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Strictly Doubly Coprime Factorizations: The Discrete Time Case
Copyright © IFAC System Structure and Control. Nantes. France. 1995
STRICTLY DOUBLY COPRIME FACTORIZATIONS: THE DISCRETE TIME CASE P. HIPPE Institut fur Regelungstechnik, Universittit Erlangen-Nurnberg, Cauerstr. 7, D-91058 Erlangen, Germany
Abstract. Doubly coprime factorizations (DCF) are defined as stable proper rational matrices, and they are used in the fractional approach to all stabilizing compensators. The DCFs related to reduced order observers are partly improper, a problem which can be overcome by introducing artificial stable dynamics, the S
1. INTRODUCTION
also the possibly improper SDCFs can be used in the parametrization of all stabilizing compensators.
'Transfer matrices can either be represented in matrix fraction descriptions (MFD) or in doubly c
Though the continuous and the discrete time cases are closely related, the continuous time results presented in Hippe (1990) cannot directly be carried over to the discrete time case. Beyond that, the use of an innovated state estimate x+ for state feedback has no correspondence in the continuous time. First, the notations for the observer based compensation scheme in discrete time are introduced, and then the SDCFs related to reduced order observer based compensators are presented for both cases, namely when the one step prediction estimate, and when the innovated state estimate are used for state feedback .
By an introduction of additional stable factors, so-called identities in the ring of stable rational proper functions, also the DCFs related to reduced order observers can be made stable and proper (Telford and Moore, 1989; Antsaklis, 1986). These cancelling factors have no meaning in an observer based compensation scheme, they are contrary to the usual notion of coprimeness, and they are superfluous. This has been demonstrated in Hippe (1992) and Hippe (1995), where strictly doubly coprime factorizations (SDCF) without identities have been defined, and where it was shown, that
2. STATE FEEDBACK AND REDUCED ORDER OBSERVERS We consider completely controllable and observable linear time invariant discrete time systems z(k+ 1) y(k)
401
=
Az(k) + Bu(k)
Cz(k)
(1)
with uRn, yeRP, ueRm . Joining the Ko outputs to be used for the reconstruction of z in Y2, the output vector y is subdivided into
Yl (k)] = [ C Cl] z () y(k) = [ Y2(k) k 2
This transfer matrix can be represented in a left coprime matrix fraction description (MFD) Fe(z) = d l (z)!1 e (z) or in a right coprime MFD FcC z) = ne (z )d 1 (z). Similarly the system transfer matrix F(z) = C(zI _A)-l B can be represented in a right coprime MFD F(z) = n(z)d-l(z) or in a left coprime MFD F(z) = J-l(z)n(z) . Correspondingly the system and the compensator transfer matrices F(s) and Fe(s) can be represented in doubly cop rime fractional representations
c
(2)
with Yl eRP-I<, Y2eR"' . A reduced order observer of order n - K.
«k+1) = F«k)+[Hl H2J
[~~~!~ ]+T Bu(k)(3)
c
F(z) = N(z)D-l(z) = b-l(z)N(z)
(10)
and
yields «k) = Tz(k) in steady state, if
(4) where the eight rational matrices N, D , N, fJ, Ne, De, Ne, be have stable denominator polynomials (with roots inside the unit circle), and the Bezout identity
holds (Luenberger, 1971). If C 2 and T are linearly independent the state estimate x is given by
In the sequel it will be of importance that
holds.
(6) 3. STRICTLY DOUBLY COPRIME FACTORIZATIONS RELATED TO REDUCED ORDER OBSERVERS
and
(7)
A natural way to obtain the eight rational matrices in the DCF is to relate the polynomial matrices of the plant and of the compensator MFDs to the (stable) polynomial matrices characterizing the controlled plant and the observer dynamics (Hippe, 1989a and 1989b). In Nett et al. (1984) the case of identity observers was considered and consequently, all eight quantities in (10) and (11) were proper, since here, plant and observer/compensator orders coincide.
are implied by (5). With Ll such that TLl = Hl the observer (3) can equally be represented by
«k + 1)
= T(A -
L ICt}8«k)
+[TL 1:T(A- L ICt} W2][
~~~!~
]+TBU(k).
(8)
In the continuous time case (Hippe, 1990) a choice of Ll = 8H l , yielding C2L l = 0, was motivated by the optimal reduced order estimation scheme (reduced order Kalman filter (Hippe, 1989c)) . However, for discrete time systems the optimal reduced order estimation scheme does not imply C2Ll = 0, (Hippe and Wurmthaler, 1990) and consequently the following developments differ from the continuous time case. Substituting in the state feedback law u = -Kz the state z by the (one step prediction) estimate x (5), the compensator transfer matrix u(z) = -FcCz)y(z) , is given by
Fc(z) = K8 [zI - T(A - LICl
-
When using reduced order observer based compensators, the rational matrix fJ (which in the . scalar case is the plant denominator polynomial divided by the characteristic polynomial of the reduced order observer) is no longer proper (Hippe, 1989b and 1990). Therefore, attempts have been made to obtain proper factorizations also for reduced order observers (Telford and Moore, 1989; Antsaklis, 1986), and this was achieved by introducing cancelling (stable) factors, the so-called identities in the ring of proper stable rational matrices. The definition of DCFs allows for such factors (Vidyasagar, 1985), but they are definitely contrary to the usual notion of coprimeness, and they are superfluous. The continuous time factorizations introduced in (Hippe, 1989b and 1990)
BK)8J- l
[TL1:T(A-LICI-BK)W2]+[0 :KW2] '
(9)
402
Then
related to reduced order observers do not contain identities, and they are consequently coprime in the usual sense. Therefore, they have been named strictly doubly cop rime factorizations (SDCF) in Hippe (1992). Though not all SDCFs are proper, they can equally be used in the parametrization of all stabilizing compensators (Hippe, 1995).
(i) all eight matrices (13) - (20) have stable denominator polynomials (ii) D(z), D(z), Dc(z) and Dc(z) are nonsingular (iii) F(z) = N(z)D-l(z)
Here the continuous time results of Hippe (1990) shall be generalized to the discrete time case. The block diagrams of Hippe (1990) remain valid, if s-1 I is replaced by z-1 I. However, the actual results of Hippe (1990) cannot simply be transferred to the discrete time case by substituting s by z, because of the fact C2 L 1 i= 0 (see above). Following the lines of Hippe (1990), but assuming CzL1 f. 0, the strictly doubly coprime factorizations related to discrete time observers of arbitrary orders are given by Theorem 1.
(iv) Fc(z)
0].
Proof of Theorem 1: Part (i) follows from the fact that the zeros of det(zl - A + BK) and of det(zI - F)= det [zl - T(A - L 1 C 1 )e] lie within the unit circle of the z-plane. Part (ii): the matrices D(z) and Dc(z) are nonsingular because of the identity matrix appearing in both expressions. Because of C2W2 = I the polynomial parts of D(z) and bc(z) have nonvanishing determinants and consequently, also these matrices are nonsingular. The proofs for (Hi) through (v) can be developed in a similar fashion as in Hippe (1990) for the continuous time case, and they have to be omitted here for brevity.
=
=
C(zI - A + BK)-l B 1 - K(zl - A + BK)-l B
(13) (14)
Dc(z)
1 + Ke(zl - F)-lTB
(15)
Nc(z)
Ke(zl - F)-l [TLl:TAlW2]
(16)
N(z) D(z)
+
The main difference between the continuous and the discrete time cases stems from the fact that C2 L 1 = 0 does not hold in the discrete time case. The optimal linear estimator (Kalman Filter) in the continuous time case (Hippe, 1989c) is characterized by an optimal feedback matrix Ll which meets the condition C 2 L 1 = O. In Hippe and Wurmthaler (1990) it was shown, that this is not the case for the optimal discrete time estimator.
[0:KW2]
Though the rational matrix b(z) is no longer proper, the above factorizations (13)-(20) can equally be used in the parametrization of all stabilizing compensators by a simple modification of the parametrizing matrix (Hippe, 1992 and 1995).
D(z) ~ [ 1- C, S(z1 - F)-'TL, -C2 [I
+ A 18(zl - F)-IT] Ll
-C, [I + S( z1 - F)-'TA,]
'Ii']
= Dcl(z)Nc(z) = Nc(z)DZ/(z)
v [Dc..(Z) f!.c(Z)][D(Z) -?c(Z)]=[Im () -N(z) D(z) N(z) Dc(z) 0 Ip
Theorem 1. Suppose F(z) C(zI - A)-l B describes an nth order completely controllable and observable system with m inputs and p outputs, and Fc(z) as defined by (9) describes a reduced order compensator of order n - Ko for this system. TAle, where Al A - L l C 1, With K and F such that A - BK and F have eigenvalues inside the unit circle, define
=
= b-1(z)N(z)
(17)
C2 [zl - Al - A1e(zl - F)-lTAdw2 N(z)
=
[ C1e(zl - F)-lTB ] C 2 [I + A1e(zl - F)-IT] B
(18)
4. DOUBLY COPRIME FRACTIONAL REPRESENTATIONS RELATED TO THE UPDATED STATE ESTIMATE In the discrete time case the optimal feedback matrix Ll of the reduced order Kalman filter is such that
Nc(z)
K(zl - A
+ BK)-l [L1 : W2]
(19)
Dc(z)
C(zI - A
+ BK)-l [L1 : W2]
(20)
+
[ lp_"
0
(21)
~" ]
o holds
403
(Hippe and Wurmthaler,
Instead of i as defined above the updated state estimate
Then
(22)
(i) all eight matrices (24)-(31) have stable denominator polynomials
is often used (Gelb, 1974). Substituting this state estimate in the state feedback u -Kz, the compensator transfer matrix is given by
(ii) D+(z), b+(z), Dt:(z) and Dt:(z) are nonsingular
=
F6"(z) = K(I - '\lCde [zI - T(A - BK)
(I -'\lC1 F>]-l T (A-BK)[,\d I -'\lCl)W2]
(iii) F(z) = N+(z)D+-l(z) = jj+-l N+(z)
(23)
+ [K'\l: K (I - '\lCl )W2] . The corresponding DCFs are then given by the following theorem.
Dt:(z) N6(z) [ -N+(z) b+(z)
Theorem 2. Given an nth order completely controllable and observable linear system with m inputs and p outputs, with its discrete transfer matrix F(z) = C(zI _A)-l B, and an (n-K)th order (optimal) observer whose updated state estimate is fed back via u = -Ki+ yielding the reduced order compensator transfer matrix (23). Now select K and F TA A6 with AA A(I - '\lCl ) such that the eigenvalues of A - BK and F lie within the unit circle of the z-plane.
=
=
+ BK)-l B
I - K(zI - A
Dt:(z) = I
=
+ K(I -
(25)
[TA'\l: TAAW2]+K['\I: (I-'\lCl)W2] (27)
:
+ AA6(zI -F)-lTA].xl :
: -Cl [I + 6(zI - F)-lT AA] W2
1
(28)
-+
_ [C 1 e(ZI-F)-lTB
N6"(z)
=
cd I + AA6(zI _ F)-IT] B
bt:(z)
] (29)
[~-~ ~~ ] .
1.
• contain polynomials of smallest possible degrees, which is desirable, as numerical problems rapidly increase with polynomial degree.
(30)
= C(zI -A+BK)-l [(A-BK».l: W2] +
0 Ip
• directly appear in the frequency domain design of observer based compensators (Hippe, 1991)
K(zI -A+BK)-l [(A-BK)'\1:W2] + [K'\l :0]
o
• are directly related to the two dual closed loop representations of observer based compensation schemes (Hippe, 1991)
: C2 [zI - AA - AA6(zI - F)-lTA A] W2
N (z) -
[Im
Starting from the continuous time results, strictly doubly coprime representations have been presented for discrete time systems. They cover the case of arbitrary observer orders n - K within the range o < K < p. For K = 0 observer and system orders coincide and consequently, all factorizations are proper and stable. For K > 0 some fractional quantities become strictly proper and others (in corresponding parts) become improper. This, however, does not cause any problems in the parametrization of all stabilizing compensators, when the parametrizing matrix is modified in an appropriate manner (Hippe, 1992 and 1995). Compared to DCFs containing identities, such SDCFs have the following benefits, as they
N6"(z) = K(I - '\lcde(zI - F)-l
-C2 [A
Dt:(z)
5. CONCLUSIONS
'\lCd6(zI - F)-lTB (26)
b+(z)= [I - Cle(zI -F)-lTA'\l
1
The proof of this Theorem goes along the same lines as the proof for Theorem 1.
(24)
+ BK)-l B
N+(z)
-N6"(z)
(v)
=
C(zI - A
1[D+(Z)
Therefore they should always be preferred to factorizations containing identities.
(31)
404
6. REFERENCES
Hippe, P. (1992) . Strictly doubly cop rime factorizations related to reduced order observers. 2nd IFAC Workshop on System Structure and Control, 3.-5.9.92, Prague. Hippe, P. (1995). Strictly doubly coprime factorizations and all stabilizing compensators related to reduced order observers. Automatica, 31, to appear. Hippe, P., and Wurmthaler, Chr. (1990). Optimal reduced order estimators in the frequency domain: The discrete time case. Int. J. Control 52, 1051-1064. Luenberger, D.G. (1971) . An introduction to observers. IEEE Trans. Automat. Contr. AC16, 596-603. Nett, C.N., Jacobson C.A. and Balas, M.J . (1984) . A Connection between state-space and doubly coprime fractional representations. IEEE Trans. Automat. Contr., AC-29, 831-832 . Telford, A.J. and Moore, J .B. (1989) . Doubly coprime factorizations, reduced order observers and dynamic state estimate feedback. Int. J. Control 50, 2583-2597. Vidyasagar, M. (1985). Control system synthesis: A factorization approach. Cambridge, Mass.
Antsaklis, P.J. (1986). Proper stable transfer matrix factorizations and internal system descriptions. IEEE Trans. Automat. ControlS1, 634638. Gelb, A. (1974). Applied Optimal Estimation. Cambridge, Mas.: The MIT Press. Hippe, P. (1989a). The computation of doubly ccr prime fractional representations directly in the frequency domain. IEEE Conference on Control and Applications - ICCON '89, Jerusalem, April 3-6, RA-2-5. Hippe, P. (1989b) . Modified doubly coprime fractional representations related to the reduced order observer, IEEE Trans . Autom. Control., AC-34, 573-576. Hippe, P. (1989c) . Design of reduced order optimal estimators directly in the frequency dcr main. Int. J. Control, 50, 2599- 2614. Hippe, P. (1990). A nonminimal representation of reduced order observers. Automatica 26, 405409. Hippe, P. (1991) . Design of observer based compensators: The polynomial approach. Kybernetika, 27, 125-150.
405
STRICTLY DOUBLY COPRIME FACTORIZATIONS: THE DISCRETE TIME CASE P. HIPPE Institut fur Regelungstechnik, Universittit Erlangen-Nurnberg, Cauerstr. 7, D-91058 Erlangen, Germany
Abstract. Doubly coprime factorizations (DCF) are defined as stable proper rational matrices, and they are used in the fractional approach to all stabilizing compensators. The DCFs related to reduced order observers are partly improper, a problem which can be overcome by introducing artificial stable dynamics, the S
1. INTRODUCTION
also the possibly improper SDCFs can be used in the parametrization of all stabilizing compensators.
'Transfer matrices can either be represented in matrix fraction descriptions (MFD) or in doubly c
Though the continuous and the discrete time cases are closely related, the continuous time results presented in Hippe (1990) cannot directly be carried over to the discrete time case. Beyond that, the use of an innovated state estimate x+ for state feedback has no correspondence in the continuous time. First, the notations for the observer based compensation scheme in discrete time are introduced, and then the SDCFs related to reduced order observer based compensators are presented for both cases, namely when the one step prediction estimate, and when the innovated state estimate are used for state feedback .
By an introduction of additional stable factors, so-called identities in the ring of stable rational proper functions, also the DCFs related to reduced order observers can be made stable and proper (Telford and Moore, 1989; Antsaklis, 1986). These cancelling factors have no meaning in an observer based compensation scheme, they are contrary to the usual notion of coprimeness, and they are superfluous. This has been demonstrated in Hippe (1992) and Hippe (1995), where strictly doubly coprime factorizations (SDCF) without identities have been defined, and where it was shown, that
2. STATE FEEDBACK AND REDUCED ORDER OBSERVERS We consider completely controllable and observable linear time invariant discrete time systems z(k+ 1) y(k)
401
=
Az(k) + Bu(k)
Cz(k)
(1)
with uRn, yeRP, ueRm . Joining the Ko outputs to be used for the reconstruction of z in Y2, the output vector y is subdivided into
Yl (k)] = [ C Cl] z () y(k) = [ Y2(k) k 2
This transfer matrix can be represented in a left coprime matrix fraction description (MFD) Fe(z) = d l (z)!1 e (z) or in a right coprime MFD FcC z) = ne (z )d 1 (z). Similarly the system transfer matrix F(z) = C(zI _A)-l B can be represented in a right coprime MFD F(z) = n(z)d-l(z) or in a left coprime MFD F(z) = J-l(z)n(z) . Correspondingly the system and the compensator transfer matrices F(s) and Fe(s) can be represented in doubly cop rime fractional representations
c
(2)
with Yl eRP-I<, Y2eR"' . A reduced order observer of order n - K.
«k+1) = F«k)+[Hl H2J
[~~~!~ ]+T Bu(k)(3)
c
F(z) = N(z)D-l(z) = b-l(z)N(z)
(10)
and
yields «k) = Tz(k) in steady state, if
(4) where the eight rational matrices N, D , N, fJ, Ne, De, Ne, be have stable denominator polynomials (with roots inside the unit circle), and the Bezout identity
holds (Luenberger, 1971). If C 2 and T are linearly independent the state estimate x is given by
In the sequel it will be of importance that
holds.
(6) 3. STRICTLY DOUBLY COPRIME FACTORIZATIONS RELATED TO REDUCED ORDER OBSERVERS
and
(7)
A natural way to obtain the eight rational matrices in the DCF is to relate the polynomial matrices of the plant and of the compensator MFDs to the (stable) polynomial matrices characterizing the controlled plant and the observer dynamics (Hippe, 1989a and 1989b). In Nett et al. (1984) the case of identity observers was considered and consequently, all eight quantities in (10) and (11) were proper, since here, plant and observer/compensator orders coincide.
are implied by (5). With Ll such that TLl = Hl the observer (3) can equally be represented by
«k + 1)
= T(A -
L ICt}8«k)
+[TL 1:T(A- L ICt} W2][
~~~!~
]+TBU(k).
(8)
In the continuous time case (Hippe, 1990) a choice of Ll = 8H l , yielding C2L l = 0, was motivated by the optimal reduced order estimation scheme (reduced order Kalman filter (Hippe, 1989c)) . However, for discrete time systems the optimal reduced order estimation scheme does not imply C2Ll = 0, (Hippe and Wurmthaler, 1990) and consequently the following developments differ from the continuous time case. Substituting in the state feedback law u = -Kz the state z by the (one step prediction) estimate x (5), the compensator transfer matrix u(z) = -FcCz)y(z) , is given by
Fc(z) = K8 [zI - T(A - LICl
-
When using reduced order observer based compensators, the rational matrix fJ (which in the . scalar case is the plant denominator polynomial divided by the characteristic polynomial of the reduced order observer) is no longer proper (Hippe, 1989b and 1990). Therefore, attempts have been made to obtain proper factorizations also for reduced order observers (Telford and Moore, 1989; Antsaklis, 1986), and this was achieved by introducing cancelling (stable) factors, the so-called identities in the ring of proper stable rational matrices. The definition of DCFs allows for such factors (Vidyasagar, 1985), but they are definitely contrary to the usual notion of coprimeness, and they are superfluous. The continuous time factorizations introduced in (Hippe, 1989b and 1990)
BK)8J- l
[TL1:T(A-LICI-BK)W2]+[0 :KW2] '
(9)
402
Then
related to reduced order observers do not contain identities, and they are consequently coprime in the usual sense. Therefore, they have been named strictly doubly cop rime factorizations (SDCF) in Hippe (1992). Though not all SDCFs are proper, they can equally be used in the parametrization of all stabilizing compensators (Hippe, 1995).
(i) all eight matrices (13) - (20) have stable denominator polynomials (ii) D(z), D(z), Dc(z) and Dc(z) are nonsingular (iii) F(z) = N(z)D-l(z)
Here the continuous time results of Hippe (1990) shall be generalized to the discrete time case. The block diagrams of Hippe (1990) remain valid, if s-1 I is replaced by z-1 I. However, the actual results of Hippe (1990) cannot simply be transferred to the discrete time case by substituting s by z, because of the fact C2 L 1 i= 0 (see above). Following the lines of Hippe (1990), but assuming CzL1 f. 0, the strictly doubly coprime factorizations related to discrete time observers of arbitrary orders are given by Theorem 1.
(iv) Fc(z)
0].
Proof of Theorem 1: Part (i) follows from the fact that the zeros of det(zl - A + BK) and of det(zI - F)= det [zl - T(A - L 1 C 1 )e] lie within the unit circle of the z-plane. Part (ii): the matrices D(z) and Dc(z) are nonsingular because of the identity matrix appearing in both expressions. Because of C2W2 = I the polynomial parts of D(z) and bc(z) have nonvanishing determinants and consequently, also these matrices are nonsingular. The proofs for (Hi) through (v) can be developed in a similar fashion as in Hippe (1990) for the continuous time case, and they have to be omitted here for brevity.
=
=
C(zI - A + BK)-l B 1 - K(zl - A + BK)-l B
(13) (14)
Dc(z)
1 + Ke(zl - F)-lTB
(15)
Nc(z)
Ke(zl - F)-l [TLl:TAlW2]
(16)
N(z) D(z)
+
The main difference between the continuous and the discrete time cases stems from the fact that C2 L 1 = 0 does not hold in the discrete time case. The optimal linear estimator (Kalman Filter) in the continuous time case (Hippe, 1989c) is characterized by an optimal feedback matrix Ll which meets the condition C 2 L 1 = O. In Hippe and Wurmthaler (1990) it was shown, that this is not the case for the optimal discrete time estimator.
[0:KW2]
Though the rational matrix b(z) is no longer proper, the above factorizations (13)-(20) can equally be used in the parametrization of all stabilizing compensators by a simple modification of the parametrizing matrix (Hippe, 1992 and 1995).
D(z) ~ [ 1- C, S(z1 - F)-'TL, -C2 [I
+ A 18(zl - F)-IT] Ll
-C, [I + S( z1 - F)-'TA,]
'Ii']
= Dcl(z)Nc(z) = Nc(z)DZ/(z)
v [Dc..(Z) f!.c(Z)][D(Z) -?c(Z)]=[Im () -N(z) D(z) N(z) Dc(z) 0 Ip
Theorem 1. Suppose F(z) C(zI - A)-l B describes an nth order completely controllable and observable system with m inputs and p outputs, and Fc(z) as defined by (9) describes a reduced order compensator of order n - Ko for this system. TAle, where Al A - L l C 1, With K and F such that A - BK and F have eigenvalues inside the unit circle, define
=
= b-1(z)N(z)
(17)
C2 [zl - Al - A1e(zl - F)-lTAdw2 N(z)
=
[ C1e(zl - F)-lTB ] C 2 [I + A1e(zl - F)-IT] B
(18)
4. DOUBLY COPRIME FRACTIONAL REPRESENTATIONS RELATED TO THE UPDATED STATE ESTIMATE In the discrete time case the optimal feedback matrix Ll of the reduced order Kalman filter is such that
Nc(z)
K(zl - A
+ BK)-l [L1 : W2]
(19)
Dc(z)
C(zI - A
+ BK)-l [L1 : W2]
(20)
+
[ lp_"
0
(21)
~" ]
o holds
403
(Hippe and Wurmthaler,
Instead of i as defined above the updated state estimate
Then
(22)
(i) all eight matrices (24)-(31) have stable denominator polynomials
is often used (Gelb, 1974). Substituting this state estimate in the state feedback u -Kz, the compensator transfer matrix is given by
(ii) D+(z), b+(z), Dt:(z) and Dt:(z) are nonsingular
=
F6"(z) = K(I - '\lCde [zI - T(A - BK)
(I -'\lC1 F>]-l T (A-BK)[,\d I -'\lCl)W2]
(iii) F(z) = N+(z)D+-l(z) = jj+-l N+(z)
(23)
+ [K'\l: K (I - '\lCl )W2] . The corresponding DCFs are then given by the following theorem.
Dt:(z) N6(z) [ -N+(z) b+(z)
Theorem 2. Given an nth order completely controllable and observable linear system with m inputs and p outputs, with its discrete transfer matrix F(z) = C(zI _A)-l B, and an (n-K)th order (optimal) observer whose updated state estimate is fed back via u = -Ki+ yielding the reduced order compensator transfer matrix (23). Now select K and F TA A6 with AA A(I - '\lCl ) such that the eigenvalues of A - BK and F lie within the unit circle of the z-plane.
=
=
+ BK)-l B
I - K(zI - A
Dt:(z) = I
=
+ K(I -
(25)
[TA'\l: TAAW2]+K['\I: (I-'\lCl)W2] (27)
:
+ AA6(zI -F)-lTA].xl :
: -Cl [I + 6(zI - F)-lT AA] W2
1
(28)
-+
_ [C 1 e(ZI-F)-lTB
N6"(z)
=
cd I + AA6(zI _ F)-IT] B
bt:(z)
] (29)
[~-~ ~~ ] .
1.
• contain polynomials of smallest possible degrees, which is desirable, as numerical problems rapidly increase with polynomial degree.
(30)
= C(zI -A+BK)-l [(A-BK».l: W2] +
0 Ip
• directly appear in the frequency domain design of observer based compensators (Hippe, 1991)
K(zI -A+BK)-l [(A-BK)'\1:W2] + [K'\l :0]
o
• are directly related to the two dual closed loop representations of observer based compensation schemes (Hippe, 1991)
: C2 [zI - AA - AA6(zI - F)-lTA A] W2
N (z) -
[Im
Starting from the continuous time results, strictly doubly coprime representations have been presented for discrete time systems. They cover the case of arbitrary observer orders n - K within the range o < K < p. For K = 0 observer and system orders coincide and consequently, all factorizations are proper and stable. For K > 0 some fractional quantities become strictly proper and others (in corresponding parts) become improper. This, however, does not cause any problems in the parametrization of all stabilizing compensators, when the parametrizing matrix is modified in an appropriate manner (Hippe, 1992 and 1995). Compared to DCFs containing identities, such SDCFs have the following benefits, as they
N6"(z) = K(I - '\lcde(zI - F)-l
-C2 [A
Dt:(z)
5. CONCLUSIONS
'\lCd6(zI - F)-lTB (26)
b+(z)= [I - Cle(zI -F)-lTA'\l
1
The proof of this Theorem goes along the same lines as the proof for Theorem 1.
(24)
+ BK)-l B
N+(z)
-N6"(z)
(v)
=
C(zI - A
1[D+(Z)
Therefore they should always be preferred to factorizations containing identities.
(31)
404
6. REFERENCES
Hippe, P. (1992) . Strictly doubly cop rime factorizations related to reduced order observers. 2nd IFAC Workshop on System Structure and Control, 3.-5.9.92, Prague. Hippe, P. (1995). Strictly doubly coprime factorizations and all stabilizing compensators related to reduced order observers. Automatica, 31, to appear. Hippe, P., and Wurmthaler, Chr. (1990). Optimal reduced order estimators in the frequency domain: The discrete time case. Int. J. Control 52, 1051-1064. Luenberger, D.G. (1971) . An introduction to observers. IEEE Trans. Automat. Contr. AC16, 596-603. Nett, C.N., Jacobson C.A. and Balas, M.J . (1984) . A Connection between state-space and doubly coprime fractional representations. IEEE Trans. Automat. Contr., AC-29, 831-832 . Telford, A.J. and Moore, J .B. (1989) . Doubly coprime factorizations, reduced order observers and dynamic state estimate feedback. Int. J. Control 50, 2583-2597. Vidyasagar, M. (1985). Control system synthesis: A factorization approach. Cambridge, Mass.
Antsaklis, P.J. (1986). Proper stable transfer matrix factorizations and internal system descriptions. IEEE Trans. Automat. ControlS1, 634638. Gelb, A. (1974). Applied Optimal Estimation. Cambridge, Mas.: The MIT Press. Hippe, P. (1989a). The computation of doubly ccr prime fractional representations directly in the frequency domain. IEEE Conference on Control and Applications - ICCON '89, Jerusalem, April 3-6, RA-2-5. Hippe, P. (1989b) . Modified doubly coprime fractional representations related to the reduced order observer, IEEE Trans . Autom. Control., AC-34, 573-576. Hippe, P. (1989c) . Design of reduced order optimal estimators directly in the frequency dcr main. Int. J. Control, 50, 2599- 2614. Hippe, P. (1990). A nonminimal representation of reduced order observers. Automatica 26, 405409. Hippe, P. (1991) . Design of observer based compensators: The polynomial approach. Kybernetika, 27, 125-150.
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