Multiplier Theory in Control

Multiplier techniques is a powerful analysis tool in analyzing the closed-loop interconnection between a linear time-invariant system in feedback with a memoryless nonlinearity that belongs to certain class. In the literature, this is also known as the Lur'e type of problem and absolutely stable if the closed-loop system is asymptotically stable for all nonlinearity in the class. Many physical systems can be modeled with this type of interconnection, such as actuator saturation control problem, Phase-Locked Loop (PLL) design, steering control in vehicles, or even uncertainties in robust control design. Then through the passivity theorem, conditions for the stability of closed-loop interconnection can be constructed based purely on its linear part. However, the passivity arguments involves the strict positive-realness of transfer functions, which are inherently conservative and may produce undesired results. This conservatism can be mitigated with the use of multiplier in the loop to achieve better results. Popular class of multipliers, such as the Circle and Popov criterion, and class of Zames-Falb multipliers, are stated in frequency domain but can also be converted into time-domain formulations through tools such as, Linear Matrix Inequalities (LMIs), and Positive-real Lemma, also known as the Kalman-Yakubovich-Popov (KYP) lemma