A linear flexible joint system using fractional order linear active disturbance rejection control is studied in this paper. With this control scheme, the performance against disturbances, uncertainties, and attenuation is enhanced. Linear active disturbance rejection control (LADRC) is mainly based on an extended state observer (ESO) technology. A fractional integral (FOI) action is combined with the LADRC technique which proposes a hybrid control scheme like FO-LADRC. Incorporating this FOI action improves the robustness of the standard LADRC. The set-point tracking of the proposed FO-LADRC scheme is designed by Bode's ideal transfer function (BITF) based robust closed-loop concept, an appropriate pole placement method. The effectiveness of the proposed FO-LADRC scheme is illustrated through experimental results on the linear flexible joint system (LFJS). The results show the enhancement of the robustness with disturbance rejection. Furthermore, a comparative analysis is presented with the results obtained using the integer-order LADRC and FO-LADRC scheme.

Due to the increasing complexity of industrial systems, it has become more difficult to analyze and control. Moreover, the non-linear behavior of physical systems produces uncertainties that cause modeling issues of those systems. There are mainly two types of uncertainties, one is internal (unknown parameters or un-modelled dynamics) and the other one is external (disturbances). Therefore, several robust control system design methods are developed for uncertain systems which has followed several directions. Sliding mode is one of those control directions which acts against external disturbances of linear or non-linear systems [

Basing on an extended state observer (ESO), a judicious feedback loop is used to estimate and cancel this generalized disruption. Experimental, as well as simulation results have been stated in several examples (see [

A suitable pole placement then makes calculating the parameters of the control law possible. Then again, in this procedure, two significant points need to be taken into consideration. First, the non-integer order of the control law is not calculated in accordance with the CL reference model. In addition, it has to be rational. Secondly, the degree of the integer order of the distinctive polynomial corresponding to the augmented model is very high. This calls for the designer to select a great number of poles while the fractional order model does not require to that extent [

However, there is another solution that involves finding a method that makes it plausible to evaluate both the parameters of the control law and the non-integer order presented by the fractional order integrator based on the preferred CL reference model. This is precisely what is presented in this paper, based on the BITF [

The novel ideas of this paper are:

A method that makes it possible to impose the iso-damping property in the CL model response. For this, a new reference model using the BITF has been proposed. It is to control systems with a high relative degree using fractional-order controllers and to enforce the planeness of the phase margin.

The LADRC control system, a control structure linked with a fractional order integral action inserted in the SPT loop, has been proposed [

The main theoretical contribution is a new design technique of the SPT controller-based design.

In order to confirm these theoretical results, numerical simulations have been offered and the robustness of the control structure with respect to the SPT controller gain variations.

The LADRC control structure which has been proposed in this paper has also been applied on an experimental testbed which consists of a linear flexible joint cart system. We have shown that it is possible to implement this structure without modeling the system. Also, this can be useful to improve the sturdiness of the LADRC control technique compared to the other existing control schemes in the literature.

In the standard formulation of LADRC method for a linear integer order system, controlling the complete model of the system is not needed. This method is based on the relative degree and the gain of the model. The controlled scheme for a second order model is shown by:

The key in the LADRC method is estimating the unknown

The structure of the full ESO is given by

The parameters

If the estimated error is ignored

As a standard form, set-point tracking is solved by state feedback. So, the control law

Non-integer order derivatives have several definitions, and these are usually not equivalent to integer-order derivatives [

The

In what follows, the method to design the coefficient of the state feedback

The closed-loop system with control plant given by

For this transfer function to be equal to the reference model in

The linear flexible joint system, which is a system of two carts sliding on a track, with one of the carts driven by a DC motor, is an interesting example that illustrates the LADRC control scheme and validates the performance of the proposed FO-LADRC. Modeling of this system is fairly complex since two systems interact. Indeed, the cart's motion affects the flexible spring movement and vice-versa. During the testing process, the goal was to determine the position of the cart, and the flexible spring was viewed as a permanent disturbance. A Linear-flexible-joint cart system (LFJ) is shown in

To evaluate and compare the performance of the two LADRC structures, the design parameters of the observer are:

A comparison of the FO-LADRC structures is also made by studying their robustness with respect to the parameter

Similar to the cart position, a comparison of the FO-LADRC structures is also made for a flexible joint cart position by studying their robustness with respect to the parameter

This paper, LADRC structure with a fractional order integral action (FO-LADRC) is proposed to control a minimum phase-stable system. Two main contributions are achieved. From the theoretical point of view, based on the BITF, a novel design method of the SPT controller is proposed for the FO-LADRC scheme. From a practical point of view, the use of LADRC control scheme is an important contribution since no modeling of the system was needed and all the parameters of the system are not accessible. Nevertheless, minimal knowledge of the system is necessary. We knew for example, that the relative degree of the model of the DC motor position is equal to 2. That is why a third order ESO is used to estimate the generalized disturbance. Through experiments on the linear flexible joint system (LFJS), the FO-LADRC scheme is shown to be effective. With disturbance rejection, the robustness of the system is enhanced. Additionally, this paper presents a comparison of results obtained by the integer-order LADRC and FO-LADRC schemes.

This research work was funded by Institutional Fund Projects under Grant No. (IFPRC-027-135-2020). Therefore, authors gratefully acknowledge technical and financial support from the Ministry of Education and King Abdulaziz University, Jeddah, Saudi Arabia.