Design of optimal and robust control with $H_\infty/\gamma_0$ performance criterion

For a double-input single-output system, this paper defines a disturbance attenuation level (called $H_\infty/\gamma_0$ norm) as the maximum-value $L_2$ norm of the output under an unknown disturbance with a bounded $L_2$ norm supplied to the first input and an impulsive disturbance in the form of the product of an unknown vector and the delta function supplied the second input, where the squared $L_2$ norm of the former disturbance plus the quadratic form of the impulsive disturbance vector does not exceed 1. Weight matrix choice in the $H_\infty/\gamma_0$ norm yields a trade-off between the attenuation level of the $L_2$ disturbance and the attenuation level of the impulsive disturbance in corresponding channels. For the uncertain systems with dynamic or parametric uncertainty in the feedback loop, a robust $H_\infty/\gamma_0$ norm is introduced that includes the robust $H_\infty$ and $\gamma_0$ norms as special cases. All these characteristics or their upper bounds in the uncertain system are expressed via solutions of linear matrix inequalities. This gives a uniform approach for designing optimal and robust control laws with the $H_\infty/\gamma_0$, $H_\infty$ and $\gamma_0$ performance criteria.