AN APPROACH FOR THE MODEL BASED MONITORING OF PIEZOELECTRIC ACTUATORS

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1 II ECCOMAS THEMATIC CONFERENCE ON SMART STRUCTURES AND MATERIALS C.A. Mota Soares et al. (Eds.) Lisbon, Portugal, July 18-1, 005 AN APPROACH FOR THE MODEL BASED MONITORING OF PIEZOELECTRIC ACTUATORS Dirk Mayer, Heiko Atrodt, Sven Herold, Martin Thomaier Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt, Germany web age: htt:// Keywords: Smart Structures, Pieoelectric Actuators, Structural Health Monitoring, Adative Filters Abstract. Pieoelectric actuators are used in a wide range from mechatronic alications like fuel injection to the field of smart structures for microositioning or noise and vibration suression. However, due to increasing demands for reliability and safety of technical systems, fault diagnosis and structural health monitoring become imortant. Therefore, the on-line diagnosis of ieo actuators is an essential challenge when develoing smart structural systems for industrial as well as automotive or aerosace alications. Model based monitoring systems can rovide detailed information about the current status of a technical system 1. In this work, an aroach for model based monitoring of ieoelectric actuators is examined. The main idea is to use the measurement of the electrical imedance of the actuator to receive information about ossible damages. Because of the electromechanical couling, this measure includes information about the electrical arameters as well as the mechanical arameters of the actuator. At hands of a simulation, basic studies on damage scenarios of electromechanically couled systems are accomlished. Methods for model based fault diagnosis of the system are develoed and imlemented with the hel of adative digital filters. These should rovide arametric models of the inut imedance function, ossibly on-line in arallel to the oeration of the ieo transducer in a vibration control system. The signal rocessing algorithms are develoed taking into account the ossibility of imlementation of a real-time caable embedded standard hardware, i.e. the comutational effort of the diagnostic system. Measurements on ieoelectric stack actuators used in smart systems for vibration decouling hel to imrove the simulation. The monitoring system is imlemented on raid rototying controller hardware and is tested in an exeriment on monitoring of the ieoelectric actuators. 1

2 1 INTRODUCTION The integration of monitoring and fault detection systems is increasingly taken into account in the develoment of mechatronic systems and smart structures. Model based fault detection methods have been roosed, e.g. in the field of automotive alications 1. They are based on tracking the changes of a arametric model of the system in case of damage. In many systems, ieoelectric actuators are alied. For examle, ieoelectric stack transducers are used to design stiff active mounts for vibration control in various alications 3. In this work, a model based monitoring system for such a mount will be examined. With the hel of the electromechanical analogy it is ossible to develo models for ieoelectric transducers 4,5. These can hel to examine the influence of the damages, i.e. changes in the hysical arameters, on the model arameters. Because of the electromechanical couling of the ieo actuator, its electrical inut imedance gives information as well about the electrical arameters as well as about the mechanical arameters and the mechanical structure couled to the actuator, and with detailed modeling it is ossible to identify also material constants 6. Therefore, the imedance is often used in structural health monitoring systems 7. Here, the electrical inut imedance of an integrated ieo actuator will be examined for its roerties as a measure towards damage diagnosis. The ossibility of monitoring mechanical systems with adative digital filters has been investigated in several alications 8. In this work, adative filters will be alied for the arameter identification of the electrical imedance. Since in smart structures, control algorithms are often based on digital signal rocessing, the imlementation of the monitoring system with similar signal rocessing methods offers the ossibility for an integration of both the vibration control and the diagnosis, which can lead to a fully integrated smart structure. The aer is organied as follows: First, the electromechanical model for a ieoelectric stack actuator will be resented and the result is comared with resective measurements. After that, the aroach of adative digital filters for model based monitoring is introduced and afterwards examined at hands of a simle examle using the develoed ieo model to simulate different damage scenarios. In section 5, the imedance monitoring method is exerimentally tested in an alication, an active ieo interface for vibration decouling with the hel of real-time signal rocessing. LINEAR DYNAMIC MODEL OF A PIEZO ACTUATOR A arametric model of the ieoelectric actuator is the basis for the develoment of a model based monitoring method. A suitable way to imlement a model of an electromechanical transducer system is the electromechanical analogy. This method imlements a linear model, yet it is including the electromechanical couling of the ieo transducer. It is assumed, that the actuator can be modeled as a lumed mass-sring-

3 system. This assumtion is valid for small signals and for the frequency range u to the first longitudinal resonance of the actuator. The method allows for reresenting a system which consists both of mechanical and electrical subsystems with electrical circuit diagrams. In the first ste, the actuator mass is neglected, while only the mechanical stiffness k, daming b, the electromechanical couling coefficient α and the electrical caacitance C are considered. Then, for the ieoelectric actuator, a reresentation by a two-ort network can be found (Figure 1). Figure 1: Electromechanical equivalent circuit of a ieo transducer This network can be described by several matrix formulations 9. Here, the imedance matrix Z is of secial interest: 1 U jωc = F α jωc k C + α jωc jωc b jωc I + α v To examine the roerties of this modeling method, the comarison to an admittance measurement of a ieoelectric stack actuator with free-free boundary conditions is used (Figure ). The model arameters of a ieo stack can be calculated from the hysical and geometrical arameters: C 33 n l (1) A = ε () k = E A l α = d 33 n k Here, A is the cross-section, l is the length of the actuator and n is the number of layers, which are stacked mechanically and electrically connected in arallel. The necessary hysical arameters are the Young s modulus E, and the ieoelectric material constants d 33 and ε 33. The mass m of the actuator is concentrated into two lumed masses in a way, that the first longitudinal resonance frequency matches the resonance of the resulting mass-sring system. In the equivalent circuit, this mass is reresented by 3

4 Figure : Pieo stack actuator, mechanical and electromechanical reresentation. terminating the two-ort network with an inductance (Figure ). With the resulting terminating mechanical admittance π Y mech ( jω) =, (3) j ω m the electrical inut admittance can be calculated: Y k dπ α π π ( jω) + ( jω) + + m mc m ( jω) = jωc. (4) d k π π ( jω) + ( jω) + m m in Figure 3: Comarison of an admittance measurement and the electromechanical model With the hel of the arameters of the actuator, which are taken from the material data sheets, the admittance can be evaluated numerically and comared to an admittance measurement (Figure 3). Because the real arameters differ slightly from the data sheet, a numerical otimiation of the model arameters is done. After the adjustment of the arameters in order to enhance the recision of the model, both the measured and the calculated admittance match quite well u to the first resonance frequency at above 4

5 40 kh. Obviously, the admittance reresents the caacitive roerty of the electrical art of the system as well as the dynamical mechanical roerties. 3 APPROACH FOR MODEL BASED MONITORING OF THE PIEZO Since the electrical admittance function includes the imortant electric and mechanical roerties of the ieo actuator, there is a motivation for using it as a monitoring arameter for damage detection in ieoelectric actuator systems. The concet is based on a standard rocedure for model based diagnosis. The system is excited by a suitable test signal, the resonse from the system is measured and a arametric model is identified from the inut and outut data. In this work, the focus is on the identification of arametric models as damage features with adative digital filters (Figure 5). Figure 5: Identification of arametric models with adative digital filters The admittance function shows a caacitive behavior over the whole frequency range. This may lead to roblems in the convergence of the adative filters regarding the modeling of the resonance of the ieo system. Therefore, instead of the admittance the monitoring system identifies a time-integrated version of the admittance: Yin ( jω) H ( jω) = (5) jω For integration of the system outut, a first-order low ass filter with a cut-off frequency far lower than the interesting frequency range is used, which can be easily added to the digital filter system (Figure 5). The adative filter will rovide a time discrete model of the transfer function H(jω). Because of its recursive characteristics, an IIR (infinite imulse resonse) filter should model the function with a minimum of arameters: 1 b b b Hˆ K + ( ) =. (6) a + K + a 1 M M N N 5

6 4 SIMULATION OF AN EXAMPLE To test the described concet in simulations, a ieo actuator system is considered, which consists of the ieo stack used before with an effective mass of m = 0. kg added to one side and fixed to the ground on the other side. The actuator mass itself is therefore small against the couled mass and can be neglected without introducing a relevant error. Because of the simulation of the comonents as one- or two-ort networks, the simulation is easily enhanced, if more comlex mechanical loads are to be considered (Figure 6). Figure 6: Test examle of a mechanically loaded ieo and network reresentation Assuming a samling interval T, the ideal model is given by the transformation of the transfer function into the time discrete domain, which is received with the hel of the imulse invariance transformation: C H ( ) = α Im{ } ( Re{ }) ω0c m 1 1 Re{ } (7) k with ω 0 = and m dt / m+ jω0t = e Therefore, an adative filter with three coefficients in the numerator and in the denominator should be able to identify the transfer function. As an adative filter algorithm, the recursive-least-squares algorithm is chosen, which ossesses fast and recise convergence roerties. Indeed, it is suitable only for a small number of coefficients for real-time imlementation because it requires a comutational effort of order I, when I is the order of the filter 10. The simulation is set u in Simulink to study all interesting effects in the time domain. It includes two different samle rates, a higher rate to simulate the electromechanical system, and a lower rate for the digital signal rocessing. Therefore, also low-ass filters for anti-aliasing and reconstruction are included (Figure 7). Because their amlitude and hase resonse affects the transfer function characteristics together with the band-limited excitation around the resonance frequency of the system, the filter order has to be chosen 6

7 State-Sace Model Ymech x' = Ax+Bu y = Cx+Du K*u xi F1 ground_dist 1/ butter Uin Iout I U butter ZOH White Noise Amlifier Imedance Matrix Interface butter ZOH -K- In Out u y e_aus [1x4] adat_iir_rls b_aus e b TP1 1 adat [1x4] a_aus b a a IIR RLS Adatation Figure 7: Time domain simulation of the ieo monitoring system higher than exected from the direct calculation of the time-discrete transfer function (equation 7). A tradeoff between a recise model and a modest comutational effort is found for orders M = 6 and N = 4 (Figure 8). With this configuration, several damage scenarios and their effect on the model arameters are studied. Probable damages in a ieoelectric actuator system are the decrease of electrical caacitance, mechanical stiffness or electromechanical couling. Therefore, these cases are examined in the simulation. Obviously, the adative filter estimates the two conjugated comlex oles and eros of the system, which are ositioned near the unit circle in the -lane, and the model is stable (Figure 8). Figure 8: FRF of the system and by the adative filter identified model, ole-ero lot of the model 7

8 Figure 9: Comarison of transfer functions for initial and damaged state To evaluate the influence of the damage cases, a gradual decrease u to 0% of each of the arameters is assumed. For the maximum damage, the transfer functions are comared to the ones of the undamaged case (Figure 9). As a simle damage index, the normalied distance between the filter arameter vectors for the initial and the damaged state is chosen. This is done searately for the numerator and denominator: ( ( i) an an ) ( i) an D = (8) D a b = ( ( i) bn bn ) ( i) bn For the three cases, the resective indices are calculated and lotted against the ercentage of arameter change (Figure 10). Obviously, the decrease of C results mainly in a change of the numerator coefficients. This matches with the exectations, because the caacitance reresents the static offset of the transfer function. The decrease of α has the smallest effect on the damage indices, but as can be seen in Figure 9, for this damage case the transfer function shows only a small variation against the initial state. The decrease of the mechanical stiffness k leads to a lower resonance frequency of the ieo system. Since not only the oles, but also the eros of the transfer function are influenced by this arameter change, there is an effect on the damage indices of both the nominator and denominator, which is in same order of magnitude. Thus, the aroach of monitoring the ieo actuator system with an adative digital filter system seems to be suitable for damage detection in different damage cases. 8

9 Figure 10: Numerator and denominator damage indices for the studied cases 5 EXPERIMENTAL STUDY ON AN ACTIVE PIEZO INTERFACE The develoed monitoring aroach is tested at hands of an active ieo interface for active vibration control (Figure 11). It is designed for automotive alications, where it can be used to decoule excitations of the car body through the engine, the gearbox, aggregates, drive shafts, or the tire/road contact. The realiation with stiff ieoelectric stack actuators makes the direct integration into different systems ossible, because the interface can bear large static loads. To make the alication in the harsh environments of automotive systems ossible, the ieo actuator system is designed to resist against oils, liquids, etc. and to withstand misuse loads in the lateral directions of about 3 kn. The active interface is able to control three degrees of freedom, one translation in -direction and two rotations about the x- and y- axis (Figure 1). Figure 11: Pieoelectric interface for active vibration control 9

10 x y x y x y x y Figure 1: Position of the actuators (left), control of the three degrees-of-freedom The control of the three degrees of freedom is realied by arranging three actuators on a circle, which can be driven searately. In the studies resented here, only the translational degree of freedom is used, thus all three actuators are connected in arallel and driven with the same voltage. The mechanical set u of the interface consists of the ieoelectric interface itself, and a mass of m add = 0.3 kg mounted to one end in order to simulate a structure couled to the system. The signal rocessing for testing of the ieo monitoring system follows the rincile shown above (Figure 5). For realisation of the adative digital filter system, a dsace raid control rototying system is used running at a samling frequency of 0 kh, which imlements the algorithm with the hel of automatic code generation from the simulation code (Figure 7). The inut current of the ieo system is measured using a shunt resistor. First, the adative system identification of the ieo interface in the initial state is tested. A roer frequency range and a number of filter coefficients of the adative IIR filter is chosen in order to get a good aroximation of the monitored transfer function with a low number of coefficients. Here, a number of M = 9 coefficients for the numerator and N = 4 for the denominator was found (Figure 11). Obviously, in the real alication the daming seems to be higher than in the receding simulations. Figure 11: Monitored transfer function and adatively identified model, ole-ero lot of the model 10

11 Figure 1: Transfer functions with added masses (left) and disconnected actuators (right) Because in this work the ieo interface should not be damaged irreversibly, some simulations for damage scenarios have to be used. Instead of decreasing the ieo stiffness, a small amount of mass (m add = 0.0 kg) is mounted to one end of the interface, similarly leading to a decrease of the resonance frequency (figure 1, left). As a second damage, the ieoelectric actuators are disconnected successively, which leads to a loss of electrical caacitance (figure 1, right). Figure 13: Damage indices for the exerimental damage cases 11

12 In every situation, the identification with the adative filter is done and the damage index (equation 8) is calculated. A decrease of the resonance frequency affects both the numerator and denominator damage index, while a loss of caacitance influences the numerator damage index more than the denominator damage index (figure 13). This matches somewhat with the results received from the simle simulation examle (figure 10), but the different damage cases are not as clearly distinguished than in the simulation. 6 CONCLUSIONS In this aer, an aroach for model based monitoring of a ieo actuator system was examined. With the hel of the electromechanical analogy, a simulation model for a ieo stack actuator was set u and the calculated electrical admittance was comared to a measurement, which lead to good results. At hands of a simle simulation built u with the hel of the actuator model, the method for monitoring of the electrical admittance with the hel of adative filters was tested. Different damage cases were studied in the simulation. At last, the monitoring system was tested exerimentally with the ieo interface. The adative filter system was imlemented on real-time caable hardware and different damage scenarios were treated. The adative filter system could detect the changes in the system characteristics, while the results showed the same characteristics like the results received in the simulations. Therefore, the method seems to be suitable for imlementation as a monitoring system for ieo actuator systems. In order to fulfill the requirements of an integrated smart structure, the monitoring algorithm has to be imlemented on low-cost embedded hardware. This will be one focus of the further work. Another imortant toic will be the refinement of the electromechanical simulation of the actuator system. Therefore, the ieo actuator model will be integrated into a multi-body-simulation to develo a dynamic model of the multile-degree-of-freedom ieo interface system under consideration of the electromechanical couling effects. Furthermore, the simulation of the actual alication scenario will be ossible by integration of dynamic models of the couled structures. This allows for simulation of the admittance monitoring system more comlex alication scenarios. When it is ossible to model realistic damage scenarios and redict the influences on the arameters of the adative filter, a detailed damage diagnosis can be imlemented using suitable algorithms. The monitoring method will have to be examined in a fatigue test of the actuator system for further imrovements, and at last tests in the actual alication will take lace. REFERENCES 1

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