Received: 21 December 2016 Revised: 10 August 2017 Accepted: 10 October 2017
Application of the state-dependent Riccati equation for
flexible-joint arms: Controller and estimator design
M. H. Korayem N. Y. Lademakhi S. R. Nekoo
Robotic Research Laboratory, Center of
Excellence in Experimental Solid
Mechanics and Dynamics, School of
Mechanical Engineering, Iran University
of Science and Technology, Tehran, Iran
M. H. Korayem, Robotic Research
Laboratory, Center of Excellence in
Experimental Solid Mechanics and
Dynamics, School of Mechanical
Engineering, Iran University of Science
and Technology, Tehran 16846 13114, Iran.
Full feedback data is mostly essential in control design. Measuring the links'
variation of a flexible-joint robot (FJR) is easy but not for its actuators. The
measurement of states and workspace are also affected by noise and dis-
turbance, respectively; hence, a state observer or a nonlinear estimator is
very helpful to improve the control performance of a dynamic system. The
state-dependent Riccati equation (SDRE) is one of the most promising meth-
ods in the field of nonlinear optimal observers for estimating variables of
multiple-input–multiple-output systems. Systematic procedure, simple struc-
ture, and incorporation of a wide range of systems (under an observability
condition) are some of the advantages of the method. Therefore, the aim of this
study is to compose the SDRE controller and estimator simultaneously to reduce
the state error of the system in the presence of external disturbance and noise.
The application of this method is shown for FJRs in the estimation of changes in
motion behind the connection of the motor and link where there is no easy way
to measure. The proposed composition of the SDRE controller and estimator was
implemented on a 6R robot to examine the various aspects of FJR systems.
flexible-joint manipulator, noise and disturbance, nonlinear optimal control, observer and estimator,
state-dependent Riccati equation
In addition to robot control, the measurement of a system's variable is important and it is usual to assume that all the
states are available in the control design; however, the measurement of each of them needs a sensor in practice. Since
physical measurement is not possible at all times, precision might not be good enough because there may be an increase
in cost. Estimating state variables from the output of the system under observable conditions is a solution to overcome
the problem. Luenberger introduced an observer for linear systems.
Thereafter, observers were explored for determin-
istic autonomous and nonautonomous systems.
Ji et al employed the linear quadratic Gaussian method to control a
utilized a class of PID controller for a robot with elastic joints and proved its semi global
Contrary to linear observers, nonlinear ones do not follow a unified structure. Some nonlinear observers are studied
as follows: observer with linearizable error dynamic,
sliding mode observer,
and state-dependent Riccati equation (SDRE) observer and estimator.
Interpolation algorithms were also used
792 Copyright © 2017 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/oca Optim Control Appl Meth. 2018;39:792–808.