Metaheuristic Based Design of Finite-level Dynamic Quantizers for Networked Control Systems


In recent years, the networked control systems or NCSs have been receiving a lot of attention from researchers and manufactures due to their many advantages and potential applications. There are two mayor problems that affect the NCSs: the quantization errors and the data rate constraints of the channel. These problems degrade the performance of the system and may lead to instability. Several studies have shown that the use of properly designed dynamic quantizers is an effective way to reduce the performance degradation. However, the design of dynamic quantizers under the channel’s data rate constraint is not an easy task. Until now, there is not a perfect design method for this type of quantizers. The existing ones give under-optimal solutions.


Figure 1: Quantized system. The quantizer transforms the continous-valued signal u into a discrete-valued one v. The quantized signal is encoded into a chain of bits and sent through the channel. In the destiny the chain of bits is decoded, reproducing the quantized signal, and applied to the plant.


The main objective of this thesis is the development of easy-to-use and powerful computational methods for the design of finite-level dynamic quantizers that:

  • Minimize the performance degradation due to quantization errors and,
  • Satisfy the communication rate constraint of the channel.

Proposed Method

The finite-level dynamic quantizer design is formulated as a non-linear and non-convex optimization problem. In this problem we minimize a parameter that measures the performance degradation of the system (performance index). Since this optimization problem cannot be solved by conventional methods like linear programing and quadratic programing we make use of metaheuristics.

The metaheuristics are strategies for exploring search spaces in order to find (near-) optimal solutions. The ones used in this study are: covariance matrix adaptation evolution strategy (CMA-ES), differential evolution (DE) and firefly algorithm (FA).


Figure 2: Two dimensional example of the operation of the CMA-ES algorithm. The black dots represent the search points (possible solutions) sampled from a multivariate normal distribution. The red dot is the mean of the normal distribution and the red circle is a section of the normal distribution with constant probability. It is possible to see how the distribution adjust itself to explore all the search space and the mean goes toward the global minima throw the generations.

Numerical examples

The effectiveness of the proposed design methods is verified by numerical examples. The simulations show that the design methods based on CMA-ES and DE works properly and that they minimize the performance degradation of the system.

Besides, we measure the performance of the design methods by comparing them among each other and with a previously developed method based on particle swarm optimization (PSO). The criteria used for comparison are the success rate, convergence behavior and the running time of the algorithm. Figures 3 and 4 show some results of these numerical examples.


Figure 3: Success rate. The ratio of the number of runs of the algorithm that finds the best solution to the total number runs is know as success rate. It give us an idea of how effective is an algorithm in finding the global optima.


Figure 4: Converge behavior of the different design methods. Each graph show the performance index versus the number of generations for a specific number search points N for the different metaheuristic based design methods. The number of generations goes from 0 to 50.

Since the success rates of the methods based on CMA-ES an DE are very high we can say that these algorithms are reliable. In the same way, the convergence behavior show us that these two methods converge very fast to the optimal solution.