In this paper, an improved high-order model-free adaptive iterative control (IHOMFAILC) method for a class of nonlinear discrete-time systems is proposed based on the compact format dynamic linearization method. This method adds the differential of tracking error in the criteria function to compensate for the effect of the random disturbance. Meanwhile, a high-order estimation algorithm is used to estimate the value of pseudo partial derivative (PPD), that is, the current value of PPD is updated by that of previous iterations. Thus the rapid convergence of the maximum tracking error is not limited by the initial value of PPD. The convergence of the maximum tracking error is deduced in detail. This method can track the desired output with enhanced convergence and improved tracking performance. Two examples are used to verify the convergence and effectiveness of the proposed method.

Generally, there are two strategies for controller design. One is based on the exact process information, and the other is based on the input-output data which is a model-free controller design strategy. Many practical industrial processes contain nonlinearity, uncertainty, and time-varying characteristics. In this case, model-based control becomes inapplicable because it is difficult to obtain an accurate mathematical model of the controlled object. In recent years, the model-free methods have attracted the attention of many researchers because of their remarkable advantages and have achieved some valuable results [

Model-free adaptive control (MFAC) is one of the typical data-driven control methods proposed by Hou [

Based on the previous research results, many researchers have further investigated the model-free adaptive iterative learning control algorithm. In [

To obtain a faster convergence, the high-order algorithm was used to estimate the value of pseudo partial derivatives, which introduced more parameters to be adjusted unexpectedly [

• An improved PPD estimation algorithm, which takes the average value of the previous PPDs can achieve faster convergence without introducing more parameters.

• A differential module is introduced into the criteria function of the controller to compensate for the dynamic performance of the system under random disturbance.

• The proposed method contains fewer parameters to design and with the number of iterations increases, the maximum error gradually converges to zero.

The rest of this paper is organized as follows:

Consider the following nonlinear discrete-time SISO system:

The following two assumptions are given for system

In order to design the control law, the following criteria function is considered:

Substituting

As with solving

Taking the derivative of

In order to make the algorithm estimate value of

However, the rapidity of convergence of the maximum error of MFAILC is related to the value of the initial PPD. The larger the value is, the slower the convergence rate is and the larger the error is. Therefore, an improved method is proposed to solve the above problem.

In the proposed method, a differential link is added to compensate for the impact of random disturbance on the system and only use input and output data to design the control law.

Consider the following criteria function:

To get the optional solution, we minimizing

Unlike other high-order algorithms designed based on control input [

If the number of iterations is less than the iteration learning

Taking the partial derivative of

Minimizing

In this section, the convergence of the proposed method is mainly proved. In order to make the following discussion more rigorous, the following assumption is given:

Assumption 3 is similar to a restriction on the direction of control input gain.

For all

When

The close-loop system is BIBO stability, for all

when

In other cases, when

Subtracting

Denote

When

From Lemma 1 we get

when

Substituting

From Lemma 1, we know

Then the convergence of tracking error and the bounded control input are proved as follows:

Denote if

Since

From

Since

According to

From

Let

Then

Let

Let

According to

Substituting

Substituting

When

Thus the following inequality is given:

Substituting

According to inequality

From inequality

This implies that

The desired output is:

k | The IATE of MFAILC | The IATE of IHOMFAILC |
---|---|---|

10 |
193.6497 |
142.9738 |

60 | 1.6799 | 1.2036 |

The maximum tracking error of IHOMFAILC and MFAILC with different initial PPDs is shown in

The mathematical model between friction force and ripple force is described as follows:

Denote

Based on the model-free adaptive iterative learning control, a novel IHOMFAILC method has been proposed in this paper based on compact form dynamic linearization. When the system is disturbed, the proposed method can still keep track of the desired output. Furthermore, this method can achieve enhanced convergence without many experiments to find a suitable PPD value. Two examples show that the proposed method has good trajectory tracking and disturbance resistance.

This work was supported by the Natural Science Foundation of China, under Grant No. 61773183.