# Robust Adaptive Control of a Class of Nonlinear Systems

## Abstract

In this chapter, we present some theoretical design tools for robust adaptive control of nonlinear systems *[84, 120, 164]* for application to various motor control problems. The development follows the lines of *[87, 88]* . First, the control design for the case that the strict matching condition is satisfied is outlined. Next, the design is extended using a combination of backstepping, nonlinear damping, and tuning functions *[120]* to a class of nonlinear systems that do not satisfy the strict matching condition, namely systems of the *special strict feedback form [87, 88].* These systems are characterized using the notion of the degree of mismatch that illustrates the tradeoff between the state equations where the uncertainty can appear, and the state variables on which the uncertain terms depend. For the case where only the outputs are measured, the adaptive design procedure applies to systems transformable to the output feedback form [89, 136] where the unknown terms are output-dependent. Results in both cases are presented for state/output tracking and regulation problems. For the case where the objective is tracking, global uniform boundedness of the tracking error is attained with respect to a compact set whose size can be made arbitrarily small *(practical* tracking) based on an appropriate choice of control gains. In the regulation case, global asymptotic regulation of all the states is achieved. Before reading this chapter, Appendices F and G provide a good basis for some of the required nonlinear design tools such as backstepping, tuning functions, and nonlinear small gain.

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## References

- 1.For simplicity, in this chapter, we focus on polynomial-bounded uncertainties. Appendix F contains a description of control design techniques in the presence of uncertainties bounded by general nonlinear functions of the states. See also Appendix G for a discussion of the notion of input-to-state stability, which is useful in the treatment of interconnected systems and systems with nonlinear dynamic uncertainties.Google Scholar
- 2.This assumption can be relaxed to ∣ĝ(z, v)∣ <
*pğ(z, v)*with*ğ(z, y)*being a known function. In fact, even this restriction can be removed if the term*Kz*,_{m}*-+ r*is omitted from the control input. This feedforward term, however, greatly improves tracking performance and is desirable to retain.Google Scholar - 4.The design can be extended to the case where
*bo, ... , b*_{np}are unknown constant parameters by using the Marino-Tomei filtering technique [136] at the expense of more dynamics in the observer.Google Scholar - 5.The design can be extended to handle the case where b
_{i}'s are unknown by utilizing MarinoTomei filters resulting in added dynamics in the observer [1361.Google Scholar