Of the four techniques presented in this section, namely bounding control, sliding mode control, Lyapunov redesign, and nonlinear damping, the first three are based on the key assumption of a known bound on the uncertainty. The nonlinear damping technique does not make this bounding assumption; however, the resulting stability property does not guarantee the convergence of the tracking error to zero, but to an invariant set whose radius is proportional to the ∞-norm of the uncertainty. Even though the residual error in the nonlinear damping design can be reduced by increasing the feedback gain parameter k, this is not without drawbacks, since increasing the feedback gain may result in high-gain feedback, with all the undesirable consequences.

In this subsection, we introduce another technique which also relaxes the assumption of a known bound. Specifically, it is assumed that η(x, u) is bounded by

where θ is an unknown parameter vector of dimension q and φ is a known vector function. Since θ^Tφ represents a bound on the uncertainty, each element of θ and φ is assumed to be non-negative. Typically, the dimension q is simply equal to one. However, the general case where both θ and φ are vectors, allows the control designer to take advantage of any knowledge where the bound changes for different regions of the state-space x. If a known function φ(x) is not available, it can be simply assumed that η(x, u)_∞ ≤ θ where θ is a scalar unknown bounding constant. The adaptive bounding control method was introduced in [215] and was later used in neural control [209].

It is worth noting that the bounding assumption of the adaptive bounding control method is significantly less restrictive than that of the Lyapunov redesign method where the bound is assumed to be known. Even though one may consider simply increasing the bound of the Lyapunov redesign method until the assumed bound holds, this is not always possible, and quite often it is not an astute way to handle the problem since it will increase the feedback gain of the system.

The adaptive bounding control technique is based on the idea of estimating online the unknown parameter vector θ. The feedback controller utilizes the parameter estimate instead of the true bounding vector θ. One of the key questions has to do with the design of the adaptive law for generating . As we will see, this is achieved again by Lyapunov analysis.

Let denote the parameter estimation error. Consider the augmented Lyapunov function

where Γ is a positive define matrix of dimension q × q, which represents the adaptive gain. By taking the time derivative of V along the solutions of (5.92), we obtain

We choose the corrective control term p_i(x) and the update law for as follows:

which implies that . Therefore, both x(t) and remain bounded and x(t) converges to zero (using Barb lat's Lemma).

The feedback control law (5.105) is discontinuous at ω_i(x) = 0.As discussed in Section 5.4.3, the discontinuous sign functions can be smoothed by using the tanh(·) function:

where ε > 0 is a small design constant. Another issue that arises with adaptive bounding control is the possible parameter drift of the bounding estimate . This may occur as a consequence of using the smooth approximation tahn(ω_i(x)/ε), which may result in a small residual error. Moreover, in the presence of measurement noise or disturbances again the bounding parameter estimate may not converge. Since the right-hand side of (5.106) is nondecreasing, the presence of such residual errors (even if small) may cause the parameter drift of the estimate, which in turn will cause the feedback control signal to become large. This can be prevented by using a robust adaptive law, as described in Chapter 4. One of the available techniques is the dead-zone, which requires knowledge of the size of the residual error. Another method is the projection modification, which prevents the parameter estimate from becoming larger than a preselected level. Yet another approach is the σ modification.

The adaptive bounding control method is also used in adaptive approximation based control in order to address the issue of having the trajectory leave the approximation region. This is illustrated in Chapters 6 and 7.