Lyapunov function in the hyper complex phase space

The paper deals with the development of background for defining Lyapunov functions for a wide range of linear dynamical objects. This background is based on assuming that the Lyapunov function is redundant energy in the considered object and this energy is dissipated only during controlled motion. We assume the full derivative of the Lyapunov function for an autonomous motion of the control objects equals zero and we use its summands to define linear algebraic equations. The solution of these equations allows us to find unknown terms of the Lyapunov function. The use of these terms, while the Lyapunov equation is being written down, shows that the left-hand expression in the Lyapunov equation is equal to the zero matrix. Thus, we avoid subjective assuming of quadratic form terms in the right-hand of the Lyapunov equation. We extend the proposed approach to the class dynamical system with uncertainty. This extension is performed by using interval methods, which allow defining object motions for minimal and maximal values of parameters. We show that for the control object, which parameters are not exactly known, one should consider two equations of object motions, which correspond to its trajectories on the boundaries of the intervals. Lyapunov functions are defined for these boundary trajectories. Since such an approach increases the number of the considered equations we offer to decrease them by using hyper-complex numbers while object equations are written down

Linear flux observers for induction motors with quadratic Lyapunov certificates

International audienceWe propose a full order Linear Time-Varying (LTV) Luenberger observer for the rotor flux estimation of an induction motor. Introducing a suitable reduced-order Linear-Time-Invariant (LTI) system that is always observable and controllable, we show that any arbitrary LTI design and its quadratic Lyapunov certificates can be lifted to the higher-order original LTV dynamics obtaining the same certificates. As a result, we show that arbitrary global uniform exponential bounds can be imposed on the estimation error, regardless of the rotor speed. Then applying a suitable order reduction technique, we build a reduced observer providing the same guarantees. We also establish interesting connections between these reduced observers and existing non-adaptive observers. We also show applications of our results when wanting to minimize the effects of measurement noise on the estimates