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The convergence of model-free adaptive control (MFAC) algorithm can be guaranteed when the system is subject to measurement data dropout. The system output convergent speed gets slower as dropout rate increases. This paper proposes a MFAC algorithm with data compensation. The missing data is first estimated using the dynamical linearization method, and then the estimated value is introduced to update control input. The convergence analysis of the proposed MFAC algorithm is given, and the effectiveness is also validated by simulations. It is shown that the proposed algorithm can compensate the effect of the data dropout, and the better output performance can be obtained.

Model free adaptive control (MFAC) is an attractive technique which has gained a large amount of interest in the recent years [

When MFAC is used in practical systems, robustness is an important aspect that should be considered. In traditional model-based control theory, robustness refers to the ability to deal with unknown uncertainties or unmodeled dynamics of the plants. However, the unmodeled dynamics has no meanings in MFAC because its controller designs without any model information. In [

It is shown that the MFAC is still convergent as long as not all the output measurement data is lost [

The paper is organized as follows. In Section

For the convenience of understanding, the MFAC algorithm is first given. Considering the following discrete-time SISO nonlinear system

The following assumptions are made about the controlled plant:

the partial derivative of

the system (

These assumptions of the system are reasonable and acceptable from a practical viewpoint. Assumption (A1) is a typical condition of control system design for general nonlinear system. Assumption (A2) poses a limitation on the rate of change of the system output permissible before the control law to be formulated is applicable. From the “energy” point of view, the energy rate increasing inside a system cannot go to infinite if the energy rate of change of input is in a finite altitude. For instance, in a water tank control system, since the change of the pump flow of water tank is bounded, the liquid level change of the tank caused by the pump flow cannot go to infinity. There exists a maximum ratio factor between the liquid level and the pump flow, just as the positive constant

The following theorem illustrates that the general discrete time nonlinear system satisfying assumptions (A1) and (A2) can be transformed into an equivalent dynamical linearized model, called CFDL model.

For the nonlinear system (

Equation (

Using assumption (A2) and the mean value theorem, (

Considering the following equation:

Equation (

Theorem

Rewritten (

For the control law algorithm, a weighted one-step-ahead control input cost function is adopted, and given by

Substituting (

The objective function for parameter estimation is used as

Using the similar procedure of control law equations, the parameter estimation algorithm can be obtained as follows:

Summarizing, the MFAC algorithm based on CFDL model for a SISO system is given as follows:

In order to make the condition

The control law (

From (

From Theorem

Then, we can give the following estimation scheme

Assuming that the probability of

Therefore, the MFAC with dropout compensation scheme can be described as

In order to obtain the convergence of the proposed MFAC algorithm, another assumption about the system should be made.

The PPD satisfies that

Most of plants in practice satisfy this condition, and its practical meaning is obvious; that is, the plant output should increase (or decrease) when the corresponding control input increase. For example, the water tanks control system, the temperature control system, and so on.

To prove our main result the following lemma is developed first.

Define that

From (

Denote that

The curve of

when

when

when

Considering

Hence, it exists constants

Using the fact that

Since

Since

Since

From the lemma, the parameters should satisfy

With the above lemma, the following result can be given.

For the system (

The estimated algorithm (

From (

Considering the SISO nonlinear system

The simulation result for dropout rate is 60%. (a) Output signal, (b) control input signal.

The simulation result for dropout rate is 80%. (a) Output signal, (b) control input signal.

The simulation result for dropout rate is 90%. (a) Output signal, (b) control input signal.

This paper proposes a robust model-free adaptive control algorithm with data dropout compensation. This algorithm first estimated the missing measurement output and then applied the estimated value into the model-free adaptive control algorithm. The convergence of the proposed MFAC algorithm is given, and effectiveness is also supported by simulations. It is shown that the proposed algorithm can compensate the effect of the data dropout, and the better output performance can be obtained.

This work is supported by State Key Program (no. 60834001) of National Natural Science Foundation of China and Program for Science & Technology Innovation Talents of Henan Province (no. 104200510021).