Gain scheduling versus robust control of LPV systems: The output feedback case

We compare robust and gain scheduling control for continuous-time LPV systems. In the literature it has been shown that some duality properties as well as a separation principle hold for gain-scheduling control. This means that an LPV plant can be stabilized by a gain scheduling control law if and only if a proper gain scheduling LPV state observer and LPV stabilizer can be designed. Moreover observer and state-feedback designs are dual problems. In this paper we show, by means of counterexamples that, in robust design neither of the above properties hold. Precisely, it is shown that, if the parameter values are not measurable on-line, then there exist robust stabilizable plants for which no state detection is possible and that there exist plants which are stabilizable via state feedback whose dual system does not admit robust observers. In passing we solve an open question. In the state-feedback design case it was previously shown the equivalence between robust and gain-scheduling control for polytopic systems, namely that state-feedback gain-scheduling stabilizability implies state-feedback robust stabilizability. We prove that this is not true anymore in the output feedback case. The discrete-time case will be briefly discussed at the end.