## 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

The control system can be modeled using the transfer function approach. This function is obtained after applying the Laplace transform on differential equation with integer order. In a similar way, the control system's dynamics can be described through a fractional order differential equation, which leads to a fractional order transfer function (after applying the classical Laplace transform). So, in general, the fractional order differential equation has the following form [1, 4, 14]:

The above Equation form is equivalent to:

By applying the Laplace transform on Eq. (2) with zero initial condition, the general fractional order transfer function form becomes:

After obtaining a fractional transfer function it is important to check the system's stability. The fractional order system can be stable, including the case when it has roots in the right half of the plan [4, 12], if the following condition is respected:

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

In the MRAS, the desired system performances are imposed by a reference model, when the plant's parameters vary in time. Therefore, the first modification which is done in the MRAS scheme is to introduce a fractional order model for the reference model and plant. To get the models, identification in the time domain using the *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.

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

Using a free identification method with some coefficient limits

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.

As it is mentioned in the introduction chapter, in a closed loop system, several situations can be tackled. Thus, by choosing the proper control law, all these cases are going to be studied. The proposed control law (for the classical case when the plant and the controller are integer order) which offers good performances in a closed loop is:

*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

For the parameters,

The whole above mentioned theory is applicable to integer order plant and integer order controller case. For other cases, when a fractional order controller is used, then a fractional order adaptive control law should be adopted. The fractional order adaptive control law is similar to the one presented in Eq. (10):

The parameter's adjustment mechanism is:

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

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

In the last case, in the MRAS scheme, a fractional order plant and an integer order controller are used. The parameter's adjustment mechanism is similar to the one used in Eqs. (11)–(13) (starting from the control law from Eq. (10)). A difference between them is due to the parameter

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.

To support the conclusions a performance index was defined (*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.

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.

## References

- [1]↑
A. Tepljakov,

*Fractional-Order Modeling and Control of Dynamic System*, Doctoral Thesis, Tallinn University of Technology, 2015. - [2]
I. Ndoye, H. Voos, M. Darouach, J. Schneider, and N. Knauf, “Hinf static output feedback control for a fractional-order glucose-insuline system,” in

*6th Workshop on Fractional Differentiation and Its Applications, 4–6 February, France*, 2013, pp. 266–271. - [3]
P. Shah, S. D. Agash, and A. Singh, “Fractional order modeling using state space theory,”

*Int. J. Eng. Technol.*, vol. 5, pp. 2891–2894, 2013. - [4]
S. Das and I. Pan,

*Fractional Order Signal Processing. Introductory Concepts and Applications*, Springer, 2012. - [5]
B. Sigh, J. Yuan, and C. Dong, “On fractional model reference adaptive control,”

*Sci. World J.*, pp. 1–8, 2014. - [6]
Y. Bensafia and S. Ladaci, “Adaptive control with fractional order reference model,”

*Int. J. Sci. Tech. Autom. Control Comput. Eng.*, vol. 5, pp. 1614–1623, 2011. - [7]
K. Bouziane, A. Djouambi, and S. Ladaci, “Fractional-order model reference adaptive controller design using a modified MIT rule and a feed-forward action for a DC-DC boost converter stabilization,” in

*5th International Conference on Electrical Engineering – Boumerdes (ICEE–B)*, 2017. - [8]↑
S. Ladaci and A. Charef, “On fractional adaptive control,”

*Nonlinear Dynam.*, vol. 43, pp. 365–378, 2006. - [9]
G. Navarro-Guerrero and Y. Tang, “Adaptive control for anesthesia based on simple fractional-order model,” in

*IEEE 54th Annual Conference on Decision and Control (CDC)*, 2015, pp. 5623–5628. - [10]
G. Navarro-Guerrero and Y. Tang, “Fractional-order closed-loop model reference adaptive control for anesthesia,”

*Algorithms*, vol. 11, pp. 1–35, 2018. - [11]
G. Kavuran, B. Baykant Alagoz, A. Ates, and C. Yeroglu, “Implementation of model reference adaptive controller with fractional order adjustment rules for coaxial rotor control test system,”

*Balkan J. Electr. Comput. Eng.*, vol. 4, pp. 84–88, 2016. - [12]↑
Y. Q. Chen, I. Petras, and D. Xue,

*Fractional Order Control – A Tutorial*, American Control Conference Hyatt Regency Riverfront, 2009, pp. 1394–1411. - [13]↑
K. J. Astrom and W. Bjorn,

*Adaptive Control*, 2nd ed., Mineola, New York, USA, Dover Publication Inc., 2008. - [14]↑
B. Vinagre, I. Petras, and Y. Q. Chen, “Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control,”

*Nonlinear Dynam.*, vol. 29, pp. 269–279, 2002. - [15]↑
S. Coman, C. Boldisor, and C. Ciusdel, “Combining the MIT and Lyapunov stability adaptive methods for second order systems,” in

*International Conference on Development and Application Systems (DAS)*, 2018, pp. 31–37. - [16]↑
A. Tepljakov, E. Petlenkov, and J. Belikov, “FOMCOM: a matlab toolbox for fractional-order system identification and control,” in

*Proceedings of the 18th International Conference Mixed Design of Integrated Circuits and Systems – MIXDES*, 2011, pp. 1–12. - [17]↑
A. Tepljakov, B. Baykant Alagoz, E. Gonzalez, E. Petlenkov, and C. Yeroglu, “Model reference adaptive control scheme for retuning method based fractional-order pid control with disturbance rejection applied to closed-loop control of a magnetic levitation system,”

*J. Circ. Syst. Comput.*, vol. 27, pp. 1–29, 2018.