Observer Design for One-Sided Lipschitz Nonlinear Systems Subject to Measurement Delays

Abstract

This paper presents a novel nonlinear observer-design approach to one-sided Lipschitz nonlinear systems in the presence of output delays. The crux of the approach is to overcome the practical consequences of time delays, encountered due to distant sensor position and time lag in measurement, for estimation of physical and engineering nonlinear system states. A Lyapunov-Krasovskii functional is employed, the time derivative of which is solved using Jensen’s inequality, one-sided Lipschitz condition, and quadratic inner-boundedness, and, accordingly, design conditions for delay-range-dependent nonlinear observer for delayed one-sided Lipschitz systems are derived. Further, novel solutions to the problems of delay-dependent observer synthesis of one-sided Lipschitz models and delay-range-dependent state estimation of linear and Lipschitz nonlinear systems are deduced from the present delay-range-dependent technique. An observer formulation methodology for retrieval of one-sided Lipschitz nonlinear-system states, which is robust againstL2norm-bounded perturbations, is devised. The resultant design conditions, in contrast to the conventional procedures, can be solved via less conservative linear matrix inequality- (LMI-) based routines that succeed by virtue of additional LMI variables, meaningful transformations, and cone complementary linearization algorithm. Numerical examples are worked out to illustrate the effectiveness of the proposed observer-synthesis approach for delayed one-sided Lipschitz systems.

Delay-range-dependent observer-based control of nonlinear systems under input and output time-delays

“…The matrix inequalities in Theorem 2 (or in Corollary 1) include nonlinear terms that can be tackled by employing cone complementary linearization algorithm approach [42][43][44]. The constraints (23) and (24) can be solved through the optimization Theorem 2.…”

“…In a recent work, an approach for observer design under measurement delays based on cone complementary linearization algorithm is developed to determine the observer gain (such as K 2 ) using LMIs. The proposed LMI-based treatment in (43)-(44) is a significant extension of the approach in [43] for the multiple time-delays at input as well as at output and for the controller gain matrix K 1 along with the observer gain K 2 . It is worth mentioning that the resultant methodology can be merely applied to solve delay-range-dependent observer-based controller synthesis problem by employing standard convex optimization routines for nonlinear systems under actuation and measurement delays.…”

Synchronization of nonlinear master–slave systems under input delay and slope‐restricted input nonlinearity

“…Hence, the nonlinear constraints in Theorem 2 can be solved using LMIs by means of the following optimization (see [30] …”

“…The inequalities in Theorem 2 and Corollaries 1 and 2 are nonlinear; however, the optimization problem in (45), containing a nonlinear objective function, can be solved using LMIs by application of the CCL algorithm (see details in [30,31] and references therein).…”

Separation principle for quasi‐one‐sided Lipschitz nonlinear systems with time delay

“…In this case, a state observer is usually employed to accurately reconstruct the state variables of the process. In recent years, a subject of tremendous research activities has been focused on observer design problem for nonlinear time‐delay systems …”