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APPEDG E
APPEDGE - Masse J. RelaxCand the other controllers 17 july 2018
RelaxCand the other controllers
Masse John ∗
∗APPEDGE, 18-22 rue d’ARRAS 92000 Nanterre
e-mail: john.masse(at)appedge.com
Abstract: First all, we recall briefly the original concept of the new controller RelaxC(Family
of Ultimate controllers).We compare RelaxCto other methods as Predictive Functional
Control(PFC), Model predictive control (MPC), Model Free-Control (MFC), ADRC (Active
disturbance rejection control) based on the linear ESO observer. We comment also two methods
Fractional PID and Sliding mode control (SMC). For the first time, we will try to explain why
we find a difference between all these methods and to connect theses methods. We remind that
in RelaxC, it has never been easier to control all the type of systems as non minimal phase,
small and large dead-time, SISO or MIMO systems and unstable processes, etc. RelaxCis a
online tuning controller.
Keywords: PID,RelaxC, Ziegler-Nichols, PID, Generating Functions, Reactivity time, Diffedge,
Real Time, Relax controller, Identification, model free control, MFC, ADRC, ESO, PFC,
MPC, IMC, SIMC, LQG, fractional PID, ( DOF) PID, Optimal Control, Appedge, Sliding
Mode, Uncertainty, Robust control, Adaptative control.
Fig. 1. The Goal of the control theory
1. INTRODUCTION
The theory of the Ultimate Controller RelaxCis described
in Masse J (2017)
In this paper, we wil try to explain the difference
and the common point between the several control law.
RelaxCseems to unify and federate different controllers
design.
RelaxCis based on two fundamental new definitions : the
generating functions (ordinary differential equations) and
the reactivity time operator. These two definitions allow :
(1) To propose the most simple and unique structure of
the controller to manage all type of the system.
(2) To establish simple rules to tune the coefficients of
the controller with reliability.
(3) To give the possibility to master the u0of the control.
(4) To give the minimum time response that we can do
with the system without overshoot.
The focus of the control theory has always been to control
the outputs of dynamic sytem via an energetic cost or
reference trajectory as Fig. 1.
2. EQUATION OF RELAXC
The main equation of RelaxCis :
−,
um=ks∗fg(yg−ym, . . . ) and u(0)m=ks∗Usp (1)
With fg(yg−ym, . . . ) = fg(yg−ym,˙yg−˙ym, . . . , yg(n)−
ym(n)). The variables yg,ymare respectively the state
space of the generating function fg(known) and ymthe
measured state space of the plant or the model for which
we don’t have the equations. The variable umis the value
of the control that we apply at the model and u(0)mis the
value at t= 0.
Definition 1. We define the ODE Generating Function
as ug=fg(yg, y0
g, y00
g, ...)that allows to generate the reference
trajectory too of the model mwith the dynamic τg.
The most useful generating function fgare given in 2.1
that allows to compute all the family of controller that
we want to build. Furthermore, the main result of the
property of generating functions allows to obtain directly
the initial condition of u0mbecause at t= 0 ym, y 0
m,· · · =
0 then u0m=Usp =fg(yg−ym, . . . ) with Usp the set
point of the reference trajectory. The factor kscontrols
the asymptotic speed convergence of the controller to the
set point.
The derivative reactivity time opererator −,
um∼u(n)−
δ(k−1−τre
T s )u(n−k)) ∼u(n)− Rτre (u(n−1)). With τre
the time constant of the filter R.
Definition 2. The reactivity time is the time-lag when
(Ordfg<Ordfm) or when we have a dead time. Thus
we can write fmis delayed of Rτre (fg)
In other words, RelaxCallows to fit fgby fmvia um
when the derivative order of fgis inferior at the order of
fm.
2.1 The most used generating functions
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APPEDG E
APPEDGE - Masse J. RelaxCand the other controllers 17 july 2018
y0
g=−yg+ug
τg
→fg=τgy0
g+yg(2)
y00
g=−2ζw0y0
g−w02yg+w02ug→fg=y00
g
w2
0
+2ζ
w0
y0
g+yg(3)
with τg=2ζ
w0
→fg=τg2y00
g
4ζ2+τgy0
g+yg(4)
(5)
The 4 is generalized by :
Remark 3.
ug=fg(yg, y0
g) = (1 + τgs)yg∼
β
X
i=0 β
i(1 + τg
βs)yg(6)
2.2 First order RelaxC
To make a RelaxCfirst order, we use Eq.2 as generating
function to realize all the examples of this paper because it
is a slow-down feature ensuring a smooth travel to the limit
stops and stable reference trajectory without overshoot, or
minimize it.
Definition 4. If fgis is a differential equation of first order
(Ordfg= 1) then ug=fg=τgy0
g+ygthus we get
−,
um=ks(τg˙e+e)(7)
With e=yg−ymthen RelaxCbecomes in the context
of ZOH (zero order hold) the following equation.
um(n) = umold +ks(τg˙e(n−1) + e(n−1))
umold =Rτre (um(n−1)) and u0m(1) = ksUsp (8)
2.3 Second order RelaxC
We can increase the order of RelaxCif we use Eq.4 (time-
based representation) or Eq.3 (frequency representation)
with w0thus we get the two following controllers Eq.9 and
Eq.10:
um(n) = umold +ks(¨e(n−1)
w02+2ζ˙e(n−1)
w0
+e(n−1))
umold =Rτre (um(n−1)) and u0m(1) = ksUsp
(9)
um(n) = umold +ks(τ2
g¨e(n−1)
4ζ2+τg˙e(n−1) + e(n−1))
umold =Rτre (um(n−1)) and u0m(1) = ksUsp
(10)
We can generalized the controller RelaxCto the higher
order of derivative with the equation Eq.6.
2.4 Setting parameters of first order RelaxC:
The impulse response seems to be better that the classic
step method to identify the fundamental characteristics of
the model. The relay feedback is a good way also. With
RelaxCit is necessary to tune 2 parameters τre,τgand
ksis a function of ( τre,τg) . They depend of the form
of the impulse response. For example, we have only one
parameter to choose if fmis a pure first order. It is only τg
because τre = 0, and ks=1
τg.( see the first order example
in Masse J (2017)).
2.4.0.1. Choice of τre: decoupling parameter The most
important parameter in RelaxCis τre. The leading
role of the filter Rτre is to decouple the integration,
or memorization of the fast fluctuations of umof the
reactivity of fm. (See section 2.5). Thus τre is the time-lag
beteewn fgand fm.
2.4.0.2. Choice of τg: the maximum bandwidth The
parameter τgis the dynamic of the reference trajectory
that depends also on τre . The parameter τgis thus the
maximum dynamic that the system can do: i.e. the limit
between the overshoot and undershoot response.
2.4.0.3. Choise of ks: the RelaxCthrottle The param-
eter ks, as the other parameters, has to be adjusted with
respect to τgand τre or by the robustness that we want to
perform and especially to control the u0.
2.5 Comment on RelaxCand fractional PID controller
For more information about the fractional order calculus
(FOC) and P IλDµ, see Sandhya (1994) and Poldluny
(1994) and the work of Alain Oustaloup.
The fractional-order differentiation and fractional-order
integrator is based on the fundamental operator aDtα
which allows to decouple the damping of the dynamic
when we use it in the control domain. The operator −,
um
of RelaxCgives a similar behavior. RelaxCis evolving
between a PD and PI controller with respect to the value
of τre. See Eq. 11.
aDtα
dα
dtαα > 0
1α= 0
Zt
a
(dτ)−αα < 0
⇐⇒
−,
uτre
u(n) = ks(τg˙e(n−1) +e(n−1) )τre =∞P D
u(n) = Rτre (u(n−1)) + ks(τg˙e(n−1) +e(n−1)) 0 < τr e <∞ RelaxC
u(n) = u(n−1) +ks(τg˙e(n−1) +e(n−1) )τre = 0 P I
(11)
2.6 Comment on RelaxCand Sliding mode control
For more information about the sliding mode control
(SMC), see Usai E (2008) and DIEE (2008), D Efimov
(2015). About this method we can read : “SMC is a non-
linear control technique featuring remarkable properties of
accuracy, robustness, and easy tuning and implementa-
tion”. “The SMC stabilizes the system in the presence of
unknown conditions and it tackles theses issues”.
The SMC uses a sliding surface function σ=σ(e, ˙e, . . . ).
The typical choice of σis for the first order σ= ˙e+p∗ethe
second order σ=¨
(e)+2p∗˙e+p2∗ewith e=ym−yr ef with
pas arbitrary choice and the control is u=−k∗sign(σ).
The control can be improved by using “Super Twisting”
that includes an integrator on the weight sign of σ. To take
account of the dead-time in the system, the sign function is
modified but that seems complicated. SMC requires only
to choose few mathematical parameters.
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APPEDG E
APPEDGE - Masse J. RelaxCand the other controllers 17 july 2018
The methodology of RelaxCcan help SMC to improve
its tuning. If we write that σ=−fg(yg−ym, . . . ) then
we have a normalized dynamic with τg. If we want to
avoid the chattering it is recommended to use directly
RelaxCbecause there are fewer parameters to tune
(and only physical parameters) and RelaxCshows that
is not necessary to use higher-order derivatives. On the
other hand, the examples 4.1 and 4.2 shows clearly that
RelaxCis able to be robust against a large variety
of disturbances and it erases them with the choice of
ks. Moreover, if we use the autotuning developped for
RelaxCthere is nothing more to do.
3. RELAXCVS PFC ,MFC, PID WITH A SECOND
ORDER
It is a classic example where the goal is to get a fast re-
sponse without overshoot. The PFC controller is adapted
at second order transfer function with dead-time. For all
the methods, we need to identify the process characteris-
tics with more and less accuracy. Through this example,
we show quickly how we tune the controllers and how the
different controller are connected.
Model equation is : Hm=kme(−2s)
(1+10s)(1+15s)with km= 0.8
and the sample time Ts= 0.1. The first disturbance is on
the output Y(load of 1 at t= 100), the second is on u(
load 10 at t= 140).
3.1 Tuning of PFC
We use the controller writing by Richalet and his setting.
We have : CLT R = 30( Control Loop Time Response),
control Horizon hi= 7.5seconds,td= 2, and the poles 10
and 15 of the model. For IMC and PFC methods, the
setting are almost similar. See also the paper J.A. Rossiter
(2012) to connect MPC to PFC. (See Fig.2)
3.2 Tuning of RelaxC
RelaxCis easy to tune for a second order with delay, we
take: τre ∼15, τg∼τre/3 + td = 7, ks= 3.6. (See Fig.2)
In the case, where we want to perform a response very
close to PFC in the goal to get a behavior similar to
PFC, we adapt the coefficients like that: hi=τg= 7.5, ks∗
τg=CLT R /3 and we deduce ks= 1.3 and τre = 16.5. The
simulation with these new values gives the Fig.1. We can
compare the energetic cost of uand especially the energy
peak of u0 = 42 for PFC with respect to RelaxCwhere
u= 13 (4 time lower) for approximatively the same time
response. The energy peak is not recommended when
we use embedded systems working on electric cells. The
disturbance rejection is always excellent for RelaxCon
the load on ueven if we choose a slow time rise.
3.3 Tuning of MFC
The MFC was introduced by Fliess (2009). There are not
rules for the choice of the parameters Kp, Ki, Kd, the
estimator F e of Fand the order (n) of the controller. For
example, for n= 1 and n= 2, we obtain the following
MFC controller :
n= 1 →u∗=−F e −˙y∗+Kp∗e
α(12)
n= 2 →u∗=−F e −¨y∗+Kp∗e+KiRe+Kd˙e
α(13)
For this example, I choose n= 1, as Fliess (2008) , we use
F e =Fτf(y(n)(n)−αu(n−1)). We set α= 0.39, Kp=α
and τf=τre = 15 and the reference trajectory is based
on τgof RelaxC. It seems difficult to do better without
overshoot with the order n= 1. The behavior of MFC is
given in Fig.2.
3.4 Tuning of PID with RelaxC
There are many rules to try to find the coefficients, but the
most simplest and easiest task is to connect the PID at the
tuning of RelaxCand we find directly a good behavior.(See
Masse pid (2017)). The equations of the PID are : u=
Kp(e+e
sTi+sTde) or u=Kpe+kie
s+skde)
Thus the tuning of RelaxCgives Kp=ksτg
τrekm= 2.10,
ki=ks
3kmτre = 0.1, and we deduce Ti=Kp/ki= 16.8 and
Td=Ti/4=5.25 (See Masse J (2017)) hence kd=Td∗
Kp= 11.0 for a well-balanced PID i.e without overshoot.
(See Fig.2 with a filter coefficient on the derivative term
equal to .0.16 ).
For more information about the other tuning of PID,
the user can read the work of Astrom (1984), Astrom
(1995), Alterton (1999), Aidan (2009), Alterton (1999),
Skogestad (2012) about SIMC method, Ziegler Nichols, ,
Loan Dor Landau (RST structure), (DOF PID): Xiang
(2004),Takahashi, ...
3.5 Analysis of results
For RelaxCwith the proposed tuning ajustment, we
obtain the best time response that the system can produce.
Furthemore RelaxCgives a very good rejection of
disturbances with respect to the other methods.
The first main result of this example are : RelaxCmini-
mizes the energetic cost of uto follow a reference trajectory
and maximizes the disturbance rejection with a small u0.
The second result is that RelaxCallows to connect almost
all controllers betweem them.
4. RELAXCAND THE NONLINEAR PLANT
To illustrate the powerful of RelaxCwe show two examples
which are used with ADRC or MFC methods. Other ex-
amples with large delay, non minimal phase and unstable
model are already realized in Masse J (2017).
There are many papers about the ADRC method ( Ac-
tive Disturbance Rejection Control) Zheng (2007), Zhao
(2014), Qing (2010) And the paper of Fliess (2018) com-
pares MFC vs ADRC.
The principle of these methods are:
•MFC uses the theorem of implicit function to find
the unknown part of the system. It is a fully implicit
problem that works well when we use a variable step
solver for a stiff problem. But in the real time, we
work with a fixed step Ts(Sample Time) and it is
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APPEDG E
APPEDGE - Masse J. RelaxCand the other controllers 17 july 2018
necessary to introduce a filter to estimate Fby Febut
this filter delayed a part of the state space whatever
the maner that we choose F e
•ADRC uses an extended linear state observer (LESO)
with a w0 the speed of convergence of the observer.
It may be the drawback of this method because all
the estimation of the state space are delayed of w0.
To eliminate this drawback for the two methods it is
necessary to increase the order of the controller or to put
other information in the controller.
For this reason and the nature of these controllers, the
time constant of filter has to be small for MFC and for
ADRC it is the inverse because it is difficult to implement
a fast observer w0→ ∞ i.e without delay.
The two examples, described below, seem to show that
where MFC works well ADRC does not work correctly
and vice versa.
RelaxChas not this inconvenience because the state space
is not delayed and to allow at it to include a wide class of
systems.
4.1 RelaxC: an Active Disturbance Rejection Control
The case studied is well described in Zheng (2007). The
goal is to get a fast response without overshoot and a good
disturbances rejection.
Model equation is given by the equation 14.
y00 =y03+y+u(14)
with the sample time Ts= 0.001. The first disturbance is
on the output Y(load of +20 at t= 4sduring 0.2s) and
the second disturbance at t= 6sduring 0.2 secondes with
a load of −20.
The authors proposes three scenarios, where the second
member of the model is totally unknown, partially known
and completely known. He uses a dynamic wc = 4.5 in the
controller and a dynamic of w0 = 20 for the observer. The
best result of the simulation is when the model is totally
or partially known and we see that the plot “LADRC
performance” has a poor disturbance rejection. The Fig. 4
shows two tuning of RelaxCwith the case (a) and case
(b). The order of RelaxCis always one and we do not
put inside the controller any more information about the
model.
•In the case (a), we tune RelaxCwith approximatively
the same dynamic of ADRC. The tuning of is τre = 5,
τg= 0.5, ks= 27. We obtain directly the same
response of ADRC when the model is completely
known. This example demonstrates that it is not
necessary to use an observer to control this system
and to know the model.
•In the case (b), we choose this tuning τre = 0.01,
τg=.32, ks= 1200 which allow to keep the same
dynamic of the case (a).
In the case (b), we see clearly that disturbances are erased
and the control uof RelaxCis very stiff and almost a
Bang Bang control. This result shows that RelaxCis
really a methodology with an Active Disturbance
Rejection control.
4.2 RelaxC: a dynamical compensation method
We have seen that RelaxCallows to perform the active
disturbance control. i.e to maintain constant steady-state
output despite variation in the plant. This behavior is
interesting for the non-linear biochemical model (see Karin
O (2016), Alon U. (2006)).
The reactivity operator and the generating function used
together lead to exact adaptation or dynamical compen-
sation. To illustrate this fact we use the nonlinear case of
Fliess (2018).
y0=−y+u3(15)
with a sample time of 0.01s.
The goal is to track a variable set point of reference
with some disturbances on Uand the output Y. For the
simulation we have a load disturbance on Yat t= 10 with
the value 2 and on Uat t= 12swith the value 3. And we
have always an oscillating load disturbance 0.2∗sin(1.25∗t)
on Uand YThe set point changes at t= 0, t= 5 and
at t= 17swe want to follow sin(2 ∗t). The tuning of
RelaxCis τre = 0, τg=.2 and ks= 0.15. The first
simulation is without any noise. See Fig. 5. The second
simulation we add a Gaussian noise of power 0.005.(See
Fig. 6)
With RelaxCwe get a very fast closed loop time response
(∼0.5s), an excellent rejection of the perturbation and a
good tracking trajectory without distortion whereas Yand
uhave a load disturbances(Note how the control is twisted
to obtain a perfect tracking). Furthemore, we see that a
large noise on the output is not a problem for RelaxC. This
example shows the following qualities of RelaxC:ensure
a robust stability, robust set point tracking and
robust adaptation or dynamical compensation.
5. CONCLUSION
RelaxCopens a new skyline in the world of the controllers.
It is easy to tune with the physical characteristics of the
plant and not with rules of thumb. It is able to control
a wide class of problems with a remarkable effectiveness.
RelaxCcan advantageously replace the PID or other
controllers, most of which are a mathematical structures
highly dependent of the process to control. Maybe, in the
short term, RelaxCwill be the future of the “PID” on the
grounds of its possibilities to control all type of processes 1,
its low computing cost (2 additions, 2 multiplications and
one filter) and especially due to its ease and flexibility of
use. Thus, the structure of RelaxCis well adapted also for
the predictive and preventive maintenance of the industrial
processes. Moreover, it is easy to apply autotuning on
RelaxC.
These last points will be presented in another paper.
1Except pathological case known
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APPEDGE - Masse J. RelaxCand the other controllers 17 july 2018
Fig. 2. Ex: 3:Comparison PFC ,RelaxC,MFC ,PID (RelaxCtuning)
Fig. 3. Ex: 3: RelaxCand PFC with approximatively the same dynamic. Comparison of the initial control.
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APPEDG E
APPEDGE - Masse J. RelaxCand the other controllers 17 july 2018
Fig. 4. Ex: 4.1: RelaxCerases the disturbance
Fig. 5. Ex: 4.2: Result without any noise
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APPEDGE - Masse J. RelaxCand the other controllers 17 july 2018
Fig. 6. Ex: 4.2: Result with noise added
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REFERENCES
John Masse (Relaxc 2017) Relaxc Controller: The ul-
timate controller. The origin CNES (version 0) 2017.
DOI10.13140/RG.2.2.15365.45287
John Masse (ppt 2017) Relaxc controller: Family of relaxc
controller, A controller almost “perfect” Presentation
CNES (version 1) 28 Nov 2017.
John Masse (PID 2017) Relaxc PID : a PID almost perfect
Presentation CNES (version 1) 28 Nov 2017.
J.Masse, C.Masse, F.Ollivier (Diffedge 2017) Automatique
differenciation of hybrid models Illustrated by Diffedge
Graphics Methodology Survey, 2017.
R D Kokate, LM Waghmare (2011), “Review of Tuning
Methods of DMC and Performance Evaluation with PID
Algorithms on a FOPDT Model” International Journal
of Control and Automation 2011
DP.Aterton, S.Majhi (1999) Modified Smith predictor and
controller for processes with time delay” IEE proceeding
Vol 146 No 5, 1999.
Astrom K. J., Hagglund T. (1984) Automatic Tuning
of Simple Regulators with Specifications on Phase and
Amplitude Margins. Automatica, 20, 645-651.
E.Haire, S.P Norsett, G Wanner (1980) Solving Ordinary
Differential Equations I. Springer-verlag 1980.
K. Astrom,T. Hgglund (1995) “pid con-
trollers,theory,Design, Tuning” Instrument society
of America.
Aidan O’Dwyer (1996) “The estimation and compensa-
tion of processes with delays” Thesis 1996
Aidan O’Dwyer (2009) “Handbook of PI and pid controller
Tuning rules” Imperial college Press 2009
Fliess (Michel), Levine (Jean), Martin (Philippe) and
Rouchon (Pierre) (1993) Flatness and motion planning:
the car with n trailers European Control Conference,
1518–1522, 1993.
Fliess (Michel), Cedric JOIN (2009) MODEL-FREE con-
trol and intelligent PID controllers: Towards a possible
trivialization of non linear control ? 2 avril 2009.
Fliess (Michel), Cedric JOIN (2018) Dynamic compensa-
tion and homeostasis: a feedback control perspective ?
ArXiv:1801.04959v1
Alon U. An Introduction to Systems Biology: Design
Principles of Biological Circuits. Chapman and Hall.
Omer Karin, Avital Swisa, Benjamin Glaser, Yuval Dor,
Uri Alon Dynamical compensation in physiological cir-
cuits DOI 10.15252/msb.20167216 — Published online
08.11.2016
Briat C., Gupta A., Khammash M. (2016). Antithetic
integral feedback ensures robust perfect adaptation in
noisy biomolecular networks Cell Syst., 2, 15-26.
Fliess (Michel), Cedric JOIN (2018) Two competing
improvements of PID controllers: A comparison ? hal-
01687952
Ollivier (Francois) and Sedoglavic (Alexandre) (2001) A
generalization of flatness to nonlinear systems of partial
differential equations. Application to the command of a
flexible rod Proceedings of the 5th IFAC Symposium
“Nonlinear Control Systems”, vol. 1, Elsevier,196–200,
2001.
Jacques Richalet (2009) Industrial Application of predic-
tive functional Control. Oxford 09/01/2009
C. Join, J. Masse, M. Fliess (2008) tude prliminaire dune
com- mande sans modle pour papillon de moteur J.
europ. syst. automat., t. 42, p. 337-354, 2008.
Chriss Grimholt, Sigurd Skogestad (2012) “Optimal PI-
Control and Verifcation of the SIMC Tuning Rule”,
IFAC Conference on Advances in PID Control,PID’12
Brescia (Italy), March 28-30, 2012
Bastogne Thierry, Thomassin Magalie, Masse John (2007)
“Selection and identifcation of physical parameters from
passive observation. Application to a winding process
Control Engineering, Practice, Elsevier, 2007, 15 (9),
pp.1051-1061.
Gilbert Jean-Charles Le Vey Georges and Masse John
(1991) La diffrentiation automatique de fonctions
reprsentes par des programmes, Tech. rep. 1557, INRIA,
1991.
Masse John and Cambois Thierry (2004) “Differentiation,
sensitivity analysis and identification of hybrid Mod-
els under Simulink” Symposium Techniques Avances
et Stratgies Innovantes en Modlisation et Commandes
Robustes des Processus Industriels , Martigues, 21, 22
septembre 2004.
DP.Aterton, S.Majhi (1999) “Modified Smith predictor
and controller for processes with time delay” IEE
proceeding 1999 Vol 146 No 5
DP.Aterton, S.Majhi (2000) “Obtaining controller pa-
rameters for a new Smith predictor using autotuning”.
Automatica 2000
Xiang Lu Yong-Sheng Yang , Qing-Guo Wang , Wei-Xing
Zheng (2004) “A double two-degree-of-freedom control
scheme for improved control of unstable delay processes”
journal of process control 2004.
Weiwen Wang and Zhiqiang Gao (2003) A Comparison
Study of Advanced State Observer Design Techniques
Proceedings of the American Control Conference July
2003
Qing Zheng, Gao (2007) On Estimation of Plant Dy-
namics and Disturbance from Input-Output Data in Real
Time DOI: 10.1109/CCA.2007.4389393 Source: IEEE
Xplore Singapore, 1-3 October 2007
Shen Zhao, Zhiqiang Gao (2014) Modified active distur-
bance rejection control for time-delay systems 2014
ZHENG Qing , GAO Zhiqiang (2010) On Practical
Applications of Active Disturbance Rejection Control
Proceedings of the 29th Chinese Control Conference
July 29-31, 2010, Beijing, China
G. Valencia-Palomo J.A. Rossiter. (2013) Comparison
between an auto-tuned PI controller, a predictive con-
troller and a predictive functional controller in elemen-
tary dynamic systems online publication research gate,
2012.
Igor Podlubny (1994) Fractional-Order SYstems and
Fractional Controllers November 2014
A.Sandhya, R.Sandhya, M.Prameela Igor Podlubny (1994)
An overview of Fractional order PID Controllers and its
Industrial applications IJIET 2016
DIEE (2008) A quick introduction to sliding Mode control
ans its Applications Universita Di CAGLARI
Usai elio (2008) Sliding mode : Basic theory and new
perspective Universita Di CAGLARI
Denis Efimov, Andrey Polyakov, Leonid Fridman, Wilfrid
Perruquetti, Jean-Pierre Richard Delayed Sliding Mode
Control Automatica, Elsevier, 2015. ¡hal-01213703¿
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