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STABILITY OF TWO-VARIABLE INTERVAL POLYNOMIALS VIA POSITIVITY
Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain
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STABILITY OF TWO-VARIABLE INTERVAL P OLYN 0 MIALS VIA P 0S ITIVITY
Dragoslav D. Siljak* and DuSan M. Stipano vi?* * Santa Clara Unauersaty, Santa Clara CA 95053, [LSA ** Stanford Universrty, Stanford CA 94305, USA
Abstract: Stability criteria are proposed for two-variable (2D) polynomials having interval parameters in polynomic uncertainty structures. Both the left-half plane and unit circle domains are considered. Save for a minor condition, the criteria reduce robust stability testing of 2D polynomials to testing positivity of only two polynomials. The appealing feature of the new robustness criteria is that positivity testing can be carried out by using the efficient Bernstein minimization algorithms. Copyright ©
2002 IFAC Keywords: Two-variable polynomials, interval parameters, stability, left-half plane, unit circle, positive polynomials, Bernstein expansion, time-delay systems.
1. INTRODUCTION
volving 1D interval polynomials (Bose and Zeheb, 1986; Basu, 1990; Rajan and Reddy, 1991; Kharitonov et al., 1997; Xiao et al., 1999). In the context, of 2D polynomials, the solution lost much of its simplicity resulting in numerically involved algorithms. This fact made the testing of 2D polynomials with interval parameters difficult, especially in the case of multiaffine and polynomic uncerta.inty structures.
Stability of two-dimensional polynomials arises in fields as diverse as 2D digital signal and image processing (Huang, 1981; Dudgeon and Mersereau, 1984; Lim, 1990) and time-delay systems (Kamen, 1982), repetitive, or multipass, processes (Rogers and Owens, 1993) and target tracking in radar systems. For this reason, there have been a large number of stabilit,y criteria for 2D polynomials, which have been surveyed and discussed in a number of papers (Jury, 1986; Premaratne, 1993; Hu, 1994; Bistriz, 1999, 2000; Mastorakis, 2000). In achieving t,he maximal efficiency of 2D stability t,ests, the reduction of algebraic complexity offered by the stability criteria in (Siljak, 1975) has been useful. Apart from some minor conditions, the criteria convert, stability test,ing of a 2D polynomial to testing of only two 1D polynomials, one for stability and the other for positivity.
The purpose of this paper is to present new criteria for testing stability of 2D polynomials with interval parameters, which are based on the criteria of (Siljak, 1975) and the positivity approach to the interval parametric uncertainties advanced in (Siljak and Stipanovid, 1999). An appealing feature of the new criteria is the possibility of using the efficient Bernstein minim i z a t i a algorithms (Garloff, 1986; hfalan et al., 1992) to carry out the numerical part of the positivity tests. Furthermore, the proposed formulation can handle the polynornic uncertainty structures having interval parameters, and can be easily extended to systems with time-delays along the lines of (Kamen, 1982).
Due to inherent uncert,aint,y of the underlying models, it has been long recognized that in practical applications it has been necessary to test robustness of stability to parametric variations (Siljak, 1989). Almost exclusively, the 2D robust stability tests have been based on the elegant Kharitonov solution of the stability problem in-
7
2. STABILITY CRITERION
Conditions (i) mea.ns that f ( s ) = 0 if and only if s E C-, while condition (ii) is equivalent to g ( w ) > 0 for all w 2 0.
Let us consider a real two-variable polynomial n
m
In stability analysis of recursive digital filters (e.g., Huang, 1981; Dudgeon and Mersereau, 1984), it is of interest to establish necessary and sufficient conditions for a polynomial h ( s ,z ) to have the stability property
3=0 k=o
where s, z E C are complex variables, and for some j , k the coefficients h,, and hmk are not both zero. We are interested in determining conditions under which the polynomial h ( z ,z ) satisfies the stability property h ( ~ , zZ) 0 ,
EC-}n { Z EC-} ,
{S
h ( s ,2) # 0 , 1s E K"}n { Z E K O )
(9) where K = {s : Is1 = l} is the unit circle, and K O = KUK' is the closure of KO = { s EC : Is1 < I}. By following Huang (1972), one can show that (9) is equivalent to
(2)
where C- is the complement of C = { s E C : Re s < 0}, the open left half of the complex plane a. As shown by Ansell ( 1 9 6 4 , property (2) is eauivalent to h ( s ,1) # 0 ,
h(s,O)# 0 , h ( e Z wz,) # 0,
Yz
(l0a)
E Ko (106)
Condition (lOa) means that h ( s , 0) = 0 if and only if s E KO. To test condition (lob), we consider
v s EC:C (3a)
h ( i w , 2) # 0,
vs E K O
Vz ECC ( 3 b )
d ( z ) = zm/i(e2",z - ' ) which we write as a polynomial
To test (3a) we can use the standard Routh test (e.g., Lehnigh, 19GG). To verify (3b) we follow Ansell's approach and consider the polynomial c ( z ) = h ( i w ,z ) ,
(11)
m
m
c(z)= C
C k Z k
,
k=O
where
(4)
with coefficients
(5)
and s = e i w . With the polynomial d ( z ) we associate the Schur-Cohn m x m matrix D = ( d j k ) specified by
n
n
ck
=
j=o
hjksi
and s = iw.With ~ ( zwe ) associate the symmetric m x m Hermite matrix C = ( c j k ) with elements c j k defined by (e.g., Lehnigh, 196G)
c(-l)"e
j djk
j
'jk
= 2(-1)(j+k)/2
(',-!+1
k 1
'm-j-k+f)
(j
>
+ k ) even
c(-1)'1m
(~m-!+1
k 1
, where
cm-j-k+f)
g ( e i " ) = det D(ei"),
f ( s ) = s%(s-1,0)
(14)
(15)
(16)
and state the following (Siljak, 1975):
Theorem 2. A two-variable polynomial h ( s ,z ) has the stability property (9) if and only if
(7)
(i) f ( s ) is Ko-stable. (ii) g ( z ) is K-positive. (iii) D(1) is positive definite.
and replace w 2 by w to get a polynomial g ( w ) . We also define a polynomial f ( s ) = h ( s , 1)
dj-edk-e),
g ( . ) is a self-inversive polynomial. We also ( j + ,q odd define the polynomial
(6) where the overbar denotes conjugacy and j 5 k . We recall that C > 0 if and only if c ( z ) = 0 implies z E C-. Since C = C ( i w ) is a real symmetric matrix, we define a real even polynomial g ( w 2 ) = det C ( i w )
-
where j 5 (e.g., Jury, 1982). The matrix O(e2") is a Hermitian matrix and we define
j
cjk = 2 ( - I ) ( j + k ) / Z
= C(dm-j+e&-k+e e=i
Condition (i) means that the polynomial f ( s ) has all zeros inside the unit circle K. Positivity ofg(z) on K can be verified by applying the methods of (Siljak, 1973).
(8)
and state the following (Siljak, 1975):
Theorem 1. A two-variable polynomial h (s,z ) has the stability property (2) if and only if
Finally, we show how the mixture of the two previous stability properties can be handled using the same tools. The desired property is defined as
(i) f ( s ) is C--stable. (ii) g ( w ) is &-positive. (iii) C ( 0 ) is positive definite.
h ( s , Z ) f 0,
8
{S
ECC} n { Z
EKO} .
(17)
tested by Theorem 4 wza bilinear transformation in pretty much the same way D-stability was tested in (Siljak and Stipanovid, 1999). We also note the structural similarity of Theorem 4 with theorems on robust SPR properties (Stipanovid and Siljak, 200l), which motivates the work presented next.
By following Ansell (1964),one can show that this property is equivalent to h ( s , O ) # 0, h ( i w , z ) # 0,
‘ds EC-
(18a)
‘dz E K O (18b)
In this case, the polynoniial d ( z ) is defined as d ( z ) = zrnh(iw,z - 1 ) ,
Condition (i) in Theorem 4 obviously means that all zeros of f ( s , p ) lie in for all p E P . To establish this type of robust stability vzu polynomial positivity, we define the magnitude function
(19) which is used to obtain the polynomial g ( . ) wza equations (12)-(15). From (18a), we get the polynomial f ( s ) = h ( s ,01, (20) then define I = { z E C : Re z = 0} and arrive at (Siljak, 1975):
n
k=n j = o
(24) where overbar denotes conjugation. We note immediat,ely that, t,he magnitude funct#ionf ( s , p ) = If(., p ) 1’ is nonnegative for all s E a. This obvious fact is essential in the following development.
Theorem 3. A two-variable polynomial h ( s ,z ) has the stability property (17) if and only if (i) f ( s ) is (I-stable. (ii) g ( z ) is I-positive. (iii) D ( 0 ) is positive definite
Let us form a family ? = { f ^ ( . , p ): p E P} and use the result of (Siljak and StipanoviC, 1999) to conclude that a family F is (€-stable if and only if the corresponding family ? is I-positive, and f ( s ,p’) is(C--stable for some p’ E P . Furthermore, from (7) it follows that positivity of det C ( 0 ; p ) is included in testing condition (ii) of Theorem 4. This means that to test condition (iii) of Theorem 4, it suffices to verify that C(0;p”) is positive definite for some p” E P . We finally arrive at
We note that I-positivity of g ( z ) can be reformulated as &-positivity (see, Siljak and Siljak, 1998).
3. UNCERTAIN POLYNOMIALS We are interested in studying stability properties of uncertain two-variable polynomials wit,h polynomic uncertainty structures. A polynomial h ( s ,z ; p ) is given as n
7n
J=o
k=O
Theorem 5. An uncertain two-variable polynomial h ( s ,z;p ) has the robust stability property 23 if and only if (i) ? is &-positive and f ( s ; p ’ ) isC--stable for some p’ E P . (ii) G is &-positive. (iii) C ( 0 ;p”) is positive definite for some p” E P .
where h j k ( p ) are polynomials themselves in uncertain parameter vector p E IR‘.We assume that p resides in a box
Example 1. To illustrate the application of Theorem 5, let us use t,he two-variable polynomial from (Xiao et al., 1999),
P = { ~ E I R ‘ :p k --k~ , bp - k ] , k ~ r } . (22) We urant to investigate the robust versions of stability properties defined in the preceding section. In the case of (a), for example, we are interested in testing the robust property
h ( s , z ; p )# 0,
{S
EC-}
n{Z
EC)
n
14% z ;P ) = h l l (p)= + h1o(p)s+ ha1 ( p ) z+ boob) ,
(25)
where
h l l ( p ) = 0.9 - 0 . 1 ~ 1- 0 . 3 ~ 2 h l o ( p ) = 0.8 - 0.5pl 0 . 3 ~ ~ (26) hOl(P) = 1 0.2Pl 0.3pz hoo(p) = 1 . G 0.5~1 -0 . 7 ~ ~
n { p E PI.
+
(23) To accommodate the uncertainty in h ( s ,z ; p ) we define the polynomial families F = { f ( . , p ) : p E P}, G = { g ( . , p ) : p E P} and state a straightforward modification of Theorem 1.
+
+
+
and
Theorem 4. An uncertain two-variable polynomial h ( s ,z;p ) has the robust stability property (23) if and only if
To test condition (i) we compute the polynomial f ( s ; p ) = (1.7-0.6pl)~$2.G+O.7pl -0.4p2
(i) F is (I-stable. (ii) G is &-positive. (iii) C ( 0 , p )is positive definite for all p E P .
(28)
and note that, in this simple case, we do not need t o compute the corresponding magnitude function f ( w ; p ) . Robust C--stability of f ( s ;y ) follows directly from positivity of its coefficients. Indeed,
Robust versions of the remaining two stability properties of the preceding section can be
9
1.7 - 0 . 6 2 ~ 1.46 ~ 2.6 0.7pl - 0.4p2 2 2.19
+
In this case, the 2 x 2 matrix C ( i w ; p )turns out t o be a diagonal matrix
(29)
C ( ~ W ;=Pdiag )
for all p E P .
+ 0 . 2 ~ +1 0.3pz)(l.6 + 0 . 5 ~ 1 0 . 7 ~ 2 ) +(0.9 0 . 1 ~ 1 0.3P2) X (0.8 0.5pl + 0 . 3 ~ 2 ) ~
Cii(iw;p) = E i i ( W 2 ; p ) = (3 - p i ) w
-
-
-
-2p;
-
2 0 . 4 4 7 3 ~+ 1.0670
$12
Our analysis is elementary when compared to the stability testing procedure of Xiao et al. (1999), which involves extensive comput'at,ionrequired by the Edge Theorem.
~OI(IJ)~
h ( P ) hzo(P) hii(P) hoi ( p ) hlZ(P) hoa(P) hlO(P) hoo(P)
+
+ + +
'
2
(31)
where
-
+
+ h1a(p)sz2+ hao(p)s Sho2(P)t2+ hll(P)SZ +hio(~)s + + hoo(P)
h ( s ,z ; p ) = s 2 z 2 +ha1(p)s2z
-
+
A two-variable polynomial is given
as
-
-
czz(iw;p) = E2z(w2;P) = ( : h p 2- d p 2 ) w 4 +(-12pipz ~OP?PZ - 6 ~ 1 ~ ; 3p:pj - $ P W + 2 p X - 2P:Pa +12PlP., - 16PTPa 12PlP2 -10P;P; 7P;Pa 3PlP; - P f P ; +2P:P; - P?Pa
Let, us consider more complex examples which well require the use of Bernstein's algorithm.
Example 2 .
+ (-12 + 10pi 6pa + 2pipz + 3p;p; P.:P;)Wa 16271 + 12p2 1 0 p m 4
+7p: 3p; - P1P; +2Pb2 - P?
(30)
is obviously &-positive.
(38)
caa(iw;p)}
and conditions (ii) and (iii) of Theorem 5 reduce to positivity of the coefficients
Since the matrix C ( i w ; p ) is a scalar, condition (iii) is included in (ii) which is satisfied because g ( w ; p ) = (1
{CII(~U;P),
(39) By using Bernstein's minimization algorithm we compute the minorizing polynomials (Siljak and StipanoviC, 1999) and establish positivity of the two polynomials cll(iw;p)and c Z % ( i w ; p )by obtaining the minima min Ell(w;p) = 1.3724 a t pl = 2,
= 3 - P1
PEP
p2
= 1,
U€R+
=PIP%
= 3PiPz - P?Pz = 6 - 5pi 3Pa - P I P Z = P1Pa = 2 - Pl Pa 2 2 = PIP2 = 2PlP2 - PTPz PlPZ
+ +
w = 0.1715 min E z z ( w ; p ) = 3.1185 a t p1 = 2, pa = 1,
+ PZ
PEP
U€R+
(32)
w = 0.2487 .
(40)
Positivity of the minima implies robust stability property 25 for the polynomial h ( s ,z ; p ) of 31.
+
and
p = { p E IRz: p1 E [1,2], Pa E [1,2]}. (33)
4. TIME-DELAY SYSTEMS
The polynomial f ( s ;p ) is computed as
+ + + + +
+
f ( s ;P ) = (4 - P I P1pa)s2 (4PlPz - P b a p b ; ) s +8 - 6 ~ 1 4Pz P1P2 P f - p f p 2 +pip; (34) The corresponding minimizing polynomial f ( s ) = 4s2
-
+ 4s + 4
n-1
(35)
~ ( ~ )+( t )
is obtained by minimizing each coefficient using Bernstein's algorithm. Obviously, F is C--stable since f ( s ) has positive coefficients.
where
+
+ " I ( P )+~ C o ( P ) ,
+
+ PZ +
m
-
(41)
where r 2 0. The coefficients h j k ( p ) are polynomials in the uncertain parameter vector p E IRL which belongs to a box P .
(36)
It is well known (Bellman and Cooke, 1963) that for a fixed pa.ramet,er p , a system (41) i s st,able if and only if
C 2 ( P ) = 2 - p1 pa - W 2 ip1paw c i ( ~1 ) 6 - 5Pi 3Pa - ~ 1 ~ -2 3 ~ ' +i(3PlPZ - P?Pa)U ' 2 2 C o ( P ) = 2p1p2 - PTP2 - P l P 2 W 2 ZPlP2U
+
F lhak(p)~(3)(t k r ) = 0 ,
i = o k=n
Next, we compute C ( z ; P )= Q ( P ) z 2
Our objective in this section is to show how the tools presented in this chapter can be applied to test robust stability of linear systems of the retarded type described by a differential-difference equation
+ PI^'
h ( s ,e F T S ; p = ) sn
(37)
+
cc
n-1
m
hjk(p)swTs
Res20.
10
#O,
j = o k=o
(42)
From (50), we compute the associated polynomial
The system is robustly stable if (42) holds for all p E P.
h ( s , 2 ; p ) = s2
The following theorem is a straightforward robustification of a theorem by Kamen (1983):
and test first the necessity of condition (43) by checking condition (44). Since
Theorem 6 . System (41) is robustly stable independent of delay if h ( s , ~ ; p#) 0,
{S
EKE} n { Z
E K}
+
(43)
h ( 0 ,z i p ) f 0, {. E K ) n { P € p > .
(44)
f ( s ; p ) = s2
To test condition (43) we first use the bilinear transformation
to define the polynomials
iw q s .i w ; p ) = (1 - i w y h ( s , 11 + - iw ~
'
p2)s
+1
- p1
+ p1p; .
(54)
+ 0.5s + 0.5
(55) implies robust stability of f ( s ; p ) , that is, condition (i) of Theorem 7 is satisfied. For testing condition (ii) we need the polynomial
h(s,iw;p)= (1- i w ) s 2
+
+
[l + p a (-1 +pa)iw]s $1 + P l p1p; +(-1 p1 - p1p;)iw . (56) Using equations (47)-(49), we compute
+
j=O 12
k=O
+
+ + + + + + +
g ( w ; p ) = q 1 - p1- 2P2 2P; 2PlP2 -2Pd PlP;)zvJ2 +8(1- 2 ~ 2 ~2 PIP$ - PIP:)^ +4(1; p1 2Pa 2p1pa P; 2PlP; +2PlP, PlP$ (57) By applying the Bernst,ein algorithm to each coefficient of g ( w ;p), we obtain the minorizing polynomial
+
With c ( s ; p ) we associate the symmetric n x n Hermite matrix C = ( C ~ I ; ) having elements c J k defined in (6), and obtain the polynomial
+
g ( w 2 ; p )= det C ( i w ; p ) . (49) Finally, with polynomials f ( s ; p ) and g ( w a ; ; p )at hand, we can imitate Theorem 5 t o state the following:
Theorem 7. System (41) is robustly stable independent of delay if
g(w) -
+
= 0.625w2+ 1 . 2 5 ~ 0.375
,
(58)
which is clearly IR-positive, and (ii) of Theorem 7 is satisfied.
(i) ? is &-positive and f ( s ; p ' ) isC--stable for some p' E P . (ii) 6 is &-positive. (iii) C(0;p") is positive definite for some p" E P .
Finally, the ma.trix of condition (iii) is comput>ed as C(0,O)= 2I7, where I2 is the identity matrix
of dimension 2 , and robust stability independent of delay of system (50) is established with respect to the uncertainty box P in (51).
It is obvious that condition (44) of Theorem G , which is included in (45), can be tested zua positivity as well. To illustrate the application of Theorem 7 let us use t,he following:
5. CONCLUSIONS
Example 3. A time-delay system (41) is given
We have shown how stability of 2D polynomials with interval parameters can be tested wia polynomial positivity. To test stability of polynomials with multiaffine and polynomic uncertainty structures, positivity of only two interval polynomials is required. A remarkable efficiency of the proposed stability criteria is due to their suitability for applications of Bernstein's expansion algorithms.
as
+ p z z ' l ) ( t - + p 1 z ( t - + .'"(t)
+
-
f ( s ) = s2
~I
(47)
+(I P l P $ ) X ( t ) = 0 with the uncertainty box
+ (1
-
(46)
n
T)
2 PlP2
T.
'
.'"(t)
+ 1+
To test robust stability of this polynomial we do It suffices t o not need t o const,ruct, tmhefamily check positivity of each coefficient, which we do by using the Bernstein algorithm. The resulting minorizing polynomial
f ( s ; p )= h ( s , - 1 ; p ) Then, by following Ansell's approach in, we coiisider the polynomial c ( s ; p )= h ( s ,i w ; p ) ,
where
h ( 0 ,2 ; P ) = P l Z
(53) and 1 p1pg > lp11, we conclude that (44) is satisfied. This implies that condition (43) is necessary and sufficient for robust stability of system (SO), and we proceed to compute the polynomial
n { p E P} .
This condition is also necessary if
+ (pas+ pi)^ + s + 1+ pip; , (52)
T)
(50)
P = { p E EL2 : p i E [-0.5,0.5], p2 E [-0.5,0.5]} (51)
11
Lim, J. S. (1990). Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: PrenticeHall.
ACKNOWLEDGMENT The research reported herein has been supported by the National Science Foundat,ion under the grant ECS-0099469. REFERENCES
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Marden, M. (1966). Geometry of Polynomials. Providence, RI: American Mathematical Society.
Basu, S. (1990). On boundary implications of stabilit,y and posit,ivity properties of multidimensional systems. Proceedings of IEEE, 78, 614626.
Mastorakis, N. E. (2000). New necessary stability conditions for 2-D systems. IEEE Transactions on Circuits and Systems, Part 1, 47, 1103-1105.
Bistritz, Y. (1999). Stability testing of twodimensional discrete linear system polynomials by a two-dimensional tabular form. IEEE Transactions on Circuits and Systems, Part I , 46, 666667.
Premaratne, K. (1993). Stability determination of two-dimensional discrete-time systems. Multidimensional Systems and Signal Processing, 4, 331-354. Rajan, P. K . , and H. C. Reddy (1991). A study of two-variable very strict Hurwitz polynomials wit8hinterval coefficient#s.Proceedings of the 1991 IEEE International Symposaum on Circuits and Systems (pp. 1093-1096).
Bistritz, Y. (2000). Immitance-type t,abular stability for 2-D LSI systems based on a zero location test for l-D complex polynomials. Circuits, System,s and Signal Processing, 19, 245-265. Bose, N. K., and E. Zeheb (1986). Kharitonov’s theorem and st,ability test of multidimensional digital filters. IEE Proceedings. 133, 187-190.
Rogers, E., and D. H. Owens (1993). Stability tests and performance bounds for a class of 2D linear systems. Multidimensional Systems and Signal Processing, 4,355-391.
Dudgeon, D., and R. Mersereau (1984). Multzdzmensional Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall.
Siljak, D. D. (1975). Stability criteria for twova.riable polynomials. IEEE Transactions on Circuits and Systems, CAS-22, 185-189.
Garloff, J . (1993). Convergent bounds for the range of multivariable polynomials, Lecture Notes in Computer Science (Vol. 212, pp. 37-56). Berlin: Springer.
Siljak, D. D. (1989). Parameter space methods for robust control design: A guided tour. IEEE Transactions on Automatic Control, 34, 674688.
Hu, X. (1994). Techniques in the stability testing of discrete time systems. In Control and Dynamic Systems (Vol. 66, pp. 153-216). C. T. Leondes (Ed.). San Diego, CA: Academic Press.
Siljak, D. D., and D. M. Stipanovid (1999). Robust D-stability via posit,ivity. Automatica, 35, 14771484.
Huang, T. S. (1981). Two-Dimensional Digital Segnal Processing I. Berlin: Springer.
Siljak, D. D. (YEAR) Algebraic criteria for positive realness relative to the unit circle. Journal of the Franklin Institute, 296, 115-122.
Jury, E. I. (1982). Inners and Stability of Dynamic Systems. (2nd ed.). Melbourne, FL:Kreiger.
Stipanovid, D. M., and D. D. Siljak (2001). SPR criteria for uncertain rational matrices via polynomial positivity and Bernstein’s expansions. IEEE Transactaons of Circuits and Systems, Part I (to appear).
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Xia.0, Y., R. Unbeha.uen a.nd X. Du (1999). Robust Hurwitz stability conditions of polytopes of bivariate polynomials. Proceedings of the 38th Conference on Decision and Control (pp. 50305035).
Kamen, E. W. (1982). Linear systems with commensurate time delays: Stability and stabilization independent of delay. IEEE Transactions on Automatic Control, 4,367-375. Lehnigh, S. H. (1966). Stability Theorems for Linear Motions. Englenrood Cliffs, NJ: PrenticeHall.
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STABILITY OF TWO-VARIABLE INTERVAL P OLYN 0 MIALS VIA P 0S ITIVITY
Dragoslav D. Siljak* and DuSan M. Stipano vi?* * Santa Clara Unauersaty, Santa Clara CA 95053, [LSA ** Stanford Universrty, Stanford CA 94305, USA
Abstract: Stability criteria are proposed for two-variable (2D) polynomials having interval parameters in polynomic uncertainty structures. Both the left-half plane and unit circle domains are considered. Save for a minor condition, the criteria reduce robust stability testing of 2D polynomials to testing positivity of only two polynomials. The appealing feature of the new robustness criteria is that positivity testing can be carried out by using the efficient Bernstein minimization algorithms. Copyright ©
2002 IFAC Keywords: Two-variable polynomials, interval parameters, stability, left-half plane, unit circle, positive polynomials, Bernstein expansion, time-delay systems.
1. INTRODUCTION
volving 1D interval polynomials (Bose and Zeheb, 1986; Basu, 1990; Rajan and Reddy, 1991; Kharitonov et al., 1997; Xiao et al., 1999). In the context, of 2D polynomials, the solution lost much of its simplicity resulting in numerically involved algorithms. This fact made the testing of 2D polynomials with interval parameters difficult, especially in the case of multiaffine and polynomic uncerta.inty structures.
Stability of two-dimensional polynomials arises in fields as diverse as 2D digital signal and image processing (Huang, 1981; Dudgeon and Mersereau, 1984; Lim, 1990) and time-delay systems (Kamen, 1982), repetitive, or multipass, processes (Rogers and Owens, 1993) and target tracking in radar systems. For this reason, there have been a large number of stabilit,y criteria for 2D polynomials, which have been surveyed and discussed in a number of papers (Jury, 1986; Premaratne, 1993; Hu, 1994; Bistriz, 1999, 2000; Mastorakis, 2000). In achieving t,he maximal efficiency of 2D stability t,ests, the reduction of algebraic complexity offered by the stability criteria in (Siljak, 1975) has been useful. Apart from some minor conditions, the criteria convert, stability test,ing of a 2D polynomial to testing of only two 1D polynomials, one for stability and the other for positivity.
The purpose of this paper is to present new criteria for testing stability of 2D polynomials with interval parameters, which are based on the criteria of (Siljak, 1975) and the positivity approach to the interval parametric uncertainties advanced in (Siljak and Stipanovid, 1999). An appealing feature of the new criteria is the possibility of using the efficient Bernstein minim i z a t i a algorithms (Garloff, 1986; hfalan et al., 1992) to carry out the numerical part of the positivity tests. Furthermore, the proposed formulation can handle the polynornic uncertainty structures having interval parameters, and can be easily extended to systems with time-delays along the lines of (Kamen, 1982).
Due to inherent uncert,aint,y of the underlying models, it has been long recognized that in practical applications it has been necessary to test robustness of stability to parametric variations (Siljak, 1989). Almost exclusively, the 2D robust stability tests have been based on the elegant Kharitonov solution of the stability problem in-
7
2. STABILITY CRITERION
Conditions (i) mea.ns that f ( s ) = 0 if and only if s E C-, while condition (ii) is equivalent to g ( w ) > 0 for all w 2 0.
Let us consider a real two-variable polynomial n
m
In stability analysis of recursive digital filters (e.g., Huang, 1981; Dudgeon and Mersereau, 1984), it is of interest to establish necessary and sufficient conditions for a polynomial h ( s ,z ) to have the stability property
3=0 k=o
where s, z E C are complex variables, and for some j , k the coefficients h,, and hmk are not both zero. We are interested in determining conditions under which the polynomial h ( z ,z ) satisfies the stability property h ( ~ , zZ) 0 ,
EC-}n { Z EC-} ,
{S
h ( s ,2) # 0 , 1s E K"}n { Z E K O )
(9) where K = {s : Is1 = l} is the unit circle, and K O = KUK' is the closure of KO = { s EC : Is1 < I}. By following Huang (1972), one can show that (9) is equivalent to
(2)
where C- is the complement of C = { s E C : Re s < 0}, the open left half of the complex plane a. As shown by Ansell ( 1 9 6 4 , property (2) is eauivalent to h ( s ,1) # 0 ,
h(s,O)# 0 , h ( e Z wz,) # 0,
Yz
(l0a)
E Ko (106)
Condition (lOa) means that h ( s , 0) = 0 if and only if s E KO. To test condition (lob), we consider
v s EC:C (3a)
h ( i w , 2) # 0,
vs E K O
Vz ECC ( 3 b )
d ( z ) = zm/i(e2",z - ' ) which we write as a polynomial
To test (3a) we can use the standard Routh test (e.g., Lehnigh, 19GG). To verify (3b) we follow Ansell's approach and consider the polynomial c ( z ) = h ( i w ,z ) ,
(11)
m
m
c(z)= C
C k Z k
,
k=O
where
(4)
with coefficients
(5)
and s = e i w . With the polynomial d ( z ) we associate the Schur-Cohn m x m matrix D = ( d j k ) specified by
n
n
ck
=
j=o
hjksi
and s = iw.With ~ ( zwe ) associate the symmetric m x m Hermite matrix C = ( c j k ) with elements c j k defined by (e.g., Lehnigh, 196G)
c(-l)"e
j djk
j
'jk
= 2(-1)(j+k)/2
(',-!+1
k 1
'm-j-k+f)
(j
>
+ k ) even
c(-1)'1m
(~m-!+1
k 1
, where
cm-j-k+f)
g ( e i " ) = det D(ei"),
f ( s ) = s%(s-1,0)
(14)
(15)
(16)
and state the following (Siljak, 1975):
Theorem 2. A two-variable polynomial h ( s ,z ) has the stability property (9) if and only if
(7)
(i) f ( s ) is Ko-stable. (ii) g ( z ) is K-positive. (iii) D(1) is positive definite.
and replace w 2 by w to get a polynomial g ( w ) . We also define a polynomial f ( s ) = h ( s , 1)
dj-edk-e),
g ( . ) is a self-inversive polynomial. We also ( j + ,q odd define the polynomial
(6) where the overbar denotes conjugacy and j 5 k . We recall that C > 0 if and only if c ( z ) = 0 implies z E C-. Since C = C ( i w ) is a real symmetric matrix, we define a real even polynomial g ( w 2 ) = det C ( i w )
-
where j 5 (e.g., Jury, 1982). The matrix O(e2") is a Hermitian matrix and we define
j
cjk = 2 ( - I ) ( j + k ) / Z
= C(dm-j+e&-k+e e=i
Condition (i) means that the polynomial f ( s ) has all zeros inside the unit circle K. Positivity ofg(z) on K can be verified by applying the methods of (Siljak, 1973).
(8)
and state the following (Siljak, 1975):
Theorem 1. A two-variable polynomial h (s,z ) has the stability property (2) if and only if
Finally, we show how the mixture of the two previous stability properties can be handled using the same tools. The desired property is defined as
(i) f ( s ) is C--stable. (ii) g ( w ) is &-positive. (iii) C ( 0 ) is positive definite.
h ( s , Z ) f 0,
8
{S
ECC} n { Z
EKO} .
(17)
tested by Theorem 4 wza bilinear transformation in pretty much the same way D-stability was tested in (Siljak and Stipanovid, 1999). We also note the structural similarity of Theorem 4 with theorems on robust SPR properties (Stipanovid and Siljak, 200l), which motivates the work presented next.
By following Ansell (1964),one can show that this property is equivalent to h ( s , O ) # 0, h ( i w , z ) # 0,
‘ds EC-
(18a)
‘dz E K O (18b)
In this case, the polynoniial d ( z ) is defined as d ( z ) = zrnh(iw,z - 1 ) ,
Condition (i) in Theorem 4 obviously means that all zeros of f ( s , p ) lie in for all p E P . To establish this type of robust stability vzu polynomial positivity, we define the magnitude function
(19) which is used to obtain the polynomial g ( . ) wza equations (12)-(15). From (18a), we get the polynomial f ( s ) = h ( s ,01, (20) then define I = { z E C : Re z = 0} and arrive at (Siljak, 1975):
n
k=n j = o
(24) where overbar denotes conjugation. We note immediat,ely that, t,he magnitude funct#ionf ( s , p ) = If(., p ) 1’ is nonnegative for all s E a. This obvious fact is essential in the following development.
Theorem 3. A two-variable polynomial h ( s ,z ) has the stability property (17) if and only if (i) f ( s ) is (I-stable. (ii) g ( z ) is I-positive. (iii) D ( 0 ) is positive definite
Let us form a family ? = { f ^ ( . , p ): p E P} and use the result of (Siljak and StipanoviC, 1999) to conclude that a family F is (€-stable if and only if the corresponding family ? is I-positive, and f ( s ,p’) is(C--stable for some p’ E P . Furthermore, from (7) it follows that positivity of det C ( 0 ; p ) is included in testing condition (ii) of Theorem 4. This means that to test condition (iii) of Theorem 4, it suffices to verify that C(0;p”) is positive definite for some p” E P . We finally arrive at
We note that I-positivity of g ( z ) can be reformulated as &-positivity (see, Siljak and Siljak, 1998).
3. UNCERTAIN POLYNOMIALS We are interested in studying stability properties of uncertain two-variable polynomials wit,h polynomic uncertainty structures. A polynomial h ( s ,z ; p ) is given as n
7n
J=o
k=O
Theorem 5. An uncertain two-variable polynomial h ( s ,z;p ) has the robust stability property 23 if and only if (i) ? is &-positive and f ( s ; p ’ ) isC--stable for some p’ E P . (ii) G is &-positive. (iii) C ( 0 ;p”) is positive definite for some p” E P .
where h j k ( p ) are polynomials themselves in uncertain parameter vector p E IR‘.We assume that p resides in a box
Example 1. To illustrate the application of Theorem 5, let us use t,he two-variable polynomial from (Xiao et al., 1999),
P = { ~ E I R ‘ :p k --k~ , bp - k ] , k ~ r } . (22) We urant to investigate the robust versions of stability properties defined in the preceding section. In the case of (a), for example, we are interested in testing the robust property
h ( s , z ; p )# 0,
{S
EC-}
n{Z
EC)
n
14% z ;P ) = h l l (p)= + h1o(p)s+ ha1 ( p ) z+ boob) ,
(25)
where
h l l ( p ) = 0.9 - 0 . 1 ~ 1- 0 . 3 ~ 2 h l o ( p ) = 0.8 - 0.5pl 0 . 3 ~ ~ (26) hOl(P) = 1 0.2Pl 0.3pz hoo(p) = 1 . G 0.5~1 -0 . 7 ~ ~
n { p E PI.
+
(23) To accommodate the uncertainty in h ( s ,z ; p ) we define the polynomial families F = { f ( . , p ) : p E P}, G = { g ( . , p ) : p E P} and state a straightforward modification of Theorem 1.
+
+
+
and
Theorem 4. An uncertain two-variable polynomial h ( s ,z;p ) has the robust stability property (23) if and only if
To test condition (i) we compute the polynomial f ( s ; p ) = (1.7-0.6pl)~$2.G+O.7pl -0.4p2
(i) F is (I-stable. (ii) G is &-positive. (iii) C ( 0 , p )is positive definite for all p E P .
(28)
and note that, in this simple case, we do not need t o compute the corresponding magnitude function f ( w ; p ) . Robust C--stability of f ( s ;y ) follows directly from positivity of its coefficients. Indeed,
Robust versions of the remaining two stability properties of the preceding section can be
9
1.7 - 0 . 6 2 ~ 1.46 ~ 2.6 0.7pl - 0.4p2 2 2.19
+
In this case, the 2 x 2 matrix C ( i w ; p )turns out t o be a diagonal matrix
(29)
C ( ~ W ;=Pdiag )
for all p E P .
+ 0 . 2 ~ +1 0.3pz)(l.6 + 0 . 5 ~ 1 0 . 7 ~ 2 ) +(0.9 0 . 1 ~ 1 0.3P2) X (0.8 0.5pl + 0 . 3 ~ 2 ) ~
Cii(iw;p) = E i i ( W 2 ; p ) = (3 - p i ) w
-
-
-
-2p;
-
2 0 . 4 4 7 3 ~+ 1.0670
$12
Our analysis is elementary when compared to the stability testing procedure of Xiao et al. (1999), which involves extensive comput'at,ionrequired by the Edge Theorem.
~OI(IJ)~
h ( P ) hzo(P) hii(P) hoi ( p ) hlZ(P) hoa(P) hlO(P) hoo(P)
+
+ + +
'
2
(31)
where
-
+
+ h1a(p)sz2+ hao(p)s Sho2(P)t2+ hll(P)SZ +hio(~)s + + hoo(P)
h ( s ,z ; p ) = s 2 z 2 +ha1(p)s2z
-
+
A two-variable polynomial is given
as
-
-
czz(iw;p) = E2z(w2;P) = ( : h p 2- d p 2 ) w 4 +(-12pipz ~OP?PZ - 6 ~ 1 ~ ; 3p:pj - $ P W + 2 p X - 2P:Pa +12PlP., - 16PTPa 12PlP2 -10P;P; 7P;Pa 3PlP; - P f P ; +2P:P; - P?Pa
Let, us consider more complex examples which well require the use of Bernstein's algorithm.
Example 2 .
+ (-12 + 10pi 6pa + 2pipz + 3p;p; P.:P;)Wa 16271 + 12p2 1 0 p m 4
+7p: 3p; - P1P; +2Pb2 - P?
(30)
is obviously &-positive.
(38)
caa(iw;p)}
and conditions (ii) and (iii) of Theorem 5 reduce to positivity of the coefficients
Since the matrix C ( i w ; p ) is a scalar, condition (iii) is included in (ii) which is satisfied because g ( w ; p ) = (1
{CII(~U;P),
(39) By using Bernstein's minimization algorithm we compute the minorizing polynomials (Siljak and StipanoviC, 1999) and establish positivity of the two polynomials cll(iw;p)and c Z % ( i w ; p )by obtaining the minima min Ell(w;p) = 1.3724 a t pl = 2,
= 3 - P1
PEP
p2
= 1,
U€R+
=PIP%
= 3PiPz - P?Pz = 6 - 5pi 3Pa - P I P Z = P1Pa = 2 - Pl Pa 2 2 = PIP2 = 2PlP2 - PTPz PlPZ
+ +
w = 0.1715 min E z z ( w ; p ) = 3.1185 a t p1 = 2, pa = 1,
+ PZ
PEP
U€R+
(32)
w = 0.2487 .
(40)
Positivity of the minima implies robust stability property 25 for the polynomial h ( s ,z ; p ) of 31.
+
and
p = { p E IRz: p1 E [1,2], Pa E [1,2]}. (33)
4. TIME-DELAY SYSTEMS
The polynomial f ( s ;p ) is computed as
+ + + + +
+
f ( s ;P ) = (4 - P I P1pa)s2 (4PlPz - P b a p b ; ) s +8 - 6 ~ 1 4Pz P1P2 P f - p f p 2 +pip; (34) The corresponding minimizing polynomial f ( s ) = 4s2
-
+ 4s + 4
n-1
(35)
~ ( ~ )+( t )
is obtained by minimizing each coefficient using Bernstein's algorithm. Obviously, F is C--stable since f ( s ) has positive coefficients.
where
+
+ " I ( P )+~ C o ( P ) ,
+
+ PZ +
m
-
(41)
where r 2 0. The coefficients h j k ( p ) are polynomials in the uncertain parameter vector p E IRL which belongs to a box P .
(36)
It is well known (Bellman and Cooke, 1963) that for a fixed pa.ramet,er p , a system (41) i s st,able if and only if
C 2 ( P ) = 2 - p1 pa - W 2 ip1paw c i ( ~1 ) 6 - 5Pi 3Pa - ~ 1 ~ -2 3 ~ ' +i(3PlPZ - P?Pa)U ' 2 2 C o ( P ) = 2p1p2 - PTP2 - P l P 2 W 2 ZPlP2U
+
F lhak(p)~(3)(t k r ) = 0 ,
i = o k=n
Next, we compute C ( z ; P )= Q ( P ) z 2
Our objective in this section is to show how the tools presented in this chapter can be applied to test robust stability of linear systems of the retarded type described by a differential-difference equation
+ PI^'
h ( s ,e F T S ; p = ) sn
(37)
+
cc
n-1
m
hjk(p)swTs
Res20.
10
#O,
j = o k=o
(42)
From (50), we compute the associated polynomial
The system is robustly stable if (42) holds for all p E P.
h ( s , 2 ; p ) = s2
The following theorem is a straightforward robustification of a theorem by Kamen (1983):
and test first the necessity of condition (43) by checking condition (44). Since
Theorem 6 . System (41) is robustly stable independent of delay if h ( s , ~ ; p#) 0,
{S
EKE} n { Z
E K}
+
(43)
h ( 0 ,z i p ) f 0, {. E K ) n { P € p > .
(44)
f ( s ; p ) = s2
To test condition (43) we first use the bilinear transformation
to define the polynomials
iw q s .i w ; p ) = (1 - i w y h ( s , 11 + - iw ~
'
p2)s
+1
- p1
+ p1p; .
(54)
+ 0.5s + 0.5
(55) implies robust stability of f ( s ; p ) , that is, condition (i) of Theorem 7 is satisfied. For testing condition (ii) we need the polynomial
h(s,iw;p)= (1- i w ) s 2
+
+
[l + p a (-1 +pa)iw]s $1 + P l p1p; +(-1 p1 - p1p;)iw . (56) Using equations (47)-(49), we compute
+
j=O 12
k=O
+
+ + + + + + +
g ( w ; p ) = q 1 - p1- 2P2 2P; 2PlP2 -2Pd PlP;)zvJ2 +8(1- 2 ~ 2 ~2 PIP$ - PIP:)^ +4(1; p1 2Pa 2p1pa P; 2PlP; +2PlP, PlP$ (57) By applying the Bernst,ein algorithm to each coefficient of g ( w ;p), we obtain the minorizing polynomial
+
With c ( s ; p ) we associate the symmetric n x n Hermite matrix C = ( C ~ I ; ) having elements c J k defined in (6), and obtain the polynomial
+
g ( w 2 ; p )= det C ( i w ; p ) . (49) Finally, with polynomials f ( s ; p ) and g ( w a ; ; p )at hand, we can imitate Theorem 5 t o state the following:
Theorem 7. System (41) is robustly stable independent of delay if
g(w) -
+
= 0.625w2+ 1 . 2 5 ~ 0.375
,
(58)
which is clearly IR-positive, and (ii) of Theorem 7 is satisfied.
(i) ? is &-positive and f ( s ; p ' ) isC--stable for some p' E P . (ii) 6 is &-positive. (iii) C(0;p") is positive definite for some p" E P .
Finally, the ma.trix of condition (iii) is comput>ed as C(0,O)= 2I7, where I2 is the identity matrix
of dimension 2 , and robust stability independent of delay of system (50) is established with respect to the uncertainty box P in (51).
It is obvious that condition (44) of Theorem G , which is included in (45), can be tested zua positivity as well. To illustrate the application of Theorem 7 let us use t,he following:
5. CONCLUSIONS
Example 3. A time-delay system (41) is given
We have shown how stability of 2D polynomials with interval parameters can be tested wia polynomial positivity. To test stability of polynomials with multiaffine and polynomic uncertainty structures, positivity of only two interval polynomials is required. A remarkable efficiency of the proposed stability criteria is due to their suitability for applications of Bernstein's expansion algorithms.
as
+ p z z ' l ) ( t - + p 1 z ( t - + .'"(t)
+
-
f ( s ) = s2
~I
(47)
+(I P l P $ ) X ( t ) = 0 with the uncertainty box
+ (1
-
(46)
n
T)
2 PlP2
T.
'
.'"(t)
+ 1+
To test robust stability of this polynomial we do It suffices t o not need t o const,ruct, tmhefamily check positivity of each coefficient, which we do by using the Bernstein algorithm. The resulting minorizing polynomial
f ( s ; p )= h ( s , - 1 ; p ) Then, by following Ansell's approach in, we coiisider the polynomial c ( s ; p )= h ( s ,i w ; p ) ,
where
h ( 0 ,2 ; P ) = P l Z
(53) and 1 p1pg > lp11, we conclude that (44) is satisfied. This implies that condition (43) is necessary and sufficient for robust stability of system (SO), and we proceed to compute the polynomial
n { p E P} .
This condition is also necessary if
+ (pas+ pi)^ + s + 1+ pip; , (52)
T)
(50)
P = { p E EL2 : p i E [-0.5,0.5], p2 E [-0.5,0.5]} (51)
11
Lim, J. S. (1990). Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: PrenticeHall.
ACKNOWLEDGMENT The research reported herein has been supported by the National Science Foundat,ion under the grant ECS-0099469. REFERENCES
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