Active vibration control using optimal LQ tracking system with additional dynamics

Transcription

1 International Journal of Control Vol. 78, No. 15, 15 October 2005, Active vibration control using optimal LQ tracking system with additional dynamics T. NESTOROVIC -TRAJKOV*, H. KÖPPE and U. GABBERT Institut fu r Mechanik, Otto-von-Guericke Universita t Magdeburg, Germany (Received 30 September 2004; in final form 10 May 2005) This paper focuses on the vibration suppression task using an appropriate controller design for active control of structures with distributed piezoelectric actuators and sensors. The problem arises from the need to control undesired vibrations caused by disturbances or excitations acting upon a structure in an efficient and at the same time a simple way. A special class of disturbances/excitations (periodical, with frequencies equal or near to the eigenfrequencies of the controlled structure) may cause undesired resonant states. In order to reject such disturbances and suppress vibrations in the presence of excitations an optimal LQ controller based on tracking systems with additional dynamics is proposed for the vibration control problem. The controller was tested in the presence of excitations with different frequencies. Controller design is model-based, where for the numeric modelling of the structure the finite element approach was used. Besides, subspace-based model identification was used as well. Controller design, testing and implementation were performed on the funnel-shaped shell structure, the inlet part of the magnetic resonance tomograph. Simulation results as well as the real-time implementation of the controller as a part of the Hardware-in-the-Loop system show considerable vibration suppression in the presence of excitations and confirm the efficiency of the controller. 1. Introduction Development of actively controlled structures involves many subsequent and complex steps. This paper focuses on the controller design problem for the active control of structures with distributed piezoelectric actuators and sensors with the aim of vibration suppression. The problem arises from the need to control undesired vibrations caused by disturbances or excitations acting upon a structure in an efficient and a simple way. Overall design of active (smart) structures represents a broad and complex field with many directions of development where, besides modelling, controller design for a specific control purpose plays an important role. Proposed controller design technique is aimed at vibration suppression but can be extended to other control problems *Corresponding author. tamara.nestorovic@mb.unimagdeburg.de in smart structures (like acoustic problems) as well. A special class of disturbances/excitations (periodical, with frequencies equal or near to the eigenfrequencies of the controlled structure) may cause undesired resonant states. In order to reject such disturbances and suppress vibrations in the presence of excitations an optimal LQ controller based tracking system with additional dynamics is proposed for the vibration control problem. Design of an optimal LQ or a feedback-gain controller has been addressed in literature as a means used in solving active control problems (Lim et al. 1999, Rao and Sana 2001, Gabbert et al. 2003, Chandiramani et al. 2004, Seeger 2004). The novel approach to optimal control presented in this paper combines a tracking system with additional dynamics (Vaccarro 1995) with the optimal LQ controller (Ogata 1995, Vaccaro 1995, Franklin et al. 1998) and Kalman filter. This approach has been successfully used for control of vibration in active structures with distributed piezoelectric actuators International Journal of Control ISSN print/issn online ß 2005 Taylor & Francis DOI: /

2 Active vibration control 1183 and sensors in the sense of considerable vibration magnitudes suppression. The advantage of the proposed method can be seen through the better vibration magnitudes reduction in the steady state compared with the standard feedback and optimal LQ controller owing to the additional dynamics incorporated in the overall design model as a part of the compensator in cascade combination with the controlled structure. The controller is especially aimed at vibration suppression in the presence of periodic excitations and disturbances with frequencies corresponding to the eigenfrequencies of the controlled structure, because they are responsible for possible resonant states, which should be prevented. An a priori knowledge about such disturbances or excitations reflecting real operating conditions is available from the modal analysis of the structure under investigation, which can be performed numerically or experimentally. An optimal LQ tracking system with additional dynamics was tested on a funnel-shaped shell structure which represents the inlet part of the magnetic resonance image (MRI) tomograph used in medical diagnostics. Since the proposed controller design is model based, as a starting point for the control system development a state space model was used, which was obtained using two approaches: (i) finite element (FE) based modelling using the FE analysis software COSAR Õ (COSAR 1992) and modal truncation and (ii) subspace based identification (Viberg 1995, Van Overschee and De Moor 1996). Controller implementation showed comparable and good results in both cases confirming at the same time the feasibility of the numeric model obtained using the FE approach. Since the model-based optimal controller design assumes state feedback, a Kalman filter was used for the modal states estimation in the case when the controller design is based on the numeric FE modelling and modal truncation and for the estimation of the state variables of the identified model. The influence of the actuator/sensor placement was also considered and optimal placement at specified regions was confirmed by simulation and experimental results. For the real-time implementation and testing of the controller Hardware-in-the-Loop (HiL) system with dspace Õ was used incorporating the funnel inlet of the MRI tomograph as a real structure. Besides the vibration control proposed controller opens the possibility of extended use in solving acoustic problems in a similar manner depending on the frequency ranges of interest aimed for control. 2. State space model for the controller design As a starting point for the controller design a state space representation of the controlled structure is used. General continuous-time or discrete-time form of the state space model used for the model based controller design can be derived using the FE approach or the subspace based identification State space model obtained using the FE approach As a result of the FE analysis, behaviour of a structure approximated by an arbitrary number of finite elements can be described with assembled equations of motion in a semi-discrete form (Gabbert et al. 2002, Nestorovic - Trajkov et al. 2003a) M q þ D d _q þ Kq ¼ F, where M, D d and K are the mass matrix, the damping matrix and the stiffness matrix, respectively. The total load vector F is divided into the vector of the external forces F E and the vector of the control forces F C F ¼ F E þ F C ¼ EwðtÞþ BuðtÞ, where the forces are generalized quantities also including electric charges. Vector w(t) represents the vector of external disturbances and u(t) is the vector of the controller influence on the structure. Matrices E and B describe the positions of the forces and the control parameters in the finite element structure, respectively. As a convenient procedure for the state space model obtaining modal truncation is adopted, since the high order of the FE model represented by equation (1) is not suitable for the controller design and the reduction of the model order is required. A decoupled system of equations (3) in modal coordinates z is obtained by performing ortho-normalization with ( T m M( m ¼ I and ( T m K( m ¼ :, where the modal matrix ( m and the spectral matrix : are obtained from the solution of the linear eigenvalue problem for (1). In decoupled system of equations ð1þ ð2þ z þ D_z þ :z ¼ ( T m F, ð3þ D ¼ ( T m D d( m represents the modal damping matrix. Generalized displacements q are related to modal coordinates z by qðtþ ¼( m zðtþ: In the modal truncation procedure the modal displacement vector z is partitioned and only a part z r corresponding to selected eigenmodes of interest for the ð4þ

3 1184 T. Nestorovic -Trajkov et al. control is retained. Introducing the modal reduced state vector xðtþ ¼ z rðtþ, ð5þ _z r ðtþ the modal reduced model is obtained in the state space form _x ¼ 0 I x þ : r D r 0 ( T uðtþþ r B 0 ( T r E wðtþ, where matrices : r, D r and ( r, and are obtained from the appropriate partition of the matrices :, D and ( m respectively. Written in a standard state space form the state equation of the model becomes _xðtþ ¼AxðtÞþBuðtÞþEwðtÞ: Using a similar procedure a state space formulation of the output equation is obtained y ¼ CxðtÞ þ DuðtÞ þ FwðtÞ, which assumes in a general case the influence of the control and external inputs on the outputs. Discrete-time equivalent of the continuous-time state space model (7), (8) is used as a starting point for the controller design where x½k þ 1Š ¼(x½kŠþ!u½kŠþew½kŠ, y½kš ¼Cx½kŠþDu½kŠþFw½kŠ, ( ¼ e AT,! ¼ ð T 0 e A Bd, and T is the sampling time. e ¼ ð T 0 e A Ed ð6þ ð7þ ð8þ ð9þ ð10þ 2.2. Subspace identification of the state space model For the identification of the state space model from the measured input-output data the subspace-based identification is used (Viberg 1995, Van Overschee and De Moor 1996, McKelvey et al. 2002). General discrete-time combined deterministic-stochastic form of the model to be identified corresponds to modified form of equations (9) x½k þ 1Š ¼(x½kŠþ!u½kŠþw½kŠ, y½kš ¼Cx½kŠþDu½kŠþv½kŠ, ð11þ where w[k] and v[k] denote the process and the measurement noise, respectively. Input-output measurement data are organized in the form of appropriate input and output block Hankel matrices (Van Overschee and De Moor 1996). Input Hankel matrix is defined as U ¼ U 02i 1 j 2 ¼ 6 4 u 0 u 1 u 2 u j 1 u 1 u 2 u 3 u j u i 1 u i u iþ1 u iþj 2 u i u iþ1 u iþ2 u iþj 1 u iþ1 u iþ2 u iþ3 u iþj u 2i 1 u 2i u 2iþ1 u 2iþj 2 3 : 7 5 ð12þ Output measurement matrix Y is defined in a similar way and the measurement data are organized in the form of the following input-output relation (Viberg 1995) Y½kŠ ¼! x½kšþ( U½kŠ, ð13þ where! represents the observability matrix for the system (11), ( is the Toeplitz matrix (Viberg 1995) of impulse responses from u to y 2 3 D 0 0 C D 0 ( ¼ ð14þ C( 2! C! D and is a specified number greater than the state dimension but much smaller than the data length. For a deterministic case (Franklin et al. 1998) the problem is simplified to determining! and ( by computing the singular value decomposition (SVD) of U in the first step " # U ¼ PRQ T Q T u1 ¼½P u1 P u2 Š½R u 0 Š : ð15þ Q T u2 If matrix U has dimension m n and rank r, then the partition in (15) is performed as follows: P ¼ p 1 p r j p rþ1 p m ¼ Pu1 P u2 ð16þ Q ¼ q 1 q r j q rþ1 q n ¼ Qu1 Q u2 ð17þ

4 Active vibration control 1185 where p i are called the left singular vectors of U. It can be shown that they are eigenvectors of UU T. Vectors q i are the right singular vectors of U. It can be shown that they are eigenvectors of U T U (Vaccaro 1995). Multiplying (13) by Q u2, matrix! can be determined from a SVD of YQ u2. Then matrix C is obtained as the first row (in a sense of a block-row) of the observability matrix!, and matrix ( is calculated from:! ¼! ( applying pseudo inverse, where! is obtained by dropping the last row of!. Matrix! represents the matrix obtained by dropping the first row of!. For the calculation of and D matrices, equation (13) is multiplied by the pseudo inverse of U on the right and by P T u2 from (15) on the left. Thus the equation is reduced to P T u2 YU 1 ¼ P T u2 ( : ð18þ After rearranging, the equation (18) can be solved for and D using the least squares. In this way the system parameters in the form of state space matrices of the model (11) are identified using the subspace based identification method. 3. Optimal LQ tracking system with additional dynamics Proposed control methodology combines tracking system with additional dynamics, optimal LQ controller and Kalman filter. The idea of introducing additional dynamics in order to make the system output track a given reference input or to reject a disturbance/ excitation relies on tracking zero-input trajectories (Vaccaro 1995, Franklin et al. 1998). Thus the tracking control problem converts to a regulation problem of tracking zero-input trajectories of a design model augmented with additional dynamics. The advantage of a tracking system with additional dynamics can be viewed in the light of the fact that once the knowledge of the specified reference input and/or the excitation/ disturbance is incorporated in additional dynamics, the designed control system can handle both types of inputs. It will be shown through the control system design procedure. Design of the optimal LQ tracking system with additional dynamics assumes inputs to the system (reference inputs and excitations/disturbances) which can be represented in the form of a rational transfer function (step, ramp, sinusoidal and exponential signals). In the vibration control of smart structures sinusoidal excitations play a significant role in the controller design procedure. The excitation frequency corresponding to an eigenfrequency of the structure under control or to a combination of eigenfrequencies can be regarded as a possible worst study case due to the resonance. The rejection of such excitations/disturbances is therefore significant for the vibration control of smart structures. For defining additional dynamics disturbance/ excitation and reference input are assumed to be the outputs of the models described with the following state space equations, respectively. Disturbance: _z d ðtþ ¼A d z d ðtþ, fðtþ ¼c d z d ðtþ: ð19þ Reference input: _z r ðtþ ¼A r z r ðtþ, rðtþ ¼c r z r ðtþ: ð20þ The following interpretation is implied by the theorem in (Vaccaro 1995, p. 332). For a system described by a discrete-time linear state space realization (,, C), see equation (11), and the system input (reference input or disturbance/excitation) expressed in the form of a rational transfer function R(z) ¼ n(z)/d(z) if at least one of input poles z i is not an eigenvalue of, then additional dynamics must be used to have a tracking system with zero steady-state error. The proof can be found in (Vaccaro 1995, p. 332). Introducing additional dynamics an augmented design plant is obtained. The regulator designed for the augmented plant achieves than exact tracking. In the case of multiple input poles additional dynamics are defined on the basis of the sets r and d, which contain the reference input and disturbance/excitation poles respectively together with their multiplicities r ¼ ð r1, m r1 Þ, ð r2, m r2 Þ,..., d ¼ ð d1, m d1 Þ, ð d2, m d2 Þ,..., ð21þ where ri is the eigenvalue of the A r matrix (20) with the multiplicity m ri and di is the eigenvalue of the A d matrix (19) with the multiplicity m di. The set is defined as the union of the sets d and r. If a common eigenvalue appears in both the sets d and r, only the one with the higher multiplicity is included in the set. Eigenvalues i in the set define the poles of additional dynamics. An important property of the control systems with additional dynamics is that the system designed to track a specified reference input rejects at the same time the disturbance of the same type and vice versa. Even if it is not required from the system to reject disturbances, by the nature of its structure it will reject the disturbances/excitations which are defined with the same poles as the specified reference input. This results from the definition of the set. If no disturbance poles are specified, the set d is an empty set. Then the set coincides with the set r since it is obtained as a union of r and the empty set. The same set is obtained if d ¼ r, i.e. if it is required that the system

5 1186 T. Nestorovic -Trajkov et al. rejects the disturbances/excitations of the same type as the reference input. Similar reasoning implies that the tracking systems designed in this way operate well in cases when the reference input and disturbance/ excitation switch the roles. The sets r and d would then reverse the elements, but their union remains the same. Mapping the poles i from the set into z-domain, the polynomial (z) in the denominator of the additional dynamics transfer function is determined in the following way: ðzþ ¼ Y i z e it mi def ¼ z s þ 1 z s 1 þ 2 z s 2 þ s, ð22þ where s ¼ P m i is the total number of the poles and m i is the multiplicity of the pole i. The additional dynamics system is then defined by the transfer function H a ðzþ ¼ zs ðzþ ¼ z s z s þ 1 z s 1, þ þ s ð23þ and implemented as a part of the compensator in a cascade combination with the plant. In this way the transfer function of the design plant H a (z)h(z) (cascade combination of the plant and additional dynamics) has poles which are at the same time eigenvalues of the state matrix for the design plant in z-domain. In order to represent the state space model of the additional dynamics in an observable canonical form, the nominator of the transfer function H a (z) is chosen as z s. Matrices a and a of the additional dynamics are then determined from the coefficients of the polynomial (z) a ¼ ,. a ¼ : s s 1 s s ð24þ In the case of multiple-input multiple-output (MIMO) systems additional dynamics has to be replicated in q parallel systems (once per each output), where q is the number of outputs. Replicated additional dynamics are then formed as a block matrix with a, a on the diagonal: def ¼ diag ð a,..., fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} a Þ, def ¼ diag ð a,..., a Þ: fflfflfflfflfflffl{zfflfflfflfflfflffl} ð25þ q times q times The discrete-time design model ( d, d ) is formed as a cascade combination of additional dynamics and the discrete-time plant model (, ) and expressed as a state-space formulation x d ½k þ 1Š ¼( d x d ½kŠþ! d u½kš, d ¼ 0 C, d ¼ 0 ð26þ, x d ¼ x½kš : ð27þ x a ½kŠ Matrices ( and! in the design model realization ð( d,! d Þ denote additional dynamics of a single-input single-output system ð( a,! a Þ or replicated additional dynamics ð, Þ of a multiple-input multiple-output system. Dimension of design model is n þ qs where n is the order of the plant model, q the number of outputs and s the order of the additional dynamics. The control system is designed for the realization ( d, d ). The feedback gain matrix L of the control law u½kš ¼ Lx d ½kŠ ð28þ is partitioned into submatrices L 1 and L 2 formed from the first n and last q s columns of the matrix L, respectively L ¼½L 1 L 2 Š: ð29þ Thus the feedback gain matrix L 1 corresponds to the state variables of the controlled structure, while the feedback gain matrix L 2 pertains to the rest of the state variables in the design state vector x d introduced by additional dynamics. Control system with additional dynamics implemented as a part of the compensator is represented in figure 1. Design of the controller involves the estimation of the state variables. If the state space model of the structure is obtained using the FE approach, state variables are modal variables which are not measurable and their estimation is therefore necessary. On the other hand in the identified state space model state variables are not measurable either, which imposes the need for their estimation. For this purpose the Kalman filter is employed (Preumont 1997, Franklin et al. 1998, Nestorovic - Trajkov et al. 2003a, b). Kalman filter design is based on the assumed plant model in the form (11). The process and measurement noise w and v are assumed to be white noise with the zero mean. The covariances of the process and measurement noise are denoted as Eðww T Þ¼Q w and Eðvv T Þ¼R v, respectively. Then the

6 Active vibration control 1187 Figure 1. Control system with additional dynamics implemented as a part of the compensator. Kalman estimator is defined by the following equations ) ^x½kš ¼x½kŠþL est ½kŠðy½kŠ Cx½kŠÞ ð30þ x½kš ¼( ^x½k 1Šþ!u½k 1Š, where the Kalman gain matrix is L est ½kŠ ¼P½kŠC T R 1 v and 9 P½kŠ ¼M k ½kŠ M k ½kŠC T CM k ½kŠC T >= 1CMk þ R v ½kŠ >; M k ½k þ 1Š ¼(P½kŠ( T þ eq w e T : ð31þ ð32þ Matrices P and M k are determined by solving equations (32). For the Kalman filter implementation the initial conditions for x½0š and M k ½0Š are required. These initial values represent the a priori estimate of the state and of the accuracy of this a priori estimate. In certain cases some test data can be used as a support for the choice of these initial values, but it is not typical. Usually it is assumed that the components of x½0š contained in y are equal to the first measurement and the remaining components are equal to zero. In a similar way the components of M k corresponding to measured component can be set to R v and the remaining components can be set to a high value. Besides the knowledge of the initial value for M k, the a priori knowledge of the process noise magnitude Q w and the measurement noise magnitude R v is also required. The value for R v in a given actual design problem can be chosen based on the sensor accuracy. Here it should also be noted that the assumption about the process noise being the white noise is introduced in order to simplify solving of the optimization problem. Physically Q w is often associated with unknown disturbances. In a case when a random disturbance is a colored noise (time correlated), it can be accurately modelled by augmenting with a coloring filter which converts a white noise input into time correlated noise (Franklin et al. 1998). Due to complexity it is often not done in practice. Rather than that the disturbances are assumed to be white and the noise intensity is adjusted to give acceptable results in the presence of expected disturbances. Incorporating the Kalman filter in the controller design, besides the necessary state estimation, a reduction of the observation spillover (the influence of the residual modes on the sensor output) can be achieved (Preumont 1988, 1997). Feedback gain matrix L in the control law (28) is determined designing the optimal LQ regulator for the design model represented by the realization ( d, d ), equation (26). The feedback gain matrix L has to be determined in such a way that the control law (28) minimizes the performance index J ¼ 1 2 X 1 k¼0 x d ½kŠ T Qx d ½kŠþu½kŠ T Ru½kŠ ð33þ respecting the constraint (26) where Q and R are symmetric, positive-definite matrices. Optimal controller design problem is solved minimizing the cost function (33) under constraint (26) using the method of Lagrange multipliers (Vaccaro 1995, Franklin et al. 1998). The choice of the weighting matrices Q and R in the performance index is designer dependant and is based on the relative importance of the various states and controls. The trade-off between the control effort and the system response determines the choice of the weighting matrices. In general, the weighting matrices are chosen in such a way that large input signals are penalized by increasing the value of the matrix R and faster response of appropriate state variables is achieved by increasing the values of appropriate elements in the weighting matrix Q. With all the steps of the optimal LQ controller design, including defining additional dynamics, estimator design and finally optimal control law design, the procedure

7 1188 T. Nestorovic -Trajkov et al. Figure 2. Optimal LQ tracking system with additional dynamics and state estimator. Figure 3. Magnetic resonance tomograph. of the model based optimal control design can be concluded resulting in a control system which can successfully be used for the vibration suppression in smart structures. Optimal LQ tracking control system with additional dynamics and Kalman estimator is represented in figure Implementation of the controller on the MRI tomograph Optimal LQ controller with additional dynamics is implemented for the vibration control (in the sense of vibration magnitudes reduction) of the funnel-shaped shell structure, the inlet part of the Siemens Õ MRI tomograph (figure 3). The aim is reduction of transmitted vibrations from the cylindrical body of the tomograph to the funnel-shaped inlet using piezoelectric actuators and sensors. The vibration control problem of a cylindrical shell used in MRI equipment was addressed in (Qiu and Tani 1995, Tani et al. 1995, Qiu and Tani 1996). This paper represents original results for the vibration control of the complex funnelshaped structure, which is excited by transmitted vibrations from the cylindrical-shell-shaped body of the tomograph. Control of these transmitted vibrations contributes to the overall vibration suppression

8 Active vibration control 1189 Figure 4. Funnel-shaped inlet of the MRI tomograph with actuator/sensor placement. Figure 5. Scheme of the experimental rig: HiL system with dspace Õ and the funnel. of the medical device and therefore plays an important role. The experimental rig includes the funnel-shaped controlled structure with piezoelectric actuators and sensors glued to the surface of the funnel and placed as shown in figure 4. They are denoted as 1L, 2L, 3L for the left-hand side actuators and sensors and 1R, 2R, 3R for the right-hand side ones. The funnel is a part of the Hardware-in-the-Loop system with dspace Õ implemented as represented by the scheme of the control experimental rig in figure 5. The periodic excitation is generated by Matlab/Simulink Õ and dspace Õ and exerted by the shaker acting at defined point on the funnel, as shown in figure 5. Applying the FE based approach for the analysis of the funnel behaviour and numeric model development (using the finite element software COSAR Õ ), the eigenfrequencies were determined and on the basis of the comparison with results of the experimental modal analysis, a reduced order state-space model of the funnel was adopted for the control of the funnel eigenmodes in the frequency range up to 35 HZ, where the eigenfrequencies of interest are f 1 ¼ Hz, f 2 ¼ Hz and f 3 ¼ Hz. Comparison of the

9 1190 T. Nestorovic -Trajkov et al. modal analysis results shows good agreement between the eigenvalues obtained with the finite element software COSAR Õ and experimentally. For the controller implementation purposes the state space model of the funnel was identified using the subspace based approach as well Optimal LQ controller with additional dynamics based on the FE model The modally reduced state space model of the funnel of order 20 is initially obtained on the basis of the first 10 eigenmodes. Since the control is aimed at the first three lowest eigenfrequencies, the state space model is further reduced to order 6 resulting in a controllable and observable realisation necessary for the controller implementation. Different periodic excitations were considered. The results are represented for the actuator/sensor pair A2R S1R. Control system is first designed assuming the sine excitation with the frequency f 1. The design model order is then equal to eight and the weighting matrices for the optimal LQ tracking system are selected as Q ¼ I 88 and R ¼ 10. In this way the first eigenfrequency is controlled. Time response of the sensor S1R and the control effort obtained experimentally with this control system in the presence of the sine excitation with frequency f 1 is shown in figure 6. The period without control is clearly indicated by the zero control input and obviously greater vibration magnitudes of the sensor response. The same control system exhibits behavior presented in figure 7 when used in the presence of the excitation obtained as a sum of the sinusoids with frequencies f 1, f 2, f 3. The control system is now designed in order to control simultaneously the first three eigenfrequencies assuming the excitation as a sum of the sinusoids with frequencies f 1, f 2, f 3. Order of the design model is 12 and the weighting matrices are selected as Q ¼ I 1212 and R ¼ 10. According to expectations this control system provides better vibration suppression (figure 8) in comparison with figure 7 using a slightly greater control effort. The magnitude of the Fast Fourier Transform (FFT) of the sensor signal is taken as the measure of the vibration suppression in the frequency domain. Results obtained from the simulated and measured responses are presented in figure 9. Finally, the results in the time domain when designed control system is used to control the first and the second eigenfrequency as well as only the first eigenfrequency are presented in figures 10 and 11 respectively. Presented results for the considered single-input single output system modeled using the FE approach show efficiency of the controller through the vibration magnitudes reduction. According to expectations the control system designed taking into account three Figure 6. Sensor response and control in the presence of the sine excitation with the frequency f 1 (FE model based control system designed to control the first eigenfrequency). Figure 7. Sensor response and control with excitation containing three eigenfrequencies (FE model based control system designed to control the first eigenfrequency).

10 Active vibration control 1191 different frequencies of the periodic excitation performs better in the presence of excitations obtained as a sum of three sinusoidal signals then the controller with additional dynamics which takes into account only one eigenfrequency in the presence of the same excitation Optimal LQ controller with additional dynamics based on the identified model This section presents the implementation results for the optimal LQ controller with additional dynamics when the state space model used for the controller design was obtained using the subspace based identification. For the actuator/sensor pair A2R S1R a good approximation of the measured frequency response function is obtained with the identified state space model of the order 60. Vibration magnitudes reduction and control in time domain for the actuator/sensor pair A2R S1R are represented in figures 12 and 13. Weighting matrices for the optimal LQ controller design are Q ¼ I and R ¼ 100. For the multiple-input multiple-output (MIMO) control, two cases were considered: one actuator two Figure 8. Sensor signal and control with excitation containing three eigenfrequencies (FE model based controller designed for simultaneous control of the first three eigenfrequencies). Figure 9. FFT magnitude of the sensor signal: (a) simulated; (b) measured.

11 1192 T. Nestorovic -Trajkov et al. Figure 9. Continued. Figure 10. Sensor signal and control with excitation containing the first two eigenfrequencies (FE model based controller designed for simultaneous control of the first three eigenfrequencies). Figure 11. Sensor signal and control with excitation containing the first eigenfrequency (FE model based controller designed for simultaneous control of the first three eigenfrequencies).

12 Active vibration control 1193 Figure 12. Actuator/sensor pair A2R S1R: sensor response and control in the presence of the sine excitation with frequency f 1. Figure 13. Actuator/sensor pair A2R S1R: sensor response and control with excitation containing three eigenfrequencies. sensors and two actuators two sensors. In the case one actuator A2R two sensors S1R, S2L, the model identified using subspace identification has order 50. Good agreement between the frequency responses obtained from measurements and from simulations based on the identified model was the criterion for the selection of the model order. Figure 14 represents the magnitudes of the frequency response in the range up to 35 HZ corresponding to the frequency range of interest. Dashed lines represent frequency response obtained as a ratio of the Fast Fourier Transforms (FFT) of the measured output-to-input (sensorto-actuator) signals. Solid lines represent the same ratio, but obtained from the simulated responses based on the identified model. Based on the identified model, optimal LQ tracking system with additional dynamics is designed. The weighting matrices of the LQ controller are: Q ¼I and R ¼ 100. Additional dynamics takes into account the first eigenfrequency of the funnel. Designed system successfully reduces oscillation amplitudes at sensors locations in the presence of the sine excitation with the frequency f 1 as well as in the presence of the excitation containing the first three eigenfrequencies (figure 15). Optimal LQ tracking system with additional dynamics was also designed in the MIMO case for the actuators A2L, A2R and sensors S1R, S2L. Additional dynamics, which takes into account the first eigenfrequency of the funnel, is replicated two times due to two outputs. Control system is designed using the weighting matrices Q ¼ I and R ¼ I Time responses of the sensors and control signals for two excitation cases (sine excitation with the frequency f 1 and the excitation containing the first three eigenfrequencies) are represented in figure 16. In this subsection the state space model used for the controller design was obtained using the subspace based identification method. The method is characterized by efficient model development based on the measured input and output signals resulting in a state space realization which is convenient for the controller design. Designed controller in turn performs successful vibration control shown by the obvious vibration magnitudes reduction in comparison with the uncontrolled case. 5. Conclusions In this paper the optimal LQ tracking system with additional dynamics was proposed as a solution for the control of vibrations caused by excitations or disturbances, in the sense of the vibration magnitudes suppression. The controller is aimed at active vibration control of structures with distributed piezoelectric actuators and sensors, but it can be widely used for different control tasks. In vibration control a special class of periodic excitations/disturbances with frequencies corresponding to the eigenfrequencies of

13 1194 T. Nestorovic -Trajkov et al. Figure 14. Frequency responses of the identified model up to 35 Hz: (a) actuator/sensor pair A2R S1R; (b) actuator/sensor pair A2R S2L.

14 Active vibration control 1195 Figure 15. Actuator A2R sensors S1R, S2L: sensor responses and control in the presence of the sine excitation f 1 (left) and with excitation containing three eigenfrequencies (right). Figure 16. Actuators A2L, A2R sensors S1R, S2L: sensor responses and control in the presence of the sine excitation f 1 (left) and with excitation containing three eigenfrequencies (right). the controlled structure is of interest, due to the need for the exclusion of the resonant states. Such excitations/ disturbances can be successfully handled using the proposed control strategy which combines tracking system with additional dynamics, optimal LQ regulator and Kalman filter. Incorporating the a priori knowledge about the frequency of the excitation (eigenfrequency of the structure) in the design model for the optimal LQ

15 1196 T. Nestorovic -Trajkov et al. controller design in terms of the excitation poles used to form additional dynamics, control problem reduces to a regulation problem for the design model. Since the controller design is model based, two approaches to state space model obtaining have been considered in this paper: FE approach based modeling and subspace identification. The first approach is convenient in the early development phases when no real structure is available and controller development is based upon the FE modeling, simulation and modal analysis. When the real structure is at disposal, the subspace based identification is convenient for the state space model development. Proposed controller is tested on the funnel shaped structure, the inlet part of the MRI tomograph. Using the state space models obtained by both mentioned approaches the optimal LQ tracking system with additional dynamics was designed and tested with periodic excitations having the frequencies corresponding to the eigenfrequencies of the funnel in the frequency range up to 35 Hz. The HiL structure with dspace Õ was used for the real-time simulations. The results showed considerable reduction of the vibration magnitudes during the time when the controller is switched on. It was noted that controller designed with additional dynamics which takes into account the first eigenfrequency can be used for the vibration magnitudes reduction in the presence of excitations obtained as a sum of three sinusoidal signals with frequencies corresponding to the eigenfrequencies of the funnel. Nevertheless, much better results are obtained in the presence of such excitations when the additional dynamics takes into account all the frequencies present in the excitation signal. In comparison with feedback control or the LQ regulator without additional dynamics implemented as a part of the design model, proposed controller results in better vibration suppression. Closely related to this issue of better vibration suppression is one of the main contributions of this work. Although the use of optimal LQ controllers is a well-known topic in the control theory, this work treats the problem of the optimal LQ controller design in combination with the additional dynamics implemented as a part of the design model. According to authors knowledge, such an approach, which combines at the same time additional dynamics, optimal LQ control and Kalman estimator, has not been treated and implemented in the literature, especially not for the vibration suppression applications yet. The advantage of the approach can be seen from the fact that the incorporated additional dynamics which take into account information about the most critical resonant states, takes care for better vibration suppression in comparison with standard LQ controller in the presence of the excitations/disturbances which can cause such resonant states. Further, the paper shows the feasibility of the numerically obtained state space model for the controller design, which has been demonstrated not only through simulations, but also experimentally. The fact that the state space model obtained using the FE approach well represents the object under investigation, justifies the application of the LQ control as a model based controller design technique. Besides the advantage that based on the model the controlled system behavior can be investigated in the early phases before the prototype has been built, a relatively simple control law (in terms of a state feedback) represents another advantage, since it is known that practical applications often require a straightforward way to the solution of vibration suppression problems. Controller design requires estimation of the state variables, which are in case of the FE based state space model modal coordinates and therefore cannot be measured. In the case of model identification, the states of the identified model are not measurable either. Nevertheless due to its relative simplicity the application of the LQ controller in combination with the Kalman estimator is viewed by the authors as an efficient and advantageous control technique for the considered class of vibration suppression problems (with periodic excitations especially) as opposed to some other alternatives which would not require state estimation (like for example direct model reference adaptive control or adaptive filters), but on the other hand do require estimation or even online identification of the model parameters, through often time consuming estimation algorithms. Kalman estimator in turn represents a means for overcoming the spillover problem (Preumont 1997, 1988) which may occur due to truncation of the higher modes in the model order reduction procedure. Experimental implementation on a complex structure demonstrated practically the vibration suppression and the effectiveness of the proposed control technique, which makes the experimental results attractive since in this area validations have mostly been performed by simulation only. Further application is directed to implementation of the controller for solving acoustic problems. In such cases the acoustic fluid interacts with the smart structure itself, which has to be taken into account in the control design technique. It can be shown that not all vibration modes of the structure have the same influence to the sound pressure in the acoustic fluid. So, it is required to estimate such behavior or to take into account additional sensors, such as microphones in the surrounding fluid. Based on the finite element method or by model identification a coupled

16 Active vibration control 1197 fluid-electro-mechanical model can be developed, which can be used, in a similar manner as presented in the paper, for a model based controller design approach (Lefe` vre and Gabbert 2004). The feasibility of the developing models in combination with the proposed control has to be proven experimentally through the further investigations. On the other hand, practical requirements impose the need for solving the acoustic problem, e.g. in MRI tomography, which is still the subject of ongoing researches as an attempt to reduce the noise accompanying the medical treatment of patients. Vibration suppression taking into account the influence of the surrounding fluid is a first step towards the solution of the noise reduction and further investigations in this field are required. Acknowledgements This work has been partially supported by the postgraduate program of the German Federal State of Saxony-Anhalt. It was motivated and supported by a cooperation with the Siemens Õ company in the frame of the German industrial Research project Adaptronik supported by the German Ministry for Education and Research. These supports are gratefully acknowledged. References N.K. Chandiramani, L. Librescu, V. Saxena and A. Kumar, Optimal vibration control of a rotating composite beam with distributed piezoelectric sensing and actuation, Smart Materials and Structures, 13, pp , COSAR, COSAR General Purpose Finite Element Package: Manual (FEMCOS mbh Magdeburg), 1992, see also G.F. Franklin, J.D. Powell and M.L. Workman, Digital Control of Dynamic Systems, 3rd ed., Menlo Park, California: Addison-Wesley Longman, Inc., U. Gabbert, H. Köppe, T. Nestorović-Trajkov and F. Seeger, Modelling, simulation and optimal design of lightweight structures, in Proceedings of the Adaptronic Congress, Wolfsburg, Germany, April 1 3, 2003, CD. U. Gabbert, T. Nestorovic -Trajkov and H. Köppe, Modelling, control and simulation of piezoelectric smart structures using finite element method and optimal LQ controller, Facta Universitatis, Series Mechanics, Automatic Control and Robotics, 3, pp , J. Lefèvre and U. Gabbert, Simulation of piezoelectric smart lightweight structures by the finite element method for acoustic control, PAMM Proc. Appl. Math. Mech., 4, pp , DOI /pamm Y.-H. Lim, S.V. Gopinathan, V.V. Varadan and V.K. Varadan, Finite element simulation of smart structures using an optimal output feedback controller for vibration and noise control, Smart Materials and Structures, 8, pp , T. McKelvey, A.J. Fleming and S.O.R. Moheimani, Subspace based system identification for an acoustic enclosure, ASME Transactions on Vibration and Acoustics, 124, 2002, newcastle.edu.au/lab/. T. Nestorović-Trajkov, U. Gabbert and H. Ko ppe, Controller design for a funnel-shaped smart shell structure, Facta Universitatis, Series mechanics, automatic control and robotics, Special issue: Nonlinear Mechanic, Nonlinear Sciences and Applications II, 3, pp , 2003a. T. Nestorovic -Trajkov, H. Ko ppe and U. Gabbert, Comparison of controller design approaches from a vibration suppression point of view, in Proceedings of SPIE, C. Ralph, Ed., Smith SPIE s 10th Annual International Symposium on Smart Structures and Materials, Conference on Modeling, Signal Processing and Control, San Diego, CA, 2 6 March, 2003b, Vol. 5049, pp K. Ogata, Discrete-time Control Systems, 2nd ed., Cliffs, New Jersey: Prentice Hall, Englewood, A. Preumont, Spillover alleviation for nonlinear active control of vibration, AIAA J. Guidance Control Dyn., 11, pp , A. Preumont, Vibration Control of Active Structures: An Introduction, Dordrecht, Boston, London: Kluwer Academic Publishers, J. Qiu and J. Tani, Vibration control of a cylindrical shell used in MRI equipment, Smart Mater. Struct., 4, pp. A75 A81, J. Qiu and J. Tani, Vibration suppression of a cylindrical shell using a hybrid control, method, J. Intell. Mater. Syst. Struct., 7, pp , V.S. Rao and S. Sana, An overview of control design methods for smart structural system, in Proceedings of SPIE, Smart Structures and Materials 2001: Modeling, Signal Processing and Control in Smart Structures, V.S. Rao, Ed., 2001, Vol. 4326, pp F. Seeger, Simulation und optimierung adaptiver schalenstrukturen, Fortschritt-Berichte VDI Reihe 20, Nr. 383, Du sseldorf: VDI Verlag, J. Tani, J. Qiu and M. Hidehisa, Vibration control of a cylindrical shell using piezoelectric actuators, J. Intell. Mater. Syst. Struct., 6, pp , R.J. Vaccaro, Digital Control: A State-space Approach, Singapore: McGraw-Hill, Inc., P. Van Overschee and B. De Moor, Subspace Identification for Linear Systems: Theory, Implementation, Applications, Boston: Kluwer Academic Publishers, M. Viberg, Subspace-based methods for the identification of linear time-invariant systems, Automatica, 31, pp , 1995.