Disturbance attenuation via state-feedback controllers (Q2721893)

The authors investigate the linear time-invariant system NEWLINE\[NEWLINE\left\{\begin{aligned}\dot x(t)& = Ax(t) + Bu(t) + Lw(t),\\ z(t)&=Cx(t)+Du(t)+Mw(t),\\ x(0)& = 0, \end{aligned}\right.NEWLINE\]NEWLINE where \(x(t)\in\mathbb R^n\) is the state, \(u(t)\in\mathbb R^m\) is the control input, \(w(t)\in\mathbb R^p\) is the disturbance input, \(z(t)\in\mathbb R^q\) is the controlled output, and \(A, B, C, D, L, M\) are constant matrices of appropriate sizes. The paper focuses on the \(H_\infty\) optimal control problem in which all the states are available for feedback. A design method for disturbance attenuation either via static or via dynamic state-feedbacks is developed. It is shown that in attenuating the disturbance, the \(H_\infty\) optimal performance of dynamic state-feedback is no better than that of static state-feedback, which generalizes current results for linear time-invariant systems with no direct transmission from the disturbance and control input to the controlled output.