Sabatier Agrawal Machado Advances in Fractional Calculus Episode 14 pptx

Section 4 proposes to make uncertain a time-delay system proposed by Wang, and then to compare the robustness and performance of Wang, Zhang The structure of the classical Smith predicto

Now, the system is studied in closed-loop so as to measure its immunity to

different disturbances applied to its input ( U ) and its output ( Y ) The

U ref

Y ref

Y U

THERMAL SYSTEM

CRONE or PID CONTROLLER

50 (dotted )

50 (dash dotted ), and G0

6 Simulation Results

control scheme is presented by Fig 9, with,

Fig 9. Closed-loop control scheme

1 2 3 4 5 6 7 8 9 10

System Input Control (V) (PID & CRONE)

1 2 3 4 5 6 7 8 9

System Input Control (V) (PID & CRONE)

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

System Input Control (V) (PID & CRONE)

Time (s)

output disturbance is applied at 1,500 s Time responses are given for different gain variations (1, 50, and 80 times as much gain)

CRONE (black), and PID (grey)

Fig 12. Simulation with disturbances and G0 50 gain variation; path (dotted ), (black), and PID (grey)

(black) and PID (grey)

Fig 10. Simulation with no disturbance and no gain variation; path (dotted ), CRONE

-1.5 -1 -0.5 0

0.5

System Input Control (V) (PID & CRONE)

Time (s)

Fig 13. Simulation with disturbances and G0 80 gain variation; path (dotted ),

Figure 10 shows the same path tracking for PID and CRONE controllers

In fact, the loop has no role in the nominal case

Figure 11 shows a good path tracking in presence of disturbances due to

the loop PID and CRONE have the same dynamic behaviour (same cg).

a better path tracking performance with the CRONE controller as well as beside the disturbances than gain variations, due to a quasi-constant phase

around cg in the Crone controller case

7 Conclusion

In this paper, a new robust path tracking design based on flatness and CRONE

systems dynamic inversion was studied Simulations with two different controllers (PID and CRONE) illustrated the robustness of the proposed path

CRONE control can also be integrated in future designs Flatness principle

conceivable

This paper is a modified version of a paper published in proceedings of

The authors would like to thank the American Society of Mechanical

definitions used in control’s theory were reminded Then, the fractional

tracking strategy The study of robust path tracking via a third-generation

application through non-linear fractional systems dynamic inversion can be

IDETC/CIE 2005, September 24–28, 2005, Long Beach, California, USA

CRONE (black) and PID (grey)

The robustness study is presented by Figs 12 and 13 We can see clearly

a fractional system: a thermal testing bench Firstly, flatness principle control approaches was presented Therefore, this method was applied to

Acknowledgment

Engineers (ASME) for allowing them to publish this revised contribution of

an ASME article in this book

3 Ayadi M (2002) Contributions à la commande des systèmes linéaires plats de dimension finie, PhD thesis, Institut National Polytechnique de Toulouse

4 Cazaurang F (1997) Commande robuste des systèmes plats, application à la commande d’une machine synchrone, PhD thesis, Université Bordeaux 1, Paris

5 Lavigne L (2003) Outils d’analyse et de synthèse des lois de commande robuste des systèmes dynamiques plats, PhD thesis, Université Bordeaux 1, Paris

6 Khalil W, Dombre E (1999) Modélisation, identification et commande des robots, 2ème édition, Editions Hermès, Paris

7 Oustaloup A (1995) La dérivation non entière: théorie, synthèse et cations, Traité des Nouvelles Technologies, série automatique, Editions Hermès, Paris

appli-8 Orsoni B (2002) Dérivée généralisée en planification de trajectoire et génération de mouvement, PhD thesis, Université Bordeaux, Paris

9 Sabatier J, Melchior P, Oustaloup A (2005) A testing bench for fractional system education, Fifth EUROMECH Nonlinear Dynamics Conference (ENOC’05), Eindhoven, The Netherlands, August 7–12

10 Cois O (2002) Systèmes linéaires non entiers et identification par modèle non entier: application en thermique, PhD thesis, Université Bordeaux 1, Paris

11 Cugnet M, Melchior P, Sabatier J, Poty A, Oustaloup A (2005) Flatness principle applied to the dynamic inversion of fractional systems, Third IEEE SSD’05, Sousse, Tunisia, March 21–24

12 Oustaloup A, Sabatier J, Lanusse P (1999) From fractal robustness to the

CRONE control, Fract Calcul Appl Anal (FCAA): Int J Theory Appl.,

2(1):1–30, January

13 Miller KS, Ross B (1993) An Introduction to the Fractional Calculus and

Fractional Differential Equations Wiley, New York

14 Podlubny I (1999) Fractional Differential Equations Academic Press, San

17 Lévine J (2004) On necessary and sufficient conditions for differential flatness, In Proceedings of the IFAC NOLCOS 2004 Conference

18 Samko SG, Kilbas AA, Maritchev OI (1987) Integrals and Derivatives of the Fractional Order and Some of Their Applications, Nauka I Tekhnika, Minsk, Russia

19 Oldham KB, Spanier J (1974) The Fractional Calculus Academic Press,

New York, London

20 Melchior P, Cugnet M, Sabatier J, Oustaloup A (2005) Flatness control: application to a fractional thermal system, ASME, IDETC/CIE 2005, September 24–28, Long Beach, California

They are often based on the use of deliberately mismatched model of the plant

predictor, IMC method

1 Introduction

In the context of the closed-loop control of time-delay systems, Smith [1] proposed a control scheme that leads to amazing performance which is impossible to obtain using common controller It is now well known that such performance can be obtained for perfectly modeled systems only When a system is uncertain or perturbed, trying to obtain high-performance, for instance settling times close to or lower than the time-delay value of the system, leads to lightly damped closed-loop system and sometimes to unstable

E-mail: {lanusse, oustaloup}@laps.u-bordeaux1.fr

proposed to enhance the robustness of Smith predictor-based controllers

and then the internal model control (IMC) method can be used to tune the troller This paper compares the performance of two Smith predictor-based controllers including a mismatched model to the performance provided by a

con-robustness and performance tradeoff It is shown that even if it can simplify the design of (robust) controller, the use of an improved Smith predictor is not necessary to obtain good performance

Keywords

Time-delay system, fractional-order controller, robust control, Smith

© 2007 Springer

511

PREDICTOR-BASED CONTROL AND

LAPS, UMR 5131 CNRS, Bordeaux I University, ENSEIRB, 351

Libération, 33405 Talence Cedex, Tel: +33 (0)5 4000 2417, Fax: +33 (0)5 4000 66 44,

fractional-order CRONE controller which is well known for managing well the

in Physics and Engineering, 511–526

J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications

system Then, many authors proposed to improve the design method of Smith predictors and provided what it is often presented as robust Smith predictors

of the time-delay system to be controlled Using such a model constrains to design the controller very carefully but does not really provide a meaningful degree of freedom to manage the robustness problem Thus, Zhang and Xu

freedom to tune the performance and the robustness of the controller Even if one degree of freedom leads to a low order, and interesting controller, it can

be thought that the performance obtained could be improved by using more degree of freedom

use of few high-level degrees of freedom CRONE is a frequency-domain design approach for the robust control of uncertain (or perturbed) plants The

Section 2 presents the classical Smith predictor design and the approaches proposed by Wang and then by Zhang

Section 3 presents the CRONE approach and particularly its third generation

Section 4 proposes to make uncertain a time-delay system proposed by Wang, and then to compare the robustness and performance of Wang, Zhang

The structure of the classical Smith predictor (Fig 1) includes the nominal

model G0 of the time-delay system G and the time-delay free model P0

G0(s) - P0(s)

+

+ - -

y u

e

[4] proposed to use the internal model control (IMC) [5] and one degree of

Fractional-order control-system design provides such further degree of freedom [6–10] For instance,

d’Ordre Non Entier which means non-integer order robust control) system design [11–18] uses the integration fractional order which permits the

control-2 Smith Predictor-Based Control-Systems

Fig 1 Smith predictor structure

[2] Wang et al [3] proposed a design method based on a mismatched model

plant uncertainties (or perturbations) are taken into account without dis- tinction of their nature, whether they are structured or unstructured Using frequency uncertainty domains, as in the quantitative feedback theory (QFT)

approach [19] where they are called template, the uncertainties are taken into

account in a fully structured form without overestimation, thus leading to cient controller because as little conservative as possible [20]

effi-The closed-loop transfer function y/e is

s G s G s P s K

s G s K s

E

s Y

0 0

If G0 models the plant G perfectly, the closed-loop stability depends on

the controller K and on the delay-free model P0 only, and any closed-loop

dynamic can be obtained As it is impossible that G0 can model G perfectly, it

has been shown that the roll-off of transfer function (1) needs to be sufficient

to avoid instability Then, it is not really important to choose a high-order an

accurate model G0 for the control of an uncertain plant G.

0

system with a delay for the mismatched model

2 m

1

e

s

k s G

1

to design a low-order PID controller K.

Using the relation between the IMC method and the Smith predictor

structure, Zhang and Xu propose an analytical way to design controller K.

Gm1(s)

K(s)

+

+ -

-y u

-du

The IMC controller Q equals:

s K s G

s K s

Q

m1

If Gm approximates well the nominal plant G0, the nominal closed-loop

transfer function y/e is close to the open-loop transfer function defined by:

s G s Q s

Wang et al propose to replace G by a deliberately mismatched model

of G Wang proposes a second-order

Figure 2 presents the Smith predictor including the IMC controller Q.

Fig 2. Smith predictor with IMC controller

Zhang proposes to choose the user-defined transfer function J as

2

1

e

s s

J

ts

, (6)

with the time constant that can be tuned to achieved performance and

robustness Then, controller K is a PID controller given by:

s s

s s

K

11

The CRONE control-system design (CSD) is based on the common

unity-feedback configuration (Fig 3) The controller or the open-loop transfer

function is defined using integro-differentiation with non-integer (or

fractional) order The required robustness is that of both stability margins and

performance, and particularly the robustness of the peak value Mr (called

resonant peak) of the common complementary sensitivity function T(s).

Three CRONE control design methods have been developed, successively

extending the application field If CRONE design is only devoted to the

closed-from the parametric variations of the plant and closed-from the controller phase

variations around the frequency cg, which can also vary The first generation

CRONE control proposes to use a controller without phase variation (fractional

differentiation) around open loop gain crossover frequency cg Thus, the

phase margin variation only results from the plant variation This strategy has

to be used when frequency cg is within a frequency range where the plant

phase is constant In this range the plant variations are only gain like Such a

y (t)

N (t)m

u(t)

+

e F

(t)

yref

3 CRONE CSD Principles

Fig 3. Common CRONE control diagram

second tracking problems

loop using the controller as one degree of freedom (DOF), it is obvious that a

Second DOF (F, linear or not) could be added outside the loop for managing

The variations of the phase margin (of a closed-loop system) come both

So the second generation must be favored

When the plant variations are gain like around frequency cg, the plant phase variation (with respect to the frequency) is cancelled by those of the controller Then there is no phase margin variation when frequency cg varies

integration) whose Nichols locus is a vertical straight line named frequency template This template ensures the robustness of phase and modulus margins and of resonant peaks of complementary sensitivity and sensitivity functions.The third CRONE control generation must be used when the plant frequency uncertainty domains are of various types (not only gain like) The vertical template is then replaced by a generalized template always described

as a straight line in the Nichols chart but of any direction (complex fractional order integration), or by a multi-template (or curvilinear template) defined by

a set of generalized templates

An optimization allows the determination of the independent parameters

powerful one, is able to design controllers for plants with positive real part

1995) Associated with the w-bilinear variable change, it also permits the

design of digital controllers The CRONE control has also been extended to linear time variant systems and nonlinear systems whose nonlinear behaviors

3.2 Third generation CRONE methodology

Within a frequency range [ A, B] around open-loop gain-crossover

frequency vcg, the Nichols locus of a third generation CRONE open-loop is

(Fig 4)

range is often in the high frequencies, and can lead to high-level control input

Such a controller produces a constant open loop phase (real fractional-order

are taken into account by sets of linear equivalent behaviors [21] For

multi-defined by a any-angle straight line-segment, called a generalized template

of the open loop transfer function This optimization is based on the mization of the stability degree variations, while respecting other specifica-tions taken into account by constraints on sensitivity function magnitude Thecomplex fractional order permits parameterization of the open-loop transferfunction with a small number of high-level parameters The optimization ofthe control is thus reduced to only the search for the optimal values of theseparameters As the form of uncertainties taken into account is structured,this optimization is necessarily nonlinear It is thus very important to limit thenumber of parameters to be optimized After this optimization, the corres-ponding CRONE controller is synthesized as a rational fraction only for the optimal open-loop transfer function

mini-The third generation CRONE system design methodology, the most

zeros or poles, time delay, and/or with lightly damped mode (Oustaloup et al

input multi-output (MIMO) (multivariable) plants, two methods have beendevelopment [22] The choice of the method is done through an analysis ofthe coupling rate of the plant When this rate is reasonable, one can opt for thesimplicity of the multi single-input single-output (SISO) approach

The generalized template can be defined by an integrator of complex

fractional order n whose real part determines its phase location at frequency

cg, that is –Re/i(n) /2, and whose imaginary part then determines its angle to

the vertical (Fig 5)

b b a

b

s s

b s

-sign i cg i cg sign

Re 2

cosh )

with n = a + ib i and w j, and where i and j are respectively

time-domain and frequency-time-domain complex planes

The definition of the open-loop transfer function including the nominal

plant must take into account:

cg

Thus, the open-loop transfer function is defined by a transfer function

s s s

m

N N k

k s s

| (j )|dB

arg (j )

0 -

0 -

cg A

B

-a f(b,a)

Fig 4 Generalized template in the Nichols plane

The transfer function including complex fractional-order integration is:

effort specifications at these frequencies

using band-limited complex fractional-order integration:

The accuracy specifications at low frequencies

The generalized template around frequency

The plant behavior at high frequencies while respecting the control

where (s) is a set of band-limited generalized templates :

with:

k k k k

k

b -q b k

k k a

k

k k b k k

s

s e

s

s C

s

sign i

1 /i

1 sign

1

11

1

0for

2 1

k

2 2 1 r 2

1

l l

-n N

s C

where h(s) is a low-pass filter of integer order nh:

h1

where Mr0 is the resonant peak set for the nominal parametric state of the

plant, while respecting the following set of inequality constraints for all plants

(or parametric states of the plant) and for +:

s G s GS s G s C

s C s CS

s G s C s S s G s C

s G s C s T

11

1

1

As the uncertainties are taken into account by the least conservative

The optimal open-loop transfer function is obtained by the minimization of

method, a nonlinear optimization method must be used to find the optimal

where (s) is an integer order n proportional integrator:

values of the four independent parameters The parameterization of the

open-loop transfer function by complex fractional order of integration, then

simplifies the optimization considerably During optimization the complex

order has, alone, the same function as many parameters found in common

rational controllers

When the optimal nominal open-loop transfer is determined, the fractional

controller CF(s) is defined by its frequency response:

j

jj

0 F

G

where G0(j ) is the nominal frequency response of the plant

The synthesis of the rational controller CR(s), consists in identifying ideal

frequency response CF(j ) by that of a low-order transfer function The

parameters of a transfer function with a predefined structure are adapted to

frequency response CF(j ) The rational integer model on which the

parametric estimation is based, is given by:

s A

s B s

where B(s) and A(s) are polynomials of specified integer degrees nB and nA.

All the frequency-domain system-identification techniques can be used An

advantage of this design method is that whatever the complexity of the

control problem, it is easy to find satisfactory values of nB and nA generally

about 6 without performance reduction

Let G be a plant whose nominal transfer function is:

z

1 mp

G s

where: Gmp (s) is its minimum-phase part; zi is one of the its nz right half-plane

zeros; is a time-delay

If (s) remains defined by (9), the use of (18) leads to an unstable

controller (whose right half-plane poles are the nz right half-plane zeros of the

nominal plant) with a predictive part e+ s Taking into account, internal

stability for the nominal plant, stability for the perturbed plants and

achievability of the controller, it is obvious that such a controller cannot be

used Thus, the definition of (s) needs to be modified by including the

nominal right half-plane zeros and the nominal time-delay:

1 z h m

s s s

where Cz ensures the unitary magnitude of (s) at frequency cg.

As frequency cg must be smaller than the smallest modulus of the right

weak, the modification of (s) does not reduce the efficiency around cg of

the optimizing parameters during the constrained minimization

A nonminimum and time-delay plant defined in [3] is used to compare the

performance of Wang and Zhang controllers (both based on the Smith

predictor structure), and CRONE controller To assess the robustness of the

controllers, a 20% uncertainty is associated with each plant parameters

Then, the uncertain plant is defined by

ds

ps

zs g s

1

1

with: g [0.8, 1.2], z [ 1 2, 0.8], p [0.8, 1.2] and d [1.6, 2.4] Its

To approximate the nominal plant, Wang proposes the mismatched model

2

07 5 m

46.1999.0

e1

s s

G

s

, (23)

s s

G

46.1999.0

1

to design a low-order PID controller KW

83.2

485.5

W

s s

s s

uovershoot is 1.28%, it reaches 40% for a parametric state of the plant The

half-plane zeros [23–25], and in a range where the effect of the time-delay is

5 Illustrative Example

and then the first-order transfer function

Figure 5 presents the response of the output y for a set of parametric states of

step disturbance d on the plant input at t = 60 s Even if the nominal percent

another plant-parametric state

nominal value given by Wang is defined by g = z = p = 1 and d = 2.

90% response time is 9.09 s for the nominal plant and can reach about 15 s for

the plant to a unit step variation at t = 0 s of the reference signal e and to a 0.1

As the responses presented by Fig 5 show that the closed-loop responses

can be very lightly damped, it is possible to use degree of freedom of the

Zhang methodology to tune a robust controller that leads to an overshoot

Fig 5. Response y(t) of the plant with the Wang controller for possible values

The controller defined by (7) is

s s

s s

K

46.12

46.1999.0

2 2

2

Taking into account the closed-loop time response obtained by

time-of the plant controlled by the optimal robust Zhang controller

The nominal and greatest values of the overshoot are respectively 0.07%

and 11.3% The nominal and greatest values of the 90% response time are

of g, z, p, and d.

values of g, z, p, and d.

respectively 24.9 s and 37.45 s

domain simulations for all the possible parametric states of the plant, an ite-

rative tuning leads to the optimal value = 5.1 Figure 6 presents the response

Fig 6. Response y(t) of the plant with a robust Zhang controller for possible

present the Bode and Nichols diagrams of the uncertain plant

Fig 7. Bode diagram of the nominal plant G0 (- - -) and lower and greatest magnitude and phase of the uncertain plant G ( _

0

1 As plant low-frequency order is 0, order nl of (12) equals 1 to reject any constant input disturbance du As the plant relative degree is 4 and as the

plane zero, order nh of (13) equals 6 to obtain a strictly proper controller

To be sure to have enough parameters to be tuned, orders N- and N+ of

The time-delay and right half-plane zero of G (22) are respectively 2 and

nominal open-loop transfer function needs to include the plant right half-

Taking into account the five sensitivity function constraints (15–16)

= 0.20, a = 9.14, b = 1.72, q = 2, = 0.38, a = 2.51, b = 1.31, q = 4, = 2.09, and = 0.0507, Y = 6.03dB Figure 9 presents the optimal open-

Then, a third generation CRONE controller is designed to control the

Fig 8. Nominal plant Nichols locus (- - -) and uncertainty domains ( _

Fig 9. Nominal open-loop Nichols locus (- - -), uncertainty domains ( _)

By minimizing the cost function (Jopt = 0.75dB), the optimal template positions the uncertainty domains so that they overlap the 0.2dB M-contour as little as possible The sensitivity functions met almost the constraints (Fig

10) Only Tl is exceeded of 0.23dB around 0.1rad/s Using zeros and poles, the rational controller CR(s) is now synthesized from (18):

s s s s s s s

s s s s s s

.

s

C

2 3

4 5

6 7

2 3

4 5

6

R

855301128243143739300704311595

0

196020607688287585262944130533585

Frequency (rad/s)

)

values of g, z, p, and d.

and 11.4% (Fig 11) The nominal and greatest values of the 90% response

(90% response time) and robustness (percent overshoot variation) obtained with the 3 controllers

Table 1. 90% response time t90%

Controller t90% nom. t90% max. Onom. Omax.

Even if the Wang controller provides short 90% response times, it also provides very long settling times (Fig 5) and great overshoots The optimized robust Zhang controller provides greater 90% response times but shorter settling times (Fig 6) and small variations of the overshoot The CRONE

controller provides small variations of the overshoot also, and shorter 90% response times than provided by the Zhang controller

6 Conclusion

In the context of the control of time-delay systems, many modifications have been proposed to enhance the performance and robustness of control-systems based on Smith predictor structure This paper has proposed to compare the

controllers) including a mismatched model of the time-delay system, to the performance provided by a CRONE controller For that comparison, a nonminimum phase plant with a time-delay is chosen To assess the robustness of the controllers some uncertainty is added on each plant parameters Even if it is more secure than a classical Smith predictor, the Wang controller reveals not to be robust enough Based on the IMC method, the Zhang controller has been optimized using one degree of freedom correlated to the settling time of the closed-loop system The time-domain optimization succeeds and provides a robust Zhang controller which provides perfectly acceptable performance Using more high-level degree of freedom, a

As the genuine plant uncertainty is taken into account without any overestimation, the CRONE controller reveals to be both robust and with higher performance

Then, it can be concluded that even if it can simplify the design of (robust) controller for time-delay system, the use of an improved Smith predictor is not necessary to obtain good performance

time are respectively 19.3 s and 24.7 s Table 1 compares the performance

performance of two Smith predictor-based controllers (Wang and Zhang

The nominal and greatest values of the overshoot are respectively 5.7%

and percent overshoot O obtained with the

3 controllers