06555780

www.ietdl.org

Published in IET Control Theory and ApplicationsReceived on 24th April 2012Revised on 6th February 2013Accepted on 11th February 2013doi: 10.1049/iet-cta.2012.0309

ISSN 1751-8644

Fractional order adaptive controller for stabilisedsystems via high-gain feedbackAbdelfatah Charef1 Mohamed Assabaa1 Samir Ladaci2 Jean-Jacques Loiseau3

1Université Mentouri de Constantine, Département d’Electronique, Route Ain El-Bey, Constantine 25000, Algeria2Université 20 Août 55 de Skikda, Département de génie électrique, BP 26, Skikda 21000, Algeria3 LUNAM Université, IRCCyN CNRS UMR 6597, 1, rue de la Noé, 44321, FranceE-mail: [email protected]

Abstract: Controllers based on fractional order calculus are gaining more and more interests from the control community.This type of controllers may involve fractional integration, fractional differentiation and/or fractional systems in their structureor implementation. They have been introduced in the control applications in a continuous effort to enhance the system controlquality performances and robustness. In this study, a new scheme of fractional order adaptive controller via high-gain outputfeedback for a class of linear, time-invariant, minimum phase and single input-single output systems of relative degree one isproposed. The basic idea of the new design is a further modification in the adaptive proportional control law by the introductionof a fractional integration besides of the regular one of the squared output of the system in the adaptation gain of the controlstrategy. An analytical stability proof of the feedback control system is presented. The control quality enhancement of theproposed control scheme compared with the classical one has been presented through the simulation results of an illustrativeexample.

1 Introduction

Lately, considerable focus on fractional calculus has beenobserved in different areas of system and control fields.Some early work was done by Bode [1] who proposed a frac-tional order open loop transfer function to maintain stableoperation of feedback amplifiers for large gain variation. Theintroduction of fractional calculus concepts in the area ofautomatic control systems was back in the early sixties [2].But, it is only in the last decades that fractional calculus-based controllers have gained more interest from the controlcommunity [3–8]. These controllers have been introducedin the control applications in a continuous effort to enhancethe system control quality performances and robustness. Thefirst who really introduced a well-established fractional ordercontroller, called CRONE controller, was Oustaloup [9].More recently, Podlubny [5] proposed a fractional PIλDμ

controller, involving an integration action of order λ anddifferentiation action of order μ. In [10], we can find a verygood tutorial on fractional calculus in controls; additionally,several typical known fractional order controllers have beenpresented and commented.

Adaptive control has been proven to be a good controlbecause of its potential application in systems with highcomplexities and uncertainties. The aim in adaptive controlis to design a controller which can achieve the pre-specifiedcontrol objectives for a given class of systems with uncertaindynamics. It is only in the last two decades that fractionalorder operators and systems have been introduced in theschemes of the adaptive control theory. In 2002, Vinagre

822© The Institution of Engineering and Technology 2013

et al. [11] have introduced for the first time the fractionalcalculus in the conventional adaptive control. They haveused a fractional order parameter adjustment rule and afractional order reference model in the conventional ModelReference Adaptive Control (MRAC). In 2006, Ladaci andCharef [12] have also used the MRAC with a fractionalorder parameter adjustment rule and a different fractionalorder reference model than the one in [11]. Besides, theyhave introduced a fractional differentiator with an appropri-ate fractional order at the output of the plant. A fractionaladaptive scheme, which combines a model reference anda fractional order adjustment rule for a feed-forward gainadjustment, has been proposed by Suárez et al. [13] forlateral control of an autonomous guided vehicle. Ladaciand Charef [14] have also introduced an adaptive fractionalPIλDμ controller based on a classical integer one. In allthe above referenced articles, the benefits, in terms of thedynamics and robustness of the control system, of the pro-posed fractional adaptive control schemes have been shownthrough illustrative examples only. But the weaknesses ofsuch works were the lack of theoretical arguments to guaran-tee the stability of such fractional adaptive control schemes.Hence, the analytical proof of stability for fractional adap-tive control schemes is up to now considered as an openproblem.

In the last years, some analytical stability proofs ofsome theoretical adaptive control schemes have been pre-sented. Li et al. [15] have studied the asymptotic stabilityof three fractional scalar systems using the universal adap-tive stabilisation technique. The stability proofs obtained

IET Control Theory Appl., 2013, Vol. 7, Iss. 6, pp. 822–828doi: 10.1049/iet-cta.2012.0309

www.ietdl.org

in [15] have been extended to MIMO systems by Li andChen [16]. Ladaci et al. [17] have shown that a frac-tional adaptive controller based on high gain output feedbackcan stabilise any given linear, time-invariant, minimumphase, single input-single output (SISO) systems of rela-tive degree one. They have also introduced a fractionalfeed-forward in the MRAC algorithm where the robuststability proof of the proposed adaptive control schemehas been derived using the almost strictly positive real-ness property of the plant [18]. In [19], a fractional orderrobust control scheme has been proposed for cogging effectcompensation on permanent magnetic synchronous motorsposition and velocity servo system. Through this range ofdesign techniques and applications, though quite far fromaiming at completeness, it is clear that fractional order adap-tive control has become an important research topic. Thegeneralisation to fractional orders of traditional controllersor control schemes translates into more tuning parame-ters and more adjustable time and frequency responsesof the control system, allowing the fulfilment of robustperformances.

In this study we present a new scheme of fractionalorder adaptive controllers via high-gain output feedbackfor a class of linear, time-invariant, minimum phase andSISO systems of relative degree one. The basic idea ofthis new concept is the introduction of a fractional integra-tion besides of the regular integration of the output of thesystem in the adaptation gain of the control strategy. Theintroduced fractional integration order can clearly improvethe behaviour of the control system. We have shown thatthis controller can asymptotically stabilise the consideredclass of systems. The proposed adaptive controller can alsodeal with the output tracking problem of a step reference.The simulation results of an illustrative example are pre-sented to show the control quality enhancement using thisproposed fractional adaptive control scheme compared withthe classical one.

2 Preliminaries

2.1 Fractional calculus

In this work Riemann–Liouville definitions of fractionalorder integration and differentiation are used [20, 21]. So,for the case 0 < α < 1, and f (t) being a causal function oft, that is, f (t) = 0 for t < 0, the fractional order integral isdefined as

I αf (t) = 1

�(α)

∫ t

0

[(t − τ)α−1f (τ )

]dτ (1)

and the fractional order derivative is also given as

Dαf (t) = d

dt

{I (1−α)f (t)

} = 1

�(1 − α)

d

dt

×{∫ t

0

[(t − τ)−αf (τ )

]dτ

}(2)

where � (.) is the Gamma function. One of the propertiesof the fractional integration operator I α = I α

0+ is its bound-edness in the space Lp(0, t1) (1 ≤ p ≤ ∞) with the norm‖f ‖p.


Lemma 1 [21]: The fractional integration operators I α0+ with

α > 0 is bounded in Lp(t0, t1) (1 ≤ p ≤ ∞)

∥∥∥I α

t+0f (t)

∥∥∥p≤ K ‖f (t)‖p with K = (t1 − t0)

α

α�(α)(3)

Proof . See Lemma 2.1 of [21] on page 72.

2.2 Adaptive high-gain control

In the continuous-time adaptive high-gain control scheme,the controller is of striking simplicity. It is not based on sys-tem identification or plant parameter estimation algorithmsor injection of probing signals [22]. It is a well-establishedfact that the adaptive high-gain controller can stabilise anySISO, minimum phase system with positive high-frequencygain and relative degree one [22–26].

Let us consider an uncertain SISO system described bythe following state space equation

dx(t)

dt= Ax(t) + Bu(t) (4a)

y(t) = Cx(t) (4b)

where t ∈ � is the time variable, x(t) ∈ �n is the state vectorwith n unknowns, u(t) ∈ � is the scalar control and y(t) ∈ �is the scalar measured output; A, B, C, are unknown matricesof appropriate dimensions.

Assumption A1: We assume that the above system is con-trollable and observable with minimum phase, positivehigh-frequency gain and relative degree one.

Theorem 1 [22–26]: Consider an uncertain system describedby (4) satisfying Assumption A1 then it is uniformly sta-bilised by the following high-gain adaptive proportionalcontrol law

u(t) = −k(t)y(t) (5)

For the proof see [22–26].

To ensure that the gain k(t) grows beyond the lowerasymptotic stability bound, k(t) is adapted by the outputas [22–27]

dk(t)

dt= y2(t) (6)

In this case, for arbitrary initial conditions x0 = x(t0) ∈ �n

and k0 = k(t0) ∈ �, the closed loop initial-value problem hasa unique solution (x(t), k(t)) with the following properties

limt→∞ x(t) = 0 and lim

t→∞ k(t) = k∞

Theorem 2 [27]: Consider an uncertain SISO system of (4),satisfying Assumption A1, then it verifies Theorem 1 withthe adaptive proportional control law of (5). The resultingfeedback control system is given as

dx(t)

dt= [A − k(t)BC]x(t) (7a)

y(t) = Cx(t) (7b)

k(.): [t0, t1) → �, is a differentiable function satisfying

dk(t)

dt≥ 0

Then there exists k ∈ � and ε > 0 such that if k(t) ≥ k forall t ≥ t0 and for x(.) [t0, t1) → �n any solution of (7a), the


www.ietdl.org

following holds

|y(t)| = |Cx(t)| ≤ Me−ε(t−t0) for all t ∈ [t0, t1)

for some M > 0.For the proof see [27].

3 Problem statement

Consider the system of (4) satisfying Assumption A1, thenit can be stabilised by the high-gain adaptive proportionalcontrol law of (5) and (6), but an annoying feature of thiscontrol law is that the gain k(t) will be unbounded in thepresence of the slightest error in the output y(t) caused, forexample, by measurement noise. In the constant output feed-back problem, this can be easily overcome by introducingan integral term to the control law. In this output feedbackcontrol problem, we intend to propose a further modificationin the adaptation law of (6) in order to keep the closed-loop system stable, to force its output to converge withoutsteady-state error and to keep the gain bounded in the caseof a noisy output; with no a priori knowledge of the plantparameters and order. So, the basic idea of this new designis the introduction of a fractional integration of the squaredoutput of the system in the adaptation gain of (6) besides ofthe regular one.

Hence, for an uncertain system described by (4) satisfyingAssumption A1, it is uniformly stabilised by the high-gainadaptive proportional control law of (5) whose adaptationlaw is given as follows

e(t) = y(t) − r(t) (8a)

u(t) = −k(t)e(t) (8b)

dk(t)

dt= γ1e2(t) + γ2Dα

[e2(t)

](8c)

where r(t) ∈ � is the reference signal, k(t) ∈ � is the adap-tive gain and e(t) = [y(t) − r(t)] ∈ � is the error signal. Forsimplicity we will consider the case where r(t) = 0 to provethe stability of the proposed control system given by

u(t) = −k(t)y(t) (9a)

dk(t)

dt= γ1

[y2(t)

] + γ2Dα[y2(t)

](9b)

The parameters γ1, γ2 and α are real numbers which satisfy

γ1 > 0, γ2 > 0 (10)

−1 < a < 1 (11)

Then, from (9b), the gain k(t) will be given as

k(t) = γ1I 1[y2(t)] + γ2I (1−α)[y2(t)] (12)

We note that if γ1 = 0 and γ2 = 0, the case will be theclassical case of [27]; and if γ1 = 0 and γ2 = 0, the casewill be the fractional case of [17]. The scheme of (12), for−1 < α < 1, does not have an equivalent scheme using onlyregular integration and differentiation.


4 Stability analysis of the feedback controlsystem

Let us consider the system of (4) which satisfies AssumptionA1 subject to the adaptive controller of (9). The resultingclosed loop system is described by

dx(t)

dt= A(k)x(t) = [A − k(t)B C]x(t) (13a)

dk(t)

dt= γ1y2(t) + γ2Dα[y2(t)] (13b)

y(t) = Cx(t) (13c)

This system can be considered as a system with the state(x, k) ∈ �n × �. The main contribution of this study willbe the stability properties of the feedback control systemwith the proposed fractional control scheme. These stabilityproperties are stated in the following theorem and lemmas.

Theorem 3: For each initial condition [t0, x0 = x(t0), k0 =k(t0)] ∈ � × �n × � of the closed loop system of (13),there exists a solution [x(.), k(.)]: [t0, t1) → �n × �. Everysolution can be continuous over [t0, ∞) and satisfies

k(t) is bounded (14)

limt→∞ x(t) = 0 (15)

Proof of Theorem 3 For the proof of Theorem 3, we willconsider two cases based on the fractional order derivativeparameter α.

Case 1: −1 < α ≤ 0 For this case we shall require thefollowing Lemma.

Lemma 2: The controller gain defined in (12) k(t): [t0, t1) →�, t0 < t1, for −1 < α ≤ 0, a differentiable function, satis-fies

k(t) ≥ 0 (16)

Proof of Lemma 2 Let α = −β, then 0 ≤ β < 1, so thecontroller gain k(t) of (13b) becomes

dk(t)

dt= γ1y2(t) + γ2I β[y2(t)] (17)

From the Riemann–Liouvillle fractional order integraldefinition of (1), we have

I β(y2

) = 1

�(β)

∫ t

t0

[(t − ξ)β−1y2(ξ)

]dξ (18)

It is obvious that the right hand side of (18) is positivebecause �(β) ≥ 1. And since we have γ1 > 0 and γ2 > 0,this implies that the right hand side of (17) is positive. So,k(t) ≥ 0 follows.

Following up with the proof of Theorem 3, we will firstdemonstrate that k(t) is bounded. The behaviour of x(t)satisfies

dx(t)

dt= [A − k(t)BC]x(t)

where from Lemma 2, k(t) ≥ 0, then, from Theorem 2, thereexists k ∈ �, ε > 0 and M > 0 such that if k(t) ≥ k for all


www.ietdl.org

t ≥ t0 and for all x(.) : [t0, t1) → �n, solution of (13a), ∀ ∈[t0, t1), the following holds

|y(t)| ≤ Me−ε(t−t0) (19)

and from Lemma 1, for 0 ≤ β < 1, we have

∥∥∥I (1+β)

t+0y2(t)

∥∥∥ ≤ K∥∥y2(t)

∥∥ (20)

Hence, (19) and (20) imply that k(t) = γ1I 1y2(t) +γ2I (1+β)

t0 [y2(t)] is bounded on [t0, t1).To prove that x(t) converges to zero when t → ∞, we

rewrite (13a) as

dx(t)

dt= Ax(t) + Bμ(t) (21)

where

A = [A − kB C] (22)

is asymptotically stable and

μ(t) = [k − k(t)]y(t) (23)

The boundedness of k(.) implies that y(.) ∈ L2 and μ(.) ∈L2. Since x(.) is the output of an asymptotically stablesystem subject to an input μ(.) ∈ L2, then x(.) ∈ L2 andx(.) ∈ L2. Therefore

limt→∞ x(t) = 0

Case 2: 0 < α < 1. Let α = (1 − β), then 0 < β < 1, sothe controller gain k(t) of (13b) becomes

dk(t)

dt= γ1y2(t) + γ2I (β−1)[y2(t)]= γ1y2(t) + γ2I β[D{y2(t)}]= γ1y2(t) + γ2I β[2y(t)y(t)] (24)

The sign of k(t) will be based on the fractional order inte-gration I β[2y(t)y(t)] of (24). Hence, we will consider twocases depending on the sign of the term [y(t)y(t)].• [y(t)y(t)] ≥ 0 on all the considered time interval:Riemann–Liouvillle fractional order integral I β[2y(t)y(t)] isgiven as

I β[2y(t)y(t)] = 1

�(β)

∫ t

t0

[(t − ξ)β−1 2y(ξ)y(ξ)

]]dξ (25)

The right hand side of this equation is positive because[y(t)y(t)] ≥ 0 and �(β) ≥ 1. And since we have γ1 > 0and γ2 > 0, this implies that the right hand side of (24) ispositive. So, k(t) ≥ 0 follows. In the same way as in theabove case (−1 < α < 0), it can be easily proven that k(t)is bounded and limt→∞ x(t) = 0.

• [y(t)y(t)] < 0 on all the considered time interval: when[y(t)y(t)] < 0, it implies straight that the function |y(t)| is


strictly decreasing function, hence we have

limt→∞ y(t) = 0 (26)

and from Lemma 1, for 0 ≤ β < 1, we also have

∥∥∥I β

t+0y2(t)

∥∥∥ ≤ K∥∥y2(t)

∥∥ (27)

Therefore (26) and (27) imply that k(t) = γ1I 1y2(t) +γ2I β

t0 [y2(t)] is bounded on [t0, t1). From (13c) we have y(t) =Cx(t) and since the system of (4) is observable because itsatisfies Assumption A1, so we can conclude that

limt→∞ x(t) = 0 (28)

5 Illustrative example

In this section, an illustrative example will be presentedto show the effectiveness and the control quality enhance-ment using the proposed control scheme. We will consider aminimum phase continuous-time unstable system of relativedegree one given by the following transfer function

G(s) = 5(s + 1)

s(s − 0.3)(29)

We can easily see that the above system satisfies Assump-tion A1 of Section 2. Then, we can apply the proposedfractional order adaptive high gain controller of (12) to thissystem. Based on the fractional order derivative α (−1 <α < 1) of (9), we will consider two cases. For both cases,all the initial conditions have been set to zeros and the timesampling period T for the numerical calculations has beenset to T = 25 ms. Besides, the values of the parameters α, γ1

and γ2 are obtained by: first, we set γ2 = 0 in (12) (thecase will be the classical one) and tune the parameter γ1 toachieve the best response in terms of the overshoot, the risetime and the settling time; then, using this obtained valueγ1, the values of the parameters α and γ2 of the proposedcontrol gain scheme are tuned to achieve the best responsealso in terms of the overshoot, the rise time and the settlingtime.

Case 1: −1 < α < 0. The parameters of the control gain k(t)of the proposed control scheme of (12) have been taken asα = −0.55, γ1 = 4.0 and γ2 = 12.0. Then the control gaink(t) is given by

k(t) = 4.0{I 1[y2(t)]} + 12.0

{I 1.55[y2(t)]} (30)

For comparison purpose, the classical case given in [27] hasalso been used. We have mentioned that if γ1 = 4.0 = 0and γ2 = 0, the proposed control scheme of (12) will bethe classical case. Hence, the classical control gain kc(t) isgiven by

kc(t) = 4.0{I 1[y2(t)]} (31)

The numerical calculation of the fractional integrator oforder 1.55 of (30) has been done using a digital FIR filterobtained from [28]. The simulations results are obtained withthe input reference signal r(t) as a unity step. Fig. 1 showsthe plots of the output responses y(t) and yc(t) of the closedloop control system with the proposed fractional controlscheme and the classical control scheme, respectively.


www.ietdl.org

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4y(t)yc(t)

Fig. 1 Outputs y(t) and yc(t) of the closed loop control systemwith the proposed fractional and the classical control schemes

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (s)

u(t)uc(t)

Fig. 2 Control signals u(t) and uc(t) of the closed loop con-trol system with the proposed fractional and the classical controlschemes

From Fig. 1, we note that the response y(t) of the closedloop control system with the proposed fractional controlscheme is faster, it has lower overshoot and a smaller risetime than the response yc(t) of the closed loop controlsystem with the classical control scheme.

The plots of the control signals u(t) and uc(t) of the closedloop control system with the proposed fractional and theclassical control schemes, respectively, are given in Fig. 2.

As shown in Fig. 2, we note that the performancesenhancement obtained by using the proposed fractional con-trol scheme requires a larger initial control signal u(t) thanthe initial control signal uc(t) using the classical controlscheme; but the control signal u(t) is as fast as the controlsignal uc(t) despite its larger initial amplitude.

The plots of the fractional and the classical control gainsk(t) and kc(t) are shown in Fig. 3.

From Fig. 3, we remark that the amplitude of the gaink(t) of the closed loop control system with the proposedfractional control scheme is bounded and has a larger steadystate amplitude than the steady state amplitude of the gain


0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

k(t)kc(t)

Fig. 3 Control gains k(t) and kc(t) of the closed loop controlsystem with the proposed fractional and the classical controlschemes

kc(t) of the closed loop control system with the classicalcontrol scheme.Case 2: 0 ≤ α ≤ 1. In this second case, the parameters ofthe control gain k(t) of the proposed control scheme of (12)have been taken as α = 0.3, γ1 = 4.0 and γ2 = 12.0. Thenk(t) is given by

k(t) = 4.0I 1[y2(t)] + 12.0I 0.7[y2(t)] (32)

The classical control gain kc(t) of (31) is also used for com-parison purpose. The numerical calculation of the fractionalintegrator of order 0.7 of (32) has been done using a digitalFIR filter obtained from [28]. The simulations results of thiscase are also obtained with the input reference signal r(t) asa unity step.

Fig. 4 shows the plots of the output responses y(t) andyc(t) of the closed loop control system with the proposedfractional and the classical control schemes, respectively. Asin the first case, the output response y(t) of the closed loopcontrol system with the proposed fractional control schemehas a lower overshoot and a smaller rise time than the clas-sical one yc(t). Also, y(t) is as fast as yc(t). However, theoutput response y(t) can be accelerated by setting a greaterγ2 of the control gain k(t) of (32).

The plots of the control signals u(t) and uc(t) of the closedloop control system with the proposed fractional and theclassical control schemes, respectively, are given in Fig. 5.As in the first case, even though a larger initial control signalu(t) compared to the initial control signal uc(t) is requiredas shown in Fig. 5, u(t) is as fast as uc(t).

The plots of the fractional and the classical control gainsk(t) and kc(t) are shown in Fig. 6. In this case, we caneasily see that, from Fig. 6, the amplitude of the gain k(t)of the proposed fractional control scheme is bounded andhas a smaller steady state amplitude than the steady stateamplitude of the gain kc(t) of the classical control scheme.

From the above example, we have remarked and notedthe following:

• the proposed fractional adaptive controller can solve thestep reference tracking problem


www.ietdl.org

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

y(t)yc(t)

Fig. 4 Outputs y(t) and yc(t) of the closed loop control systemwith the proposed fractional and the classical control schemes

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5–0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (s)

u(t)uc(t)

Fig. 5 Control signals u(t) and uc(t) of the closed loop con-trol system with the proposed fractional and the classical controlschemes

• the tuning parameters γ1 > 0 and γ2 > 0 affect only thetransient performances not the stability of the feedbackcontrol system• the value of the parameter γ1 of the classical control gainscheme has been tuned to achieve a ‘good’ response in termsof overshoot, rise time and settling time. Using the obtainedparameter γ1, the values of the parameters α and γ2 of theproposed control gain scheme have been tuned to achieve a‘good’ response, also, in terms of overshoot, rise time andsettling time• the introduction of the fractional integration in the adap-tation gain of the control strategy has improved the risetime and the settling time and lowered the overshoot of theresponse but this performances enhancement has required alarger initial control signal• the output response can be accelerated by setting a greatertuning parameter γ2 of the control gain of the proposedfractional adaptive control scheme


0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

k(t)kc(t)

Fig. 6 Control gains k(t) and kc(t) of the closed loop controlsystem with the proposed fractional and the classical controlschemes

• for any given tuning parameter γ1 of the classical controlgain scheme, there is always tuning parameters α and γ2 ofthe control gain of the proposed fractional adaptive controlscheme for performances enhancement.

6 Conclusion

In this work, we have presented a concept of fractional orderadaptive controllers via high-gain output feedback for a classof linear, time-invariant, minimum phase, SISO systems ofrelative degree one. The basic idea of this concept is theintroduction of a fractional integration besides of the reg-ular integration of the squared output of the system in theadaptation gain of the control strategy. The introduction ofthe fractional order operator in the adaptation gain of thecontrol strategy has provided supplementary tuning parame-ters which have enhanced the performances of the closedloop control system. The stability analysis of the closedloop control system with the proposed fractional controlscheme has been performed analytically. Simulation resultsof an illustrative example have been presented to showthe control quality enhancement using this proposed con-trol scheme compared with the classical one. The resultsobtained have opened some horizons to propose other frac-tional order adaptive control schemes that do not exist in theregular case.

7 References

1 Bode, H.W.: ‘Network analysis and feedback amplifier design’ (VanNostrand, New York, 1945)

2 Manabe, S.: ‘The non-integer integral and its application to controlsystems’, ETJ Jpn., 1961, 6, (3–4), pp. 83–87

3 Oustaloup, A.: ‘Systèmes Asservis Linéaires d’Ordre Fractionnaire:Théorie et Pratique’ (Editions Masson, Paris, 1983) (in French)

4 Axtell, M., Bise, E.M.: ‘Fractional calculus applications in control sys-tems’. Proc. IEEE Nat. Aero. and Electronics Conf., NY, USA, 21–25May, 1990

5 Podlubny, I.: ‘Fractional order systems and PIλDμ controllers’, IEEETrans. Autom. Control, 1999, 44, (1), pp. 208–214

6 Valério, D., Sáda Costa, J.: ‘Time-domain implementation of fractionalorder controllers’, IEE Proc. – Control Theory Appl., 2005, 152, (5),pp. 539–552


www.ietdl.org

7 Charef, A.: ‘Analogue realization of fractional order integrator, differ-entiator and fractional PIλDμ controller’, IEE Proc. Control TheoryAppl., 2006, 153, (6), pp. 714–720

8 Monje, C.A., Ramos, F., Feliu, V., Vinagre, B.M.: ‘Tip position con-trol of a lightweight flexible manipulator using a fractional ordercontroller’, IET Control Theory Appl., 2007, 1, (5), pp. 1451–1460

9 Oustaloup, A.: ‘La commande CRONE’ (Hermès, Paris, 1991) (inFrench)

10 Chen, Y.Q., Petras, I., Xue, D.: ‘Fractional order control – a tutorial’.Proc. 2009 Amer. Con. Conf., St. Louis, MO, USA, 10–12 June 2009,pp. 1397–1411

11 Vinagre, B.M., Petras, I., Podlubny, I., Chen, Y.Q.: ‘Using frac-tional order adjustment rules and fractional order reference models inmodel-reference adaptive control’, Nonlinear Dyn., 2002, 29, (1–4),pp. 269–279

12 Ladaci, S., Charef, A.: ‘On fractional adaptive control’, NonlinearDyn., 2006, 43, (4), pp. 365–378

13 Suárez, J.I., Vinagre, B.M., Chen, Y.Q.: ‘A fractional adaptationscheme for lateral control of an AGV’, J. Vibrat. Control., 2008, 14,(9–10), pp. 1499–1511

14 Ladaci, S., Charef, A.: ‘An adaptive fractional PIλDμ controller’. Proc.Int. Symp. series on Tools and Methods of Competitive Engineering,TMCE’2006, Ljubljana, Slovenia, 18–22 April 2006, pp. 1533–1540

15 Li, Y., Chen, Y.Q., Cao, Y.C.: ‘Fractional order universal adaptive sta-bilization’. Proc. Third IFAC Workshop on Fractional Differentiationand its Applications, Ankara, Turkey, 05–07 November 2008

16 Li, Y., Chen, Y.Q.: ‘Fractional order universal adaptive stabilizationfor fractional order MIMO system’. Proc. Fourth IFAC Workshop onFractional Differentiation and its Applications, Badajoz, Spain, 18–20October, 2010

17 Ladaci, S., Loiseau, J.-J., Charef A.: ‘Fractional order adaptive high-gain controllers for class of linear systems’, Commun. Nonlinear Sci.Numer. Simul., 2008, 13, (4), pp. 707–714


18 Ladaci, S., Charef, A., Loiseau J.J.: ‘Robust fractional adaptive controlbased on the strictly positive realness condition’, Int. J. Appl. Math.Comput. Sci., 2009, 19, (1), pp. 69–76

19 Luo, Y., Chen, Y.Q., Ahn, H.S., Pi, Y.G.: ‘Fractional order robustcontrol for cogging effect compensation in PMSM position servo sys-tems: stability analysis and experiments’, Control Eng. Pract., 2010,18, (9), pp. 1022–1036

20 Petras, I., Podlubny, I., O’Leary, P., Dorcak, I., Vinagre, B.M.: ‘Ana-logue realization of fractional order controllers’ (TU Kosice, FakultaBerg, 2002)

21 Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: ‘Theory and applicationsof fractional differential equations’ (Elsevier, Amsterdam, Netherlands,2006)

22 Willems, J.C., Byrnes, C.I.: ‘Global adaptive stabilization in theabsence of information on the sign of the high frequency gain’.Springer, Berlin, 1984 (LNCS, 62), pp. 49–57

23 Corless, M.: ‘First order adaptive controllers for systems which arestabilizable via high gain output feedback’, in Byrnes, C.I., Martin,C.F., Seaks, R.E. (Eds): ‘Analysis and control of nonlinear systems’(Elsevier Science Publishers B. V., North-Holland, 1988), pp. 13–16

24 Ilchmann, A.: ‘Non-identifier-based high-gain adaptive control’,Springer-Verlag, Berlin, Germany, 1993 (LNCS, 189).

25 Polderman, J.W., Mareels, I.M.Y.: ‘High gain adaptive control revis-ited: first and second order case’. Proc. 38th Conf. Decision & Control,Phoenix, Arizona, USA, December 1999, pp. 3329–3333

26 Ilchmann, A., Ryan, E.P.: ‘On tracking and disturbance rejection byadaptive control’, Syst. Control Lett., 2004, 52, (2), pp. 137–147

27 Corless, M.: ‘Simple adaptive controllers for systems which are sta-bilizable via high-gain feedback’, IMA J. Math. Control Inf., 1991, 8,(4), pp. 379–387

28 Charef, A., Bensouici, T.: ‘Design of digital FIR variable fractionalorder integrator and differentiator’, Signal Image Video Process., 2012,6, (4), pp. 679–689


06555780

Documents