10 Vibration Control derivatives of the flat output required for implementation of the controller (20): ˙ η 1 = η 2 ˙ η 2 = η 3 ˙ η 3 = η 4 ˙ η 4 =  1 + bu ˙  1 =  2 ˙  2 =  3 . . . ˙  r−1 =  r ˙  r = 0 y = η 1 = L (22) where  1 = ,  2 = ˙ ,  3 = ¨ , ···,  r =  ( r−1 ) . A Luenberger observer for the system (22) is then given by  ˙ η 1 =  η 2 + λ r+3 ( y −  y )  ˙ η 2 =  η 3 + λ r+2 ( y −  y )  ˙ η 3 =  η 4 + λ r+1 ( y −  y )  ˙ η 4 =   1 + bu + λ r ( y −  y )  ˙  1 =   2 + λ r−1 ( y −  y )  ˙  2 =   3 + λ r−2 ( y −  y ) . . .  ˙  r−1 =   r + λ 1 ( y −  y )  ˙  r = λ 0 ( y −  y )  y =  η 1 (23) The estimation error dynamics e 1 = y −  y satisfies the following dynamics: e (r+4) 1 + λ r+3 e ( r+3 ) 1 + λ r+2 e ( r+2 ) 1 + λ r+1 e ( r+1 ) 1 + λ r e (r) 1 +λ r−1 e (r−1) 1 + ···+ λ 2 ¨ e + λ 1 ˙ e 1 + λ 0 e 1 = 0 (24) Therefore, the design parameters λ i , i = 0, ··· , r + 3, can be chosen so that the output estimation error e 1 exponentially asymptotically converges to zero. On the other hand, it is assumed that the perturbation input signal  (t) can be locally approximated by a family of Taylor polynomials of fourth degree. Therefore, the characteristic polynomial for the dynamics of output observation error (24) is given by p o2 = s 9 + λ 8 s 8 + λ 7 s 7 + λ 6 s 6 + λ 5 s 5 + λ 4 s 4 + λ 3 s 3 + λ 2 s 2 + λ 1 s + λ 0 (25) It is then proposed the following Hurwitz polynomial to compute the proper gains for the observer: p o2 ( s ) = ( s + p 2 )  s 2 + 2ζ 2 ω 2 s + ω 2 2  4 (26) 140 Vibration Analysis and Control – New Trends and Developments Control of Nonlinear Active Vehicle Suspension Systems Using Disturbance Observers 11 One then obtains that λ 0 = p 2 ω 8 2 λ 1 = ω 8 2 + 8p 2 ζ 2 ω 7 2 λ 2 = 8ω 7 2 ζ 2 + 24p 2 ω 6 2 ζ 2 2 + 4p 2 ω 6 2 λ 3 = 24ω 6 2 ζ 2 2 + 4ω 6 2 + 32p 2 ω 5 2 ζ 3 2 + 24p 2 ω 5 2 ζ 2 λ 4 = 32ω 5 2 ζ 3 2 + 24ω 5 2 ζ 2 + 16p 2 ω 4 2 ζ 4 2 + 48p 2 ω 4 2 ζ 2 2 + 6p 2 ω 4 2 λ 5 = 16ω 4 2 ζ 4 2 + 48ω 4 2 ζ 2 2 + 6ω 4 2 + 32p 2 ω 3 2 ζ 3 2 + 24p 2 ω 3 2 ζ 2 λ 6 = 32ω 3 2 ζ 3 2 + 24ω 3 2 ζ 2 + 24p 2 ω 2 2 ζ 2 2 + 4p 2 ω 2 2 λ 7 = 24ω 2 2 ζ 2 2 + 4ω 2 2 + 8p 2 ω 2 ζ 2 λ 8 = p 2 + 8ω 2 ζ 2 with p 2 , ω 2 , ζ 2 > 0. From the practical viewpoint, main advantage of this high-gain observer is that it could be employed for hydraulic or electromagnetic active vehicle suspension systems, requiring only information of the stiffness constant of tire k t and the unsprung mass m u . In addition, it can be shown that the proposed observer design methodology is quite robust with respect to parameter uncertainty and unmodeled dynamics, by considering the parameter variations into the perturbation input signal  ( t ) . In fact, in (Sira-Ramirez et al., 2008a) has been presented through some experimental results that the polynomial disturbance signal-based GPI control scheme, implemented as a classical compensation network, is robust enough with respect to parameter uncertainty and unmodeled dynamics in the context of an off-line and pre-specified reference trajectory tracking tasks. It is important to emphasize that, the proposed results are now possible thanks to the existence of commercial embedded system for automatic control tasks based on high speed FPGA/DSP boards with high computational performance operating at high sampling rates. The proposed observer could be implemented via embedded software applications without many problems. 5. Simulation results Some numerical simulations were performed on a nonlinear quarter-vehicle suspension system characterized by the following set of realistic parameters (Tahboub, 2005) to verify the effectiveness of the proposed disturbance observer-control design methodology (see Table 1): Parameters Values Sprung mass (m s ) 216.75 [kg] Unsprung mass (m u ) 28.85 [kg] Spring stifness (k s ) 21700 [ N m ] Damping constant (c s ) 1200 [ N·s m ] Tire stifness (k t ) 184000 [ N m ] nonlinear spring stiffness (k ns ) 2170 [ N m ] nonlinear damping constant (c ns ) 120 [ N·s m ] Table 1. Parameters of the vehicle suspension system. 141 Control of Nonlinear Active Vehicle Suspension Systems Using Disturbance Observers 12 Vibration Control Fig. 2 shows some schematic diagram for the implementation of the proposed active vibration controllers based on on-line disturbance estimation using a flatness-based controller and GPI observers. Fig. 2. Schematic diagram of the instrumentation for active vehicle suspension control implementation. The following trajectory was utilized to simulate the unknown exogenous disturbance excitations due to irregular road surfaces (Chen & Huang, 2005): z r ( t ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ f 1 ( t ) + f ( t ) for t ∈ [3.5, 5) f 2 ( t ) + f ( t ) for t ∈ [5, 6.5) f 3 ( t ) + f ( t ) for t ∈ [8.5, 10) f 3 ( t ) + f ( t ) for t ∈ [10, 11.5 f ( t ) else with f 1 ( t ) = − 0.0592 ( t − 3.5 ) 3 + 0.1332 ( t − 3.5 ) 2 f 2 ( t ) = 0.0592 ( t − 6.5 ) 3 + 0.1332 ( t − 6.5 ) 2 f 3 ( t ) = 0.0592 ( t − 8.5 ) 3 − 0.1332 ( t − 8.5 ) 2 f 3 ( t ) = − 0.0592 ( t − 11.5 ) 3 − 0.1332 ( t − 11.5 ) 2 f ( t ) = 0.002 sin ( 2πt ) + 0.002 sin ( 7.5πt ) Figs. 3-9 describe the robust performance of the controller (7) using the observer (14). It can be seen the high vibration attenuation level of the active vehicle suspension system compared with the passive counterpart. 142 Vibration Analysis and Control – New Trends and Developments Control of Nonlinear Active Vehicle Suspension Systems Using Disturbance Observers 13 Moreover, one can observe a robust and fast on-line estimation of the disturbance ξ(t) as well as the corresponding time derivatives of the flat output up to third order. Similar results on the implementation of the controller (20) with disturbance observer (23) for estimation of the perturbation  ( t ) and time derivatives of the flat output are shown in Figs. 10-23. In the computer simulations it is assumed that the perturbation input signals ξ (t) and  ( t ) can be locally approximated by a family of Taylor polynomials of fourth degree. The characteristic polynomials for the ninth order observation error dynamics were all set to be of the following form: p o ( s ) = ( s + p o )  s 2 + 2ζ o ω o s + ω 2 o  4 with p o = ω o = 300rad/s and ζ o = 20. The characteristic polynomials associated with the closed-loop dynamics were all set to be of the form: p c ( s ) =  s 2 + 2ζ c ω c s + ω 2 c  2 ,withω c = 10rad/s and ζ c = 0.7071. 0 5 10 15 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 t [s] Displacement [m] Active Passive Road profile Fig. 3. Sprung mass displacement response using controller (7) and observer (14). 0 5 10 15 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 t [s] Acceleration [m/s 2 ] Active Passive Fig. 4. Sprung mass acceleration response using controller (7) and observer (14). In general, the proposed active vehicle suspension using a flatness-based controller and GPI observers for the estimation of unknown perturbations yields good attenuation properties and an overall robust performance. 143 Control of Nonlinear Active Vehicle Suspension Systems Using Disturbance Observers 14 Vibration Control 0 5 10 15 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 t [s] Deflection [m] Active Passive Fig. 5. Suspension deflection response (x 1 − x 3 ) using controller (7) and observer (14). 0 5 10 15 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x 10 −3 t [s] Deflection [m] Active Passive Fig. 6. Tire deflection response (x 3 − z r ) using controller (7) and observer (14). 0 5 10 15 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x 10 7 t [s] Perturbation Signal Estimated perturbation Actual perturbation Fig. 7. Perturbation estimation ξ(t) using observer (14). 144 Vibration Analysis and Control – New Trends and Developments Control of Nonlinear Active Vehicle Suspension Systems Using Disturbance Observers 15 0 5 10 15 −10 0 10 t [s] First derivative 0 5 10 15 −200 −100 0 100 200 t [s] Second derivative 0 5 10 15 −2 0 2 x 10 4 t [s] Third derivative Estimate Actual value Estimate Actual value Estimate Actual value Fig. 8. Estimation of time derivatives of the flat output using the observer (14). 0 5 10 15 −2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500 t [s] u [N] Fig. 9. Control force using the observer (14). 0 5 10 15 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 t [s] Displacement [m] Active Passive Road profile Fig. 10. Sprung mass displacement response using controller (20) and observer (23).). 145 Control of Nonlinear Active Vehicle Suspension Systems Using Disturbance Observers 16 Vibration Control 0 5 10 15 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 t [s] Acceleration [m/s 2 ] Active Passive Fig. 11. Sprung mass acceleration response using controller (20) and observer (23). 0 5 10 15 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 t [s] Deflection [m] Active Passive Fig. 12. Suspension deflection response (x 1 − x 3 ) using controller (20) and observer (23). 0 5 10 15 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x 10 −3 t [s] Deflection [m] Active Passive Fig. 13. Tire deflection response (x 3 − z r ) using controller (20) and observer (23). 146 Vibration Analysis and Control – New Trends and Developments Control of Nonlinear Active Vehicle Suspension Systems Using Disturbance Observers 17 0 5 10 15 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x 10 7 t [s] Perturbation signal Estimated perturbation Actual perturbation Fig. 14. Perturbation estimation  ( t ) using observer (23). 0 5 10 15 −10 −5 0 5 10 t [s] First derivative 0 5 10 15 −200 −100 0 100 200 t [s] Second derivative 0 5 10 15 −1 −0.5 0 0.5 1 x 10 4 t [s] Third derivative Estimate Actual value Estimate Actual value Estimate Actual value Fig. 15. Estimation of time derivatives of the flat output using the observer (23). 0 5 10 15 −2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500 t [s] u [N] Fig. 16. Control force using the observer (23). 147 Control of Nonlinear Active Vehicle Suspension Systems Using Disturbance Observers 18 Vibration Control 6. Conclusions In this chapter a robust active vibration control scheme, based on real-time estimation and rejection of perturbation signals, of nonlinear vehicle suspension systems is described. The proposed approach exploits the structural property of differential flatness exhibited by the suspension system fot the synthesis of a flatness based controller and a robust observer. Therefore, a perturbed input-output differential equation describing the dynamics of the flat output is obtained for design purposes of the control scheme. The exogenous disturbances due to irregular road surfaces, nonlinear effects, parameter variations and unmodeled dynamics are lumped into an unknown bounded time-varying perturbation input signal affecting the differentially flat linear simplified dynamic mathematical model of the suspension system. A family of Taylor polynomials of (r-1)th degree is used to locally approximate this perturbation signal. Hence the perturbation signal is described by a rth-order mathematical model. Then, the perturbed suspension system model is expressed as a (r+4)th-order extended mathematical model. The design of high-gain Luenberger observers, based on this kind of extended models, is proposed to estimate the perturbation signal and some time derivatives of the flat output required for implementation of differential flatness-based disturbance feedforward and feedback controllers for attenuation of vibrations in electromagnetic and hydraulic active vehicle suspension systems. Two high-gain disturbance observer-based controllers have been proposed to attenuate the vibrations induced by unknown exogenous disturbance excitations due to irregular road surfaces, which could be employed for nonlinear quarter-vehicle active suspension models by using hydraulic or electromagnetic actuators. Computer simulations were included to show the effectiveness of the proposed controllers, as well as of the disturbance observers based on Taylor polynomials of fourth degree. The results show a high vibration attenuation level of the active vehicle suspension system compared with the passive counterpart and, in addition, a robust and fast real-time estimation of the disturbance and time derivatives of the flat output. 7. References Ahmadian, M. Active control of vehicle suspensions. In: Encyclopedia of Vibration, Edited by Braun, S.G., Ewins, D.J. & Rao, S.S. (2001), Vols. 1-3, Academic Press, San Diego, CA. Basterretxea, K., Del Campo, I. & Echanobe, J. (2010). A semi-active suspension embedded controller in a FPGA, 2010 IEEE International Symposium on Industrial Embedded Systems, pp. 69-78, Trento, July 7-9. Beltran-Carbajal, F., Silva-Navarro, G., Blanco-Ortega, A. & Chavez-Conde, E. (2010a). Active Vibration Control for a Nonlinear Mechanical System using On-line Algebraic Identification, In: Vibration Control, M. Lallart, (Ed.), 201-214, Sciyo, Rijeka, Croatia. Beltran-Carbajal, F., Silva-Navarro, G., Sira-Ramirez, H. & Blanco-Ortega, A. (2010b). Application of on-line algebraic identification in active vibration control, Computación ySistemas, Vol. 13, No. 3, pp. 313-330. Cao, J., Liu, H., Li, P. & Brown, D. (2008). State of the Art in Vehicle Active Suspension Adaptive Control Systems Based on Intelligent Methodologies, IEEE Transaction on Intelligent Transportation Systems, Vol. 9, No. 3, pp. 392-405. 148 Vibration Analysis and Control – New Trends and Developments [...]... (2002) MR damper and its application for semi-active control of vehicle suspension system, Mechatronics, Vol 12, pp 963- 973 152 Vibration Analysis and Control – New Trends and Will-be-set-by-IN-TECH Developments 2 R R 0 S I 1 Semi-Active Control of Civil Structures Based on the Civil Structures Based on the Structural Structural Response: Integrated Design Approach Approach Semi-active Control of Prediction... International Conference on Electrical and Electronics Engineering (ICEEE), pp 306-309, Mexico City, Mexico, September 5 -7 Sira-Ramirez, H & Agrawal, S.K (2004) Differentially Flat Systems, Marcel Dekker, New York 150 20 Vibration Analysis and Control – New Trends and Developments Vibration Control Shoukry, Y., El-Kharashi, M W & Hammad, S (2010) MPC-On-Chip: An Embedded GPC Coprocessor for Automotive Active... () = > z( ) z( ) Semi-Active Control of Civil Structures Based on the Civil Structures Based on the Structural Structural Response: Integrated Design Approach Approach Semi-active Control of Prediction of the Prediction of the Response: Integrated Design () = 1 57 7 = α α < ()= = α= () = α – – α = α< 158 Vibration Analysis and Control – New Trends and Will-be-set-by-IN-TECH Developments 8 = = = ( ) (... 159 9 ⎞ ⎠ ⎞ ⎠ = ( = < λ < = = = = = ) = 160 Vibration Analysis and Control – New Trends and Will-be-set-by-IN-TECH Developments 10 = = = = () = D D = βK β > β> = = ( ) = [ [ ] = ( ) = = = = = ] = = = Semi-Active Control of Civil Structures Based on the Civil Structures Based on the Structural Structural Response: Integrated Design Approach Approach Semi-active Control of Prediction of the Prediction of... Approach Approach Semi-active Control of Prediction of the Prediction of the Response: Integrated Design () = ( )= = Q 155 5 ⎧ ⎨ = ⎩ z ( + )Q z( + ) > =Q z( ) z( + ) = C x( + x( x( u ( ) x( ) + ) x( )+ )+D u ( + + ) x( ) u ( + z( u ( )+ ) z( u ( ) ) + = x ( ) = Ax ( ) + Bu ( ) ) + ) u ( ) x( ) 156 Vibration Analysis and Control – New Trends and Will-be-set-by-IN-TECH Developments 6 x ( ) u ( ) w( )... without asymptotic observers and generalized PID regulators, Nonlinear Control in the Year 2000, Lecture Notes in Control and Information Sciences, Vol 258, pp 3 67- 384, Springer, London Fliess, M., Lévine, J., Martin, Ph & Rouchon, P (1993) Flatness and defect of nonlinear systems: introductory theory and examples, International Journal of Control, Vol 61, No 6, pp 13 27- 1361 Gysen, B.L.J., Paulides,... ( ) = b +b ( ) b = b +b R M D K S b R = ( )) 154 Vibration Analysis and Control – New Trends and Will-be-set-by-IN-TECH Developments 4 M b b () D b b M b () D D x ( ) = Ax ( ) + Bu ( ) z( ) = C x( ) + D u ( ) ⎡ x( ) = A = z( ) 0 M R A B C ⎤ w( ) u ( ) = ⎣ w( ) ⎦ u ( ) = w( ) q( ) q( ) I K M D( ) B = M w( ) w( ) 0 b b ()b z( ) () D () = = Semi-Active Control of Civil Structures Based on the Civil Structures... ) + D u ( ) q( ) 161 11 162 Vibration Analysis and Control – New Trends and Will-be-set-by-IN-TECH Developments 12 q( ) r( ) ⎡ ⎤ ⎡ () ⎢ ()⎥ ⎥ ⎢ ⎥ r( ) = ⎢ ⎦ ⎣ () ⎢ ⎢ q( ) = ⎢ ⎣ ⎤ () ()⎥ ⎥ ⎥= ⎦ () ⎡ ⎢ ⎢ ⎢ ⎣ () D C ⎡ C = ⎣0 R ⎤ 0 R ⎡ ⎦ D = ⎣0 I s ⎤ 0 s ⎦ 0 ⎤ ⎡ ⎥ ⎥ ⎥ ⎥ s= ⎥ ⎦ ⎢ ⎢ ⎢ R=⎢ ⎢ ⎣ = ⎤ () () ⎥ ⎥ ⎥ ⎦ () () () 0 () = z( ) Q Q Q = 1 1 1 1 1 1 1 1 1 1 1 1 > = Semi-Active Control of Civil Structures... pp 172 6- 173 1, Saint Petersburg, Russia, July 8-10 Chen, P & Huang, A (2005) Adaptive sliding control of non-autonomous active suspension systems with time-varying loadings, Journal of Sound and Vibration, Vol 282, pp 1119-1135 Fliess, M., Marquez, R., Delaleau, E & Sira-Ramirez, H (2002) Correcteurs Proportionnels-Integraux Généralisés, ESAIM Control, Optimisation and Calculus of Variations, Vol 7, ... 25- 27 Sira-Ramirez, H., Barrios-Cruz, E & Marquez-Contreras, R.J (2009) Fast adaptive trajectory tracking control for a completely uncertain DC motor via output feedback, Computación y Sistemas, Vol 12, No 4, pp 3 97- 408 Sira-Ramirez, H., Silva-Navarro, G & Beltran-Carbajal, F (20 07) On the GPI balancing control of an uncertain Jeffcot rotor model, 20 07 4th International Conference on Electrical and . (2002). MR damper and its application for semi-active control of vehicle suspension system, Mechatronics, Vol. 12, pp. 963- 973 . 150 Vibration Analysis and Control – New Trends and Developments 2. and observer (23). 146 Vibration Analysis and Control – New Trends and Developments Control of Nonlinear Active Vehicle Suspension Systems Using Disturbance Observers 17 0 5 10 15 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x. active vehicle suspension system compared with the passive counterpart. 142 Vibration Analysis and Control – New Trends and Developments Control of Nonlinear Active Vehicle Suspension Systems Using