Third Generation Crone Control of Continuous Linear Time Periodic Systems

Copyright Cl IFAC System Structure and Control, Nantes, France, 1998

THIRD GENERATION CRONE CONTROL OF CONTINUOUS LINEAR TIME PERIODIC SYSTEMS Jocelyn Sabatier, Aitor Garcia Iturricha, Alain Oustaloup and

Fran~ois

Levron

Equipe CRONE - LAP - ENSERB - Universite Bordeaux J 351, cours de la Liberation - 33405 Talence cedex - FRANCE Tel :+33 (0) 5 56842418 - Fax: +33 (0) 5 56 8466 44 email: {sabaJier - oustaloup}@/ap.u-bordeauxfr Abstract: Third generation CRONE control based on complex non integer differentiation results from an optimal approach of control. Its aim is to ensure the robustness of the stability degree of the control loop at the time of a reparametration of the plant In this paper, third generation CRONE control is extended to the control of linear time periodic systems. This extension is possible through a generalisation of the Nyquist theorem, and through the use of transfer functions for linear time periodic system modelling : time varying transfer function and harmonic transfer function. Copyright © 1998 IFAC Resume: La commande CRONE de troisieme generation est basee sur la derivation non entiere complexe et resulte d'une approche optimale de la commande. Son objectif est d'assurer la robustesse du degre de stabilite de la boucle de commande lors d'une reparametrisation du procecte. Dans cet article, la commande CRONE de troisieme generation est etendue a la commande de systemes lineaires non stationnaires acoefficients periodiques. Cette extension est rendue possible grace a une generalisation du theoreme de Nyquist et grace a l'utilisation deux extensions de la notion de fonction de transfert : les fonctions de transfert non stationnaires elles fonctions de transfert harmoniques. Keywords: CRONE control, linear lime periodic systems, harmonic transfer functions, time varying transfer function Section 2 presents some general points concerning LTP systems including two extensions of the transfer function notion: - time varying transfer function (TVTF) of Zadeh (Zadeh, 1950; Zadeh, 1961) which can be extended to the general case of linear time varying systems; - harmonic transfer function (HTF) of Wereley and Hall (Wereley 90, Hall 90) limited to L TP systems but permitting a generalisation of the Nyquist criterion (Wereley and Hall, 1991; Hall and Wereley, 1990). Finally, an extension of the Nyquist criterion for L TP systems is summarised. Section 3 gives an extension of third generation CRONE control to L TP system control. This extension allows the synthesis of control laws which ensure: - an almost stationary behaviour of the control loop for the nominal parametric state of the plant; - designer specified performances of the control loop for the nominal parametric state of the plant ; - robustness of stability and of dynamic performances when the plant is reparametrated, - some immunity to the time varying character of the planL. These objectives are reached by the computation of an optimal open loop behaviour based on complex non integer differentiation which: - respects the extended Nyquist criterion even with parametric variations of the plant;

1. INTRODUCTION

This paper deals with the robust control of a particular class of continuous linear time varying systems: linear time periodic (LTP) systems. In practice, numerous systems prove to be L TP. They are mostly mechanical systems in rotation (Chassande, 1981; Sheu, 1978; Kern, 1980; McKillip, 1985). Although there is abundant literature on analysis and control of LTP systems mainly based on the works of Floquet (Floquet, 1883), robust control of LTP systems is in fact rarely evoked, especially for continuous systems. In spite of some developments concerning LQG control (Rabenasolo, 1992; Wereley, 1991), robust control of LTP systems, and time varying systems in general, is far from reaching the same maturity as robust control of stationary linear systems. The control strategy proposed in this paper auempts to close this gap. It is a direct extension of third generation CRONE control which has already proved its efficiency in the stationary case (Oustaloup, et al., 1995a, b; Mathieu, et al., 1996) . This extension is possible through the use of transfer functions for LTP system modelling, and a generalisation of the Nyquist criterion. The frequency approach proposed in this paper is particularly attractive as it uses well known scalar and multi variable stationary linear system control principles. This paper is organised as follows.

299

- minimises, at reparametration, variations of the resonance ratio of the stationary part of the TVTF of the control loop, and the time varying part of this TVTF. In section 4, the proposed synthesis method is applied to the position control of a testing bench with two D.C. motors.

2. LINEAR TIME PERIODIC SYSTEMS

2.1. Definitions

contrary to stationary systems, LTP systems map sinusoidal input signals into periodic and not purely sinusoidal output signals. To extend the notion of transfer function to the LTP system case, Wereley and Hall (Wereley and Hall, 1990) have used this characteristic to defme a new symbolic test input signal, known as exponentially modulated periodic (EMP) signal, defined by : u(t) =

=A(e. t+T)

(1)

yet) =

seC. (7)

L

'V t

yke'k',

~

o.

(8)

'kE~

Using this property; Wereley and Hall have defined the HTF as an operator, :J-f(s), connecting the harmonics of the EMP output signal to the harmonics of the EMP input signal, namely:

(2)

,

'V t ~ 0, and Sk = S + jkroo,

,

At steady state, an asymptotically stable LTP system maps an EMP input signal to an EMP output signal of the same frequency, but with different amplitude and phase (Wereley and Hall, 1990), namely:

where u(t) E R, yet) E I., x(t) E Rnxl and where A(.), B(.) and C(.) are real-valued matrices of appropriate dimension. Matrix A(.) is both a function of an uncertain but stationary parameter vector e, and a periodic function of time variable t, namely: A(e. t)

lkl

ute

kEZ

L TP systems considered in this paper have a state space description of the fonn : x(t)= A(9.t) x(t) + B(9.!) u(t) yet) = C(9.t) x(t)

L

y= :H(s) u ,

(9)

and similarly for B(e, t) and C(e, t) . Period T represents the smallest value satisfying relation (2).

with

Zadeh (Zadeh, 1961) has demonstrated that linear time varying systems can be characterised by TVTFs H(jm, t), linked to the impulse response of the system, het, ~) (both function of the time variable t and of the moment of application of the impulse ~) by the relation:

For the LTP system given by relation (1), where matrix A(e, t) can be expanded in a Fourier series of the form

yT =[ ... , Y.I. Yo, YI, ...)

A(e.

and

t}= L

uT

AJek

= [ ...•

k6IQt

u..],

Uo. UI, ...).

(10)

,

kEZ

and similarly for B(e, t) and C(e, t), the HTF :Ji(s) is given by :

(3)

JK..s)

This representation of time varying systems is particularly attractive in the area of automatic control because it allows the computation of the steady and transitory states of the system. Indeed, if yet) is the output of a system characterised by the TVTF H(jw, t), then: yet) =

If-

27t _

H(jro, t) U(jro) ei""dro ,

.!'l =

(4)

where U(jw) denotes the Fourier transfonn of the input

L J{Uro)cos(kroot) + H~(jro)sin(krool),

or using the complex exponential form of sinus and cosine functions:

L kEZ

HkUc.o) e(jkl»ot}

where

roo =2x. T

... A 0(9) A .1(9)

A .2(e)

A 1(9) A 0(9)

A .1(9)

... A 2(9) A 1(9)

A 0(9)

...

(12)

As shown in the next paragraphs, this particularly useful representation of LTP systems allows the generalisation of : - the fonnulas resulting from the association of several transfer functions of stationary systems (Sabatier, 1998) ; - the Nyquist theorem.

(5)

k:1

HUro, t) =

(11)

and \Vith a similar de(inition for 'B and C in term of (BJ eJ, ke Z) and (Ck(9), ke Z }. I denotes the doubly infinite identity matrix and 'J{ = blkdiag (jkwOI) , I E R nxn is the identity matrix.

Zadeh has also demonstrated that LTP system (1) can be represented by a TVTF, H(jw, t), of the fonn : HUc.o, t) =Ho(jc.o) +

= C[sf - ( .!'l- ~)JI'1l ,

with

(6)

TVTF of system (1) can also be deduced from (11) with the fonn :

These representations highlight that LTP systems can be decomposed in a stationary part characterised by HOUw) and in a time varying part characterised by the other terms of relations (5) and (6). They also allow a generalisation of for example, initial and final value theorems, and a prediction of the existence of limit cycles (Sabatier, 1998). Relations (5) and (6) show that,

Ho(jro-jroo) ?fjc.o) = [ ••• HIUc.o-jroo) ... H2Uc.o-jroo)

. ·1

H.IUc.o) H. 2(jc.o+jroo) .•. Ho(jc.o) H.I(jc.o+jroo) ... , HI(jc.o) Ho(jc.o+jroo) .. .

.

300

.

(13)

(transmittances HkGW), k E Z of relation (6) are thus given by the central column of matrix (13». 2.2. Nyquist criterion (Hall and Wereley, 1990)

This generalisation of the Nyquist criterion is based on the inspection of the eigenvalue locus in the complex plane of the open loop HTF of the closed loop system whose stability is being studied. If :Ji(s) denotes the HTF of the open loop, then, the closed loop system is stable if, and only if, the eigenvalue locus of 9iGw), with CIlE[ -COtI2, IDo/2], encircles the critical point of coordinates (-1, 0) as many times counterclockwise as the open loop has characteristic coefficients (Rabenasolo, 1992) with a positive real part.

where

(15) K ensures the open loop unit gain frequency rou set by the designer. 00 'b , 00 b, 00 hand 00 'h are transitional frequencies. nb and Oh are respectively the asymptotic behaviour orders in open loop at the low (00 < W'b) and the high (00 > W'h) frequencies. a and b are the real and imaginary orders of integration. wT is the resonance frequency close to rou.

3. THIRD GENERATION CRONE CONlROL OF LTP SYSTEMS 3.1 . Introduction

3.3. Optimisation of the open loop behaviour

Over the last ten years , CRONE control and the mathematical tools that it uses have emerged in the area of robust control of stationary linear systems (Oustaloup, and al., 1995a). Furthermore, several applications of CRONE control on real systems demonstrate its reliability (Oustaloup, and al., 1995b).

The optimisation of the open loop behaviour consists in determining the seven optimal parameters of the nominal open loop transmittance pes) : - optimal real integration order, Clopt, and optimal gain, Kopt - optimal imaginary integration order, bopt - optimal, transitional frequencies, ro'bopt, robopt, Whopt and 00 hopt.

At present, CRONE control is used on the following uncertain linear systems: - non minimum phase (Mathieu, and al., 1996), - unstable (Mathieu, 1997), - resonant mode (Oustaloup, and al., 1995b), - multi variable (Mathieu, 1997).

The unit gain frequency and the tangency to an isoovershoot contour are chosen by the designer, so, only five independent parameters need to be considered.

In practice however, most systems to control are rarely purely linear. Thus, synthesis techniques allowing CRONE control to take into account other classes of systems, and notably LTP system, need to be developed.

The open loop behaviour which satisfies the objectives defined in section 3.2 can be computed by solving a constrained optimisation problem. The performance criterion and constraints which provide optimal open loop behaviour respecting the objectives of section 3.2 thus comprise terms which guarantee :

3.2. Objectives

Within the context of an extension of the CRONE control to the control of LTP plants, the objective is to describe an open loop behaviour for the nominal parametric state of the plant which:

- robustness of the stability degree of the control - immunity of the control to the time varying character of the plant - the performance objectives set by the designer.

- ensures an almost stationary behaviour of the closed loop system (given the truncation of HTF and TVTF) ; - ensures performances set by the designer such as the rapidity and the resonance ratio in tracking of the closed loop system ; - takes into account the behaviour of the plant at the low and the high frequencies to ensure satisfactory accuracy of steady state, and immunity of the plant input to measurement noise.

To define the constrained optimisation problem analytically, the standard control scheme of fig. 1 is considered, in which CGw, t) and PGw, t) denote the TVTFs of the optimal controller and of the plant to be controlled.

Whenever the plant is reparametrated, namely if the plant P is element of the description family JP, this open loop must also ensure:

r{ t}

- robust closed loop stability and performances; - a certain immunity of the closed loop to the time varying character of the plant.

Nm{t} Fig. 1. Control scheme

As in the stationary case, the behaviour thus defined can be described for stable LTP plants, by transmittance based on the frequency limited complex non integer integration (Oustaloup, et al., 1995c), namely:

We take PGw, t) to denote the TVTF of the open loop. So for the nominal parametric state of the plant: PGw,t)

301

=:

PGw).

(17)

where Sadm(W) is the maximum admissible value of the modulus of S(jw, t).

We also consider that T(jw, t), R(jw, t) and S(jw, t) are respectively TVTFs connecting:

Our approach pays particular attention to the shaping of function TO(jw). Fig. 2 illustrates the shaping constraints on this function.

- the reference input r(t) or the measurement noise Nm(t) to the output y(t) ;

- the output disturbance Dy(t) or the measurement noise Nm(t) to the plant input u(t) ; - the output disturbance Dy(t) to the output y(t). TVTF T(jw, t) can be written under the form : T(jOO,t)=To(jOO) +L Tk(joo)cos(kroot)

+T;(joo)sin(kroot),

(18)

pI

and with a similar definition for R(jw, t) and S(jw, t).

log

00

Robustness of stability. and immunity to the time varying character of the plant

In the time domain, the stability degree can be estimated by the first overshoot of the step response in tracking. In the frequency domain, if the closed loop is near-stationary, this first overshoot can be estimated from the resonance ratio in tracking, Q, deduced from the frequency response of TO(jw). This extension of third generation CRONE control thus ensures the robustness of the stability degree through the minimisation of the resonance ratio variations of the stationary part of T(jw, t), To(jw), at the time of the reparametration of the plant, and through the minimisation of the time varying part of T(jw, t) whenever the plant is reparametrated. This lead to minimisation of the criterion:

Fig. 2. Diagram of cons'traints on transmittance TO(jw) The three hachured zones define the exclusion zones of the nominal and reparametrated function TO(jw). Each have an effect on the time performances of the control in tracking and in regulation: - zones 1 and 2 limit the effects of hauling on the step response in relation to its value in steady state; - zone 3 limits the high frequency oscillatory effects. These constraints can be written as : Sup

J = (Qa-x-Qdf-+{QmiD-Qdf+

0 Sup

tr..{joo).

ITJjoo) < hI ,

Pe P, 0
(19)

(22)

Pep,CdEa+,kEz·

Sup ITJjoo) < h3. PeP,,,,:!_

With a judicious choice of weighting coefficient 0, the minimisation of criterion (19) ensures the minimisation of the resonance ratio variations of TO(jw) and the minimisation of the time varying part of the TVTF T(jw, t).

Optimisation of the open loop behaviour

The optimisation of the open loop behaviour consists in determining the five optimal parameters of the nominal open loop transmittance ~(s) which minimise the criterion (19) and satisfy the constraints (20), (21) and (22). The optimisation algorithm is based on the non linear simplex (Subrahmanyam, 1989).

Performance objectives .' shaping of the control loop TVTFs

Some open loop transmittances ~(s), which produce undesirable closed loop behaviours for one or more plant P of the description family lP, need to be eliminated. We thus define how each function R(jw, t), S(jw, t) and T(jw, t) should be shaped.

3.4. Optimal controller

The optimal controller is computed by pseudo inversion so that the cascade connection of the controller and the nominal plant can be described by transmittance ~(jw). The resulting controller can be described by a TVTF C(jw, t) of the form :

The solicitation level of the plant input is taken into account through limitation of the TVTF R(jw, t), namely:

C(joo,t) = Co(jOO) +L C~(joo)cos(kO>ot) + C~(joo)sin(kroot). (23)

(20)

Sup IR{joo, tl < R.dm PeP, lEa+

pI

The synthesis of the controller thus consists in the approximation of transmittances CO(s), C'k(S) and C"k(S) by transmittances of the form :

where Radm(W) is the maximum admissible value of the modulus of R(jw, t). In regulation, the quality of a control resides in its ability to reject plant output disturbances taken into account by the shaping constraint:

C.(s) =

~ bis i/

±

aisi ,

(24)

i~

Sup IsUoo, tl < Sacm,(oo),

'V

00 E

R+ ,

(21)

where degrees nand d are set by the designer. Two techniques can be used to determine coefficients ai and bi of relation (24). The first is a non iterative synthesis

Pe p. IER+

302

The optimal open loop behaviour which minimises the criterion

method based on the elementary symmetrical functions of Vietes roots (Oustaloup, et al., 1995c) and the second is based on the resolution of a linear programming problem (Oustaloup, et al., 1995c). In general, to allow the implantation of the C(jw, l) .controller, relation (23) must be truncated.

J =(QDWl-1.5f~QmiD-1.5f+

Sup

tr..(jw]

(28)

PeP. • a+.kEz·

is determined under constraints

4. APPLICATION

tro(jwj < 1.26 ,

Sup

4.1. Presentation of the plant

~ p.

am

The extension of third generation CRONE control previously de~ribed is applied to the position control of a testing bench with two DC motors coupled by a rigid connection (fig. 3).

O-Cll rdII

~

Inf

p. 0c:tIK3 rdII

troUwj > 0.9

(29) tro(jwj < 0.79.

Sup ~ p.

200 rdII

This optimisation gives a = 1.4516, b = 1.0694, K = 33.1214, W'b =0.2658 rd/s, Wb = 7.697 rd/s, Wh = 95.381 rd/s, W'h = 96.968 rd/s, and then permits the synthesis of an optimal controller which has the form :

connection

lI2(t)

(30)

Fig. 3. Synoptic diagram of the plant

To improve traCking, a transmittance prefilter F(P) is placed at the input of the closed loop system. Transmittance F(p) is defined by:

Each motor rotates a disk on which equal weight loads are mounted. By varying the number of loads, inertia Jm driven by the two motors varies. For the testing bench to behave as an L TP system, motor 2 is supplied by periodic voltage of the form

i=2

I. aip i F(p)= i~

(31)

,

I. bip i

uit) = - (Ao + A1COS(~t)} 8(t) ,

i=O

(25)

with :

with Ao =0.333 , AJ = 0.1 and Wo = 7 rd/s, where EX t} denotes the angular speed of two driving shafts. The plant to be controlled thus admits a state space representation of the form : 0 1 x(t}= 0 0 [

x{t}

with

=[

o an 8(t}

bO = 1 ; bl =4.3426; b2 = 1.3768; b3 = 2.5806E-2 .

aO = 1 ; al = 4.2626 ; a2 = 1.3098 ;

o

Fig. 4 presents the eigenvalue locus of the open loop

1

HTF for the nominal and the reparametrated plants. It shows that the optimal open loop behaviour previously computed ensures the robust stability of the closed loop. As shown in fig . 5, this open loop behaviour also ensures the minimisation of : - the time varying part of the closed loop TVTF - the variations of the resonance ratio of transmittance TO(jw).

2E-3 _ 333.3 Jm

e( t} e( t)

r

(26) and

y(t} =e{t},

a32=[- 0.66 - 716 .67(Ao+A1COs(~t))JlJm .

(27)

Parametric variations of this plant result from variations of the number of loads thus the inertia. Five parametric states have been considered :

1·./·\0·. ;/j . .. 1j !.

- minimal load : J m = 0.024 kg.m2 - 25% of the maxirnalload : Jm =0.066 kg.m 2 - 50% of the maximal load : Jm = 0.108 kg.m2(nominal plant) - 75% of the maxirnalload : Jm = 0.150 kg.m2 - maximal load : Jrn = 0.192 kg.m2

... ... . . . ... . .

.

i !

i

iI

.i

1

Control the angular position of the driving shaft of motor 1 must now be achieved. 4.2. Synthesis of the control law

=

The unit gain frequency has been fixed at mu 10 rd/s for the nominal behaviour of the plant, and the asymptotic behaviour orders in open loop at low and at high frequencies have been fixed at nb = 2 and nh = 4.

·40 · ,'80

.170

, '60

,'50

·'40

.'30

.'20

·110

·'00

·90

·80

Phase (0)

Fig. 4. Nichols eigenvalue locus of the open loop HTF : - - nominal plant; -reparametrated plants. 303

iii

::.

next objective is to achieve synthesis in which this parameter is considered as uncertain. Moreover, the extension proposed in this paper does not permit the control of ~s~ble plants, or plants with positive real part transmission zeros. Our current research aims to suppress this restriction. We also aim to control discrete and multivariable LTP systems and non LTP time varying systems using the strategy developed in this paper.

[':

:

REFERENCES Chassande J. P. (1981) - Etude analytique complete d'une machine synchrone autopilote a caracreristiques bilineaires et a commutation naturelle de courant These de Doctorat es Sci. Phys., Grenoble Floquet G.(1883) - Sur les equations differentielles lineair'es acoefficients periodiques, Annales Sc. de IE.N.S., tome, XII, No 47. Hall S. R. and Weret.ey N. M (1990), Generalized Nyquist Stability Criterion for linear time periodic systems, Proc. of the 1990 Am. Contr. Conf., pp 1518-1525 Kern G. (1980) - Linear closed loop control in linear periodic systems with application to spinstabilized bodies - 1nl J. of Control, Vo1.31, No. 5,905-916 Mathieu B., Oustaloup A. and Lanusse P. (1996) - Third generation CRONE control: generalized template and curvilinear template, Int. Congo IEEE-SMC CESA'96 IMACS Multiconference - Symposium on control, optimiztion and supervision, Lille, Frace, July 9-12 McKillip R. M. (1985) - Periodic control of the individual-blade-control helicopter rotor - Veruca, Vol 9, No 2, pp 199-225. Oustaloup A., Mathieu B. et Lanusse P (1995a) - un tour d'horizon sur la commande CRONE, RAlRO-APII Oustaloup A., Mathieu B. and Lanusse P. (1995b) - The CRONE control of resonant plants: application to a flexible transmission, Eur. J. of Con., Vol.l, pp.113-121 Oustaloup A., Lanusse P. and Mathieu B. (1995c) Robust Control of SISO Plants : the CRONE Control - ECC'95 - P 1423 - Rome Rabenasolo A. Besoa (1992) - Analyses etcommande des systemes lineaires a coefficients periodiques, These de Doctorat d'Universite, Universite des Sciences et techniques de Lille Aandres Artois Sabatier J. (1998) - Approches non entieres en modelisation et commande - These de Doctorat d'Universire, Univ. Bordeaux I - to appear. Sheu D. L. (1978) - Effects of tower motion on the dynamic response of windmill rotors - Tech. Rep. Vol. VII, Dept of Aeron. and Astronautics, MIT Subrahmanyam M. B. (1989) - An extension of the simplex method to constrained nonlinear optimization - J. of Optim. Theory and Applic. Vol. 62, pp 311-319 Wereley N. M. and Hall S. R. (1990) - Frequency response of linear time periodic systems, Proc. of the 29th con£. on decis. and contr., pp. 3650-3655 Wereley N. M. (1991) - Analysis and Control of Linear Periodically Time Varying Systems - PhD thesis, Dept. of Aeronautics and Astronautics, M.LT. Zadeh A. Lotfi (1950) - Circuit Analysis of linear Varying-Parameter Networks, Journal of Applied Physics, Vol. 21, pp. 1171-1177 Zadeh A. Lotfi (1961) - Time-Varying Networks, Proceedings of the I.R.E .• pp. 1488-1503

10' Frequency (r
Fig. 5. Gain diagrams of ToGw) (-), TIGW) (- -) and T"lGW) (_._) for the nominal and reparametrated plants

1 i

oL-____L -__

o

~-L

__

~

________________

3 Time (5)

Fig. 6. Time responses corresponding to the extreme parametric states of the plant to the step inputs r(t)=U(t-'t), with 't = 0, 't = T +T/4, 't = 2T+5T/8

4.3. Results Controller CGw, t) and prefilter F(p) have been implemented using the delta transform s = (l-z-l)/Te with T e = 1 ms. Fig. 6 shows the responses of the closed loop with two extreme parametric states of the plant to the step function r(t)=U(t-'t) with 't=0, 't=T+T/4, 't=2T+5T/8 (V(t) denotes the Heaviside function) and demonstrates the efficiency of the synthesis method in spite of plant uncertainties and time varying character of the plant. 5. CONCLUSION The third generation CRONE control has been extended to the control of LTP systems. This extension uses two generalisations of the transfer function notion and a generalisation of the Nyquist criterion. The synthesis method proposed in this paper provides good position control of a testing bench constituted of two DC motors coupled by a rigid connection and driving variable inertia disks. In this extension of CRONE control, the period T which characterises the non stationary is supposed known and unvarianl However, T can take any value: small or large in relation to the dynamic of the loop. One of our

304