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A new solution to the identification problem
Systems & Control Letters 2 (1983) 267-270 North-Holland Publishing Company
February 1983
A new solution to the identification problem Abduladhem ABDULKAREEM R. N A G A R A J A N
and
Department of Electrical Engineering, University of Basrah, Iraq Received 12 June 1982 Revised 8 September and 19 November 1982 A new parameter identification scheme for SISO discretetime plants is proposed. The scheme has a set of high-speed relays introduced with an objective to reduce on-line computational complexity. The stability of the scheme is ensured by hyperstability criteria. The proposed scheme is extended to MIMO plants.
require the availability of all state variables, with an a priori knowledge of their bounds, for implementation. The proposed scheme in this paper successfully extends the principle suggested in [3] for identifying the parameters of SISO linear noise-free discrete-time plants. The stability of the identification scheme is analysed through hyperstability criteria. The proposed scheme does not require a state vector for feedback, neither bounds of any state variable. The developed identification procedure is found applicable to a class of MIMO plants.
Keywords: Adaptive systems, Algorithm, Identification, MIMO plants, Model reference, Modelling.
2. Statement of identification problem I. Introduction A number of asymptotically stable parameter modification algorithms suitable for identification of multivariable systems is available in the current literature. The stability of these algorithms is ensured either through Lyapunov's direct method [ l] or through hyperstability criteria [2]. Most of these algorithms require accurate measurement of signals from plant and model. They also demand many on-line computations. Implementation of these algorithms requires complex hardware and large memory space of the on-line computer for software. Reduced on-line computations in the identification algorithm enable the plant signals to be sampled at a faster rate. Thus the computer with the same computing speed can be used for identification of faster responding plants. In addition the computer can be effectively time shared with a larger number of real time processes. Recently a new.class of adaptive systems useful for identification of discrete-time plants has been suggested [3]. These systems have a set of highspeed relays introduced with an objective to reduce the on-line computational complexity. The stability of these adaptive schemes has been guaranteed by Lyapunov's direct method. They
An n th order discrete-time completely controllable SISO plant is described by the time-series representation
y(k)=~aiy(k-i)+~biu(k-i) iffil
(1)
iffi0
where y(k) and u(k) are respectively the scalar output and input, al, i = 1, 2 . . . . . n, and b,., i = 0, 1, 2 . . . . . n, are the unknown parameters to be identified. Defining ~r=[a,
a 2 ...
a. bo b, . . .
b.]
(2)
as the (1 x f ) , f = 2n + 1, plant parameter vector, and zT(k)=[y(k-1)
,.,(k)
y(k-2) -..
,(k-,,)]
... y(k-n)
(3)
as the (1 x / ) information vector, (1) can be written as
y( k ) = gOVz(k ).
(4)
A series-parallel identification model is considered for determining 4' as
.~(k) -- ~r(k - 1)z(k)
0167-6911/83/0000-0000/$03.00 © 1983 North-Holland
(5) 267
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
It can be easily shown that the inequality (11) is satisfied when kl
I I o.l e r l. I I
b,~k
" E s*(k)s(k) <~A2o
I-
Fig. I. Autonomous nonliriear system.
where h 0 is a constant for all k~, and s*(k) is the sign function on s(k) such that
where 6 ( k ) and ~(k) are respectively the estimates of ~ and y(k). The dynamics of the output error
e( k ) = f i ( k ) - y ( k )
(7)
(8)
It is proposed to update 6 ( k ) from
O ( k ) = O ( k - I) + A O ( k - I)
(9)
so that 6 ( k ) is driven towards ~ as k increases. Using the parameter estimate at k, let us construct an estimate, s(k), on the output error as s(/,) = $ ~ ( k ) z ( k ) - y ( k ) = 0T(k)z(k).
-
i f s ( k ) > O' ifs(k) =0, ifs(k) <0.
(13)
Let us assume that O(k) is updated by (9) with
wherein Or(k) is the parameter-error matrix defined as O(k) = $ ( k ) - , .
(+i s*(k) =
(6)
is then characterised by
e( k ) = OX(k - 1)z(k)
(12)
k=O
(10)
It is noted that s(k) converges to zero as ¢ ( k ) is forced towards 0. Equation (10) can now be considered as a SISO autonomous nonlinear system as given in Fig. 1, with G ( z ) = 1. G(z) is a class of strictly positive real transfer functions. Now, it is required to determine an algorithm, (9), of the form proposed in [3], and to ensure that the overall identification scheme is hyperstable.
AO( k ) = --q ® [ s*( k )z( k )]
(14)
where
q=[q:>O],
j=l,2
..... :,
(15)
is an ( f x 1) gain vector, and ® represents the 'element by element matric product' operator [5]. Substitution of (10) in (12) leads to kl
~'~ s*( k )OX( k )z(k ) <~h2o.
(16)
k=O
The inequality (16) is arranged to be of f inequalities of the form kt
~_, ~.(k)s*(k)zj(k) <~X~
(17)
k=0
where X: is a constant for all k,; zj(k) and Oj(k) are respectivel3, the j t h components of z(k) and O(k),j-- l, 2 ..... f. Using (14) and (9), and knowing that A0S(0), the j t h element of A0(0), is zero, the inequality (17) can be simplified to be
E
l qJ k=O
h=O
3. The hyperstable algorithm One case of the Popovian theorem [4] states that, for the system presented in Fig. 1 to be hyperstable, the following inequality is to be valid:
-
[
Ag.(k) +¢(0)
?
(18)
kl
E s2(k) ~< ao2 k--0
where ao is a constant for all k,. 268
(11)
The inequality (18) is satisfied for any hj such that
¢2(0) 2qj
(19)
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
wherein 0(0) is an arbitrary value of O(k) at k = 0. The inequalities (17) and (19) hence ensure hyperstability of the proposed identification scheme.
4. Relation between
s(k) and e(k)
Determination of s*(k) from (10) requires a substantial number of computations at every k. An alternative way of obtaining s*(k) is as follows: Consider (1), (9), (10) and (14), from which one obtains / s(k) = e ( k ) - ~_, qjs*(k)z2(k). (20)
(21)
where ~(k) =
Is(k)l Is(k)l + E f_ lqjzf(k)"
(22)
Since a(k) is a positive sequence, s*(k) is easily determined from (21) as
s*(k) = e*(k).
(23)
The identification algorithm which updates ,#(k) is therefore given by 6(k+
1)=6(k)-q
® [e*(k)e(k)].
(24)
It is essential to note that the orthogonality between O(k - 1) and z(k) makes
e(k) = OT(k -- 1)z(k) = 0.
M I M O plant in state representation
x( k + 1) = Fx( k ) + gu( k ), y( k ) = Hx( k ),
(26) (27)
where u(k), y(k) and x(k) are respectively the (r X l) input vector, the (m × 1) output vector and the (n × 1) state vector. The matrices F, G and H are appropriately dimensioned. It is assumed that the triplet (F, G, .H) it completely controllable and completely observable, and hence the state representation is irreducible. For the case when rn = r > 1, the system description can be represented by A R M A form [7] y(k)-- ~Aiy(k-i)+ iffil
~'~Biu(k-i ).
i~0
(28)
It is further assumed that the system (28) is identifiable so that unique parameterisation is possible.
Rearrangement of (20) results in a relation
s(k) = a(k)e(k)
February 1983
(25)
From (25), e(k) is hence forced to be zero even when O(k)* O. This makes the scheme uniformly stable in the identification error (e, 0) space. When the plant input contains sufficient frequency components, uniform asymptotic stability of the scheme in (e, 0)-space is achieved, which ensures e(k) = 0 only when O(k) = 0 [1,6].
The structure of (28) is identical to that of (1). The suggested identification procedure can hence readily be extended to the M I M O plant,, (28). The parameter modifying algorithm, in this case, is obtained as ~ ( k + I) = ~ ( k ) - a ® [ e*(k)zT(k)]
(29)
where the e.~fimated parameter matrix ~ T ( k ) = [.~T( k ) 1 " " I-~( k)lBoT( k ) l . . . IBm( k)],
(30.) the information matrix zT(k) = [ y T ( k - - 1)1... lyT(k--n)[
uT(k)l ' - I : ( k -
.)],
(31)
every element of e*(k) is a sign function on the corresponding element of
e(k) = ~ ( k - 1 ) z ( k ) - y ( k )
(32)
and Q is an (m × f ) gain matrix, f = (2n + 1)m, having all its elements greater than zero.
6. Conclusion 5. MIMO plants
The proposed identification procedure is applied to a class of M I M O plants. Consider a
A new identification scheme for estimating the parameters of discrete-time plants is presented and its global st.ability is ensured by a hyperstability criterion. The proposed scheme employs a set of 269
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
high-speed relays which can be s i m u l a t e d b y software of the on-line c o m p u t e r . I n t r o d u c t i o n of relays in the i d e n t i f i c a t i o n scheme a n d the a b s e n c e o f division in the a l g o r i t h m result in a significant r e d u c t i o n in the n u m b e r of c o m p u t a t i o n s at every k, which in turn reduces the soft-ware complexity. It is expected that the suggested r e l a y - b a s e d algor i t h m will find a p p l i c a t i o n s where r e d u c t i o n in c o m p u t i n g b u r d e n a n d c o n s e r v a t i o n of m e m o r y space are of p r i m e concern, p a r t i c u l a r l y when mic r o c o m p u t e r s are e m p l o y e d in identifying a large n u m b e r of p a r a m e t e r s .
270
February 1983
References [1] P. Kudva and K.S. Narendra, An identification procedure for discrete multivariable systems, 1EEE Trans. Automat. Control 23 (1978) 726-729. [2] I.D. Landau, Synthesis of discrete model reference adaptive systems, I E E E Trans. Automat. Control 16 (1971) 507-508. [3] A. Abdulkarim, R. Nagarajan and K.B. Mirza, Discrete time MRAS with reduced implementation complexity, Electronics Letters 17 (1981) 809-810. [4] J.M. Martine-Sanchez, A new solution to adaptive control, Proc. I E E E 64 (1976) 1209-1218. [5]" R.L. Carroll, New adaptive algorithms in Lyapunov's synthesis, I E E E Trans. Automat. Control 21 (1978) 246-249. [6] J.M. Mendel, Discrete Techniques for Parameter Estimation (Marcel Dekker, New York, 1973). [7] G.C. Goodwin, P.J. Ramadge and P.E. Caines, Discrete time multivariable adaptive control, I E E E Trans. Automat. Control 25 (1980) 449-456.
February 1983
A new solution to the identification problem Abduladhem ABDULKAREEM R. N A G A R A J A N
and
Department of Electrical Engineering, University of Basrah, Iraq Received 12 June 1982 Revised 8 September and 19 November 1982 A new parameter identification scheme for SISO discretetime plants is proposed. The scheme has a set of high-speed relays introduced with an objective to reduce on-line computational complexity. The stability of the scheme is ensured by hyperstability criteria. The proposed scheme is extended to MIMO plants.
require the availability of all state variables, with an a priori knowledge of their bounds, for implementation. The proposed scheme in this paper successfully extends the principle suggested in [3] for identifying the parameters of SISO linear noise-free discrete-time plants. The stability of the identification scheme is analysed through hyperstability criteria. The proposed scheme does not require a state vector for feedback, neither bounds of any state variable. The developed identification procedure is found applicable to a class of MIMO plants.
Keywords: Adaptive systems, Algorithm, Identification, MIMO plants, Model reference, Modelling.
2. Statement of identification problem I. Introduction A number of asymptotically stable parameter modification algorithms suitable for identification of multivariable systems is available in the current literature. The stability of these algorithms is ensured either through Lyapunov's direct method [ l] or through hyperstability criteria [2]. Most of these algorithms require accurate measurement of signals from plant and model. They also demand many on-line computations. Implementation of these algorithms requires complex hardware and large memory space of the on-line computer for software. Reduced on-line computations in the identification algorithm enable the plant signals to be sampled at a faster rate. Thus the computer with the same computing speed can be used for identification of faster responding plants. In addition the computer can be effectively time shared with a larger number of real time processes. Recently a new.class of adaptive systems useful for identification of discrete-time plants has been suggested [3]. These systems have a set of highspeed relays introduced with an objective to reduce the on-line computational complexity. The stability of these adaptive schemes has been guaranteed by Lyapunov's direct method. They
An n th order discrete-time completely controllable SISO plant is described by the time-series representation
y(k)=~aiy(k-i)+~biu(k-i) iffil
(1)
iffi0
where y(k) and u(k) are respectively the scalar output and input, al, i = 1, 2 . . . . . n, and b,., i = 0, 1, 2 . . . . . n, are the unknown parameters to be identified. Defining ~r=[a,
a 2 ...
a. bo b, . . .
b.]
(2)
as the (1 x f ) , f = 2n + 1, plant parameter vector, and zT(k)=[y(k-1)
,.,(k)
y(k-2) -..
,(k-,,)]
... y(k-n)
(3)
as the (1 x / ) information vector, (1) can be written as
y( k ) = gOVz(k ).
(4)
A series-parallel identification model is considered for determining 4' as
.~(k) -- ~r(k - 1)z(k)
0167-6911/83/0000-0000/$03.00 © 1983 North-Holland
(5) 267
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
It can be easily shown that the inequality (11) is satisfied when kl
I I o.l e r l. I I
b,~k
" E s*(k)s(k) <~A2o
I-
Fig. I. Autonomous nonliriear system.
where h 0 is a constant for all k~, and s*(k) is the sign function on s(k) such that
where 6 ( k ) and ~(k) are respectively the estimates of ~ and y(k). The dynamics of the output error
e( k ) = f i ( k ) - y ( k )
(7)
(8)
It is proposed to update 6 ( k ) from
O ( k ) = O ( k - I) + A O ( k - I)
(9)
so that 6 ( k ) is driven towards ~ as k increases. Using the parameter estimate at k, let us construct an estimate, s(k), on the output error as s(/,) = $ ~ ( k ) z ( k ) - y ( k ) = 0T(k)z(k).
-
i f s ( k ) > O' ifs(k) =0, ifs(k) <0.
(13)
Let us assume that O(k) is updated by (9) with
wherein Or(k) is the parameter-error matrix defined as O(k) = $ ( k ) - , .
(+i s*(k) =
(6)
is then characterised by
e( k ) = OX(k - 1)z(k)
(12)
k=O
(10)
It is noted that s(k) converges to zero as ¢ ( k ) is forced towards 0. Equation (10) can now be considered as a SISO autonomous nonlinear system as given in Fig. 1, with G ( z ) = 1. G(z) is a class of strictly positive real transfer functions. Now, it is required to determine an algorithm, (9), of the form proposed in [3], and to ensure that the overall identification scheme is hyperstable.
AO( k ) = --q ® [ s*( k )z( k )]
(14)
where
q=[q:>O],
j=l,2
..... :,
(15)
is an ( f x 1) gain vector, and ® represents the 'element by element matric product' operator [5]. Substitution of (10) in (12) leads to kl
~'~ s*( k )OX( k )z(k ) <~h2o.
(16)
k=O
The inequality (16) is arranged to be of f inequalities of the form kt
~_, ~.(k)s*(k)zj(k) <~X~
(17)
k=0
where X: is a constant for all k,; zj(k) and Oj(k) are respectivel3, the j t h components of z(k) and O(k),j-- l, 2 ..... f. Using (14) and (9), and knowing that A0S(0), the j t h element of A0(0), is zero, the inequality (17) can be simplified to be
E
l qJ k=O
h=O
3. The hyperstable algorithm One case of the Popovian theorem [4] states that, for the system presented in Fig. 1 to be hyperstable, the following inequality is to be valid:
-
[
Ag.(k) +¢(0)
?
(18)
kl
E s2(k) ~< ao2 k--0
where ao is a constant for all k,. 268
(11)
The inequality (18) is satisfied for any hj such that
¢2(0) 2qj
(19)
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
wherein 0(0) is an arbitrary value of O(k) at k = 0. The inequalities (17) and (19) hence ensure hyperstability of the proposed identification scheme.
4. Relation between
s(k) and e(k)
Determination of s*(k) from (10) requires a substantial number of computations at every k. An alternative way of obtaining s*(k) is as follows: Consider (1), (9), (10) and (14), from which one obtains / s(k) = e ( k ) - ~_, qjs*(k)z2(k). (20)
(21)
where ~(k) =
Is(k)l Is(k)l + E f_ lqjzf(k)"
(22)
Since a(k) is a positive sequence, s*(k) is easily determined from (21) as
s*(k) = e*(k).
(23)
The identification algorithm which updates ,#(k) is therefore given by 6(k+
1)=6(k)-q
® [e*(k)e(k)].
(24)
It is essential to note that the orthogonality between O(k - 1) and z(k) makes
e(k) = OT(k -- 1)z(k) = 0.
M I M O plant in state representation
x( k + 1) = Fx( k ) + gu( k ), y( k ) = Hx( k ),
(26) (27)
where u(k), y(k) and x(k) are respectively the (r X l) input vector, the (m × 1) output vector and the (n × 1) state vector. The matrices F, G and H are appropriately dimensioned. It is assumed that the triplet (F, G, .H) it completely controllable and completely observable, and hence the state representation is irreducible. For the case when rn = r > 1, the system description can be represented by A R M A form [7] y(k)-- ~Aiy(k-i)+ iffil
~'~Biu(k-i ).
i~0
(28)
It is further assumed that the system (28) is identifiable so that unique parameterisation is possible.
Rearrangement of (20) results in a relation
s(k) = a(k)e(k)
February 1983
(25)
From (25), e(k) is hence forced to be zero even when O(k)* O. This makes the scheme uniformly stable in the identification error (e, 0) space. When the plant input contains sufficient frequency components, uniform asymptotic stability of the scheme in (e, 0)-space is achieved, which ensures e(k) = 0 only when O(k) = 0 [1,6].
The structure of (28) is identical to that of (1). The suggested identification procedure can hence readily be extended to the M I M O plant,, (28). The parameter modifying algorithm, in this case, is obtained as ~ ( k + I) = ~ ( k ) - a ® [ e*(k)zT(k)]
(29)
where the e.~fimated parameter matrix ~ T ( k ) = [.~T( k ) 1 " " I-~( k)lBoT( k ) l . . . IBm( k)],
(30.) the information matrix zT(k) = [ y T ( k - - 1)1... lyT(k--n)[
uT(k)l ' - I : ( k -
.)],
(31)
every element of e*(k) is a sign function on the corresponding element of
e(k) = ~ ( k - 1 ) z ( k ) - y ( k )
(32)
and Q is an (m × f ) gain matrix, f = (2n + 1)m, having all its elements greater than zero.
6. Conclusion 5. MIMO plants
The proposed identification procedure is applied to a class of M I M O plants. Consider a
A new identification scheme for estimating the parameters of discrete-time plants is presented and its global st.ability is ensured by a hyperstability criterion. The proposed scheme employs a set of 269
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
high-speed relays which can be s i m u l a t e d b y software of the on-line c o m p u t e r . I n t r o d u c t i o n of relays in the i d e n t i f i c a t i o n scheme a n d the a b s e n c e o f division in the a l g o r i t h m result in a significant r e d u c t i o n in the n u m b e r of c o m p u t a t i o n s at every k, which in turn reduces the soft-ware complexity. It is expected that the suggested r e l a y - b a s e d algor i t h m will find a p p l i c a t i o n s where r e d u c t i o n in c o m p u t i n g b u r d e n a n d c o n s e r v a t i o n of m e m o r y space are of p r i m e concern, p a r t i c u l a r l y when mic r o c o m p u t e r s are e m p l o y e d in identifying a large n u m b e r of p a r a m e t e r s .
270
February 1983
References [1] P. Kudva and K.S. Narendra, An identification procedure for discrete multivariable systems, 1EEE Trans. Automat. Control 23 (1978) 726-729. [2] I.D. Landau, Synthesis of discrete model reference adaptive systems, I E E E Trans. Automat. Control 16 (1971) 507-508. [3] A. Abdulkarim, R. Nagarajan and K.B. Mirza, Discrete time MRAS with reduced implementation complexity, Electronics Letters 17 (1981) 809-810. [4] J.M. Martine-Sanchez, A new solution to adaptive control, Proc. I E E E 64 (1976) 1209-1218. [5]" R.L. Carroll, New adaptive algorithms in Lyapunov's synthesis, I E E E Trans. Automat. Control 21 (1978) 246-249. [6] J.M. Mendel, Discrete Techniques for Parameter Estimation (Marcel Dekker, New York, 1973). [7] G.C. Goodwin, P.J. Ramadge and P.E. Caines, Discrete time multivariable adaptive control, I E E E Trans. Automat. Control 25 (1980) 449-456.