- Email: [email protected]
Parameter identification for models with bounded errors in all variables
Systems & Control Letters 19 (1992) 425-428 North-Holland
425
Parameter identification for m o d e l s with b o u n d e d errors in all variables * G.
Belforte
Dipartimento di Automatica e lnformatica Politecnico di Torino, Italy
Received 30 January 1992 Revised 19 June 1992 Abstract: In this paper, the problem of parameter identifica-
tion for models with bounded measurement errors both on the input and on the output is addressed and some corrections to previously published results are presented. In particular, it is shown that only parameter overbounds can in general be computed for systems of the form y = (qb + 6q~)0 + 6y when the bounded measurement errors fiq) and 6 y are correlated. Since ARMAX and bilinear systems can be represented in this form, it turns out that tight parameter bounds are in general not available for these systems. Finally, we show that it is possible to check a posteriori whether the obtained bounds are tight or not. Keywords: Robust identification; set-theoretic estimation; un-
known but bounded errors; error correlation; parameter estimation algorithms; confidence regions; parameter overbounding.
I. Introduction I n t h r e e c o n f e r e n c e p a p e r s [3-5] t h a t have a p p e a r e d in 1991, C e r o n e a d d r e s s e s t h e p r o b l e m of p a r a m e t e r b o u n d e v a l u a t i o n for m o d e l s with b o u n d e d e r r o r s in all t h e variables. In [3] he gives a p r o o f of a p r o p o s i t i o n a l r e a d y s t a t e d (but not p r o v e d ) by Belforte, B o n a a n d C e r o n e [1,2]; while in [4] a n d [5] he uses t h e p r o v e d r e s u l t to assess tight p a r a m e t e r b o u n d s for A R M A X a n d b i l i n e a r systems respectively. It t u r n s o u t that, for t h e s e systems, t h e p r o v e d result allows for p a r a m e t e r o v e r b o u n d i n g only; t h e o c c u r r e n c e o f tight p a r a m e t e r b o u n d s b e i n g a
p a r t i c u l a r l y lucky case t h a t can b e c h e c k e d only a This fact was a l r e a d y m e n t i o n e d by Belforte, B o n a a n d C e r o n e [2] b u t t h e p r o c e d u r e t h a t t h e y give for testing t h e b o u n d t i g h t n e s s is incorrect. A i m of this p a p e r is to a d d r e s s t h e p r o b l e m o f p a r a m e t e r b o u n d e s t i m a t i o n for m o d e l s with b o u n d e d e r r o r s in all t h e v a r i a b l e s clearly stating w h e n tight p a r a m e t e r b o u n d s c a n b e o b t a i n e d a n d w h e n only o v e r b o u n d s can b e in g e n e r a l c o m p u t e d . In this last case the c r i t e r i o n for c h e c k i n g a posteriori w h e t h e r t h e o b t a i n e d b o u n d is tight or n o t is given.
posteriori.
2. Problem statement and notation L e t a system b e d e s c r i b e d by y* =q~*0
(1)
w h e r e y * ~ R m is t h e v e c t o r o f all the available n o i s e - f r e e o u t p u t samples, @ * ~ R mxp is t h e n o i s e - f r e e system m a t r i x ( r e g r e s s o r matrix) a n d 0 ~ R p is t h e u n k n o w n p a r a m e t e r vector. D u e to m e a s u r e m e n t e r r o r s t h e actually o b s e r v e d variables a r e y =y* + 6y, =
-
(2) (3)
w h e r e 6 y ~ R m a n d 6q~ ~ R m / p a r e t h e m e a s u r e m e n t e r r o r s on t h e o u t p u t s a m p l e s a n d on t h e r e g r e s s o r s respectively. F r o m (1), (2) a n d (3) o n e gets y = ( ~ + 6q~)O + 6 y .
(4)
This can b e r e w r i t t e n as Correspondence to: Prof. G. Belforte, Dipartimento di Auto-
matica e Informatica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Email: [email protected]. * This work was partially supported by the grants MURST40% and MURST-60% of the Italian Ministry of University and Scientific Research.
Y, = (4ff + ~b/T)o + ~SYi, i = 1 . . . . . m ,
(5)
w h e r e ~b/a"a n d 6~b/T d e n o t e t h e i-th r o w of q~ a n d 6q) respectively. It is a s s u m e d t h a t t h e available i n f o r m a t i o n o n t h e e r r o r is s e t - t h e o r e t i c , t h a t is 6 y a n d 6q~ a r e
016%6911/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
G. Belforte / Parameter identification for models with bounded errors
426
supposed to belong to given error sets sly and S2, 6y e sly,
6~ •
slo.
(6)
In this context, the identification p r o b l e m consists in finding the feasible p a r a m e t e r region D D={O•RP:
(12) and for the properties of the set D, the interested reader should refer to [2] and [3]. T h e i m p o r t a n c e of computing the p a r a m e t e r bounds (12) lies in the fact that they define a tight orthotopic o u t e r b o u n d B D to the set D, Bt~= { O e R P : omin ~ o i ~ O max, / = 1 . . . . . p}, (13)
y=(O+8@)O+Sy;
8y e Sly, 8@ • sl,~},
(7)
that is the set of all p a r a m e t e r s consistent with the system (4) and the m e a s u r e m e n t error information (6). No exact solution is available for this general formulation and even approximate solutions are hard to find [9]. However, useful results are available for the special case in which the error sets sly. and slm are equal to o r t h o t o p e s centred in the origin of the error space [2,6,8,10], so that
fly=Bv={SyeRm: laY~l<--Ay i,
i = 1 . . . . . m},
(8) s l , = B~
= {80 e R m × p : 184,ii i=l ..... m;j=l,...,p}
(9)
where 8yi, i = 1. . . . , m , are the c o m p o n e n t s of 8y and 84,ij, i = l . . . . , m , j = l . . . . . p, are the elements of 84, and Ayi, A4,ij, i = 1 , . . . , m, j = 1 . . . . , p, are known quantities. F r o m (5), (8) and (9) it can be derived that the feasible p a r a m e t e r region D is described by the following set of 2 m inequalities:
that is simpler to describe than the set D itself and yet proves to be highly informative and useful [7,10]. It results that 0~m a x (0 im i n ) is achieved on a vertex of D where at least p out of the 2m inequalities (10) are satisfied sharply. Using a terminology introduced in [7] we will refer to these p inequalities as the active constraints of max min Oi (0 i ). So far nothing new has been reported. T h e main points were already published in [1] and [2] and C e r o n e in [3] has given a formal p r o o f of the necessity and sufficiency of relation (10) for defining the set D for a system of the form (4) when the error sets are orthotopic according to (8) and (9). It is worth noting that the obtained results are no more valid if there is correlation a m o n g the errors 8y i, i = 1 , . . . , m, a n d / o r 84,ij, i = 1. . . . . m, j = 1. . . . . p, so that the corresponding error sets sly and slm are not equal to o r t h o t o p e s although they are b o u n d e d by orthotopes. In this case, only p a r a m e t e r overbounds can be c o m p u t e d in general and the occurrence of tight p a r a m e t e r bounds is a particular lucky case that can be only checked a posteriori. This fact is discussed in the next section where the criterion for checking the par a m e t e r ' s tightness is given.
- 44,7T)0 _Yi_AYi,
i= 1,...,m,
(10)
where /t4,TT = [d4,ilsgn(O1) ".. A4,ipSgn(Op) ] .
(11)
Using these inequalities it is possible to compute the p a r a m e t e r bounds O,~ x and 0 im i n , i = 1 . . . . . p, defined as max _
O~
-maxOi, OeD
min
O~
= m i n O i, O~D
3. The case of correlated errors
In this section, we consider the case in which all the errors are unknown but b o u n d e d and there is linear correlation a m o n g some error elements. Let 6e • R m×(p+l) be the vector of all the errors 6y i i = 1 . . . . . m and 84,ij i = 1 , . . . , m , j = 1. . . . . p. T h e error set D E = .Oy × s l , is defined as
OE = { 8 8 c R rnX(p+l)"
i= l ..... m.
(12) For the derivation of (10), for computational details about the evaluation of the p a r a m e t e r bounds
[Sy i] < d y i, i = l . . . . . m , 184,ijl _< a4,ij, i = 1 . . . . . m , j = 1 . . . . . p, b =Ae}
(14)
G. Belforte / Parameter identification for models with bounded errors
427
it results that D c D therefore B~ is an orthotopic overbound to D and B D C_B~. []
where b is a known vector and A is a given matrix of suitable dimensions describing all the existing linear correlations among the different errors. We notice that a special case of linear correlation is that of A R M A X and bilinear systems for which the system matrix q~ is constituted by the input and the output sample vectors suitably shifted. Let B e =By X Bq, be the tight orthotopic bound to /2 E defined as
Therefore the results presented in [4] and [5] where Cerone claims that tight p a r a m e t e r bounds for A R M A X and bilinear systems can be obtained are incorrect. In fact, only overbounds can be in general obtained for these systems. This fact can be easily illustrated by the following simple example consisting of an M A model described by
BE = {6e ~ R m x ( p + l ) "
y(k) = u(k)O, + u(k-
16Yil
i = 1. . . . . m,
1)0 2.
For this model three inputs are available: u ( 1 ) = 20, u ( 2 ) = 30 and u ( 3 ) = - 2 2 0 and the two output measurements are y ( 2 ) = 190 and y ( 3 ) = - 5 1 0 . The errors on the inputs are au(1) = 6u(2) = 3 u ( 3 ) = 5 and the errors on the outputs are 6y(2) = 6 y ( 3 ) = 5. In Figure 1 are reported both the true set D (obtained with a gridding of the errors) and the lines relative to relation (10) defining the set D. It can be seen that D c D so that B o c B ~ and in particular t h e 0~nin and 0~ ax relative to D do not belong to the D set. In [2] the occurrence of overbounding in presence of error correlation was already mentioned, but it was also stated that it is possible to determine a posteriori whether the obtained bound is tight or not only looking at the active constraints
] 6~bijl < A ~ i j , i = 1 . . . . . m , j = 1 . . . . , p}. (15) It turns out that 12e c B e ; /2 E C B e. It is then possible to state the following proposition:
Proposition 1. Let a system be described by (4) and let ~2e and B E be described by (14) and (15). Let D be the feasible parameter set obtained if I2~ = B e and let B~ be its orthotopic bound. Then B o G B~. Proof. B~ is, by construction, a tight orthotopic outbound to D; since D E c B E from definition (7)
8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.6
2.7
2.8
2.9
3
3.1 Fig. 1.
3.2
3.3
3.4
3.5
3.6
428
G. Belforte / Parameter identification for models" with bounded errors
of 0/rain (0imax) relative to the set D. it is indeed true that it is possible to check a p o s t e r i o r i the bound tightness, that is to state whether Bt) = B~ or B D c B~. However to answer this question all the available measurements must be used and not the active constraints only. The result can be obtained according to the following procedure. S t e p 1: Compute 0/rain (0/max) relative to the error orthotope B E (i.e. neglect the error correlation). S t e p 2: Substitute the parameter vector 0 for which 0rain (0 max) was obtained into relation (5). This leads to m equations in the 6 y i, i = 1 . . . . . m , and 649ii, i = 1 . . . . . m , j = 1 . . . . . p , variables. S t e p 3: Check whether the m equations obtained at Step 2 together with those defining the .OF. set in (14) have any admissible solution. I f yes." the obtained 0rain (0 max) is a tight bound to D; I f n o : 0~ in (0~ a~) is not a tight bound to D. In order for Bo to be tight to D so that B ~ = B o it is necessary that all the 0irain and 0 max, i = 1 , . . . , p , are tight to D. R e m a r k again that all the m equations obtained at Step 2 must be used at Step 3 and not only those relative to the active constraints.
4. Conclusive remarks
It is hard to assess, in general, whether the occurrence of tight p a r a m e t e r bounds is, in practical applications, more or less frequent. Clearly it depends on the error realization and therefore and analysis of this aspect, at this stage, seems to require a simulation study that is beyond the scope of this p a p e r and that could hardly give definite answers for the intrinsic nature of any simulation study. Also the influence of the number of measurements could, in principle, be relevant and could be investigated.
Finally it must be remarked that the exact solution could be numerically seeked solving a constrained minimization problem. However, since the solution domain is, in general, not convex and could even be not connected, since the number of involved variables (the parameters 0, the errors 6~bij and 6y i) can be quite large, it turns out that the numerical computation could be difficult and highly time consuming thus preventing any practical use.
References [1] G. Belforte, B. Bona and V. Cerone, A bounded error approach to the tuning of a digital voltmeter, 12th IMACS Congress on Scientific Computation, Paris, France (1988) 498-501. [2] G. Belforte, B. Bona and V. Cerone, Identification, structure selection and validation of uncertain models with set-membership error description, Math. and Comput. in Simulation 32 (1990) 561-569. [3] V. Cerone, Parameter bounds for models with bounded errors in all variables, Proc. 9th IFAC/IFORS Symposium on Identification and System Parameter Estimation Budapest, Hungary (1991) 1518-1523. [4] V. Cerone, Parameter bounding in A R M A X models from records with bounded errors in variables, Proc. 9th IFAC / IFORS Symposium on Identification and System Parameter Estimation, Budapest, Hungary (1991) 1518-1523. [5] V. Cerone, Feasible parameter set for bilinear systems from records with bounded output errors, Proc. 1991 American Control Conference Boston, MA (1991) 37-42. [6] E. Fogel and Y.F. Huang, On the value of information in system identification bounded noise case, Automatica 18 (1982) 229-238. [7] M. Milanese and G. Belforte, Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors. Linear families of models and estimators, IEEE Trans. Automat. Control 27 (1982) 408-414. [8] J.P. Norton, Identification and application of boundedparameter models, Automatica 23 (1987) 497-597. [9] F.C. Schweppe, Uncertain Dynamic Systems (Prentice Hall, Englewood Cliffs, NJ, 1973). [10] E. Walter and H. Piet-Lahanier, Estimation of parameter bounds from bounded-error data: a survey, Math. and Comput. in Simulation 32 (1990) 449-468.
425
Parameter identification for m o d e l s with b o u n d e d errors in all variables * G.
Belforte
Dipartimento di Automatica e lnformatica Politecnico di Torino, Italy
Received 30 January 1992 Revised 19 June 1992 Abstract: In this paper, the problem of parameter identifica-
tion for models with bounded measurement errors both on the input and on the output is addressed and some corrections to previously published results are presented. In particular, it is shown that only parameter overbounds can in general be computed for systems of the form y = (qb + 6q~)0 + 6y when the bounded measurement errors fiq) and 6 y are correlated. Since ARMAX and bilinear systems can be represented in this form, it turns out that tight parameter bounds are in general not available for these systems. Finally, we show that it is possible to check a posteriori whether the obtained bounds are tight or not. Keywords: Robust identification; set-theoretic estimation; un-
known but bounded errors; error correlation; parameter estimation algorithms; confidence regions; parameter overbounding.
I. Introduction I n t h r e e c o n f e r e n c e p a p e r s [3-5] t h a t have a p p e a r e d in 1991, C e r o n e a d d r e s s e s t h e p r o b l e m of p a r a m e t e r b o u n d e v a l u a t i o n for m o d e l s with b o u n d e d e r r o r s in all t h e variables. In [3] he gives a p r o o f of a p r o p o s i t i o n a l r e a d y s t a t e d (but not p r o v e d ) by Belforte, B o n a a n d C e r o n e [1,2]; while in [4] a n d [5] he uses t h e p r o v e d r e s u l t to assess tight p a r a m e t e r b o u n d s for A R M A X a n d b i l i n e a r systems respectively. It t u r n s o u t that, for t h e s e systems, t h e p r o v e d result allows for p a r a m e t e r o v e r b o u n d i n g only; t h e o c c u r r e n c e o f tight p a r a m e t e r b o u n d s b e i n g a
p a r t i c u l a r l y lucky case t h a t can b e c h e c k e d only a This fact was a l r e a d y m e n t i o n e d by Belforte, B o n a a n d C e r o n e [2] b u t t h e p r o c e d u r e t h a t t h e y give for testing t h e b o u n d t i g h t n e s s is incorrect. A i m of this p a p e r is to a d d r e s s t h e p r o b l e m o f p a r a m e t e r b o u n d e s t i m a t i o n for m o d e l s with b o u n d e d e r r o r s in all t h e v a r i a b l e s clearly stating w h e n tight p a r a m e t e r b o u n d s c a n b e o b t a i n e d a n d w h e n only o v e r b o u n d s can b e in g e n e r a l c o m p u t e d . In this last case the c r i t e r i o n for c h e c k i n g a posteriori w h e t h e r t h e o b t a i n e d b o u n d is tight or n o t is given.
posteriori.
2. Problem statement and notation L e t a system b e d e s c r i b e d by y* =q~*0
(1)
w h e r e y * ~ R m is t h e v e c t o r o f all the available n o i s e - f r e e o u t p u t samples, @ * ~ R mxp is t h e n o i s e - f r e e system m a t r i x ( r e g r e s s o r matrix) a n d 0 ~ R p is t h e u n k n o w n p a r a m e t e r vector. D u e to m e a s u r e m e n t e r r o r s t h e actually o b s e r v e d variables a r e y =y* + 6y, =
-
(2) (3)
w h e r e 6 y ~ R m a n d 6q~ ~ R m / p a r e t h e m e a s u r e m e n t e r r o r s on t h e o u t p u t s a m p l e s a n d on t h e r e g r e s s o r s respectively. F r o m (1), (2) a n d (3) o n e gets y = ( ~ + 6q~)O + 6 y .
(4)
This can b e r e w r i t t e n as Correspondence to: Prof. G. Belforte, Dipartimento di Auto-
matica e Informatica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Email: [email protected]. * This work was partially supported by the grants MURST40% and MURST-60% of the Italian Ministry of University and Scientific Research.
Y, = (4ff + ~b/T)o + ~SYi, i = 1 . . . . . m ,
(5)
w h e r e ~b/a"a n d 6~b/T d e n o t e t h e i-th r o w of q~ a n d 6q) respectively. It is a s s u m e d t h a t t h e available i n f o r m a t i o n o n t h e e r r o r is s e t - t h e o r e t i c , t h a t is 6 y a n d 6q~ a r e
016%6911/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
G. Belforte / Parameter identification for models with bounded errors
426
supposed to belong to given error sets sly and S2, 6y e sly,
6~ •
slo.
(6)
In this context, the identification p r o b l e m consists in finding the feasible p a r a m e t e r region D D={O•RP:
(12) and for the properties of the set D, the interested reader should refer to [2] and [3]. T h e i m p o r t a n c e of computing the p a r a m e t e r bounds (12) lies in the fact that they define a tight orthotopic o u t e r b o u n d B D to the set D, Bt~= { O e R P : omin ~ o i ~ O max, / = 1 . . . . . p}, (13)
y=(O+8@)O+Sy;
8y e Sly, 8@ • sl,~},
(7)
that is the set of all p a r a m e t e r s consistent with the system (4) and the m e a s u r e m e n t error information (6). No exact solution is available for this general formulation and even approximate solutions are hard to find [9]. However, useful results are available for the special case in which the error sets sly. and slm are equal to o r t h o t o p e s centred in the origin of the error space [2,6,8,10], so that
fly=Bv={SyeRm: laY~l<--Ay i,
i = 1 . . . . . m},
(8) s l , = B~
= {80 e R m × p : 184,ii i=l ..... m;j=l,...,p}
(9)
where 8yi, i = 1. . . . , m , are the c o m p o n e n t s of 8y and 84,ij, i = l . . . . , m , j = l . . . . . p, are the elements of 84, and Ayi, A4,ij, i = 1 , . . . , m, j = 1 . . . . , p, are known quantities. F r o m (5), (8) and (9) it can be derived that the feasible p a r a m e t e r region D is described by the following set of 2 m inequalities:
that is simpler to describe than the set D itself and yet proves to be highly informative and useful [7,10]. It results that 0~m a x (0 im i n ) is achieved on a vertex of D where at least p out of the 2m inequalities (10) are satisfied sharply. Using a terminology introduced in [7] we will refer to these p inequalities as the active constraints of max min Oi (0 i ). So far nothing new has been reported. T h e main points were already published in [1] and [2] and C e r o n e in [3] has given a formal p r o o f of the necessity and sufficiency of relation (10) for defining the set D for a system of the form (4) when the error sets are orthotopic according to (8) and (9). It is worth noting that the obtained results are no more valid if there is correlation a m o n g the errors 8y i, i = 1 , . . . , m, a n d / o r 84,ij, i = 1. . . . . m, j = 1. . . . . p, so that the corresponding error sets sly and slm are not equal to o r t h o t o p e s although they are b o u n d e d by orthotopes. In this case, only p a r a m e t e r overbounds can be c o m p u t e d in general and the occurrence of tight p a r a m e t e r bounds is a particular lucky case that can be only checked a posteriori. This fact is discussed in the next section where the criterion for checking the par a m e t e r ' s tightness is given.
- 44,7T)0 _
i= 1,...,m,
(10)
where /t4,TT = [d4,ilsgn(O1) ".. A4,ipSgn(Op) ] .
(11)
Using these inequalities it is possible to compute the p a r a m e t e r bounds O,~ x and 0 im i n , i = 1 . . . . . p, defined as max _
O~
-maxOi, OeD
min
O~
= m i n O i, O~D
3. The case of correlated errors
In this section, we consider the case in which all the errors are unknown but b o u n d e d and there is linear correlation a m o n g some error elements. Let 6e • R m×(p+l) be the vector of all the errors 6y i i = 1 . . . . . m and 84,ij i = 1 , . . . , m , j = 1. . . . . p. T h e error set D E = .Oy × s l , is defined as
OE = { 8 8 c R rnX(p+l)"
i= l ..... m.
(12) For the derivation of (10), for computational details about the evaluation of the p a r a m e t e r bounds
[Sy i] < d y i, i = l . . . . . m , 184,ijl _< a4,ij, i = 1 . . . . . m , j = 1 . . . . . p, b =Ae}
(14)
G. Belforte / Parameter identification for models with bounded errors
427
it results that D c D therefore B~ is an orthotopic overbound to D and B D C_B~. []
where b is a known vector and A is a given matrix of suitable dimensions describing all the existing linear correlations among the different errors. We notice that a special case of linear correlation is that of A R M A X and bilinear systems for which the system matrix q~ is constituted by the input and the output sample vectors suitably shifted. Let B e =By X Bq, be the tight orthotopic bound to /2 E defined as
Therefore the results presented in [4] and [5] where Cerone claims that tight p a r a m e t e r bounds for A R M A X and bilinear systems can be obtained are incorrect. In fact, only overbounds can be in general obtained for these systems. This fact can be easily illustrated by the following simple example consisting of an M A model described by
BE = {6e ~ R m x ( p + l ) "
y(k) = u(k)O, + u(k-
16Yil
i = 1. . . . . m,
1)0 2.
For this model three inputs are available: u ( 1 ) = 20, u ( 2 ) = 30 and u ( 3 ) = - 2 2 0 and the two output measurements are y ( 2 ) = 190 and y ( 3 ) = - 5 1 0 . The errors on the inputs are au(1) = 6u(2) = 3 u ( 3 ) = 5 and the errors on the outputs are 6y(2) = 6 y ( 3 ) = 5. In Figure 1 are reported both the true set D (obtained with a gridding of the errors) and the lines relative to relation (10) defining the set D. It can be seen that D c D so that B o c B ~ and in particular t h e 0~nin and 0~ ax relative to D do not belong to the D set. In [2] the occurrence of overbounding in presence of error correlation was already mentioned, but it was also stated that it is possible to determine a posteriori whether the obtained bound is tight or not only looking at the active constraints
] 6~bijl < A ~ i j , i = 1 . . . . . m , j = 1 . . . . , p}. (15) It turns out that 12e c B e ; /2 E C B e. It is then possible to state the following proposition:
Proposition 1. Let a system be described by (4) and let ~2e and B E be described by (14) and (15). Let D be the feasible parameter set obtained if I2~ = B e and let B~ be its orthotopic bound. Then B o G B~. Proof. B~ is, by construction, a tight orthotopic outbound to D; since D E c B E from definition (7)
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G. Belforte / Parameter identification for models" with bounded errors
of 0/rain (0imax) relative to the set D. it is indeed true that it is possible to check a p o s t e r i o r i the bound tightness, that is to state whether Bt) = B~ or B D c B~. However to answer this question all the available measurements must be used and not the active constraints only. The result can be obtained according to the following procedure. S t e p 1: Compute 0/rain (0/max) relative to the error orthotope B E (i.e. neglect the error correlation). S t e p 2: Substitute the parameter vector 0 for which 0rain (0 max) was obtained into relation (5). This leads to m equations in the 6 y i, i = 1 . . . . . m , and 649ii, i = 1 . . . . . m , j = 1 . . . . . p , variables. S t e p 3: Check whether the m equations obtained at Step 2 together with those defining the .OF. set in (14) have any admissible solution. I f yes." the obtained 0rain (0 max) is a tight bound to D; I f n o : 0~ in (0~ a~) is not a tight bound to D. In order for Bo to be tight to D so that B ~ = B o it is necessary that all the 0irain and 0 max, i = 1 , . . . , p , are tight to D. R e m a r k again that all the m equations obtained at Step 2 must be used at Step 3 and not only those relative to the active constraints.
4. Conclusive remarks
It is hard to assess, in general, whether the occurrence of tight p a r a m e t e r bounds is, in practical applications, more or less frequent. Clearly it depends on the error realization and therefore and analysis of this aspect, at this stage, seems to require a simulation study that is beyond the scope of this p a p e r and that could hardly give definite answers for the intrinsic nature of any simulation study. Also the influence of the number of measurements could, in principle, be relevant and could be investigated.
Finally it must be remarked that the exact solution could be numerically seeked solving a constrained minimization problem. However, since the solution domain is, in general, not convex and could even be not connected, since the number of involved variables (the parameters 0, the errors 6~bij and 6y i) can be quite large, it turns out that the numerical computation could be difficult and highly time consuming thus preventing any practical use.
References [1] G. Belforte, B. Bona and V. Cerone, A bounded error approach to the tuning of a digital voltmeter, 12th IMACS Congress on Scientific Computation, Paris, France (1988) 498-501. [2] G. Belforte, B. Bona and V. Cerone, Identification, structure selection and validation of uncertain models with set-membership error description, Math. and Comput. in Simulation 32 (1990) 561-569. [3] V. Cerone, Parameter bounds for models with bounded errors in all variables, Proc. 9th IFAC/IFORS Symposium on Identification and System Parameter Estimation Budapest, Hungary (1991) 1518-1523. [4] V. Cerone, Parameter bounding in A R M A X models from records with bounded errors in variables, Proc. 9th IFAC / IFORS Symposium on Identification and System Parameter Estimation, Budapest, Hungary (1991) 1518-1523. [5] V. Cerone, Feasible parameter set for bilinear systems from records with bounded output errors, Proc. 1991 American Control Conference Boston, MA (1991) 37-42. [6] E. Fogel and Y.F. Huang, On the value of information in system identification bounded noise case, Automatica 18 (1982) 229-238. [7] M. Milanese and G. Belforte, Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors. Linear families of models and estimators, IEEE Trans. Automat. Control 27 (1982) 408-414. [8] J.P. Norton, Identification and application of boundedparameter models, Automatica 23 (1987) 497-597. [9] F.C. Schweppe, Uncertain Dynamic Systems (Prentice Hall, Englewood Cliffs, NJ, 1973). [10] E. Walter and H. Piet-Lahanier, Estimation of parameter bounds from bounded-error data: a survey, Math. and Comput. in Simulation 32 (1990) 449-468.