Observer-Based Sequential Control of a Two Time Scale Spring-Mass-Damper System

Robustness analysis of nonlinear observers for the slow variables of singularly perturbed systems

“…Consider the variablê = [ , ] , it follows from Lemma 3.2 in Khalil 29 that for any Δ 1 > 0 and Δ 2 > 0 there exists 0 > 0 so that when | ℎ ∕ ̂ | ≤ for all and such that | | ≤ Δ 1 and | | ≤ Δ 2 , there exists > 0 such that | ℎ ∕ | ≤ . We have verified that Assumption 5 holds for reduced-order 2 , full-order [1][2][3][4][5][6][7] , and higher-order 8 observers.…”

“…While Kazantzis et al 1 considers nonlinear plants where the slow part of the model satisfies a Lipschitz condition and the fast dynamics and the output of the system are linear, here we deal with a more general class of nonlinear plants. Moreover, whilst Kazantzis et al 1 only works with a nonlinear Luenbergertype observer that exhibits a linear error dynamics for the reduced system, our results can cover a number of nonlinear observers including reduced-order 2 , full-order [1][2][3][4][5][6][7] , and higher-order 8 observers.…”

“…Nevertheless, those observers are valid only locally and only work well when the system is evolving close to the equilibrium point considered for the linearisation. To counteract this problem, a wide variety of nonlinear observer design methods have been developed [1][2][3][4][5][6][7][8][9][10] , but these approaches may lead to ill-conditioned gains when used for systems exhibiting multiple time scales.…”

“…The authors in Kazantzis et al 1 only consider a particular class of plants and a specific nonlinear observer that exhibits linear error dynamics for the slow system. An observer for spring-mass-damper systems with singularly perturbed structure is given in Saha and Valasek 6,7 ; however, those results apply to a very specific class of systems and observers. Other results on observer design for nonlinear singularly perturbed systems are available in Wang and Liu 20 , and Darrogheh et al 21 .…”

“…Then, in Section 5 we illustrate and demonstrate the applicability of our results by presenting three classes of plants and observers for which our results hold. Although not presented here, the results cover the situation when the reduced (slow) system is such that reduced-order 2 , full-order [1][2][3][4][5][6][7] , and higher-order 8 observers can be used to estimate the slow variables. Finally, to demonstrate the theoretical findings, we test the results on a simulation of a suspension system.…”