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Adaptive Estimation and Control of Magneto-Rheological Damper for Semi-Active Suspensions  303


                            1) Ride comfort: This needs to design the input current I of MR damper
                               to create appropriate force F to regulate the vertical displacement of
                               vehicle body under the road shocks, i.e., x 1 = z s → 0.
                            2) Road holding: The firm uninterrupted contact of wheels to road should
                               be ensured for the safety of passengers, that is

                                                       |F t | <(m s + m us )g         (19.20)

                            3) Suspension movement limitation: The suspension space should not exceed
                               the allowable maximum, i.e., the difference z s −z us should be bounded
                               by the maximum suspension space z max as

                                                                                      (19.21)
                                                       |z s − z us | ≤ z max
                               The suspension performance 1), 2) and 3) will be studied by introducing
                            an adaptive control, where the unknown parameters will be also online
                            estimated.

                            19.3.2 Adaptive Control Design With Parameter Estimation
                            To address the regulation of vertical displacement x 1 of system (19.19), we
                            first define the filtered error variable as

                                                                   T
                                                    s 1 =[ ,1][x 1 ,x 2 ]             (19.22)
                            where  > 0 is a positive constant. Thus, s 1 is bounded as long as the

                            filtered error s 1 is bounded. In particular, |x 1 | ≤ |s 1 |   and |x 2 | ≤ 2|s 1 | are
                            true for zero initial condition.
                               Furthermore, we can obtain the time derivative of s 1 as


                               s ˙ 1  =  x 2 +  1  [(c 0 − b e )(x 2 − x 4 ) + (k 0 − k s )(x 1 − x 3 )
                                            m s                                       (19.23)
                                                  3
                                      −k sn (x 1 − x 3 ) + θ I I tanh c 1 (x 2 − x 4 ) + k 1 (x 1 − x 3 ) ]
                               In this section, the coefficients of springs, the mass of car body and
                            the parameters of hyperbolic model are all unknown. We will present an
                            online estimation algorithm to obtain these unknown parameters. Hence,
                            we denote the system dynamics as a more compact form as


                                        1                                               3
                                T(Z) =     c 0 − b e (x 2 − x 4 ) + k 0 − k s (x 1 − x 3 ) − k sn (x 1 − x 3 )
                                       m s
                                         T
                                     = W φ 1 (Z 1 )                                   (19.24)
                                         1
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