P. A. Ioannou and F. Ahmed-Zaid are with the Electrical Engineer-. Park, Los ..... Engineering, Florida Institute of Technology, Melbourne, FL 32901. IEEE Log ...
166
IEEE TRANSACTIONSON AUTOMATIC
A Stable Indirect Adaptive Control Schemefor First-Order Plants with no Prior Knowledge
CONTROL, VOL.38,NO. 5, MAY 1993
mum phase or relative degree assumptions are made. However, these schemes suffer from a major drawback referred to as the stabilizability problemof the estimated plant. In other words, for on the Parameters the APP or ALQ scheme to be stable, the estimated plant has to satisfy the usual conditions of stabilizability and/or detectability F. Giri, P. A. Ioannou, andF. Ahmed-Zaid at each time t. Since the adaptive laws for estimating rhe plant is a Abstnret-An indirect adaptive control scheme for a first-order linear parameters do not guarantee such properties, stabilization time-invariant plant with unknown parameters is presented and ana- serious problem in almost every APP or ALQ control scheme. lyzed. The scheme requires no apriori knowledge on the sign or bounds In recent years, various solutions to the stabilizability problem of the plant parameters. A discontinuous control law, involving the use have been proposed. In [1]-[5] for example, the adaptive law is of pmbing,is utilized toavoidany sinplarities caused by the lack of modified by incorporating a projection algorithm (onto a convex stabilizabilty of the estimated model. The scheme guarantees signal compact subset B of the parameter space) to ensurestabilizabilboundedness and zero residual tracking error. ity of the estimated model. However, this approach relies on the I. INTRODUCTION B is known.Another ratherstrong assumptionthattheset approach [6], [7] is the addition of an external signal rich enough The design of most adaptive control schemes is based on the in frequencies so that the parameter estimates converge to the certaintyequivalenceapproach.According to thisapproach,a control law that can meet the control objective is first developed true ones, for which stabilizability is guaranteed by assumption. an However, this approach requires that the reference or external assumingtheplantparametersareexactlyknown.Then, adaptive law for estimating the unknown parameters of a certain signal be applied all thetime in additiontobeingsufficiently rich, which implies that accurate regulationor tracking of signals plant parameterization in chosen to generate on-line estimates which are then used to form the control law. When the plant is that are not rich is not possible. of the More recently, new solutionsfeaturingmodification minimum phase, it is possible to parameterize it in terms of the controller parameters for some control objectives such as model certainty equivalence control in order to overcome the loss of reference following. In this case, the controller parameter esti- stabilizability have been proposed for first-order plants.In [8] for example, the certainty equivalence control is modified in order matesaregenerateddirectlyfromtheadaptivelawandno to allow the estimated high-frequency gain to go through zero intermediate calculations take place. This adaptive control scheme is referred to in the literature as direct adaptive control. (in case the initial guess is chosen with the wrong sign) without A very common example of such a scheme is model reference affecting the boundedness of the control input. In [9],a similar adaptivecontrol (MRAC), whichiswidely used and analyzed, idea, i.e., discontinuity in thecontrol lawwith respect tothe parameter estimates, is used to achieve global stability. In both see [l]and list of references therein. When the plant is nonminimum phase, its parameterization in studies, the knowledge of a lower bound on the high-frequency terms of the controller parameters of a general pole placement gain magnitude is assumed to be known. to the above problem for In this note we present a solution control law, even though possible, is not useful from the estimation point ofview. This is due to the nonlinear appearance of first-order continuous-time plants without requiring any a priori the unknown parameters in the plant parametric model. In this knowledge on the high-frequency gain. We propose an adpative controller in Section I1 that performs a continuous comparison case, a plant parameterization which is linear in the unknown parameters is chosen. An adaptive law is then selected to gener- of themagnitude of thehigh-frequencygainestimatewitha moving referencelevel;wheneverthemagnitude of thehighatetheestimatedparameters which are used to computethe controller parameters. The resulting control scheme is referred frequencygain is greater,thestabilizingcertaintyequivalence control lawis used,othenvise,anappropriatecontrolinput is to as indirectadaptivecontrol. The class of indirectadaptive The new inputguarantees,afterafinite control includes adaptive pole placement (APP), adaptive linear usedfortheplant. transient period, that the high-frequency gain estimate remains etc. quadratic (ALQ), lawis All three control schemes mentioned earlier, namely MRAC, bounded away fromzero.Thisdiscontinuouscontrol of theclosed-loopsignalsand APP, and ALQ have a number of advantages and disadvantages. showntoensureboundedness zero residual tracking error. The idea of an updated threshold For example, in MRAC, @e plant is assumed to be minimum wasfirst exploited in disphase with known relative degree and sign of the high-frequency for the high-frequency gain estimate gain. While the assumption about the knowledge of the sign of crete-time systems, see[10]-[12] and the references therein. The the high-frequency gain can be relaxed asin [2]-[4], the algo- advantages of our solution as compared to the ones described earlier are twofold: i) neither the sign nor a lower bound on the rithms used may cause high gain or bad oscillatory transient. In high-frequencygainmagnitude areneeded, ii) zeroresidual APP and ALQ control schemes, no high-frequency gain, minitracking error is achieved. We should mention however that our approachdoesnotguaranteeconvergence of theestimated ManuscriptreceivedMarch15,1991;revisedNovember8,1991and ones.This is due to the factthatthe May 18, 1992. This work was supported in part by the National Science parameters to the true FoundationunderGrantDMC-8452002 and in part by the General probing control input is used when “near” loss of stabilizability MotorsFoundation.Thefvstauthor wassponsored by theFullbright occurs in the estimated model and is switched off after a finite scholarship program for university lecturers and scholars while visiting time period. the University of Southern California. F. Giri is with Laboratoire d’automatique( M I ) , Ecole Mohammadia 11. h m m CONTROL SYSTEM d’hginieurs, Rabat, Morocco. P. A. Ioannou and F. Ahmed-Zaid are with the Electrical EngineerA. Problem Statement ing-Systems Department, University of Southern California, University In this section, we consider the control problem of a singlePark, L o s Angeles, CA 90089-2563, U.S.A. 1inputsingle-outputplantthat is described by alineartimeIEEE Log Number 9204988. 0018-9286/93$03.00 0 1993IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL. 38, NO.5 , MAY 1993
invariant differential equation of the form y
= a*y
+ b*u,
y(0) = y o
C. Adaptive Control Law
(2.1)
where u and y are the plant input and output, respectively. a* and b* are the unknown plant parameters. For the plant to be stabilizable, it is further assumed that: Assumption A): b* # 0. Notice that neither the sign of b*, nor a lower bound on Ib*l are assumed to be known a priori. Our objective is to design an adaptive controller for the plant (2.1) such that: a) A l l the closed-loop signals remain bounded and a with y,,, satisfying the following b) ly - y,I + 0 as t differential equation: --f
Y,,
=
-amy,,,
+ b,r,
y d 0 ) =Y,,
767
(2.2)
where r , the reference input, is a known piecewise continuous bounded signal; a,,, and b,, are real design constants such that a,,, > 0.
B. Parameter Estimation L e t e = y - y, denotetheoutputtrackingerror.Then, it follows from (2.1) and (2.2) that e satisfies the following differ-
The certainty equivalence control law -iy - a m y uce
=
+ b,r
(?,lo)
6
where a" and 6 are generated from the adaptive laws(2.? and (2.9), could achieve the control objective provided that b # 0. This condition amounts to requiring that the estimated model be stabilizable. Unfortunately, algorithm (2.81, (2.91, as well as other existing estimationalgorithms,cannotbeshowntoguarantee such a property. Consequently, the riskofdivision by zero in (2.10) is real,andcould,eventuallyleadtoanunbounded control signal. As pointed out in the introduction, such a problem is encountered, in one form or another,eachtimeanadaptivecontrol scheme,designedaccordingtothecertaintyequivalenceapkg., those proach, uses a prediction-type estimation algorithm belonging to the family of gradient and least squares). In order to deal with such a problem, the certainty equivalence control law is modified as follows: Modified CE Control Law
ential equation e
+ a,e
=
a*y
+ b*u + a m y - b,r,
e(0) = e , = y o - ymo
(2.3) which is rewritten in the form
e
-
amyf + y,
=
0*'4
(2.4)
where,without loss of generality,theexponentiallydecaying terms, due to non zero initial conditions, are ignored. Throughout the note, the subscript "f" denotes filtering by F ( s ) = l/s + a,,,, i.e., for any piecewise continuous scalaror vector function
(2.11) where {t&, k = 1,2,3, ..* is thesequence defined by
(2.12)
2
fk-1
+T
and
I 6 ( t k ) l 5 E(k
- 1).
(2.13)
{ e ( k ) )is the decreasing positive real sequence, updated at the time instants t k and defined by
In (2.4), 4 is the information vector defined by (2.6)
(2.14)
(2.7)
(or ~ ( k=) Z, if a lower bound E > 0 on the size of b* is available, Le., Ib*l > Z). Ilylllris defined by
and O * , w are the following vectors: w = [y,~]'.
2)is the first time instant such that fk
(2.5)
fJ* = [ a * ,b*]',
tk
t , is the first time instant such that
tk(k 2
X
of timeinstants
Since e , yf, and 4 can be measured, (2.4)is asuitable parameterizationforestimatingtheunknownplantparameter vector e*. The estimate 0 or fJ* may be generated by using the following normalized gradient algorithm:
llytllr
:=
SUP
(2.15)
lY(T)I.
01TSf
c and T are any real constants chosen such that
T>O
(2.16) (2.17)
c2c,>o
where
co = In [2(3 -
A,)/Ai]
and
A,
=
1-
where e,
=
0'4 - [ e - amyf + b,,,rf]
(2.9)
and N O ) = [a"(O), $a)]' is arbitrary but $0) # 0. Let us note that the choice of the estimation algorithm is not unique. For instance, a normalized least squares algorithm could be used instead of (2.8) and (2.9), see [13]. Actually, any algorithm possessing, as (2.8) and (2.9) does, the properties pointed out later in Lemma 3.1 may be used.
We see from (2.13) that {tk), k 1,2, ... indicatesthe sequence of timeinstantsat which Ib(t)l reaches the threshold ~ ( -k 11, indicating "near" loss of stabilizability. At that point, the control pointinput is switchedfromthecertaintyequivalencecontrol u = u,, tothe probingsignal u = 4eck[l + ~ ~ y , ~ ~ ~ ] / I bfor ( Oa) lfixed * chosen time period T . At time t = t, + T , if theestimate b(t, T ) isbelow thethreshold ~ ( k in) magnitude, then t,, = tk T , and the probing signal is applied
+ +
for another time period T. Otherwise, the certainty equivalence controlinput is useduntilthestabilizabilitytest fails and the procedure described earlier is repeated. In the following section we show that switching in the control law occurs only during a finite transient pe!iod, after which the estimated model is always stabilizable, i.e., Ib(t)l 2 E* for some E* > 0, and the certainty equivalence control is used for the rest of the time to achieve the control objective. 11. STABILITY ANALYSIS OF THE ADmm CONTROL SYSTZM
The following lemma, which constitutes the crucial part of the stability proof, merely states that after a finite transient period, the estimated model is always stabilizable and no further switching in the control law is needed. Lemma 3.2: Consider the closed-loop adaptive control system A), the consisting of the plant (2.1) subjecttoAssumption law (2.11). estimationalgorithms (2.8), (2.9), andthecontrol Then thereAexist finite positive constants T* and e* such that: V t 2 T*, Ib(t)l 2 E* > 0 and u ( t ) = uce. Proof See Appendix.
In thissection, we analyze the closed-loopcontrol system Now, based pn Lemmas 3.1 and 3.2, for t 2 T*, wehave consisting of the plant (2.1), the adaptive law (2.81, (2.91,and the u ( t ) = u&), Ib(t)l 2 E* and all signals in the closed-loop syscontrol law (2.11). The following theorem establishes the stabiltem can be shown to be boundedand the tracking error e ( t ) + 0 ity andtrackingproperties of theproposedadaptivecontrol ad t + 3~ by following the same arguments asin [l], [14]. This system. 0 proof completes the of Theorem 3.1. Theorem 3.1: For any fixed T > 0 and c chosen as in (2.17), Since the sign of the big!-frequency gain is not available, the all signals in the closed-loop (2.11, (2.81, (2.9), and (2.11) remain sign of theAinitial estimate b(0) could be “wrong.” Therefore, the =. bounded and the tracking error ly(r) - y,(t)l -+ 0 as t estimate b is requiredtogo throughzero,renderingthecerProof The proof will becarriedoutusingthefollowing taintyequivalencecontrol law useless.However,thissituation lemmas. does not affect the systemstability. Indeed, the proppsed control Lemma 3.1: The gradient algorithm (2.8), (2.9) guarantees the scheme is designedsuchthatwhilethe es:imate b is crossing following properties; zero, the control signal is independent of b. Note that convergence of theestimatedparameterstothetrueones is not guaranteed. This is not needed to achieve the tracking objective andnotsurprisingsincetheprobingsignal is usedonlyfora finite transient period. Proof of Lemma 3.1: The proof of a), b), and c) can be IV. CONCLUSION found in many references, see for instance [l]. Part d): For any An indirectadaptivecontrollerthatdoesnotrequireany fixed T > 0 and any t > 0, we have knowledge about the sign or bounds of the high-frequency gain is proposed for a scalar plant. The control law used is a disconI l e ( t ) - e ( t - 7111 r - r ~ ( v ) d v l l[ s- / 1 6 ( v ) 1 1 d v tinuous one and allows the estimate of the high-frequency gain to go throughzerowithoutaffectingtheboundedness of the control input. We show that after a finite time, no switching of the controllaw is required and the estimateof the high-frequency (Schwartzinequality). (3.1) gain remains bounded away from zero for the rest of the lime. Research is currentlyunder way to extendtheseresults to Since 6 E L,, for any fixed real T , we have higher order plants. -+
=l l / t
lim r-w
I16(v)l12
dv
=
(3.2)
0
f - 7
which together with (3.1) yields, for any fixed limsupIIO(t) -
T,
e(r - 7111 = 0.
(3.3)
1-z
This concludes the proof of Lemma 3.1. Remark 3.1: The properties listed in Lemma 3.1 are guaranteed independently of the boundedness of the various signals. However, it is implicitly required that all solutions of the overall differential equation describing the adaptive control system exist. This is actuallythecaseandcan be formallyestablished based on the following observations: i) Over the interval [tkr tk + T ) , u = 4eCk(l+ ~ ~ y ~ l l ~ ~ where all terms are constant within this interval, except forIIyrllm which can at most vary asfast as ly(t)l; Since the closed-loop system over [ t k , t k + T ) behaves as a linear time-invariant system,theexistenceanduniqueness of solutions in theinterval follows. ii) Over the interval [ t k + T , t k + u = uce.We can conclude that all solutions exist and are unique in this interval by following the analysis and arguments given in [l]. In viewof i) and ii)we can conclude that all signals of the control system belong to L,.
A~PENDIX PROOF OF LEMMA 3.2
Since the sequence { t k } is strictly increasing by construction, Lemma 3.2 holds provided that I t k ) is bounded. In order to prove, by contradiction, that Itk) is bounded, let us suppose that: lim k-
t k = 30.
(A.1)
x
In the following, wewillshow that the effect of the control law, especiallywithintimeintervals of ?he form [ t , , tk + T ) , make it asymptoticallyimpossiblefor Ib(f)l toarbitrarily approachzero,afact which contradicts (A.1) due to (2.11). The / proof ~ ~ ~will O ~be1 carried 2, outin three main steps, where thefollowing notation will be used: I, is the time interval Ik = [ t , + T / 2 , t k + TI. ( a ( k ) )is the sequence defined by a ( k ) = aoeck
(A.2)
for c as in (2.17) and a,, = 4/l6(O)l2.
p is a real function defined by P(t) =
1
+ Ily,llx.
(A.3)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO.5 , MAY 1993
Therefore, with these notations, the sequence ( 4 ~ ) is) given by: E(k) =
Step 1: Inthisstep sufficiently large,
1 -
that V k
which together with (A.3) and (2.11) yields: V J t 2, V t E [O, t,)U [ T + ti, ti+ 1
(A.4)
m.
wewillshow
769
forsome
2
2
lu(t)l I l J m [ k oP ( t )
for some positive real constants increasing, (A.17) implies lu(t)l
+ E(t)p(t) 2 d
+ k11
(A.17)
k , and k , . Since a and fi are
m [ k , P ( t , )
+ k,l
+ a ( k - l)p(t,)
(A.18)
for V t E [0, t , ) u [ T + r,, t j t , ) and V j , 2 I j < k - 1. Since a ( k ) diverges, it follows from (A.18) that an integer k , exists such that: V k > k,, V t E [O,t,)U[T + t,,t,+,) lu(t)l
+ a ( j ) p ( t ) 5 2a(k - l)p(tk)
(A.19)
which together with (A.15) yields
(A.20) for all t E [t,, t, + T ) and all k > k,. Furthermore, using (A.11) and (A.121,we have V k , V tE [f,, t, +T)
( Z p ) f ( t ) = e"m('~-')(EP)f(t,) + /'e"-('.-'&(v)p(v)dv fk
=
(A.9)
enm('k-')(CUP) f ( t k )
+ a ( k ) [pf(t) - e a , ( r k - t ) p f ( t k ) ] .
Moreover, V k , and with an appropriate choice of initial conditions, Le., ( E p ) f ( 0 ) = 0, one has
Step 2: In this step we will show that the sequence
(Zp)f(t,)
From (A.21, (A.3), and (2.11) we have, v k , vr
E
(A.21)
[tkrf k
+T)
= ~"(u)P(u)eam(u-'i)du.
(A.22)
Since Z, p are increasing, (A.22) yields: V k
(A.11)
u(t) = z(t)P(t)
where 5 is the function defined by
Vr ~ [ t , , t , + , )V, k .
32) =dk),
From (A.111, we also get that: V k , V tE [t,, tk
(A.12)
+ T)
u f ( t ) = ( Z p ) f ( t ) + s,(t)
(A.13)
where 15,is the term due to initial conditions, at t = t k . Since F ( s ) = l/(s + a,,,), it canbe easily checkedthat: V k , V t E [t,, 1, T ) , and with an appropriate choice of initial conditions at t = 0 for u,(r) and (Zp,Xt), i.e., uf(0) = (apfX0)
+
I6,,(t)l
Ie
I
' m ( ' ~ - t ) ~ ~ a ~ (-u E- fp ~ l du ) l u.
(A.14)
Using (A.11) and (A.12) it follows from (A.14) that: V k , V t E [ t k ,t, + T ) ~s,(t)l2 /'leam('-tk)IU - (~(0)pI dv 0
k- 1
e
' J + I a,,,("-'
+ ,?I
k)lu
- a(j)pldu.
(A.15)
j T t t,
Since p is increasing, I pf(tk)l/l pf(r)l I 1, for all r E [t,, tk T ) and all k. Also, since a ( k ) = ageck,we have: a ( k - l)/a(k) = e-', for all k. Theseobservations,together with (A.7)-(A.9), yield: V k > k , = max k,], V r E I, +
On the other hand, since V t E [t,-, + T, f k ) : 161 > c ( k - l), it follows from (A.4) that: V j t 2, V t E [O,t , ) U [ T + r,, ti+
,)
[x,
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO.5. MAY 1993
770
Now, if c
L co, where
Now squaringboth sides of (A.37) and integrating over the interval [ t k + T/2, t,, + T), yields: V k > k ,
(A.28)
T(b* )2 2
-=
then it follows from (A.27) that: V k > k,, V t TI IS,(t)l
I1
E [fk
+ T/2, tk +
A, -2
/Ik fk
+
’ Ib* 1’
+ r/2
dt
(A.29)
which together with (A.25) yields: V k > k ,
(A.30) Step 3: The contradiction to (A.1). From the equality
From (2.13) and (A.4) we have
Oi.43) wherethe
second inequality was obtained using thefactthat p(tb) 2 1. Since O(t)T4(t)/lt Il+(t)li is in L,, (A.43) implies that O(t)?$(t)/iu,(t)l is also in L,. Hence,
lirnsup( sup k-m
I&t)
- 6 ( t k ) l ) = 0.
(A.34)
(A.35)
I(?) is bounded and
]
i(t)T+(t)
rksr k3
Using the fact that from (A.35) that:
2 a, > 0 and
&)
Also, from Lemma 3.1 [part dl] we get
klim - x / I1k ~ + tT/2(
uf(t)
dt = 0.
Oi.44)
But (A.44) and (A.41) mean, due to (A.401, that b* should be equal to zero, which contradicts Assumption A). Now since the sequence I t k } is bounded, let k* be the largest integer such that tk* E { t k } , and define T* = tt* + T and e* = d k * ) . This completes the proof of Lemmas 3.2.
REFERENCES
a ( k ) diverges, it follows
S . Sastry and M. Bodson, AdaptUte Control: Stability, Confiergence and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989.
From (A.32) we obtain: V k > k , , V t E [ t k + T/2, tk
+ T) (A.37)
J. C. Willems and C. I. Bymes, “Global adaptive stabilization in theabsence of information on the sign of the high-frequency gain,” Anaiysis and Optimization of Systems, Part 1. New York: Springer-Verlag, 1984. R. D. Nussbaum, “Someremarks on aconjecture in parameter adaptive control,” Syst. Contr. Lett., vol. 3, pp. 243-246, 1983. A. S. Morse, “On adaptivecontrolfor globally stabilizing linear systems with unknown high-frequency gains,” Anaiysk and Optimization of Systems, Part 1. New York: Springer-Verlag, 1984. R. H.Middleton, G.C.Goodwin, D. Q. Mayne, and D. J. Hill, “Design issues in adaptive control,” IEEE Tmns. Automat. Con@., V O ~ 33, . no. 1, pp. 50-58, 1988. H. Elliot. R. Cristi. and M. Das. “Global stabilitv of adaptive Dole placement algorithms,” IE€€ Tram. Automat. kontr., v h Ad-30, no. 4, pp. 348-356, 1985. B. D. 0. Anderson and R. M. Johnstone, “Global adaptive pole positioning,” IEEE Trans. Automat. Contr., vol. AC-30, no. I, pp. 11-22, 1985. R. H. Middleton and P. V. Kokotovic, “Boundedness properties of simple adaptive control systems,” in Proc. 1991 Amer. Conm. Con$, vol. 2, Boston, MA, 1991, pp. 1216-1220. F. M. Pait and A. S. Morse, “Global tunability of one dimensional SISO systems.” IEEE Trans. Automat. Contr., to appear. ~~
From (A.33), (A.341, and (A.36) it becomes clear that lim sup 6 J k ) k+
x
=
0.
(A.39)
~
~I
IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL. 38, NO. 5, MAY 1993
[lo] F. Giri, J. M. Dion, M. MSaad,and [ll]
[12] [13] [14]
L. Dugard, “A globally convergent pole placement indirect adaptive controller,” in Proc. 1987 Conf. Decision Contr., vol. 1, Los Angeles, CA, 1987,pp. 1-6. -, “Robustpoleplacementdirectadaptivecontrol,” in Roc. 1987 Conf. Decision Contr.,vol. 1, Los Angeles, CA, 1987, pp. 1-6. J. W. Polderman,“Astatespaceapproachtotheproblem of adaptivepoleplacementassignment,” Moth Confr. Signals.Syst., V O ~ 2, . pp. 71-94, 1989. E. W.Bai and S. Sastry, “Globalstability proofs for continuous-time indirect adaptive conirol schemes,” I€E€ Trans. Automat. Confr., vol. AC-32, no. 6, pp. 537-543, 1987. K. S . Narendraand A. M. Annaswamy, Stable Adaptbe Systems. Englewood Cliffs, NJ: Prentice-Hall, 1989.
A System-Theoretic Appropriate Realizationof the Empty Matrix Concept C. N. Nett and W. M. Haddad Abstract-Inthis note we propose an algebraic realization of the empty matrix concept which is appropriate for system-theoretic applications. This realization differs considerably from the realization currently implemented by the Mathworks, Inc. within their MATLAB program. We demonstrate by repeated example the utility of our realization of the empty matrix concept, and through these same examples indicate the deficiencies of the current MATLAB realization of this concept. These examples fully delineate how the empty matrix concept can be utilized to transparently handle static and /or single vector input, single vector output systems within the more general context of dynamic, two vector input, two vector output systems.
I. INTRODUCTION For many system-theoretic matrix formulas there exist special cases for which oneormore of the matrices involved in the formula do not exist, and hence theformula is inappropriate. One illustration of this point can be given by considering the formula for the transfer matrix of a continuous-time, lumped, linear, time invariant, state-space system:
G(s)
=
C(s1 - A ) - ’ B
+ D.
(1)
This formula involves the matrices A , B , C, and D.However, A , B , and C do not exist in the special case where the underlying state-space system is static,for in this case the system has no states. Correspondingly, in this special case theformula given above is clearly inappropriate. In special cases of the type described above, one can often simply notethe special case, andthen invoke analternative formula. Indeed, in the above example, one can simply note the special case of a static system, and then invoke the alternative formula C(s) = D. An elegant alternative to the procedure described above can be advanced by introducing theconcept of an empty matrix. Intuitively, an empty matrix has noentries,andhence can be Manuscript received July 12, 1992; revised April 17, 1992. The work of the first author was supported by the NSF and ONR. The work of the second author was supported by the AFOSR. of C.N.Nett is with theLaboratoryforIdentificationandControl ComplexHighlyUncertainSystems(LICCHUS),School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150. W.M. Haddad is with the Department of Mechanical and Aerospace Engineering, Florida Institute of Technology, Melbourne, FL 32901. IEEE Log Number 9204989.
771
substituted in a matrix formula for any matrix which does not exist. An algebraic realization of the empty matrix concept is specified toindicate how the usual rules of matrix addition, matrix multiplication, etc., are extended toapply to empty matrices. If this algebraic realization is chosenappropriately,the matrix formula can be manipulated, using the algebraic properties of empty matrices, to arrive at the desired result without ever having to explicitly notethe special case or invoke an alternative formula. The eleganceafforded by the empty matrix approach can potentially be exploited togreatadvantage in the context of large-scale software development.Indeed,thetraditionalapproach to special cases one must include in the software explicit checks for special cases along with appropriate provisions for each special case. This would be done for each matrix formula implemented in the software with potential for special cases of the type described above. Using the empty matrix approach, one would simply introduce the empty matrix as an object, or data type, and then code its algebraic realization. Thiswould be done only once. Subsequent code would then be devoid of cumbersome explicit checks and provisions for special cases of the type described above. A. Previous Work
The authors were first exposed to the empty matrix concept through their use of the MATLAB program developed by The Mathworks, Inc. [l].The empty matrix was first introduced into MATLAB several years ago. A specific realization of the empty matrix concept is implemented in the current version of MATLAB (Version 3.5). Quoting from [l]: “We’re not sure we’ve done it correctly, or even consistently, but we have found the idea useful.” MATLAB realization of the Ouropinion is thatthecurrent empty matrix concept is neither correct, consistent, or useful, at least not for system-theoretic applications. To give some indication ofwhywe have formed this opinion, consider once again the matrix formula (1). Let [ ] denote an empty matrix. Substituting A = [ 1, E = [ 1, and C = [ ] into (1) and using the algebraic realization of the empty matrix concept currently implemented within MATLAB, one obtains G(s) = [ I, as opposed to the desired result G(s) = D. A more fundamental illustration of the deficiencies inherent in the current MATLAB realization of the empty matrix concept can be given by considering the matrix formula below:
A=UN+VD.
(2)
Since we may write
in the case where V = [ ] and D = [ 1, one desires that A = UN. However, upon substituting V = [ ] and D = [ ] in (2) and using the algebraic realization of the empty matrix concept currently implemented within MATLAB, one obtains A = [ 1. We could continue here, and indeed many other deficiencies inherent in the current MATLAB realization of the empty matrix concept are expounded upon in the sequel. The above examples should, however, provide ample evidence tosupporttheopinionput forth above.
0018-9286/93$03.00 0 1993 IEEE
