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This paper introduces a new approach to design Model-Free Adaptive Controller (MFAC) using adaptive fuzzy procedure as a feedback linearization based on output error. The basic idea is to transfer the control signal to an appropriate surface and then, depending on the output error of system, the control signal changes around this surface. Some examples are provided as well to illustrate the efficiency of the proposed approach. The obtained simulation results have shown good performances of the proposed controller.

Classical control method is based on mathematical equations of the system; however, this method suffers from some drawbacks. For example, in this method the performance of system can be affected by unmodeled dynamics of system and/or by large delays. Model-Free Adaptive Control (MFAC), as a part of modern control theory, shows superiorities compared to model-based methods. MFAC is an adaptive control method and needs no information about system model. It only uses

In 1994, Han and Wang introduced model-free topic and proved the stability of MFAC controllers. In 1993-1994, Hou Zhongsheng represented applications of these controllers. With using pseudo-Jacobian matrix, nonlinear systems are replaced to near the rail line of controlled system and then the

In 1995, Jagannathan suggested a fuzzy stable controller for a limited class of nonlinear system in the form of

Another idea is introduced in 2000 [

Compared with other adaptive control schemes, the MFAC approach has several advantages, which make this method suitable for control applications. First, MFAC just depends on the real-time measurement data of the controlled plant. Second, MFAC does not require any other testing signals and any training process. Third, MFAC is simple and easily implementable with small computational burden and has strong robustness. Fourth, MFAC approach does not need a specific controller for each specific process. Finally, the MFAC has been successfully implemented in many practical applications, for example, chemical industry, linear motor control, injection modeling process, PH value control, and so on [

The main contribution of this work is to introduce a new MFAC approach based on output error and feedback linearization. The proposed model has rapid monotonic tracking error convergence, robust stability, and good disturbance rejection.

The rest of the paper is organized as follows: in Section

In feedback linearization method, control law is determined in such a way that nonlinear terms eliminate the system dynamic and replace it with appropriate reference input as seen below: (for simplifying

Notice that the initial values of

Adaptive-fuzzy control method has been divided to two parts: direct and indirect approachs; this paper uses indirect method.

First, consider the affine system equations given below:

Control law

This method is suitable for minimum phase plants and also has the following limitation for

In the previous section, an adaptive fuzzy controller using fuzzy basis functions is described for stabilizing the controlled system. In our introduced method, the same approach has been employed but our method does not need to utilize fuzzy basis functions for estimating system parameters. Instead, three proposed rules based on output errors are considered to control the plant.

Consider a system with general form

As it is seen from (

If

The major principles of proposed MFA controller are constructed based on the three following experimental rules, which make controller produce appropriate control signal:

Regarding adaptive fuzzy controllers [

Let the Lyapunov function candidate

(a) For the first region,

At first we should obtain the difference of Lyapunov function in

(b) For the second region,

In this region the proposed Lyapunov functions

Consider MIMO system with 2 input-2 output system state equations which are given below [

Simulation results of MIMO system without disturbances

Simulation results for MIMO system without disturbance. (a), (b) system outputs 1 and 2. (c), (d) control signals

Simulation results for MIMO system with disturbance. (a), (b) system outputs 1 and 2. (c), (d) control signals

Simulation results for MIMO system with the measurement noise and disturbance. (a), (b) system outputs 1 and 2. (c), (d) control signals

The aim of using this MIMO system which is brought by [

The second system under study is Maglev train which is shown in Figure

System state equations are described as follows:

Vertical slice of magnetic levitation and rail in a magnetic train [

Consider the state vector as below; thus the above equations convert to new state equations:

Maglev trains work in one of two ways; both methods are based on the same concept but involve different approaches.

Electromagnetic Suspension is based on magnetic attraction; it is very complex and somewhat unstable.

Electrodynamic Suspension is based on the repulsion of magnets. The magnetic levitation force balances the weight of the car at a stable position. Controlling EMS is more difficult than EDS train, because normality of this dynamic state is unstable.

Magnetic trains have two important issues, levitation and propulsion; the target of controller is first: goes up train to desired level along with guarantee stabilizing against some uncertainty such as wind and changing train mass in boltroads. And second adjust train speed at the working frequency of magnets. In this paper just levitation part (important section of train in controlling) is discussed. The train in this example goes from 10 mm to 16 mm level. By considering these assumption values for train and controller as follows,

The result of simulating without considering disturbance has been displayed in Figure

Simulation result for Maglev without disturbance (in which it goes from 10 to 16 mm). (a) Distance between train and rail, (b) control signal.

Simulation result for Maglev with disturbance. (a) Distance between train and rail, (b) control signal.

Simulation result for Maglev with sinuous measurement noise at 15 Hz and disturbance. (a) Distance between train and rail, (b) control signal.

Simulation result for Maglev train with white Gaussian measurement noise along with disturbance. (a) Distance between train and rail, (b) control signal.

Model-free adaptive controller is an adaptive controller that just uses system outputs and does not require another system states (so does not need observer) and with this, as model-free controller, it performs good and also has some merits such as good stability, appropriate tracking, robust against uncertainty, disturbance rejection, and good decoupling and the biggest advantage of MFA controller is no requiring to system dynamic and even does not need to any prior experiment about system.

For each time

Let

According to (

With regards to that, the proposed control algorithm is divided into two regions

(a) First region:

In consideration of Lemma

As regards

(b) Second region:

In consideration of error definition

In consideration of (

By utilizing (