On the Necessity of the Circle Criterion

According to the well-known circle criterion developed by Bogiorno (1962) (see also Venkatesh(1977)) the system (1),(2) is absolute stable in the class M if 11 W 1100= max{IW(iw)1 :;w E R} < 1. Here it is proved that any system is not absolute stable in the class M provided that the Hoo-norm 11 W 1100 is sufficiently large (and of course, bigger than 1). A.Megretski formulated the problem: whether there exist a universal constant C such that any feedback system with stable transfer function W

249

2. PRELIMINARY RESULTS Let there exists a sequence of stable transfer functions {Wn~l such that systems with such transfer functions are absolute stable in the class M and 11 Wlll oo --+ 00 as j --+ 00. For any positive a denote by Wa (s) the function W(as) which is a rational function of the form a- n,B(as)/(a- n<5(as)) with a- n<5(as) = sn + ... . Then 11 W 1100=11 Wa 1100 for any a > O.

For system (1),(2) this change corresponds to the change of time scale: t ----> t/a and has no impact on the absolute stability property of this system. If a ----> 00, then coefficients of nominator a- n{3( as) and roots of denominator a- n 8(as) of W(as) tend to zero, and hence IWa(iw) 1 tends to zero as a ----> 00 uniformly with respect to W E [1',00) for any positive number t. We apply this transformation for any W jO with constants aj which tend to infinity sufficiently fast. Functions (Wl)aj we denote again by Wl (with some evident abuse of notation).

each j there exists at least one pole on the unit sphere. The system with transfer function (Wj )aj is stable in the class M. Denote by {3jkl, 8jkl , {3jl, 8j1 the coefficients of rational functions (Wjk)aj and (Wj)aj:

Therefore, without loss of generality we assume that all coefficients of numerators and all poles of the rational functions WjO tend to zero as j ----> 00 and for any positive number I' the set {IWl(iw)1 : j ~ 1,w E [t,oo)} is bounded.

Condition 2 implies the existence of numbers C k such that for I {3jkl l:s Ck 8jkO, 1 8jkl 1:S Ck 8jkO , for alIi = 0, ... , njk - 1, k = 1, ... , r, j ~ 1. The systems with transfer functions Wj are absolute stable in the class M, Therefore 1 {3jl 1< 8jl for all l = 1, ... , nj - 1, j ~ 1. Thus, there exists a constant C such that I {3jOl 1< C8j01 for all l = 1, .. , , njO, j ~ 1.

njk-l (Wjk)aj (s) = ( L {3jkl SI )/(Sn jk 1=0 nj-l

(Wj)aj (s)

p

where p

~

+L

k=l 0 and following conditions hold:

1) functions Wj,k are bounded uniformly with respect to j, k, Le. there exists a constant R such that 11 Wj,klloo :s R for all j,k;

----> 00

as j

----> 00,

b) all coefficients of transfer functions (Wj,k) l/Cj,k (s) are bounded uniformly with respect to j, k,

Gj(s)

c) and all poles of functions (Wj,k h/Cj,k (s) lie outside some open ball centered at the origin, d) Cj,k+l/Cj,k

:s p -

nj-l

{3jlsl)/(sn j

+

L 8j1 s l ). 1=0

Now assume that {max{1 Gj(iw) I : wE [t,oo)}} is bounded for all I' >0. Without loss of generality we assume that the sequences of poles of functions G j are convergent. Then we may extract the additive part of functions G j which has poles convergent to nonzero values:

2) for each k = 1, ... ,p there exists a sequence {Cj,d such that

a) {Cj,d

njk- l L 8jk1 SI), 1=0

If there exists positive I' such that the sequence {max{1 Gj(iw) 1 : w E [I', oo)}} is not bounded, then there exists a sequence {Wj} such that Wj ~ I' and {I Gj(iwj) I} is unbounded sequence. All claims of lemma hold for Cj = CjrWj / aj'

(3)

Wj,k

= (L 1=0

Lemma 1. A subsequence {Wj } of the initial sequence {Wn of rational functions has a representation

W j = WjO

+

=

{3j (s) 8j (s)

= G jl (S)+G j2 (s) =

{3jl (s) {3j2(S) 8jl (s) + 8j2 (s) (5)

0, W jO = Wr

where deg({3jl) < deg(8j I), deg({3jI) < deg(8j d, all roots of polynomials 8jl tend to zero as j ----> 00, all roots of polynomials 8j2 have absolute values bigger than some positive number I' for all j. We have {3j = {3j1 8j2 + {3j28jl. All coefficients of {3j are bounded, the distance between poles of polynomials 8jl and 8j2 is more than 1'/2 for all sufficiently large j. Therefore all coefficients of polynomials {3jl, {3j2 are bounded. For sufficiently large j functions G j and G jl are bounded on the interval [1'/2,00). Therefore the same holds for the function G j2 . On the interval [0,1'/2] these functions are bounded because denominator is bigger than some positive number and the numerator is bounded. Thus, 11 G j 2 lloo:S const for all j. Denote Wj(r+l) = (G j2 h/aj' WjO = (Gjlh/aj' Cj(r+l) = l/aj. Then we have representation (4) with (r + 1) terms in the sum.

Denote by aj the maximal absolute value of poles of the function W jO ; Gj = (WjO )aj' Then ajCjr ----> 0, all poles of G j are in the unit ball and for

The number of poles of the functions Wj is finite, Therefore in several steps we get representation (3) with function W jO satisfying condition 3. 0

----> 00

as j

----> 00

for all k

1;

3) there exists a sequence {Cj } such that Cj/Cj,p ----> 00 as j ----> 00 if p > 0, all coefficients of the transfer functions (Wjoh/c j are bounded uniformly with respect to j ~ 1, and {I (Wjoh/cj(i) I} ----> 00 as j

----> 00.

Proof We will use the induction in p. Let there exists a representation r

W j = WjO

+ LWjk

(4) k=l where functions Wjk satisfy conditions 1,2, {max{1 (Wjoh/cj)iw) 1 w E [I', oo)}} < 00 for all I' > 0; all poles offunction Gj = (WjOh/Cjr and all coefficients of numerators of G j tend to zero as j ----> 00. The base for induction is r =

250

Now we'll show that the unit impulse responses of functions Wjk are uniformly bounded in £1 [0, 00) and in C[O, 00).

for all z ERn, TJ E R, j

o

1) polynomials 8j (s) are Hurwitz, coefficients of 8j (s) tend to coefficients of a polynomial 80 (s) = sn + 1'lSn-l + ... + 1'0, 1'0 > 0;

since pairs (A j , Cj) are observable. The sequence of matrices A j has a limit: A j -+ G where G has the same form as A j and the last column contains numbers 1'0, ... , 1'n-l. The pair (G, c) is observable. Thus, there exists a positive number f such that H j ~ d for all j ~ 1.

2) all coefficients of polynomials !3j are uniformly bounded;

3) 11 W j 1100:::; K < 00 for some number K and all

The next step is to show that all solutions of systems (6) with uniformly bounded initial data Xj(O) and functions f,j : I f,j(t) I:::; 1 "it ~ 0 are uniformly bounded. Assume on the contrary that it is not a case. Then there exist a sequence of initial data XjO and function f,j such that the following conditions hold:

j~1.

Let {fj} be corresponding unit impulse responses:

~ 0,

j

~

1. Setting TJ=O we get

00 H J -> jeA;tcoc*.eAjtdt >0 J J

Lemma 2. Consider the sequence of rational transfer functions Wj(s) = !3j(s)j8j (s), such that

00 fj(t) = ~ j eiwtWj(iw)dw, t 21T -00

~

1.

Then there exist constants Cl, C 2 such that

00

JI

fj(t) 1dt :::; Cl,

for all t E [0,1], and

I fj(t) I:::; C 2

o for all t

~

0, j

~

1

1.

j(ciXj(t»2dt:::; K 2,

Proof Consider the Frobenius state space realization of systems with transfer functions W j :

where

1 -r~jO 1

(0 0 0 -8-5,. 1 0 ... 0 1 ... 0

Aj =

o Cj =

C

=

0

OJ

1

jl

-8 j2

,

!3jl

T

!3j(n-l)

and

Wj(s) = ci(sI - Aj)-lbj = n-l

1=0

1=0

00.

t) I dt = Jo I rj(t) 1 dt. Therefore, 11 r j IILdO,Tj:::; C(O). But constant C(O) does not depend on T. Thus, 11 r j IIL I [O,oo):::; C(O).

!3j(n-2)

n-l

-+

Fix positive T. Consider the solution Xj(·) of the system (6) with zero initial data and f,j(t) = sign(fj(T - t). Then cjxj(T) = JoT 1 rj(T-

(7)

bj =

(L !3jlsl)j(sn + L

as j

Define Yj(t) = Xj(t)j 11 XjO 11. Without loss of generality assume that the sequence {Yj(O)} is convergent: Yj(O) -+ y. Then Yj(t) -exp(Gt)y -+ 0 and ciy(t) -+ 0 as j -+ 00 for all t E [0,1]. But Cj == c and the pair (G, c) is observable. We've got a contradiction. Therefore for any positive number R there exists positive number C(R) such that 11 Xj(t) 11:::; C(R) for all solutions of systems (6) with 11 Xj(O) 11:::; R, I f,j(t) I:::; 1 "it ~ o.

(6)

o

11 XjO 11-+ 00

o

8jls l ).

The set of vectors bj is bounded: 11 bj 11:::; R for some number R and all j ~ 1. Solutions of systems (6) with f,(t) = 0 and initial conditions Xj(O) = bj lie in the ball {x : 11 x 11:::; C(R)} and coincide with functions r j' Thus, the derivative of functions j are uniformly bounded: 1drj(t)jdt I:::; Q and I fj(t) I:::; (QC(0»1/2 for all t ~ 0, j ~ 1. 0

r

The pairs (A j , bj ) are stabilizable since A j are Hurwitz. Therefore according to Kalman-Yakubovich lemma there exist Hermitian matrices H j such that

3. MAIN RESULT Now we formulate the main result.

251

Theorem 1. For any integer n there exists a constant C(n) such that any n-dimensional system with stable rational transfer function W is not absolute stable in the class M if 11 W 11002: C(n).

Multiply this equation bye-it and integrate it over the interval [0, T]:

J T

Proof Assume that this statement is wrong. Then there exists an integer n and a sequence {Wj } of stable transfer functions of n-dimensional systems which are absolute stable in the class M. According to lemma 1 without loss of generality we assume that functions W j have a form (3) and conditions 1-3 of lemma 1 hold. Then the unit impulse response functions corresponding to transfer functions Wjk, k = 1, ... , p, j 2: 1, are uniformly bounded in LIlO, 00) and in C[O, 00). Now divide the numerators of functions Wjk by the numbers I Wjo(i) 11/ 2 and denote new functions by the same notation. Then the norms in L 1 [0,00) and in C[O, 00) of unit impulse respond functions of systems with transfer functions Wjk are bounded by constj I Wjo(i) 11/ 2 and tend to zero as j - 00 while the systems with these transfer functions are absolute stable in the class

aj(t)e-itdt = cjo(A - iI)-l e (Ajo-iI)TXjO+

o

JOT L~=1 rjk(t)e-itdt, H jT = J: rjo(t)e-itdt. Then 1 GjT 1- 0 as j - 00 uniformly with respect to T E [0,00) and HjT Wjo(i) as T - 00 for any j 2: l. Denote GjT

Fix integer j and positive T. Consider function ~(T)

=1 a(T) I sign{Re[e-iT(Gj(T_T)+Hj(T_T»)]}'

Then

M.

J T

For any j consider the state space representation of the system with transfer function W j associated with the decomposition (3):

aj(t) cos(t)dt-Re{c;o(A-iI)-l e (A jO -iI)TXjo} =

o T

dXjojdt = Ajoxjo + bjo~ dXjI/dt = Aj1 xjl + bjl~ { .............

=

J

I aj(T) 11 Re[e-iT(Gj(T_T)

(11) We will show further that for sufficiently large j and T this equality doesn't hold. Represent the function aj in the form aj = ajO +... + ajp, where

(9)

where Wjk(S) = c;k(s! - A jk)-lbjk, k O, ... ,p, j 2: 1 and triples (Ajk,bjk,Cjk) have a form (7). Then functions rjk(t) = CjkeAjktbjk tend to zero in L 1 [0,00) and C[O, 00) as j - 00; {I Wjo(i) I} - 00 as j - 00; coefficients of matrices A jo are uniformly bounded and coefficients of vectors bjo tend to zero as j - 00 .

t

ajk(t) =

t

P

rjk(t -

rjk(t -

T)~(T)dT,

k

= 1, ... ,p.

Denote fj = max{L~=1 11 r jk IIc[o,oo), L~=1 11 rjkIlLdo,oo)}' Then fj - 0 as j - 00. Since 11 ~(t)1I = lIa(t)1I for all t > 0 we have

Choose a vector XjO such that cjo(Ajo-iI)-IXjO = #- O. Such a vector exists because the pair (A jo , Cjo) is observable and matrix A jo is Hurwitz. System (8),(9) is absolute stable in the class M. Therefore, as it was proved by Barabanov (1989), there exists a constant A such that for all solutions x(·) of system (8),(9) with ~(t) = cp(a, t), cp E M and initial conditions x(O) = AXjO the inequality 11 x(t) II~ 1 holds for all t 2: O. Further without loss of generality assume A = l. Consider the solutions Xj(-) of the system (8), (9) with initial conditions col(xjo, 0, ... ,0). Then

JL

J o

0, CjOXjO

+

I dT.

o

(8)

dXjpjdt = Ajpxjp + bjp~

aj(t) = c;oeAjotxjO

+ Hj(T-T»]

p

L

k=1

11

ajk IILdO,tl~ fj 11 aj 11 LdO,tj

and p

11 ajO IIL I [o,tj2:11 aj IILIIO,t] -

L

11 ajk IILdo,tj2:

k=l

2: (1 - fj)

11

aj 11 LdO,tj

The system (8),(9) is absolute stable in the class M. Therefore, as it was shown by Barabanov (1989), it is exponentially stable in this class, i.e. there exists positive numbers C, 0 such that 11 x(t) 11 ~ Ce M for all solutions x(·) of the

T)~(T)dT.

o k=O

252

system (8),(9) with ~(t) = O. Thus, for any positive K.j there exists positive T jl such that for all t ~ T jl

T jl

~ ,(1 Wjo(i)

I

-Vj)

J J

1 ajo(r) 1 dr-

o

T,I

-(I Wjo(i) 1 +Vj)€j

1

ajo(r) 1 dr-

o

Tjl

- (1 - K.j)(1 1 _ €j) T jl

~ Pj

JI

J

1

a JO( r ) 1 dr> -

o ajo(r) I dr

o

For a small positive number Vj fix positive number T j2 such that

with some Pj which is positive for sufficiently big + T j2 . But the matrix A jo is Hurwitz. So, this inequality is wrong for sufficiently large T. Theorem is proven. 0 j and is the same for all T ~ T jl

t--r

I

J

rjo(s)e-iSds - Wjo(i)

1< Vj,

o 4. CONCLUSION

I d(arg(Gj(t_-r) + Hj(t_T»)/dr 1< 1/2

It's shown that for any integer n there exists constant C(n) such that if 11 W lloo~ C(n) then the system (1),(2) is not absolute stable in the class of functions

for all t ~ T jl + T j2 , r E [0, T jl ]. Then there exists positive number, which does not depend on Vj and j ~ 1 and such that Tjl

J

lI a jo(r) 11 11 cos[r - arg(Gj(T_-r) +

o 5. REFERENCES Tjl

+Hj(T--r»)llldr

~,

J

Barabanov N.E. (1989). On the absolute characteristic index of systems of linear differential equations. Siberian Mathematical Journal, No.2, pp.5-14.

(12)

ajo(r) 1 dr.

I

o ~

Now assume T get

T jl

+ T j 2.

Use equation (11) to

Bongiorno J.J. (1963). An extension of the NyquistBarkhausen stability criterion to linear lampedparameter systems with time-varying elements". IEEE Trans., Autom. Contr., AC-8, No.2, pp.136154.

IIRe{c;o(A - iI)-le(Ajo-iI)T XjO} 11 ~

J Tjl

~

iT lI a j(r) 11 11 Re[e- (Gj(T--r)

o

Venkatesh Y.V. (1977). Energy methods in timevarying stability and instability analysis. Berlin: Springer-Verlag.

+ H j(T--r»)J1l dr -

T

-J

1

aj(r) I dr

~

o Tjl

~

J

1

ajo(r)

11

Re[e-iT(Gj(T_T)

+ Hj(T--r»)]

1

dr-

o p

-JL Tjl

o

lIajk(r)III1Re[e-i-r(Gj(T_T)+

k=l

T

+Hj(T--r»)ll dr -

J

1

aj(r) I dr

~

o

253