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Robust stabilization for composite observer-based control of discrete systems
Automatica. Vol.30. No. 5. pp. 877-881, 1994
Copyright(~) 1994ElsevierScienceLtd Printed in Grcat Britain. All rights reserved (NXDS-1098194 $7.00+ 0.00
Pergamon
Brief Paper
Robust Stabilization for Composite Observer-based Control of Discrete Systems* VAN-TSAI LIU~" and CHUN-LIANG LIN~ Key Woods--Discrete time systems; model reduction; observers; robust control; stability criteria.
Aladraet--Design criteria of robust observer-based controllers for linear, discrete-time, singularly perturbed systems with slow and fast modes are considered. The developed stability measure complements the quantitative information for unmodeled fast modes that cannot be provided by the usual qualitative robustness analyses. A numerical example is provided to show the analytical results obtained.
controller. New robust stability conditions used for estimating quantitatively permissible perturbation bounds are derived by applying the perturbation theory. A numerical example is presented to confirm our proposed approach. 2. System models
The linear, shift-invariant, discrete-time singularly perturbed system under consideration is described in a slow-time-scale as follows:
1. Introduction
SINGULARPERTURBATIONAPPROACHplays a powerful role in practical system designs, since many physical systems possess multiple time scales, and a simple control structure is obtainable through this method. With this approach, the high dimensionality and ill-conditioning resulting from the interaction of slow and fast dynamic modes are alleviated. The standard model reduction procedure for a singularly perturbed system reduced to a lower order system is to discard its fast mode subsystem. However, as a feedback controller is designed with respect to the reduced-order model, neglected parasitic modes may sometimes excite the nominal design model to cause a net destabilizing effect. This is known as the robustness problem, and has been discussed previously (Khalil, 1981; O'Relly, 1986). The stability issue of singularly perturbed systems has been extensively investigated in the literature (Kokotovic and Khalil, 1986). Recently, there has been interest in studying robust stability of the control of multiple-time-scale systems (Sandell, 1979, Khalil, 1981; Abed and Tits, 1986; O'Relly, 1986; Esfandiari and Khalil, 1989). For the discrete case, a general discussion for the stability of observer-based control was considered by Mahmoud (1982a) and OIoomi and Sawan (1987). The robustness of output feedback control methods to unmodeled high-frequency dynamics was studied by Mahmoud and Singh (1985) and Esfandiari and Khalil (1989). Among the papers cited, most of the methods used were focused on the qualitative aspects of stabilizing controller designs. Quantitative results regarding this issue are still lacking. With regard to quantitative robustness analyses, it is necessary to have a mechanism to measure the size of the uncertainty. This paper provides such a mechanism. We consider here a general situation where a high-order system is in closed-loop with a reduced- or full-order observer-based
x ( k + l ) = A t x ( k ) + A 2 z ( k ) + B t u ( k ),
x ( O ) = x o (la)
z(k + 1) = A3x(k) + A4z(k) + B2u(k),
z(0) = zo (lb)
y(k) = C t x ( k ) + C2z(k).
(lc)
where x(k)¢ R "l is the slow state vector, z(k)¢ R "2 is the fast state vector, u(k) ~ R m is the control vector, y(k) 6 R p is the output vector. In general, system (1) has a typical two-time-scale property, i.e. its eigenvalues consist of n I eigenvalues that are large in magnitude, and n2 eigenvalues that are small in magnitude with respect to the unit circle. Therefore, after a short initial period, the influence of the fast modes will be negligible and the behavior of the system will be dominated by the slow modes alone. It is known that while the following matrix inequality holds O(A4) << _a(A, + A2(1,2 - A4)-'A3),
(2)
where O(-) denotes the largest singular value; ~(-), the smallest singular value, then system (1) has the timeseparation property. For the case of systems formulated by a fast-time-scale this is equivalent to making the singular perturbation parameter E sufficiently small. Now let In2- A4 be invertible, the separated slow and fast subsystems can be obtained as follows (Mahmoud, 1982b; Fernando and Nicholson, 1983): xs(k + 1) = Aoxs(k) + Bous(k )
(3a)
y~(k) = Cox,(k) + D0u.~(k)
(3b)
zf(k + 1) = A4zt(k ) + B2ut(k) yt(k) = C2z~k),
(4a) (4b)
where A o = A 1 + m2(/n2 - A4)- IA3, B 0 = B I + A2(1.2 - A4)- In 2 Co = CI + C2(I,2 - A4)-IA3, Do = C2(1.2 - A4)-IB2
*Received 22 November 1991; revised 28 October 1992; revised 25 June 1993; received in final form 20 July 1993. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor M. Ikeda under the direction of Editor A. P. Sage. Corresponding author V.-T. Liu. Tel. +886 56329643-651; Fax +886 56328863; E-mail [email protected]. t Department of Electrical Engineering, National Yunlin Polytechnic Institute, Huwei, Yunlin, Taiwan 632, R.O.C. ~tChung Shan Institute of Science and Technology, Lungtan, Taoyuan, Taiwan 325, R.O.C.
u(k) = us(k) + uf(k). The control vector u(k) has a slow component u.~(k) which drives the slow subsystem (3), and a fast component uf(k) which drives the fast subsystem (4). Note that for asymptotically stable fast modes, the assumption for/,2 - A4 being invertible is always valid. 3. Reduced-order observer-based control
Under the assumption that the fast modes being asymptotically stable, a reduced-order observer-based 877
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Brief Papers
controller designed for the reduced system (3) is given by ~,(k + 1) = , 4 o ~ ( k ) +
Bou,(k)
+ Ko[y~(k) - Co$,(k) - Dou~(k)l,
$,(0) = 0 (5a)
u~(k) = Go~(k)
(5b)
e~(k) = x,(k) - $~(k),
(5c)
where $,(k) is the estimate of x,(k), e~(k) is the state reconstruction error, the (nt x p ) observer gain matrix Ko and the (m x n0 control gain matrix Go are to be determined. The corresponding governing equation for the reduced-order, composite, closed-loop system is w(k + 1) = Aow(k)
Theorem 1. Consider the actual closed-loop system (7) and the reduced-order, closed-loop system (6) with the matrices Ao and A4 being stable, the actual system under control will be asymptotically stable, if
IlqJ(z)ll IIAn(z) - AH(1)I[ < 1,
(10a)
V0 • [0, 2:t],
(10b)
or equivalently, II~(ei°)ll IIAH(ej°) - AH(1)II < 1, where [ (zlnl - A t ) - t(zl,,t - Ac)(zl,1 - A,~)-' ql(z ) = L Go(zlnl - A,.)- IgoCo(zlnl - A e)- 1
(6)
(zl,.,, - a , ) - ' B o G o ( z l , , - A ¢ ) - t K , , ] Go(zl, t - A , ) - ' ( z l , , - Ao)(zln, - A ¢ ) - ' g , , J
where
A, - o ol and w(k) = [x~(k) eT(k)] T • g 2hI, A r A o + BoGo, A e = Ao - KoCo. If I;t(A,)l < 1 and I~.(Ae)l < 1 then matrix Ao is obviously stable. It should be mentioned that as Go and Ko are designed to stabilize system (3), they may not guarantee the actual dused-loop stability. In the actual system, x,(k), u~(k), y~(k), and e~(k) in (5) should be replaced, respectively, by x(k), u(k), y(k) and e(k). Therefore, the composite actual closed-loop system can be obtained as follows:
Vz • D
with A,: = A o + B o G o - K o C o . matrix
(1 la)
and the perturbation transfer
=
.~(k + 1) = AH~(k ) + A t2~.(k)
(7a)
~.(k + 1) = A2t.~(k ) + A22z(Ik),
(7b)
where A 11~
['AI + B1G° L
-F
-BIG°
1
A,-KoC~+F]'
A2! -- [A 3 + B ~ G o - B2Gol.
[
A2
1,
A~2=LA2-KoCzJ
Proof. Taking z-transform of the actual closed-loop system
(7), yields
.Z(z ) = (zl.2 - A22)- ' A 2 , f ( (z ) + z(zl.2 - Az2)- I£'(0), (12b)
where $(0) and $(0) are. respectively, the initial states of ~(k) and ~(k). Substituting (12b) into (12a) then solving for X(z). we have f ( ( z ) = zK-'(z).~(O) + z K - l ( z ) A , 2 ( z l . z - A22)-1~.(0),
A H + A 12(1.2 - A22 ) - IA2, = Ao.
r = (A2 - KoCz)(l,2 - A4)-I(A3 + B2Go)
and $(k)= [xT(k)eT(k)] T, ~'(k)= z(k). With regard to the closed-loop stability of (7), the following lemma is recalled from Mahmoud (1982a). (a) 12(A4)1<1, (b) (Ao, Bo) is a stabilizable pair, (c) (Ao, Co) is a detectable pair, then controller (5) is a stabilizing one for system (1). The gains Go and Ko are any matrices such that A~ and A~ are stable. • The usefulness of Lemma 1 lies in identifying qualitative stability conditions for the closed-loop systems having strong mode-separation properties. For systems lack of strong mode-separation feature, the subsequent theorem will give an explicit quantitative condition to ensure the loop stability. Before stating the main result, some technical lemmas forming the foundation of later development will be given first. The maximum modulus principle gives the following results (Vidyasagar, 1985). L e m m a 2. Let a matrix E ( z ) • ~ × "
with ~ × " denoting the set of m x n matrices whose elements are proper stable rational functions [i.e. the poles of all elements of E ( z ) are within the unit disc], then
K ( z ) = zlz, t - A i i
(14)
- A I:,(l,-, - A:,..)-IA21
+ A lz(l,2 -- A22 ) - IA., 1 - A i.,(zl.., - A-,.,)-IA21 = (zi2. t - Ao){/2.1 + (z12., - Ao)-'Ai2[(l.2 - A22)-I - (zl,-, - A-,-,)-']A2, } = (z12,,, - Ao)A,(z),
(15)
where At(z ) =/2,, + (zl':,.i - Ao)-'At2[(l.2 - A:,2) -I - ( z l , , : A22)-I]A21. Substitution of (15) into (13) results in X(z) = Ar'(z)(zl=., - Ao)-'z~(O)
+ A t I(z)(zl-,,,t - A~,)-IA t2z(zl,,2 - Az,)-1£(0),
(16)
where A ~ t ( z ) = [Iz,, + (zI2,.,,- A o ) - t [ A 2 ? K o C . , ]
x [(In: - A4)-'(zln2 - A4)- '] x [A3+ B 2 G o - B.,Go]] =[12,1+(z12,
sup IIE(eJ°)ll o~1o, 2.-r]
(13)
Using (14) and by some algebraic rearrangements. K ( z ) in (13) can be further expanded as
L e m m a 1. If system (1) exhibits the following properties:
=
(12a)
z f ( ( z ) = A H f ( ( z ) + zi(O) + A , 2 Z ( z )
in which K ( z ) = z12. i - AII -- A,z(zI,,2 - A22)- IA2,. It is easy to show that the following matrix identity holds:
A22 = A 4,
su~,~IIE(z)ll = I~l~,supIIE(z)ll
(lib)
AH(z)=[A:](zI~2-A4)-'[A3B2].
_1~,)_1[~:
_OKo][C;] ][m'l
(8) (9)
where D ~ {z = re j° • C, 0 • [0, 2~], Irl ~> 1}. Since E ( z ) is analytic for z ¢ D, this norm is well defined. • The discrete version of the small gain theorem can be described by: L e m m a 3. If E ( z ) ¢ ~t"~×~ and IIE(z)ll < 1, '¢ Izl ~> 1, then [1. - E ( z ) ] - ' • ~ ' ~ " .
x [(1,2- A4)-' - ( z l , 2 - A 4 ) ~1 0 -I x J a ~ B2].[ 1"' - G o ] ] "
tGo
(17)
Since (zlz,,t - Ao) tz • ~ , × 2 , , , and z(zl,, e - A 4 ) - ' • ~2×,,,,, therefore, for bounded initial states $(0) and £(0) in (13), to stabilize .~(k) we only need to find conditions which ensure A [ I ( z ) ( z l 2 , , - A o ) - 1 • ~ 2n i ×2n I . Using the following matrix inversion identity !1 + Z(z)Q]-' = t - Z(z)[t + Q:~(z)l-'~
Brief Papers
whereX(z) = (z/znl - Ao)_|[/I::
(zl.2-A4)-t][a~B2]
_0K.o][A;][(In2_ A 4 ) - I o] -G/
and t3= [1"' [Go
879
If the maximum perturbation bound is known in advance, i.e. s~p IIAH(ej") AH(1)ll -< ~', -
then Art(z) then (21) can be expressed in a standard form of H®-norm inequality as follows:
can be expanded as A?t(z) = 12.t - (z12. ~- 1%)-'A~2[(I... - A4) -l
sup IIW(ei°)ll = sup II-=(eJ~/2., - Ao)-'rlll < 1 . Y
- (zl.2 - A , ) - qIa~Bd •
By the result given by Xie et al. (1991). this inequality will hold. if and only if there exists a positive definite matrix X = X T such that
where ~ ( z ) = =-(zl~., - Ao)- ~rl with
A~(X- ~- }'2rll-lr)- tAo - X + -=v.X< 0
r,., _k], i,=r'o, -- --
_°o]
,,.,
and
or equivalently A~XAo - X + r ~ x n ( t . , + p
A H ( z ) - AH(1)= [A:](zl.2- A4)-'[A3B2]
- ),2IItXl-l)-'If"XAo + ='r= < 0 for InI+p-y2HIXH>O. The above algebraic Riccati equation provides a useful way to test the proposed stability inequality.
Using the matrix inversion formula to evaluate (zl2.t/%)-', it is straightforward to show that ql(z) is equivalent to • • (lla). Smce • ~2nl X2n t and the- form g,ven m (z/2.,- Ao)--I e ~® (zl.2-A4)-~e~t"= 2"n2, our attention is focused on finding conditions such that the term {Im+.l-Ud(z)[AH(z) AH(I)]} -j is analytic for all z • D. By Lemma 3 we know that if the following inequality holds: IIW(z)ll IIAH(z) - A n ( I ) , < 1, Vlzl >/1
as follows: (a) If a unit step signal excites the perturbation transfer matrix AH(z), the corresponding steady state output (bias) can be obtained by the final value theorem as lim ___~z. (z - 1)- All(z) = AH(1).
Z~ I Z--I
(19)
or, by Lemmas 2 and 3. IIW(ei°)ll IIAH(eis) - AH(1)II < 1, V0 • [0, 2~], (20) then ATl(z)(zlzm- Ao) -t • ~ , × 2 . , . This implies that ~(k) is stable. From ('12b). since (zl.2- A4)-t e ~,,2×-2 and .~(k) has been stabilized, thus, ~.(k) should be stabilized as well. These results also imply that the perturbed eigenvalues of A4 and Ao will not be brought out the unit circle. The proof is thus completed. • Note that if the chosen gains Go and Ko satisfy inequality (10), the overall system will be stabilized in the presence of the high-frequency parasitics (fast modes). Since no approximation was used in the proof above, Theorem 1 is more general than Lemma 1, in the sense that it can handle the class of discrete systems in which the controlled slow and uncontrolled fast modes are not strongly separable. Furthermore, the gains Go and Ko do not couple with the high-frequency perturbation AH(z), thus, # ~ - I I ~ ( z ) l l can be viewed as a robustness measure of the nominal design system. With this specific advantage, one can use inequality (10) to estimate a permissible perturbation upper bound. To do this, express the proposed stability condition (10b) in an alternative form as follows: IIAH(eja) - AH(1)II < ll~(eJ°)ll-' - #t,
Remark. Asymptotic properties of I}AH(ei°) -AH(1)II are
V0 • [0, 2~]. (21)
One proceeds to plot a frequency-dependent singular value curve of #~ based on the chosen control gains. The envelope of #~ then indicates the permissible upper bound of undesired perturbations under which the overall system could not be destabilized. The result of Theorem 1 is also attractive because in contrast to the qualitative analyses, it can lead to an optimal robust control problem. If Go and Ko are chosen to maximize
Therefore, the perturbation bound IIAH(ei°)-AH(1)II is unbiased. (b) IIAH(eJO) - AH(I)II---~0 as 0---*0 and 2dr. Also due to symmetry with respect to 0, the stability inequality (10b) will naturally hold within certain finite frequency band (0, ~r).
4. Full-order observer-based control In this section the problem of constructing observers to estimate the slow and fast states is considered. The fast dynamics is now treated as a modeled subsystem. A full-order observer-based controller designed for system (1) is given by £(k + I) = A l.~(k) + Az~(k ) + Btu(k ) + r,[y(k) - C : ( k ) - C : ( k ) l
(22a)
t(k + 1) = A3~(k) + A4£(k) + B2u(k) + K2[y(k) - CrY(k) - C2~(k)] u(k) = Gt~(k ) + G2~(k),
(22h) (22c)
where K~ and K s a r e the observer gains; Gj and G2, the regulator gains. In terms of the observation error vectors e,(k) = x(k) -,e(k) and el(k) = z(k) - $.(k), the actual closedloop system can be partitioned as
where
A,~ = [A, +:~G~ ~,2=[A2+:tG2
A 2 - K, C2/
/i, -= [s~p II~(eJ~)ll]-' subject to
fi,2, = I t 3 +0B2G,
(a) lit(Ao)l < 1. (b) s~p IIAH(e i#) - AH(1)II < Or,
0
•
[0, 2~r],
the resulting system will achieve maximum robustness under our design constraints.
-B~G~ ], A t - K , CtJ -BIG2 ]
A22 = [ A4 "1":2G2
-B.G,
]
A3- k,.clj' --
B2G,
]
A , - K2CzJ and .~(k)= [xT(k) eT(k)]T, ~(k)= [zT(k) efT(k)]T. If (/n2A4 )- i exists,and the composite regulator/observer gains are
880
Brief Papers
Gt = [/,,1 - Gt(l,2-A4)-IBz]Go- Gf(I,2- A4)-IA3, G2 = Gf (24a) and
proposed to construct a stabilizing composite observer. We have now considered a more general situation where a composite feedback controller is also involved in the loop. In addition, Theorem 2 provides a numerically checkable condition to identify the overall system stability.
K, =/Co[/p+ C2(I.2 - Aa)- 'Kf] - A2(I.2 - A4)- 'Kf, K2= Kf.
Remark. (a) If A4 is a stable matrix, then Gf = 0 and Kf = 0
chosen as (OIoomi and Sawan, 1987)
are admissible choices. For this case, one should use Theorem 1 to check the loop stability. (b) The separated subsystem matrices A~ and Af can be obtained via the usual quasisteady state approach or matrix decoupling technique (Mahmoud, 1982b). Since the actual closed-loop eigenvalues of (23) are perturbations from ).(A~) and ~,(Af), the closed-loop system will be asymptotically stable for the corresponding time separation ratio being small. To ensure the loop stability, inequality (26) can be used to replace this small time separation ratio assumption.
(24b) then the separated slow subsystem matrix As of (23) is given by .~ll+/~12(12n2--.~22)-12d[21=[ Mr
-B°(~°I=A, A, J
(25a)
where A r and A~ are defined as in (6), and
Go = Go + GtO- Gf(I,2- Aa - B2Gt)-'(A.s- B2GtO) with
5. Example
0 = (/.2 - A4 + KfC2)-I(A3 -- KfCI).
To confirm the main result, we present a simple example in which a reduced-order observer-based controller is adopted to control a two-time-scale system. Consider a fourth-order discrete model described by
The separated fast subsystem matrix Af is
t{22 = [ A4 +oB2Gf
A4
- _ B2Gf ] KfC2 j =
(25b)
Af.
Sufficient conditions for the stabilization of the closed-loop system (23) under this composite controller are established in the following theorem.
a'=[~i~
~i~]'
f0.15 A4=[0.0 ~i~]'
Theorem 2. If the gains (G I , (];2) and (Kp g2) are designed
A2=[00:005 -00:11]' m3=[O.0i~ Bt=[~I~
0.0 1.0]'
B2=[01~
0.030"0],
001~]'
such that As and Af are stable in the discrete sense, then the actual closed-loop system (23) will also be asymptotically stable, provided II(z12,I - As)-IA,2[O(z) - ~(1)]-1¢i2111 < 1,
Vz ~ D
Choosing the following control gains
(26a)
or equivalently,
G,,=
II(ei°12,1 A~)-1.412[~(ei° ) - *(1)]- IA2111< 1, V0 • [0, 2~t] (26b) where
11--0'3238 -0.0032] [-0.0050
Ko = [3.70620.1488] 1_1.1424 3.05271
-0.02591'
-
(I)(z)
=
(ZZ2n 2 -- A22)
the controlled poles will be assigned at 0.4715 and 0.6700, i.e. both lie within the unit circle. Using these results, the robustness condition (21) can be check to satisfy. We find
-I
is the resolvent matrix of ,,{22.
intlt4,
Proof. From (25), it is easy to see that 3.(A~) ~ MAr) O X(Ae) and X(Af)=-X(A4+BzGt)U~.(A4-KtC2). As the gain matrices G~, G2, K,, K2 are chosen as indicated in (24) such
(zt2.,
-
&)-'A,2[~(z)
at 0 = 0.6912 (rad).
See Fig. 1 for a graphical singular valued illustration of the robustness test. Therefore, the chosen set of control gains is permissible, i.e. robust stability of the overall closed-loop system could be ensured. Output responses (with initial states of the controlled subsystem setting to ones and the remaining states setting to zeros) for the system are shown in Figures 2 and 3.
that I~.(A~)I< 1 and I;t(Ae)l < 1 then the developed analysis procedure in Theorem 1 can be applied. The system matrices All , AI2 , m2t , A22, and ~ correspond, respectively, to Au, At2, A2~, A~. and Ao in Theorem 1. It directly follows from (12)-(17) that system (23) will be stable if, and only if, a ~ ' ( z ) = (12., -
- O(AH(e j°) - AH(1))] = 0.0893,
- ¢(1)1-'A2,)-'
• ~,x2.~.
6. Conclusions New frequency domain stability conditions for a high-order discrete model controlled by a reduced- or full-order observer-based controller are derived. The present approach allows one to estimate a permissible upper bound of the high-frequency residuals under which the controlled system
By Lemmas 2 and 3, the stability inequalities (26a) and (26b) are sufficient conditions to guarantee this requirement. • The quantitative result of Theorem 2 complements the one given in Mahmoud (1982a) in which a qualitative approach is
0.B
,
,
,
,
,
c~ >
0.4 ~D c~
0.2 ~ 0
G-[AH(e )-AH(1)]
.......
....- . - ~ _ _ ~
~
0
1
2
3 Complex
4 Angle
(rad)
FIG. 1. Robustness test.
5
6
Brief Papers 0.1 e~ o
l',,, ~ c o n t
0.o5
~
0
rolled controlled
0
50 Step FIG. 2. Yl response.
0.1
0 e~
ee
0.05
controlled
0 controlled
-0.05 0
50 Step FIG. 3. Y2 response.
will remain stable. Additional topics remain to be investigated including extension of the current results to systems formulated by a fast-time-scale, and existence of stabilizing controllers which make the proposed stability criteria hold.
881
References Abed, E. H. and A. L. Tits (1986). On the stability of multiple time-scale systems. Int. J. Control, 44, 211-218. Esfandiari, F. and H. K. KhalU (1989). On the robustness of samplod-data control to unmodeled high-frequency dynamics. IEEE Trans. Aut. Control, AC-34, 900-903. Fernando, K. V. and H. Nicholson (1983). Singular perturbational approximations for discrete-time balanced systems. IEEE Trans. Aut. Control, AC-28, 240-242. Khalil, H. K. (1981). On the robustness of output feedback control methods to modeling errors. IEEE Trans. Aut. Control, AC-26, 524-526. Kokotovic, P. V. and H. K. Khalil (1986). Singular Perturbations in System and Control. IEEE Press. Mahmoud, M. S. (1982a). Design of observer-based controllers for a class of discrete systems. Automatica, 18, 323-328. Mahmoud, M. S. (1982b). Order reduction and control of discrete systems. Proc. lEE, 129, 129-135. Mahmoud, M. S. and M. G. Singh (1985). On the use of reduced-order models in output feedback design of discrete systems. Automatica, 21, 485-489. OIoomi, H. and M. E. Sawan (1987). The observer-based controller design of discrete-time singularly perturbed systems. IEEE Trans. Aut. Control, AC-32, 246--248. O'Relly, J. (1986). Robustness of linear feedback control systems to unmodeled high-frequency dynamics. Int. J. Control, 44, 1077-1088. Sandell, N. R., Jr (1979). Robust stability of systems with application to singular perturbations. Automatica, 15, 467-470. Vidyasagar, M. (1985). Control System Synthesis: A Factorization Approach. MIT Press, Cambridge, MA. Xie, L., C. E. de Souza and M. Fu (1991). H~ estimation for discrete-time linear uncertain systems. Int. J. Robust Nonlinear Control, 1, 111-123.
Copyright(~) 1994ElsevierScienceLtd Printed in Grcat Britain. All rights reserved (NXDS-1098194 $7.00+ 0.00
Pergamon
Brief Paper
Robust Stabilization for Composite Observer-based Control of Discrete Systems* VAN-TSAI LIU~" and CHUN-LIANG LIN~ Key Woods--Discrete time systems; model reduction; observers; robust control; stability criteria.
Aladraet--Design criteria of robust observer-based controllers for linear, discrete-time, singularly perturbed systems with slow and fast modes are considered. The developed stability measure complements the quantitative information for unmodeled fast modes that cannot be provided by the usual qualitative robustness analyses. A numerical example is provided to show the analytical results obtained.
controller. New robust stability conditions used for estimating quantitatively permissible perturbation bounds are derived by applying the perturbation theory. A numerical example is presented to confirm our proposed approach. 2. System models
The linear, shift-invariant, discrete-time singularly perturbed system under consideration is described in a slow-time-scale as follows:
1. Introduction
SINGULARPERTURBATIONAPPROACHplays a powerful role in practical system designs, since many physical systems possess multiple time scales, and a simple control structure is obtainable through this method. With this approach, the high dimensionality and ill-conditioning resulting from the interaction of slow and fast dynamic modes are alleviated. The standard model reduction procedure for a singularly perturbed system reduced to a lower order system is to discard its fast mode subsystem. However, as a feedback controller is designed with respect to the reduced-order model, neglected parasitic modes may sometimes excite the nominal design model to cause a net destabilizing effect. This is known as the robustness problem, and has been discussed previously (Khalil, 1981; O'Relly, 1986). The stability issue of singularly perturbed systems has been extensively investigated in the literature (Kokotovic and Khalil, 1986). Recently, there has been interest in studying robust stability of the control of multiple-time-scale systems (Sandell, 1979, Khalil, 1981; Abed and Tits, 1986; O'Relly, 1986; Esfandiari and Khalil, 1989). For the discrete case, a general discussion for the stability of observer-based control was considered by Mahmoud (1982a) and OIoomi and Sawan (1987). The robustness of output feedback control methods to unmodeled high-frequency dynamics was studied by Mahmoud and Singh (1985) and Esfandiari and Khalil (1989). Among the papers cited, most of the methods used were focused on the qualitative aspects of stabilizing controller designs. Quantitative results regarding this issue are still lacking. With regard to quantitative robustness analyses, it is necessary to have a mechanism to measure the size of the uncertainty. This paper provides such a mechanism. We consider here a general situation where a high-order system is in closed-loop with a reduced- or full-order observer-based
x ( k + l ) = A t x ( k ) + A 2 z ( k ) + B t u ( k ),
x ( O ) = x o (la)
z(k + 1) = A3x(k) + A4z(k) + B2u(k),
z(0) = zo (lb)
y(k) = C t x ( k ) + C2z(k).
(lc)
where x(k)¢ R "l is the slow state vector, z(k)¢ R "2 is the fast state vector, u(k) ~ R m is the control vector, y(k) 6 R p is the output vector. In general, system (1) has a typical two-time-scale property, i.e. its eigenvalues consist of n I eigenvalues that are large in magnitude, and n2 eigenvalues that are small in magnitude with respect to the unit circle. Therefore, after a short initial period, the influence of the fast modes will be negligible and the behavior of the system will be dominated by the slow modes alone. It is known that while the following matrix inequality holds O(A4) << _a(A, + A2(1,2 - A4)-'A3),
(2)
where O(-) denotes the largest singular value; ~(-), the smallest singular value, then system (1) has the timeseparation property. For the case of systems formulated by a fast-time-scale this is equivalent to making the singular perturbation parameter E sufficiently small. Now let In2- A4 be invertible, the separated slow and fast subsystems can be obtained as follows (Mahmoud, 1982b; Fernando and Nicholson, 1983): xs(k + 1) = Aoxs(k) + Bous(k )
(3a)
y~(k) = Cox,(k) + D0u.~(k)
(3b)
zf(k + 1) = A4zt(k ) + B2ut(k) yt(k) = C2z~k),
(4a) (4b)
where A o = A 1 + m2(/n2 - A4)- IA3, B 0 = B I + A2(1.2 - A4)- In 2 Co = CI + C2(I,2 - A4)-IA3, Do = C2(1.2 - A4)-IB2
*Received 22 November 1991; revised 28 October 1992; revised 25 June 1993; received in final form 20 July 1993. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor M. Ikeda under the direction of Editor A. P. Sage. Corresponding author V.-T. Liu. Tel. +886 56329643-651; Fax +886 56328863; E-mail [email protected]. t Department of Electrical Engineering, National Yunlin Polytechnic Institute, Huwei, Yunlin, Taiwan 632, R.O.C. ~tChung Shan Institute of Science and Technology, Lungtan, Taoyuan, Taiwan 325, R.O.C.
u(k) = us(k) + uf(k). The control vector u(k) has a slow component u.~(k) which drives the slow subsystem (3), and a fast component uf(k) which drives the fast subsystem (4). Note that for asymptotically stable fast modes, the assumption for/,2 - A4 being invertible is always valid. 3. Reduced-order observer-based control
Under the assumption that the fast modes being asymptotically stable, a reduced-order observer-based 877
878
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controller designed for the reduced system (3) is given by ~,(k + 1) = , 4 o ~ ( k ) +
Bou,(k)
+ Ko[y~(k) - Co$,(k) - Dou~(k)l,
$,(0) = 0 (5a)
u~(k) = Go~(k)
(5b)
e~(k) = x,(k) - $~(k),
(5c)
where $,(k) is the estimate of x,(k), e~(k) is the state reconstruction error, the (nt x p ) observer gain matrix Ko and the (m x n0 control gain matrix Go are to be determined. The corresponding governing equation for the reduced-order, composite, closed-loop system is w(k + 1) = Aow(k)
Theorem 1. Consider the actual closed-loop system (7) and the reduced-order, closed-loop system (6) with the matrices Ao and A4 being stable, the actual system under control will be asymptotically stable, if
IlqJ(z)ll IIAn(z) - AH(1)I[ < 1,
(10a)
V0 • [0, 2:t],
(10b)
or equivalently, II~(ei°)ll IIAH(ej°) - AH(1)II < 1, where [ (zlnl - A t ) - t(zl,,t - Ac)(zl,1 - A,~)-' ql(z ) = L Go(zlnl - A,.)- IgoCo(zlnl - A e)- 1
(6)
(zl,.,, - a , ) - ' B o G o ( z l , , - A ¢ ) - t K , , ] Go(zl, t - A , ) - ' ( z l , , - Ao)(zln, - A ¢ ) - ' g , , J
where
A, - o ol and w(k) = [x~(k) eT(k)] T • g 2hI, A r A o + BoGo, A e = Ao - KoCo. If I;t(A,)l < 1 and I~.(Ae)l < 1 then matrix Ao is obviously stable. It should be mentioned that as Go and Ko are designed to stabilize system (3), they may not guarantee the actual dused-loop stability. In the actual system, x,(k), u~(k), y~(k), and e~(k) in (5) should be replaced, respectively, by x(k), u(k), y(k) and e(k). Therefore, the composite actual closed-loop system can be obtained as follows:
Vz • D
with A,: = A o + B o G o - K o C o . matrix
(1 la)
and the perturbation transfer
=
.~(k + 1) = AH~(k ) + A t2~.(k)
(7a)
~.(k + 1) = A2t.~(k ) + A22z(Ik),
(7b)
where A 11~
['AI + B1G° L
-F
-BIG°
1
A,-KoC~+F]'
A2! -- [A 3 + B ~ G o - B2Gol.
[
A2
1,
A~2=LA2-KoCzJ
Proof. Taking z-transform of the actual closed-loop system
(7), yields
.Z(z ) = (zl.2 - A22)- ' A 2 , f ( (z ) + z(zl.2 - Az2)- I£'(0), (12b)
where $(0) and $(0) are. respectively, the initial states of ~(k) and ~(k). Substituting (12b) into (12a) then solving for X(z). we have f ( ( z ) = zK-'(z).~(O) + z K - l ( z ) A , 2 ( z l . z - A22)-1~.(0),
A H + A 12(1.2 - A22 ) - IA2, = Ao.
r = (A2 - KoCz)(l,2 - A4)-I(A3 + B2Go)
and $(k)= [xT(k)eT(k)] T, ~'(k)= z(k). With regard to the closed-loop stability of (7), the following lemma is recalled from Mahmoud (1982a). (a) 12(A4)1<1, (b) (Ao, Bo) is a stabilizable pair, (c) (Ao, Co) is a detectable pair, then controller (5) is a stabilizing one for system (1). The gains Go and Ko are any matrices such that A~ and A~ are stable. • The usefulness of Lemma 1 lies in identifying qualitative stability conditions for the closed-loop systems having strong mode-separation properties. For systems lack of strong mode-separation feature, the subsequent theorem will give an explicit quantitative condition to ensure the loop stability. Before stating the main result, some technical lemmas forming the foundation of later development will be given first. The maximum modulus principle gives the following results (Vidyasagar, 1985). L e m m a 2. Let a matrix E ( z ) • ~ × "
with ~ × " denoting the set of m x n matrices whose elements are proper stable rational functions [i.e. the poles of all elements of E ( z ) are within the unit disc], then
K ( z ) = zlz, t - A i i
(14)
- A I:,(l,-, - A:,..)-IA21
+ A lz(l,2 -- A22 ) - IA., 1 - A i.,(zl.., - A-,.,)-IA21 = (zi2. t - Ao){/2.1 + (z12., - Ao)-'Ai2[(l.2 - A22)-I - (zl,-, - A-,-,)-']A2, } = (z12,,, - Ao)A,(z),
(15)
where At(z ) =/2,, + (zl':,.i - Ao)-'At2[(l.2 - A:,2) -I - ( z l , , : A22)-I]A21. Substitution of (15) into (13) results in X(z) = Ar'(z)(zl=., - Ao)-'z~(O)
+ A t I(z)(zl-,,,t - A~,)-IA t2z(zl,,2 - Az,)-1£(0),
(16)
where A ~ t ( z ) = [Iz,, + (zI2,.,,- A o ) - t [ A 2 ? K o C . , ]
x [(In: - A4)-'(zln2 - A4)- '] x [A3+ B 2 G o - B.,Go]] =[12,1+(z12,
sup IIE(eJ°)ll o~1o, 2.-r]
(13)
Using (14) and by some algebraic rearrangements. K ( z ) in (13) can be further expanded as
L e m m a 1. If system (1) exhibits the following properties:
=
(12a)
z f ( ( z ) = A H f ( ( z ) + zi(O) + A , 2 Z ( z )
in which K ( z ) = z12. i - AII -- A,z(zI,,2 - A22)- IA2,. It is easy to show that the following matrix identity holds:
A22 = A 4,
su~,~IIE(z)ll = I~l~,supIIE(z)ll
(lib)
AH(z)=[A:](zI~2-A4)-'[A3B2].
_1~,)_1[~:
_OKo][C;] ][m'l
(8) (9)
where D ~ {z = re j° • C, 0 • [0, 2~], Irl ~> 1}. Since E ( z ) is analytic for z ¢ D, this norm is well defined. • The discrete version of the small gain theorem can be described by: L e m m a 3. If E ( z ) ¢ ~t"~×~ and IIE(z)ll < 1, '¢ Izl ~> 1, then [1. - E ( z ) ] - ' • ~ ' ~ " .
x [(1,2- A4)-' - ( z l , 2 - A 4 ) ~1 0 -I x J a ~ B2].[ 1"' - G o ] ] "
tGo
(17)
Since (zlz,,t - Ao) tz • ~ , × 2 , , , and z(zl,, e - A 4 ) - ' • ~2×,,,,, therefore, for bounded initial states $(0) and £(0) in (13), to stabilize .~(k) we only need to find conditions which ensure A [ I ( z ) ( z l 2 , , - A o ) - 1 • ~ 2n i ×2n I . Using the following matrix inversion identity !1 + Z(z)Q]-' = t - Z(z)[t + Q:~(z)l-'~
Brief Papers
whereX(z) = (z/znl - Ao)_|[/I::
(zl.2-A4)-t][a~B2]
_0K.o][A;][(In2_ A 4 ) - I o] -G/
and t3= [1"' [Go
879
If the maximum perturbation bound is known in advance, i.e. s~p IIAH(ej") AH(1)ll -< ~', -
then Art(z) then (21) can be expressed in a standard form of H®-norm inequality as follows:
can be expanded as A?t(z) = 12.t - (z12. ~- 1%)-'A~2[(I... - A4) -l
sup IIW(ei°)ll = sup II-=(eJ~/2., - Ao)-'rlll < 1 . Y
- (zl.2 - A , ) - qIa~Bd •
By the result given by Xie et al. (1991). this inequality will hold. if and only if there exists a positive definite matrix X = X T such that
where ~ ( z ) = =-(zl~., - Ao)- ~rl with
A~(X- ~- }'2rll-lr)- tAo - X + -=v.X< 0
r,., _k], i,=r'o, -- --
_°o]
,,.,
and
or equivalently A~XAo - X + r ~ x n ( t . , + p
A H ( z ) - AH(1)= [A:](zl.2- A4)-'[A3B2]
- ),2IItXl-l)-'If"XAo + ='r= < 0 for InI+p-y2HIXH>O. The above algebraic Riccati equation provides a useful way to test the proposed stability inequality.
Using the matrix inversion formula to evaluate (zl2.t/%)-', it is straightforward to show that ql(z) is equivalent to • • (lla). Smce • ~2nl X2n t and the- form g,ven m (z/2.,- Ao)--I e ~® (zl.2-A4)-~e~t"= 2"n2, our attention is focused on finding conditions such that the term {Im+.l-Ud(z)[AH(z) AH(I)]} -j is analytic for all z • D. By Lemma 3 we know that if the following inequality holds: IIW(z)ll IIAH(z) - A n ( I ) , < 1, Vlzl >/1
as follows: (a) If a unit step signal excites the perturbation transfer matrix AH(z), the corresponding steady state output (bias) can be obtained by the final value theorem as lim ___~z. (z - 1)- All(z) = AH(1).
Z~ I Z--I
(19)
or, by Lemmas 2 and 3. IIW(ei°)ll IIAH(eis) - AH(1)II < 1, V0 • [0, 2~], (20) then ATl(z)(zlzm- Ao) -t • ~ , × 2 . , . This implies that ~(k) is stable. From ('12b). since (zl.2- A4)-t e ~,,2×-2 and .~(k) has been stabilized, thus, ~.(k) should be stabilized as well. These results also imply that the perturbed eigenvalues of A4 and Ao will not be brought out the unit circle. The proof is thus completed. • Note that if the chosen gains Go and Ko satisfy inequality (10), the overall system will be stabilized in the presence of the high-frequency parasitics (fast modes). Since no approximation was used in the proof above, Theorem 1 is more general than Lemma 1, in the sense that it can handle the class of discrete systems in which the controlled slow and uncontrolled fast modes are not strongly separable. Furthermore, the gains Go and Ko do not couple with the high-frequency perturbation AH(z), thus, # ~ - I I ~ ( z ) l l can be viewed as a robustness measure of the nominal design system. With this specific advantage, one can use inequality (10) to estimate a permissible perturbation upper bound. To do this, express the proposed stability condition (10b) in an alternative form as follows: IIAH(eja) - AH(1)II < ll~(eJ°)ll-' - #t,
Remark. Asymptotic properties of I}AH(ei°) -AH(1)II are
V0 • [0, 2~]. (21)
One proceeds to plot a frequency-dependent singular value curve of #~ based on the chosen control gains. The envelope of #~ then indicates the permissible upper bound of undesired perturbations under which the overall system could not be destabilized. The result of Theorem 1 is also attractive because in contrast to the qualitative analyses, it can lead to an optimal robust control problem. If Go and Ko are chosen to maximize
Therefore, the perturbation bound IIAH(ei°)-AH(1)II is unbiased. (b) IIAH(eJO) - AH(I)II---~0 as 0---*0 and 2dr. Also due to symmetry with respect to 0, the stability inequality (10b) will naturally hold within certain finite frequency band (0, ~r).
4. Full-order observer-based control In this section the problem of constructing observers to estimate the slow and fast states is considered. The fast dynamics is now treated as a modeled subsystem. A full-order observer-based controller designed for system (1) is given by £(k + I) = A l.~(k) + Az~(k ) + Btu(k ) + r,[y(k) - C : ( k ) - C : ( k ) l
(22a)
t(k + 1) = A3~(k) + A4£(k) + B2u(k) + K2[y(k) - CrY(k) - C2~(k)] u(k) = Gt~(k ) + G2~(k),
(22h) (22c)
where K~ and K s a r e the observer gains; Gj and G2, the regulator gains. In terms of the observation error vectors e,(k) = x(k) -,e(k) and el(k) = z(k) - $.(k), the actual closedloop system can be partitioned as
where
A,~ = [A, +:~G~ ~,2=[A2+:tG2
A 2 - K, C2/
/i, -= [s~p II~(eJ~)ll]-' subject to
fi,2, = I t 3 +0B2G,
(a) lit(Ao)l < 1. (b) s~p IIAH(e i#) - AH(1)II < Or,
0
•
[0, 2~r],
the resulting system will achieve maximum robustness under our design constraints.
-B~G~ ], A t - K , CtJ -BIG2 ]
A22 = [ A4 "1":2G2
-B.G,
]
A3- k,.clj' --
B2G,
]
A , - K2CzJ and .~(k)= [xT(k) eT(k)]T, ~(k)= [zT(k) efT(k)]T. If (/n2A4 )- i exists,and the composite regulator/observer gains are
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Gt = [/,,1 - Gt(l,2-A4)-IBz]Go- Gf(I,2- A4)-IA3, G2 = Gf (24a) and
proposed to construct a stabilizing composite observer. We have now considered a more general situation where a composite feedback controller is also involved in the loop. In addition, Theorem 2 provides a numerically checkable condition to identify the overall system stability.
K, =/Co[/p+ C2(I.2 - Aa)- 'Kf] - A2(I.2 - A4)- 'Kf, K2= Kf.
Remark. (a) If A4 is a stable matrix, then Gf = 0 and Kf = 0
chosen as (OIoomi and Sawan, 1987)
are admissible choices. For this case, one should use Theorem 1 to check the loop stability. (b) The separated subsystem matrices A~ and Af can be obtained via the usual quasisteady state approach or matrix decoupling technique (Mahmoud, 1982b). Since the actual closed-loop eigenvalues of (23) are perturbations from ).(A~) and ~,(Af), the closed-loop system will be asymptotically stable for the corresponding time separation ratio being small. To ensure the loop stability, inequality (26) can be used to replace this small time separation ratio assumption.
(24b) then the separated slow subsystem matrix As of (23) is given by .~ll+/~12(12n2--.~22)-12d[21=[ Mr
-B°(~°I=A, A, J
(25a)
where A r and A~ are defined as in (6), and
Go = Go + GtO- Gf(I,2- Aa - B2Gt)-'(A.s- B2GtO) with
5. Example
0 = (/.2 - A4 + KfC2)-I(A3 -- KfCI).
To confirm the main result, we present a simple example in which a reduced-order observer-based controller is adopted to control a two-time-scale system. Consider a fourth-order discrete model described by
The separated fast subsystem matrix Af is
t{22 = [ A4 +oB2Gf
A4
- _ B2Gf ] KfC2 j =
(25b)
Af.
Sufficient conditions for the stabilization of the closed-loop system (23) under this composite controller are established in the following theorem.
a'=[~i~
~i~]'
f0.15 A4=[0.0 ~i~]'
Theorem 2. If the gains (G I , (];2) and (Kp g2) are designed
A2=[00:005 -00:11]' m3=[O.0i~ Bt=[~I~
0.0 1.0]'
B2=[01~
0.030"0],
001~]'
such that As and Af are stable in the discrete sense, then the actual closed-loop system (23) will also be asymptotically stable, provided II(z12,I - As)-IA,2[O(z) - ~(1)]-1¢i2111 < 1,
Vz ~ D
Choosing the following control gains
(26a)
or equivalently,
G,,=
II(ei°12,1 A~)-1.412[~(ei° ) - *(1)]- IA2111< 1, V0 • [0, 2~t] (26b) where
11--0'3238 -0.0032] [-0.0050
Ko = [3.70620.1488] 1_1.1424 3.05271
-0.02591'
-
(I)(z)
=
(ZZ2n 2 -- A22)
the controlled poles will be assigned at 0.4715 and 0.6700, i.e. both lie within the unit circle. Using these results, the robustness condition (21) can be check to satisfy. We find
-I
is the resolvent matrix of ,,{22.
intlt4,
Proof. From (25), it is easy to see that 3.(A~) ~ MAr) O X(Ae) and X(Af)=-X(A4+BzGt)U~.(A4-KtC2). As the gain matrices G~, G2, K,, K2 are chosen as indicated in (24) such
(zt2.,
-
&)-'A,2[~(z)
at 0 = 0.6912 (rad).
See Fig. 1 for a graphical singular valued illustration of the robustness test. Therefore, the chosen set of control gains is permissible, i.e. robust stability of the overall closed-loop system could be ensured. Output responses (with initial states of the controlled subsystem setting to ones and the remaining states setting to zeros) for the system are shown in Figures 2 and 3.
that I~.(A~)I< 1 and I;t(Ae)l < 1 then the developed analysis procedure in Theorem 1 can be applied. The system matrices All , AI2 , m2t , A22, and ~ correspond, respectively, to Au, At2, A2~, A~. and Ao in Theorem 1. It directly follows from (12)-(17) that system (23) will be stable if, and only if, a ~ ' ( z ) = (12., -
- O(AH(e j°) - AH(1))] = 0.0893,
- ¢(1)1-'A2,)-'
• ~,x2.~.
6. Conclusions New frequency domain stability conditions for a high-order discrete model controlled by a reduced- or full-order observer-based controller are derived. The present approach allows one to estimate a permissible upper bound of the high-frequency residuals under which the controlled system
By Lemmas 2 and 3, the stability inequalities (26a) and (26b) are sufficient conditions to guarantee this requirement. • The quantitative result of Theorem 2 complements the one given in Mahmoud (1982a) in which a qualitative approach is
0.B
,
,
,
,
,
c~ >
0.4 ~D c~
0.2 ~ 0
G-[AH(e )-AH(1)]
.......
....- . - ~ _ _ ~
~
0
1
2
3 Complex
4 Angle
(rad)
FIG. 1. Robustness test.
5
6
Brief Papers 0.1 e~ o
l',,, ~ c o n t
0.o5
~
0
rolled controlled
0
50 Step FIG. 2. Yl response.
0.1
0 e~
ee
0.05
controlled
0 controlled
-0.05 0
50 Step FIG. 3. Y2 response.
will remain stable. Additional topics remain to be investigated including extension of the current results to systems formulated by a fast-time-scale, and existence of stabilizing controllers which make the proposed stability criteria hold.
881
References Abed, E. H. and A. L. Tits (1986). On the stability of multiple time-scale systems. Int. J. Control, 44, 211-218. Esfandiari, F. and H. K. KhalU (1989). On the robustness of samplod-data control to unmodeled high-frequency dynamics. IEEE Trans. Aut. Control, AC-34, 900-903. Fernando, K. V. and H. Nicholson (1983). Singular perturbational approximations for discrete-time balanced systems. IEEE Trans. Aut. Control, AC-28, 240-242. Khalil, H. K. (1981). On the robustness of output feedback control methods to modeling errors. IEEE Trans. Aut. Control, AC-26, 524-526. Kokotovic, P. V. and H. K. Khalil (1986). Singular Perturbations in System and Control. IEEE Press. Mahmoud, M. S. (1982a). Design of observer-based controllers for a class of discrete systems. Automatica, 18, 323-328. Mahmoud, M. S. (1982b). Order reduction and control of discrete systems. Proc. lEE, 129, 129-135. Mahmoud, M. S. and M. G. Singh (1985). On the use of reduced-order models in output feedback design of discrete systems. Automatica, 21, 485-489. OIoomi, H. and M. E. Sawan (1987). The observer-based controller design of discrete-time singularly perturbed systems. IEEE Trans. Aut. Control, AC-32, 246--248. O'Relly, J. (1986). Robustness of linear feedback control systems to unmodeled high-frequency dynamics. Int. J. Control, 44, 1077-1088. Sandell, N. R., Jr (1979). Robust stability of systems with application to singular perturbations. Automatica, 15, 467-470. Vidyasagar, M. (1985). Control System Synthesis: A Factorization Approach. MIT Press, Cambridge, MA. Xie, L., C. E. de Souza and M. Fu (1991). H~ estimation for discrete-time linear uncertain systems. Int. J. Robust Nonlinear Control, 1, 111-123.