Digital multirate control for a class of non-linear

INT. J. CONTROL,

1999, VOL. 72, NO. 10, 851± 865

Digital multirate control for a class of non-linear singularly perturbed systems M. DJEMAI² , J. P. BARBOT³ and H. K. KHALIL§ This paper deals with the digital control of a particular class of non-linear singularly perturbed (NLSP) systems. Three distinct multi-rate composite control strategies are discussed. In the ® rst case, it is assumed that measurements of the slow and fast variables are scheduled at the same fast sampling period nd . Then, multi-rate measurements are considered and two di€ erent strategies are presented. The paper ends with an academic example and simulations.

1.

Introduction The paper deals with the digital control of a particular class of non-linear singularly perturbed (NLSP) systems. This particular class of systems is not very restrictive, and many systems can be represented by this model (see for example Spong et al. 1987, Hernandez and Barbot 1996). Moreover, it has a very interesting structural property, i.e. the preservation of linearity in u after reduction and easy computation of the ® rst term of the invariant manifold. In this paper, we use the same composite control strategy as used in Litkouhi and Khalil (1985) for the linear case. We highlight some di culties which arise in the non-linear case, as for example the predictor of the slow variable. Moreover, Theorem 1 may be interpreted as a non-linear version under sampling of Tikhonov’s theorem for the particular class considered in this paper. The proof of this theorem is realized thanks to the work of Litkouhi and Khalil (1985) in the linear case and the previous work of Monaco and Normand-Cyrot (1990) on non-linear systems under sampling. The paper is organized as follows. In § 2, we present the class of NLSP systems considered, followed by the fast discretization scheme of such a class in § 3. The two time-scale nature of the system is exploited to discretize the real system and to decompose it into two lowerorder subsystems. In § 4, we develop the composite control strategy. Three distinct multirate composite control strategies are discussed. In the ® rst case, we assume that measurements of the slow and fast variables are scheduled at the same fast sampling period nd . Then, multirate measurements are considered and two di€ erent strategies are presented. Received 8 April 1997. Revised 16 November 1998. ² Author for correspondence. ENSIB, 2 Bv. Lahitolle, BP229, Bourges Cedex, France. e-mail: [email protected] ³ Laboratoire des Signaux et SysteÁmes CNRS± ESE, 91192 Gif-sur-Yvette, France. e-mail: [email protected]. Also at: Equipe Commande des SysteÁmes (ECS), ENSEA, 6 Avenue du Ponceau, 95014 Cergy Cedex, France. § Department of Electrical Engineering, Michigan State University, East Lansing, MI 48824-1226, USA. e-mail: khalil @ee.msu.edu

2.

Some recalls on NLSP systems In the following, some recalls on the integral manifold approach for continuous time NLSP systems are given (see, for example, Fenichel (1971), Saksena et al. (1984), Naidu and Rao (1985), Kokotovic et al. (1986), Spong et al. (1987), Naidu (1988), Lennartson (1989), Khorasani (1990), Barbot et al. (1992), Khalil (1996) for further details). Consider the singularly perturbed nonlinear system e :

xÇ = f ( x, z, u) = a1 ( x) + a2 ( x) z + b1( x) u

e zÇ = g(x, z, u) = a3 (x) + a4 (x)z + b2(x)u

(1) (2)

where the state vectors x and z belong respectively to Mns and Mnf which are su ciently smooth real manifolds of dimensions ns and nf , respectively, the control vector u belongs to < p, the output vector y belongs to q < , and e > 0 is the small perturbation parameter. The functions f , g, h and u are assumed to be su ciently smooth in their arguments, and bounded. Denoting by D the region of interest, the following assumption is made throughout the paper: Assumption 1: The matrix a4 ( x) is non-singular for all x 2 < ns \ D, and D is bounded. D is supposed to be bounded because the considered system is non-linear. This is not necessary when the system is linear or globally Lipschitz. If the fast dynamics are stable, then, after a small time period, the fast variables z converge to a manifold, called the slow manifold. This manifold is characterized by the integral manifold condition (Fenichel 1971). Otherwise, one looks for a control law, denoted by uf , which brings z to the slow manifold. For this reason, the control vector u in equations (1)± (2) can be decomposed as

u = us + uf

(3)

where u, called the composite control (Saksena et al. 1984, Litkouhi and Khalil 1985, Kando and Iwazumi

International Journal of Control ISSN 0020± 7179 print/ISSN 1366± 5820 online Ñ http://www.tandf.co.uk/JNLS/con.htm http://www.taylorandfrancis.com/JNLS/con.htm

1999 Taylor & Francis Ltd

852

M. Djemai et al.

1986, Kokotovic et al. 1986, Barbot et al. 1992), is composed of two terms: the fast control uf , which equals zero on the slow manifold, and the slow control us, which is computed with respect to the slow dynamics. Let M be the ns -dimensional slow manifold M = f ( x, z) : z = © ( x, us, e uÇ s , e uÈ s, . . . ) g characterized from (2) by the invariance condition Ç g( x, © , us ) = e ©, i.e.

¶ © ¶ ©Ç g( x, © , us ) = e f ( x, © , us) + e u + O( e ¶ x ¶ us s

2

) ( 4)

Roughly speaking, condition (4) ensures that if the system is initialized at any point on the manifold M then, under the control (3), the state evolution remains on M. Solving (4) exactly with respect to © ( x, us, e ) is, in general, a di cult task. A classical method is to consider the Taylor series expansion of © with respect to e , i.e. © ( x, us, e

) = © 0(x, us ) + e

© 1( x, us, uÇ s ) + O( e

2

)

( 5)

Moreover, if e is close to zero, with e 2 [0, e 0 ], one can approximate (4) by the quasi-steady state solution © 0 ( x, us) which veri® es (4) for e = 0, i.e. g( x, © 0 , us) = 0

( 6)

which gives for the speci® ed dynamics (1)± (2) © 0 = - a-4 1( x) f a3( x) + b2( x) usg

( 7)

To describe the behaviour of z in a fast time scale, it is usual to set t - t0 ¿= ¿ = 0 at t = t0 ; ´ = z - © 0 e so that dx = e f (x(¿), © 0(t0) + ´(¿), us(¿) + uf (¿)) + O(e d¿

2

) ( 8)

d´ ( 9) = g(x0, ´(¿) + © 0(t0), us (0) + uf (¿)) + O(e ) d¿ The solution ´( ¿) of (8) and (9) to this initial value problem is used as a boundary layer correction for a possibly uniform approximation of z with respect to e z = ´ + © 0 + O( e

)

(10) Thus, © 0 represents the slow transient of z whereas ´( ¿)

is the fast transient of z. To ensure the convergence of z( t) in (10) to the slow approximation ( z = © 0) , equation (7), the corrective term ´( ¿) must decay as ¿ ! 1 to some O( e ) quantity; this is ensured by the following assumption (Tikhonov 1952, Kokotovic et al. 1986). Assumption 2: The equilibrium ´ = 0 of (9) is exponentially stable, uniformly in x0 and t0 , and ´(t0 ) = z0 - © 0 (t0 ) belongs to its domain of attraction.

Substituting © 0 for z in (1), one obtains the behaviour of x( t) on the slow manifold xÇ s = f ( x, © 0 ( xs, us) , us)

( 11)

which describes the slow dynamics of the system. According to (11), and taking into account the linearity in u, the reduced slow system 0, approximated at the order 0 in e , is described by xÇ = f 0( x) + g0( x) us

( 12)

where xs is replaced by x, with f 0( x)

= a1(x) - a2(x)a-4 1(x)a3(x)

g0 ( xc)

= b1(x) - a2(x)a-4 1(x)b2(x)

( 13)

2.1. Examples 2.1.1. Flexible joints manipulator. In Hernandez and Barbot (1996), the authors use the model of ¯ exible joints manipulator in singular perturbation form. Denoting by qa 2 < n and qm 2 < n the vectors of link positions and actuator positions respectively, the Lagrangian model of a manipulator consisting of n + 1 links interconnected by n ¯ exible revolute joints with an actuator on each joint can be written as (Spong 1987): q K m - qa = A( qa) qÈ a + Fa qÇ a + H( qa , qÇ a ) qÇ a N

+ G(qa ) K (Nqa - qm) + u = MqÈ m + Fm qÇ m N2

( 14) ( 15)

This model satis® es 2n second-order di€ erential equations and is 4n-dimensional in state space representation, with N, the gear ratio, A( qa ) and M 2 < n n , the links and actuators inertia positive-de® nite matrices respectively, Fa and Fm 2 < n n , the joints and actuators viscous friction constant matrices respectively, H( qa, qÇ a ) 2 < n n , the coriolis/centrifugal matrix, G( qa) 2 < n, the gravity vector, u 2 < n , the torque/force delivered by the actuators, K( Nqa - qm ) 2 < n , the vector of elastic forces at the joints, where K 2 < n n is the joint sti€ ness (or elasticity) diagonal constant matrix. Remark 1: In the case of rigid joints, then, 8 i, K(i,i) ! 1 and the model reduces to a rigid one satisfying n equations, in which Nqa = qm .

To obtain the singular perturbed model, assuming now, without any loss of generality, that all the sti€ ness constants are of the same large order of magnitude k, one can write K as kIn where In 2 < n n is the identical matrix and where the positive parameter e = 1/ Ö k is such that e 1. The link positions and velocities are

853

Digital control for NL SP systems

a2 ( x) =

where

a4 ( x) =

b= s( z) =

0

-1 - A (x1) 0

0 NM- 1 Fm x2 + Na ( x1 , x2 ) 0

In

- [A- 1( x1) + (N2 M)- 1] 0

x1 sin ( px2 )

p© f J

- cos (px2) - sin (px2) cos (px2) 0

0

3.

0

- M- 1 0

- M- 1Fm z2

with

a (x1, x2 ) = - A- 1(x1 )[Fa x2 + H(x1, x2)x2 + G(x1 )]

Fast sampled schemes for NLSP system In this section, the fast sampling of the non-linear dynamics e is tackled (see Barbot et al. (1996) for details). Assuming the sampling period d is su ciently close to e , one writes d as a e , where a is a real positive number, keeping a e small enough. The fast rate sampled model may be used to assign the fast closed-loop behaviour and to stabilize the fast state component z (or z - © 0 ). For computing convenience, one rewrites system , given in (1)± (2), in the following form:

YÇ 2.1.2. Synchronous motor. The reader may refer to Leonhard (1985) for a general description of synchronous motors. Based on the well known two-phase equivalent ( a - b ) representation, a suitable choice for the state variables is x = ( ,µ) T for the slow variables (mechanical speed and position), z = ( ia , ib ) T for the fast variables (currents) and u = ( va , vb ) T for the control vector (the converter output voltage), and is given by

with

Cr J

I2 denotes the identity matrix on < . The parameter e is the singular perturbation parameter expressing the speed of the slow versus the fast system dynamics; it represents the inductance, which is supposed to be known and constant. In the model, the parameters are represented by © f : ¯ ux created by the magnet of the rotor; p: number of pairs of poles; Rs: stator resistance; L : inductance; J: inertia; Cr : load torque; f : viscous friction coe cient. Both cases, the synchronous motor and the robot, are represented in the form given in equations (1) and (2).

x2 a (x1, x2 )

N

a3 ( x) =

b = I2

2

0 a2 ( x) =

x1-

a3 ( x) = p© f x1

e zÇ = a3 (x) + a4(x)z + bu + e s(z)

a1 ( x) =

f J

-

a1 ( x) =

xÇ = a1 ( x) + a2( x) z :

a4 = - RsI2 ;

e = L;

the slow variables and the elastic forces at the joints and their time derivatives are the fast variables; the coordinate changes zt = [zt1 , zt2 ] and xt = [xt1, xt2], where z1 = k( Nqa - qm ) , z2 = Ö k( NqÇ a - qÇ m ) x1 = qa , and x2 = qÇ a , allow us to express (14)± (15) as a singularly perturbed model of the form

xÇ = a1 ( x) + a2( x) z

(16)

e zÇ = a3 (x) + a4z + bu

(17)

1

( 18)

= S+e F

with Y

x

= z

;

S

f

= 0

;

0 g

F=

Let d be a ® xed sampling period. For a piecewise constant control over time intervals of amplitude d u( t)

:

= u(n), nd

t < ( n + 1)d ,

n

0

( 19)

If d = a e , then at any sampling instant t = ( n + 1) d the solution to (18) is given by (Barbot et al. 1996): d L Y ( n + 1) + e (S+Fe ) Idj Y (n)

( 20)

where Id represents the identity function (for explicit computations see Monaco and Normand-Cyrot 1990, Barbot et al. 1996).

854 e- a

M. Djemai et al.

Setting d = a e in (20) and reminding oneself that e = Id, equation (2) can be rewritten as: ²

LF a LF

Y ( n + 1)

= f ea

L e S+F - a L F

e

a LF

g e

(21)

Idj Y (n)

where Id represents the identity function. The following Proposition 1 gives the ® rst-order approximation (for x and z) in e of (20), which may be su cient for control purposes.

(ii) I designates indi€ erently the identity matrix or operator and Idns and Idnf designate respectively the identity function for the slow and fast components.

(23)

The above expressions con® rm the well known fact that the evolution of the slow state component x (given in (23)) is close to O( e ) while the evolution of the fast state component z (given in (23)) is close to O( 1) in this sampling time scale. This may be compared with the continuous time case in the fast time scale ¿ expressed in (8)± (9). The main advantage of the previous fast discrete-time solutions is their homogeneity in e . Thus, the in¯ uence of terms predominating in e is underlined and simpli® cation with respect to e is much easier.

where Idns and Idnf denote the identity projections Idns : Rn ! Rns and Idnf : Rn ! Rnf , respectively.

Remark 3: In the linear context (Litkouhi and Khalil 1985, Kando and Iwazumi 1986) a is considered equal to 1 for reason of simplicity.

Proposition 1 (Barbot et al. 1996) : The ® rst-order approximation in e of the fast discrete-time solution (20) is given by

x( n + 1)

~ = x(n) + e E1(a F, a S )Idnsj Y (n)

z( n + 1)

= f ea

LF

~ Idnf + e E1 ( a F, a S ) ea

(22) LF

Idnf g j

Y (n)

In this case, one has E1 ( a F, a S ) Idns j Y (n) ~

ea

LF

Idnf j Y (n)

=

a i 1

i

i!

L ig- 1 ( f ) j Y (n)

= ea L g (z)j Y (n)

Applying this formula to (1) and (2), one gets x( n + 1)

= x(n) + e f a (a1 + a2z(n) + b1u(n)) +g

z( n + 1)

1

(a3 + a4z(n) + b2u(n))g

g

and

Y (n)

+ O( e

2

) (24)

= ea a4 z(n) + g 2(a3 + b2u(n)) + e f E~ 1( a F, a S ) ea

with

j

1

= a2(ea g

2

LF

Idnf g

Y ( n) j

+ O( e

2

) (25)

- Id - a a4)[a4]-j x2(n)

a4

= (ea

a4

- Id)[a4]-j x1( n)

Remark 2: (i) In the text ai denotes ai ( x( n)) and bi denotes bi ( x( n)) . ² One explains the notaion used in (20): d (Lf )

e

( Id) j x(k) = x(k) + d d

L f (Id)j x(k) +

d

2

2!

Lf

L f ( Id )j x(k)

+

S

=

a1 ( x) + a2 ( x) z 0

;

F=

0 a3 ( x) + a4 z + bu

Equation (20) is the key point of the discrete-time solution, because the Campbell± Baker± Hausdor€ exponent naturally appears in the exponential between the parentheses. Specifying (24) and (25) on the basis of the synchronous motor dynamics, (16)± (17), one gets x( n + 1)

= x(n) + e f a [a1(x(n)) + a2(x(n))z(n)] +g

z( n + 1)

1

(a3(x(n)) + a4z(n) + bu( n))g

( 26)

= e- a Rs z(n) + g 2[a3(x(n)) + bu(n)]

with

p

L op (I ) + p! f d x(k) where Id is the identity function, ` ’ represents the composition of the functions and `j ’ represents the evaluation of the function f at this point. The operator L f when applied to the function h gives n ¶ h( x) ¶ h(x) Lf h = f (x) = f ( x) ¶ x ¶ xi i i=1

+

3.1. Example: Synchronous motor In this subsection, the fast sampling of the non-linear dynamics is tackled (see Barbot et al. (1992), Djemai and Barbot (1995) for details). Assuming the sampling period d su ciently close to e , it is reasonable to write d as a e where a is positive real, such that a e is small enough. The fast rate sampled model may be used to assign the fast closed-loop behaviour and to stabilize the fast state component ´. In the sequel, one rewrites system (16)± (17) in the form (18) with

g

1

= a2(x(n))( e- a

Rs

- 1 + a Rs)a-4 2

and

g

2

= (e- a

Rs

- 1)a-4 1

The dynamic of ´ is equal to

´(n + 1) = e- a

Rs

´(n) + g 2 a-4 1 b(- us(n) + u(n))

( 27)

855

Digital control for NL SP systems One notes that for the fast variables z, only the predominant term f ea L F Idnf g j Y ( n) is considered, while the ~ term f e E1 ( a F, a S ) ea L F Idnf g j Y (n) is neglected, for computational reasons. We see in (27) that the stability of the fast variable ´ is only a function of the stator resistance Rs if uf = 0.

Design of digital composite control In the feedback design of systems having slow and fast variables it is intuitively clear that lower measurement rates can be tolerated for the slow variables compared to those of the fast variables. This leads naturally to two time-scale measurements and the design of multirate sampling schemes design, which have been used in many applications. Three distinct feedback control designs, which represent gradual deviations from the single rate control, will be given:

© d0( n)

= © 0(n) ( 29) We de® ne the fast variable as ´( n) = z( n) - © d0 ( n) ; the

In the following we develop each of these three approaches. 4.1. Single rate design with single rate implementation In this section we discuss the problem of composite control in order to stabilize the real system for the case that fast and slow subsystems evolve in the fast timescale n and the measurements of all variables are available at all values of n. We start from the fast sampling form (24)± (25) under the composite control u( n) = us( n) + uf ( n)

(28) where us( n) = g s ( x( n)) , with us( n + 1) - us( n) = O( e ) and uf ( n) = 0 for z( n) = © d0 ( n) , where © d0 is the slow manifold for the sampled dynamic. In our case, and for the class of systems considered, one has

g j x(n)

dynamic of the fast decoupled variable is equal to

´(n + 1) = f ea

a4

´(n) + g 2 b2 uf (n) + O(e )g j x(n)

( 30)

and the slow reduced dynamics are given by x( n + 1) = x( n) + e f a ( a1 + b1 us( n))

4.

(i) Firstly, assuming that x( n) and z( n) are measured for every n, we develop a single rate composite control on the basis of the fast sampled model. Hereafter, we note this controls scheme: single rate design with single rate implementation. (ii) In the second approach, we assume that only the fast variable z( n) is measured for every n, whereas the slow variable x( n) is measured at a lower rate and we develop a single rate control with predictor for this case. Hereafter we note this control scheme: multirate implementation. (iii) The third approach consists in calculating the slow control on the basis of the slow sampling model of slow dynamic. We note this control scheme as: multirate design with multirate implementation.

= - f a-4 1f a3 + b2us(n)g

+ a2 f - [a4 ]- 1 (a3 + b2us (n)) g

g j x(n)

( 31)

with

g

1

= a2(ea

a4

- Id - a a4)[a4]-j x2(n)

and

g

2

= (ea

a4

- Id)[a4]-j x1(n)

Let us make the following assumptions: Assumption 3: The matrix ( ea a4 - Id ) is regular for any x 2 D \ < n . n Remark 4: The ( ea a4- Id) j x(n) regularity implies the regu-

larity of a4 ( x( n)) (Assumption 1) but the inverse proposal is false. The fast system

Assumption 4:

´(n + 1) = f ea

a4

´(n) + g 2b2 uf (n)g j x(n)

is uniformly experientially stabilizable by a smooth fast control uf (n) . Remark 5: As equation (30) is linear (because x is considered constant in the fast time-scale, and ea a4 , g 2 and b2 , which are functions of x only, are supposed to be constant), the previous Assumption 4 ensures that ´ is bounded during the sampling period for e su ciently small ( O( e ) is very small) .

From this assumption, we can choose one fast control of the form uf ( n) = b ( x( n) , us ( n)) ´( n)

( 32)

where b is computed in order to obtain k

´(n)k

[k0]n Mk ´(0)k

+ k1e

()

8 x 0 , us

( 33)

with M > 0 and 1 > k0 0, where M, k0 and k1 are real constants. From (24)± (25) and (32) we rewrite the discrete-time dynamics in the fast time-scale as x( n + 1)

= f x(n) + e

´(n + 1) = f ea

a4

f C

+ K ´(n)g + O(e

)g j

x( n)

( 34)

´(n) + g 2b2 uf (n) + O(e )g j x(n)

= X ´( n) + O(e )g where

2

j

x(n)

( 35)

856

M. Djemai et al. j

x(n)

= f a (a1 + a2©d0 + b1g s)g

j

x(n)

= f g 1(a4 + b2b ) + a b1b

j

x(n)

= f ea

(36)

Now, in order to characterize the evolution of the slow component in an appropriate sampling period, we set the slow sampling period ¢ equal to [1/ e ]d . From the two previous lemmas, we get

In all the following, we assume that the control law uf is chosen in order to obtain (33).

Theorem 1: Under Assumptions 3± 5 and the two timescale composite control (28), where uf ( n) is given in (32), the dynamics (1)± (2) have the following behaviour at each slow sampling time k

C K X

a4

+ g 2 b2b

j

x(n)

g j x( n)

g j x(n)

Assumption 5: All the derivatives of C and K are bounded in the study domain (i.e. a neighbourhood of z = © d0 M0, where M0 is an open set).

The condition of C implies that the reduced continuous time system is smooth, whereas the condition on K means that the fast dynamics act smoothly on the slow discrete-time dynamics. Before we give the main result of this section, we need to recall some mathematical tools introduced by Monaco and Normand-Cyrot (1990). Lemma 1: For any analytic function h( x) 2 Rns and with respect to the slow discrete-time dynamics (24) we have

h( x( n + 1)) = hj x(n) + e L C

+ O( e

+K ´(n) hj x( n)

2

)

(37)

where ´ is considered as an input for these dynamics. Proof:

= (Idns + e L C

+K ´(0)

2

x( n) =

+K ´( j)

. . . (Idns + e L C

+K ´(n- 1)

+ O( e

+

+ LC

. . . + LC

+K ´(0)

+ O( e

+K ´(n- 1) g j x(0)

+ O( e

2

1

7 ( Id + e la C

+K ´(0)

+ O( e

i=0

)

(38)

2

j

x(0)

L K ´( i) ( I + e L C

+ e L K ´( 0) (I + e L C

+K ´

+ O( e

+K ´

)n- i- 1Idn j x(0) s

)n- 1Idn j x( 0) s

+ (I + e L C ) e L K ´(1) (I + e L C +

+ (I + e L C

)ie

+

+ (I + e L C

)n- 1e

+K ´

)n- 2Idn j x(0) s

L K ´(i) ( I + e L C

+K ´

)n- i- 1Idn j x(0) s

L K ´(n- 1) Idns j x( 0)

where LC

+K ´

)n- i = (I + e

LC

+K ´(i)

) . . . (I + e

+K ´(n- 1)

LC

+K ´(i+1)

)

)

LC

)nIdn j x(0) k s

e > 0 and 8 a 0 > a > 0, Bd is a matrix with all eigenvalues j ¸i j < 1. Consequently, for e a 2 [0, e 0 a 0 [, the sampled solution of (41) with a slow continuous time control and Z.O.H. is bounded

()

k xn k

k Bndk k x0 k

which implies that the slow dynamics are stable (limn! 1 x(n) = 0) because all the eigenvalues of Bd are inside the unit circle. Lemma 3: Under Assumption 6, there exist e 0 and a 0 such that, 8 e 2 ]0, e 0 ] and 8 a 2 ]0, a 0 ], the dynamic (41) controlled by the sampled continuous-time control us( n) = g c ( x( t = nd )) = g s ( n) , with sampling period d = a e , is uniformly exponentially stable.

(B) Digital slow control This paragraph deals with the same problem as the previous one, but the slow control does not employ Z.O.H. but us( n) is directly calculated on the basis of the slow reduced discrete-time expressed in the fast-time scale. Let us consider the following assumption: Assumption 7: There exists a digital smooth slow control us( n) = g d ( x( n)) which uniformly exponentially stabilizes (34) in the discrete-time sense. The design method is not crucial.

For both types of single rate control strategy we have: Theorem 2: (i) Under Assumptions 3± 6, there exist a 0 > 0, e 0 > 0, such that for any 0 < e a < e 0a 0 the composite control (28), with uf de® ned in (32) and holding constant the slow control us( t) = g c ( x( t)) during the fast sampling period d such that us(n) = g c (x(t - nd )) for t 2 [nd , (n + 1)d [, ensures the exponential stability of the dynamic (1)± (2) under sampling. (ii) Under Assumptions 3± 5 and 7, there exist a 0 > 0, e 0 > 0, such that for any 0 < e a < e 0 a 0 the composite control (28), with uf de® ned in (32) and the digital slow control us( n) = g d ( x( n)) , ensures the exponential stability of the dynamic (1)± (2) under sampling.

Proof: (i) For the slow digital control, the proof results directly from Theorem 1 and the fact that the O( e ) term in (39) is equal to zero when x and ´ are equal to zero. This is because this term is a composition of L ie brackets constituted with bounded functions f and g which are equal to zero for x = 0 and z = 0. (ii) For the slow continuous-time control which is maintained constant during the fast sampling period, the proof results from Theorem 1, L emma 3 and the same argument on O( e ) . h

4.2. Multirate implementation In this section, we assume that only the fast variable z is available for every fast sampling instant, while the slow variable x is available only for every slow sampling period k. Starting from the result of the previous section, where x and z are available for every n, the slow measurement of x gives a crucial problem and the composite control, presented for the previous case, cannot be implemented directly because of the lack of the measurement of x at times between n = k[1/ e ] and n = ( k + 1)[1/ e ]. The di culty generated by this problem may be overcome by using an estimator or predictor to estimate the unmeasured values of x during the slow intersampling. Let x^ be the estimate of x, and assume that z( n) has reached its steady state © d0; we have from (39) x^ ( n + 1)

= ee L Idnsj x(n) C

^

with x^ ( n = k[1/ e 8 n2

]) = x(n = k[1/ e ]) [k[1/ e ], (k + 1)[1/ e ]- 1]

Consequently, a slow state predictor is given by x^ ( n) = ee (n- k[1/e ]) L C Idnsj x^ (k[1/e ]) 8 n2

( 42)

( 43)

[k[1/ e ], (k + 1)[1/ e ]]

The validity of (43) is deduced immediately from Theorem 1 for n ! [1/ e ] and from (39). Contrary to the case for the linear systems, the predictor (43) is completely computed at each fast sampling time (i.e. we do not deduce x^ ( k[1/ e ]+ l) from x^ ( k[1/ e ]+ l - 1) , for l 2 [0, [1/ e ]], without a total formal derivation of the previous result). To illustrate the di€ erence between the two case (linear and non-linear), let us consider the linear case, and more particularly a slow linear subsystem xÇ = C

= Ax + Bu + O(e )

859

Digital control for NL SP systems we see immediately that we have LC C

= C (C ) = A(Ax + Bu)

(44)

and, more generally L jC C

= C (C

j

) = Aj (Ax + Bu)

(45)

j

where C denotes the j composition of C . Thus, in the linear case we can substitute the exponential Lie derivative by the exponential composition function, which gives an exponential matrix equation x^ ( k[1/ e ]+ l) = ee ( l) L C Idns j x^ (k[1/ e ])

=

1 i=0

(e l)i Ai- 1(Ax + Bu) i!

(46)

Therefore, from (37) we deduce that the classical recursive predictor for slow reduced dynamics x^ ( k[1/ e ]+ l)

= Id + L e C x^ (l - 1 + k[1/ e ])j x^ (k[1/ e ])

[0; 1/ e ] an x^ ( k[1/ e ])

8 L 2

where x^ ( k[1/ e ]+ l - 1) is considered as function which may not be easily computed in general but only for the linear case. From equation (44) we ® nd x^ ( k[1/ e ]+ n) = Id + e Ax^ ( l - 1 + k[1/ e

]) + e

Bu( k)

which is easily computed as in Litkouhi and Khalil (1985). The feedback composite control is given by us ( n)

= g s( . )j x(n)

uf ( n)

= b ( . , us (n)) j x( n)

^

^

f or k[1/ e

]

n < ( k + 1)[1/ e ], k

0

where x( n) is given by (43); thus the prediction error is bounded by ^

( ) - x^ (n)k

k xn

When n !

n- ( k[1/ e

])

i=k[1/ e ]+1

Oi ( e

(k + 1)[1/ e ]- , one obtains lim k x( n) - x^ ( n) k = O( e ) n! 1

2

Sketch of proof: As the error due to estimation of x is O( e ) and introduces an O( e 2 ) error in the control, the proof is similar to that of Theorem 1. h

4.3. Multi-time design of multirate implementation The composite control designed in the previous sections exploits the two-time-scale nature of the system to decompose the overall design problem into slow and fast designs. However, both the slow and fast design problems are solved in fast time-scale. This implies that slow and fast variables are available at each nd . Because the behaviour of the slow variables increases from n to (n + 1) with an error of O(e ), it may be unnecessary to design the slow subsystem in a fast time-scale. An alternative method is to design the slow control in the slow time-scale k. In this section we apply a multirate stabilizing composite control (Litkouhi and Khalil 1985) to the system (24)± (25) by supposing the measurement of the slow variables available at slow time-scale k[1/ e ], for k = 0, 1, 2, . . . . Consider the system (24)± (25). We assume that the composite control is given by u( n) = us( n) + uf ( n) ,

us( n) = us( k) ; k[1/ e

(47)

= u(n)j x(n) + O(e )

Corollary 1: (i) Under Assumptions 3± 6, there exist a 0 > 0, e 0 > 0, such that for any 0 < e a < e 0 a 0 , the composite control (28), with uf de® ned in (32) and holding constant the slow control us( t) = g c( x^ ( t)) during the fast sampling period d such that u( n) = g c ( x^ ( t = nd )) for t 2 [nd , ( n + 1)d [,

( 48)

n = 0, 1, 2, . . .

where the slow control us ( n) is kept constant over the cycle k[1/ e ] n < ( k + 1)[1/ e ], i.e.

)

Now, as us and uf are smooth, we have u( n) j x^ (n)

ensures exponential stability of the dynamics (1)± (2) under sampling. (ii) Under Assumptions 3± 5 and 7, there exist a 0 > 0, e 0 > 0, such that for any 0 < e a < e 0 a 0 , the composite control (28), with uf de® ned in (32) and the digital slow control us( n) = g d ( x^ ( n)) , ensures exponential stability of the dynamics (1)± (2) under sampling.

]

n < ( k + 1)[1/ e

]

( 49)

4.3.1. Slow sybsystem. Assuming that the fast dynamics z( n) have reached their steady-state solution © d0 ( n) , this implies that uf ( n) = 0. Then, starting from the reduced slow dynamics expressed in the fast timescale x( n + 1) = x( n) + e C

x(n) j

+ O( e

2

)

the corresponding discrete time slow dynamics expressed in terms of x( k[1/ e ]) and us ( k[1/ e ]) are de® ned as x(( k + 1)[1/ e Now let

]) = eL C

Idns j x( k[1/e ])

( 50)

860

M. Djemai et al. u( k)

= us (k[1/ e ])

x( k)

= x( k[1/ e ])

is exponentially stable in the discrete-time sense, with initial conditions x( 0)

Now we make the following assumption:

^

´^( 0) = z(0) - © d0 (0)

Assumption 8: Suppose that there exists a state feedback control law us ( k) which exponentially stabilizes (50).

4.3.2. Fast subsystem. We de® ne the fast subsystem with respect to the estimated fast variable, de® ned as ^

´^ = z - © d0

´^ (k[1/ e

^d

© 0 ( k[1/ e

^

= f - a-4 1f a3 + b2us(n)g

])

g j x^ (n)

(52)

because the slow manifold depends on x( n) . Since ^ © d0 ( 0) = © d0 ( 0) , we have ^

´(0) = z(0) - © d0 (0) = ´^ (0)

(53)

and, from equations (47), (51) and (52), we obtain immediately

´(n) = ´^ (n) + O(e

)

and ^

© d0 ( n) = © d0 ( n) + O( e

)

We recall that the measurements of x are available at slow cycles [1/ e ], the error between x^ ( n) and x( n) stays in O ( e ) and the measurements are `reinitialized’ at each slow sampling cycle, which makes the error equal to zero at these times. Remark 4: The initial conditions for the fast subsystem depend on the slow control. This means that any change in the slow control will excite the fast modes and cause a fast transient for a short period during the beginning of the slow sampling period. This remark is very important in our case, since for every [1/ e ] time-intervals there is an abrupt change in the slow control, and also in the quasi-steady solution, so that the fast subsystem has to be solved at the beginning of each cycle of the slow control. Theorem 3: Under Assumptions 3± 5 and 8 with the composite control (48)± (49), for su ciently small e , the closed-loop system

x( k + 1)

= el Idnsj x(k) C

´(n + 1) = f e ^

´(n) + g 2 b2 uf ( n) + O(e )g j x^ (n)

a a4 ^

( 54)

^

)

( 55)

Remark 5: We see that the error ´^ ( n) - ´( n) + O( e generates in (38) an additional error in O( e 2 ) .

and © d0( n)

(k[1/ e ])

j x^

]) = x(k) + O( e )

z( n) = ´^ ( n) + © d0( n) + O( e

(51)

with

]) = z(k[1/ e ]) -

a-4 1f a3 + b2[u( k - 1) - u( k)]g g

x( n = k[1/ e

´^(n + 1) = f ea a4 ´^( n) + g 2 b2 uf (n) + O(e )g j x^( n) ´(k[1/ e

]) = - f

for k > 0. Moreover, the state trajectories can be approximated as:

and we get

^

= x(0)

)

Proof: Under Assumptions 3, 5 and 7 we have, from Theorem 1

x( k + 1) = ee

LC

Idns j x(k) + O( e

)

( 56)

From Assumption 8, the dominant part of the slow dynamics is exponentially stable. Moreover, the term O( e ) is equal to zero at x = 0. This is due to the fact that the O( e ) term is composed of Lie brackets constituted only of functions f and g which are equal to zero for x = 0 and z = 0. This implies that there exist e 0 , such that 8 e < e 0, equation (56) is exponentially stable. Consequently, the slow state is exponentially stable for n such that k[1/ e ] n < ( k + 1)[1/ e ] Moreover, from the de® nition of ´^ , (55) is proved. h 5.

Illustrative academic example Now, for the sake of simplicity, let us use the following second-order example to illustrate the results

xÇ = x2 + z

e zÇ = x + 2z + u The objective of control is x ! xd = 1 - e- at . This system is given in the form (1)± (2) with a1 ( x) = x2 ;

a2 = 1;

b1 = 0

and a3( x) When e

= x; a4 = 2; b2 = 1

= 0 is veri® ed, one obtains z0 = 12 ( - x - u)

( 57)

The slow dynamics are xÇ = x2 + 12 ( - x - us )

( 58)

and the dynamics of the fast variable ´ = z - z are 0

861

Digital control for NL SP systems

Figure 1. Simulation obtained with us and uf in continuous case.

(59)

e ´Ç = 2´ + uf

The fast sampling model, obtained by applying (24) and (25) to our system, is given by x( n + 1)

= x(n) + e f a (x2 + z) + 14 (e2a - 1 + 2a )(x(n) + 2z(n) + u(n)) g

z( n + 1)

(60)

= e2a z(n) + 12 (e2a - 1)(x(n) + u(n))

(61)

and the discrete time dynamics of the fast variable (the equivalent of (59)) are given by

´(n + 1) = e2a ´(n) + 12 (e2a - 1)uf ( n)

(62)

5.1. Continuous control For the purpose of comparison, we start with the well known continuous-time composite control. The slow control is given by us = 2[v( t) -

1 + 2 x 2x

]

where v( t) = xÇ d - Ks( x - xd) yields the error dynamics

(63)

d (x - xd) = - Ks(x - xd) dt Ks is chosen positive so that the error dynamics are stable. The fast control, computed from the dynamics of ´ given in (59), is given by uf

= - (Kf + 2)´

( 64)

We use the composite control uc = uf + us and the simulation results are given in ® gure 1. 5.2. Digital control (single rate design and implementation) The objective of control on ´ is usually ´ ! can suppose that

0, so we

´(n + 1) = - Kfd ´(n) where Kfd is chosen inside the unit circle. Then, from equation (62), one obtains uf ( n)

2

= (e2a - 1) f - Kfd - e2a g ´(n)

( 65)

862

M. Djemai et al.

Figure 2. Continuous us control with ZOH (with fast sampling period) and digital uf .

5.2.1. Slow continuous time control with Z.O.H. As the evolution of x is O( e ) , we suppose it to be continuous (see equation (40)). The slow control is calculated on the basis of slow continuous dynamics (58) to obtain the continuous time control (63). The composite control uc is the sum of the fast control (65) and a zeroorder-hold implementation of the slow control us uc( t = nd

) = uf (n) + us(t = nd )

Simulation results are given in ® gure 2. 5.2.2. Digital single rate control. The objective of control on the slow variable x is represented by x( n + 1) = v( n) where v( n) = xd( n + 1) + Ksd ( x( n) - xd( n)) and in our case = xd( n + 1) = xd( n) = 1. The slow control is computed from the reduced fastsampled dynamics of x, obtained by substituting z by its slow manifold z0 x( n + 1)

d

d

= 1 - 2 x(n) + d x(n)2 - 2 us( n) - v(n)

This gives the following slow control us ( n) =

2

d

- v(n) + 1 - x(n) + d x(n)2 d 2

(66)

and the simulation results are given in ® gure 3. 5.3. Multirate implementation As was presented in § 4.2, we suppose that the fast variable z is available at each nd while the slow variable x is available at each k¢. In order to apply the slow control at each nd we need to estimate the variable x between each k¢ and ( k + 1) ¢. Let x^ be the estimate of x, and assume that z( n) has reached its steady state © d0 . We have, from (39) and (43), the following predictor for x x^ ( n) = f x^ ( n) + d [x^ 2 ( n) -

( (n) + u(n))]g j x(k[1/e ])

1 ^ 2 x

^

[k[1/ e ], (k + 1)[1/ e ]] ( 67) with x^ ( n = k[1/ e ]) = x( n = k[1/ e ]) . Using x^ ( n) instead of x( n) in the digital control (66), 8 n2

one obtains the digital composite control uc ( n) = us( x^ ( n)) + uf ( n) and the simulation results in ® gure 4.

5.4. Multirater design of multirate implementation The method presented in § 4.3 will be tested in this case. The fast control is the same one as computed in (65), whereas the slow control is computed from the

Digital control for NL SP systems

Figure 3. Simulation of single rate design of single rate implementation.

Figure 4. Simulation by using estimator of x in between successive slow sampling points.

863

864

M. Djemai et al.

Figure 5. Simulation of multirate design of multirate implementation.

x( 0) = 5 and z( 0) = 6: the initial conditions; h = 0. 0001: a fixed step of integration; d = 0. 005: a fast sampling period; ¢ = 10 d : a slow sampling period; e = 0. 01: a singular perturbation parameter; Kf = Ks = 10: parameters in continuous time;

reduced slow-sampled dynamics of x obtained by substituting z by its slow manifold z0 x( k + 1) = 1 -

¢

2

x( k) + ¢x( k) 2 -

¢

2

us( k)

= v(k)

This gives the slow control us( k)

2

=¢

- v(k) + 1 -

¢

2

x( k) + ¢x(k) 2

(68)

where ¢ is the slow sampling period and v( k) is the desired dynamics, given by v( k)

Kfd

= xd (k + 1) + Ks¢ ( xd (k) - x(k))

and the desired objective is usually xd( k + 1) = xd( k) = 1. It is important to remark in this ® gure that uf is updated at the slow sampling points. Parameters of simulation The parameters of simulations are the following:

6.

= 0. 01, Ksd = 0.5 and Ks¢ = 0. 8: feedback gains.

Conclusion In this paper a multirate composite control has been adopted to drive systems described by a particular class of non-linear singularly perturbed equations under sampling. The strategy used in this paper is a non-linear extension of the composite technique used in the linear case by Litkouhi and Khalil (1985). A theoretical framework has been presented for studying such a class of systems. We highlight the fact that some classical linear techniques are not available in the non-linear context, as is particularly evident in the second case with predictor. The system studied in the paper conforms with the hypothesis d / e ’ 1 (d is the fast sampling period and e is the singular perturbation parameter). We note that the

Digital control for NL SP systems choice between the di€ erent methods presented in the paper is a function of two important parameters: e and the cost which we can use to realize the objective of control in terms of acquisition of measurements and time needed for the calculation. However, using these parameters, we can choose one of the three di€ erent methods to calculate the control needed. In fact, when e is very small, the state vector x is supposed to be continuous (because it is supposed to be constant during the fast sampling period d ); thus, us can be calculated as continuous control. In the same way, us can be calculated at every fast sampling period nd . This allows us to take into account the small variation of the system. Also, if the x varies slowly, it is possible to calculate us every slow sampling period ¢ = Ndd , whereas the fast control is calculated at every fast sampling period and the composite control is implemented over the same period. This technique is useful when x and us vary slowly, because the time of calculation becomes small, as does the number of iterations. It is necessary to use a fast system of acquisition or measurement for the fast variables to apply this technique in good conditions. To conclude the paper, we can say that d / e , e , the cost in terms of time of calculation and acquisition are the important parameters needed to choose the appropriate technique of control for NLSP systems. References Barbot, J. P., Djemai, M., Monaco, S., and Normand Cyrot, D., 1996, Analysis and control of nonlinear singularly perturbed systems under sampling. In C. T. Leondes (ed.) Control and Dynamic Systems: Advances in Theory and Application (San Diego, CA: Academic Press), Vol. 79, pp. 203± 246. Barbot, J. P., Pantalos, N., Monaco, S., and Normand Cyrot, D., 1992, On the control of singularly perturbed nonlinear systems. IFAC± NOL COS Symposium, Bordeaux, France, pp. 453± 458. Djemai, M., and Barbot, J. P., 1995, Singularly perturbed method for the control design of a synchronous motor with its PWM inverter. In IEEE Conference on Control Application, Albany, NY, pp. 789± 804.

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Fenichel , N., 1971, Persistence and smoothness of invariant manifold for ¯ ows. Indiana University Mathematics Journal, 21, 193± 226. Hernandez , J., and Barbot, J. P., 1996, Sliding observerbased feedback control for ¯ exible joints manipulators. Automatica, 32, 1243± 1254. Kando, H., and Iwazumi, T., 1986, Multirate digital control design of an optimal regulator via singular perturbation theory. International Journal of Control, 44, 1555± 1578 Khalil, H. K., 1996, Nonlinear Systems, 2nd edition (New Jersey: Prentice Hall). Khorasani, K., 1990, Nonlinear interacting control with stability: A high-gain control approach. Proceedings of the IEEE CDC, Hawaii, pp. 3385± 3387. Kokotyovicí , P. V., Khalil , H. K., and O’ Reilly, J., 1986, Singular Perturbation Method in Control: Analysis and Design (New York: Academic Press). Lennartson, B., 1989, Multirate sampled-data control of two-time-scale systems. IEEE Transactions on Automatic Control, 34, 642± 644. Leonhard , W., 1985, Control of Electrical Drives (Berlin: Springer-Verlag). Litkouhi, B., and Khalil, H. K., 1985, Multirate and composite control of two-time-scale discrete-time systems. IEEE Transactions on Automatic Control, 30, 645± 651. Monaco, S., and Normand-Cyrot, D., 1990, A combinatorial approach of the nonlinear sampling problem. Lecture Notes in Control and Information Sciences. Proceedings of the 9th International Conference, Antibes, France (Berlin: Springer-Verlag). Naidu, D. S., 1988, Singular perturbation methodology: in control systems. IEE Control Engineering Series 34 (London: Peter Peregrinus). Naidu, D. S., and Rao, A., 1985, Singular Perturbation Analysis of Discrete Control Systems (Berlin: SpringerVerlag). Lecture Notes in Mathematics. Saksena , V., O’ Reilly, J., and Kokotovic, P. V., 1984, Singular perturbations and time-scale methods in control theory: Survey 1976± 1983. Automatica, 20, 273± 293. Spong, M., 1987, Modeling and control of elastic joints robot. Journal of Dynamic Systems Measurement and Control, 109 , 310± 319. Spong, M., Khorasani, K., and Kokotovicí , P. V., 1987, An integral manifold approach to the feedback control of ¯ exible joints robot. IEEE Journal of Robotics and Automation, 3, 273± 290. Tikhonov , A. N., 1952, Systems of di€ erential equations containing small parameters multiplying some of derivatives. Mathematica Sborniki, 31, 575± 586.