In networked control systems, continuous-valued signals are compressed to discrete-valued signals via quantizers and then transmitted/received through communication channels. Such quantization often degrades the control performance; a quantizer must be designed that minimizes the output difference between before and after the quantizer is inserted. In terms of the broadbandization and the robustness of the networked control systems, we consider the continuous-time quantizer design problem. In particular, this paper describes a numerical optimization method for a continuous-time dynamic quantizer considering the switching speed. Using a matrix uncertainty approach of sampled-data control, we clarify that both the temporal and spatial resolution constraints can be considered in analysis and synthesis, simultaneously. Finally, for the slow switching, we compare the proposed and the existing methods through numerical examples. From the examples, a new insight is presented for the two-step design of the existing continuous-time optimal quantizer.

With the rapid network technology development, the networked control systems (NCSs) have been widely studied [

Motivated by this, researchers [

Two systems.

Discrete-valued input system

Usual system

When we consider controlling a mechanical system with an on-off actuator, first the controlled object and its uncertainties are usually modeled in the continuous-time domain. Second, the model and its uncertainties are discretized to apply the above dynamic quantizer. However, the discretization sometimes results in uncertainties more complicated than those in the original model and creates undesirable complexity in robust control. The continuous-time setting quantizer is more suitable for the robust control of the quantized system than discrete-time one. Thus, our previous works [

On the other hand, the above assumption is essentially weak in the case of the slow switching such as the mechanical systems with on-off actuators [

We propose a numerical optimization method for the continuous-time dynamic quantizer under switching speed and quantized accuracy constraints. To achieve the method, this paper solves the design problem via sampled-data control framework that has so far provided various results for networked control problems [

Consider the discrete-valued input system

For the system

Continuous-time dynamic quantizer with switching speed

Midtread quantization.

In synthesis, our previous works [

For the system

On the other hand, the simultaneous consideration of the temporal and spatial resolution constraints is the problem we address in this paper. To consider the temporal resolution constraint caused by the operator

Motivated by the above, our objective is to solve the following continuous-time dynamic quantizer synthesis problem

This paper proposes continuous-time quantizers in terms of solving the problem

The cost function setting of this paper is more complicated than the existing continuous-time and discrete-time cases [

The plant

In this subsection, we consider the system expression for the quantizer analysis. Define the quantization error

Denote by

See Appendix

We focus on

The quantization error

Define the reachable set of the system (

Define the invariant set of the system (

The analysis condition can be expressed in terms of matrix inequalities as summarized in the following proposition [

Consider the system (

Note that the ellipsoidal set

Then, this paper utilizes the reachable set to estimate the influences of the quantization error and the invariant set to characterize the cost function

For the inequalities (

The inequalities (

For the matrix

Since

By using Lemma

Consider the system

See Appendix

Denote by

In numerical computation, it is appropriate to fix the structure of

The problem

From the matrix product such as

Under some circumstances (

Consider the system

There exist matrices

There exist matrices

We fix

In the limit of

To consider numerical optimization analysis or synthesis of a quantizer as shown in

For the slow switching, we compare the proposed method and existing continuous-time quantizer [

For the comparison, we set the switching speed

Time responses of

Time responses of

Next, we consider the case

Time responses of

Time responses of

In the above numerical experiments, the proposed quantizer is designed and the quantizer

Here, we focus on the eigenvalues of

From the above results, the existing continuous-quantizer in [

Focusing on the broadbandization and the robustness of the networked control systems, this paper has dealt with the continuous-time quantized control. We have proposed numerical optimization methods analyzing and synthesizing the continuous-time dynamic quantizer on the basis of the invariant set analysis and the sampled-data control technique. The contributions of the proposed method can be summarized as follows.

Both the temporal and spatial resolution constraints can be simultaneously considered, whereas Ishikawa et al. [

The maximum output difference for each sampling interval is proven to be evaluated numerically via the matrix uncertainty approach, while the existing results [

The analysis and synthesis conditions are given in terms of BMIs. However, the quantizer analysis and synthesis problems are reduced to tractable optimization problems.

The new insight is presented for the existing continuous-time quantizer design [

Because of the feedforward structure, the plant is restricted to be stable. To address unstable systems, we need to propose design methods for feedback control systems.

The sensors and actuators are distributed in the networked control system [

The class of exogenous signals evaluating the cost function is restricted. To avoid this conservativeness, it is necessary to propose the equivalent discrete-time expression instead of (

For networked control applications, it is important to consider the time-varying sampling period, time delay, packet loss, and so on similar to [

The proof of Lemma

For the proof of Theorem

For the real matrices

Then, the proof of Theorem

We use Lemmas

For the system

The authors would like to thank the reviewers for their valuable comments. This work was partly supported by Grant-in-Aid for Young Scientists (B) no. 24760332 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.