Design and Simulation on PID Variable Damping Ratio Controller of Second-order System

Bin Zhong Equipment Transportation Department

Engineering College of Chinese Armed Police Force Xi’an, China

LianSheng Tang

College of Business Guangxi University for Nationalities Nanning, China

zhongbinchina@163.com 940554@163.com

Abstract—Generally, in order to improve dynamic performance, second-order systems generally decrease oscillation and overshoot of output response by increasing equivalent damping ratio with velocity feedback. But output response will be dull and system’s speediness property will be worse because of damping ratio’s increment. Variable damping ratio control by fuzzy adaptive PID may adjust system’s damping ratio in real-time on line on the base of output response’s different phases, so improve system’s dynamic property. It is proposed variable damping though establishing PID’ s three parameters’ fuzzy logic control rules based on property of PID control and fuzzy logic control. Simulation results verify that the method is feasible, dynamic property’s improvement is prominent, and disturbance rejection property and robust is excellent.

Keywords- fuzzy PID; second-order system; adaptive control; variable damping ratio; simulation

I. INTRODUCTION The system described by second-order differential equation

is called second-order system. The second-order system is widely applied in control engineering. The second-order system’s response characteristics are often regarded as a kind of reference for third-order or high-order system’s design while analyzing and designing system. Because the system’s dynamic performance is improved by introducing second-order system output variable’s derivative, the second-order systems generally decrease oscillation and overshoot of output response by increasing equivalent damping ratio with velocity feedback [1]. Although the second-order system response’s stationarity may be improved, the system’s response will be dull, regulating time may be prolonged and rapidity will be slow. On the other hand, the system’s equivalent damping ratio is also fixed value because the velocity feedback coefficient is fixed value. The second-order system with fixed damping ratio is difficult to obtain optimal dynamic performance, so this system is not suitable for the situation that demands better rapidity and stationarity.

Proportion integral differential (PID) control structure is simple, easy to realize and can improve the system’s dynamic and static performance [2], [3] and [4]. But because conventional PID controller can not adjust the fixed PID parameters on line and on real time according to the system’s actual work situation, the system’s stationarity and control precision will not have satisfactory result [5], [6]. So other

compensation methods will be used for practical application [7], [8] and [9], and the PID parameters can be adjusted on line by fuzzy reasoning method for these compensation methods [10], [11], [12] and [13]. Fuzzy adaptive PID variable damping ratio controller was designed in this paper considering damping ratio’s effect on the second-order system’s performance and combining advantages of simple PID control structure, easy realization and fuzzy control’s strong robustness and adaptability. The fuzzy adaptive PID variable damping ratio controller’s inputs are system output’s error and output error’s derivative. The controller’s outputs are three parameters: proportional gain, integral gain and differential gain. The real-time adjusting PID gains can be obtained on line according to system’s rise and settling stage’s response characteristics. PID’s adaptability could be realized through constructing fuzzy control rules and changing velocity feedback gain and system’s equivalent damping ratio. So the second-order system’s performance demands could be satisfied.

II. SECOND-ORDER SYSTEM’S FUZZY ADAPTIVE ADJUSTING PID VARIABILE DAMPING CONTROLLER

Figure 1 describes the second-order system with velocity feedback. τ is the velocity feedback. Obviously, the system’s closed-loop transfer function is described as

220

2

2

)2()()()(

nnn

n

sssRsYsΦ

ωτωωξω

+++== (1)

)2( 0

2

n

n

ss ωξω

+

sτ

)(sY)(sR

Figure 1. Second-order system with velocity feedback

Comparing typical second-order system’s standard closed-loop transfer function, we define

2/0 nτωξξ += (2) as equivalent damping ratio (hereinafter referred to as damping ratio for short). The parameter nω is called the system’s undamped natural oscillation angle frequency. 0ξ is the typical second-order system’s damping ratio.

According to (1) and (2), the second-order system’s undamped natural oscillation angle frequency will not be

978-1-4244-7941-2/10/$26.00 ©2010 IEEE

changed though introducing velocity feedback, but the system’s damping ratio is changed and damping ratio’s increment is 2/nΔ τωξ = . The damping ratio’s effect on the second-order system’s dynamic performance is obvious when

nω is determinate: the system’s step response will have sustaining undamped oscillation and large overshoot when the system works in negative damping ratio ( 0<ξ ) state; the system’s step response will have oscillating attenuation when the system works in under damping ratio ( 10 << ξ ) state; the system’s step response will not have overshoot when the system works in critical damping ratio ( 1=ξ ) and over damping ratio ( 1>ξ ) state, and although no overshoot response’s system has favorable stability, the rise time will be obviously prolonged with the damping ratio’s increase, the system’s response is blunt and the rapidity becomes worse. So, the response’s rapidity with negative damping ratio is obviously superior to other work state on the system output response’s rise stage. Consequently, when nω is known, in order to decrease second-order system response’s overshoot and improve system’s stability, the damping ratio must be increased, but the system’s rise time will be prolonged, response will become blunt and the rapidity become worse with damping ratio’s increase; on the other hand, in order to shorten the rise time, the damping ratio must be decreased, but the system response will have overshoot, system’s settling time will be prolonged and the system’s stability will be worse with damping ratio’s decrease. So the rise time and overshoot are contradictory one another. But the two parameters are important specifications that measure system’s dynamic performance. So, in order to resolve this contradictory problem, generally 707.0=ξ is regarded as optimum damping ratio by splitting the difference or the other compensation methods are adopted in engineering practical application.

The second-order system fuzzy PID variable damping ratio control can improve system’s dynamic performance based on velocity feedback control. It is known that 2/)()( 0 ntKt ωξξ += may be described as system’s damping ratio if gain )(tK is introduced into the velocity feedback channel. So the system’s variable damping control will be realized if the velocity feedback channel gain )(tK is adjusted on line according to system response’s output error e and output error’s derivative ec.

The curves of second-order system’s step response corresponding output error e and output error’s derivative ec are shown in figure 2. From this figure, in the system’s rise time interval 1~0 t , e gradually becomes zero from positive maximum; ec gradually becomes negative maximum from zero, and then gradually decreases. A positive and negative sharp pulse interference signal )(tx are forced on the system’s step response respectively at 2t and 5t . 32 ~ tt and 65 ~ tt time interval respectively are processes deviating stable state just under the positive and negative sharp pulse interference signal, and the sign of e and ec is uniform; 43 ~ tt and 76 ~ tt time interval respectively are processes coming back to stable state

after the positive and negative sharp pulse interference signal, and the sign of e and ec is contrary. The rise stage of system response is research emphasis for the second-order system variable damping control. The system’s damping ratio )(tξ is set to negative value through changing )(tK , so the system’s rise time may be shortened. The system’s damping ratio is adjusted to maximum by variable damping controller when the system deviates stable state (the sign of e and ec is uniform), so the deviation phenomenon will be effectively decreased or restrained, and the system’s capacity of resisting disturbance will be enhanced. At the same time, the overshoot phenomenon will be effectively decreased or avoided on the system’s regulation stage, the system’s regulation time will be shortened and the system’s stability will be improved.

0 2 4 6 8 10 12 14-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

t7t6

t5

t4

t3

t2

time (s)

y

x(t)

e

ec

t1

Figure 2. Disturbance’s effect on second-second system’s step response

sτ)(tK

)(sY)(sR

dtde

)2( 0

2

n

n

ss ωξω

+

Figure 3. Variable damping ratio controller’s structure of fuzzy adaptive adjust PID

However, for a practical second-order system it is difficult to determine the quantitative relation between gain )(tK and e or ec, construct this exact math model and realize system’s variable damping control according to practical system output response’s rise and stable stage. In order to adapt controlled plant’s uncertainty, nonlinearity, time-variability and enhance system’s adaptability and robustness, it is necessary that PID controller’s three parameters should be regulated on line. So, combining the advantages of PID control and fuzzy control, in order to obtain the variable damping control goal, the system’s damping ratio should be changed on line using fuzzy adaptively regulating pK , iK and dK according to second-order

system output response’s change law of e and ec. Based on damping ratio ξ ’s effect on the second-order system’s performance and structure of fuzzy adaptive regulating PID variable damping controller, the basic control thought may be described:(figure 3) on the system output response’s rise stage, in order to quicken system’s response velocity and shorten rise time, the proportional gain pK should be increased because the output error’s derivative gradually increase; in order to avoid integrator windup and bigger overshoot, integral gain iK should not be too big; so the PID controller’s output μ is a negative value and the system works in the negative damping ratio state. In order to shorten the regulation time with the error e’s decrease, pK should be properly decreased, μ should be gradually increased, damping ratio ξ should be gradually increased from negative to positive and the system works in the under damping ratio state. In order to decrease oscillation and bigger overshoot of the under damping system, the system output’s rise velocity should be decreased, so pK should be decreased and iK should be increased. μ and ξ should be bigger when e approximates to zero, so the system stably works in over damping ratio state and the system’s output will reposefully reach steady-state value. When the system stably works, it is important to enhance the system’s performance of disturbance rejection and restrain the change of error and error’s derivative, so the differential gain dK should be properly increased, forecast ec and ξ should be maximum. e and ec are fuzzy controller’s input variables, pK , iK and dK are the controller’s outputs. These input and output variables’ fuzzy set is {NB, NM, NS, ZO, PS, PM, PB}. The characters N, P, B, M, S denote negative, positive, big, medium and small, respectively, and ZO represents zero. Because e and ec are big on the system response’s initial stage and under the interference, in order to enhance fuzzy control system’s robustness, gently changing Gaussian membership function should be adopted; in order to enhance control system’s stability high sensitive triangle membership function should be adopted on other stage of transient response. The same control strategies and fuzzy control rules are adopted for PID controller’s three parameters pK , iK and dK and e and ec. The universe of discourse and membership functions of e, ec and pK , iK , dK is shown in figure 4. The fuzzy control rules is shown in table 1.

TABLE I. FUZZY CONTROL RULES OF pK , iK AND dK

TABLE II. COMPARSION OF SIMULATION RESULTS

Regulation time

Peak value time

Overshoot

0.8362 s

1.091 s4.2%

0.2986 s

0.2986 s

0

＝0.707 Variable damping ratio

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

0.060.040.020-0.02-0.04-0.060.30.20.10-0.1-0.2

Ki

Kp

PBPMPSZONSNM

Mem

bers

hip

func

tions

NB

e/ec/Kd -0.3

Figure 4. Membership functions of e, ec, pK , iK and dK

III. SIMULATION EXAMPLE In order to test and verify the second-order system fuzzy

adaptive adjust PID variable damping ratio control scheme’s feasibility and the control system’s performance, the second-order system )1(/16)( += sssG is simulated and studied. The velocity feedback coefficient is 1=τ . The damping ratio change region is ]6.1,2.0[ +− . The simulation results of second-order system’s dynamic characteristics are shown in figure 5 when ξ is 0.707, 1.5, 0.2 and variable damping ratio respectively. The comparative simulation results are shown in table 2 when ξ is 0.707 and variable damping ratio, respectively. The comparative simulation results of system’s disturbance rejection capacity are shown in figure 6. At the same time, we have a system’s robustness simulation experiment shown in figure 7 when the second-order system’s undamped natural oscillation frequency nω is increased positive 15% and negative 15%. According to these simulation results, the system’s dynamic performance and disturbance rejection capacity are obviously improved when the system adopts fuzzy adaptive adjust PID variable damping control.

0 2 4 6 80.0

0.5

1.0

1.5

2.0

time (s)

ξ < 0 ξ = 1 0 < ξ < 1 ξ > 1

y

Figure 5. Simulation results of ξ ’s effect on second-order system’s

dynamic performance

0 3 6 9 12 15-0.5

0.0

0.5

1.0 ξ = 1.5 Pulse interference Variable damping ratio ξ = 0.707

time (s)

1.3y

Figure 6. Simulation results on disturbance rejection property

0.0

0.2

0.4

0.6

0.8

1.0

16.0/( s2+s) 18.4/( s2+s) 13.6/( s2+s)

time (s)

y

0.30.20.10.0

Figure 7. Simulation results on robustness property

IV. CONCLUSION The second-order system fuzzy adaptive PID variable

damping ratio controller was designed according to change of the second-order system output response’s output error and error’s derivative. The PID controller can adjust PID’s three gain parameters on line, change the system’s damping ratio on the different stages of system output response. So the second-order system’s rapidity and stability can be obviously improved. The simulation results showed that the fuzzy adaptive adjust PID variable damping ratio control scheme was feasible, the system output response was rapid, has no overshoot, and disturbance rejection capacity and robustness were very strong.

REFERENCES

[1] K. H. Ang, G. Chong, Y.Li, “PID control system analysis, design, and technology ,” IEEE Transactions on Control Systems Technology, vol.13, pp.559-576,2006.

[2] M. H. Moradi, “New techniques for PID controller design ,” Control Applications, Proceedings of 2003 IEEE Conference on Digital Object Identifier, vol.2, pp.903-908, 2003.

[3] Y. Huang, S. Yasunobu, “A general practical design method for fuzzy PID control from conventional PID control,” Fuzzy Systems, The Ninth IEEE International Conference, vol.2,pp. 969-972, 2006.

[4] A. Rubaai, M.J. Castro-Sitiriche, A. Ofoli,” DSP-Based Implementation of Fuzzy-PID Controller Using Genetic Optimization for High Performance Motor Drives,” Industry Applications Conference, the 42nd IAS Annual Meeting. Conference Record of the 2007 IEEE, pp.1649-1656, 2007.

[5] P. H. Chang, J. H. Jung, “A Systematic Method for Gain Selection of Robust PID Control for Nonlinear Plants of Second-Order Controller Canonical Form ,” Control Systems Technology, IEEE Transactions on, vol.17, pp.473-483, 2009.

[6] M. Li, Y. L. Sun, M. Qin, H. G. Zhang, “GFHM-PID Controller Analysis and Design,” Intelligent Control and Automation, The Sixth World Congress on Intelligent Control and Automation, vol. 1, pp. 4011-4015, 2006.

[7] Z. L. Ding, C. D. Wang, G. M. Tan, G. H. Guan, “The Application of the Fuzzy Self-Adaptive PID Controller to the Automatic Operation Control of Water Transfer Canal System ,” Intelligent Computation Technology and Automation, International Conference on Intelligent Computation Technology and Automation, vol. 2, pp. 822-825, 2009

[8] Z. H. Xiu, W. Wang, “A Novel Nonlinear PID Controller Designed By Takagi-Sugeno Fuzzy Model ,” Intelligent Control and Automation, the sixth world congress on Intelligent Control and Automation, vol.1, pp. 3724-3728, 2006.

[9] J. J. Wang, C. F. Zhang, Y. Y. Jing, “Study of Neural Network PID Control in Variable-frequency Air-conditioning System ,” Control and Automation, IEEE International Conference on Control and Automation, pp. 317-322, 2007.

[10] H. C. Chen, “Optimal fuzzy pid controller design of an active magnetic bearing system based on adaptive genetic algorithms ,” Machine Learning and Cybernetics, vol. 4, pp. 2054-2060, 2008.

[11] B. Z. Jia, G. Ren, G. Long, “Design and Stability Analysis of Fuzzy Switching PID Controller,” Intelligent Control and Automation, the Sixth World Congress on Intelligent Control and Automation, vol. 1, pp. 3934-3938, 2006.

[12] H. Wang, Y. B. Yang, M. Y. Liu, “Fuzzy-PID Control in the Application of Multi-purpose Vehicles of the Road Snow Plowing ,” International Conference on Web Information Systems and Mining, pp. 246-250, 2009.

[13] X. K. Wang, X. H. Yang, G. Liu, H. Qian, “Adaptive Neuro-Fuzzy Inference System PID controller for SG water level of nuclear power plant ,” Machine Learning and Cybernetics, 2009 International Conference on Machine Learning and Cybernetics, vol.1, pp. 567-572, 2009.