## Abstract

The main idea of the current paper consists in introducing the fractional order calculus in a control system. To control the system, an adaptive control technique with reference model is used. The fractional order models for the plant and reference model are obtained. To achieve the performances imposed by the fractional order reference model, a fractional order adaptive control law is proposed, which is a combination of two methods (MIT and Lyapunov stability). The original contribution in this paper is the use of fractional order calculus in the combined MIT and Lyapunov stability method and showing the dynamic behavior of the system. Several simulations are used to emphasize the effectiveness and benefits of the proposed method.

## 1 Introduction

In the specialized literature, there are many papers which present how the fractional order calculus can be used to control a dynamical system [1–4] (see the Model Reference Adaptive System (MRAS) [5–11]). Introducing the fractional order calculus in the adaptive control system leads to good performances. More precisely, they are given by the fractional order systems over integer order systems [8].

Starting from the general idea mentioned in [12], in a closed loop system, one of the following situations can be found: 1) the integer order plant and integer order controller; 2) the fractional order plant and fractional order controller; 3) the fractional order plant and integer order controller; 4) the integer order plant and fractional order controller. For the aforementioned cases, a comparison is done by applying the MRAC (Model Reference Adaptive Control) theory, which is the most known and easy to apply adaptive method, to show the benefits of using the fractional order calculus.

So, in some cases, the MRAS basic scheme (for more details, refer to [13]) is modified by a fractional order reference model, and by the usage of the fractional order plant and fractional derivatives [14]. The fractional order reference model and fractional order plant are introduced in the adaptive scheme because they offer better representation of the system's dynamics. Also, these models improve the system's performances, such as response time and overshoot. The fractional derivatives in the chosen models ensure the system's stability.

The proposed control law is characterized by several adjustable parameters. The adaptation mechanism used to adjust the parameters is a combination between the MIT and Lyapunov stability methods [15], which will be modified using the fractional order calculus [1].

The paper's structure looks for: basic definitions on fractional order dynamic systems (Section 2). Then, how the fractional order models for the plant and reference model are obtained (Section 3). In Section 4, the theory about the combining MIT and Lyapunov stability method is extended using the fractional order calculus. Basically, a new fractional order control law was developed, which is the main contribution of the paper. Different scenarios are simulated in Section 5 and, finally, a few remarks are taken in Section 6.

## 2 Fractional order calculus in control systems

## 3 Identification of the fractional order model for the plant and reference model

*fotfid*tool is needed [16]. The identification is carried out (both for the reference model and plant), starting from the following integer order models:

- for the plant, the nominal transfer function is:

- for the reference model (the transfer function is according to module criterion, i.e. damping ratio equal to 0.7):

The procedure continues with storing the data for each model (a special data structure was created), after the output was obtained considering the input a step signal.

*fotfid*tool, a first guess model must be set up: an integer order model is obtained by choosing the characteristics

The models introduced in (8) and (9) are a better representation of the imposed dynamics of the control system.

By conducting a comparison with a square wave as an input signal, the fractional order models are confirmed and as it can be seen (Figs. 1 and 2), the fractional order models are accurate enough. The new transfer functions must be stable, property checked in the time domain. If the relation from Eq. (4) is satisfied, then the system is stable [12]. The property is illustrated in Figs. 3 and 4 (for the plant and reference model too): the poles are outside of the shaded area, so the models are stable.

## 4 Obtaining the fractional order adjustment mechanism

To achieve the performances, imposed by the reference model, the controller's parameters must be adjusted/adapted (using an adjustment/adaptation mechanism), in such a way that the error between the plant output and reference model output is reduced to zero or exceedingly small. In literature, there are two approaches which can be used to derive an adaptation mechanism: the first one is the gradient method (or the MIT rule) and the second one is the Lyapunov stability method [13]. So, by modifying the parameter's adjustment mechanism using the fractional derivative [14], a second modification is introduced in the MRAS scheme.

*k*

_{2},

*k*

_{3}correct the system dynamics. The adjustment mechanism used for these parameters is a combination between two methods: MIT rule and Lyapunov stability, with the purpose to use the same value for the adaptation gain [15]. So, to adapt the parameter

- in case of fractional order plant and fractional order controller:

- and for the integer order plant and fractional order controller case:

It can be seen that when a fractional order plant is used, in the MRAS scheme a fractional order reference model must be used, too.

## 5 Simulations and results

All scenarios already presented in the earlier section are simulated and analyzed, using different values such as 0.9 and 1.05 for alpha.

The simulations are done for

- Case 1: integer order plant and integer order controller (Fig. 5);
- Case 2: fractional order plant and fractional order controller (Fig. 6);
- Case 3: fractional order plant and integer order controller (Fig. 7);
- Case 4: integer order plant and fractional order controller (Fig. 8).

The simulations are presented in the same area (60–70s). By using a fractional order control with

When comparing the resulted performances with the ones from the other two cases (see Figs. 5 and 7), where a classical adaptive control is used, it can be concluded that the MRAS has more benefits when using fractional order control.

Also, the error variation was examined and becomes exceedingly small much faster only for cases two and four.

*IAE*– Integral of Absolute Error) [17]. All simulations were done over 100 seconds, using

*Te*= 0.001 fixed-step size, so the output signals of the plant and reference models are saved as Matlab vectors with

*N*= 100.001 samples. The performance index is computed from the saved samples

*y*[

*t*] as:

Better performance means smaller value of the *IAE*. The results for all cases in Figs. 5–8 are presented in Table 1. Compared to the conventional control scheme with integer-order plant and reference model (Case 1), the fractional-order control law (Cases 2 and 4) led to approximately 10–20% improvement.

The *IAE* values

Case 1 | IAE = 3,325.6 |

Case 2, with alpha 0.9 | IAE = 2,902.7 |

Case 2, with alpha 1.05 | IAE = 5,085.0 |

Case 3 | IAE = 4,137.6 |

Case 4, with alpha 0.9 | IAE = 2,621.0 |

Case 4, with alpha 1.05 | IAE = 4,031.2 |

As a remark, when using in the MRAS scheme a fractional order reference model, the tracking performances are carried out with fractional order plant.

## 6 Conclusions

In this paper, through examples, the benefits of the fractional order adaptive control are shown. The method used to adapt the parameters is a combination of two adaptive methods, MIT and Lyapunov stability, and implies a reference model.

By using the fractional order calculus, the combining MIT and Lyapunov stability method is extended. More precisely, the fractional order derivative and fractional order reference model are introduced in the parameter's adjustment mechanism. To achieve good performances, besides using a fractional reference model, a fractional order plant was used as well. All the cases supported by MRAS are studied and the benefits are highlighted.

Finally, it can be concluded that it is easy to apply this method on any second order plant.

## Acknowledgments

This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CCCDI – UEFISCDI, project number ERANET-FLAG -ITFoC (2), within PNCDI III.

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