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 5.4.1 - Bounding Control
Bounding control is one of the simplest approaches for dealing with unknown nonlinearities. Here, we consider a simple scalar system with one unknown nonlinearity, which lies within certain known bounds. This approach can be extended to more complex systems. In Chapter 6, we will revisit bounding control as a way of motivating adaptive approximation of the unknown component of nonlinear systems. Consider the scalar nonlinear system  where the objective is to design a control law such that y(t) = x(t) tracks a desired signal yd (t). Let e(t) – y(t) –yd (t) be the tracking error. We assume that the function ƒ is unknown but belongs to a certain known range as follows:  where ƒL and ƒUare known lower and upper bounds, respectively, on the unknown function ƒ. In general, the bounds ƒL and ƒU may be positive or negative, or their sign may change as x varies. Consider the following control law:  where am> 0. Using the above control, it is easy to see that the tracking error dynamics satisfy  Now, let V = ½e2 be a Lyapunov function candidate. The time derivative of V satisfies  Therefore, me tracking error converges to zero exponentially fast. It is noted that, in general, the control law (5.82) is discontinuous at e = 0. This may result in the trajectory x(t) going back and forth between The chattering can be remedied by using a smooth approximation to the control law of the form  where δ > 0 is a small design constant. Exercise 5.18 asks the reader to prove that the closed-loop system with the above smooth approximation of the discontinuous bounding control achieves convergence to the set x < δ in finite time. |
, causing the control law to be switching, thus creating chattering problems. By 
we denote a value of the trajectory y(t) which is slightly larger than yd (t), thus causing the tracking error e to be slightly positive, and correspondingly, 
denotes a value of the trajectory which is slightly smaller than yd (t). 
