An Application of Fractional Intelligent Robust Controller for Electromechanical Valve Tribeni Prasad Banerjee1, Suman Saha2 , Swagatam Das1, and Ajith Abraham3, 4 1

Dept. of Electronics and Telecommunication Engineering, Jadavpur University, Kolkata 700 032, India CSIR-Central Mechanical Engineering Research Institute, Drives and Control System Laboratory, Durgapur713209, West-Bengal, India 3 Machine Intelligence Research Labs (MIR Labs) Scientific Network for Innovation and Research Excellence, WA, USA 4 IT For Innovations, EU Center of excellence, Faculty of Electrical Engineering and Computer Science VSB - Technical University of Ostrava, Czech Republic e-mails: [email protected], [email protected] , [email protected], [email protected] 2

Abstract- The paper analyzed and model of a fault tolerant electromechanical controlled worm gear driven fuel shut off valve for aerospace application. The analysis is mainly on design a reduced order fractional controller. This is for controlling the velocity of worm gear so that the friction torque due to uneven rotational speed can be avoided. Because with the friction and stiction and backlash the valve system performance is reduced and more and more unstable this leads the system to failure. The proposed controller can provide robustness against the uncertain loading torque and system parameter.

important for the control engineer also when the system is purely design of electromechanical drive system. II. SYSTEM HARDAWRE DESCRIPTION The worm gear coupled with motor is standard for typical electromechanical system. Practically it has been observed that the vibratory problem due to friction is a critical issue to control the worm gear operated system.

Keywords: Mechatronics, Worm Gear, BLDC, Friction, FOPI, Robust Controller Torque, Torsional system.

I. INTRODUCTION The research results within gear faults due to friction, back lash and stiction variation methods have not yet gained wide industrial acceptance. The main reason that this has not yet happened is believed to be the scarcity of well defined and realistic example for use within the research community working with electromechanical drive system. Friction occurs in all electromechanical system, e.g. gear, breaks, actuator, actuator coupled with motor, valves and also pneumatic and hydraulics system. There is, thus, a need for real applications with genuine real world problem. This paper alleviates this circumstance by introducing an electromechanical position actuator coupled with DC motor with ball valve, used in the fuel flow control and injection system in main flight engine. The BLDC motor operates a worm gear to rotate a ball valve and it is obvious that the quantization problem in low position resolution to control the valve smoothly has a crucial role. Any fault in gear transmission will affects the smooth and precise velocity control of torsional system. Especially this nonlinearity shows between the teeth of the gear. It causes delay, vibration and speed inaccuracy that degraded the overall performance [1]. In extreme case, the friction creates severe vibration due to uneven control, which can cause of catastrophic failure of the system. Friction was studied extensively in classical mechanical engineering and it is very

c 978-1-4673-0126-8/11/$26.00 2011 IEEE

Fig. 1. Schematic proposed control system structure

The torque which is required to overcome by the motor includes the torque required against the inertia and the torque required to rotate the ball of the valve. The packing force and the sheer force for the fluid are acts on the system. The torque TV is required to rotate the ball valve. We can write from above equations T = Ieqv Įm + Tv => T = Ieqv

dw + Tv dt

(1)

From Fig.1 we can write

ω T + =1 ω 0 T0

(2)

where Ȧ & T are the angular velocity & torque respectively at a particular time, and T0 & Ȧ0 are the constant term taken from the motor characteristic curve and combining equation (1) & (2)

dω

368

dt

T + v

I

T = 0 eqv

I

T − v eqv

I

ω eqv

ω

0

Ÿ dω

dt

+

T0

I eqv

ω

ω0

=

T0 − Tv

(3)

I eqv

The general solution of above equation is I t I t 0 0 I ω I ω T T − ω e eqv 0 = ³ 0 v e eqv 0 dt + c I eqv I t I t 0 0 I ω T −T ω I ω eqv 0 eqv 0 Ÿ ωe = 0 v 0e +c T 0 Where, c is the integral constant.

(

)

(4)

The constant c may be calculated by putting the initial conditions in equation (4) at time t =0 the angular velocity Ȧ =0

c=−

(T0 − Tv )ω0

Motor switched off at time t1, so the torque produced by the motor will be zero and the inertia torque will act in the opposite direction. As shown in Fig. 3 the electromechanical actuation system includes actuator coupled with controller and the hydraulic valve, Current to pressure and pressure to current converter as signal conditioner (I/P & P/I), encoder, differential pressure transducer, and the motor inside the actuator to rotate the valve. The pink line shows the flow line and black line shows the control or signal line. The electro mechanical valve dispenses the fluid in realtime according to the control input. The fluid that comes from the main supply line (shown in pink line) passes through this valve and according to the engine requirement discharges the flow. In the real time operation the differential pressure transducer measure the input and output flow through the valve gives the control feedback to the controller. The Motor and the gear arrangement have shown in the Fig. 4.

(5)

T0

Putting the value of constant ‘c’ in equation (4) an rearranging the terms I t I t 0 0 I ω T −T ω I ω T −T ω ω e eqv 0 = 0 v 0 e eqv 0 − 0 v 0 T T 0 0 I t I t § · 0 0 ¨ ¸ I ω T −T ω ¨ I ω ¸ eqv 0 eqv 0 v 0 0 ¨e = − 1¸ ωe T ¨ ¸ 0 ¨ ¸ ¨ ¸ © ¹

(

)

(

(

)

)

(T − T )ω § − I ω = 0 v 0 ¨1 − e T0

¨ ©

I 0t eqv ω 0

· ¸ ¸ ¹

Fig. 3. Schematic diagram of the experimental setup of the plant

(6)

The velocity profile as shown in the Fig. 2 from a to c govern by the equation (6). After reaching time t1 motor is switched off.

Fig. 2. Velocity profile of the ball valve

At point c for the time t1 the cutoff velocity was ωc , now the torque equation (7) for the motor for the time t'. dω ′ + Tv (7) T = −I eqv dt ′

Fig. 4. Motor connection with the gear

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III. MODEL REDUCTION Model reduction technique was introduced by Davison in 1966. The method introduces was to reduce the dimension of the coefficient matrix of the system while preserving some of the dominant eigen-values or more influential states of the original system. Routh approximation method proposed by Hutton, this is to find a stable reduced order model for the original asymptotically stable model.

P(s) =

5240( s + 1.18 ± 43.1 j ) s ( s + 0.133 ± 42.7 j )( s + 260 )( s + 112 )

(8) The plant transfer function as shows in equation (8), has two complex zeros, poles at origin, two poles on negative real axis, and two complex poles. G

r

=

.0 1 8 0 3 s + 1 4 .2 s 2 + 7 7 .4 s

(9) Now with proposed reduced order model it has two poles and a zero, one pole at origin and one pole and zero at negative real axis shown in equation (9). IV. THE PROBLEM STATEMENT The BLDC motor operates a worm gear to rotate a ball valve and it is obvious that the quantization problem in low position resolution to control the valve smoothly has a crucial role. Any fault in gear transmission will affects the smooth and precise velocity control of tensional system. Especially nonlinearity in between the teeth. It causes delay, vibration and speed inaccuracy that degraded the overall performance [2], to overcome this friction issue and improve the robustness and of course the performance of the actuator controlled by the motor we modeled a controller. V. THE CONTROLLER MODELING From the basic studies of the feedback control actions behavior in the frequency domain of proportional, integral, derivative and effects over the controlled system behavior are such that, for proportional action, increase the speed of the response (lower rise time) with decrease steady-stateerror and relative stability. In derivative action increase relative stability and sensitivity to noise and for integral action steady-state error eliminated but decrease relativity. In the frequency domain π 2 phase is introduced by derivative action, positive for that increase relative stability, negative for that increased sensitivity to high frequency noise as gain is increasing with slope of 20 db/decade. As same for integral action π 2 phase lag is introduce in the system, for that decrease relative stability is the negative effect and positive is that eliminate steady state error by infinite gain at zero frequency. Considering this all, we can easily conclude that by introducing more general control actions such s n , n ∈ ℜ or 1 , n∈ℜ . sn

We can achieve more satisfactory balance between positive and negative effects of all control actions, and combining effect must satisfy the controlled system

370

specifications. By using this notation of fractional order may be a more realistic step than integer one. As all the real processes are fractional [1]. The most common form of noninteger model is PI λ D μ controller. λ Is the order of the integrator and, μ is the order of differentiator. The transfer functions of such a controller written as: K c(s ) = K p + λi + K d s μ (10) s Or in time domain controllers output written as: u (t ) = K p e(t ) + K i D −λ e(t ) + K d D μ e(t )

(11)

With different values of the parameter and the estimated structure of the controller is such that, for a classical PID controller the value of λ and, μ =1, then equation (10) will be, K (12) c (s ) = K p + i + K d s s For, a classical PI controller K d = 0, λ = 1 K (13) c(s ) = K p + i s For, a non-integer PI λ controller K d = 0

c(s ) = K p +

Ki

(14) sλ Non-integer or Fractional PI λ D μ controller has the most important advantage lies in the fact that this type of controllers are less sensitive to change of parameters of a controlled system. This is due to the extra degrees of freedom to better adjust the dynamical properties of a noninteger order control system K c(s ) = K p + λi = K p + K i s − λ (16) s Transfer function of the ª § λπ = « K p + K i w − λ cos¨ © 2 ¬«

PI λ controller is

ª ·º § λπ ·º −λ (17) ¸» − j « K i w sin¨ ¸» ¹¼ © 2 ¹¼» ¬ (Here λ is a non-integer number, by Euler’s theorem). Magnitude of its is

=

ª « K + K w−λ cos §¨ λπ i « p © 2 ¬

2 ª ·º § λπ −λ ¸ » + « Ki w sin ¨ ¹¼ © 2 ¬

2 º ·» ¸ (18) ¹» ¼

And Phase § § λπ · · ¨ K i w −λ sin ¨ ¸ ¸ ¨ © 2 ¹ ¸ (19) = − tan −1 ¨ § λπ · ¸ −λ ¨¨ K p + K i w cos¨ ¸ ¸¸ © 2 ¹¹ © Let ωc is the gain crossover frequency and φ is the m phase margin; to solve three unknown K , K , λ , we need P i three equations, or three specifications.

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A. SPECIFICATION

1-

FOR

PHASE

MARGIN

SPECIFICATION.

= −π + PM Arg ª«C ( jw ) G ( jw ) º» p ¬ ¼w − w PM is phase margin of the system. B. SPECIFICATION SPECIFICATION.

G ( jw )

w− w

2-

FOR

GAIN

= 1 , or, C ( jw) × G ( jw )

C. SPECIFICATION SPECIFICATION.

3-

FOR

CROSS-OVER

w−w

=1

ROBUSTNESS

K 1 = K + i = 1.3 + PI p s s K 1 and, C = K + i = 1.3 + FOPI p s s 0.34 C

when desired cross over frequency,

ω

π

= 1, PM = gc 3 To make system robust against gain variation or other parameter variation, we have to keep phase constant around gain crossover point. It means

d ( Arg (G ( jw))) =0 dw w = wc where phase of the system is ϕ =ϕ +ϕ plant controller In the Fig .5 blue graph for FOPI controller and pink line for PI controller has been shown. It is clearly shown that by using another additional tuning parameter, we got almost constant phase around gain crossover point; means system became more robust for parameter uncertainty. Than controlled by conventional PI controller. For hardware realization of corresponding FOPI model, we use Carlson’s base realization approach. And the corresponding performance is compared in the Fig. 6. VI. SIMULATION AND DISCUSSION In Fig. 5, open loop frequency response indicates that for PI controller phase under gain cross over point is not flat in nature, where as FOPI produces constant phase under cross over point, which provides additional robustness against parametric uncertainties. In step response also conventional PI controller produces large overshoot than FOPI controller shows in Fig.6.

Fig. 5. Comparison of bode plot using pi and FOPI controller for same reduced order plant

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Step Response 1.6 Reduced plant model tuned by PI controller only Reduced plant model tuned by FOPI controller 1.4

1.2

Amplitude

1

0.8

0.6

0.4

0.2

0

0

200

400

600 Time(sec)

800

1000

1200

Fig. 6. Comparison between two tuning method for same reduced order plant

VII. CONCLUSION In this paper a new robust fractional order control tuning method has been proposed and it’s applied into electromechanical primary flight fuel off valve actuator, from all simulation result we can easily conclude that these results show evidence that the use of a detailed and validated model of the system can produce a good performance. VIII. ACKNOWLEDGEMENT The authors gratefully acknowledge the experiment support provided by CMERI (Central Mechanical Engineering Research Institute), Durgapur, India. REFERENCE

[1]

R.M.Phelan, Fundamentals of mechanical Design, McGraw Hill, New-York, 1970. [2] Suman Saha, Saptarshi Das, Ratna Ghosh, Bhaswati Goswami, R. Balasubramanian, A.K. Chandra, Shantanu Das and Amitava Gupta, “Fractional order phase shaper design with Bode’s integral for iso-damped control system”, ISA Transactions, Volume 49, Issue 2, pp. 196-206, April 2010. [3] Ma, C. Hori, Y. Backlash Vibration Suppression Control of Torsional System by Novel Fractional Order PID^k Controller in IEE J.Trans. Ind. Application., vol. 124, no.3, pp.312–317, 2004. [4] G. E. Carlson and C. A. Halijak, “Approximation of fractional capacitors Approximation of fractional

capacitors (1 s ) by a regular newton process”, IEEE Transactions on Circuit Theory, Volume 11, Issue 2, pp. 210-213, June 1964. [5] S. Vohnout, D. Goodman, J. Judkins, M. Kozak, K. Harris, “Electronic prognostics system implementation on power actuator components”, Proc. IEEE Aerospace Conference, 2008. 1n

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G. Vachtsevanos, F. L. Lewis, M. Roemer, A. Hess, and B. Wu, Intelligent Fault Diagnosis and Prognosis for Engineering Systems: Methods and Case Studies. Hoboken, NJ: John Wiley & Sons, 2006. [7] L. C. Puttini, K. T. Fitzgibbon, R. H. K. Galvão, T. Yoneyama, “Prognostics and health monitoring solution applied to aeronautic heat exchangers”, Proc. 11th Joint NASA/FAA/DoD Conference on Aging Aircraft, 2008. [8] C. S. Byington, M. Watson, D. Edwards, P. Stoelting, “A model-based approach to prognostics and health management for flight control actuators”, Proc. IEEE Aerospace Conference, 2004. [9] J. J. Gertler, Fault Detection and Diagnosis in Engineering Systems. New York: Marcel Dekker, 1998. [10] D. S. Bodden, N. S. Clements, B. Schley, G. Jenney, “Seeded failure testing and analysis of an electro-mechanical actuator”, Proc. IEEE Aerospace Conference, 2007. [11] M. J. Roemer, C. S. Byington, G. J. Kacprzynski, G. Vachtsevanos, “An overview of selected prognostic technologies with reference to an integrated PHM architecture”, Proc. IEEE Aerospace Conference, 2005. [12] M. A. O. Alves Jr, D. Cabral, K. T. Fitzgibbon, “Model-based failure severity prediction based on robust residual design”, Proc. 15th Mediterranean Conference on Control and Automation, 2007. [13] Arijit Biswas, Swagatam Das, Ajith Abraham, Sambarta Dasgupta, Design of fractional-order PIlambdaDμ controllers with an improved differential evolution. Eng. Appl. of AI 22(2): 343350 (2009). [6]

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