In parallel with developments in adaptive nonlinear control, there has been a tremendous amount of activity in neural control and adaptive fuzzy approaches. In these studies, neural networks or fuzzy approximators are used to approximate unknown nonlinearities. The input/output response of the approximator is modified by adjusting the values of certain parameters, usually referred to as weights. From a mathematical control perspective, neural networks and fuzzy approximators represent just two classes of function approximators. Polynomials, splines, radial basis functions, and wavelets are examples of other function approximators that can be used and have been used in a similar setting. We refer to such approximation models with adaptivity features as adaptive approximators, and control methodologies that are based on them as adaptive approximation based control. Adaptive approximation based control encompasses a variety of methods that appear in the literature: intelligent control, neural control, adaptive fuzzy control, memory-based control, knowledge-based control, adaptive nonlinear control, and adaptive linear control. |
During the last few years there have been significant developments in the control of highly uncertain, nonlinear dynamical systems. For systems with parametric uncertainty, adaptive nonlinear control has evolved as a powerful methodology leading to global stability and tracking results for a class of nonlinear systems. Advances in geometric nonlinear control theory, in conjunction with the development and refinement of new techniques, such as the backstepping procedure and tuning functions, have brought about the design of control systems with proven stability properties. In addition, there has been a lot of research activity on robust nonlinear control design methods, such as sliding mode control, Lyapunov redesign method, nonlinear damping, and adaptive bounding control. These techniques are based on the assumption that the uncertainty in the nonlinear functions is within some known, or partially known, bounding functions.
TABLE OF CONTENTS Chapter 5.4.4 - Nonlinear Damping
The Lyapunov redesign approach developed in Section 5.4.3 is based on the principle of first designing a nominal controller u = p0(x), with a Lyapunov function such that the nominal system satisfies some desirable stability properties, and then augmenting the control law using u = p0(x) + p*(x) such that the corrective term p*(x) is designed (using the same nominal Lyapunov function) to address a matched uncertainty term (x, u). One of the key assumptions made in the design methodology described in Section 5.4.3 is that the uncertainty term (x, u) is bounded by a known bounding term Consider again the system described by (5.92); i.e.,  The uncertainty function n(x, u) is assumed to be of the form  where the m × m matrix Φ is known, and 0 is unknown but uniformly bounded (i.e., 0(x, u)∞ < M for all (x, u)). In this case the bound M does not need to be known. Again, the objective is to design a "corrective" control law p*(x) that stabilizes the closed-loop system. Following the same procedure as in Section 5.4.3, we consider a nominal Lyapunov function V0(x) that satisfies (5.93), (5.94) for some class  where (x) is the same as defined in (5.96). Now, let us select p*(x) as  where k > 0 is a scalar. By substituting (5.104) in (5.103) we obtain  Since 0(x, u) is uniformly bounded in (x, u),  The term  is of the form Q(α) = –kα2 + αM, where α = (x)2 • Φ(t, x)2; therefore, Q attains the maximum value of M/4k at α = M/2k. Therefore,  Since α3(x) is strictly increasing and approaches ∞ as x → ∞, there exists a ball  where can be made smaller by increasing the feedback gain k or by decreasing the infinity norm of the model error. ■ EXAMPLE 5.10 Consider the nonlinear model of Example 5.9. In this case, instead of assuming that The corrective control term obtained using the nonlinear damping method is given by
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. The nonlinear damping method developed in this section relaxes somewhat this assumption by not requiring that the bounding term 
is known.
∞ functions α1, α2, α3. The time derivative of V0 along the solutions of (5.101), (5.102) is given by
of radius such that 
for x outside 
is known, we assume that z(z) = Φ(z) 0(z) where Φ is known, while 0 is unknown but uniformly bounded.
