Fault Diagnosis of Buck Converter Driven DC Motor Based on a Hybrid Unknown Input Observer

Asma EL MEKKI and Kamel BEN SAAD LARA Laboratory, University of Tunis El Manar National Engineering School of Tunis, ENIT BP 37, 100 Tunis, Tunisia [email protected] [email protected]

Abstract— In this paper a model based approach is presented between the measured and the estimated outputs is used to for the diagnosis of parametric and discrete faults of a buck generate residuals that are sensitive to faults. converter driven DC motor. A hybrid unknown input observer The paper is organized as follow; section II presents the is designed based on a hybrid model of the buck converter DC hybrid unknown input observer design method. In section III motor combination. The observer parameters are synthesized using Linear Matrix Inequalities techniques. Simulation results the hybrid model of the studied converter-motor system and are given to demonstrate the efficiency of the proposed method. the proposed diagnosis method is presented. The simulation results and the conclusion are given in sections IV and, V respectively. Keywords— Dynamical hybrid model, buck converter DC motor combination, Unknown Input Observer, Fault detection, II. HYBRID UNKNOWN INPUT OBSERVER DESIGN Linear Matrix Inequalities Consider the linear switching system given by:

I. INTRODUCTION x˙ = Al(t ) x + B l ( t ) u + D l ( t ) v (1 With the increasing demand for reliability and efficiency y= Hl (t ) x ) of power converters and electrical machines, fault diagnosis is becoming a very active area of research. Many researchers investigated DC motors fault diagnosis where x n is the state vector, u m denotes the known using different approaches. In [1] parity equations are used input vector, v m represents the unknown input vector and for fault detection of multiple sensors. In [2] a combination of the parity equation and parameter estimation approach is y p is the output vector. l(t ) :�+ � q{ 1, , N} is the used in the diagnosis of parametric and additive faults in a switching signal deciding which mode is active from a finite

DC motor. In [3] authors designed a Luenberger observer set of discrete states q. The matrices Al (t ) , Bl (t ) Dl (t ) and based method for sensor fault detection. In [4] authors H developed a model based method combining parity l (t ) are known constant matrices with appropriate equations for residual generation with fuzzy logic for dimensions. decision making. Under certain conditions a full order unknown input Like most complex systems, power System exhibit a observer [11] could be designed as follow : hybrid behavior which involve both continuous and discrete z˙ = Nl(t ) z + G l ( t ) u + L l ( t ) y (2 event dynamics [5]. In fact, electrical currents and voltages xˆ = z - El(t ) y ) evolve according to the laws of electromagnetism and also change in a discontinuous manner according to the state of the switches. The diagnosis problem of such systems, The estimation error is governed by the following classified as switched systems, was investigated by many equation: researches. In fact, faults and external disturbances are e˙=x ˙ - xˆ˙ = x ˙ - z˙ + Ey ˙ coupled which increases the rate of false alarms and makes l (t ) the diagnosis process more challenging. This problem was =(Ml(t ) A l ( t ) - K l ( t ) H l ( t )) x - N l ( t ) xˆ (3 tackled in the literature for instance, by using unknown input +M B - G u ) observers [6], fault detection filter [7]-[9] and optimal fault ( l(t ) l ( t ) l ( t ) ) detection filter [10]. +Ml(t ) D l ( t ) In this paper, we propose an observer based method for the where fault detection in a buck converter driven DC motor. The (4) diagnosis method is based on a hybrid unknown input Ml(t )= I + E l ( t ) H l ( t ) observer that provides an estimation of the measurable output K= L + N E (5) and decouple the effect of the disturbances. The difference l(t ) l ( t ) l ( t ) l ( t ) existence of a common Lyapunov function for all subsystems or the existence of multiple Lyapunov functions for each Furthermore if subsystem. N= M A - K H l(t ) l ( t ) l ( t ) l ( t ) l ( t ) (6 Consider a Lyapunov function candidate for (9) defined by ) : (14 G= M B l(t ) l ( t ) l ( t ) (7 V= eT P e ) )

M D = 0 V is a common Lyaunov function if there exist l(t ) l ( t ) (to decouple disturbance from the (8 symmetric positive definite matrices P and Q such that observer error) ) (15 T then equation (3) of the error dynamics becomes as follows : NqP+ P N q � Q ) (9 e˙ = Nl (t ) e ) By substituting the matrix Nq by its expression into (15), The observer described by (2) could be designed if for it yields : every q there exist an UIO of the following form : T I+ U H + Y V H A - K H P (16 (1 (( q q q q q) q q q ) z˙ q= N q z q + G q u + L q y ) 0) +PI + Uq H q + Y q V q H q A q - K q H q + Q 0 xˆq= z q - E q y (( ) ) Inequality (16) could be reformulated as the following where matrices Nq , Gq , Lq and Eq are chosen such that linear matrix inequalities (LMI's): Mq= I + E q H q , T ((I+ Uq H q) A q) P + P(( I + U q H q) A q ) Kq= L q + N q E q , (17 T T ) +(Vq H q A q) Y q + Y q( V q H q A q ) Nq= M q A q - K q H q with Nq is Hurwitz "q = {1, , N} ,

G= M B T T q q q , -Kq H q - H q K q + Q 0 and Mq D q = 0 . (18 -1 The necessary and sufficient conditions for the existence Yq= P Y q ) of the qth UIO (10) are given by : rank H D= rank D = m p (19 i. ( q q) ( q ) , this condition is to -1 Kq= P K q ) ensure the existence of a matrix Eq such that (11

Eq H q D q= - D q for all q= {1, , N} ) III. DIAGNOSIS OF BUCK CONVERTER DRIVEN DC MOTOR In this section the model of the buck converter driven DC motor is described. And the proposed diagnosis method is ii. all the pairs (Mq A q, H q ) are detectable. presented. Furthermore the continuity of e( t) must be guaranteed by the following condition [11] A. Model of the Buck Converter DC Motor Combination (12 The buck converter driven DC motor shown in Fig. 1 is considered in this study. iii. E1 H 1= E 2 H 2 =... = EN H N )

d R L i i R L The general solution of (11) is given by L L a a a (13 E= U + Y V q q q q ) i c with + , + , is an V + ωK U= - D H D V= I - H D H D Y in C e q q( q q ) q( q q)( q q ) q - arbitrary matrix of suitable dimensions and + -1 ( X) = ( XT X) X T is the Moore-Penrose pseudo inverse. The error convergence could be seen as a stability problem Fig. 1 Buck converter DC motor combination of a hybrid dynamical system. And thus it could be The electromechanical dynamics of the buck converter demonstrated by one of the concepts of the stability analysis driven DC motor can be modeled by the following of hybrid systems [12]. Based on Lyapunov approach, the differential equations : stability of the error dynamic could be guaranteed by the di R V d Y= P-1 Y L= - L i -c + V dt LL L L in K= P-1 K The residuals are generated using the difference between dVciL i a = - (2 the measured and the estimated output of the system. dt C C T 0) r=轾 V - Vˆ i - iˆ w - wˆ (22) dia V c R a K e 臌c c a a = -ia - w dt La L a L a

dw KT B The residual evaluation function is chosen as the L2 norm =mi -L - v w dt Ja J J of the residual vector r p (23) and the threshold is chosen as a constant positive value (24). where ia is the armature current, iL the inductor current,Vin 1 nr 2 (23 the input voltage,Vc the converter output voltage and w the 骣 r t= r2 t motor angular velocity. ( ) 琪 i ( ) 桫i=1 ) The variable d depends on the state of the switch and is defined as follow : 0 when the switch is closed, T= sup r( t) (24 d = t 0 1 when the switch is open. ) The buck converter DC motor combination can be considered as a dynamical hybrid system without state A fault can be detected if the residual exceed the jumps. Thus its model can be written in the form of (1) where predefined threshold T . the state vector is defined as x= ( i V i w)T , the L c a r( t) � T f 0 (2 torque load T is considered as a disturbance and the L r( t) > T f 0 5) matrices Aq , Bq , Dq and Hq , such that q = {1,2} , are given by:

骣 RL 1 IV. SIMULATION RESULTS 琪- - 0 0 琪 L L The proposed method for the fault detection of the buck 琪 1 1 converter DC motor combination was implemented and 0- 0 琪C C demonstrated by numerical simulation. 琪 A1= A 2 = , The LMIs were solved using MATLAB LMI Toolbox. A 琪 1 Ra K e 0 - - buck converter driven DC motor converter and its 琪 L L L 琪 a a a corresponding UIO observer were simulated on MATLAB 琪 K B Simulink Stateflow. The system parameters are listed in 琪 0 0 m- v 桫 J J Table 1. T T 骣1 TABLE I B1 = (0 0 0 0) , B2 = 琪 0 0 0 , 桫L Parameter Value

RL 0.2 Ω 骣0 1 0 0 T buck coil resistance 琪 骣 1 L buck coil inductance 1.33 mH H1= H 2 = 琪0 0 1 0 , D = 琪0 0 0 琪 桫 J C filter capacitor 470 µF 桫0 0 0 1 L 8.9 mH The considered faults consist of parametric faults a armature inductance characterized by changes in the armature resistance and Ra armature resistance 6Ω inductance, and discrete fault caused by open switch fault in Ke back EMF constant 0.0517 V. s/rad the buck converter. Km motor torque constant 0.0517 N.m/A B. Fault Diagnosis of the Converter -Motor System J moment of inertia 7.95 10-6 Kg.m2 It can be easily demonstrated that the necessary and B viscous friction 0.001 v sufficient conditions for this system are satisfied. Moreover coefficient N.m/rad/s A1= A 2 , H1= H 2 and D1= D 2 = D , thus the LMI's can be simplified to : The buck converter operates with a switching frequency of T (21 V= 24 V ((I+ UH) A) P + P(( I + UH) A) 45kH. And the input voltage is in . ) T T T T Different cases were simulated considering different kinds +(VHA) Y + Y( VHA) - KH - H K + Q 0 of faults, both converter and motor related. The considered faults are as follows: - parametric faults caused by a change in the armature Case 3 Armature resistance variation at t=0.3s resistance of the DC-motor or a change in both the armature 8 0 resistance and inductance due to inter-turn short circuit fault 6 0 - and discrete fault due to open switch fault in the buck w 4 0 2 0 converter. 0 Case 1 no faults, torque load change at t=0.2s 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 3

1 0 0 2 a

i 1

0

w 5 0

- 1 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 0 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 3 0 4 2 0 c V 1 0 a

i 2

0 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 0 T i m e [ s ] 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 2 0 Fig. 6 Measured and estimated outputs in case of a resistance variation

0 . 2

c 1 0 V 0 . 1 5

0 r 0 . 1 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 T i m e [ s ] 0 . 0 5 0 Fig. 2 Measured and estimated outputs 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 T i m e [ s ] - 3 x 1 0 3 Fig. 7 Residual signal 2

r Case 4 Inter-turn short circuit fault at t=0.3s 1 1 0 0 0 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 Time [s]

w 5 0 Fig. 3 Residual signal 0 Case 2 open switch fault at t=0.3s 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 1 0 0 3 Estimated output 2 Measured output a

w 5 0

i 1

0 0 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 - 1 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 2 0 0 2 0 a

i 0 1 5

c 1 0 V - 2 0 0 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 5

1 0 0 0 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 T i m e [ s ]

c 0 V Fig. 8 Measured and estimated outputs in case of an inter-turn fault - 1 0 0 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 0 . 2 5 T i m e [ s ] 0 . 2

Fig. 4 Measured and estimated outputs in the presence of an open switch 0 . 1 5

fault r 0 . 1 1 5 0 0 . 0 5

1 0 0 0 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5

r T i m e [ s ] 5 0 Fig. 9 Residual signal 0 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 5 T i m e [ s ] For each case, the measured and the estimated outputs are illustrated in Fig.2, Fig.4, Fig.6 and Fig.8. Fig. 5 Residual signal In the first case the change in the torque load impact the detection: Online application to DC motor,” Turkish J. Electr. measurable output of the system Fig.2. However, the residual Eng. Comput. Sci., vol. 22, no. 2, pp. 363–370, 2014. [4] L. J. De Miguel and L. F. Bla, “Fuzzy logic-based decision- Fig.3 remains unchanged. This demonstrates the robustness making for fault diagnosis in a DC motor,” vol. 18, pp. 423–450, of the used method against the disturbance which is here the 2005. Torque load. When a fault occurs residuals change [5] I. Hiskens and M. Pai, “Hybrid systems view of power system significantly as illustrated in Fig.5, Fig.7 and Fig.9. modeling,” in 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353), 2000, vol. 2, pp. 228– 231. V. CONCLUSIONS [6] J. Wang, Y. Shen, and M. Zhang, “Fault detection for linear switched systems using unknown input observers,” Proc. 33rd In this paper, the problem of fault diagnosis of a DC motor Chinese Control Conf., pp. 3219–3223, 2014. combined with a buck converter is investigated. A Hybrid [7] A. Abdo, S. X. Ding, W. Damlakhi, and J. Saijai, “Robust fault UIO is designed to generate residuals that are sensitive to detection filter design for uncertain switched systems with faults and indifferent to torque load disturbance. adaptive threshold setting,” 50th IEEE Conf. Decis. Control Eur. Control Conf., pp. 5467–5472, 2011. The convergence analysis of the observer was carried out [8] D. W. P. Shi and W. Wang, “Robust fault detection for using common Lyapunov function approach. The parameters continuous-time switched delay systems : an linear matrix of the observer were designed based on a LMI formulation. inequality approach,” vol. 4, no. November 2008, pp. 100–108, The effectiveness of the proposed method in parametric and 2010. [9] D. E. C. Belkhiat, N. Messai, and N. Manamanni, “Design of a discrete fault detection was demonstrated by simulations. robust fault detection based observer for linear switched systems with external disturbances,” Nonlinear Anal. Hybrid Syst., vol. 5, REFERENCES no. 2, pp. 206–219, May 2011. [10] K. Iftikhar, A. Q. Khan, and M. Abid, “Optimal fault detection filter design for switched linear systems,” Nonlinear Anal. Hybrid [1] C. Chan, S. Hua, and A. Hong, “Application of fully decoupled Syst., vol. 15, pp. 132–144, 2015. parity equations in fault detection and identification of dc motors,” [11] W. Chen and M. Saif, “Unknown Input Observer of A Class of IEEE Trans. Ind. Electron., vol. 53, no. 4, pp. 1277–1248, 2006. Switched Control Systems,” pp. 1173–1178, 2005. [2] R. Isermann, Fault Diagnosis Applications : model-based [12] D. Liberzon and A. S. Morse, “Basic problems in stability and condition monitoring: actuators, drives, machinery, plants, design of switched systems,” IEEE Control Syst. Mag., vol. 19, no. sensors, and fault-tolerant systems., Springer S. Berlin Heidelberg, 5, pp. 59–70, 1999. 2011. [3] A. Alkaya and I. Eker, “Luenberger observer-based sensor fault