ROBUST ADAPTIVE CONTROL USING A FRACTIONAL

FEEDFORWARD BASED ON SPR CONDITION

Samir Ladaci, Jean Jacques Loiseau

IRCCyN, Ecole Centrale de Nantes, 1, rue de la No

¨

e, BP: 92101 Nantes, 44321, France

Abdelfatah Charef

D

´

epartement d’Electronique, Universit

´

e Mentouri, Route de Ain Elbey, Constantine 25000, Algeria

Keywords:

Robust adaptive control, Fractional Adaptive Control, Model Reference Adaptive Control, Feedforward, Frac-

tional order systems.

Abstract:

This paper presents a new approach for robust adaptive control, using fractional order systems as parallel

feedforward in the adaptation loop. The basic adaptive algorithm used here is Model Reference Adaptive

Control (MRAC), which do not require explicit parameter identiﬁcation. The problem is that such a control

system may diverge when confronted with ﬁnite sensor and actuator dynamics, or with parasitic disturbances.

One of the classical robust adaptive control solutions to these problems, makes use of parallel feedforward and

simpliﬁed adaptive controllers based on the concept of positive realness.

This control scheme is based on the ASPR property of the plant. We show that this condition implies also

robust stability in case of fractional order controllers. A simulation example of a SISO robust adaptive control

system illustrates the interest of the proposed method in the presence of disturbances and noises.

1 INTRODUCTION

Adaptive control has proven to be a good control so-

lution for the partially unknown systems or varying

parameter systems. In this domain Model reference

adaptive control (MRAC) became very popular since

it presents a very simple algorithm with easy im-

plementation and does not require identiﬁers or ob-

servers in the control loop (Astrom and Wittenmark,

1995; Landau, 1979). However such algorithm shows

its limits in noisy or disturbed environment, which

may make it inefﬁcient or uncompetitive. Unfortu-

nately very few industrial control processes are not

subject to theses practical problems, which can dam-

age the quality of product and the good process oper-

ating.

The use of simple parallel feedforward in the adap-

tation loop appeared as a robust solution since the

80’s. Many works have used this approach towards

robust control systems (Bar-Kana, 1987; Naceri and

Abida, 2003). In the last decade a great interest was

given to fractional order systems, which have shown

good robustness performances, several robust control

methods based on these systems have been developed,

like CRONE Control (Oustaloup, 1991) and frac-

tional adaptive control (Vinagre et al., 2002; Ladaci

and Charef, 2006; Ladaci et al., 2007).

In this paper we present a fractional robust adaptive

control solution for disturbed applications, based on

the idea of Bar-kana (Bar-Kana, 1987), which uses the

basic stabilizability property of the plant and simple

parallel feedforward in order to satisfy the desired ”al-

most positive realness” condition that can guarantee

robust stability of the nonlinear adaptive controller.

The main contribution of this work is to improve the

feedforward approach robust performances by using

fractional order ﬁlters. This result is illustrated by a

simulation example of a test in bad realistic condi-

tions like ﬁnite bandwidth of actuators, input and out-

put disturbances and no assumed natural damping.

This paper is structured as follows:

In section 2 deﬁnitions of fractional order systems are

presented. Section 3 introduces the principles of ro-

bust adaptive control based on the concept of ’pos-

itive realness’ condition and then the main result in

fractional order case is presented in section 4. The

implementation in Model Reference Adaptive Con-

trol scheme is introduced in section 5 and a simulation

414

Ladaci S., Jacques Loiseau J. and Charef A. (2007).

ROBUST ADAPTIVE CONTROL USING A FRACTIONAL FEEDFORWARD BASED ON SPR CONDITION.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 414-420

DOI: 10.5220/0001636904140420

Copyright

c

SciTePress

example is given in section 6. The paper is concluded

in section 7.

2 FRACTIONAL ORDER

SYSTEMS

The analysis in Bode plot of many natural pro-

cesses, like transmission lines, dielectric polarisation

impedance, interfaces, cardiac rhythm, spectral den-

sity of physical wave, some types of noise (Van-

DerZiel, 1950; Duta and Hom, 1981), has allowed

to observe a fractional slope. This type of process

is known as 1/ f process or fractional order system.

During the last decade, a great interest was given by

researchers to the study of these systems (Sun and

Charef, 1990) and their application in control systems

(Oustaloup, 1991; Hotzel and Fliess, 1997; Ladaci

and Charef, 2006; Ladaci et al., 2007).

A SISO fractional order system can be represented by

the following transfer function,

X(s) =

b

m

s

β

m

+ b

m−1

s

β

m−1

+ ...+ b

0

s

β

0

a

n

s

α

n

+ a

n−1

s

α

n−1

+ ... + a

0

s

α

0

(1)

Where,

• α

i

, β

j

: real numbers such that,

0 ≤ α

0

< α

1

< ... < α

n

0 ≤ β

0

< β

1

< ... < β

m

• s: Laplace operator.

for the purpose of this work, let us introduce the fol-

lowing deﬁnitions,

Deﬁnition 1 The fractional order transfer function

X(s) given in (1) is called proper if: β

m

≤ α

n

It is called stricly proper if: β

m

< α

n

Deﬁnition 2 (Desoer and Vidyasagar, 1975) The

fractional order transfer function Matrix M

X

(s)

whose elements are of the form (1) is proper (strictly

proper) if and only if all elements of M

X

(s) are

bounded at ∞ (tend to zero at ∞, resp.).

We use in the sequel a description equation into

frequency domain of a single pole fractional order

process, given as follows:

Y(s) =

1

(s+ p

T

)

α

(2)

with

• α: fractional exponent, 0 ≤ α ≤ 1

• p

T

: fractional pole which is the cut frequency.

Many previous works have shown that fractional

systems present best qualities, in response time and in

transition dynamic stability (Sun and Charef, 1990).

All the control theory developed by Oustaloup es-

pecially on CRONE control was based on fractional

order systems robustness in presence of uncertainties

and perturbations (Oustaloup, 1991).

3 CONCEPT OF POSITIVE

REALNESS CONDITION

Robustness is deﬁned relatively to a certain property

and a set of models. A property (generally stability

or performance level) is said to be robust if all

the models belonging to the set satisfy it. Robust

adaptive stabilization means that all values involved

in the adaptation process namely, states, gains and

errors are bounded in the presence of any bounded

input commands and input or output disturbances

(Bar-Kana and Kaufman, 1985; Kwan et al., 2001).

In this paper we are interested by a particular con-

ﬁguration of feedforward controllers combined with

MRAC control and fractional order systems giving a

fractional robust adaptive control method.

The use of a simple feedforward in the adaptation

loop (see Figure 4) improves the robust stability of the

control system. This approach is based on the con-

cept of the ”positive realness” condition (Bar-Kana,

1989); witch can guarantee stable implementation of

adaptive control conﬁguration. Let us present these

deﬁnitions:

Deﬁnition 3 The m × m transfer function matrix

G

s

(s) is called strictly positive real (SPR) if (Landau,

1979; Bar-Kana, 1989):

1. All elements of G

s

(s) are analytic in ℜ(s) ≥ 0.

2. G

s

(s) is real for real s.

3. G

s

(s) + G

T∗

s

(s) > 0 for ℜ(s) ≥ 0 and ﬁnite s.

We also show that (Shaked, 1977) for a fractional or-

der transfer function matrix G

s

(s),

G

s

(s) is SPR ⇔ G

−1

s

(s) is SPR (3)

Indeed, by using the SPR property if we write (Bar-

Kana, 1989),

G

s

(s) = A+ jB ⇒ G

T∗

s

(s) = A

T

− jB

T

Since by deﬁnition

G

s

(s) + G

T∗

s

(s) = A+ A

T

+ j(B− B

T

) > 0

ROBUST ADAPTIVE CONTROL USING A FRACTIONAL FEEDFORWARD BASED ON SPR CONDITION

415

we get B = B

T

and A > 0 (not necessary symmetric).

Then whenever

ℜ[G

s

(s)] = A > 0

we get

G

−1

s

(s) = (A+ BA

−1

B

T

)

−1

− jA

−1

B(A+ BA

−1

B

T

)

−1

and

ℜ

G

−1

s

(s)

= (A+ BA

−1

B

T

)

−1

> 0

which proves (3).

Deﬁnition 4 (Bar-Kana, 1987)

Let G

a

(s) be a m× m transfer matrix. Let us assume

that there exists a positive deﬁnite constant gain ma-

trix,

˜

K

e

such that the closed-loop transfer function

G

c

(s) =

I + G

a

(s)

˜

K

e

−1

G

a

(s) (4)

is SPR. G

a

(s) is called ”almost strictly positive real

(ASPR)”.

Now if we consider a fractional order proper or

strictly proper ASPR transfer matrix G

s

(s).

Then the following statements are equivalent,

G

s

(s) = [I + G

a

(s)K

e

]

−1

G

a

(s) is SPR (5)

G

s

(s) = [I + G

a

(s)K

e

]

−1

is SPR (6)

G

−1

s

(s) = G

−1

a

(s) + K

e

is SPR (7)

ℜ

G

−1

a

(s) + K

e

ℜ(s)≥0

> 0 (8)

G

−1

s

(s) is asymptotically stable and

K

e

is sufﬁciently large (9)

Because ∃M such that ℜ

G

−1

a

(s)

ℜ(s)≥0

> M > −∞,

and then any K

e

> −M will do (Bar-Kana, 1989).

G

a

(s) is strictly minimum phase and

K

e

is sufﬁciently large (10)

All the above algebraic manipulation, as done to

obtain (3) and deﬁnitions 3 and 4, apply to fractional

systems as well. Here we can generalize as fellows

the result of (Bar-Kana, 1989) to the fractional order

case.

Lemma 1 Let a fractional order transfer function

matrix G

a

(s) be ASPR and let

˜

K

e

be any gain that

satisﬁes (4). Then G

a

(s) is SPR for any gain K

e

that

satisﬁes K

e

>

˜

K

e

.

It is obvious that ASPR fractional order systems,

which are minimum phase proper systems maintain

stability with high gains. The high gain stability is

important when nonstationary or nonlinear (adaptive)

control is used, because the robustness of the control

system is maintained if, due to speciﬁc operational

conditions, the time-varying gains become too large.

Remarks

1. The ASPR plant must also be proper.

2. The open loop is not necessarily stable (the plant

will actually be stabilized by the ﬁctitious gain

K

e

), however all the zeros must be placed in the

left half plane. The plant must be minimum phase

to obtain positivity.

3. We can easily show (Bar-Kana, 1987) that if a sys-

tem is ASPR, then it can be stabilized by any con-

stant or time variable output gain K

e

, if it is large

enough, i.e. K

e

>

˜

K

e

.

But in this method, instead of using high gain regula-

tion we will use a simple parallel feedforward conﬁg-

uration which can by a similar way satisfy the positive

realness conditions.

The idea of using feedforward in parallel with the

controlled plant is based on the following Lemma of

Bar-Kana,

Lemma 2 (Bar-Kana, 1989) Let the plant be de-

scribed by the m× m transfer function G

p

(s) of order

n. Let C(s) be any dynamic stability output feedback

controller. Then

G

a

(s) = G

p

(s) +C

−1

(s) (11)

is ASPR if C

−1

(s) is proper or strictly proper.

We can adapt the proof of (Bar-Kana, 1989; Bar-

Kana, 1986)) to the fractional case.

4 MAIN RESULT

At this stage we propose a fractional order feedfor-

ward conﬁguration of the form:

F(s) =

F

p

1+

s

s

0

α

(12)

with a real fractional power 0 < α < 1, to improve the

robustness of the adaptive algorithm, in presence of

perturbations, as such systems do not amplify much

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

416

G(s)K(1+ qs

α

)

u(s)

y(s)

G

CL

(s)

u

c

(s)

−

+

Figure 1: Closed-loop system.

G(s)

K

−1

1+qs

α

y

s

(s)

y

a

(s)

y

p

(s)

G

a

(s)

G

s

(s)

K

e

u

c

(s)

+

−

Figure 2: The ﬁctitious SPR conﬁguration.

these random signals. This conﬁguration could be

considered as the inverse of an improper fractional

PD

µ

controller, which was used in control systems

with good proven performances (Oustaloup, 1983;

Hotzel and Fliess, 1997; Podlubny, 1999).

We can formulate the main result of this paper in the

following theorem.

Theorem 1 Let G(s) be any m × m strictly proper

transfer matrix of arbitrary MacMillan degree. G(s)

is not necessarily stable or minimum phase. Let

H

f

(s) = K(1+ qs

α

) (13)

be some stabilizing controller for G(s). Then the aug-

mented controlled plant

G

f

a

(s) = G(s) + H

−1

f

(s) = G(s) +

K

−1

1+ qs

α

(14)

is ASPR.

Proof of Theorem 1:

From deﬁnition 4, if G

a

(s) is ASPR then the closed-

loop transfer function

G

c

(s) =

I + G

a

(s)

˜

K

e

−1

G

a

(s)

is ASPR.

Since H

−1

f

(s) from (13) is strictly proper (relative

degree α > 0), then Lemma 2 implies that the

augmented system G

f

a

(s) as deﬁned in (14) is ASPR,

which proves Theorem 1.

The stabilizing controller H

f

(s) can also be mod-

elized as follows,

H

f

(s) = K(1+ qs)

α

(15)

Figure 1 represents the feedback control system cor-

responding to the control (13).

From Deﬁnition 4 and the fact that the transfer func-

tion G

f

a

(s) is ASPR, we know that it can be stabilized

by a gain

˜

K

e

. Figure 2 illustrates the feedforward con-

ﬁguration. In addition, the stabilization is robust, it

holds for any gain K

e

>

˜

K

e

.

Many previous works (Hotzel and Fliess, 1997;

Podlubny, 1999) have proposed PD

µ

improper con-

trollers of the form (13):

C(s) = K

p

+ K

i

s

α

(16)

which can stabilize many realistic plants for sufﬁcient

high values of K.

A feedforward of equivalent effect is chosen as fol-

lows:

F(s) = C

−1

(s) =

F

p

1+

s

s

0

α

(17)

Where F

p

= K

−1

, such that the augmented plant be-

comes:

G

a

(s) = G

p

(s) + F(s) (18)

As K should be very large, so F

p

are small coefﬁ-

cients, guaranteeing that G

a

(s) be ASPR. And dur-

ing the control design we can take G

a

(s) ≈ G

p

(s) as a

practical approximation.

5 IMPLEMENTATION IN MRAC

SCHEME

Model Reference Adaptive Control (MRAC) is one

of the more used approaches of adaptive control, in

which the desired performance is speciﬁed by the

choice of a reference model. Adjustment of param-

eters is achieved by means of the error between the

output of the plant and the model reference output.

Let us introduce the basic ideas of this approach rep-

resented in Figure 3.

We consider a closed loop system where the con-

troller has an adjustable parameter vector θ. A model

which output is y

m

speciﬁes the desired closed loop

response. Let e be the error between the closed loop

system output y and the model one y

m

, one possibility

is to adjust the parameters such that the cost function:

J(θ) =

1

2

e

2

(19)

ROBUST ADAPTIVE CONTROL USING A FRACTIONAL FEEDFORWARD BASED ON SPR CONDITION

417

y

m

u

y

u

c

Adjustment

Mechanism

Process

Controller

Reference

Model

Parameters of the Controller

Figure 3: Direct Model Reference Adaptive Control.

be minimised. In order to make J small it is reason-

able to change parameters in the direction of negative

gradient J, so:

dθ

dt

= −γ

δJ

δθ

= −γe

δe

δθ

(20)

or

dθ

dt

= γϕe (21)

where ϕ = −

δe

δθ

is the regression (or measures) vector

and γ is the adaptation gain. This aproach is called

M.I.T. rule.

The introduction of a simple feedforward in the

MRAC adaptation loop us represented in ﬁgure 4

improves the robust stability performance against

the controller gain ﬂuctuations in presence of per-

turbation and noises (Naceri and Abida, 2003).

Previous works (Sobel and Kaufman, 1986), showed

that the ASPR property of a process, allows the

implementation of very simple adaptive controllers

that garantee robust stability of the closed loop in

presence of bounded input or output disturbances.

The feedforward transfer function is choosen like in

(12) where the gain F

p

is a small coefﬁcient.

F

ProcessActuator

Controller

u

c

y

m

Reference

Model

Adjustment

Mechanism

y

u

Figure 4: Simple feedforward in MRAC scheme.

6 SIMULATION EXAMPLE

Without any loss of generality we will apply this ro-

bust adaptive control method, both in the case of inte-

ger and fractional order feedforward, to a SISO model

of a DC motor controlled in respect of velocity, given

by:

G

p

(z) =

0.8513z+ 5.099 10

−6

z

2

+ 2.442 10

−7

z+ 1.37 10

−11

(22)

with a sampling period T

s

= 0.3sec, and an actuator

model of the form:

A(z) =

0.007667z+ 0.007049

z

2

− 1.763z+ 0.7772

(23)

The plant is subject to random input and output per-

turbations of amplitudes 2 and 0.05 respectively.

The reference model G

m

is given by:

G

m

(z) =

0.9411z+ 0.1208

z

2

+ 0.05679z+ 0.005092

(24)

6.1 Integer Order Feedforward Case

The feedforward trunsfer fuction F is given by:

F(z) =

3.2394 10

−7

z− 0.9997

(25)

with a regulation parameter γ = 0.001 we obtain the

results of Figure 5.

6.2 Fractional Order Feedforward Case

The fractional order feedforward trunsfer function F

is given in Laplace domain by:

F(s) =

0.001

(s+ 500)

0.6

(26)

For the purpose of our approach we need to use an

integer order model approximation of the fractional

order feedforward model in order to implement the

adaptation algorithm, for this aim we have used the

so-called singularity function method (Charef et al.,

1992).

The fractional transfer function (26) is approximated

to a linear transfer function and sampled to give the

following formula:

ˆ

F(z) =

0.001(z− 4.78 10

−97

)

z

2

− 2.407 10

−96

z+ 1.001 10

−207

(27)

with a regulation parameter γ = 0.005, we obtain the

results of Figure 6.

6.3 Remarks

• The command signal u is more polish in the frac-

tional case witch is a very useful property in reg-

ulation problem.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

418

(a)

(b)

(c)

Figure 5: Process output with integer feedforward

(a) Process output y(t), (b) Control signal u(t), (c) Error

signal e(t).

• The proposed fractional order conﬁguration of

feedforward maintains stability and at less the

same level of performances, witch conﬁrms the

interest of integrating fractional strategy in robust

adaptive control.

(a)

(b)

(c)

Figure 6: Process output with fractional feedforward

(a) Process output y(t), (b) Control signal u(t), (c) Error

signal e(t).

7 CONCLUSION

In this paper we have presented a new robust adap-

tive control strategy, by introducing simple fractional

feedforward conﬁguration in the MRAC algorithm.

The concept of positive realness condition which is

the basis of this robust control strategy is extended to

fractional order control systems. The idea was to take

beneﬁt of the high performance quality of fractional

ROBUST ADAPTIVE CONTROL USING A FRACTIONAL FEEDFORWARD BASED ON SPR CONDITION

419

order systems conﬁrmed in many precedent research

works. The stability proofs of this adaptive control

scheme developed for integer order ﬁlters in control

literature still holds for such systems. Simulation re-

sults have shown a better ﬁltering ability of command

and output signals, and more robustness against ad-

ditive perturbations, than in the integer order feedfor-

ward conﬁguration case.

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