One major approach to dealing with model uncertainty is the robust control. Broadly speaking, robustness is a property which guarantees that essential functions of the designed system are maintained under adverse conditions in which the model no longer accurately reflects reality. In modeling for robust control design, an exactly known nominal plant is accompanied by a description of plant uncertainty, that is, a characterization of how the true plant might differ from the nominal one. This uncertainty is then taken into account during the design process (Freeman & Kokotovic, 1996).

For simplicity, we explain the method for a single-input system. The extension to multi-input systems is straight forward, as will be illustrated in the AUV example. A more detailed discussion of this method is given by (Slotine, 1985), (Slotine & Sastry, 1983), and (Slotine & Li, 1991). Consider the dynamic system

where u(t) is the control input and X = [* (n-1)]r is the state vector. It is assumed that the generally nonlinear function f (X;t) is not exactly known, but the extent of imprecision on f is upper-bounded by a known continuous function of X and t . Similarly the control gain b (X;t) is not exactly known, but is of constant sign and is bounded by known continuous functions of X and t . The control problem is to track the desired trajectory

Xd = [xd ,Xd ,---,x(f 1)]r in the presence of model imprecisions on f and b . Defining the tracking error as usual, X = X-Xd , we assume that

A time-varying sliding surface S (t) is defined in the state space Rn as S (t): s (X;t) = 0 , with s (X;t) = jd + xc, X>0, (17)

where X is a positive constant. Given the initial condition (16), the problem of tracking X d is equivalent to that of remaining on the surface S (t) for all t >0 . Now a sufficient condition for such positive invariance of S (t) is to choose the control law u of Eq. (15) such that outside of sliding condition S (t), the following holds:

2 dt where k is a positive constant. Sliding condition (18) constraints state trajectories to point toward the sliding surface S (t). Geometrically, it looks like the trajectories are sliding down S (t) to reach the desired state. Satisfying Eq. (18) guarantees that if condition (16) is not exactly verified, the surface S (t) will nonetheless be reached in a finite time, while definition (17) then guarantees that X ^ 0 as t (Slotine, 1985).

The controller design procedure in the suction control method, consists of two steps. First, a feedback control law u is selected so as to verify sliding condition (18). Such a control law is discontinuous across the surface, which leads to control chattering. Chattering is undesirable in practice because it involves high control activity and further may excite high-frequency dynamics neglected in the course of modeling. Thus in a second step, discontinuous control law u is suitably smoothed to achieve an optimal trade-off between control bandwidth and tracking precision. While the first step accounts for parametric uncertainty, the second step achieves robustness to high-frequency unmodeled dynamics. Construction of a control law to verify the sliding condition (18) is straight forward, and will be illustrated in section 3.4 through an example.

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