Vr Vd

are the reference signals. For notational simplicity, we define the vectors S = [s1,s2]r , and Nr = [yr,Wr] . Considering the equality of the sliding condition (18), one can write sisi = -ki |si l>

Defining a vector Ksgn(S), with the elements ki sgn(si), the sliding condition will be S = -Ksgn(S) . Differentiation of S yields

Substitution of the dynamic equation and solving the result for U, the control law is found to be

The above control law is discontinuous across the sliding surface. Since the implementation of the associated control law is necessarily imperfect (for instance, in practice switching is not instantaneous), this leads to chattering. Chattering is undesirable in practice, since it involves high control activity and further may excite high frequency dynamics neglected in the course of modeling (such as unmodeled structural modes, neglected time-delays, and so on). Thus, in a second step, the discontinuous control law is suitably smoothed. This can be achieved by smoothing out the control discontinuity in a thin boundary layer neighboring the switching surface (Slotine & Li, 1991):

B (t )= {x,| s (x;t )|<0} 0>0, where O is the boundary layer thickness. In other words, outside of B (t), we choose control law u as before (i.e. satisfying the sliding condition); all other trajectories starting inside B (t =0) remain inside B (t) for all J > 0 . The mathematical operation for this to occur is to simply replace sgn(s) with sat I O I, with the saturation function defined as:

The control law derived by this method is robust in nature; therefore, insensitive to uncertainties and disturbances. One can adjust the robustness of the system by selecting proper control gains. When the upper bounds and lower bounds of uncertainties and/or disturbances are known, one can include these bounds in the control law design, to assure the robustness of the system. See (Slotine & Sastry, 1983) for more information.

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