2026.3692992 (Open Access, CC BY 4.0).

Co-authors: Shun Shirahama, Tatsunori Hayashi, and Prof. Nobutomo Matsunaga — all at Kumamoto University, Japan.

Self-Introduction

This article was written by Hiroshi Okajima, Associate Professor at Kumamoto University, Japan. My research focuses on control engineering, particularly tracking control, analysis of non-minimum phase systems, model error compensators, state estimation, and quantized control for networked systems. I have been working in control engineering research for 20 years. For more on my work, visit www.control-theory.com (research), blog.control-theory.com (blog), YouTube channel, video portal, and GitHub.

If you found this article helpful, please consider bookmarking or sharing it.

SystemIdentification #PolytopicUncertainty #CyclicReformulation #RobustControl #HInfinityControl #LMI #SubspaceIdentification #N4SID #ControlEngineering #IEEEAccess

*1:Nn)^{3})], coming from subspace identification and coordinate transformation on the lifted state space of dimension $Nn$ . The total complexity therefore scales as $O((Nn)^{3})$ , and the data length must be sufficiently larger than $(Nn)^{2}$ for the vertex estimates to be reliable. This trade-off is examined quantitatively in the numerical examples below.

Working Hypothesis and the Best In-Polytope Point

A natural question at this point is whether the constructed polytope actually contains the true plant. The paper is explicit that this property is not claimed:

A rigorous theoretical guarantee of this containment property remains an open problem, so the existence of such interior points is treated as a hypothesis and examined numerically.

The working hypothesis is weaker but practical:

The convex hull of the $N$ vertices contains models that, taken as a set, capture meaningful uncertainty information about the true plant.

To examine this hypothesis quantitatively, the paper introduces the best in-polytope point

$\displaystyle \lambda^{\ast} = \arg \min_{\lambda \in \varepsilon} \sum_{k=1}^{N_{\mathrm{val}}} \| y_{\mathrm{val}}(k) - y(k; \lambda) \|^{2}$

where $y_{\mathrm{val}}(k)$ is the noise-free output of the true plant on a validation input, and $y(k; \lambda)$ is the prediction from the in-polytope model [tex: (A(\lambda), B(\lambda), C(\lambda), D(\lambda