A tracking problem, time-delay, uncertainty and stability analysis of a predictive control system are considered. The predictive control design is based on the input and output of neural plant model (NPM), and a recursive fuzzy predictive tracker has scaling factors which limit the value zone of measured data and cause the tuned parameters to converge to obtain a robust control performance. To improve the further control performance, the proposed random-local-optimization design (RLO) for a model/controller uses offline initialization to obtain a near global optimal model/controller. Other issues are the considerations of modeling error, input-delay, sampling distortion, cost, greater flexibility, and highly reliable digital products of the model-based controller for the continuous-time (CT) nonlinear system. They are solved by a recommended two-stage control design with the first-stage (offline) RLO and second-stage (online) adaptive steps. A theorizing method is then put forward to replace the sensitivity calculation, which reduces the calculation of Jacobin matrices of the back-propagation (BP) method. Finally, the feedforward input of reference signals helps the digital fuzzy controller improve the control performance, and the technique works to control the CT systems precisely.

During the past decade, many fuzzy theories [

In addition, neural-network- (NN-) based modeling has become an active research field because of its unique merits in solving complex nonlinear system identification and control problems (see [

The study of CT control of CT time-delay systems has received considerable attention in recent years since delay is a major cause of poor performance in many important engineering systems [

Based on the timer of the micro-controller, the effect of delay in neural system identification can be approximated by many tape-delay terms. This reduces the difficulty of delay identification. The DT NARMAX model is generally sufficient to approximate an unknown, nonlinear, dynamical, and delayed CT system by selecting an appropriate sampling time. Moreover, the measured modeling error between the model and the physical system is considered in the theorems by Lyapunov functions. In Remark

The feedforward term in [

Conventional optimization methods assume that all parameters and goals of a model are precisely known [

Inspired by the DT neural controller of [

The remainder of this paper is organized as follows. In Section

First, the conventional PWM buck converter by using AM-OTS-DS [

The nonlinear, uncertain, hotter circuit’s components, time-delay, and digital control problems of PWM buck converter CT system renders a tracking control problem difficult to analyze. A simulation system in (

Then, consider a general nonlinear system with delays described as follows:

With the understanding that normal physical systems are low-pass filter and smooth nonlinearity, the upper bound, UBSP (shown in Figure

UBSP of (

In this paper, an adaptive prediction control structure is proposed, as shown in Figure

Two-stage adaptive prediction structure of DT-CT control system. (a) Two-stage block diagram and (b) online adaptive prediction block diagram.

The offline training input of controller is

The proposed digital fuzzy controller

If the amount of neurons of the neural model is sufficient and the appropriate sampling time

First, consider the following ideal Lyapunov candidate [

Then, the following Lyapunov candidate of the controller part is designed:

Furthermore, the following theorem for the convergence of the controller is obtained by the same procedure as the above proof.

If Theorem

Hence, the dynamic response of the system

The digital feedback controller includes a delay block

If Theorem

The tracking error is

Figure

In the first stage, the measured data is used to train the global optimal NARMAX plant and the fuzzy controller by the training-data-shuffle method. This method shuffles the training data to avoid most of the local optimal solutions obtained by the offline training procedure in the next section. The measured data is divided into a training data and other testing data. This testing data is not used for training the NN. However, the final performance of the NN is decided by the testing data and the training data.

In the second stage, the global optimal NARMAX plant model and fuzzy controller is adapted. The two stages are divided into the following five steps.

First, the reference signal

The feedforward structure model

In practice, according to the exchanged output/input pairs’ data from Step

If (

Update the online weights and biases

Adapt the digital fuzzy controller for the modeling error and tracking error by using Theorems

To make sure of the robustness of the control system, the convergence to the global optimal solution of parameters of the model/controller has to be guaranteed. Hence, some random initial weights and biases of the model are designed by particle swarm optimization (PSO) [

The PSO supplies random initial parameters, hence, it is an initial parameters’ conductor. These initial parameters are then converged locally by the BP method and the best solution for the initial model/controller is chosen. Finally, the global optimal solution of parameters can be found every time. Hence, this idea has been named the random-local-optimization (RLO) algorithm. The RLO algorithm is a composite of the BP algorithm and a random initialization procedure of evaluating fitness value

Back propagate through

Back propagate through

Compute

To clarify this method, in [

First, the conventional PWM buck converter, by using AM-OTS-DS [

Referring to Figure

To compare with other methods, the following cases are introduced.

This case is in [

This case is in [

This is the control method presented here, and the proposed neural-model-based fuzzy controller is adaptive, predictive, and globally optimal.

The detail designs of the Cases

Case

Case

And the design parameters

Case

First, the sampling time of the model/controller is set for

The adaptive weights and biases,

(a) The learning curve of the summation of

And the tracking control performance of Case

(a) The tracking control performance and (b) the parameters

(a)(b) Comparison of the output trajectories for Cases

Finally, the control performances of Cases

It is clear that the two-stage scheme, Case

The proposed two-stage adaptive prediction control converges very fast, works highly effective, and precise. It simplifies the complex model-based adaptive control design, and works for nonlinear delayed plants with uncertainty. The proposed recursive and feedforward control scheme is partitioned into two stages that can be independently optimized. First, an offline neural model of continuous-time (CT) nonlinear power plant is made. Second, a constrained offline digital fuzzy controller is generated; then, an adaptive plant model is made, and an adaptive NARMAX prediction tracker is generated. Finally, all processes may continue concurrently, and robustness and adaptive prediction design with DT-CT problems for a power plant are solved. Although this power system is only a simulation, the control strategy can be extended to LED dimmer systems and time-delay robot systems based on visual servo, and is within my plans of future research.

The author would like to thank the National Science Council and Asia University for the support under Contracts nos. NSC-97-2218-E-468-009, NSC-98-2221-E-468-022, NSC-99-2628-E-468-023, NSC-100-2628-E-468-001, NSC-101-2221-E-468-024, 98-ASIA-02, 100-asia-35, and 101-asia-29.

_{∞}fuzzy output-feedback control with multiple probabilistic delays and multiple missing measurements

_{∞}control of uncertain fuzzy dynamic systems with time-varying delay