This article provides a detailed explanation of dynamic quantizer design for networked control systems under communication rate constraints. Related articles, related papers, and MATLAB 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, 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)

MATLAB Code (GitHub)

This paper is co-work with Prof. Kenji Sawada (Professor, Osaka University) and Prof. Nobutomo Matsunaga (Kumamoto University).

Contents

Why Dynamic Quantizer Design under Communication Rate Constraints Matters

In networked control systems (NCS), data must be transmitted through communication channels with limited capacity. Because these channels can only carry a finite number of bits per sampling period, control signals must be compressed using quantizers. This quantization inevitably introduces performance degradation.

Feedback-type dynamic quantizers, such as delta-sigma modulators, are known to be effective for encoding high-resolution data into lower-resolution data while minimizing quantization error. A dynamic quantizer consists of a filter and a static quantizer. The filter utilizes previous quantization error information to generate better quantizer output — a technique widely used in signal processing (AD/DA converters, audio data compression, switched-mode power supplies) and increasingly in control engineering.

However, when the communication channel has a limited data rate of $M$ bits per sampling, the number of quantization levels must satisfy $N \leq 2^{M}$ . This means the quantization interval in the static quantizer must be chosen carefully — it is directly linked to the number of output levels. Prior works on dynamic quantizer design (e.g., the optimal dynamic quantizer by Azuma and Sugie, 2008) focused on filter design while assuming the quantization interval was given. The number of output levels had not been analyzed explicitly.

This paper addresses the complete design of a feedback-type dynamic quantizer — including both the filter parameters and the quantization interval — to minimize a performance index while satisfying communication rate constraints. The design method combines an invariant set analysis with a particle swarm optimization (PSO) algorithm.

Control System with a Communication Channel

Consider a SISO discrete-time stable plant:

$\displaystyle x_p(k+1) = A_p x_p(k) + B_p u_p(k), \quad y_p(k) = C_p x_p(k)$

where $x _p \in \mathbb{R}^{n _p}$ is the state, $u _p \in \mathbb{R}$ is the control input, and $y _p \in \mathbb{R}$ is the control output.

The system structure is a feedforward control system with a communication channel between the quantizer output and the plant input. An outer signal $u$ (e.g., a command or operating signal) is transformed by the quantizer $Q$ into a lower-resolution signal $v$ , which is then encoded and transmitted through the channel. After decoding, the plant receives $u _p = v$ as its input.

The number of quantization levels $N$ depends on the channel data rate. When $M$ bits are transmitted per sampling period:

$\displaystyle N \leq 2^M$

The outer signal $u$ is bounded: $u(k) \in U = \lbrack u _{\min}, u _{\max} \rbrack$ for all $k$ .

Dynamic Quantizer Form

The feedback-type dynamic quantizer $Q$ is defined as:

$\displaystyle \xi(k+1) = \mathcal{A}\xi(k) - \mathcal{B}u(k) + \mathcal{B}v(k)$

$\displaystyle v(k) = Q_{st}[\mathcal{C}\xi(k) + u(k)$ ]

where $Q _{st}$ is a mid-riser type uniform static quantizer with saturation, defined by a quantization interval $d \in \mathbb{R} _+$ and a center point $c \in \mathbb{R}$ . The matrices $\mathcal{A} \in \mathbb{R}^{n _q \times n _q}$ , $\mathcal{B} \in \mathbb{R}^{n _q \times 1}$ , and $\mathcal{C} \in \mathbb{R}^{1 \times n _q}$ are the filter parameters, and the initial state is $\xi(0) = 0$ .

The quantizer output $v(k)$ is obtained by static quantization of $\mathcal{C}\xi + u$ . The key difference from existing approaches is that this paper designs both the filter parameters $\{\mathcal{A}, \mathcal{B}, \mathcal{C}\}$ and the quantization interval $d$ together, subject to the constraint that the total range $Nd$ covers the signal range.

Design Problem Based on Error System

The design goal is formulated using an error system. Let $y _r(k)$ be the desired output obtained by applying $u(k)$ directly to the plant, and $y(k)$ be the actual output when the quantized signal $v(k)$ is applied. The error $e(k) = y(k) - y _r(k)$ should be minimized.

The performance index is:

$\displaystyle E(Q) = \sup_{u(k) \in U} \lVert Y - Y_r \rVert$

where $Y$ and $Y _r$ are the actual and desired output time series, respectively. The infinity norm is used: the performance index represents the worst-case output error over all admissible input signals. If $E(Q)$ is small, the actual output closely approximates the desired output despite quantization.

Minimum Quantization Interval for Given Dynamic Quantizers (Theorem 1)

Assuming the filter parameters $\{\mathcal{A}, \mathcal{B}, \mathcal{C}\}$ are given, the paper derives the minimum quantization interval that satisfies the communication rate constraint.

The key observation is that the input to the static quantizer is $\bar{u} = u + \mathcal{C}\xi$ , and the range of $\bar{u}$ depends on both the signal range $U$ and the range of the internal variable $\phi = \mathcal{C}\xi$ . Finding the range of $\phi$ is equivalent to finding the $l _1$ -norm of the impulse response of a related linear system.

Through a coordinate transformation, the problem reduces to finding a parameter $\psi$ that characterizes the signal amplitude caused by dynamic quantization.

Theorem 1: The minimum quantization interval and optimal center point are:

$\displaystyle d^{opt} = \frac{u_{\max} - u_{\min}}{N - \psi^{opt}}, \quad c^{opt} = \frac{u_{\max} + u_{\min}}{2}$

where $\psi^{opt}$ is the optimal solution of a minimization problem related to the reachable set of the internal state. If $N - \psi^{opt} \leq 0$ , no valid quantization interval exists, and the filter parameters must be redesigned.

This result establishes a direct relationship between the filter parameters (through $\psi^{opt}$ ) and the minimum quantization interval. The value $\psi^{opt}$ serves as an index of usability of the dynamic quantizer for signal communication.

Estimation of Optimal Quantization Parameter

The value $\psi^{opt}$ can be estimated using an $l _1$ -norm computation. For a positive integer $L$ :

$\displaystyle \psi^{opt,L} = \sum_{i=0}^{L} \lVert \mathcal{C}(\mathcal{A} + \mathcal{B}\mathcal{C})^i \mathcal{B} \rVert$

An upper bound is obtained by combining this partial sum with a residual term $\psi^{\ast,L}$ derived from an LMI problem based on invariant set analysis:

$\displaystyle \psi^{opt,L} < \psi^{opt} \leq \psi^{opt,L} + \psi^{\ast,L}$

For large $L$ , the residual $\psi^{\ast,L}$ approaches zero, giving a tight estimate.

Remark 1: Combining this with existing results, the performance index can be expressed as:

$\displaystyle E(Q) = \left(\sum_{i=0}^{\infty} \lVert \bar{C}\bar{A}^i\bar{B} \rVert \right) \frac{d^{opt,L}}{2}$

where $\bar{A}$ , $\bar{B}$ , $\bar{C}$ are augmented system matrices combining the plant and dynamic quantizer. This means that for given filter parameters, the control performance under communication rate constraints can be analyzed explicitly.

Design of Dynamic Quantizer Using PSO

The complete design of the dynamic quantizer is formulated as a two-step design method.

Step 1: Iterative Design Based on Invariant Set Analysis

An iterative algorithm using LMI optimization is used to obtain initial quantizer parameters. The design problem minimizes a combined objective:

$\displaystyle \Gamma(\gamma, \psi) = \gamma \cdot \frac{u_{\max} - u_{\min}}{2(N - \psi)}$

subject to LMI constraints from invariant set analysis. Because the evaluation function $\Gamma$ is nonlinear in $\gamma$ and $\psi$ , it is replaced by a linear substituted function $J = a\gamma^{2} + b\psi^{2} + g$ with appropriately updated coefficients at each iteration step. The coefficients $a$ and $b$ are chosen so that minimizing $J$ approximates minimizing $\Gamma$ in the neighborhood of the current solution. The iteration alternates between updating filter parameters and updating the Lyapunov-like matrices until convergence.

Step 2: Particle Swarm Optimization (PSO)

The iterative algorithm provides good initial particles for a PSO algorithm that searches for the globally optimal filter parameters. PSO is a population-based optimization method. Each particle represents a candidate quantizer $p _i = \{\mathcal{A} _i, \mathcal{B} _i, \mathcal{C} _i\}$ , and particles update their positions based on personal best and global best solutions:

$\displaystyle p_i^{t+1} = p_i^t + \Delta p_i^{t+1}$

$\displaystyle \Delta p_i^{t+1} = \omega_0 \Delta p_i^t + \omega_1 r_1 (p_{pbest,i}^t - p_i^t) + \omega_2 r_2 (p_{gbest}^t - p_i^t)$

The communication rate constraint $\psi^{opt,L}(p) < N$ is enforced using a penalty function. The quantizers obtained from the iterative algorithm serve as initial particles (with $m _r \geq 1$ of the $m$ total particles), providing better starting points than random initialization.

Numerical Examples

The paper evaluates the method using three plants derived from continuous-time transfer functions with sampling time $\Delta t = 0.1$ :

$P _1(s) = (s+20)/(s^{2}+3s+2)$
$P _2(s) = 1/(s^{2}+3s+2)$
$P _3(s) = (s-5)/(s^{2}+3s+2)$ (non-minimum-phase)

The signal range is $U = \lbrack -1, 1 \rbrack$ , and PSO parameters are $m _r = 1$ , $m = 1000$ , $t _{\max} = 300$ , $L = 100$ .

For $P _1$ with $N = 2$ , the proposed algorithm yields a dynamic quantizer with $\psi^{opt} = 0.937$ , satisfying the communication rate constraint $N - \psi^{opt} > 0$ , and achieves $E(Q) = 0.757$ .

Comparison with Iterative Algorithm (Ref. [16])

Plant	E(Q) Proposed (N=2)	E(Q) Ref.[16] (N=2)	E(Q) Proposed (N=8)	E(Q) Ref.[16] (N=8)
P1	0.757	3.27	0.316	0.329
P2	0.0378	0.246	0.0016	0.0552
P3	0.309	0.747	0.0303	0.0530

The proposed method outperforms the iterative algorithm for all plants and all values of $N$ . The improvement is especially large when $N$ is small (severe communication rate constraint).

Comparison with Optimal Quantizer (Ref. [11])

The optimal dynamic quantizer by Azuma and Sugie (Ref. [11]) does not explicitly account for communication rate constraints. For plant $P _1$ :

N	E(Q) Proposed	E(Q) Ref.[11]
2	0.757	— (infeasible)
4	0.1088	0.1116
8	0.0316	0.0316
16	0.0130	0.0130

For $N = 2$ , the optimal quantizer from Ref. [11] yields $\psi^{oq} = 2.4176$ , which does not satisfy the communication rate constraint. For larger $N$ (= 8, 16), the proposed method recovers nearly the same performance as the unconstrained optimal quantizer.

This work is part of a broader research program on quantized control and networked control systems:

Unilateral Control Under Communication Rate Constraint — H. Okajima, Y. Minami and N. Matsunaga, A Control Structure for Unilateral System with Communication Rate Constraint, SICE JCMSI, Vol. 11, No. 6, pp. 510–516 (2018). Extends the dynamic quantizer framework to unilateral control structures, which are practically important for teleoperation and remote control applications.

MIMO Dynamic Quantizer Design — K. Sawada, H. Okajima, N. Matsunaga and Y. Minami, Dynamic quantizer design for MIMO systems based on communication rate constraint, IECON 2011. Extends the dynamic quantizer design to multi-input multi-output systems, broadening the applicability of the communication-constrained design framework.

Periodically Time-Varying Dynamic Quantizer — H. Okajima, Y. Hosoe, T. Hagiwara and Y. Minami, Basic Idea of Periodically Time-Varying Dynamic Quantizer in Networked Control Systems, Proc. SICE Annual Conference 2019. Proposes a periodically time-varying extension of the dynamic quantizer concept, connecting the quantizer design with multi-rate system theory.

Model Error Compensator (MEC) — The dynamic quantizer can be combined with the Model Error Compensator to achieve robust control in networked environments with both quantization and model uncertainty.

MATLAB Code

MATLAB File Exchange: Dynamic Quantizer for NCS (example in IEEE TAC note)
GitHub: MATLAB Dynamic Quantizer 01

Blog Articles (blog.control-theory.com)

Research Web Pages (www.control-theory.com)

Dynamic Quantizer / Publications / LMI / MEC

Video

YouTube: Control Engineering Channel / Portal

Paper Information

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. DOI: 10.1109/TAC.2015.2509438. MATLAB Code

Co-authors: Kenji Sawada (Professor, The University of Osaka), Nobutomo Matsunaga (Kumamoto University)

Earlier conference paper: R. Yoshino, H. Okajima, N. Matsunaga and Y. Minami, "Dynamic quantizers design under data rate constraints by using PSO method", SICE Annual Conference 2014, pp. 1041–1046 (2014)

Self-Introduction

Hiroshi Okajima — Associate Professor, Graduate School of Science and Technology, Kumamoto University. Member of SICE, ISCIE, and IEEE.

www.control-theory.com / Blog / YouTube / GitHub

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DynamicQuantizer #NetworkedControlSystems #CommunicationRateConstraint #QuantizerDesign #PSO #ParticleSwarmOptimization #InvariantSet #DiscreteTimeControl #MATLAB