Less conservative performance bounds with the LMI method for a class of semi-active control problems with nonlinear actuator dynamics

The problem of establishing less conservative performance bounds relative to a classical technique is investigated for a class of semi-active control problems with nonlinear actuator dynamics and discontinuous feedback control. The performance problem is formulated as a linear matrix inequality problem and it is shown by way of a 3D example that this approach provides several orders of magnitude performance bound improvement over that obtainable by a classical technique. However, the LMI bound is still one order of magnitude larger than that provided by simulation. It is interesting to note that the LMI bound is the same order of magnitude as that provided by the L₁-system norm of the linear system that results from the case where the valve on the semi-active element remains open on the whole space (equivalent to a passive damper). Even with the large improvement over the classical technique, the LMI bound did not differentiate between various competing Lyapunov type controllers for the 3D example. The LMI method is only used here to establish an upper bound on the system performance. The control design here is based on the steepest descent Lyapunov formulation, not on the LMI treatment.