Vibration Analysis and Control New Trends and Developments Part 7 docx

One then obtains thatcan be shown that the proposed observer design methodology is quite robust with respect toparameter uncertainty and unmodeled dynamics, by considering the parameter

derivatives of the flat output required for implementation of the controller (20):

approximated by a family of Taylor polynomials of fourth degree Therefore, the characteristicpolynomial for the dynamics of output observation error (24) is given by

One then obtains that

can be shown that the proposed observer design methodology is quite robust with respect toparameter uncertainty and unmodeled dynamics, by considering the parameter variations

presented through some experimental results that the polynomial disturbance signal-basedGPI control scheme, implemented as a classical compensation network, is robust enough withrespect to parameter uncertainty and unmodeled dynamics in the context of an off-line andpre-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/DSPboards with high computational performance operating at high sampling rates The proposedobserver could be implemented via embedded software applications without many problems

5 Simulation results

Some numerical simulations were performed on a nonlinear quarter-vehicle suspensionsystem characterized by the following set of realistic parameters (Tahboub, 2005) to verifythe effectiveness of the proposed disturbance observer-control design methodology (see Table1):

Table 1 Parameters of the vehicle suspension system

Fig 2 shows some schematic diagram for the implementation of the proposed active vibrationcontrollers based on on-line disturbance estimation using a flatness-based controller and GPIobservers.

Fig 2 Schematic diagram of the instrumentation for active vehicle suspension controlimplementation

The following trajectory was utilized to simulate the unknown exogenous disturbanceexcitations due to irregular road surfaces (Chen & Huang, 2005):

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 comparedwith the passive counterpart

Moreover, one can observe a robust and fast on-line estimation of the disturbanceξ(t)as well

as the corresponding time derivatives of the flat output up to third order Similar results onthe implementation of the controller (20) with disturbance observer (23) for estimation of the

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)s2+2ζ o ω o s+ω24

The characteristic polynomials associated with the closed-loop dynamics were all set to be of

t [s]

Active Passive Road profile

Fig 3 Sprung mass displacement response using controller (7) and observer (14)

t [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 GPIobservers for the estimation of unknown perturbations yields good attenuation properties and

an overall robust performance

0 5 10 15

−0.15

−0.1

−0.05 0 0.05 0.1 0.15

t [s]

Active Passive

Estimate Actual value

Estimate Actual value

Estimate Actual value

Fig 8 Estimation of time derivatives of the flat output using the observer (14)

t [s]

Active Passive Road profile

Fig 10 Sprung mass displacement response using controller (20) and observer (23).)

t [s]

2 ]

Active Passive

Fig 11 Sprung mass acceleration response using controller (20) and observer (23)

−0.15

−0.1

−0.05 0 0.05 0.1 0.15

t [s]

Active Passive

Estimate Actual value

Estimate Actual value

Estimate Actual value

Fig 15 Estimation of time derivatives of the flat output using the observer (23)

t [s]

Fig 16 Control force using the observer (23)

6 Conclusions

In this chapter a robust active vibration control scheme, based on real-time estimationand 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 robustobserver Therefore, a perturbed input-output differential equation describing the dynamics

of the flat output is obtained for design purposes of the control scheme The exogenousdisturbances due to irregular road surfaces, nonlinear effects, parameter variations andunmodeled dynamics are lumped into an unknown bounded time-varying perturbationinput 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 tolocally 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, isproposed to estimate the perturbation signal and some time derivatives of the flat outputrequired for implementation of differential flatness-based disturbance feedforward andfeedback controllers for attenuation of vibrations in electromagnetic and hydraulic activevehicle suspension systems

Two high-gain disturbance observer-based controllers have been proposed to attenuate thevibrations induced by unknown exogenous disturbance excitations due to irregular roadsurfaces, which could be employed for nonlinear quarter-vehicle active suspension models byusing hydraulic or electromagnetic actuators Computer simulations were included to showthe effectiveness of the proposed controllers, as well as of the disturbance observers based onTaylor polynomials of fourth degree

The results show a high vibration attenuation level of the active vehicle suspension systemcompared with the passive counterpart and, in addition, a robust and fast real-time estimation

of the disturbance and time derivatives of the flat output

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