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A necessary and sufficient condition for robust BIBO stability
Systems & Control Letters 11 (1988) 271-275 North-Holland
271
A necessary and sufficient condition for robust BIBO stability M u n t h e r A. D A H L E H
and Y o s h i t o O H T A
Laboratoryfor Information and Decision Systems, Massachusetts Institute of Technolog),, Cambridge, MA 02! &9, U.S.A. Received 18 March 1988 Revised 29 June 1988
An operator F : ~'~--} [ ~ is called proper (causal) if P N F = PjvFP,v and strictly proper if PNF = PjvFPN-]. Note that the set of strictly proper operators is an ideal of the ring of proper operators. Note also that if F is a strictly proper operator then ( I + F ) is invertible as an operator
Abstract: This paper is concerned with obtaining necessary and
sufficient conditions for robustly stabilizing a class of plants characterized by a known linear shift invariant plant with additive perturbations of the form of possibly nonlinear, time varying ¢®-stable operators. We will show that, in some sense, the small gain theorem is necessary and sufficient even when the perturbations are restiricted to a class of linear operators.
Keywords: Robust BIBO stability, Time varying perturbations,
Small gain theorem.
I. Introduction
The basic motivation of this work is to incorporate robustness in the ¢Lbased design methodology [1], and highlight its conservatism. Define the class of plants
{e=eo+ Notation
R: Real numbers. ~'P (1 _
where Po is LSI (linear shift invariant), W is a stable LSI filter and A is an arbitrary ¢~°-stable, strictly proper operator (i.e. possibly nonlinear time varying) with gain
,.,.=(Zi.,l')'"<
goo( A ) =
~'~: One-sided sequence of real numbers u = (Uo, ul, u2,...) with the norm
sup
/~:=\{o)
I l a ( f ) lifo < 1 Ilfll~ "
It was shown in [2] that a LSI compensator C that stabilizes P0 will stabilize all plants P ~ f / i f
II ull ~o = s u p l u , I < oo.
t'~: Extension space of ¢oo. d : Algebra of BIBO linear, shift-invariant, causal operators on ¢~o. Hoe: Bounded analytic functions on the unit disc with the norm II f II oo - ess sup l f ( e io) I < o0.
gp (1 _
sop /~:\to}
IlZX(f)llp Ilfllp
PN: Truncation operator
e,,: (Uo,
II c ( I
+ Ceo)-'w
II ~ , ~ 1,
(C1)
where li H II ~,= II h Ill and h is the pulse response of H and z¢ denotes the algebra of BIBO linear, shift-invariant, causal operators on ¢,~o (see [1,2]). We note that the conservatism of (C1) was not discussed. The main result of this paper is to show that (C1) is also necessary. By this we mean that if (C1) is not satisfied, then there exists an admissible A such that the closed loop system is not BIBO stable. At this point it is interesting to note that results parallel to the above have been derived for the Hoo-problem [4]. There it was shown that
.,,
-'(Uo, ul,...,u~-l, O, 0 , . . . ) .
IIC(l + c P 0 ) - ' w II . * < 1
0167-6911/88/$3.50 © 1988, ElsevierScience Publishers B.V. (North-Holland)
(c2)
M.A. Dahleh: Y. Ohta / Condition for robust BIBO stabifity
272
is necessary and sufficient for robustly stabilizing the class 12'= { P = P o +
WA, g2(A) < 1}.
Also, it was shown that if (C2) is not satisfied then there exists a LSI A, II A II ~ < 1, such that the closed loop system is not d2-stable.
2. M a i n result
The main result of this paper is the following theorem concerning the small gain theorem:
Step I Lemma. Suppose q ~ d I is a convolution operator with II q Ill > 8 > 1. Then there exists an input e
g'~ \ t '°° and an integer N O such that for N > No,
IIPN-~(q*e) ll°° >__8>
1.
II PN e II oo
Proof. The proof is by construction. Since II q Ill > 8, there exists an e o E doo, %(0) = 0, II eo II oo = 1, such that II q * eo II ~ > 8. Because q is causal, given M with IIq * eo II ~ > M > 8, there is an NO such that
Theorem. Let Q ~dd and suppose A is an t'°°-sta-
II PNo(q * PNeo) II oo >-- M.
ble, strictly proper operator with goo(A) < 1. Then the operator I + QA has an d°°-stable inverse with bounded gain for all A if and only if II Q II ~,=
Given m > 0, there is an N ! such that
Ilqll~
oo
Iq(k) I
< 1. k=Nt+l
From this theorem, we obtain the result on robust stability: Corollary. Condition (C1)
is both necessary and
such an N ! can be found since q ~ ¢1. We define e ~ de°° as follows:
e(k(No + NI) + i)
sufficient.
0_
= ( a(k)e°(i)O ', Proof. Immediate from [2] and the above theorem. Proof of main theorem. The sufficiency is im-
mediate from the small gain theorem [3]. For necessity, we assume that II q ][ t > 1 and construct a A such that (I + QA)-I is not d~-stable. The proof will consist of two major steps: Step h In the followifng lemma, we show that if II q II! > 8 > 1, then there exists an input e ~ d ~ such that
IIPN-I(Qe) II~ >-8> 1 V N > No II PN e II and lim II PN e II oo = oo. N---b oo
Step II: Use the above e to construct an admissible linear shift varying A such that if u = Qe, then ( I + QA )u ~ d :¢. Noting that u ~ d ~ \ d ~, this proves that ( I + QA)- 1 is not /~-stable.
for k = 0, 1, 2 , . . . , where a(k+ 1)=~-'~(k),
fl(k + 1 ) = m a ( k + 1) - ¥ ( k ) , v ( k + 1)=m~x(k + 1)+ lt(k), a(0)-l,
fl(0)-M,
V(0) f m .
Notice that a, r , ¥ have the following interpretations. a(k): the magnitude of the k-th chunk of input. fl(k): a lower bound of the magnitude of the output during the k-th excitation. ~,(k): an upper bound of the accumulation of the fringe effect of the p t h ( j = 0 . . . . . k) input. Now we claim that for m small enough and 8 close enough to 1, we have e ~ d~ \ d °° and
IIPN-I(q*e) II°° >--8>1,
N > N o.
II eNe II oo
Henceforth we assume that m is small enough and 6 is close enough to 1.
M A . Dahleh, Y. Ohta / Condition for robust BIBO stability
(i) We will show that e ~ fe~ \ foo: Let tj(k) - [ fl(k), 7(k)] T. Then we have the following difference equation: ~(0) = [ M, m ]T,
/j(k + 1) = A ~ ( k ) ,
a ( k + 1) = [8 -1, 0 ] ~ ( k ) ,
Affi[MS-1 m8 -1
a(0) = 1,
For 8 ffi 1, m ffi 0, A has eigenvalues {1, M }. Furthermore the unstable eigenvalue M is controllable with the input vector [M, 0] T and observable with the output vector [1,0]. Hence by the continuity argument we see that A is unstable and the unstable eigenvalue is controllable with the input vector [M, m] T and observable with the output vector [1,0]. Hence it follows that a ( k ) ~ ao as k "-* ~ . This proves e ~ ~ \ t '~. (ii) Now we show that
IIPNelI~
>_8> 1,
This is a consequence of (ii.2) and the definition
o f a , B, 7. (ii.4) For N > No,
IIPN-l(q*e) ll~ >_8. II Pive II Indeed, 8 = f l ( k ) / a ( k + 1) and fl(0)/a(0) = M > & Hence if k ( N o + N ~ ) < N < ( k + i ) ( l v o + N ~ ) , k >__1, then
-1] 1 "
II/'~,_l(q* e) II ~,
273
N>
N o,
which can be proven as follows: (ii.1) First recall that the input PNoeO has the following property: II Pno(q * Plvoeo) II ,o > M ,
H(1 - P~o+lV,)(q* P~oeo) II ~ < m .
IIP~-l(q*e)lloo > f l ( k - 1) --8. II P~ve IIoo a(k) If NO< N < NO+ N1, then
IIe~-l(q*e)ll,o > B(O) liege II~ - a - ~ >- 8.
I-I
Step H: Construction of the A (linear shift varying) By the lemma there is an e ~ f e ~ \ f °° such that
IIP~-l(Qe) ll~ > 8 > 1 II PNe i1~ -'
N>No.
Using e, we define A in the following way: let A N be a bounded linear functional defined on P N - i ~°° "- { Il E fl~z I Ilk "- 0 V k >_ N
(ii.2) { a(k)}, { fl(k)}, { 7(k)} are monotone increasing sequences. To see that, consider the (fl, Y)-plane and let K be the cone generated by {[1,01 T, [1, M S - ' I X } . Notice that A~ > ~ > 0 if and only if ~ ~ K. Since the initial state ~(0)--[M, ml'r~K, then it is clear {fl(k)} and {7(k)} are increasing if At'~(0) K for any k. Hence, it suffices to show that there is an A-invariant cone in K which contains ~(0). Let ~e >--0 be an eigenvector of the largest eigenvalue of A. Then a straightforward calculation shows that the cone generated by {[1,0] T, ~Je} is a desired one. (ii.3) For k ( N o + N1) < N < (k + 1)(No + N~), II P~e II o, -< a ( k ) , and for No + k ( N o + N1) < N_< (k + 1)(No + N,), II P~_l(q*e)II oo -> # ( k ) .
(1~
},
i.e.,
A~ : P~_tfo* --* R for all N. Then A is defined as
[a(w)] (N) =
(2)
where [A(o)I(N) is the N-th entry of the sequence a(,~).
Now, we set A N = 0 for any N < N0. For N > No, consider the linear functional fN defined on the one-dimensional subspace { PN- I(Qe) } as
fN" { PN-,(Qe)} --* R, fN(PN_,(Qe)) -- - e ( N ) .
(3)
Let A N be a Hahn-Banach extension [5] of fN for all N > No. Then it can be shown that A is admissible, i.e., f~-stable, strictly proper and of gain less than I,
M.A. Dahleh, Y. Ohta / Condition for robust BIBO stability
274
and that (I + QA)-1 is not f~-stable. Indeed (3) implies that le(N)I <~-1 < 1 . II zl ~, II = II f II = II PN- 1(Qe) II ~ -
(4) II Q II ~o- II A I! ~o -< 1.
Hence we have Ilall=
sup
IIa(w)ll
Ilwll®=l
=
sup
supl[a(w)l(N)l
Ilwll®=l N
_< ~-1 < I.
which is precisely the condition for invertibility of an element in Hoo. Hence, the operator I + QA is invertible in ~ for all LSI A, iff
(5)
Moreover (2) implies A is strictly proper. Finally to show that (I + QA)-! is not d°°-stable, it suffices to show that ( I + QA)(u)~ foo where u = Qe, since from (1), u ¢ ¢ ~ \ d °°. Note that (2) and (3) imply
Suppose II Q II d > 1 but II Q II oo < 1; then the destabilizing LSI A has to satisfy IIA il~o >-1/ll Q II ~ > 1 and hence !l a II ~ , > 1. Thus, we cannot destabilize the system with an admissible LSI a. Also, note that a nonlinear shift invariant A can be constructed to make (I + Q A ) - 1 unstable. Let e, u ~ f ~ be as in the proof of the main theorem. A possible a is
if N ( w ) 4=0,
a
= / s,,,-,a
if K ( w ) # O , otherwise,
A(u)=PNe--e.
where N(w) is the smallest integer N such that w(N)4=u(N), K(w) is the smallest integer K such that w(K)4= O, and S is the shift operator, i.e.
Hence we have
<; +
e]
[Sw](N)=w(N-1), as desired because PNe ~ foo.
[Sw](O)=O.
ra
Remark. One possible A consists of time-varying gain and delay. To see this, let M(N), N > N 0, be an integer such that
I Qe ( M ( N ) ) I - II PN_ l ( Qe ) ll oo.
It is straightforward to verify that g~(A) < 1 and hence A is admissible. This construction was used in [6] in a different context.
Conclusions
It is easy to see that
e(N) w(M(N)) AN(PN-IW) ---- -- Q e ( M ( N ) ) satisfies equations (2)-(5).
3. Discussion of the results The theorem shows that A can always be constructed as a linear time (shift) varying operator. It should be noted that it is generally not possible to construct a linear shift invariant operator. It is well known that ' ~ " ' is a subalgebra of H~, i.e. ~ r = H~. Also, if H ~.~¢ then H-~ ¢ . ~ iff inf IH(z) l > 0
Izl
'The main objective of this paper is to demonstrate that (C1) is both necessary and sufficient for robustly stabilizing all P ~ 1~. This is equivalent to showing that I + QA has a stable inverse for all A with gain less than unity if and only if II Q I1~,-< 1. Hence, if II Q II > 1, we are able to construct either a linear shift varying A, or a nonlinear shift invariant A such that (I + QA)-1 is unstable. Considering only linear perturbations, the resuits have an interesting interpretation. Given a linear A, I + QA has a stable inverse if and only if - 1 is not in the spectrum of QA, whose spectral radius p(QA) is always bounded by II Q II" II a II. The result of this paper shows that sup p (QA) = II Q II Ilall_
M.A. Dahleh, Y. Ohta / Condition for robust BIBO stability
where A is linear, but possibly shift varying. Also, we have shown that in general
sup 0 (QA) < IIQ II Ilall <1 if A is linear shift i n v a r i a n t . T h i s i n t e r p r e t a t i o n highlights the c o n s e r v a t i o n o f (C1) as a r o b u s t n e s s measure.
Acknowledgement This research was supported by the Army Research Office, Center for Intelligent Control Systems, under grant DAAL03-86-K-0171.
References [1] M.A. Dahleh and J.B. Pearson, e'! optimal feedback controllers for MIMO discrete-time systems, IEEE Trans. Automat. Control 32 (1987) 314-323.
275
[2] M.A. Dahleh and J.B. Pearson, Optimal rejection of persistent disturbances, robust stability and mixed sensitivity minimization, IEEE Trans. Automat. Control. 33 (1988) 722-731. [3] C.A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties (Academic Press, New York, 1975). [4] J.C. Doyle and G. Stein, Multivariable feedback design: concepts for a classical/modern synthesis, IEEE Trans. Automat. Control 26 (1981) 4-16. [5] D.G. Lunenberger, Optimization by Vector Space Methods (John Wiley and Sons, New York, 1969). [6] K. Poolla and T. Ting, Nonlinear time-varying controllers for robust stqbilization, IEEE Trans. Automat. Control 32 (1987) .195-200.
271
A necessary and sufficient condition for robust BIBO stability M u n t h e r A. D A H L E H
and Y o s h i t o O H T A
Laboratoryfor Information and Decision Systems, Massachusetts Institute of Technolog),, Cambridge, MA 02! &9, U.S.A. Received 18 March 1988 Revised 29 June 1988
An operator F : ~'~--} [ ~ is called proper (causal) if P N F = PjvFP,v and strictly proper if PNF = PjvFPN-]. Note that the set of strictly proper operators is an ideal of the ring of proper operators. Note also that if F is a strictly proper operator then ( I + F ) is invertible as an operator
Abstract: This paper is concerned with obtaining necessary and
sufficient conditions for robustly stabilizing a class of plants characterized by a known linear shift invariant plant with additive perturbations of the form of possibly nonlinear, time varying ¢®-stable operators. We will show that, in some sense, the small gain theorem is necessary and sufficient even when the perturbations are restiricted to a class of linear operators.
Keywords: Robust BIBO stability, Time varying perturbations,
Small gain theorem.
I. Introduction
The basic motivation of this work is to incorporate robustness in the ¢Lbased design methodology [1], and highlight its conservatism. Define the class of plants
{e=eo+ Notation
R: Real numbers. ~'P (1 _
where Po is LSI (linear shift invariant), W is a stable LSI filter and A is an arbitrary ¢~°-stable, strictly proper operator (i.e. possibly nonlinear time varying) with gain
,.,.=(Zi.,l')'"<
goo( A ) =
~'~: One-sided sequence of real numbers u = (Uo, ul, u2,...) with the norm
sup
/~:=\{o)
I l a ( f ) lifo < 1 Ilfll~ "
It was shown in [2] that a LSI compensator C that stabilizes P0 will stabilize all plants P ~ f / i f
II ull ~o = s u p l u , I < oo.
t'~: Extension space of ¢oo. d : Algebra of BIBO linear, shift-invariant, causal operators on ¢~o. Hoe: Bounded analytic functions on the unit disc with the norm II f II oo - ess sup l f ( e io) I < o0.
gp (1 _
sop /~:\to}
IlZX(f)llp Ilfllp
PN: Truncation operator
e,,: (Uo,
II c ( I
+ Ceo)-'w
II ~ , ~ 1,
(C1)
where li H II ~,= II h Ill and h is the pulse response of H and z¢ denotes the algebra of BIBO linear, shift-invariant, causal operators on ¢,~o (see [1,2]). We note that the conservatism of (C1) was not discussed. The main result of this paper is to show that (C1) is also necessary. By this we mean that if (C1) is not satisfied, then there exists an admissible A such that the closed loop system is not BIBO stable. At this point it is interesting to note that results parallel to the above have been derived for the Hoo-problem [4]. There it was shown that
.,,
-'(Uo, ul,...,u~-l, O, 0 , . . . ) .
IIC(l + c P 0 ) - ' w II . * < 1
0167-6911/88/$3.50 © 1988, ElsevierScience Publishers B.V. (North-Holland)
(c2)
M.A. Dahleh: Y. Ohta / Condition for robust BIBO stabifity
272
is necessary and sufficient for robustly stabilizing the class 12'= { P = P o +
WA, g2(A) < 1}.
Also, it was shown that if (C2) is not satisfied then there exists a LSI A, II A II ~ < 1, such that the closed loop system is not d2-stable.
2. M a i n result
The main result of this paper is the following theorem concerning the small gain theorem:
Step I Lemma. Suppose q ~ d I is a convolution operator with II q Ill > 8 > 1. Then there exists an input e
g'~ \ t '°° and an integer N O such that for N > No,
IIPN-~(q*e) ll°° >__8>
1.
II PN e II oo
Proof. The proof is by construction. Since II q Ill > 8, there exists an e o E doo, %(0) = 0, II eo II oo = 1, such that II q * eo II ~ > 8. Because q is causal, given M with IIq * eo II ~ > M > 8, there is an NO such that
Theorem. Let Q ~dd and suppose A is an t'°°-sta-
II PNo(q * PNeo) II oo >-- M.
ble, strictly proper operator with goo(A) < 1. Then the operator I + QA has an d°°-stable inverse with bounded gain for all A if and only if II Q II ~,=
Given m > 0, there is an N ! such that
Ilqll~
oo
Iq(k) I
< 1. k=Nt+l
From this theorem, we obtain the result on robust stability: Corollary. Condition (C1)
is both necessary and
such an N ! can be found since q ~ ¢1. We define e ~ de°° as follows:
e(k(No + NI) + i)
sufficient.
0_
= ( a(k)e°(i)O ', Proof. Immediate from [2] and the above theorem. Proof of main theorem. The sufficiency is im-
mediate from the small gain theorem [3]. For necessity, we assume that II q ][ t > 1 and construct a A such that (I + QA)-I is not d~-stable. The proof will consist of two major steps: Step h In the followifng lemma, we show that if II q II! > 8 > 1, then there exists an input e ~ d ~ such that
IIPN-I(Qe) II~ >-8> 1 V N > No II PN e II and lim II PN e II oo = oo. N---b oo
Step II: Use the above e to construct an admissible linear shift varying A such that if u = Qe, then ( I + QA )u ~ d :¢. Noting that u ~ d ~ \ d ~, this proves that ( I + QA)- 1 is not /~-stable.
for k = 0, 1, 2 , . . . , where a(k+ 1)=~-'~(k),
fl(k + 1 ) = m a ( k + 1) - ¥ ( k ) , v ( k + 1)=m~x(k + 1)+ lt(k), a(0)-l,
fl(0)-M,
V(0) f m .
Notice that a, r , ¥ have the following interpretations. a(k): the magnitude of the k-th chunk of input. fl(k): a lower bound of the magnitude of the output during the k-th excitation. ~,(k): an upper bound of the accumulation of the fringe effect of the p t h ( j = 0 . . . . . k) input. Now we claim that for m small enough and 8 close enough to 1, we have e ~ d~ \ d °° and
IIPN-I(q*e) II°° >--8>1,
N > N o.
II eNe II oo
Henceforth we assume that m is small enough and 6 is close enough to 1.
M A . Dahleh, Y. Ohta / Condition for robust BIBO stability
(i) We will show that e ~ fe~ \ foo: Let tj(k) - [ fl(k), 7(k)] T. Then we have the following difference equation: ~(0) = [ M, m ]T,
/j(k + 1) = A ~ ( k ) ,
a ( k + 1) = [8 -1, 0 ] ~ ( k ) ,
Affi[MS-1 m8 -1
a(0) = 1,
For 8 ffi 1, m ffi 0, A has eigenvalues {1, M }. Furthermore the unstable eigenvalue M is controllable with the input vector [M, 0] T and observable with the output vector [1,0]. Hence by the continuity argument we see that A is unstable and the unstable eigenvalue is controllable with the input vector [M, m] T and observable with the output vector [1,0]. Hence it follows that a ( k ) ~ ao as k "-* ~ . This proves e ~ ~ \ t '~. (ii) Now we show that
IIPNelI~
>_8> 1,
This is a consequence of (ii.2) and the definition
o f a , B, 7. (ii.4) For N > No,
IIPN-l(q*e) ll~ >_8. II Pive II Indeed, 8 = f l ( k ) / a ( k + 1) and fl(0)/a(0) = M > & Hence if k ( N o + N ~ ) < N < ( k + i ) ( l v o + N ~ ) , k >__1, then
-1] 1 "
II/'~,_l(q* e) II ~,
273
N>
N o,
which can be proven as follows: (ii.1) First recall that the input PNoeO has the following property: II Pno(q * Plvoeo) II ,o > M ,
H(1 - P~o+lV,)(q* P~oeo) II ~ < m .
IIP~-l(q*e)lloo > f l ( k - 1) --8. II P~ve IIoo a(k) If NO< N < NO+ N1, then
IIe~-l(q*e)ll,o > B(O) liege II~ - a - ~ >- 8.
I-I
Step H: Construction of the A (linear shift varying) By the lemma there is an e ~ f e ~ \ f °° such that
IIP~-l(Qe) ll~ > 8 > 1 II PNe i1~ -'
N>No.
Using e, we define A in the following way: let A N be a bounded linear functional defined on P N - i ~°° "- { Il E fl~z I Ilk "- 0 V k >_ N
(ii.2) { a(k)}, { fl(k)}, { 7(k)} are monotone increasing sequences. To see that, consider the (fl, Y)-plane and let K be the cone generated by {[1,01 T, [1, M S - ' I X } . Notice that A~ > ~ > 0 if and only if ~ ~ K. Since the initial state ~(0)--[M, ml'r~K, then it is clear {fl(k)} and {7(k)} are increasing if At'~(0) K for any k. Hence, it suffices to show that there is an A-invariant cone in K which contains ~(0). Let ~e >--0 be an eigenvector of the largest eigenvalue of A. Then a straightforward calculation shows that the cone generated by {[1,0] T, ~Je} is a desired one. (ii.3) For k ( N o + N1) < N < (k + 1)(No + N~), II P~e II o, -< a ( k ) , and for No + k ( N o + N1) < N_< (k + 1)(No + N,), II P~_l(q*e)II oo -> # ( k ) .
(1~
},
i.e.,
A~ : P~_tfo* --* R for all N. Then A is defined as
[a(w)] (N) =
(2)
where [A(o)I(N) is the N-th entry of the sequence a(,~).
Now, we set A N = 0 for any N < N0. For N > No, consider the linear functional fN defined on the one-dimensional subspace { PN- I(Qe) } as
fN" { PN-,(Qe)} --* R, fN(PN_,(Qe)) -- - e ( N ) .
(3)
Let A N be a Hahn-Banach extension [5] of fN for all N > No. Then it can be shown that A is admissible, i.e., f~-stable, strictly proper and of gain less than I,
M.A. Dahleh, Y. Ohta / Condition for robust BIBO stability
274
and that (I + QA)-1 is not f~-stable. Indeed (3) implies that le(N)I <~-1 < 1 . II zl ~, II = II f II = II PN- 1(Qe) II ~ -
(4) II Q II ~o- II A I! ~o -< 1.
Hence we have Ilall=
sup
IIa(w)ll
Ilwll®=l
=
sup
supl[a(w)l(N)l
Ilwll®=l N
_< ~-1 < I.
which is precisely the condition for invertibility of an element in Hoo. Hence, the operator I + QA is invertible in ~ for all LSI A, iff
(5)
Moreover (2) implies A is strictly proper. Finally to show that (I + QA)-! is not d°°-stable, it suffices to show that ( I + QA)(u)~ foo where u = Qe, since from (1), u ¢ ¢ ~ \ d °°. Note that (2) and (3) imply
Suppose II Q II d > 1 but II Q II oo < 1; then the destabilizing LSI A has to satisfy IIA il~o >-1/ll Q II ~ > 1 and hence !l a II ~ , > 1. Thus, we cannot destabilize the system with an admissible LSI a. Also, note that a nonlinear shift invariant A can be constructed to make (I + Q A ) - 1 unstable. Let e, u ~ f ~ be as in the proof of the main theorem. A possible a is
if N ( w ) 4=0,
a
= / s,,,-,a
if K ( w ) # O , otherwise,
A(u)=PNe--e.
where N(w) is the smallest integer N such that w(N)4=u(N), K(w) is the smallest integer K such that w(K)4= O, and S is the shift operator, i.e.
Hence we have
<; +
e]
[Sw](N)=w(N-1), as desired because PNe ~ foo.
[Sw](O)=O.
ra
Remark. One possible A consists of time-varying gain and delay. To see this, let M(N), N > N 0, be an integer such that
I Qe ( M ( N ) ) I - II PN_ l ( Qe ) ll oo.
It is straightforward to verify that g~(A) < 1 and hence A is admissible. This construction was used in [6] in a different context.
Conclusions
It is easy to see that
e(N) w(M(N)) AN(PN-IW) ---- -- Q e ( M ( N ) ) satisfies equations (2)-(5).
3. Discussion of the results The theorem shows that A can always be constructed as a linear time (shift) varying operator. It should be noted that it is generally not possible to construct a linear shift invariant operator. It is well known that ' ~ " ' is a subalgebra of H~, i.e. ~ r = H~. Also, if H ~.~¢ then H-~ ¢ . ~ iff inf IH(z) l > 0
Izl
'The main objective of this paper is to demonstrate that (C1) is both necessary and sufficient for robustly stabilizing all P ~ 1~. This is equivalent to showing that I + QA has a stable inverse for all A with gain less than unity if and only if II Q I1~,-< 1. Hence, if II Q II > 1, we are able to construct either a linear shift varying A, or a nonlinear shift invariant A such that (I + QA)-1 is unstable. Considering only linear perturbations, the resuits have an interesting interpretation. Given a linear A, I + QA has a stable inverse if and only if - 1 is not in the spectrum of QA, whose spectral radius p(QA) is always bounded by II Q II" II a II. The result of this paper shows that sup p (QA) = II Q II Ilall_
M.A. Dahleh, Y. Ohta / Condition for robust BIBO stability
where A is linear, but possibly shift varying. Also, we have shown that in general
sup 0 (QA) < IIQ II Ilall <1 if A is linear shift i n v a r i a n t . T h i s i n t e r p r e t a t i o n highlights the c o n s e r v a t i o n o f (C1) as a r o b u s t n e s s measure.
Acknowledgement This research was supported by the Army Research Office, Center for Intelligent Control Systems, under grant DAAL03-86-K-0171.
References [1] M.A. Dahleh and J.B. Pearson, e'! optimal feedback controllers for MIMO discrete-time systems, IEEE Trans. Automat. Control 32 (1987) 314-323.
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[2] M.A. Dahleh and J.B. Pearson, Optimal rejection of persistent disturbances, robust stability and mixed sensitivity minimization, IEEE Trans. Automat. Control. 33 (1988) 722-731. [3] C.A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties (Academic Press, New York, 1975). [4] J.C. Doyle and G. Stein, Multivariable feedback design: concepts for a classical/modern synthesis, IEEE Trans. Automat. Control 26 (1981) 4-16. [5] D.G. Lunenberger, Optimization by Vector Space Methods (John Wiley and Sons, New York, 1969). [6] K. Poolla and T. Ting, Nonlinear time-varying controllers for robust stqbilization, IEEE Trans. Automat. Control 32 (1987) .195-200.