**Robust Control**

**Chapter 1**

(1)

is one disturbance in‐

is the measured

**Stochastic Mixed LQR/H∞ Control for Linear**

Mixed *H*<sup>2</sup> / *H∞* control has received much attention in the past two decades, see Bernstein & Haddad (1989), Doyle et al. (1989b), Haddad et al. (1991), Khargonekar & Rotea (1991), Doyle et al. (1994), Limebeer et al. (1994), Chen & Zhou (2001) and references therein. The

mixed *H*<sup>2</sup> / *H∞* control problem involves the following linear continuous-time systems

*x*˙(*t*)= *Ax*(*t*) + *B*0*w*0(*t*) + *B*1*w*(*t*) + *B*2*u*(*t*), *x*(0)= *x*<sup>0</sup> *z*(*t*)=*C*1*x*(*t*) + *D*12*u*(*t*) *y*(*t*)=*C*2*x*(*t*) + *D*20*w*0(*t*) + *D*21*w*(*t*)

is another disturbance input that belongs to*<sup>L</sup>* <sup>2</sup> 0,*∞*), *y*(*t*)∈*<sup>R</sup> <sup>r</sup>*

Bernstein & Haddad (1989) presented a combined LQG/*H<sup>∞</sup>* control problem. This problem is defined as follows: Given the stabilizable and detectable plant (1) with *w*0(*t*)=0 and the

> © 2012 Xu; licensee InTech. This is an open access article 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.

© 2012 Xu; 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.

where,*x*(*t*)∈*<sup>R</sup> <sup>n</sup>* is the state, *u*(*t*)∈*<sup>R</sup> <sup>m</sup>*is the control input, *w*0(*t*)∈*<sup>R</sup> <sup>q</sup>*<sup>1</sup>

**Discrete-Time Systems**

http://dx.doi.org/10.5772/51019

Additional information is available at the end of the chapter

Xiaojie Xu

**1. Introduction**

put, *w*(*t*)∈*R <sup>q</sup>*<sup>2</sup>

expected cost function

output.
