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【正文】 he mass of the actuator and f d is the equivalent viscous friction. Consider the nonlinear system described by the equations: _ x_ 188。 g240。x240。t222。 u240。t222。222。 x240。0222。 188。 xo Defining the state vector i i o o x 188。 189。 1x x2 x3 T x 4_ 188。 189。v P i T P o x sp_ 240。 5222。 240。 6222。 Eqs. (1)(5) are rearranged as (x sp P 0 and x sp 0): 1042 H. Khan et al. / Mechatronics 15 (2021) 10371059 1 188。 v 188。 189。_ df x 1 254。 A ix 2 _ A ox 3_ m 8 x1_ 8 x2_ : x2_ 8 x_ 3 _ iP 188。__ _ iP 188。__ _ oP 188。最 _ b V i b V i b V o b V o 189。_A ix 1 254。 k dxx4 189。_A ix 1 254。 k dxx4 189。A ox 1 _ k dxx4 189。A ox 1 _ k dxx4 pffffffffffffff p s _ x 2_。 pffffffffffffff x 2 _ p r_。 x sp P 0 xsp 0 x sp P 0 xsp 0 240。 7222。 : x3_ : x_ 4 188。 xsp_ _ oP 昀 _ 1 pffffffffffffff x 3 _ p r_。 pffffffffffffff p s _ x 3_。 188。 189。k spu _ x 4_ s Combining the driving force in Eq. (5) with the continuity equation (3), it can eas ily be shown that the process described in Eq. (6) can be brought in the following form by suitable transformations: x_ 188。 g240。x240。t222。 u240。t222。222。 188。 Ax240。t222。 254。 f 240。x240。t222。 u240。t222。222。 y240。t222。 188。 Cx240。t222。 240。 8222。 where matrices A and C and the nonlinear terms f(x, u) of the state model, as follows: 2 3 fd 6 m ? 6 bAi 6 6 _ 6 A 188。 6 6 6 6 6 4 V i bAo V o 0 Ai m 0 0 0 ? 0 0 0 Ao m 0 0 0 ? 1 s 7 7 7 7 7 7 7 7 7 7 5 and 240。 9222。 f (x, u) is defined as follows: 8 f 240。x 1。 u222。 188。 0 8 f 240。x 。 u222。 188。 f 240。x 。 u222。 188。 2 : 8 2 b V i b V i k xx d pffffffffffffff p s _ x 。 pffffffffffffff x 2 _ p r。 pffffffffffffff 2 4 k dxx4 b x P 0 sp xsp 0 x sp P 0 xsp 0 f 240。x 3。 u222。 188。 _ : f 240。x 3。 u222。 188。 _ : f 240。x 。 u222。 188。 4 ksp s u k dxx4 k dxx4 V o b V o x 3 _ p r。 pffffffffffffff p s _ x 3。 240。10222。 T y 188。 189。 1 0 0 0_189。x 240。11222。 1 x2 x3 x 4_ H. Khan et al. / Mechatronics 15 (2021) 10371059 3. Development of nonlinear observer . Nonlinear observer design 1043 Fig. 3 shows the structure of the nonlinear observer used in this paper. It is essen tially direct extension of Luenberger_s linear observer to the nonlinear case [3]. We therefore take advantage of the structure of the system model to design the observer that is not in a strict canonical form. As a result, we have separated the linear behav ior from the nonlinear system dynamics, to achieve a plete stability design pro cedure. This approach satisfies a fairly general class of observers, and covers the nonlinear system under consideration. However, it is not easy to design the constant observer gain K such that it sufciently guarantees system stability. From the system, the observer makes use of one measurement (for instance, the velocity). The velocity and the control signal u are used as observer inputs, whose dynamics are of the form _^x 188。 u240。^x240。t222。 y240。t222。 u240。t222。222。 240。12222。 Based on these inputs, the observer estimates the states of the system, and the devi ation between the observed and measured states, called the residual, is evaluated. Note that the desired active force of the actuator can be tracked by using a non linear control methodology which guarantees asymptotically stable tracking, but needs access to all states. It should be noted that Lyapunov has stated conditions under which a system is stable. Although the conditions ensure only local asymptotic stability, global asymptotic stabilities can be established and the extent of asymptotic stability can also be obtained. Under these conditions, the state deviations or resid uals, e(t) approach zero, as shown in Fig. 3. An observer for the system (7) is a dynamic system with inputs y(t), u(t) and out put z(t) such that ^x240。t222。 is an estimate of x(t) in the sense that kx240。t222。 _ ^x240。t222。k ! 0 as t ! 1 . When system (8) is linearly observable, there are many approaches to the ob server design, most prevalent among which is the constant gain observer [4]. Since x C System y (t) u (t) g + z (t) x? ??? C e (t) Observer Fig. 3. Nonlinear observer structure. 1044 H. Khan et al. / Mechatronics 15