# Fractional order PID controller improves motor velocity control

## Compare the integer order proportional-integral-derivative (PID) controller (IOC) with the fractional order PID controller (FOC) for improving the velocity control loop of the permanent magnet switched motor (PMSM), according to research highlighted in Control Engineering Poland. See supporting equations, diagrams, and graphs.

For the velocity control loop of a permanent magnet switched motor (PMSM), compare the integer order PID controller (IOC) with the fractional order proportional-integral-derivative (PID) controller (FOC).

**Fractional order calculus**

Today’s increased computing power displaces tools that have been used for hundreds of years in research and modeling of physical phenomena. Such modeling can use fractional order calculus with integrals and derivatives of real order instead of integer order. The result is a significantly more comprehensive description of a given phenomenon, and this, in turn, leads to better results. Such an approach may have some implications in automation and control theory. Because objects controlled by engineers are of fractional order, it seems logical to apply fractional order controllers (with adequate properties) to control them.

**Fractional order PID controller**

There are many ways to describe fractional order calculus. To solve the problem of describing the elements of the fractional order PID controller, the Grünwald–Letnikov definition was used (1).

(1)

where

Formula (1) for α<0 shows the fractional integration process, whereas for α>0 shows the fractional differentiation process. It is possible to clearly present the elements of the fractional order PID controller. Equation (2) shows the discrete fractional integration term and equation (3) the discrete fractional differentiation term.

(2)

(3)

The main difference between the IOC and FOC is the sum in equations (2) and (3), which determines the memory of the fractional order elements. In a classical PID controller the current value of the state x(t+1) is dependent only on the previous value of the state x(t) and the current input value u(t), whereas in the fractional order PID controller the current value of the state x(t+1) depends on all the previous states. The coefficients and are directly dependent on the order of the element and bind together all the memory states.

The general definition of the PID controller is assumed and described by equation (4)

where e(t) is a control error at a given moment t, and T_{s} is a sampling time, whereas Iα and Dβ symbolize a fractional order integral and a derivative, respectively. By choosing α=1 and β=1 the classical PID controller of integer order is achieved.

**Test stand**

Figure 1 presents the test stand where the experiment was performed. It is the rapid prototyping platform, which integrates the designing stage, complex simulation analyses, and prototyping of newly designed algorithms in the target control system.

The general scheme used on the control system is shown in Figure 2. A cascade control system with velocity control loop and torque (current) was applied. The parameters of the current controllers selected for the simulation are predefined as the default set by the servo-drive producer for a given motor. In this case the following parameters are set:

K_{v}=54.587318,

T_{i}=0.00088248722.

**IOC and FOC, compared**

During the research, the optimal settings of the controllers were selected. Equations (5) and (6) present the FO and IO controllers, respectively.

(5)

(6)

Figure 3 shows the response to reference velocity signal for these settings.

The experiment confirms the assumption. The best FO PI controller obtains significantly better results than the classical IO PI controller. The response of the motor controlled by the FOC is noticeably closer to the reference signal and has faster reaction to its changes.

To objectively assess control quality the obtained results were supported by the values of integral criteria ITAE (Integral of Time multiplied by Absolute value of Error): ITAE_IO=1005.04 and ITAE_FO=806.19, which shows that the improvement of control quality exceeds 19%.

**Differences shown**

FOC provides significant improvement when compared to IOC. The research also proves that for an FO object, the FO controller should be selected. Also, settings of the optimal controllers (IO and FO) differ only in parameter α responsible for the order of integral term. This offers the possibility to implement a new algorithm to the target control system, take the settings from the controller already working in the system, and then fine-tune the order of the implemented controller to achieve the optimal performance of the system.

The research is part of the project, “Development of the construction and experimental tests of a mechatronic machine tool feed unit with a drive controlled by an intelligent modular actuator” (MNiSW Project No. N 502 336936, code-name M.A.R.I.N.E. (multivariable hybrid ModulAR motIon coNtrollEr)).

- Artur Kobyłkiewicz and Krzysztof Pietrusewicz are with West Pomeranian University of Technology, Szczecin. They contribute to *Control Engineering Poland*. Edited by Mark T. Hoske, content manager, CFE Media, *Control Engineering, *mhoske(at)cfemedia.com.

**Additional explanation from Mr. Pietrusewicz based on comments below:**

*Thank you Larry for your comment. It put an interesting point of view into our work, and we should consider such a research in further exploration. However, this paper was prepared mostly to show the rapid prototyping test stand with automatic code generation and its readiness level in the real case problem with not a typical controller just to show its value.*

*We are in the beginning of work on FOC for motion control applications – we know it pros and cons according to the implementation issues. It will be scope of more in-depth research within the PhD thesis of Artur.*

*Application of FOC is not so popular in industrial application on the market. We have shown, that You can take off-the-shelf digital servodrive from B&R and test Your own algorithm -> even FOC -> without time consuming development of the hardware where You can implement FOC controller. *

*Best regards,*
*Krzysztof*

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Thanks

The advantage of the method is that more gain and phase margin may be obtained. The drawback is the high number of parameters that one has to define in order to tune the control optimally.

There are some related patents existing, for example US5371670 and WO 02/097546