Compressing Control Signals by Design: Dynamic Quantizer Optimization for Level-Constrained Communication
H. Okajima, K. Sawada, and N. Matsunaga, "Dynamic Quantizer Design Under Communication Rate Constraints," IEEE Transactions on Automatic Control, Vol. 61, No. 10, pp. 3190–3196, 2016.
- Paper (IEEE Xplore): https://doi.org/10.1109/TAC.2015.2509438
- Paper (Kumamoto University Repository, PDF): https://kumadai.repo.nii.ac.jp/
- Support Page: https://sites.google.com/view/quantizer-communication-rate
- MATLAB Code (GitHub): Hiroshi-Okajima/MATLAB_Dynamic_Quantizer01
- MATLAB Code (Code Ocean): https://doi.org/10.24433/CO.1974157.v1
Extended Abstract
Background: The Quantization Bottleneck in Networked Control
In networked control systems (NCS), control signals and sensor data must be transmitted over digital communication channels with limited capacity. Before transmission, continuous-valued signals are quantized into a finite set of discrete levels. The fewer the levels, the less data needs to be transmitted — but coarser quantization introduces larger quantization errors, which can degrade control performance or even destabilize the system.
This tension between data compression and control performance is a central challenge in NCS design. It becomes particularly acute in scenarios such as IoT-based control, remote operation over bandwidth-limited links, and embedded systems with constrained memory for storing signal histories.
A naive static quantizer (e.g., uniform rounding) offers no mechanism to manage this trade-off. Dynamic quantizers, such as delta-sigma ($\Delta\Sigma$) modulators, take a fundamentally different approach: they incorporate internal feedback to shape the quantization error, pushing its energy away from the frequency bands that matter most for control. This is the same noise-shaping principle used in high-fidelity audio AD/DA conversion — but here it is applied to control signal compression.
The Problem: Complete Design Under Level Constraints
A feedback-type dynamic quantizer consists of two parts: a linear filter and a static quantizer with a fixed number of output levels. The filter processes the input signal and feeds back the quantizer output, effectively distributing the quantization error over time.
The design challenge is that all quantizer parameters — filter coefficients, quantization interval (step size), and the number of output levels — are tightly coupled. Changing one affects the feasibility and optimality of the others. Previous works (e.g., Azuma & Sugie, Automatica 2008) designed the filter assuming a fixed quantization interval and did not explicitly account for the number of output levels. When such a quantizer is placed under a level constraint, the internal state may exceed the representable range, causing saturation and performance breakdown.
This paper provides the first complete, simultaneous design of all dynamic quantizer parameters under an explicit quantization level constraint $N \leq 2^M$.
The Key Relationship: Theorem 1
The central result connects the minimum quantization interval $d^{opt}$, the number of levels $N$, the signal range $[u_{min},, u_{max}]$, and a quantity $\psi^{opt}$ that captures the internal signal amplification of the filter:
$$
d^{opt} = \frac{u_{max} - u_{min}}{N - \psi^{opt}}
$$
Here, $\psi^{opt}$ is determined by the filter parameters ${\mathcal{A}, \mathcal{B}, \mathcal{C}}$ and corresponds to the $\ell_1$-norm of the impulse response of the internal dynamics:
$$
\psi^{opt} = \sum_{i=0}^{\infty} \left| \mathcal{C}(\mathcal{A} + \mathcal{B}\mathcal{C})^i \mathcal{B} \right|
$$
This equation reveals the fundamental trade-off: $\psi^{opt}$ represents how much the feedback filter expands the signal range internally. If $\psi^{opt}$ is large, the quantizer needs a wider range to avoid saturation, forcing a larger step size $d^{opt}$ and thus worse performance. If $N - \psi^{opt} \leq 0$, no valid quantization interval exists — the filter simply cannot operate within the given number of levels.
In other words, the filter must be designed so that its internal dynamics fit within the budget of quantization levels. This is the constraint that prior designs did not address explicitly.
Achievable Performance Under the Constraint (Remark 1)
Combining Theorem 1 with the $\ell_1$-norm characterization of the overall error system, the worst-case output error $E(Q)$ is expressed in closed form. Let $\bar{A}, \bar{B}, \bar{C}$ be the augmented matrices of the plant $P$ and the quantizer filter:
$$
E(Q) = \left( \sum_{i=0}^{\infty} \left| \bar{C}\bar{A}^{i}\bar{B} \right| \right) \frac{d^{opt}}{2}
$$
where $d^{opt} = (u_{max} - u_{min}) / (N - \psi^{opt})$ from Theorem 1. This is the key equation: it gives the exact achievable control performance under the level constraint as a product of two factors — the $\ell_1$-gain of the cascaded plant-quantizer system and the minimum feasible quantization interval. When $N$ is small (severe constraint), $d^{opt}$ grows and $E(Q)$ increases; a well-designed filter minimizes both the $\ell_1$-gain and $\psi^{opt}$ simultaneously. For the full derivation and the augmented system matrices, see Section III of the paper.
Design Method: Two-Step Optimization via Invariant Set + PSO
Because the coupled optimization over ${\mathcal{A}, \mathcal{B}, \mathcal{C}, d}$ is non-convex, the paper employs a two-step approach to minimize $E(Q)$ above:
Step 1: Iterative LMI-based design. An iterative algorithm alternates between optimizing the filter parameters ${\mathcal{A}, \mathcal{B}, \mathcal{C}}$ and the Lyapunov matrices ${Z_p, Z_d}$ using invariant set analysis. Each sub-step reduces to an LMI problem solvable by standard interior-point methods. This produces a high-quality initial quantizer together with a search direction (parameter velocity).
Step 2: PSO refinement. The quantizer from Step 1 seeds one particle in a Particle Swarm Optimization (PSO) population. The remaining particles are initialized randomly. PSO then searches the combined parameter space to minimize the performance index $E(Q)$ subject to the level constraint $\psi^{opt,L} < N$. Infeasible particles (those violating the level constraint) receive a large penalty.
This hybrid approach combines the structural insight from control-theoretic invariant set analysis with the global search capability of PSO. The initial quantizer from Step 1 ensures that PSO starts from a feasible, high-quality solution rather than a random point — a critical advantage given the non-convex landscape.
Beyond Communication Rate: Data Compression Perspective
Although the paper formulates the constraint in terms of communication rate ($N \leq 2^M$), the framework applies more broadly. From Theorem 1, reducing $N$ directly increases the minimum quantization interval $d^{opt}$ — but a well-designed filter keeps $\psi^{opt}$ small, mitigating this penalty. This means the dynamic quantizer acts as a designed signal compressor: it minimizes the number of discrete levels needed to represent a control signal while preserving the information content essential for control performance.
Related Works on Dynamic Quantizers by the Authors
The TAC paper represents the culmination of a sustained research program on dynamic quantizer design. Below is the development trajectory:
| Year | Title | Key Contribution | Ref |
|---|---|---|---|
| 2010 | Optimal quantization interval design of dynamic quantizers which satisfy the communication rate constraints | First formulation of quantization interval design under rate constraints; invariant set-based approach | IEEE CDC 2010 |
| 2011 | Design of Dynamic Quantizers for 2-DOF IMC and Its Application to the Temperature Control of a Heat Plate | Application to dead-time systems — dynamic quantizer for internal model control under memory constraints; experimental validation on thermal plant | JCMSI (OA) |
| 2011 | Integrated Design of Filter and Interval in Dynamic Quantizer under Communication Rate Constraint | Joint filter + interval design via IFAC World Congress | IFAC WC 2011 |
| 2011 | Dynamic quantizer design for MIMO systems based on communication rate constraint | Extension to MIMO — multi-input multi-output quantizer design | IECON 2011 |
| 2014 | A design method of delta-sigma data conversion system with pre-filter | Pre-filter + post-filter co-design for AD/DA conversion; PSO-based optimization for voice signal compression | JCMSI (OA) |
| 2014 | Dynamic quantizers design under data rate constraints by using PSO method | PSO method introduced for quantizer parameter search | SICE 2014 |
| 2016 | Dynamic Quantizer Design Under Communication Rate Constraints (this paper) | Complete simultaneous design of all parameters (filter, interval, levels) via PSO + invariant set; IEEE TAC | IEEE TAC |
| 2016 | Color Quantization and Optimization of Luminance for DMD-Based Projector | Cross-domain application — quantization framework applied to image color quantization in DLP projectors | IEEE Trans. CE |
| 2018 | A Control Structure for Unilateral System with Communication Rate Constraint | Unilateral control — quantizer design for one-way communication systems | JCMSI (OA) |
| 2019 | Basic Idea of Periodically Time-Varying Dynamic Quantizer in Networked Control Systems | Periodic quantizer — extending dynamic quantizers to time-varying structures (Finalist of SICE poster award) | SICE 2019 |
Development Arc
The research evolved through three phases: (1) foundational theory (2010–2011) establishing the invariant set-based analysis and the rate constraint formulation; (2) complete optimization (2014–2016) introducing PSO for simultaneous parameter design, culminating in the IEEE TAC paper; and (3) extensions and applications (2016–2019) broadening the framework to MIMO systems, unilateral control, periodic quantizers, and cross-domain applications including image processing and voice signal compression.
Reproducibility
MATLAB implementations are available from three sources:
- GitHub: Hiroshi-Okajima/MATLAB_Dynamic_Quantizer01
- Code Ocean: https://codeocean.com/capsule/1974157/tree/v1
For full derivations including the invariant set conditions and PSO initialization procedure, see the original paper.