ProGIIcllnpo 01 tho 30th Conference on DIchlon and Control erlghton, England a December 1991 W2-3 - 150

Abstract

Discretization Schemes for Nonlinear Singularly Perturbed Systems

J.P. Barbot', S. Monaco.', D. Normand-Cyrot* and N. Pantalos'

'Laboratoire des Signaux et SystBmes, CNRS/ESE, Plateau de Moulon 91190 Gif sur Yvette, France

** Dipartimento di Informatica e Sistemistica, Universitl di Roma "La Sapienza" 18 Via Eudossiana, 00184 Roma, Italy

This paper deals with the sampling problem of nonlinear singula.rly perturbed systems. Using combinatorial equal- ities, related to the Baker-Campbell-Hausdorff formula, discretization schemes for such systems are proposed. It is shown as in the linear context that slow and fast sampling result in two structurally different discrete-time systems.

1 Introduction In recent years, analysis and control of linear singularly perturbed discrete-time systems have been of great inter- est (see for example the survey [ll] and the references contained therein). In particular, when sampling singular perturbation differential equations, the multi-time scale property is preserved and the resulting sampled models contain both fast and slow dynamics also. As already mentioned by many authors [8,6,11], sampling a singularly perturbed system is strongly dependant on the choice of the sampling rate; namely, slow and fast rates leads to different discrete-time models. This paper studies the effect of slow and fast sampling and analyses the structure of the resulting sampled mod- els in the case of nonlinear singularly perturbed differen- tial equations. Strongly inspired by previous works on the sampling problem of nonlinear systems [9,10], explicit for- mulae for the resulting sampled models are obtained. The proposed results allow to establish some structural similar- ities with the linear case which, in this paper, is recovered as a case study. The outline of this paper is as follows. The second section proposes some combinatorial results useful for the analysis of the sampling problem. The third section presents the considered class of systems and recalls some results of the singular perturbation theory. The fourth and fifth sections deal with fast and slow sampling, respectively, while the last section comments the above results. The linear case as

well as a simple nonlinear system are analysed throughout this paper as case studies.

2 Mathematical tools Given twosmooth Rn-valued vector fields P and Q: R" -+

U , where U is an open set of R", e L p will denote the ex- ponential Lie series operator given by e L p = Cklo $L$ where L p is the usual Lie derivative operator associated to the vector field P. From [4], the following formal series equalities hold:

I t = O = -CO) d d d ( e L P + d Q , - L P ( l d ) l @ ( p , Q ) - d t

(2.1) where Id represents the identity function, and 1 the iden- tity operator. The quotient appearing in the right-hand side of (2.1) means the cancellation of the numerator by the denominator. Generalizing (2.1) the following operators are iteratively defined as in [lo] :

~ i + l ( p , = d ( E ' ( P + t Q , Q ' ) ) d t l f = O ' i > l (2.2)

@+l(p, Qi+l) = d ( E " ( P + t Q , Q ' ) f i 2 1 dt lt=O'

where Q' denotes the presence of Q at i-times, in the right- hand side of (2.2). Expanding the expressions (2.2), we easily verify the following relations:

which are useful1 to explicitely compute E ' ( P , Q ) and W P , 9).

CH3076-7/91/0000-0443$01 .OO 0 1991 IEEE 443

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According to these notations, the next theorem can be s- tated. Theorem 2.1: The following expansion holds true:

eLp+loe-Lp(Id) = ~ d +

c e(im, ..., i l ) ~ i m ( P , Q i m ) . . . ~ i l ( p , Q i l ) k > l ( i m + ..+il)=k

where the coefficients e(i,, ..., i l ) are given, recursively, b y

C ( i ) = 1 Vi 2 1

E ( i m , ..., i z , ~ ) = C(im, ..., iz)+ C(i,, ..., iz - 1, l ) + ... + C(im - 1, ..., iz, 1)

with E(im, ..., i z , i l ) = 0 Vi, 5 0 Moreover, for all il such that il 2 2, we have :

e(im, ..., iz, i l ) = e(i,, ..., iz, i l - I)+ e ( i m , ..., i2 - 1, i l ) + ... + C ( i m - 1, ..., iz, i l )

Proof: The proof is similar to the one proposed in theo- rem 2 in [lo]. Denoting T any R" valued vector field, a straightforward application of the following well-known equality derived from the Baker-Campbell-Hausdorf formula, that is:

eadpLT = e L p LT e - L p

results in the following expression relating E ' ( P , Q ) and E 1 ( P , Q ) :

E 1 ( P , Q ) = e a d p ( E ' ( P , Q ) ) (2 .4) Using (2.4) and theorem 2.1, one can easily prove the fol- lowing corollary. Corollary 2.1 : The following expansion holds true:

e-LPeLP+lO(Id) = ~ d +

tk - k! &, ..., i m p i m ( p , Q ~ ~ ) . . . E ~ ~ ( P , Q")

k > 1 (i,+ ...+ i l ) = f

wh.ere the coeficients e ( i 1 , ..., i m ) are defined in theorem 2.1. On the other hand, it can be shown, in terms of integral forms, that:

(2.5) and iteratively

Lemma 2.1: The following equality h.olds true:

E 2 ( P , Q 2 ) + E ' ( P , Q ) E ' ( P , Q ) = 2 J: J," e-oLpL Q e (o - f )Lp~Qe 'LP(zd )d tda (2.7)

We note that Lemma 2.1 can be extended to multiple in- tegrals of any order.

3 Nonlinear singularly perturbed systems

Consider the class of nonlinear singularly perturbed sys- tems described by the following equations:

E = { j. = f l ( X , 2) + 91(x, 2) . x ( 0 ) = 2 0

z ( 0 ) = 20 E i = f z ( x , z ) + gz(x , z ) u

with x E M , , z E M J and U ER", where M , and M J are manifolds of dimension n, and n~ respectively. The functions f i , g i are assumed to be real, analytic of appro- priate dimensions and the parameter E denotes the singular perturbation parameter expressing the speed of the slow versus the fast system dynamics. Suppose, that system (E) conforms to the following as- sumptions [7]. Assumption A.l: The Jacobian J,( f2 + gzu) i s non- singular f o r all x , z , U E ?), wh.ere ?) is the region of inter- est. Assumption A.2: For al l Z J , U E ?), the real parts of the eigenvalues of J , (f2 + gzu) are smaller than a f i x e d negative number -A, that is

Under assumptions A . l and A.2, the reduced slow dynani- ics is obtained by setting E=O in (E) and solving with respect to z the algebraic-type vector equation:

0 = h(x, 2 ) + g z ( 2 , Z b (3.1)

Then, substituting to z in (E), the fast quasi steady state E (solution of (3.1)), results to the reduced slow dynamics

i = ti(., E ) + g1(5, E)"

The next proposition gives an explicit formal expression of 5 , in terms of the inverse series associated to (3.1) (see [l] for details). Proposition 3.1: A n explicit expression of the solution Z of (3.1) is the following one:

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where D' denotes the i ih column of the inverse of the Ja- cobian of the vector function (f2(z, t ) + g2(z, z)u), that is

D d&' [ J z ( f z + g z ~ ) ] - ~ and (fi(z, t o ) + g2(z, Z ~ ) U ) ~ denotes the i ih component of the vector function (fz(z, t o ) + gz(z, z o ) ~ ) . The proof is based on formal series inversion proposed in

In the sequel, for computing convenience, set system (E) in the following form:

PI.

(3.3) F Y = (S + T ) where

y = ( ~ ) , S = ( f l ~ g l u ) a n d F = ( 0 ) f 2 + QZU

At the sampling instant t = ( p + 1)6, where 6 is the sam- pling period, the solution to (3.3) is given by :

Y ( ( p + l)6) = e a L ( ' + f ) I d IY(P6) (3.4)

Hereafter, we will denote, as usual [8,6], by n and k, the fast and the slow sampling points, respectively.

4 Fast Sampling In this section, the fast sampling of the nonlinear dynam- ics C (or equivalently (3.3)), is addressed. Assuming the sampling period 6 and E sufficiently smalls, it is reason- able to write 6 as LYE where a is a real, keeping OE small enough. In the linear context [8,6] o is equal to 1 for sim- plicity reasons; this particular choice is not made in the present study. The fast rate sampled model may be used to assign the fast closed-loop behaviour and to stabilize the fast state component t when assumption A.2 is not satisfied. Setting 6 = LYE and reminding that e-QLFeaLF = I d , e- quation (3.4) can be rewritten in the form:

Y((n + 1)5) = {eaL*s+Fe-uLF}eaLF Idly(na)

Because of Theorem 2.1, the following expansion gives the exact form of the discrete-time solution under fast sam- pling. Theorem 4.1: The ezact fast discrete t ime dynamics as- sociated io @), is given by:

E k Y ( ( n + 1)6) = {Id+ mn, ..., il) k > l (i,+ ...+ i l ) = k

@ m ( F , S ' m ) . . . @ a ( F, S ' l ) } e a L F l d l y ( , q (4.1)

where ihe E'(F, s') are defined in(2.2). Proof: The proof follows directly from theorem 2.1 by

Next corollary gives the first-ordcr approximation in E of (4.1), useful for control purposes. Corollary 4.1: The first order approzimation in E of the fast discrete-time solution (4.1) is given by:

Y ( ( n + 1)6) = e u L F I d p ( , a ) + E&'(oF, aS)euLF2dly( , ,a )

which can be decomposed as follows:

~ ( ( n + 1)6) = z(n6) + E&'((;L.F, OS)IdnsIy(na)

1 I Y(n6 1 z ( (n + 1)6) = {eaLFIdn, + e.@(cuF,cuS)eaLFId,j

Id , , and Idn/ denote the identity projections: Id,, : R" In this case, we have:

Rns and Id , j : Rn -., RnJl respectively.

The above expressions show that the evolution of the slow state component P is close to O(E) while the fast state component is close to O( 1). Hereafter, we apply the above results to the case of linear singularly perturbed systems. Example 4.1 : Consider the linear singularly perturbed system (Cl) defined by the following equations:

X = Allx + A122 + Blu ~i = A Z ~ Z + A222 + & U

where the matrix A22 is assumed to be stable. In this case,

Step One : computation of the term euLF I d l y ( , q . Since the first row element of the vector F is equal to zero, it easily results that:

Idn8lY(n6) = g ( n 6 ) e a L ~

After some algebraic manipulations, one verifies that:

which can be rewritten formally as:

substituting a-& to t , aF to P and Q to S. "44

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To summarize, the first-order approximation of the discrete-time version of ( E l ) obtained under fast sampling, takes the form:

+ ( 2 ) u ( n 6 )

where

Remark : The present method does not require the in- vertibility of the matrix A22. We nr >e that the above solution is exactly the same as the one proposed in [8,6] which is deduced using a state space decoupling transfor- mation actuatelly not available in a nonlinear context. Example 4.2: Consider the following simple nonlinear system:

x * = z E Z = -x - tan z + U

ster, One: we find

a2 -(-z(n6) - tan z(n6) + u ( n 6 ) ) + O(a3) 2!

Rewriting these approximations in a compact form yields the following fast discrete-time dynamics:

z((n + 1)6) = z(n6) + €[ar(n6)+

CY2 -(-z(n6) - tan z(n6) + u ( n 6 ) ) + 0(a3) ] + 0(2) 2!

z ( ( n + 1)6) = z ( n 6 ) + + (tan z (n6 ) ) ' ) ] }

(-z(ns) - tan z ( n 6 ) + u(n6) + O ( a 3 ) ) + O(E)

5 Slow Sampling In this section, since the slow sampling period 6 is such that 6 >> E , let us formally write 6 as PE such that lim,,oP = CO. The slow time scale model is useful to assign the slow state z, provided that the state z is stable. Since 6 = P E , relation (3.4) becomes:

Y ( ( L + 116) = e L ( 6 s + ~ ~ ) I d 1 y ( k 6 ) (5.1)

which can be rewritten as follows:

The next proposition gives an explicit expression of the solution (5.1) by means of the integral forms given in (2.5), (2.6) and (2.8) which are convenient to study the behaviour of the solution as E goes to zero. Proposition 5.1: The following equal i ty holds:

P l

Lse'PLFldly(k6)dadt + O(b3)

Moreover, under assumptions A . l and A.2 , as E + 0 , one gets :

Y ( ( k + 1)6) = Y ( k 6 ) + 6(LsY)lr(k6)+

6 2 - 2! L s ( L s y I y ( k 6 ) ) l y ( b 6 ) f 0(63>

where Proof: The first part of the theorem results by substitut- ing t by PE, P by PF and Q by S in theorem 2.1 and using (2.5), (2.6) and (2.8).

= ( e T , fT)T and f is the solution of (3.1).

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In the sequel, we study the asymptotic behaviour as E -+ 0 . We have

6 2 lim 1' lt e(l-t)PLF(l - t)LSe(t-g)PLF P-w

~ LseUPLFI+(k,qdudt + 0 ( b 3 )

The first term ePLFIdnf in the right-hand side of the above expression represents the solution to the differential equa- tion:

i = h(., z ) + g z ( 2 , t.)u where the state I is considered to be constant. There- fore, under assumptions A . l and A.2 and as p + 03, this solution converges to Z, that is,

lim e P L F I d n f l y ( k a ) = f ( z ( k 6 ) , 1 ( k 6 ) , u(k6)) P-00

(5.2)

Because of ePLFIdnSIY(k,q = r (k6 ) and relation (5.2), the first term is compactly given by:

For the second and the third term, one gets:

1 ZLs(Ls(~)ly(ka)),P(ka)

respectively and the proof is complete. Corollary 5.1: Decomposition into the slow and fasi parts of the solution given in Proposition 5.1, up to order two in 6 , gives:

1 t ( ( k + 1>6) = + ( ~ 6 ) + 6 J e('-t)PLF(fI + g lu) l y (ka )d t

+62 1' 1' e(1-t)PLF Lse('-")PLF (fi + glu>ly(k6)dadt

0

Moreover, as E 4 0 , one has:

Proof: All the results follow from the fact that:

ePFIdnSly (kq = z ( k 6 ) and LsePFIdnsly(ka) = (fi + glu)ly(ka)

The slow-time scale model exhibits the slow dynamics in I which will be used to compute a slow controller as well as the evolution of the quasi-steady state. Example 5.1: Consider again the linear singularly per- turbed system defined in example 4.1. We obtain:

~ ( ( k + 1)6) = 1 ( k 6 ) + 6 { A l l ~ ( k 6 ) + Blu(kS)+ A12(ePA2zz(k6)+

As p = :, we can set, as in [6,8], ePA2a = &A22 and thus:

~ ( ( k + 1)6) = z ( k 6 ) + 6{Ai i1 (M) + Biu(kC)+

A ~ ~ ( E A ~ ~ Z ( R S ) + A T i ( A ~ i ~ ( f f 6 ) + &u(W))}+

0 ( b 2 > .((k + i)a) = ~ A ~ ~ ~ ( j 1 " 6 ) + A ; ; ( A ~ ~ I ( ~ ~ ) + ~ ~ u ( k b ) ) ) +

O(6) We note that the above system is exactly the C-model given in [ l l ] . Moreover, in this case, the fast quasi-steady state is given by:

E = - A , - , ' ( A ~ ~ I + & U )

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and consequently: account the evolution of the slow component 2 during the sampling period and consequently this approximation is valid only for small sampling periods. Therefore, it would z ( ( k + ll6) = z(k6)+6(Allz(k6) +A122(k6) B l u ( k 6 ) ) +

6 2 ~ ( A i i - Ai2A;~A2i)(Allz(k6) + Alzt(k6) + B1u(k6))+

be preferable to design digital control on the slow time scale model defined in section 5.

W 3 ) where the fast state dynamics is described by:

f ( ( k + l)6) = -A,;I.(A21~(k6) + Bzu(k6))-

6A;iAzi(Aii2(k6) + Ai22(k6) + Biu(k6))- 62 -A;iA21(A11 2! - A12A&421)

(Allz(k6) + AlzE(k6) + Blu(k6)) + O(b3) Example 5.2: For the nonlinear system defined in ex- ample 4.2, the quasi steady fast state solution 2 is obsious- ly given by:

2 = -arctg(u - z)

and the slow-time scale model is given by:

z ( ( k + 1)6) = z ( k 6 ) + 6 f ( k 6 )

1 6 z ( ( k + 1)6) = Z(kb)[l + 1 + ( u ( k 6 ) -

References [l] J . P. Barbot, N. Pantalos. Using symbolic calcu-

lus for singularly perturbed nonlinear systems, Euro- pean Conference on Algebraic Computation in Con- trol, Paris, Springer-Verlag, 1991.

[2] J . P. Barbot, N. Pantalos. A sampled-data control scheme for stabilization of systems with unmodeled high-frequency dynamics. To Appear in IEEE Trans. on Aut. Control, 1991.

[3] E'. Esfandiari, H. Khalil. On the robustness of sampled-data control to unmodeled hight-frequency dynamics. IEEE, Trans. on Aut . Control, AC-34, pp.900-903, 1989.

[4] R. Goodman. Lifting vector fields to nilpotent Lie groups. J . Math. Pures et A p p l , 57, pp, 77-86, 1978.

[5] W. Grobner, Serie d i Lie e Lor0 Applicarioni, Poliedro, Gremonese, Roma, 1973.

[GI H. Kando, T. Iwazumi. Multirate digital control de- sign of an optimal regulator via singular perturbation theory. Int. J . Control, Vo1.44, No.6, pp.1555-1578, 1986.

6 Concluding Remarks Some comments and remarks concerning with both fast and slow sampling are in order. In order to find an approximated fast time scale model, a direct way could be to consider the Euler approximation of the full system. For the linear case, this results to

[7] P. V. Kokotovic, H. I(. Khalil, J . O'Reilly. Singular Perturbation Methods in Control: Analysis and De- sign. New York: Academic Press, 1986.

[8] B . Litkouhi, H. Khalil. Multirate and composite con- trol of two-time-scale discrete-time systems. IEEE z((n + 1)6) 21 z(n6) + b(Allz(n6) + A12~(n6) + B i ~ ( n 6 ) ) Trans. on Aut. Control, AC-30, No.7, pp. 645-651, 1985.

Comparison with the first order approximation given in this paper (see example 4.1) shows that Euler approxima- tion exhibits less informations about the inherent struc- ture of the system. This is due to the fact that Euler a,)proximation does not take into account the two-time- scale property of the initial system. Similarly, a direct way to compute a slow time scale model could be to sample the quasi-steady state dynamics (re- duced dynamics). Such procedure does not result to the equations of corollary 5.1. This illustrates the fact that re- duction and discretization do not commute as pointed-out in [3] and [2] for the linear case. In fact, this corresponds to neglect the dynamics on the slow manifold by setting f ( ( k + 1)6) = Z ( k 6 ) . This simplification does not take into

[9] S. Monaco, D. Normand-Cyrot. On the sampling of linear control system. Proc. of the 24-th IEEE CDC, Fort Lauderdale, USA, pp. 1457-1462, 1985.

[lo] S. Monaco, D. Normand-Cyrot. A combinatorial ap- proach of the nonlinear sampling problem, Lecture Notes in Control and Information Sciences, No.144, Proc. of the 9'h International Conjerence, Antibcs, 1990.

[11] D. S . Naidu, D. B. Price, J . L Hibey. Singular pertur- bations and time scales in discrete control system-An overview. Proc. of the 26-2h IEEE CDC, Los Angeles, USA, 1987.

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