adaptive model predictive vibration control with state

ADAPTIVE MODEL PREDICTIVE VIBRATION CONTROL WITH STATE AND PARAMETER ESTIMATION USING EXTENDED KALMAN FILTERING Gergely Takács, Tomáš Polóni and Boris Rohal’-Ilkiv Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering, Institute of Automation, Measurement and Applied Informatics, Nám Slobody 17, 812 31 Bratislava, Slovakia, e-mail: [email protected] This paper presents an adaptive-predictive vibration control system using extended Kalman filtering for the joint estimation of system states and model parameters. A fixed-free cantilever beam equipped with piezoceramic actuators is used to test the proposed control strategy. Deflection readings taken at the end of the beam are used to reconstruct position and velocity information for a second order state-space system. In addition to the system states, the dynamic model is augmented by the unknown model parameters: stiffness, damping constant and a voltage/force conversion constant; characterizing the actuating effect of the piezoceramic transducers. The states and parameters of this augmented system are estimated in real-time, using a continuous-discrete hybrid extended Kalman filter. The estimated model parameters are used to reconstruct the continuous state-space model of the vibrating system, which in turn is discretized for the predictive controller. The model predictive control algorithm generates state predictions and infinite-horizon quadratic cost prediction matrices based on the updated discrete state-space models. The resulting cost function is then minimized using quadratic programming to find the sequence of optimal but constrained control inputs, from which only the first is utilized in a form of a control voltage send to the piezo actuators, the rest is discarded and the procedure is repeated at the next sampling instant. The proposed active vibration control system is implemented and evaluated experimentally to validate the viability of the control method.

1.

Introduction

Undesirable mechanical and structural vibrations may often cause mere discomfort in humans, but in certain engineering applications can even lead to catastrophic failure or other extreme consequences. With the advent of new actuator and sensor types and the availability of cheap computing technology active vibration control (AVC) has become an important tool in managing excessive vibration levels 1,5 . When designing the algorithmic support for AVC systems a frequent assumption to make is that the controlled structure maintains its dynamic properties throughout the control procedure. This assumption enables the straightforward tuning of some controllers like positive position feedback 1 (PPF), while in model based algorithms like linear quadratic 5 (LQ) or model predictive ICSV20, Bangkok, Thailand, July 7-11, 2013

1

20th International Congress on Sound and Vibration (ICSV20), Bangkok, Thailand, 7-11 July 2013 control 12 (MPC) it allows the use of a relatively precise nominal model. However, not every AVC application fits this convenient assumption. A time-varying system behavior may simply de-tune the controller causing it to opearate sub-optimally, but it may also affect the stability of the closed-loop system. In addition to robust controller design practices, this problem is often addressed by introducing a level of adaptivity into the control loop. Possibly one of the best known and most widely used adaptive method applied for AVC and active noise control (ANC) is least mean square (LMS) feedforward control 2 where changes in system dynamics are translated into coefficient variations in an adaptive finite impulse response filter (FIR). A frequent choice of adaptive AVC encountered in literature is model reference adaptive control (MRAC) which does not require the use of explicitly defined system parameters 13 . Other possible online identification methods for AVC include subspace identification or autoregressive moving average (ARMAX) models, while variants of extended Kalman filtering (EKF) have been used for real time modal parameter diagnostics applications 14 . Instead of identifying coefficients in FIR filters or using models without explicit parametrization, in this work we apply extended Kalman filtering 6 to estimate system states and the physically interpretable parameters of a vibrating mechanical system, such as stiffness and damping coefficients. The estimated parameters are then used to reconstruct a continuous model, which is in turn discretized to create state predictions for an input constrained infinite-horizon dual-mode MPC algorithm 7,10 . The resulting adaptive EKF–MPC algorithm is then applied to a clamped aluminum cantilever beam which is equipped with piezoceramic transducers and its deformation is measured at the free end using a laser triangulation system. Let us assume that the dynamics of this beam can be modeled using a second order differential equation describing the motion of the point-mass-damper system. Furthermore, let us assume that the force contribution from the actuators can be modeled by the input voltage multiplied by a conversion constant, and that the beam is excited by an unknown external disturbance force. The aim of the adaptive EKF–MPC control algorithm then is to attempt to keep the beam position near its equilibrium using constrained inputs and despite of external disturbances, while continually adapting to possible changes in the dynamic behavior by observing the unknown system parameters in addition to the system states.

2.

Methodology

2.1

Modeling: system dynamics augmented by unknown parameters

Let us assume that the vibration dynamics of the cantilever beam can be approximated by a second order differential equation describing the motion of a point-mass-damper with an external force input 1,5 and an unknown external disturbance: m¨ q (t) + bq(t) ˙ + kq(t) = cu(t) + Fe (t)

(1)

where m (kg) is the equivalent mass of the beam and the actuators, b (Ns/m) the equivalent viscous damping coefficient, k (N/m) the equivalent stiffness, let c (N/V) represent the scalar coefficient transforming input voltage to force F (t) = cu(t) (N) and Fe (t) (N) is a disturbance. The input to this system is a voltage to the actuators u(t) (V) and the single measured output is the position of the beam q(t) (m). If we choose position and velocity as state variables (x1 (t) = q(t), x2 (t) = q(t)), ˙ it is possible to express this equation in a continuous, linear, state-space form: x(t) ˙ = Φx(t) + Γu(t) + ΘFe (t)

and

y(t) = Cx(t)

(2)

Matrix Φ is a R2×2 state transition matrix, Γ is a R2×1 input matrix and Θ is a R2×1 matrix which for the assumed model take the following form 8 : ] [ ] [ ] [ 0 1 0 0 Γ= c Θ= 1 (3) Φ= b k −m −m m m ICSV20, Bangkok, Thailand, July 7-11, 2013

2

20th International Congress on Sound and Vibration (ICSV20), Bangkok, Thailand, 7-11 July 2013 [ ] Furthermore, C = 1 0 outputs the deformation y(t) = q(t) (m), measured at a single point at the beam tip, while direct input feed-through is omitted in this case. Assuming that the equivalent mass m is known and constant but other parameters may vary with time, the system dynamics noted as Φ(t) and Γ(t) will be time variable and will be continually updated in the EKF–MPC algorithm. The continuous state x(t) describing only the system dynamics as defined in Eq. (2) can be aug[ ]T mented by a vector of unknown parameters p(t). In this case p(t) = k(t) b(t) c(t) Fe (t) is expressing the unknown and potentially changing damping, stiffness, force conversion parameter and an [ ]T external disturbance. The new augmented state therefore is xa (t) = x(t) p(t) . The system dy˙ namics augmented by the unknown parameters can be expressed as x(t) = f˜c (x(t), p(t), u(t))+ws (t) ˙ and p(t) = wp (t) where ws (t) is the system process noise and wp (t) is the parameter process noise. Even though the original model was linear in the states, the inclusion of unknown model parameters leads to a nonlinear filtering problem 6 . Combining the state and parameter equations results in x˙ a (t) = fc (xa (t), u(t))+w(t) and y(t) = h(xa (t), u(t))+v(t), where function fc represents the con[ ]T tinuous augmented dynamics, hc is a continuous measurement function and w(t) = ws (t) wp (t) expresses system and process noise. The process and measurement noise have the properties of white noise, in other words, they have a sequentially uncorrelated Gaussian distribution with a zero mean: w(t) ∼ N (0, Qf t ) and v(t) ∼ N (0, Rf ), where Qf t is a process noise covariance matrix and Rf is a measurement noise covariance matrix. Even though the real system dynamics is continuous in nature, the output can be only observed at discrete sampling times t = T (k), k = 1, 2, 3, . . ., therefore we shall consider a numerically simulated nonlinear system described and propagated by 6 xa(k+1) = f (xa(k) , u(k) ) + w(k) y(k) = h(xa(k) , u(k) ) + v(k) 2.2

(4) (5)

Observer: continuous–discrete extended Kalman filter

The EKF algorithm summarized here is known in literature as the continuous-discrete or hybrid ˆ+ EKF 6 . The EKF is initialized with the initial estimate of the augmented state x a0 = E [x [ ] a0 ] and the + + T + ˆ a0 )(xa0 − x ˆ a0 ) . To obtain covariance matrix of the initial state estimate error P 0 = E (xa0 − x the a priori state estimate (denoted with the − subscript) in the continuous part of the filter, the state estimate of the augmented dynamic system (4) is simulatively propagated from time t = (k − 1) one step ahead to the next time instant t = (k) ˆ− x+ x a(k) = f (ˆ a(k−1) , u(k−1) )

(6)

using the system dynamics of the mass-spring-damper augmented by the assumption that the change [ ]T in parameters p(t) = k(t) b(t) c(t) Fe (t) in between samples t = (k − 1) and t = (k) is zero: x˙ 1 (t) = x2 (t) 1 1 1 1 x˙ 2 (t) = − x3 (t)x1 (t) − x4 (t)x2 (t) + x5 (t)u(t) + Fe (t) m m m m ˙ p(t) = 0 The time update of the a priori covariance matrix estimate is given by the equation P˙ (t) = Z(ˆ xa (t))P (t) + P (t)Z T (ˆ xa (t)) + Qf t where the Jacobian of the state equation Eq. (7) is expressed by and re-computed online by   0 1 0 0 0 0 − 1 x3 (t) − 1 x4 (t) − 1 x1 (t) − 1 x2 (t) 1 u(t) 1  m m m m m  m  ∂fc (xa ) 0 0 0 0 0 0   = Z(ˆ xa ) =  0 0 0 0 0 0 ∂xa xa =ˆxa     0 0 0 0 0 0 0 0 0 0 0 0 xa =ˆxa ICSV20, Bangkok, Thailand, July 7-11, 2013

(7)

(8)

(9)

3

20th International Congress on Sound and Vibration (ICSV20), Bangkok, Thailand, 7-11 July 2013 Concluding the continuous part of the hybrid EKF, the covariance matrix estimate is obtained by simulative propagation of Eq. (8) from time t = (k − 1) to the sample at t = (k): ( ) + + P− = g P , Z(ˆ x ) (10) a(k−1) (k) (k−1) The discrete-time of the filter initiates with the update of the Kalman gain matrix at sample (k): [ ]−1 T − T T K f (k) = P − L LP L + M R M (11) f (k) (k) The augmented state estimate is updated to obtain the a posteriori estimate (denoted with the + subscript) along with the a posteriori update of the error covariance matrix: [ ] ˆ+ ˆ− x (12) xa(k) ) a(k) = x a(k) + K f (k) y(k) − h(ˆ [ ] [ ] T P+ I − K f (k) L P − + K f (k) M Rf M T K Tf(k) (13) (k) = (k) I − K f (k) L using the Jacobians of the measurement equation h(xa(t) ) that in our case are time-invariant [ ] ∂h(xa(t) ) ∂h(xa(t) ) = 1 0 0 0 0 0 M (k) = ∂va(t) =1 L(k) = ∂xa(t) − − xa(t) =ˆ xa(t)

2.3

xa(t) =ˆ xa(t)

(14)

Model update and discretization

Following a run of the EKF introduced in Sect. 2.2, the states xk and the parameters k(k) , b(k) and c(k) are updated and the continuous model in Eq. (2) is re-assembled at each sample (k) to get the actual Φ(k) and Γ(k) . The MPC algorithm used here expects a discrete-time model: x(k+1) = A(k) x(k) + B (k) u(k)

and

y(k) = Cx(k)

(15)

2×2 state transition matrix, B (k) is a R2×1 input matrix and as previously, C = [Matrix] A(k) is a R 1 0 outputs the deformation y(k) = x1(k) = q(k) (m). Using a discrete sampling time T (s), the continuous-time state and input matrices Φ(k) and Γ(k) are discretized using 4 :

A(k) = I + Φ(k) T Ψ(k)

and

B (k) = Ψ(k) T Γ(k)

where Ψ(k) can be approximated using successive iterations j of ( ( ( )) ) Φ(k) T Φ(k) T Φ(k) T Φ(k) T Φ(k) T Ψ(k) = I + + ... ≈ I + I+ ... I+ ... 2! 2 3 j−1 j 2.4

(16)

(17)

Controller: model predictive control

The on-line MPC optimization problem involves minimizing a cost function subject to constraints on the voltage input. The particular MPC method considered here is known as the infinitehorizon constrained dual-mode MPC algorithm 7,10 . After choosing a horizon of n steps, we may → predict the evolution of the future states—compactly denoted by − x (k) —based on the estimate of the + → ˆ 0(k) (excluding parameters p(k)) and the sequence of future inputs − current state x0(k) = x u (k) , at 7 time (k) by the recursive substitution of states using Eq. (15). We may express this in a compact → → x (k) = M (k) x0(k) + N (k) − u (k) , where matrices M (k) and N (k) for step (k) are 10 : notation by 7,10 −   0   A(k) 0 0 ... 0 0 0   A1(k)  B (k) 0 ... 0 0 0       A(k) B (k)    .. B . . . 0 0 0 (k) M (k) =  .  (18)  N (k) =    n−1  .. .. .. .. ..  .. .    A . . . . .  (k)

An(k)

n−2 2 An−1 (k) B (k) A(k) B (k) . . . A(k) B (k) A(k) B (k) B (k)

ICSV20, Bangkok, Thailand, July 7-11, 2013

4

20th International Congress on Sound and Vibration (ICSV20), Bangkok, Thailand, 7-11 July 2013 → Let us define a linear, quadratic cost function, expressing the contribution of future states − x (k) and → − future inputs u (k) into a numerical control quality indicator J (-). This cost function uses the dualmode paradigm, utilizing free inputs for the first n steps and a fixed feedback controller afterwards 11 : J(k) =

n−1 ∑ (

) xT(k+i) Qx(k+i) + uT(k+i) Ru(k+i) + xT(k+n) P f (k) x(k+n)

(19)

i=0

where Q = QT ≥ 0 is a user determined penalization matrix for states and R = RT ≥ 0 is a penalization matrix for inputs. Furthermore, P f (k) is the solution of the unconstrained, infinitehorizon quadratic regulation problem 7,10,11 at sample (k) for the updated K (k) and A(k) , B (k) : P f (k) − (A(k) + B (k) K (k) )T P f (k) (A(k) + B (k) K (k) ) = Q + K T(k) RK (k)

(20)

where K (k) is a stabilizing feedback law (linear-quadratic (LQ)) updated for time (k) by the actual discrete dynamics A(k) , B (k) . The cost function in (19) can be compactly denoted as 7,10 → → → J(k) = − u T(k) H (k) − u (k) + 2xT0(k) GT(k) − u (k) + xT0(k) F (k) x0(k)

(21)

→ Since the part xT0(k) F (k) x0(k) does not depend on the optimization variable − u (k) and contributes only a fixed value to the cost at each iteration, we may omit it from the optimization procedure. For the variable model structure, H (k) and G(k) is evaluated on-line at each step (k) by 7,10 H (k) = G(k) =

n−1 ∑ i=0 n−1 ∑

N Ti(k) QN i(k) + N Tn(k) P f (k) N n(k) + R

(22)

N Ti(k) QM i(k) + N Tn(k) P f (k) M n(k)

(23)

i=0

where the index i denotes the i-th block row, respectively n denotes the last block row of N (k) and M (k) identified, discretized and computed for time (k). Matrix R is a block matrix with the input penalty R on its main diagonal. Upper and lower constraints on all future input variables → u≤− u (k) < u may also be compactly expressed 7 . In this case the constraints define limits on the allowable voltage potential supplied to the actuators. 2.5 1. 2. 3. 4. 5. 6. 7. 8. 9.

The resulting adaptive-predictive EKF–MPC control strategy Algorithm: Perform the following set of operations at each sampling instant (k): Propagate the state (6) and covariance matrix (8) in simulation to obtain the a priori estimate. Sample the actual deflection y(k) filtered by a low-pass and running mean filter. Compute the Kalman gain K f (k) (11), then to get a posteriori estimates, based on the measure+ ˆ+ ment sample and gain update the state x a(k) (12), then update covariance matrix P (k) (13). Based on the new parameters p(k) , re-assemble the continuous system model Φ(k) and Γ(k) (3). Approximate Ψ(k) (17) to discretize the system matrices A(k) and B (k) (16). Use the discretized model to compute the state prediction matrices M (k) and N (k) by (18). Use the state prediction matrices to compute the cost prediction matricesH (k) and G(k) by (21) Minimize the quadratic MPC cost function J(k) in Eq. (21) subject to the following input constraints: u ≤ u(k+i) ≤ u, i = 0, 1, 2, . . . , n − 1. → Apply the first element of the vector of optimal control moves − u (k) to the controlled system.

ICSV20, Bangkok, Thailand, July 7-11, 2013

5

20th International Congress on Sound and Vibration (ICSV20), Bangkok, Thailand, 7-11 July 2013

Figure 1. Schematic representation of the experimental system

3.

Experiments

The experimental hardware (Fig. 1) consists of a cantilever beam fixed at one end and free at the other. The beam is made of commercially available pure aluminum in the dimensions of 550 × 40 × 3 mm. The actuating elements are MIDÉ QP16n piezoelectric transducers made of PZT5A piezoceramic material. Actuators are connected counter-phase to a MIDÉ EL-1225 amplifier and receive an analog control signal. The sensor is a Keyence LK-G82 laser triangulation system, connected to a Keyence LK-G3001V filtering and processing unit, providing an analog voltage signal to the controller. External disturbance—resembling the transient vibration effects in aerospace constructions 9 —is emulated using a stinger mechanism, providing bursts of force at the free end of the beam using a linear motor controlled by a digital input signal. The controller algorithm is implemented on a personal computer running the Matlab xPC Target rapid control software prototyping platform, connected to hardware components with a National Instruments PCI-6030E measurement card. The control system is developed in the Matlab and Simulink software environment, then loaded onto the Target platform via TCP/IP. To evaluate the control strategy presented in Sect. 2, four experimental scenarios were considered. In all of these scenarios, during the first 30 s an initial identification procedure was performed using a fixed LQ gain based on the initial parameter estimates and the settings of the MPC algorithm, excited by disturbance bursts spaced 4 s apart. This was followed by switching to the adaptive MPC control scheme and a 8 s pause, then the disturbances spaced at 4 s apart continued. Experiment 1 assumed the original beam with the controllers switched off, Exp. 2 assumed a small weight added to the end of the beam during the whole experiment, Exp. 3 assumed that the weight was added during the pause after ∼30 s and finally Exp. 4 assumed no added weight, just the original beam. Global sampling times were set T = 0.01 s for all experiments, while the simulation step for the continuous part of the EKF algorithm was dt = T /500 s. The EKF was set with a fixed mass of [ ]T ˆ+ m = 0.178 g, the initial state estimate x a0 = 0 0 450 0.1 1E − 3 0 ] , initial error covariance [ estimate P + 0 = diag 1E − 5 1E − 2 1E + 4 1E − 2 1E − 5 1E − 1 , measurement noise covariance for the [ single position output Rf = Rf = 1E − 9 and the ]process noise covariance matrix Qf t = diag 1E − 10 1E − 4 1E − 2 1E − 8 1E − 12 E − 2 . To prevent the estimation of negative parameters, an ad hoc clipping strategy 8 was used in the EKF. The continuous model was discretized with an approximation of Ψ with j = 10 iterations. The MPC algorithm was set up using → an input penalty R = R = 1E − 10, state penalty Q = C T C, −100 ≤ − u (k) < 100 input constraints, no state constraints and a n = 10 steps prediction horizon. The constrained quadratic programming problem was solved online using the qpOASES solver 3 . The measurement chain included a low-pass FIR filter with a passband edge at 30 Hz and stopband edge at 50 Hz, and a windowed running mean filter with a 0.1 Hz fundamental frequency. Figure 2 shows a detail view of the measured position y(k) , estimated velocity x2(k) , input u(k) and disturbance force Fe(k) for a time window of 15 s. The scheme is switched to adaptive MPC before the second disturbance occurs. All controlled experiments (Exp. 2–4) attenuate the vibration levels very effectively, in fact, settling times are shorter by approximately a factor of ten (Fig. 2(a)). Figure 2(c) shows that the MPC algorithm respects process inputs, moreover, that the larger mass in ICSV20, Bangkok, Thailand, July 7-11, 2013

6

20th International Congress on Sound and Vibration (ICSV20), Bangkok, Thailand, 7-11 July 2013 −3

0.3 dq(t)/dt= x2(t) (ms−1)

q(t)=x1(t) (m)

x 10 5 0 −5 2500

3000 3500 Time (samples)

0.2 0.1 0 −0.1 −0.2

4000

2500

(a) Position

4000

(b) Velocity

100 Fe(t)=x6(t) (N)

2

50 u(t) (V)

3000 3500 Time (samples)

0 −50

1 0 Exp. 1

−1

Exp. 2

−2

−100 2500

3000 3500 Time (samples)

4000

(c) Input voltage

2500

3000 3500 Time (samples)

Exp. 3 4000 Exp. 4

(d) Disturbance

Figure 2. Estimated system states, disturbance force and control input

Exp. 2 results in more aggressive inputs during the whole duration of the experiment than in Exp. 4, while the effects of the mass increase in Exp. 3 are demonstrated after a short adaptation period. The estimated force disturbance is somewhat distorted by the low-pass and running mean filtering on the measured inputs. Figure 3 shows the estimated parameters and an equivalent LQ gain for the whole duration of the experiments. All parameters demonstrate change to compensate for the mass added to the system, since the mass itself is not identified but fixed and appears in the formulation for every unknown parameter in the state-space formulation. The equivalent stiffness, force conversion constant and the K 1,1 component of the LQ gain drop with increased mass, while damping and increases. All parameters adapt to the changes in the system dynamics, while the force conversion constant tends to zero in the uncontrolled case.

4.

Conclusion

Four experimental scenarios were used to test the proposed EKF–MPC adaptive control algorithm. Closed-loop settling times were reduced by a factor of ten, while the controller promptly reacted to changes in the beam dynamics: an added mass and therefore a harder to control system resulted in an adapted controller with more aggressive control moves. Certain aspects of the unmodeled dynamics, namely the effect of the outside disturbance, may cause the identification algorithm to diverge (not shown in the experiments here). These drawbacks will be addressed in upcoming work by selectively disabling the updates of the error covariance matrix during disturbances.

Acknowledgements The authors would like to gratefully acknowledge the financial support granted by the Slovak Research and Development Agency (SRDA) under the contracts APVV-0090-10, APVV-0131-10, APVV 0280-06 and by the Scientific Grant Agency (VEGA) of the Ministry of Education, Science, Research and Sport of the Slovak Republic under the contract 1/0138/11.

ICSV20, Bangkok, Thailand, July 7-11, 2013

7

20th International Congress on Sound and Vibration (ICSV20), Bangkok, Thailand, 7-11 July 2013

b(t)=x4(t) (Nsm−1)

k(t)=x3(t) (Nm−1)

500 450 400 350 300 250

0.4 0.3 0.2 0.1 0

0

1000

2000 3000 4000 Time (samples)

5000

6000

0

1000

(a) Stiffness K1,1,K1,2 (Vm−1,Vsm−1)

c(t)=x5(t) (NV−1)

x 10

0.5

0 0

1000

2000

3000 4000 Time (samples)

3000 4000 Time (samples)

5000

6000

(b) Damping

−3

1

2000

5000

6000

(c) Force conversion

Exp. 2 0

Exp. 3 Exp. 4

−5000 −10000 −15000 0

1000

2000 3000 4000 Time (samples)

5000

6000

(d) Equivalent unconstrained LQ gain

Figure 3. Estimated model parameters and the equivalent unconstrained LQ gain

REFERENCES 1

Benaroya, H. and Nagurka, M. L. (2010). Mechanical Vibration: Analysis, Uncertainities and Control. CRC Press, Taylor&Francis Group, Boca Raton, FL, 3. edition. 2 Elliott, S. J. and Nelson, P. A. (1993). Active noise control. IEEE Signal Proc Mag, 8(10):12–35. 3 Ferreau, H. J., Bock, H. G., and Diehl, M. (2008). An online active set strategy to overcome the limitations of explicit MPC. International Journal of Robust and Nonlinear Control, 18(8):816–830. 4 Franklin, G. F., Powell, J. D., and Workman, M. L. (1997). Digital Control of Dynamic Systems. Addison-Wesley, Boston, MA, 3. edition. 5 Fuller, C. R., Elliott, S. J., and Nelson, P. A. (1996). Active Control of Vibration. Academic Press, San Francisco, CA, United States of America, 1. edition. 6 Gelb, A. (1974). Applied optimal estimation. The M.I.T. Press, Cambridge, MA. 7 Maciejowski, J. M. (2000). Predictive Control with Constraints. Prentice Hall, Upper Saddle River. 8 Poloni, T., Eielsen, A. A., Rohal’-Ilkiv, B., and Johansen, T. (2012). Moving horizon observer for vibration dynamics with plant uncertainties in nanopositioning system estimation. In American Control Conference (ACC), 2012, pages 3817–3824. 9 Richelot, J., Bordeneuve-Guibe, J., and Pommier-Budinger, V. (2004). Active control of a clamped beam equipped with piezoelectric actuator and sensor using generalized predictive control. In 2004 IEEE International Symposium on Industrial Electronics, volume 1, pages 583 – 588 vol. 1. 10 Rossiter, J. A. (2003). Model-Based Predictive Control: A Practical Approach. CRC, Boca Raton. 11 Scokaert, P. O. M. and Rawlings, J. B. (1996). Infinite horizon linear quadratic control with constraints. In Proceedings of IFAC’96 World Congress, volume 7a-04 1, pages 109–114. 12 Takács, G. and Rohal’-Ilkiv, B. (2012). Predictive Vibration Control: Efficient constrained MPC vibration control for lightly damped mechanical systems. Springer, London, United Kingdom. 13 Trajkov, T. N., Koppe, H., and Gabbert, U. (2008). Direct model reference adaptive control (MRAC) design and simulation for the vibration suppression of piezoelectric smart structures. Communications in Nonlinear Science and Numerical Simulation, 13(9):1896 – 1909. 14 Zghal, M., Mevel, L., and Moral, P. D. (2012). Modal parameter estimation using interacting kalman filter. Mechanical Systems and Signal Processing, (0):– (Preprint.).

ICSV20, Bangkok, Thailand, July 7-11, 2013

8