New results in optimal quadratic supply rates

This paper concerns studying dissipativity of a system with supply rates that depend on one or more parameters. We show that suitable choice of supply rate turns out to make dissipativity equivalent to traditional gain/phase margin conditions for stability. Further, the well-known circle criterion corresponds to a different supply rate, and here optimizing the supply rate is nothing but finding the largest circle such that circle criterion implies absolute stability for time-varying nonlinearities. L_¿-control is another example of dissipativity with respect to a relevant supply rate, and here we show that, in fact, improper L_¿-controllers are easily dealt with using our approach (unlike the standard state space methods). We formulate and prove necessary and sufficient conditions for L_¿-control, and then conclude that optimal controllers always exist (under suboptimal solvability conditions).