VCC 2010 Program

Overall Time Table

8.00-9.00: Opening and keynote

9.00-12.00: first session (6 papers)

12.00-13.00: lunch break

13.00-16.30: second session (7 papers)

16.30-17.00: coffee break

17:00-20.00: third session (6 papers)

20.00-20.15: practical informations and closing

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Titles and authors

8.00: Opening and welcome

Professor Jakob Stoustrup, Aalborg University, Denmark

Keynote: 8.15

Professor Karl-Erik Årzen, Lund University, Sweden:

"Control for Embedded Systems"

Abstract: The strong focus on cyber-physical systems and the increased
demands on power efficiency have led to new challenges for control in
embedded systems. In this talk I will discuss the background for this
and focus on two research topics. The first is co-design of controllers
and task scheduling and the second is the use of feedback techniques in
order to increase the flexibility and adaptivity of embedded systems.

___________________________________________________________________________________________________

Session 1: 9.00
Robust Control, Linear Systems, Optimization, Proces Control

Reference Tracking and Profit Optimization of a Power Plant

Martin Kragelund
Email: mkr@es.aau.dk
John Leth
Email: jjl@es.aau.dk
Rafal Wisniewski
Email: raf@es.aau.dk
Section for Automation and Control, Dept. for Electronic Systems, Aalborg University Denmark

Modeling and Control of an Experimental pH Neutralization Plant using Neural Networks based Approximate Predictive Control
Ireneus Wior, Institute of Automation Technology, Helmut-Schmidt-University, Germany, Email: ireneuswior@arcor.de
Sudchai Boonto, Hossam Seddik Abbas and Herbert Werner
Institute of Control Systems, Hamburg University of Technology, Germany

Sensitivity Analysis of the LMI-based H-inf Control Problem
A. S. Yonchev*, P. Hr. Petkov*, N. D. Christov** and M. M. Konstantinov***
*Department of Systems and Control, Technical University of Sofia
1000 Sofia, Bulgaria; Email: ajonchev@mail.bg, php@tu-sofia.bg
**Laboratory of Automatics, Computer Engineering and Signal Processing
Lille University of Science and Technology, 59655 Villeneuve d’Ascq
France; Email: Nicolai.Christov@univ-lille1.fr
**Department of Mathematics, University of Architecture, Civil Engineering
and Geodesy, 1046 Sofia, Bulgaria; Email: mmk fte@uacg.bg

Revisited H-inf control with transients: LMI based time-invariant output-feedback controllers
Dmitry V. Balandin
Department of Numerical and Functional Analysis
Nizhny Novgorod State University
Gagarin ave., 23, Nizhny Novgorod, 603950 Russia
Email: dbalandin@yandex.ru
Mark M. Kogan
Department of Mathematics
Architecture and Civil Engineering University
Il’yinskaya str., 65, Nizhny Novgorod, 603950 Russia
Email: mkogan@nngasu.ru

Robust H-inf Output Feedback Sliding Mode Control With Applications
Dr.-Ing Bader W. Juma and Prof. Dr. Herbert Werner
Institute of Control Systems, Hamburg University of Technology
Email:baderwady@yahoo.com, h.werner@tu-harburg.de
Telephone: +49-40-42878-3215, Fax: +49(0)40 42 878-2112

Quadratic estimation of discrete-time signals using observations with multiple packet dropouts
R. Caballero-A´ guila
Dpto. de Estad´istica e I.O., Universidad de Ja´en,
Paraje Las Lagunillas s/n, 23071 Ja´en, Spain
Email: raguila@ujaen.es
A. Hermoso-Carazo and J. Linares-P´erez
Dpto. de Estad´istica e I.O., Universidad de Granada,
Avda. Fuentenueva s/n, 18071 Granada, Spain
Email: ahermoso@ugr.es, jlinares@ugr.es

________________________________________________________________________

Lunch Break: 12.00

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Session 2: 13.00
Control Applications and Modelling

Modelling of A Helicopter System
K.K.T. Thanapalan
Faculty of Engineering Sciences, University College London
London WC1E 6BT, U.K
E-mail: kary.thanapalan@ucl.ac.uk

Robust Design of Terminal Iterative Learning Control with µ-synthesis Approach
Applied for thermoforming oven control
Gauthier, Guy Department of automated production engineering, École de technologie supérieure
1100 Notre-Dame Ouest, Montreal, Canada
guy.gauthier@etsmtl.ca
Boulet, Benoit, Department of electrical and computer engineering, McGill University
3480 University, Montreal, Canada
benoit.boulet@mcgill.ca

Gain-scheduled H-inf control of a robotic manipulator with nonlinear joint friction

Hashemi, Seyed Mahdi, Email: seyed.hashemi@tu-harburg.de
Herbert, Werner, Email: h.werner@tu-harburg.de

Institute of Control Systems, Hamburg University of Technology, Germany

An Observer Based Scheme for Adapt to Blade Aerodynamic Parameters for Power Control of Wind Turbines
Peter Fogh Odgaardpeodg@kk-electronic.com
Rasmus Nielsenranie@kk-electronic.com
Chris Damgaardchdam@kk-electronic.com
kk-electronic a/s
Viby J, Denmark

Critical fault detection, by measured current on electromechanical hydraulic valves
Lasse Skovlasko@kk-electronic.com
Peter Fogh Odgaardpeodg@kk-electronic.com
Rasmus Nielsenranie@kk-electronic.com
kk-electronic a/s
Ikast, Denmark

Unknown Input Observer Based Detection Scheme for Faults in Hydraulic Valves
Peter Fogh Odgaardpeodg@kk-electronic.com
Lasse Skovlasko@kk-electronic.com
Rasmus Nielsenranie@kk-electronic.com
kk-electronic a/s
Ikast, Denmark

A Simulation Based Investigation of Interactions between VVA and Idle Control for SI Engines
Antonio Palma and Ferdinando De Cristofaro
Systems and Controls, Gasoline EMS, Elasis S.C.p.A. Via ex Aeroporto snc,
80038, Pomigliano d’Arco (NA) Italy.
Email: antonio.palma@fptpowertrain.elasis.it
ferdinando.decristofaro@fptpowertrain.elasis.it
Angelo Palladino and Giovanni Fiengo
Dipartimento di Ingegneria, Universit`a degli Studi del Sannio,
Piazza Roma 21, 82100 Benevento, Italy.
Email: angelo.palladino@unisannio.it,
gifiengo@unisannio.it
Alessandra Guzzo
Universit`a della Calabria, Rende (CS), Italy
Email: guzzo.alessandra@gmail.com

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Coffee Break: 16.30
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Session 3: 17.00
Nonlinear Systems, Hybrid Systems, Autonomous Systems, System Identification, Adaptive and Learning Systems

Optimal Control Design for Polynomial Nonlinear Systems using Sum of Squares Technique with Guaranteed Local Optimality

N. Boonnithivoraku, nattpong@siu.edu
F. Pourboghrat, Senior Member, IEEE, pour@siu.edu
Southern Illinois University, Carbondale, United States

Switching boundary feedback stabilization for a star-shaped network of strings
Martin Gugat
Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Lehrstuhl 2 f¨ur Angewandte Mathematik, Martensstr. 3, 91058, Erlangen, Germany
Email: gugat@am.uni-erlangen.de
Mario Sigalotti
INRIA Nancy – Grand Est and Institut ´Elie Cartan
UMR CNRS/INRIA/Nancy Universit´e BP 239, 54506
Vandoeuvre-l`es-Nancy, France
Email: mario.sigalotti@inria.fr

Singular Perturbation Approach to Pulse-Width Modulated Control in Nonlinear Dynamical Systems
Valery D. Yurkevich
Automation Department, Novosibirsk State Technical University
Novosibirsk, Russia, 630092
Email: yurkev@ac.cs.nstu.ru

Robust Formation Controller Synthesis with Different Time Delays
Ulf Pilz, Herbert Werner
Hamburg University of Technology, Institute of Control Systems
Eissendorfer Str. 40, D-21073 Hamburg, Germany
Email: {ulf.pilz, h.werner}@tu-harburg.de

On adaptive control problems of continuous-time stochastic systems
Uwe K¨uchler
Institute of Mathematics, Humboldt University Berlin
Unter den Linden 6, D-10099, Berlin, Germany
Email: kuechler@math.hu-berlin.de
Vyacheslav A. Vasiliev
Department of Applied Mathematics and Cybernetics,
Tomsk State University,
Lenina 36, 634050 Tomsk, Russia
Email: vas@mail.tsu.ru

Fractional Order Ultra Low-Speed Position Servo: Improved Performance via Describing Function Analysis
Ying Luo*, YangQuan Chen** and YouGuo Pi***
*Dept. of Automation Science and Engineering, South China University of Technology, Guangzhou, P. R. China
Email: ying.luo@ieee.org
**Center for Self-Organizing and Intelligent Systems (CSOIS),
Dept. of Electrical and Computer Engineering, Utah State University, 4120 Old Main Hill, Logan, UT 84322, USA.
Email: yqchen@ieee.org
***Dept. of Automation Science and Technology, South China University of Technology, Guangzhou, P. R. China
Email: auygpi@scut.edu.cn

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Practical informations and closing 20.00-20.15
__________________________________________________________________________________________

Abstracts

Session 1
Robust Control, Linear Systems, Optimization, Proces Control

Reference Tracking and Profit Optimization of a Power Plant

Kragelund, Martin, Email: mkr@es.aau.dk, Leth, John, Email: jjl@es.aau.dk, Wisniewski Rafal, Email: raf@es.aau.dk
Section for Automation and Control, Dept. for Electronic Systems, Aalborg University

In this paper we discuss two different
methods for implementing reference tracking in a
profit optimization problem of a power plant. It is
shown that tracking included as a side constraint
results in an significant tracking error only when the
reference gradient is large. When tracking is included
in the cost function, as a quadratic term, the reference
is tracked with a small accumulated error. Finally, the
two methods are compared both in terms of tracking
performance and computational burden. (paper ((pdf))

Modeling and Control of an Experimental pH Neutralization Plant using Neural Networks based Approximate Predictive Control Ireneus Wior, Institute of Automation Technology, Helmut-Schmidt-University, Germany, Email: ireneuswior@arcor.de
Sudchai Boonto, Hossam Seddik Abbas and Herbert Werner
Institute of Control Systems, Hamburg University of Technology, Germany

A nonlinear experimental pH neutralization plant is
controlled using a neural networks based Approximate Predictive
Control (APC) strategy. First a closed-loop identification is
performed, further, using neural networks, a black-box modeling
of the experimental plant is conducted. Then the approximate predictive
controller is realized, where a linear model of the plant is
extracted at each sampling period from the neural network model.
This strategy is used to control the experimental neutralization
plant for set point tracking and disturbance rejection. (paper ((pdf))

Local perturbation bounds are obtained for the
continuous-time H-inf control problem based on linear matrix
inequalities (LMI). The sensitivity analysis of the perturbed LMI
is done by introducing a suitable slightly perturbed right-hand
part. This approach leads to tight, condition number based
perturbation bounds for the LMI solutions to the H-inf control
problem. (paper ((pdf))

Attenuating both exogenous signals and initial disturbances
caused by unknown initial conditions is considered
in the framework of a so called problem of H-inf control with
transients. Applying an LMI approach, instead of the Riccati
equations one, in characterizing the performance measure that is
the worst-case norm of the regulated output over all exogenous
signals and initial states allows one to synthesize a time-invariant,
instead of the time-varying, output-feedback controller for which
the performance measure of the closed-loop system is less than
a prescribed number. State-space formulae for all time-invariant
state- and output-feedback controllers in the problem of the H-inf
control with transients are also presented. (paper ((pdf))

This paper presents an output feedback sliding
mode control scheme for uncertain dynamical systems. The design
problem is solved in two steps, involving first a state feedback
and then an output feedback problem. First, using the null space
dynamics, the sliding surface for the unmatched uncertainty is
designed. Then, by tuning the sliding surface a robust controller
is constructed for the whole uncertainty; this problem takes the
form of static output feedback. Based on this, a dynamic output
feedback controller for the system augmented with the sliding
surface is designed. The synthesis involves the solution of an
LMI and a BMI problem; the BMI problem is solved iteratively.
The proposed approach is illustrated by applying it to a wellknown
robust benchmark problem, and also experimentally on
a spring mass system with variable stiffness. Simulation and
experimental results show that the proposed method outperforms
previous approaches in terms of robust performance. (paper ((pdf))

The least-squares quadratic filtering and fixed-point
smoothing problems of discrete-time stochastic signals from
observations with multiple packet dropouts are addressed. The
random dropouts are modelled by introducing a sequence of
Bernoulli random variables with known distributions in the
observation model. A recursive estimation algorithm is deduced
without requiring full knowledge of the state-space model generating
the signal process, but only information about the dropout
probabilities and the moments of the processes involved. Defining
suitable augmented signal and observation vectors, the quadratic
estimation problem is reduced to the linear estimation problem
of the augmented signal based on the augmented observations,
which is solved by using an innovation approach. (paper ((pdf))

________________________________________________________________________

Session 2
Control Applications and Modelling

Modelling of A Helicopter System
K.K.T. Thanapalan
Faculty of Engineering Sciences, University College London
London WC1E 6BT, U.K
E-mail: kary.thanapalan@ucl.ac.uk

This paper considers modelling and
simulation study of a helicopter system – UH-60 Black
Hawk helicopter. Mathematical model of single main rotor
helicopters is presented in this paper. For the convenience of
presentation, force and moment expressions of the various
helicopter components are given in the paper to bridge a
generic model to the model of UH-60 Black Hawk
helicopters. For simulation study a UH-60 like Flightlab
GRM model (Generic Rotorcraft Model) is used.
Comparisons are made between the simulation results and
flight test data. A general agreement exits but where
disagreements and anomalies occur, clues are gathered to
give explanation. Overall the model represents the UH-60
Black Hawk helicopter. This model can be used for
controller development to improve flight handling quality
and performances. (paper ((pdf))

Robust Design of Terminal Iterative Learning Control with µ-synthesis Approach
Applied for thermoforming oven control
Gauthier, Guy
Department of automated production engineering, École de technologie supérieure
1100 Notre-Dame Ouest, Montreal, Canada
guy.gauthier@etsmtl.ca
Boulet, Benoit
Department of electrical and computer engineering, McGill University
3480 University, Montreal, Canada
benoit.boulet@mcgill.ca

This paper presents a robust design approach for
the Terminal Iterative Learning Control (TILC) algorithm
based on the µ-synthesis approach. TILC is used to control
the reheat phase of plastic sheets in a thermoforming oven.
TILC adjusts the heater temperature setpoints so that the
plastic sheet temperature measured at the end of the reheat
cycle converges to a desired temperature after a few cycles.
Simulation results are included to show the effectiveness of
this robust TILC algorithm. (paper ((pdf))

Gain-scheduled H-inf control of a robotic manipulator with nonlinear joint friction
Hashemi, Seyed Mahdi, Email: seyed.hashemi@tu-harburg.de
Herbert, Werner, Email: h.werner@tu-harburg.de Country: Germany
Institute of Control Systems, Hamburg University of Technology

This paper presents the LPV modelling and control
of a robotic manipulator with a nonlinear joint friction
model. A nonlinear dynamic model of the manipulator including
viscous and Coulomb friction terms is obtained and the
signum function in the friction model is approximated by a
hyperbolic function in order to smooth such hard nonlinearity.
A quasi-LPV model is derived and since it has a large number
of affine scheduling parameters and a large overbounding,
parameter set mapping is used to reduce conservatism and
complexity in controller design by finding tighter parameter
regions with fewer scheduling parameters. Then, a polytopic
LPV gain-scheduled controller is synthesized and implemented
experimentally on an industrial robot for a trajectory tracking
task. The experimental results illustrate that the designed LPV
controller outperforms a similar LPV controller based on a
linear friction model, a model-based inverse dynamics and a
decentralized PD controller in terms of tracking performance. (paper ((pdf))

An Observer Based Scheme for Adapt to Blade Aerodynamic Parameters for Power Control of Wind Turbines
Peter Fogh Odgaard
kk-electronic a/s
An Observer Based Scheme for Adapt to Blade
Aerodynamic Parameters for Power Control of Wind
Turbines
Peter Fogh Odgaard
peodg@kk-electronic.com
Rasmus Nielsen
ranie@kk-electronic.com
Chris Damgaard
chdam@kk-electronic.com
kk-electronic a/s
Viby J, Denmark

This paper presents an observer based scheme for
adapting the power control of wind turbines to the actual
power coefficients of the blades mounted on the wind turbine.
Normally it is assumed that the power coefficients for one
turbine in a production series are valid for all the turbines in
that production series. An unknown input observer is used to
estimate the actual table of power coefficients depending on
blade pitch angle and tip speed ratio. If the actual table is
much different from the initial assumed table, the actual table
is found by iterations. A simulation is used to illustrate the
schemes potential to estimate the power coefficients and to see
the gained potential of the use of this scheme compared with a
non-corrected situation. (paper ((pdf))

Critical fault detection, by measured current on electromechanical hydraulic valves
Lasse Skov
lasko@kk-electronic.com
Peter Fogh Odgaard
peodg@kk-electronic.com
Rasmus Nielsen
ranie@kk-electronic.com
kk-electronic a/s
Ikast, Denmark

Fault detection in large scale industrial systems are
of importance in order to detect and accommodate eventual
faults and failures in the system. Such an industrial system is a
hydraulic system which typically consists of a high number of
control valves. It would be beneficial to detect eventually
faults in these valves. In computer controlled systems these
valves are often electromagnetic actuated. In this paper a
model of an electromagnetic normally-closed hydraulic valve
is completed. This model of the valve is used to test a scheme
to detect blocked gliders in the electromagnetic actuated
hydraulic valve. The scheme tested in this paper only detects
when a valve is blocked. This is a very important feature to
have in a system with many hydraulic valves, since it can take
a lot of time to find the valve with the failure in such a system.
In a situation like this it is critical to detect and isolate a
blocked valve; however, it is not necessary to detect the fault
at the instant it occurs. The algorithm used in this paper find
100% faulty cases, without any false positive detection. (paper ((pdf))

Unknown Input Observer Based Detection Scheme for Faults in Hydraulic Valves
Peter Fogh Odgaard
peodg@kk-electronic.com
Lasse Skov
lasko@kk-electronic.com
Rasmus Nielsen
ranie@kk-electronic.com
kk-electronic a/s
Ikast, Denmark

In this paper an observer based scheme is proposed
to detect blocked gliders in electromagnetic actuated hydraulic
valves. Detection of blocked gliders in electromagnetic
actuated hydraulic valves is of large importance in large
hydraulic systems which contain a large number of control
valves. If the glider position was measured this detection
would be simple, however, in many cases only the coil current
is measured. In case of a blocked glider, it can be viewed as
the introduction of an extra force to keep the glider in position.
This extra force can be viewed as an unknown input, and can
be estimated by the use of an unknown input observer. Using
this estimated fault signal gives correct fault detection in case
of 96.6% of 1500 faulty cases and only 4% in case of no
faults, these numbers are found using Monte Carlo
simulations. The detection time is as fast as the step response
of the current changes in the valve. (paper ((pdf))

A Simulation Based Investigation of Interactions between VVA and Idle Control for SI Engines
Antonio Palma and Ferdinando De Cristofaro
Systems and Controls, Gasoline EMS, Elasis S.C.p.A. Via ex Aeroporto snc,
80038, Pomigliano d’Arco (NA) Italy.
Email: antonio.palma@fptpowertrain.elasis.it
ferdinando.decristofaro@fptpowertrain.elasis.it
Angelo Palladino
and Giovanni Fiengo
Dipartimento di Ingegneria, Universit`a degli Studi del Sannio,
Piazza Roma 21, 82100 Benevento, Italy.
Email: angelo.palladino@unisannio.it,
gifiengo@unisannio.it
Alessandra Guzzo
Universit`a della Calabria, Rende (CS), Italy
Email: guzzo.alessandra@gmail.com

Automotive idle speed control is a critical issue in
engine control fields. Essentially it is a highly nonlinear and timevarying
problem. Its performance has a significant impact on fuel
economy and emission levels. In this paper, the authors present
a complete and coherent engine model, aimed at the challenging
purpose of the analysis of the interaction between the idle speed
control and variable valve actuation system. The model is based
on an innovative approach for engine dynamics conceived mainly
on the analogy with electric systems. Firstly, the behavior of a
relatively simple and well known control, named Mid-Ranging
scheme has been tested on a complete ”in-cylinder” engine model,
after a modified control is proposed, in order to test the modern
engine performances improvement due to the synergy between
variable valve actuation devices and idle speed control strategy. (paper ((pdf))

______________________________________________________________________________

Session 3:
Nonlinear Systems, Hybrid Systems, Autonomous Systems, System Identification, Adaptive and Learning Systems

Optimal Control Design for Polynomial Nonlinear Systems using Sum of Squares Technique with Guaranteed Local Optimality
N. Boonnithivoraku, nattpong@siu.edu

F. Pourboghrat, Senior Member, IEEE, pour@siu.edu

Southern Illinois University, Carbondale, United States

This paper deals with a computational approach to find the optimal control for nonlinear systems with polynomial vector fields. The approach involves four steps to find global optimality. First, local optimal control is found for the linearized part of the system and the quadratic part of the given performance index. Second, the density function method is used to find a stabilizing polynomial control for the nonlinear system. Third, the corresponding Lyapunov function is found for the control. Finally, the pair of control and its Lyapunov function are iteratively updated, using SOSTOOLS, for global optimal control. Numerical examples illustrate the effectiveness of the design approach. (paper ((pdf))

We consider a star-shaped network with N strings
that are coupled at one end and subject to a feedback control at
the other end. Each feedback control is switched on and off by
a time-dependent switching signal. We provide conditions on the
switching laws that guarantee the exponential decay to zero of
the system velocity. These sufficient conditions for stability ensure
that at each moment in time only one wave arrives at the coupling
note of the network or, alternatively, that at each moment in time
only one wave is reflected at the controlled ends. (paper ((pdf))

The problem of robust controller design under
PWM feedback is discussed in terms of Filippov’s average model
where control variable is a duty ratio function. The proposed
controller is an extension of PI/PID control scheme under PWM
feedback. The presented design methodology guarantees desired
output transient performance indices by inducing of two-timescale
motions in the closed-loop system. Stability conditions
imposed on the fast and slow modes and sufficiently large mode
separation rate between fast and slow modes can ensure that
the full-order closed-loop nonlinear system achieves the desired
properties in such a way that the output transient performances
are desired and insensitive to external disturbances and plant’s
parameter variations. The method of singular perturbations is
used throughout the paper in order to get explicit expressions
for evaluation of the controller parameters. Simulation results of
tracking control for magnetic levitation system are presented as
an example of the application for the discussed PWM control
design methodology. (paper ((pdf))

In this paper we consider the robust controller
synthesis problem for a group of identical agents which have
to fulfill a common goal. To achieve this, the agents have the
capability to communicate with each other. The communication
structure is modeled as a directed graph and is affected by time
delays and changes in the communication topology. These time
delays differ depending on whether they affect also an agent‘s own
states and not only the communicated states of the other agents.
For both cases, a synthesis method that guarantees stability of the
formation is proposed. The design of a controller that guarantees
stability in the face of self-delays is the main contribution of
this paper and an extension to previous work. A comparison of
the controller design method for communication delays with and
without self-delays has been carried out by a simulation of a
formation flight of quad-rotor helicopters. (paper ((pdf))

Let $x=(x_t)_{t\geq0}$ be a scalar observed process with control $u=(u_t)_{t\geq0}$ described by the stochastic
differential equation
$$
dx_t= \vartheta x_tdt + u_tdt + dw_t,\quad t\geq0,
$$
driven by the standard Wiener process $(w_t)_{t\geq0}.$ Assume the parameter $\vartheta$ to be unknown.
The problem solved in this paper is to approximate the process $x$ to the stable Ornstein-Uhlenbeck process
$x^0=(x^0_t)_{t\geq0}$ with the given dynamic parameter $a<0,$ satisfying the equation
$$
dx^0_t= a x^0_tdt + dw_t,\quad t\geq0,
$$
by choosing the control process $u.$
More precisely, based on continuous observation of $x,$ for
given $\varepsilon>0$ an adaptive control law
$u^\varepsilon=(u^\varepsilon_t)_{t\geq0}$ is constructed, such that
the corresponding observed process
$x^\varepsilon=(x^\varepsilon_t)_{t\geq0},$
$x^\varepsilon_t=x_t(u^\varepsilon)$ satisfies the following relations
$$
\sup\limits_{t\geq 0}\ E(x^\varepsilon_t)^2\leq L
$$
and
$$
\sup\limits_{t\geq t_\varepsilon}\ E(x^\varepsilon_t-x^0_t)^2\leq
\varepsilon,
$$
where $L$ is some constant (independent of $\varepsilon\mbox{)}$ and $t_\varepsilon$ is an unboundedly increasing as
$\varepsilon\to 0$ non-random function.
The first relation ensures the stability of the object $x^\varepsilon.$ %for all $t\in[0,\infty).$
The rate $\varepsilon^{-1}\ln \varepsilon^{-1}$ of increase of $t_\varepsilon$ is obtained.
A similar problem is solved for a stochastic delay differential equation with an unknown parameter. (paper ((pdf))

In [1][2], a new systematic design method for
fractional order proportional and derivative (FOPD) controller
is proposed for a class of typical second-order plants. Simulation
and experimental results show that the dynamic performance and
robustness of the position ramp response at normal speed with
the designed FOPD controller outperforms that with the ITAE
optimized traditional integer order proportional and integral
(IOPI) controller. Furthermore, we found that, for the ultra low
speed position tracking with significant friction effect in the same
experimental system in [2], the tracking performance using the
designed FOPD controller is much better than that using the
optimized IOPI controller. In this paper, using the describing
function method and the Bode plots analysis, the observed
advantage of the designed FOPD controller for the nonlinear
low speed position tracking system with friction effect over the
optimized IOPI controller is explained, which is consistently
demonstrated by our extended experimental results. (paper ((pdf))