A Microcontroller Implementation of Fractional Order Controller

Transcription

1 A Microcontroller Imlementation of Fractional Order Controller Aymen Rhouma and Hafsi Sami Université de Tunis El Manar, Ecole Nationale d Ingénieurs de Tunis, LRES Laboratoire Analyse, Concetion et Commande des Systèmes,, Tunis, Tunisie; Université de Tunis El Manar, Faculté des Sciences de Tunis, 9, Tunis, Tunisie; aymenrh@yahoo.fr, samihafsi@ymail.com Abstract: The idea of this aer is to imlement a Fractional Order roortional integral PI on a real electronic system by using the STM microcontroller. This controller is founded on an extended version of Hermite-Biehler theorem to determine the comlete set stabilizing. The STM keil starter kit based on a JTAG interface and the STM board was used to imlement the roosed fractional controller. So, this controller is imlemented using Keil develoment tools designed for ARM rocessor-based microcontroller devices and working with C/C++ langage. The erformances and the efficiency of the develoed strategy are illustrated with simulation results. Keywords: Fractional-order calculus, fractional order PI controller, STM microcontroller, Keil.. Introduction Fractional order calculus is a mathematical disciline with a -years-old history []. In recent years the fractional calculus has attracted the attention of researchers in many fields such as biology, economics and engineering [-5]. The fractional order calculus has been found that many hysical systems have a dynamic behavior non-integer order. The first noninteger order system to be widely recognized is the thermal system; other fractional order systems that are known are the electromagnetic wave systems, electrode-electrolyte olarization, viscoelastic, and many others [6]. Also, the fractional dynamic system aears in the system industries, such as control alication [7]. The first fractional order control to regulator the hysical systems is roosed by Oustalou in 988 [8, 9]. Later, Podlubny develoed the fractional PID controller in 994 []. Recently, Rhouma is roosed the fractional model redictive control to fractional order system [8]. In [9], the authors roose a new PI tuning method for first order systems with time delay. Some results on the control of integrating systems with time delay using fractional order PD controllers were obtained. Recently, Monje et al. [] give a new tuning method called F-MIGO for PI extended from the MIGO method, this tuning rules is used to determine the best fractional and the best PI gains. In [], the authors roose two sets of tuning rules for fractional PID similar to those of the first set of Ziegler Nichols. A frequency aroach for the auto-tuning of fractional-order PID is roosed in [], where PI is used to cancel the sloe of the curvehase of a osition servo system with time delay around a frequency oint and the PI D controller is designed to fulfill the secifications of gain crossover frequency. A microcontroller is described as a comuter on a chi because it contains all the features of a full comuter including central rocessor outut and inut orts with secial features, serial communication, digital-analog conversion, analog-digital conversion and signal rocessing. In this aer, we roose a Fractional Order roortional integral PI controller for a erformed STMicroelectronics microcontroller (STM). The control alication could benefit from the ower features and flexibility of the STMFxB devices. The roosed framework was develoed using Keil develoment tools designed for ARM rocessor-based microcontroller devices. The outline of this aer is organized as follows. In section, a roblem formulation and some definitions of fractional order systems are introduced, and the G-L definition used to aroximate the fractional order system is detailed. Section states the hardware as well as the software develoment tools used in this alication. The section 4 is reserved to focus on the ISSN: Volume, 7

2 necessary stes in finding the design method roosed for the PI. Simulation results are given in the section 5. Finally, a conclusion is given.. Preliminary of fractional calculus Fractional calculus is a generalization of differentiation and integration to non-integer orders fundamental oerator D f t which is defined as: t d dt Dt f t = () t ( d) < Where is the order, R, t and t are the uer and lower limits of the oeration, resectively. There are several definitions for fractional order calculus, the most oular definitions used are the Grunwald- Letnikov (G-L) and Riemann-Liouville (R-L) definitions []. The R-L definition of function f() t is defined as: n t d f () t n n D f t d, n n () ( n ) dt ( t ) The G-L definition of function f() t is defined as: ( t / h) D j ( ) t f ( t ) lim h ( ) ( ) ( j ) ( j ) f t jh h j () Where ( ) is the Gamma function, h is the samling eriod, n is an integer. The lalace transform of R-L and G-L definitions for zero initial conditions can be given as: L Dt f ( t) s Lf ( t) s F( s) (4) In general, a fractional model can be described by a fractional differential equation characterized of the following form: L M al l m l m bm a D y( t) b D u( t) (5) Using the lalace transform in equation (5), the fractional-order system can be reresented by the following transfer function: Where Ys () Gs () Us () M bm s m L a as l l l bm (6) l m al bm a, b R,, R, l=,, L, m=,, M. Fractional-order PI design The aim of this section is to resent the system which will be controlled by a fractional order PI controller and to resent the design of the fractional controller. This controller is based on the unit return configuration shown in the following figure. Figure: Closed-loo fractional system The first fractional order systems can be described by: K G( ) (7) T Our tuning strategy, is based on Hermite-Biehler theorem and the Pontryagin condition to determine the k and k i arameters. The fractional PI controller transfer function C(s) is given by the following equation: Ki C( ) K (8) The control inut of the PI controller is: u t K r t y t K D r t y t (9) i t Where r(t) is the reference inut or the setoint signal, e(t) is the error, u(t) is the control, y(t) is the outut signal and D t is the fractional differential/integral oerators. The control design method roosed in this aer is based on a Hermite-Biehler and Pontryagin theorem which consist on interlacement roerty of the real roots of the olynomial characteristic. The closed-loo characteristic olynomial of a first order system is given by: ( ) ( KK KK ) ( T) i However, we resent our theorem [9] useful to comute the stability region of a first order system. Based on the first roerty of Hermite-Biehler [] ISSN: Volume, 7

3 which consist that all the roots of the olynomial characteristic of the closed loo equation are real. Theorem: [4] We consider a first order lant given by the following transfer function: K Ls G() s e Ts where the arameters T, L and K are ositive. We can determine the set of all stabilizing ( k, k i ) values for the given lant using the fractional order controller PI Ki C() s K s 4. Microcontroller STM Star Kit In this section, an overview of the hardware and the software develoment tools is resented. The choice to adot STM is based on tradeoffs between rice and erformance. So, the STM erformance line family incororates the higherformance ARM Cortex-M4, 7 MHz frequency, an extensive range of enhanced I/Os and high-seed embedded memories. the devices of STM offer two -bit ADCs. Moreover, the STM has configurable and flexible ower management features that allow to choose the ower otion to fit alication [5-6]. The STM starter kit resented in Figure was used to imlement the Fractional PI rogram and The STMF architecture is resented in Figure. Figure. STMF4 architecture. Keil is a software develoment tool includes C/C++, simulation models, Real-Time Oerating System, debuggers, integrated environments and evaluation boards for ARM, Cortex-M, Cortex-R4 families. The used version, in this aer, is the µvision 4. This version screen sulies a menu bar for command entry, windows for source files, and a toolbar where we can choose command buttons dialog boxes. The µvision 4 has two oerating modes: Build Mode and Debug Mode. - Build Mode: It s the standard working mode. It lets us to convert all the alication files and to generate executable rograms. - Debug Mode: It sulies an effective debugger fo testing alications. 5. Exeriment results In order to illustrate the effectiveness and erformances of the fractional order controller develoed in this aer, we make a ractical imlementation of the roosed controller on an electronic system by using an STM microcontroller. The ractical system used in this aer is a first order system as shown in Figure 4. Figure. STMF4 starter kit ISSN: Volume, 7

4 Figure 4. First order system. In this work we choose R=.K Ω and C=47µF. So, by adoting these values, the system transfer function will be given by: Y( ) G ( ) U( ).54 () The outut of the system will be injected into the STM card through in 6 of ort A (PA6) which will be used as the inut of the analog converter "ADC". This converter consist to converts the analog value of the outut voltage to a digital value which can be used in the rogram of the Fractional PI regulator already imlemented on the STM board. Then, the control delivered by the corrector asses through the in PA4 of the STM card (figure 5). The designed PI arameters are fixed as follows: K, K.6 and.9 i The evolutions of the outut, the set oint and the control signal, obtained by alying the Fractional Order roortional integral PI controller to the real system, are shown in Figures 6 and 7 resectively. Based on these results, we note that the outut signal meets the desired requirements and the control signal obtained rovide a small variation. Tension (v) Figure 6. Set oint and outut signals with fractional PI.5 Control Signal Outut Set oint Tension(v) Figure 7. Control signal with fractional PI Figure 5. Connecting the STM with the system To imlement the fractional-order PI of the thermal system we have used the model given by equation (6). So, the thermal system is defined as a first order system with time delay. Therefore, we roceed the design of the controller by exloiting the aroach exosed in section 4. A comarison of the closed-loo erformances of the ro-osed aroach is established with classic Proortional Integral (PI) controller. The designed PI arameters are fixed as follows: K, K.6 i Figures 8 and 9 exhibits the outut, the set oint and the control signal obtained by alying the classic PI controller. ISSN: Volume, 7

5 Tension (v) Figure 8. Set oint and outut signals with classic PI Tension (v) Figure 9. Control signal classic PI Comaring the results obtained by the roosed PI and the classic PI, we deduce that the first controller reaches the desired reference desite the change of the system dynamic whereas the second controller resents oscillations at the set oint variations. We have also remark that the control law obtained by the PI is smoother than one obtained by PI. 6. Conclusion Outut Set oint Control signal The imlementation of frcational PI controller by the use of the STM microcontroller has been the object of this work. This controller is designed so as to ensure certain closedloo erformances. The classic PI controller has been tested with the same microcontroller and comared with the fractional PI controller. Practical results show the effectiveness of the latter. REFERENCES [] Podlubny I. Fractional Differential Equations, Academie Press, New York, 999. [] Mainardi F., Raberto M., Gorenflo R., Scalas E. Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A: Statistical Mechanics and its Alications,, 87: [] Yuste S., Abad E., Lindenberg K. Alication of fractional calculus to reaction-subdiffusion rocesses and morhogen gradient formation. Arxiv rerint arxiv, [4] Bagley R., Calico R. Fractional order state equations for the control of viscoelastically damed structures. Journal of Guidance, Control, and Dynamics, 99, 4 :4. [5] Zhang Y., Tian Q., Chen L., Yang J. Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods. Multibody System Dynamics, 9, :8. [6] Sun H.H., Charef A., Tsao Y., Onaral B. Analysis of olarization dynamics by singularity decomosition method. Annals of Biomedical Engineering, 99, :-5 [7] Shantanu D. Functional fractional calculs for system identification and controls. Sringerverlag, Berlin, 8. [8] Rhouma, A. and Bouani, F. Robust model redictive control of uncertain fractional systems: a thermal alication IET Control Theory and Alications. 4, Vol. 8, Iss. 7, [9] Sundaravadivu K., Arun B., and Saravanan K. Design of fractional order PID controller for liquid level control of sherical tank. IEEE Int. Conf. Control Syst. Comut. Eng., Penang, Malaysia,, [] Monje C, A., Chen Y. Q. Vinagre, B. M, Dingyu X. and Vicente F. Fractional-order Systems and Controls: Fundamentals and Alications (Advances in Industrial Control).. London: Sringer. [] Valério D, and Costa J. S. Tuning of fractional PID controllers with Ziegler-Nichols-tye rules. Signal Processing, 6, 86: [] Sundaravadivu K., Arun B., and Saravanan K. Design of fractional order PID controller for liquid level control of sherical tank. IEEE Int. Conf. Control Syst. Comut. Eng., Penang, Malaysia,, ISSN: Volume, 7

6 [] Oustalou A. From fractality to non-integer derivation through recursivity, a roerty common to these two concets: a fundamental idea from a new rocess control strategy. th IMACS World Conf., Paris, 988, 8. [4] Hafsi.S, Laabidi.K and Farkh.R. A New Tuning Method for Stabilization Time Delay Systems Using PI D Controllers. Asian Journal of Control, 5 Vol. 7, No.,. -. [5] Ben Hariz M. Bouani F. Imlementation of a fixed low order controller on STM microcontroller, International Conference on Control, Engineering & Information Technology (CEIT 4), Tunisie 4. [6] Kherigi A. Bouani F. Ksouri M. ben Ahmed M. A Microcontroller Imlementation of Model Predictive Control. International Journal of Electrical, Comuter, Energetic, Electronic and Communication Engineering Vol:5, No:5,. ISSN: Volume, 7