ISSN 0005-1179, Automation and Remote Control, 2018, Vol. 79, No. 3, pp. 406–424.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
A.I. Malikov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 3, pp. 21–43.
Synthesis of State Unknown Inputs Observers
for Nonlinear Lipschitz Systems
with Uncertain Disturbances
A. I. Malikov
Tupolev Kazan National Research Technical University,
Kazan, Tatarstan, Russia
Institute of Mechanics and Engineering, Kazan Science Center, Russian Academy of Sciences,
Kazan, Tatarstan, Russia
Received December 7, 2016
Abstract—We propose methods to synthesize observers for the state and unknown input in-
ﬂuences that ensure that estimation error is ﬁnite time bounded with respect to given sets of
initial states and admissible trajectories or suppress initial deviations and uncertain bounded
-norm external disturbances for time-varying continuous Lipschitz systems. Here gain
coeﬃcients of the observers depend on time and are determined based on numerical solutions
of optimization problems with diﬀerential linear matrix inequalities or numerical solutions of
the corresponding matrix comparison system. With the example of an electric drive system
with elastic transmission of motion we show that their application for state estimation and
unknown inputs for time-invariant systems proves to be more eﬃcient (with respect to con-
vergence time and accuracy of the resulting estimates) compared to observers with constant
coeﬃcients obtained based on numerical solutions of optimization problems with linear matrix
Keywords: time-varying systems with Lipschitz nonlinearities, unknown input inﬂuences, un-
certain disturbances, measurement errors, observer synthesis
Guaranteed estimation problems were studied in the late 1960s and early 1970s in the works [1, 2]
and others. General approaches to controllability and observability of systems with uncertainties
based on convex analysis were proposed in the works of N.N. Krasovskii  and A.B. Kurzhanskii 
who initiated the theory of minimax guaranteed estimation for linear control systems . Ellipsoidal
technique for guaranteed state estimation in dynamical systems was developed in [6, 7]. A survey
of results in this ﬁeld can be found in [8–14].
The state estimation problem for nonlinear systems has been a subject of study over several
decades; see, e.g., a brief survey of this direction given in . The work  distinguishes four
diﬀerent approaches to synthesis of nonlinear observers. In the ﬁrst approach [17–19], a nonlinear
system with a nonlinear state transform is represented in the so-called observable normal form.
This special normal form lets one, during the synthesis of a nonlinear observer, reduce the problem
to fully linear error dynamics. It remains, however, a nontrivial problem to ﬁnd such a nonlinear
state transformation. The work  proposes an approach of stepwise linearization of the original
system and then applying linear estimation theory.
In the second approach, system dynamics is divided into a linear and nonlinear parts. The
linear part is assumed to be observable, and the nonlinear part is assumed to be locally or globally