**Game Theory in Engineering**

104 Game Theory Relaunched

[25] Eaton J and Grossman GM (1986). Optimal trade and industrial policy under oligopoly*.*  In: Schmalense R and Williy RD (Eds.) (1989). *HandBook of Industrial Organization*

(Vol.11., pp. 1181-1223). Elsevier Science Publishers.

**Chapter 0**

**Chapter 5**

**A Game Theoretic Approach Based Adaptive**

**SISO Linear Systems**

http://dx.doi.org/10.5772/50488

**1. Introduction**

control in the 1990s.

disturbance inputs.

properly cited.

Sheng Zeng and Emmanuel Fernandez

Additional information is available at the end of the chapter

**Control Design for Sequentially Interconnected**

Adaptive control has attracted a lot of research attention in control theory for many decades. In the certainty equivalence based adaptive controller design [4, 5], the unknown parameters of the uncertainty system are substituted by their online estimates, which are generated through a variety of identifiers, as long as the estimates satisfy certain properties independent of the controller. This approach leads to structurally simple adaptive controllers and has been demonstrated its effectiveness for linear systems with or without stochastic disturbance inputs [10] when long term asymptotic performance is considered. Yet, the certainty equivalence approach is unsuccessful to generalize to systems with severe nonlinearities. Also, early designs based on this approach were shown to be nonrobust [13] when the system is subject to exogenous disturbance inputs and unmodeled dynamics. Then, the stability and the performance of the closed-loop system becomes an important issue. This has motivated the study of robust adaptive control in the 1980s and 1990s, and the study of nonlinear adaptive

The topic of adaptive control design for nonlinear systems was studied intensely in the last decade after the celebrated characterization of feedback linearizable or partially feedback linearizable systems [7]. A breakthrough is achieved when the integrator backstepping methodology [8] was introduced to design adaptive controllers for parametric strict-feedback and parametric pure-feedback nonlinear systems systematically. Since then, a lot of important contributions were motivated by this approach, and a complete list of references can be found in the book [9]. Moreover, this nonlinear design approach has been applied to linear systems to compare performance with the certainty equivalence approach. However, simple designs using this approach without taking into consideration the effect of exogenous disturbance inputs have also been shown to be nonrobust when the system is subject to exogenous

> ©2013 Zeng and Fernandez, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

©2013 Zeng and Fernandez, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.
