Applied Mathematics and Mechanics (English Edition, Vol. 14, No. 12, Dec. 1993)

Published by SUT, Shanghai, China

ON ROBUST STABILITY OF A CLASS OF POLYNOMIAL FAMILIES*

Wang Yong ( ~ ~ ) Yu Nian-cai (~:~lz?j-)

(Dept. of Mechanics, Peking University, Beijing)

(Received July. 13, 1992; Communicated by Zhu Zhao-xuan)

A b s t r a c t In this paper, we discuss the robust stability of a class of polynomial families

more general than the interval polynomial family and diamond polynomial family. We

prove that the Hurwitz stability of some special cases of this class of polynomial

families can be determined by checking finite polynomials. We also give an example to

illustrate that it is not always possible to de}ermine the Hurwitz stability of all this

class of polynomial families by checking finite polynomials.

Key words value mapping, H-stability, H-equivalence

I. I n t r o d u c t i o n

In 1978, the Russian scholar Kharitonov EIJ studied the H-stability of the interval polynomial family

. <.> -- {: <.,> , <(.>= v,.,.', ,_-o,,,...,.} i=O

and got the celebrated Kharitonov's Theorem in order to determine the H-stability of (1.1), it is sufficient to check four special polynomials of the polynomial family.

In the years that followed, many scholars made extensive and unceasing studies on the H- stability of polynomial families. R. Tempo studied the diamond polynomial family

n n

F@){f(s), f ( s ) = y ~ a , s ' , ~'-~,lcz,-#,l~r } (t.2) f - O ( - 0

which is dual to the interval family and got the Diamond Theorem that in order to determine the H-stability of (1.2) it is sufficient to check eight special polynomials.

As an extension of the interval polynomial family and the diamond polynomial family, we study the following n-th order real polynomial family

n m r lk

= = E E f , - O k ~1 f=O

n k

~ . I ~ t m - a . m l ~ r k , k = l , 2 , - - - , r a } (1.2,) / , = 0

*Project supported by National Natural Science .Foundation of China. 1095

1096 Wang Yoag and Yu Nian-cai

where { Jk0, jh~ , . . - , ]knh} (k = 1 , 2 , . . . ,m} are subsets o f {0,1 , . - . , n } and have the property

h~ { 1,0,1,, , ' . . , /h, ,} ={0,1_ ,... ,n}

J' 0,1'. ,"",/h.,} N { 1.0, h i ,"" ,11.: } =r V l~- k

Apparently when m=n+l, (1.3) reduces to (l.1); when m= 1, (1.3) reduces to 0.2). Since (1.3) is a convex polytope in coefficient space R "+1 , according to the Edge

Theorem E31, the H-stability of (1.3) is determined by the H-stability of its exposed edges, and this is a one-dimensional test. In this paper, our final point is whether the polynomial family (1.3) has finite test polynomials, like the interval polynomial family and diamond polynomial family.

II . Some Examples That Have Finite Test Polynomials

We can give many special cases of polynomial family (1.3) that have finite test polynomials. In this paper, we will give three of them and write in the "form of theorems, which'are proved using the concepts and theories such as H-equivalence c4~, parametrization c4~, Boundary Theorem rs] and Edge Theorem.

Based on the Edge Theorem, we get a one-dimensional test. In order to get finite tests, the following two results are useful.

Proposition 2.1 E61 If the n-th order polynomials f t ( s ) a n d fz(s) have the same

coefficients of either all even or all odd powers of s, then the convex combination

f ( s , A ) = ( 1 - A ) A ( s ) + 2 f ~ ( s ) , ,~e[0,1] (2 .1)

is H-stable if and only if f l (s)and f2 (s) are H-stable.

Proposition 2.2 E7~ For the n-th order polynomials f0 (s) and fo (s) + (a+f ls ) f l (s) , with the coefficients of all even (or all odd) powers of s in f l (s) are zero. Then the convex combination

f (A,s) =fo(-~)+Z(a+f ls ) f~(s ) , ).E [0 ,1] (2 .2)

is H-stable if f (0 , s ) = f0 (s) and f (1 ,s) = f0 (s) W (aW/Ss) f l (s) are H-stable. Suppose the n-th order real polynomial family is given by

n r . n / 2 l

F(s)= f(~)l f(s)=Y~,a,s', ~ l a2,-~z,l<r,, It=0 f = 0

n + l ) t ' 2 ]

(z. 3) I I= l J

It is a special case of (1.3) when m=2 and the subsets of {0, 1 . . . . . n} are { 0 ,2 , . . . ,2 [n/ 2] } and {] ,3 , . . . ,2[ (n+1) / 2] - - 1 } .

It

Let f0(s) = ~ ~ s i , then we have I I=0

Theorem 2.1 (2.3) is H-stable if and only if the following eight polynomials

fo(s) +rl~_+rzs , fo(s) +_rlsn'Jr'rzsn-i~ n even

o r

are H-stable.

fo(s) +rl++_r_zs, fo(s) +rzs"+rts n-', n odd

( 2 4a)

( 2 . 4 b )

�9 i

P r o o f

On Robust Stability of a Class of Polynomial Families

Su~ pose n is even, !:t

Pt(s) = { f ( s ) j f (s) =/o ( s ) +2rl"at'#rzs~ / ] . ,#E[ - I ,1.] } P2(s) = { / ( s ) ~ f (s) = f0 ( s ) -{-;~rls"Wl~rzs"-~ ,~,/zE[.-- ! ,1] }-

P(s) =P1(s) U Pa(s)

It is easy to verify that p (s) c F (s) and

F (./co) =P~ (jw) ~Pz(jco), F (j0~) =P~ (jco) ~PI ( j o ) ,

O~co~~ (Fig i.)

Hence, F(s) and P(s) are H-equivalent.

/:~OoJ)- r, + r,jco

f I

fo(jo~)--r,--rjo9

Im [00o9) + r, -~-r~]o9

Re [oqco). 1

[oqco)§

1097

(2.sa) (2.5b)

Fig. 1 F(jrco) in the complex plane for even n, with 0<,0~1

F(s) is H-stabler P(s) is H-stable<CO>the exposed edges of P(s) are H-stable. In the complex plane the four vertices of P~(s) are

Pl (s) -=fo (s) +rl+rzs~ P2(s) =f0( s ) --rl+rzs Ps (s) =/0 (s) - r. - rzs, p,(s) =/o (s) + rl - rzs

while the four exposed edges are

2pl (s) + (i -,~) P2 (s), 2p~ (s) + (1 - 2) P3 (s) ~p3(s) + (1 -~t)p,(s), ~p,(s) + (i ~-~)pl(s) Z~[0,i]

Based on proposition 2.1, the four exposed edges are H-stable if and only if the four

exposed vertices are H-stable. As for Pz(s), we have the same result.

Hence, F (s) is H-stable if and only if (2.4.a) is H-stable.

When n is odd, the proof is similar The n-th order polynomial family

fl

F ( s ) = ( f (s)l f (s) = y-]. a,s', f l , ~ a , ~ ] , , , i = 0 , , 1 , . . . , k , ~ = 0

71

l a , - c~,l ~ r} (2.6) ~ = k + l

is another special case of (1.3). We rewrite (2.6) as 11

F(s) ={/(s)j I (s )=fo(S) + 5-2, a*s', - 3,<~aT<~d,,

1098 Wang Yong and Yu Nian-cai

where

Let

i = o , 1 , . . . , t ~ . ~ l a ? l < ~ r }

n

:o(,> = ~ ~ (:,+,,,>."+ i ~ b §

I l 6, = - ~ ' - ( ~ ' , - f l , ) , * . . . . . . (fl ,- 'k~) i=O,~l,. . . ,k

, d = a ~ - a ~ , i = k + l , . . . , n

hi (2) = ~o - 5z3.+ 6,33 . . . .

~1 ('~) = 5~ = 5~A+ 552 ~ . . . . rk+l ]

o, (s) = fo (s) +h~ (s ~) +sg~ (s ~) + ( - 1 ) L-2-Jrs~+'

p=(6) = fo (s) +h~ (s ~) + s ~ (s ~) + (-1,) [ ~ r s ~+~

p, (s) --- f o (~) - h~ (s ~) + s ~ (s ~) + ( - ~ ) [ ~ - ~ rs *+:

p,(s) =/o(s) -h~ (r +so~ (s ~) - ( - 1 ) [~-~rs T M

p, (.~) = fo( , ) -h , (,,) - , 9 , (, ') - ( - ~,) [ ~ - ~ - ' + :

p, (,) = 1o (~) + h, (,:) - ~g, (,~-) - ( - ' ), [ ~ ] , - , * + '

p,( , ) =.fo(~) +h, (~') -sg, (.~') + { - 1 ) [ -~ ] , - , *+ '

p,(~) =/0(s) +& (~=) +sa, (s9 + ( -1 ) [-~]rs"

~,, (.i = fo(.) -h, ( . ,)+,, , (..) - ( - , ) E ~ ] . . - ,

p,,(,)=lo(,)-h,(,,)+,g,(,,) + (- ~)["--~-]~,.

p,, (s) = f , ( , ) - h , (sZ)-sfl, (s') ~ ( - -1) ? - ' ~ ] r , " - '

p~,(s) =/o(s) + h~ (s') -sa~ (s') + ( - t ) [-~-]r~"-'

p,,(S) -= /0 (s) -l-h, (s =) --so, (s z) - - ( - - 1 ) ["2-~-~rs"

(2.T)

(z .s)

On Robust Stability of a Class of Polynomial Families 1099

T h e o r e m 2.2 Polynomial family (2.6) is H-stable if and only if the 16 polynomials of (2.8) are H-stable.

P r o o f When 0 < c o ~ 1 , the boundary of F (jco) is an octagon, whose vertices are p, (jco) ,i = 1 , 2 , . . . , 8 , Fig. 2.

When 1<c.0<-l-oo , the boundary of "F(jco) is also an octagon, whose vertices are

p, (ja)), i =9,10,. . . , 16.

Im

p~q~)

p+qw)

p2(jw)

fo(ja))

J p'q~)

p,(j~o)

Re �9 i t

p.(/~)

Fig. 2 P(jca) in complex plane for 0 ~ 1 , k----2m+l Based on the Boundary Theorem F(s) is H-stable if and only if all the polynomials

corresponding the convex combinations of the neighboring vertices are H-stable, while these convex combinations satisfying either the conditions of proposition 2.1 or those of proposition 2.2. Hence, F(s) is H-stable if and only if the 16 polynomials of (2.8) are H-stable.

Since n-th order polynomial

f (s) =ao+a,s+... + a . _ l S " - ' + a . s " (2.9)

is H-stable if and only if the n-th order polynomial

/',(s) = a . + a . _ , s + ... + a l s ' - ' +aGs"

is H-stable, we can conclude tilat the polynomial family

n k

F(s)=(f(s)~ f ( s ) = Y - ~ a , s ' , ~ la,-~,l~r, tt-O i = O

~,<~a,<~v,, i=/,+1, ..., n} (2.10)

is H-stable if and only if the 16 special vertex polynomials are H-stable. The n-th order polynomial family

F (s) = ( f (s) ~ f(.,)=y-~,ct,s', Y~, l a , - ~ , l ~ < r , , i - 0 I I -O

i - k § (2.11)

1100 Wang Yong and Yu Nian-cai

can be rewritten as

n b m r . . , } a i s , ~ la~l<r , , ~ l a ? l < r , , - . , -~ ,-.+t

II "

where f o (s ) = ~--~. ~.ls', a'~ =a , - -~ , . g-O

A group of vertex polynomials of F(s) are

fo (s) +rl +rts *§ fo (s)'+rl +r,s "+z "t

re(s) • k+l, re(s) '~-rl*-at-rz$/~+z f

f o (s) -kr~s ~-' +r~s'-' , f o (s) -4-rts*-' q-rzs"

fo(s)+rtsk+r:s "-1, fo(~)+rise+rzs" }

T h e o r e m 2.3 The polynomial family (2.12) is H-stable if and polynomials of (2.13) are H-stable.

P r o o f Let Pl (s) = { / ( s ) I f (s) = f0 (s) +:tt~l ( s + l ) +Azrt ( s - 1 )

+ s ~+I [~lr~:(s.+ 1) +l~r2(s - ' ! ) ] ,

At, Az, Pl, p~E[.I/2, 1/2]} P~(s) = { / ( s ) s - f (s) =i~(s )+~lr~s '+S ' - ' )

+ , ~ r t (s ~ - s ~-1) + Iz~r. (~s § + # , r , (s" - s ' - 1)

/~1, A S ' ~ l , b g Z ~ [ - - 1 ) / 2 ~ �9 l / Z ] }

P (s) =Pt (s) U P,(s)

It can be verified that

P (s) = F (s), F (/co) =Pt (1co) ~ P , ( / c o ) , F (/co) -- 'P, (ice) D P I ( joe) ,

0 < c o < l

(2.12)

(2.13a)

(2.13b)

only if the 32 vertex

(2.14a)

(2.14b)

hence F(s) and P(s) are H-equivalent. F(s) is H-stable<=~ P(s) is H-stable<e~ the exposed edges of P(s) are H-stable.

The polynomials in (2.13a) and (2.13.b) are the vertices o f P t ( s ) a n d P2(s), while the exposed edges of PI (s) and Pffi(s) , which are the convex combinations Of the neighboring vertex polynomials in the complex plane, satisfy Proposition 2.2. Therefore F(s) is H-stable if and only if the 32 polynomials in (2.13) are H-stable.

R e m a r k s on T h e o r e m 2.3:

In factPl(jco)andPz(jco)orF(jco)are at most either octagons or diamonds in the complex plane. It can be verified that the number of test polynomials depends on n and k as follows

1) n odd, k even: 16 test polynomials 2) n odd, k odd: 8 test polynomials 3) n even: 12 test polynomials We will not trouble ourselves in our limited space to write the test polynomials explicitly

and give the proof of those cases.

On Robust Stability of a Class of Polynomial Families 1101

III. E x a m p l e s T h a t Have Not F in i te Tes t Polynomials

In Section II we have given several special cases of (1.3) that have finite test polynomials. Unfortunately not all of the cases of polynomial family (1.3) have finite test polynomials.

The exposed edges of polynomial family (1.3) can be written as

/ ( ~ , , l ) =/o(~) + ~ ( a s . , + # s . , ) , / ~ [o ,1 ] ( 3 . 1 )

where f~ (s)is an n-th order polynomial and nl, n2~n . Suppose f (s, 4) is n-th order for ;t~ [o,I ] .

Whether the polynomial family (1.3) has finite test polynomials: or not depends on whether or no t / ( s ,A) is H-stable from the H-stability o f / ( s , 0) and [ (s, 1).

If n , - n ~ = + 1 or n~--n2= an even number, then the convex combination off(s,0) and f(s, 1) is H-stable from the H-stability off(s , 0)andf(s ,1) , otherwise if n~--n2 = an odd number (not + 1), then the above results are not true.

E x a m p l e 1 Let

f ( s , 2 ) ='(. ls3-1-OsZ+6s+5+~(--2. ts""}-2.1) , 21E[0,1] (3.2)

then f(s, O)= 7.1sS +6s2 +6s+ 5, f(s, 1)=5sS +6s2 +6s+ 7.1, f(s, 0.5)=6.0/Ss~ +6s2 +6s+6.05. It is easy to verify that f(s, 0) andf(s, l) are H-stable, but f(s, 0.5) is unstable. E x a m p l e 2 Suppose polynomial family is

F (s) = { / ( s ) t / (s) = f 0 (s) + a o + a l s + a ~ s ~ + a 3 s ~,

la01 + las l~<rl , lall + la, l~<rz}

wheref(s)=5s3+6s~+6.01s+5, r1=2.1, r2=0.01. The vertices of F(s) in complex plane are

1'1 (s) = f o (~) + r i - r ~s ~ ,

p, (s) = / 0 (s) + r~ + r , s ,

#, ( , ) = / o (s) - ris 3 + r , , ,

S', (s) = f o ( , ) - ri + r~s,

p5 (s) = f0 (s) - ri + r~s~

p8 (s) - - f0 (-~) - rl - rzs

p , ( s ) = f o (s) q-rls s - rz8

P8 (s) = / o (s) + r~ - r~s

P , ( J ~ ) ~ 2 ( f l o )

Ps(Je~ C _ , ~ Re

t'o(/<J>) J s,,6"<o) p,fjw) pdjog)

Fig. 3 F(jW) in complex plane for 0~c0~1

1102 Wang Yong and Yu Nian-cai

It can be verified that p~(~)and (i= 1, 2,.... ,8) are H-stable. The convex combination of pffs) and ps(s) is

p (s, 2) = ( I - - I ) P, (s) +,~pa (s) = 7. IsS+ 6s 2 + 6s+ 5-F 2 ( -- 2. i ss+ 2. i )

From Example 1, p(s,O. 5) is unstable.

Therefore, some polynomial families denoted by (1.3) have finite test polynomials, while others have not.

A c k n o w l e d g m e n t We are grateful to Prof. Huang Lin for his valuable suggestions and to Dr. Wang Long for his great help during the preparation of the present paper.

References

[ 1 ] Kharitonov, V. L., Asymptotical stability of a,l equivilibrium positions of a family of systems of differential equations. Differential" ,ye Uravneniva, 14 (1978), 2086- 2088.

[ 2 ] Tempo, R., A dual result to Kharitonov's theorem, IEEE Trans. AC, 134 (1990). [ 3 ] Bartlett, A. C., C. V. Hollot and L. Huang, Root locations of an entire polytope of

polynomials: It suffices to check the edges, Math. Control. Signals Systems, 1 (1988), 61 -71.

[ 4 ] Huang Lin and Wang Long, Value mapping and parametrization in robust stability analysis, Scientia Sinica, Ser. A, 8 (1991). (in Chinese)

I 5 ] Yu Nian-cai and Huang Lin, Boundary theorems for robust stability of complex polynomial families, Scientia Sinica, Ser. A., I 1 (1992), 1177- 1182. (in Chinese)

[ 6 ] Bilas, S. and J. Garioff, Convex combinations of stable polynomial, J. of the Franklin Institute, 139, 3 (1985), 3737-3739.

[ 7 ] Hollot, C. V., F. J. Kraus, R. Tempo and B. R. Barmish, Extreme point results for robust stabilization of interval plants with first order compensators, Proc. American Control Conference (1990), 2533 - 2538.