LIMB DYNAMIC FUNCTION MODEL SIMULATION BASED ON THE IDEAL BODE SYSTEM TRANSFER FUNCTION Текст научной статьи по специальности «Техника и технологии»

UDK 681.53

©Ji Zou, Ping Song, Kaiwen Sun, Zhao Gang Sheng, 2021

LIMB DYNAMIC FUNCTION MODEL SIMULATION BASED ON THE IDEAL BODE SYSTEM TRANSFER FUNCTION

Ji Zou - PhD, Assistant Professor, Electronic Information Engineering College; Ping Song - PhD, Assistant Professor, Electronic Information Engineering College, e-mail: 7317421@qq.com; Kaiwen Sun - PhD, Assistant Professor; ZhaoGang Sheng - PhD, Assistant Professor, Electronic Information Engineering College (Changchun University, Changchun, China)

The fractional order calculus theory and its modeling methods have been widely applied in control field especially in Limb Dynamic Function Model Simulation. The design of fractional order control systems has become urgent in recent years. This paper establishes fractional order systems which parameters are obtained by Bode's ideal transfer function method to get the desired frequency response. A new control structure of fractional order pseudo-derivative feedback (FOPDF) is suggested to apply this method. Control methods are designed for integral first order system and for fractional first order system, they can be applied to a real high power semiconductor laser diodes constant temperature controlling system, as well as to the hydraulic servo systems and the electric drive systems, the simulation results indicate the effectiveness and validity of this method.

Keywords: Limb Dynamic Function, Bode's ideal transfer function, high power Semiconductor Laser Diode, robustness, Rehabilitation.

Introduction

Fractional calculus deals with derivatives and integrals to an arbitrary order. There are several definitions of fractional derivatives and integrals, the most fundamental definition of a fractional derivative and integral of order a is given by Grunwald-Letnikov definition [1, 3, 5], Grunwald-Letnikov (GL) definition is given as:

t-a

Df (t) = limh- X (-1)J ^ J jf (t - jh)

a}_a(a + Y)...(a + J-1) _ a! J J~ J! " J!(a-j)! ,

ВЕСТНИК ТОГУ. 2021 № 2 (61)

BECTHHK TOry. 2021. № 2(61)

when a>0 means that a is a derivative order of f(t); when a<0 means that a is an integral order off(t). This definition is wildly used in control field [2].

Design the fractional controller is a main researching area of fractional calculus applying in the control field. And there are many kinds of methods to design a fractional order PID(FOPID) controller, including the phase margin, the amplitude margin, dominant pole method, optimization method, and the Bode's ideal transfer function method. The Bode's ideal transfer function method is easy and effective. It suggested an ideal shape of the open-loop transfer function of the form [4]:

opi\

1 S

Gopi(s) = 1 — 1 «e R

a

where Cc is the gain crossover frequency, In fact, the transfer function Gopi(s) is a fractional-order differentiator for a>0 and a fractional-order integrator for a<0. Its Open-loop characteristics are as follows:

Amplitude-frequency curve is a straight line of constant slope-20adB/dec; Phase curve is a horizontal line at -an/2 rad;

The Nyquist curve consists, simply, on a straight line through the origin with argGopi(jœ)=- an/2 rad.

a) Frequency and Phase Curves When Fix b) Frequency and Phase Curves when Fix a=11/9 and Changing rac rac =10rad/s, and Changing a

Fig. 1. Frequency and Phase Curves When Parameter Changing

Fig. 1 indicate that if the gain changes Oc, will changes together but the phase margin of the system remains as an independent value as ^ = (n-2 aln) rad [4]. In

this paper we applied the Bode's ideal transfer function method to determine the parameters of the FOPID system for integral first order system and fractional first order system. The procedure is of design FOPID system. Choose the controlled plant and its transfer function Gp (S).

Assume gain crossover frequency 0)c, the phase margin Om. Work out the transfer function of FOPID controller as:

Gc(s) =

Gopi(s) Gp(S)

Using impulse invariance method get the discretization model of Gc (s) and the simulation block of FOPID controller, Schematic Diagram [4] is as Fig. 2.

f initialization \

r=r+1,u=u+1

Ï

Take the sampling time Ts

Caculate ( | & 1 8 J ( ^ I

„ , ( Tsr*n(r-1) |' Caculate I I & ^ gamma(r) J ( Tsu*n(u-1) |U ^ gamma(u) J

I

Obtain

Gc( z )

I

End

iteration, obtain s & S

Fig. 2. Schematic Diagram of the controller discretization process

Fractional

5 1

Int s^-D .22222} Der 3ftD.311

Fig. 3. The Simulation Block of Fractional pi*d"Controller

Simulate base on MATLAB software, and get the unit step responses of the system, analysis dynamic performances.

This system is simple yet effective. PDF structure provides all the control aspects of PID control, but without system zeros that are normally introduced by a PID compensator. Phelan named this structure "Pseudo-derivative feedback (PDF) control from the fact that the rate of the measured parameter is fed back without having to calculate a derivative. The PDF structure internalizes a pre-filter, one would apply to cancel the zeros introduced in the PI (or PID) equivalent system [6]. The PDF structure is usually introduced into the design of electro-hydraulic servos [7], automatic control systems of electric traction [8], it offers a good disturbance rejection performance and promotes the response speed. In this paper the fractional PID algorithm is introduced into FOPDF systems to

BECTHHK TOry. 2021. № 2(61)

promote the property of fractional PID control system, the simulation results to a first-order controlled plant illustrate that the FOPDF control method has superior performances to IOPDF structure. The basic structure of IOPDF is as Fig. 4.

Fig. 4. The basic structure of IOPDF System

And using Bode's ideal transfer function for FOPDF system should use step (5) to determine the control structure and parameters.

According to the basic structure of IOPDF, establish the FOPDF control system model to a first-order controlled plant 1/(Ts+1), 1/(Tsy+1), the model for integer plant is shown in Fig. 5. We suggest a basic structure of FOPDF system like Fig. 5, where, the FOPI maybe a single FOPI controller or combined by FOPI controllers. The parameter Kd can be tuned of Chenliu method [9], as a basic tuning parameter of design FOPDF control structure. The Kd value is 5.4. Then the internal-loop can be treated as a whole controlled object. The equivalent open-loop block diagram of Fig. 5 is shown in Fig. 6. From Fig.6 the transfer function of FOPI controller designed based on Bode's ideal transfer function can be expressed as:

GcFOfI (s) = ^^ = V S ^ = fa T • [Ts1'" + ( 1 + Kd) • s- ]

Ts + 1 + Kd

a

Fig. 5. Block Diagram of FOPDF Control System Aimed to Integer First-order Control

Object

Fig. 6. Open-loop Block Diagram of FOPDF Control System Aimed to Integer First-order

Control Object

Establishing Fractional Order Control System Model

1. Establishing FOPID Controller Model Aimed at Integer First-Order Control Plant

Assume the open-loop transfer function of control object is as here T=0.4s, hypothesis ®m=700, Cc =10rad/s.

1

GP1(s) =

Ts +1

As procedures, the transfer function of controller aimed at integer first-order control plant based on Bode's ideal transfer function can be get from formula:

Gci(s) =

Gopi(s) Gp(S)

( Ii V 10s

0.4s 9 + s 9

The simulation model of FOPID controller aimed at integer first-order control plant is shown in Fig. 7.

Fig. 7. Simulation Model of FOPID Controller Aimed at Integer First-order Control Plant

2. Establishing FOPDF Controller Model Aimed at Integer First-Order Control Plant

Assume T is 0.4s, the expected crossover frequency Cc is 10 rad/s, the phase margin Om is 700, then equation of the controller can be determined as:

11 _11 Gc1FoPi(s) = (10)9 -[0.4s 9 + 6.44s 9 ] Establishing FOPDF control system simulation model as Fig. 8.

2

BECTHHK Tory. 2021. № 2 (61)

Fig. 8. The Simulation Block of FOPDF Aim to Integer First-oder Plant Control System

3. Establishing FOPDF Control System Model Aimed at Fractional FirstOrder Control Plant

Assume the open-loop transfer function of control object is:

Gp/s) =

1

Ts7 +1

here T=0.5s, r=0.5 hypothesis 0m=700 0)c =10rad/s.

As procedures, the transfer function of controller aimed at fractional firstorder control plant based on Bode's ideal transfer function can also be get from formula:

i "V 13

=^ GP(S)

10s

0.5s

+ s

IA

9

The simulation model of FOPID controller aimed at fractional first-order control plant is shown in Fig. 8. While the subsystem block of FOPID controller and Fractional Order system are designed of the simulation blocks shown in Fig. 3 applying the parameters in equation [10].

Fig. 9. Simulation Model of FOPID Controller Aimed at Fractional First-order

Control Plant

18

ВЕСТНИК ТОГУ. 2021. № 2 (61)

4. Establishing FOPDF Control System Model Aimed at Fractional FirstOrder Control Plant

For 1/(Tsy+1), assume yis 0.5, T is 0.4s, (the other parameters are same as integer first-order control plant), the FOPDF structure is the same as integer one, but the equation of the controller is expressed as:

13 11

G

11

,(s) = (10)^ • [0.4s 18 + 6.44s 9]

Fig. 10. The Simulation Block of FOPDF Aim to Fractional First-order Plant Control

System

5. Simulation and Result Analysis

Unit Step Responses with open-loop system gain K varying (0.9K,K,1.1K) are shown in Fig. 11. Fig. 11 shows that the controllers designed based on Bode's ideal transfer function have both robustness anti-to the variation of system open-loop gain.

Fig. 11. Unit Step Responses of FOPID Controller

The parameter of Kd can be tuned to a suitable value to get a desirable step response. The bode diagram of open-loop FOPDF system both for integer or fractional first-order plant are the same, is shown in Fig.12. a). Fig. 12, b) is partial enlarged unit step responses of FOPDF control system of integer first-order plant,

BECTHHK TOry. 2021. № 2 (61)

when changing the system gain K. And Fig.12 is unit step responses of FOPDF control system of fractional first-order plant, when changing the system gain K.

Fig. 12. Unit Step Responses of FOPDF Control Systems

6. Summary

With the Bode's ideal transfer function method, we obtain closed-loop systems robust to gain variations and the step responses indicating an iso-damping property. And the calculating process is fairly easy to traditional fractional order controller design methods. But, it must be noted that the characteristic has limitation to the gain crossover frequency Cc and that the phase margin of the resulting closed-loop system is not exactly identical to the prescribed value defined by the slope a at that frequency. This is due to realize an arbitrary order by physical facilities are usually not easily. Simulations illustrate that the order a can be confirmed easily, but the select of Cc must consider a lot. To obtain a good performance, Cc must be as high as possible. And if too high this will hardly to get the PID parameters. And in this paper, we also have presented a new strategy of FOPDF control structure and apply Bode's ideal transfer function method, Chenliu method for parameters tuning. The numerical performances indicate that FOPDF control may result in a more rapid, more accurate and more robust control system. FOPDF method which has iso-damping property is of the practicability and effectiveness. FOPDF control method is likely applied in first order or first order time lag system like the hydraulic servo systems and the electric drive systems, high power semiconductor laser diodes constant temperature controlling systems.

7. Acknowledge

Research on multimodal rehabilitation training system of upper limb motor function based on finite state machine model (2020LY702L03)

Research on multimodal rehabilitation training method of upper limb motor function (2019JBC02L08)

Gait kinematics and plantar pressure assessment system for rehabilitation diagnosis (2020LY502L11)

References

1. Ying Luo, Yang Quan Chen, Chunyang Wang, Youguo Pi. Tuning Fractional Order Proportional Integral Controllers for Fractional Order Systems. Journal of Process Control.

- Volume 20, Issue 7, August 2010. - 823-831. [10] Chunyang Wang, Yongshun Jin, YangQuan Chen Auto-tuning of FOPI and FO[PI] Controllers with Iso-damping Property. 48th IEEE Conference on Decision and Control and 28th Chinese Control Confer-ence(CDC/CCC), Shanghai, China, Dec. 2015. - P 16-18.

2. Chen Y. Q., Hu C. H., Moore K. L. 'Relay feedback tuning of robust PID controllers with iso-damping property', in theProceedings of the 42nd IEEE conference on Decision and Control // Maui, Hawaii, December 9-12, 2003. - P. 2180-2185.

3. Chen Y. Q., Bhaskaran T, Xue D. Y. Practical tuning rule development for fractional order proportional and integral controllers[J] // Journal of Computational and Nonlinear Dynamics, 2008, 3(2). - 020201.1-021404.7.

4. Xue D.Y., Zhao C.N., Chen Y.Q. Fractional order PIDcontrol of a DC-motor with elatic shaft: A case study, InProc. of American Control Conference (ACC). - 2006. - Minnesota, USA. - P. 3182-3187.

5. Lee Ching-Hung Chang Fh-Kai. Factional-order PID controller optimization via improved electroma gnetism like algorithm [J] // Expert Systems with Applications, 2010, 37(12). - 8871-8878.

6. Podlubny, I.,Fractional Differential Equations,Academic Press, San Diego. - California, 2009.

7. Oustaloup A., Melchior P. The great principles of CRONE control[C]//International Conference on Systems, Man and Cybernetics: vol.2. Piscataway. - NJ. USA: IEEE, 2008.

- P. 118-129.

8. Podlubny I. Fractional-order systems and PIXD^controllers[J].IEEE Transactions on Automatic Control. - 1999, 44(1). - P. 208-214.

9. Barbosa R.S. Tenreiro Machado J A, Ferreira I M. Tuning of PID controllers based on Bode's ideal transfer function[J] // Nonlinear Dynamics. - 2004, 38(1/2). - P. 305-321.

10. Chunyang Wang,Ying Luo, YangQuan Chen An Analytical Design of Fractional Order Proportional Integral and [Proportional Integral] Controllers for Robust Velocity Servo // In Proc. of The 4th IEEE Conference on Industrial Electronics and Applications. -Xi'an, China, 25-27 May, 2009.

Заглавие: Моделирование динамики конечностей на основе передаточной функции передачи системы идеального тела

Авторы:

Цзи Цзоу - Чанчуньский университет, Чанчунь, КНР Пинг Сонг - Чанчуньский университет, Чанчунь, КНР Кайвен Сун - Чанчуньский университет, Чанчунь, КНР Чжаоган Шэн - Чанчуньский университет, Чанчунь, КНР

Аннотация: Теория исчисления дробного порядка и методы его моделирования широко применяются в области управления, особенно в имитационной модели динамической функции конечностей. Разработка систем управления объектами дробного

ВЕСТНИК ТОГУ. 2021. № 2(61)

порядка стала актуальной в последние годы. В этой статье рассматриваются системы дробного порядка, параметры которых определяются методом идеальной передаточной функции Боде для получения желаемой частотной характеристики. Для применения этого метода предлагается новая структура управления обратной связи с псевдопроизводной дробного порядка (FOPDF). Методы управления разработаны для интегральной системы первого порядка и для дробной системы первого порядка и применимы к реальной системе контроля постоянной температуры полупроводниковых диодов большой мощности, а также к гидравлическим сервосистемам и системам электропривода, результаты моделирования показывают эффективность и обоснованность этого метода.

Ключевые слова: динамическая функция конечностей; идеальная передаточная функция Боде, полупроводниковый лазерный диод высокой мощности; надежность, реабилитация.