On the Relationship Between Continuous- and Discrete-Time Control Systems

Transcription

1 On the Relationhip Between Continuou- and Dicrete-Time Control Sytem V.M. Veliov Reearch Report November, 21 Operation Reearch and Control Sytem Intitute of Mathematical Method in Economic Vienna Univerity of Technology Reearch Unit ORCOS Argentiniertraße 8/E15-4, 14 Vienna, Autria

2 On the Relationhip Between Continuou- and Dicrete-Time Control Sytem V.M. Veliov Thi paper i dedicated to the 7-th anniverary of Gutav Feichtinger Abtract Building on previou reult of the author thi paper preent two new error etimate for the reachable et of an affine control ytem if only piece-wie contant admiible control on a uniform meh are ued intead of all meaurable admiible control. It i natural to expect that the reulting hrinkage of the reachable et i of the order of the meh ize. In thi paper it i proved that under certain reaonable condition the error i of higher than firt order. Keyword: control ytem, dicretization, error analyi 1 Introduction Control theory play a ubtantial role in the mathematical economic and in the invetigation of typical dynamic problem of operation reearch uch a optimal production planning, exploitation of renewable reource, labour allocation, et. Both continuou- and dicrete-time control model are ued in the literature, and the debate about the advantage and hortcoming of thee two type of model are peritently preent in meeting and publication. In many practical problem dicrete-time control model eem to be more appropriate at leat for the following reaon: (i) tate meaurement become available at dicretetime intance; (ii) control deciion are taken at dicrete time. The main advantage of the continuou-time model i that one can employ more conveniently the technique of the claical and the modern mathematical analyi. In addition, in many area the preent computing and communication facilitie are o fat that monitoring and deciionimplementation take very hort time. The appropriate time-cale i alway relative, therefore continuou-time model can be relevant even if the obervation/deciion time-quantum i large, provided that the dynamic i ufficiently low. ORCOS, Intitute of Mathematical Method in Economic, veliov@tuwien.ac.at Vienna Univerity of Technology, 1

3 The relation between continuou-time dynamic ytem decribed by differential equation and their dicrete-time verion (involving a mall time-tep) i profoundly invetigated tarting from the beginning of the 18-th century. The approximation theory for differential equation by dicrete-time dynamic equation and the correponding numerical method are till a hot topic, although they have already been rather well developed. In our opinion thi i not the cae with dicrete-time approximation of control ytem. The principle reaon i that a good approximation require good propertie of the olution, while in the cae of a controlled ytem uch propertie often cannot be enured a priori. Sometime uing dicontinuou or highly ocillating control bring advantage and, repectively, create trouble. In thi paper we addre only the following quetion: given a continuou-time control ytem, what, quantitatively, i the diadvantage of uing only piece-wie contant control on a given time-meh intead of all admiible control. On one hand uing piece-wie contant control i a common engineering practice, which make the quetion meaningful. On the other hand, retricting the cla of admiible control to piece-wie contant one open the door for complete dicretization of the control ytem by uing ingle-tep dicretization cheme and make the tandard error analyi for ODE applicable. A it i well-known, uing richer finitely parameterized et of admiible control (ay, one having one free jump in every meh interval) may bring qualitative advantage concerning the approximation rate (Doitchinov and Veliov 1993; Baier and Lempio 1994; Kratanov and Veliov 21). Piece-wie linear control are alo widely ued in the context of dicretization of optimal control problem (Schwartz and Polak 1996; Dontchev et al. 2; Hager 2), although they may be advantageou only under regularity condition for the optimal control. We do not touch thee richer clae of dicrete admiible control in thi paper. Moreover, to avoid (ometime ubtantial) technical complication we retrict the conideration to ytem that are affine with repect to the control. The main meage of the paper i that the cla of piece-wie contant control on the uniform time-meh with tep h on a finite time interval i (omewhat urpriingly) capable to provide higher than firt order approximation (relative to h) of the reachable et of affine commutative control ytem, together with the uual firt approximation of the trajectorie. Moreover, we introduce and invetigate the notion of information pattern of the approximation and how that the information pattern of the higher than firt order approximation i anticipative. The time horizon i finite in the preent paper. The important cae of infinite horizon (ee e.g. Caulkin et al. 25; Gra et al. 28) i rather challenging and almot no reult are available. 2

4 2 The approximation problem and ome known reult We conider an affine control ytem m ẋ(t) = f (t, x(t)) + f i (t, x(t))u i (t), x() = x R n, t [, T ], (1) i=1 where u U = L ([, T ] U), U R m i convex and compact. For u U we denote by x[u] the olution of (1) that correpond to u (auming exitence and uniquene). Control theory and et-memberhip etimation theory raie two main problem: (i) approximate the et of trajectorie, X = {x[u] : u U}, of (1); (ii) approximate the reachable et, R = {x[u](t ) : u U}, of (1). Since the et of admiible control U contain rather irregular function 1 it i natural to plit the approximation problem of (1) into two part: (P1) Replace the et of admiible control U by a finitely parameterized ubet V N coniting only of function u for which (1) can be dicretized efficiently; (P2) Apply a dicretization cheme for olving (1) for u V N. The requirement that V N i a finitely parameterizable et (ay, with a degree of freedom proportional to N) i needed to make the approximation computable. Moreover, for each u V N equation (1) hould be well dicretizable, that i, the function from V N hould be ufficiently regular. 2 Then the error analyi of the dicretization can be carried out in the uual way a for differential equation. Therefore, a mentioned in Introduction we focu on the approximation quetion in problem (P1): what i the approximation error if the et of admiible control U i replaced with the et V N = {u U : u( ) i contant on each (t k 1, t k ))}, (2) where t k = kh, h = T/N, N i a natural number. That i, we want to etimate the uniform error H C (X, X N ) = up inf x[v] x[u] C[,T ] (3) u U v V N and the terminal error H(R, R N ) = up u U inf x[v](t ) x[u](t ), (4) v V N 1 The reachable et R i uually not generated by continuou control. Even more, control function of unbounded variation or non-integrable in Riemann ene may generate point of R that are not reachable by other control, a in Fuller phenomenon or a in Silin (1981). Thi create the difficulty of approximating (1) by dicrete-time dynamic ytem. 2 Of coure, there i a trade-off in chooing V N : the larger V N, the better the approximation to X and R by control from V N ; on the other hand, the lower i the accuracy of dicretization. 3

5 where X N and R N are the et of trajectorie and the reachable et correponding to the et V N of admiible control. We mention that Problem (P1) make ene and i tudied in the literature alo for richer finitely-parameterized clae of admiible control, ee e.g. Baier and Lempio (1994), Doitchinov and Veliov (1993), Ferreti (1997), Kratanov and Veliov (21). Thi paper addree only the implet and mot often ued cae (2). Under tandard aumption the mapping u x[u] i continuou in L 1 and ince V N i compact in the ame pace, the infimum in (3) and (4) i achieved. Then there exit a mapping π N : U V N uch that or up x[π N (u)] x[u] C[,T ] = H C (X, X N ), (5) u U up x[π N (u)](t ) x[u](t ) = H(R, R N ), (6) u U repectively (the mapping π N need not be the ame in the two equalitie). Thu the quetion of accuracy of approximation can formulated in term of the mapping π N : V N provide approximation of order α to X if there exit a mapping π N : U V N (called further approximation mapping) uch that x[π N (u)] x[u] C[,T ] cont. N α for every u U. Similarly for the reachable et. Thi reformulation of the approximation problem ha an advantage: one can tudy the information pattern of the mapping π N that provide a given approximation rate. Namely, we can ditinguih the following cae: Definition 1 (i) The mapping π N : U V N i called local if for every k =,..., N 1, and for every u, u U with u (t) = u (t) on [t k, t k+1 ] it hold that π N (u )(t) = π N (u )(t) on [t k, t k+1 ]; (ii) The mapping π N : U V N i called non-anticipative if for every k = 1,..., N, and for every u, u U with u (t) = u (t) on [, t k ] it hold that π N (u )(t) = π N (u )(t) on [, t k ]; (iii) The mapping π N : U V N i called anticipative if it i not non-anticipative. The above notion are adapted from the theory of differential game (Varaiya-Roxin- Elliot-Kalton trategie). A we hall ee, it may happen that a certain order of approximation can be achieved by anticipative approximating mapping π N but cannot be achieved by non-anticipative (rep. local) mapping. The information pattern of π N may play a role for the order of the accuracy. It i to be treed that in different problem related to the control ytem (1) one may need to retrict the choice of the approximation mapping to a precribed information pattern: local or non-anticipative. Thi i the cae, for example, if one ha to imulate a real 4

6 ytem modeled by (1) only knowing the current, or the pat information about the input u. For other problem, ay for an optimal open-loop control problem one can freely employ anticipative approximation mapping to pa directly to mathematical programming. Below we recall a few known approximation reult in the light of the above concept of information pattern. One commonly ued approximation mapping π N : U V N i defined a π N (u)(t) = 1 h k t k 1 u() d for t (t k 1, t k ). (7) Obviouly it i local and even more, it i independent of the pecific form of the equation (1). Let u conider firt a linear control ytem in (1): ẋ = Ax + Bu. From a general reult in Dontchev and Farkhi (1989) it follow that for the local approximation mapping π N : U V N defined by (7) enure x[π N (u)] x[u] C[,1] ch u U. (8) In the ame time the reult in Veliov (1992) and Doitchinov and Veliov (1993) imply that there exit an anticipative approximation mapping π N : U V N (which i not explicitly defined in thee paper) uch that x[π N (u)](t ) x[u](t ) ch 2 u U. (9) We mention that the reult hold for an arbitrary convex and compact et U, therefore it applie alo to the pathological example in Silin (1981) mentioned in footnote 1. A econd order approximation a in (9) cannot be achieved by uing local approximation mapping. An important extenion i proved in Pietru and Veliov (29): there exit an anticipative approximation mapping π N : U V N that enure imultaneouly (8) and (9). Thi reult open the door to error etimate for non-linear ytem by local linearization. The approach will be followed in Section 4. There are only few higher than firt order approximation reult concerning the nonlinear cae (1). 3 The firt i that in Veliov (1989), where a econd order approximation of X i proved (or an approximation of order 3/2 for a more general form of the right-hand ide in (1)) auming, however, that the et f(t, x)u := (f 1,..., f m )U i uniformly trongly convex, which i a rather trong aumption for many application. The implicitly involved approximation mapping π N in thi paper i local. 3 Higher than firt order approximation to optimal control problem are known. However, mot of thee reult are baed on a priori aumption that the optimal control i ufficiently regular (i.e. Lipchitz continuou with the firt derivative having bounded variation), ee e.g. Dontchev et al. (2) and Hager (2). The reult recalled or obtained in the preent paper are applicable in the optimal control context without uch aumption. 5

7 Another group of reult concern the cae of commutative affine ytem, i.e. uch that the Lie bracket [g i, g j ] are all zero for i, j 1. A rather general indirect (variational) etimation of H(R, R N ) in the cla V N i obtained in Veliov (1997). It allow to obtain a econd order etimation of H(R, R N ) provided that the et R N have (uniformly) the o called exterior ball property. Thi i done in Section 3. The lat iue we briefly recall i that of approximation uing the cla V extr N where V extr N := {u : [, T ] U extr : u(t) i contant on each (t i 1, t i ))} of control, and U extr i the et of all extreme point of U. Thi iue i important for numerical treatment of optimal control problem for witching ytem, ee e.g. Sager (29). The following etimation i obtained, eentially, in Donchev (21) and Grammel (23): for the approximating cla of control VN extr, H C (X, X extr N ) Ch 1/2. (1) Thi etimation i proved for more general ytem than (1) under Lipchitz continuity of f. In Veliov (23) the author of the preent paper conjectured that a firt order etimation hold in (1) and proved thi in everal particular clae of ytem. The paper by Pietru and Veliov (28) alo contain a mall contribution in thi direction. A ubtantial progre in proving the conjecture i done in Sager (29), where however, U i aumed to be a polyhedral et. In all the above contribution the (implicitly or explicitly) involved approximation mapping π N i non-local and non-anticipative. Alo, it i quite clear that local approximation mapping cannot provide even (1). In general, the problem of firt order approximation i till open. 3 A econd order approximation reult under exterior ball property In thi ection we prove that the et of piece-wie contant admiible control U N i powerful enough to enure a econd order approximation to the reachable et of the control ytem (1), provided that a certain condition known a exterior ball property hold (ee e.g. Nour et al. 29 for a recent dicuion of thi property). In particular, thi property i trivially atified if the approximating reachable et R N are convex, thu the next reult extend thoe in Veliov (1992), Doitchinov and Veliov (1993) devoted to linear ytem and Theorem 4.1 in Veliov (1997), where the approximating reachable et are aumed convex. We tart with a lit of aumption for ytem (1). Aumption 1. There i a convex and compact et S R n uch that (i) the function f i : [, T ] S, i =,..., m are differentiable and the firt derivative are Lipchitz continuou; 6

8 (ii) for every u U the olution x[u] of (1) exit in the interior of S on [, T ]; (iii) the ytem i commutative: the Lie bracket of the controlled vector field f 1,..., f m, [f i, f j ] := f i x f j f j x f i, (11) are identically equal to zero for every i, j = 1,..., m. Below we hall employ the following reformulation of Theorem 2.2 in Veliov (1997). Theorem 1 Let Aumption 1 be fulfilled. Then there exit a contant C uch that for every natural number N and for every function g : S R which i differentiable in the interior of S, g/ x i bounded by a contant L g and i Lipchitz continuou with a Lipchitz contant L g at each point of R, the following etimation hold: inf x R N g(x) inf x R g(x) C L g + L g N 2. (12) Before continuing with the exterior ball property we briefly dicu the above reult. The mapping π N that enure econd order accuracy of approximation in the above theorem i anticipative. We mention that the contant C doe not depend on the propertie of the control at which inf x R g(x) = inf u U g(x[u](t )) i achieved. A hown in Silin (1981), thee control can be of unbounded variation, non-integrable in Riemann ene and dicontinuou almot everywhere (Aumption 1 doe not exclude thee poibilitie) even for linear ytem. We alo mention that Aumption 1 (iii) i retrictive for many application. However, the problem of higher than firt order approximation of non-commutative control ytem (even uing larger finitely parameterized clae of admiible control) i till open. The reult in Kratanov and Veliov (21) concerning a rather pecific cla of problem how that the iue i probably complicated. The lat two remark apply alo to the reult preented in the next ection. Definition 2 The compact et Q R n ha the exterior r-ball property (with a poitive real number r) if for every x R n \ Q and y P Q (x) ( y + r x y ) x y + r IB Q =, where P Q (x) i the projection of x on Q and IB i the unit ball in R n. Thi mean that the exterior of Q i a union of ball of radiu r (ee Nour et al. 29) for more detail). 7

9 Theorem 2 In addition to Aumption 1, let there be r > uch that for all (ufficiently large) N the et R N have the exterior r-ball property. Then there exit a contant C uch that H(R, R N ) Ch 2. Proof. Denote ρ N = H(R, R N ). From Dontchev and Farkhi (1989) we know that ρ N. Since R i compact, there exit a point x N R uch that dit(x N, R N ) = ρ N. We may aume ρ N > ince the alternative cae i trivial. Let y N P RN (x N ). According to the exterior ball property it hold that ( y N + rl N + r IB ) R N =, where l N = x N y N x N y N. (13) For a ufficiently large fixed N, o that ρ N r/2, we define the function g N (x) = y N + rl N x. According to (13) there are no point of R N in the open ball with radiu r around y N +rl N. Then the definition of ρ N implie that there are no point of R in the open ball with radiu r ρ N centered at the ame point. Hence, for x R we have that y N + rl N x r ρ N r/2. Then the derivative g (x) = y N + rl N x y N + rl N x exit for x R and it norm i L g = 1. Moreover, g i Lipchitz with ome contant L g at point x R, where L g depend on S and r but not on N. Thu the condition in Theorem 1 are fulfilled and it implie that Ch 2 inf x R N g(x) inf x R g(x), where C i independent of N. Due to (13) the firt infimum i attained at x = x N, while the econd infimum i not larger than g(x N ). Hence, Ch 2 y N + rl N y N y N + rl N x N = r (r ρ N ) = ρ N. Thi complete the proof. Q.E.D. The above theorem ha the drawback that the exterior ball property i difficult to check in the non-linear cae and even more, it may fail to hold. An example where it fail i provided by the following imple two-dimenional bilinear commutative ytem: ẋ = u 1 A 1 x + u 2 A 2 x, x R 2, u 1 + u 2 1, u i [, 1], (14) 8

10 with ( A 1 = 1 1.3π 1.3π 1 ) (.7 π, A 2 = π.7 ). (15) The reachable et at time T = 1 i plotted on Figure 1. The exterior r-ball property fail at one point (whatever i r > ). Since R N converge to R, the exterior r-ball property fail alo for R N if N i ufficiently large. Figure 1: The reachable et in the example (14), (15). In the next ection we conider the general cae of a commutative affine ytem, not relying on the exterior ball property. 4 Approximation of a ingle input ytem In thi ection we conider the affine ytem (1) in the cae m = 1: ẋ = f (x) + f 1 (x)u, x() = x R n, u [, 1]. (16) Aumption 2. There i an open et S R n uch that (i) the function f i : S R n, i =, 1 are time-invariant, twice differentiable and the econd derivative are Lipchitz continuou; (ii) for every u U the olution x[u] of (1) exit in S on [, T ]; The next theorem extend Theorem 2 in Pietru and Veliov (29) for non-linear ytem and, in addition, add the tatement of imultaneou firt-order approximation of the trajectory bundle. Theorem 3 Let Aumption 2 hold. Then there exit a contant C uch that for every N there exit (an anticipative) mapping π N : U V N for which x[π N (u)] x[u] C[,T ] Ch, (17) x[π N (u)](t ) x[u](t ) Ch 1.5, (18) 9

11 where h = T/N. The key tool for proving the above theorem i provided by the following lemma Lemma 1 (Lemma 2 in Pietru and Veliov 29) Let Aumption 2 hold and let N and M [2, N] be natural number and h := 1/N. Then there exit (an anticipative) mapping π N : U V N uch that for every u U (u() π N (u)()) d h t [, Mh], Mh (u() π N (u)()) d 1 Mh 2 h2, (u() π N (u)()) d 1 2 h2. Proof of Theorem 3. The idea of the proof i the ame a in Pietru and Veliov (29). Without any retriction we aume T = 1 (a different finite T > affect only the contant in etimation (17), (18)). Let M be the larget natural number uch that M 2 N and let t = Mh. Let u 1 U be arbitrarily fixed and let u h 1 V N be defined a u h 1 = π N (u 1 ), where π N i the mapping defined in Lemma 1. Define, in addition u (t) = u h (t) 1. The third-order Volterra expanion of the olution of (1) for initial point x give a repreentation of the olution x[u 1 ] (ee e.g. Grüne and Kloden 26) of the following form: x[u 1 ](t) = x + + i= f i (x ) i= j= k= u i () d + τ i= j= L j f i (x )u j (τ) dτu i () d (19) L k L j f i (x )u k (θ) dθu j (τ)u i () d + O(t 4 ), where by definition L i = f i and a uual O()/ i bounded, uniformly in u x 1 U. The above repreentation hold alo for x[u h 1]. We hall etimate x[u 1 ](t) x[u h 1](t) by conidering each of the four term in (19) eparately. Obviouly the difference of the firt term i zero. Alo f i (x ) u i () d f i (x ) u h i () d i= i= = f i (x ) (u i () u h i ()) d C 1h 2 C 1 M 2 h 3, i= (where C 1 i independent of t) according to the econd inequality in Lemma 1, which obviouly hold alo for u and u h. 1

12 The term with L f 1 and L 1 f in x[u 1 ](t) x[u h 1](t) can be etimated by C 3 h 2 C 3 M 2 h 3 a above. To etimate the ret of the third term we notice that u 1 (τ) dτ u 1 () d = 1 ( ) u 1 () d, 2 which implie that [ L1 f 1 (x ) u 1 (τ) dτ u 1 () d L1 f 1 (x ) h 2 (2Mh) = C 2 Mh 3. 2 u h 1(τ) dτ u h 1() d] To etimate the difference of the triple integral we ue the repreentation τ u() u() u() u(τ) u(θ) dθ dτ d = t 2 u() d t τ τ u(τ) τ τ dθ dτ d = 2 dθ dτ d = t2 2 τ dθ dτ d = 1 2 u(θ) dθ dτ d = u(τ) dτ + t u() d u() d u(τ) dτ, u() d t 2 u() d u(τ) dτ, ( 2 u(τ) dτ) d u() d (t )u() d u(τ) u(θ) dθ dτ d = 1 ( 2 u(τ) dτ) d 2 τ u() u(τ) u(θ) dθ dτ d = 1 ( 3 u() d). 6 ( 2 u(τ) dτ) d Then one can eaily etimate the third order integral in the difference x[u 1 ](t) x[u h 1](t) by CM 2 h 3. For example, ( 2 ( 2 u 1 (τ) dτ) d u h 1(τ) dτ) d (u 1 (τ) u h 1(τ)) dτ (u 1 (τ) + u h 1(τ)) dτ d Mh (.5h2 + h) 2Mh CM 2 h 3, 11

13 ince t (u 1 (τ) u h 1(τ)) dτ = (u 1 (τ) u h 1(τ)) dτ + (u 1 (τ) u h 1(τ)) dτ.5h2 + h according to Lemma 1. Thu we obtained that the local error x[u 1 ](Mh) x[u h 1](Mh) i of order M 2 h 3. Then repeating thi on every ubinterval [imh, (i+1)mh], i = 1,..., M we obtain by a tandard propagation of error argument that x[u 1 ](T ) x[u h 1](T ) CM 2 h 3.(M + 1) C M 3 h 3 C h 1.5. The firt order etimation for the trajectorie follow in a tandard way from the firt inequality in Lemma 1. Q.E.D. Having in mind Theorem 2 one can ak if the above etimation i harp. Thi challenging quetion i till open even without the requirement that the approximation map π N provide a firt order approximation of the trajectory bundle, that i, it i not know whether the etimation H(R N, R) Ch 2 doe not hold under Aumption 2. Reference Baier R, Lempio F (1994) Computing Aumann Integral. In: Modeling Technique for Uncertain Sytem, Kurzhanki A, and Veliov VM (Ed.), PSCT, 18:71 9 Caulkin JP, Feichtinger G, Gra D, Tragler G (25) A model of moderation: Finding Skiba point on a lippery lope. Central European Journal of Operation Reearch 13(1):45 64 Doitchinov BD, Veliov VM (1993) Parametriation of integral of et-valued mapping and application. J Math Anal and Appl 179(2): Donchev T (21) Approximation of lower emicontinuou differential incluion. Numer Funct Anal and Optimiz 22(1&2):55 67 Dontchev A, Farkhi E (1989). Error etimate for dicretized differential incluion. Computing 41(4): Dontchev AL, Hager WW, Veliov VM (2) Second-order Runge-Kutta approximation in control contrained optimal control. SIAM J Numerical Anal 38(1): Ferretti R (1997) High-order approximation of linear control ytem via Runge-Kutta cheme. Computing 58(4):

14 Grammel G (23) Toward fully dicretized differential incluion. Set-Valued Analyi 11(1):1 8 Gra D, Caulkin JP, Feichtinger G, Tragler G, Behren DA (28) Optimal Control of Nonlinear Procee. With Application in Drug, Corruption, and Terror. Springer, Berlin Grüne L, Kloeden PE (26) Higher order numerical approximation of witching ytem. Sytem Control Lett (9): Hager WW (2) Runge-Kutta method in optimal control and the tranformed adjoint ytem. Numeriche Mathematik 87: Kratanov M, Veliov VM (21) High-order approximation to multi-input non-commutative control ytem. To appear in Large-Scale Scientific Computation, Springer, Berlin, Heidelberg Nour C, Stern RJ, Takche J (29) The union of uniform cloed ball conjecture. Control and Cybernetic 38(4): Sy- Pietru A, Veliov VM (29) On the Dicretization of Switched Linear Sytem. tem&control Letter 58: Schwartz A, Polak E (1996) Conitent approximation for optimal control problem baed on Runge-Kutta integration. SIAM J Control and Optim 34(4): Silin D (1981) On the Variation and Riemann Integrability of Optimal Control of Linear Sytem. Dokl. Akad. Nauk SSSR 257: Sager S (29) Reformulation and Algorithm for the Optimization of Switching Deciion in Nonlinear Optimal Control. Journal of Proce Control, to appear. (URL Veliov VM (1989) Second order dicrete approximation to trongly convex differential incluion. Sytem & Control Letter 13: Veliov VM (1992). Second order dicrete approximation to linear differential incluion. SIAM J Numer Anal 29(2): Veliov VM (1997). On the time-dicretization of control ytem. SIAM J Control Optim 35(5): Veliov VM (23) Relaxation of Euler-Type Dicrete-Time Control Sytem. Reearch Report No 273, ORCOS, TU-Wien, 23. ( Report/Re Rep till 29/RR273.pdf) 13